An introduction to logic, by H.W.B. Joseph ..

J

THE LIBRARY

OF

THE UNIVERSITY

OF CALIFORNIA

LOS ANGELES

Oxford University Press, Ely House, London W. I

GLASGOW NEW YORK TORONTO MELBOURNE WELLINGTON

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BOMBAY CALCUTTA MADRAS KARACHI LAHORE DACCA

KUALA LUMPUR HONG KONG TOKYO

AN INTRODUCTION

TO

LOGIC

Br
H. W. B. JOSEPH

FELLOW AND TUTOR OF NEW COLLEGE

SECOND EDITION, REVISED

OXFORD
AT THE CLARENDON PRESS

FIRST EDITION I906

SECOND EDITION I916

REPRINTED I925, 1931, I946, 1950, 1957,

1961, I966, I967

PRINTED IN GREAT BRITAIN

College
library

377

TO

J. E. J.

1G1 i

PREFACE TO THE SECOND EDITION

The present edition, the preparation of which has been some-
what delayed by reason of the war, has been carefully revised
throughout, and is enlarged to the extent of some 40 pages.
Though it does not differ in its main teaching from the first edition,
there are very few pages without some slight alteration, if only
for the sake of greater clearness, or of more consistency in phrase-
ology. I hope the alterations are improvements ; I must confess
to some dissatisfaction at finding how many I desired to make.
The following list of some of the principal alterations and additions
may be of use to any reader who is interested in a comparison.

c. i. On p. 5, n. 1 (upon the distinction between the form and
matter of thought), I have endeavoured to show that the
matter of thought, in this antithesis, is not the same as its
4 subject-matter ' ; and I have altered the language of
many passages accordingly. The latter part of p. 10, n. 1,
and the last page and a half of the chapter, are new.

c. ii. Much of this chapter has been rewritten. In particular,
the general discussion of the nature of a term on pp. 14-28,
and that of the distinction between abstract and concrete
terms, on pp. 32-35, are largely new. The former includes
some discussion of concepts ; and both have involved small
consequential alterations at many subsequent points.

c. iii. The note on p. 52, on the position of ' first substances '
in Aristotle's doctrine of categories, dissents from p. 39, n. 1, of
the first edition.

c. v. In this chapter I have laid more stress on the makeshift
character of most classification, and have done more justice
to the use of negative differentiae.

c. vi. This chapter has been largely rearranged and rewritten.
I now prefer, in order to express the truth in the contention

PREFACE TO THE SECOND EDITION

of those who have denied connotation to proper names, to
distinguish between intension and connotation, allowing
them the former, but not the latter.

c. viii. This chapter also has been largely rewritten, parti-
cularly in the discussion of the modality of judgements, and
of the distinction between analytic and synthetic judge-
ments. The close print on pp. 192-195, 201-205, 214-215,
is new matter. In this and subsequent chapters I think
that much which in the first edition was said about judge-
ments should properly have been said about propositions,
and correction has been made accordingly. I have made use
in certain places of expressions borrowed from Prof. Cook
Wilson.

c. ix. On p. 228, and also on p. 120, n. 1, 1 have inserted a few
remarks on Symbolic Logic, which will in some measure
explain why the book does not deal more fully with that
subject.

c. x. The discussion of the inferential character of so-called
Immediate Inference (pp. 240-247) has been enlarged and
recast ; pp. 241-242 are new matter.

c. xii. The discussion of the Fourth Figure of Syllogism,
pp. 280-284, has been largely rewritten.

c. xiii. In the note beginning on p. 296, the discussion of the
passage Cat. iii. lb 10-15 (pp. 298-299) has been emended.

c. xiv. In pp. 310-311 I have emphasized the subsumptive
and therefore inferior character of syllogistic thinking. The
close-print discussion on pp. 331-334 is new.

c. xviii. The attempt in the last three pages, 397-399, to
characterize the difference between inductive and deductive
reasoning is new.

c. xix. The new matter in this chapter, which has also been
considerably rewritten, is chiefly in pp. 403-404, 410-413,
418-419, 421, n.l.

c. xx. I have corrected language which spoke of an event
causally connected with another as its cause, both here and
subsequently ; the point is discussed on pp. 426-428.

PREFACE TO THE SECOND EDITION

c. xxi. In pp. 475-476 I have dwelt on one or two further
matters belonging to the pursuit of the inductive sciences
which are no part of their reasoning.

c. xxii. A rather obscure passage in the previous edition
(pp. 459-460) is replaced by pp. 495-496.

c. xxiii. I have tried to improve the statement of what explana-
tion is on pp. 502, 521-522 ; and p. 523, n. 1, is new. The
close print on pp. 524-527 is an answer to a criticism which
Dr. Bosanquet has made upon the view of induction taken
in this book.

c. xxv. I have on pp. 546-549 rewritten and to some extent
altered the tenour of what was said before (on pp. 506-509
of the first edition) concerning the foundation of our power
to generalize in mathematics.

c. xxvii. Some additional matter is contained in the first
notes on pp. 572, 580, 582, 589, 595.

I have taken account of such criticisms as I have seen in print,
though I have not thought all equally well founded ; for these,
and also for various criticisms privately communicated, I desire to
express my thanks. I should like here to name again Prof. J. A.
Smith, Mr. H. H. Joachim, and Mr. H. A. Prichard, who
were all good enough to send me comments on divers points,
Besides these, Prof. W. G. de Burgh, of University College,
Reading, very kindly helped me with a list of criticisms and
suggestions based on his use of the book with students ; and
Miss Augusta Klein sent me a series of most careful notes upon
the first eight chapters. These were particularly helpful upon
points of science referred to by way of illustration, and upon the
theory of classification, with the logical doctrines on which it
rests ; and the principal changes which I have introduced on
those topics are due to her criticism, though not involving a
full acceptance of it.

But chiefly here I desire to put on record the debt which I owe,
in common with so many other of his older or younger pupils,
to Prof. J. Cook Wilson, whose death occurred while these sheets
were passing through the press. Various footnotes will show

PREFACE TO THE SECOND EDITION

the use that I have made of his unpublished teaching ; but his
illness prevented me from submitting to him what I have written,
and his authority must be made responsible for no errors that
I may have made. His few and scattered publications can do
little to convey to strangers the power and stimulus of his personal
teaching. And there are subjects on which, by his combina-
tion of scholarly and mathematical with philosophic insight,
he was qualified as few have been to produce new work of real
value. The hope has vanished that he might put in permanent
form the full results of his thinking. But those who knew him
well will not misjudge this failure. For they will remember him
as not more patient and eager in philosophic reflection than in his
devotion during many years to a suffering wife and in his endur-
ance afterwards of his own wasting and fatal illness.

H. W. B. J.

Advantage has been taken of reprinting to correct a number of
small errors, mostly typographical, to which my attention has been
called by the kindness of readers.

H. W. B. J.

January, 1925.

PREFACE TO THE FIRST EDITION

If an apology that precedes it could mitigate an offence, 1
should be inclined to convert my preface into an apology for
publishing this book. Progress, and the hope of progress, in
logical investigations, have lain perhaps during the last three
generations chiefly in two directions, either of analysing more
closely the processes of thought exhibited in the sciences, or of
determining what knowledge is, and the relation of the knowing
mind to what it knows. Though I have been compelled to deal
in some degree with the first of these questions, I am well aware
that it demands a scientific knowledge which I do not possess ; the
second I have not attempted systematically to discuss. The aim
of the following book is more modest. There is a body of what
might be called traditional doctrine in Logic, which is not only
in fact used by itself as an instrument of intellectual discipline,
but ought also to be in some degree mastered by those who would
proceed to the higher and abstruser problems. It is of this
traditional doctrine that Benjamin Jowett is recorded to have
said, that Logic is neither a science, nor an art, but a dodge.
I could perhaps best describe the motive with which this work
was begun, as the desire to expound the traditional Logic in a way
that did not deserve this accusation. The accusation was doubtless
provoked by the attempt to force into a limited number of forms
processes of thought, many of which can only with pretence and
violence be made to fit them : an attempt, it may be added, at
least as characteristic of ' Inductive Logic ' as of any other.

In the course of centuries, the tradition has become divergent,
and often corrupt. In this difficulty, I have ventured, like one or
two other modern writers, to go back largely to its source in
Aristotle. Problems of thought cannot in any case be studied
without careful regard to their terminology, and their terminology

PREFACE TO THE FIRST EDITION

cannot be understood without reference to its history. The
terminology of Logic owes more to Aristotle than to any one else ;
but there is this further reason for attention to what he said, that
much prevalent falsehood or confusion in the tradition is a corrup-
tion of truths expressed by him. At the same time, I have not
pretended to believe in the verbal inspiration of his writings.

I have in particular been anxious to teach nothing to beginners
which they should afterwards have merely to unlearn. They may
of course come to dissent from the positions here taken up ; but
only, I hope, because they think I have the worst of the argument
on a proper issue, and not because, as meat for babes, I have been
dogmatically expounding acknowledged fictions.

While dealing largely with the more technical parts of logical
tradition and terminology, I have done my best to avoid a super-
fluity of technical terms ; and the subjects discussed have been for
the most part discussed in detail, and the principles involved in
them debated. The dryness with which the more formal branches
of Logic are often charged springs, I think, in part from their
being presented in too cut-and-dried a manner ; those who go
beyond the jejune outline, and get into an argument, often find
the subject then first begin to grow interesting. At any rate
I have tried to secure this result by greater fullness, and attention
to controversial issues. In every study there must be something
to learn by heart ; but Logic should appeal as far as possible to
the reason, and not to the memory. Thus such a question as the
' reduction ' of syllogisms has been dealt with at length, not from
any wish to overrate the importance of syllogistic reasoning, or
burden the student with needless antiquarianism, but because the
only thing of any real value in the subject of reduction is just that
investigation of the nature of our processes of thinking which ia
involved in asking whether there is any justification for reducing
all syllogisms to the first figure.

Topics whose main interest is obviously historical or antiquarian
have been either relegated to footnotes or placed in closer type and
between brackets ; and as I have followed the advice to translate
what Greek I quote, I do not think that there is anything in these

PREFACE TO THE FIRST EDITION

discussions which a reader need be altogether precluded from
following by ignorance of that language. I have also put between
brackets in closer type other passages which, for one reason or
another, might be omitted without spoiling the argument ; among
the matters so treated is the fourth figure of syllogism ; for I have
reverted to the Aristotelian doctrine of three figures, with the
moods of the fourth as indirect moods of the first.

I hope that I have sufficiently acknowledged all detailed obliga-
tions to previous writers in the places where they occur. But I owe
here a more comprehensive acknowledgement both to the published
work of Sigwart, Lotze, Mr. F. H. Bradley, and Professor Bosan-
quet, and to the instruction received in private discussion with
various friends. Among these I should like to mention in particular
Mr. J. Cook Wilson, Fellow of New College, Wykeham Professor
of Logic in the University of Oxford, whose reluctance to write
is a source to many of serious disappointment and concern ;
Mr. J. A. Smith, Fellow of Balliol College ; Mr. C. C. J. Webb,
Fellow of Magdalen College ; Mr. H. H. Joachim, Fellow of
Merton College ; and Mr. H. A. Prichard, Fellow of Trinity
College, Oxford. To the last three of these, and also to Mr. C.
Cannan, Secretary to the Delegates of the University Press, I am
further indebted for the great kindness with which they read large
portions of the work in MS. or in proof ; without their suggestions
and corrections it would be even more imperfect than it is.
Lastly, I have to thank my sister, Miss J. M. Joseph, for the
help she gave me in reading the whole of the proof-sheets and in
undertaking the laborious and ungrateful task of checking the
index.

CONTENTS

CHAPTER PAGE

I. Of the General Character of the Enquiry . 1

II. Of Terms, and their Principal Distinctions . 14

III. Of the Categories ...... 48

IV. Of the Predicables ..... 66
V. Of the Rules of Definition and Division :

Classification and Dichotomy . .111

VI. Of the Intension and Extension of Terms and

OF THEIR DENOTATION AND CONNOTATION . 136

VII. Of the Proposition or Judgement . . . 159
VIII. Of the Various Forms of the Judgement . 171
IX. Of the Distribution of Terms in the Judge-
ment : AND OF THE OPPOSITION OF JUDGE-
MENTS ....... 216

X. Of Immediate Inferences .... 232

XL Of Syllogism in General .... 249

XII. Of the Moods and Figures of Syllogism . . 254

XIII. Of the Reduction of the Imperfect Syllo-

gistic Figures ...... 287

XIV. Of the Principles of Syllogistic Inference . 294
XV. Of Hypothetical and Disjunctive Reasoning . 335

XVI. Of Enthymeme, Sorites, and Dilemma . . 350

XVII. Of the Form and Matter of Inference . . 366

XVIII. Of Induction 378

XIX. Of the Presuppositions of Inductive Reason-
ing : the Law of Causation . . . 400
XX. Of the Rules by which to judge of Causes and

Effects ...•••• ^26
XXI. Of Operations preliminary to the Application

of the Foregoing Rules ...» 458

CONTENTS

CHAPTEB PAGB

XXII. Op Non-eeciprocating Causal Relations . . 478

XXIII. Of Explanation 502

XXIV. Op Induction by Simple Enumeration and the

Argument from Analogy . . . 528

XXV. Of Mathematical Reasoning .... 543

XXVI. Of the Methodology of the Sciences . . 554

XXVII. Appendix on Fallacies ..... 566

INDEX 600

CHAPTER I

OF THE GENERAL CHARACTER OF THE ENQUIRY

It is a common practice to begin a treatise on any science with
a discussion of its definition. By this means the reader's attention
is directed to the proper objects, and to those features of them, with
which the science is concerned ; a real advantage, when, as with
Logic, those objects are not apprehended through the senses, and
for this reason ordinarily attract little notice. But the same reason
which makes a definition of Logic at the outset useful, makes con-
troversy about its definition comparatively useless at such an early
stage. The reader is too unfamiliar with the subject-matter of his
science to be able to judge what definition best indicates its nature ;
he cannot expect thoroughly to understand the definition that is
given, until he has become familiar with that which is defined. The
definition will at first guide more than enlighten him ; but if, as he
proceeds, he finds that it helps to shew unity in the different en-
quiries upon which he successively enters, it will so far be justified.

Logic is a science, in the sense that it seeks to know the principles
of some subject which it studies. The different sciences differ in
the subjects which they so study ; astronomy studies the nature,
movements, and history of the heavenly bodies, botany the structure,
growth, history, and habits of plants, geometry the properties and
relations of lines, surfaces, and figures in space ; but each attempts
to discover the principles underlying the subjects with which it has
to deal, and to explain their great variety by the help of one set
of principles. These principles are often spoken of as laws ; and in
the physical sciences that deal with change, as ' laws of nature '.
The phrase may suggest that ' nature ' is not the sum of things and
of events in the physical universe, but a sort of power prescribing
to these the rules which they are to follow in their behaviour ; as
the King in Parliament prescribes rules of conduct to his people.
That, however, is not what we have to understand in science by
a ' law ' ; a law in science is not, like human laws, a rule enjoined but
sometimes disregarded ; it is a principle illustrated — and existing

1770 B

2 AN INTRODUCTION TO LOGIC [chap.

only in the necessity of its being illustrated — in the department of
fact to which it belongs. There are therefore no breaches of scien-
tific law, or of a law of nature x ; if events are observed which do
not conform to what we have hitherto called a law, we conclude not
that the law is broken, but that we were ignorant of the true law ;
if water, for example, were observed to boil on the top of Mont Blanc
at a lower temperature than 212° Fahr., we should infer not that the
law that water boils at 212° Fahr. was broken but that it is not a
law of nature that water boils at 212° Fahr., — that there are other
conditions which have to be fulfilled, if water is to boil at that tem-
perature ; and the ' law ' is that it should boil only when those
conditions are fulfilled. Such laws, the general principles to which
things in their properties and their behaviour do actually conform,
are what the physical sciences seek to discover, each in its own
department, and if Logic is a science, it must have a subject of its
own, in which it seeks for principles and laws.

That subject is thought, but thought is always thought about
something ; and thinking cannot be studied in abstraction from
anything thought about. But yet in the same way that we may
study the laws of motion, as they are exemplified in the movement
of all bodies, without studying all the bodies that ever move, so we
may study the laws of thought, as they are exemplified in thinking
about all subjects, without studying all the subjects that are ever
thought of. This comparison may be pushed further. Just as we
must have experience of moving bodies, before we can investigate
the laws of their motion, so we must have experience of thinking
about things, before we can investigate the principles of thinking ;
only this means, in the case of thinking, that we must ourselves
think about things first, for no one can have experience of thinking
except in his own mind. Again, although, in studying the laws
of motion, we do not study every body that moves, yet we must
always have before our minds some body, which we take as repre-
senting all possible bodies like it ; and in the same way, when we
investigate the principles that regulate our thinking, though we do
not need to study all things ever thought of, we must have before
our minds something thought of, in order to realize in it how we
think about it and all possible things like it. For example, it is a
general principle of our thought, that we do not conceive of qualities

1 The question of the possibility of a ' miraculous ' breach of natural law
need not be considered here; something is said of it inc. xix, infra, pp. 417-421,

I] GENERAL CHARACTER OF THE ENQUIRY 3

except as existing in some substance ; and that nevertheless the
same quality is thought to exist in many substances ; green is
a quality, which exists not by itself, but in grass and leaves of trees
and so forth ; at the same time, green may exist in many different
leaves or blades of grass. The general principle which is thus
illustrated in the quality green is readily understood to be true of
all possible qualities ; but unless we were able to think of some
particular quality to illustrate it, we could not understand the
general principle at all.

What has been now said will serve to remove an objection which
Locke brought against the study of Logic. 'God', says Locke1,
' has not been so sparing to men, to make them barely two-legged
creatures, and left it to Aristotle to make them rational.' He is
urging that men thought rationally, or logically, i. e. in accordance
with the principles that Logic discovers to regulate all sound thought,
long before those principles were recognized ; and that this is still
so with each of us ; we do not therefore need Logic to teach us how
to think. That is quite true, and would be a pertinent criticism
against any one who pretended that no one could think rationally
without studying Logic ; but it is not the business of Logic to make
men rational, but rather to teach them in what their being rational
consists. And this they could never learn, if they were not rational
first ; just as a man could never study (say) the principles of volun-
tary motion, if he was not first accustomed to move his limbs as
he willed. Had God made men barely two-legged creatures, Aris-
totle would in vain have taught them to be rational, for they would
not have understood his teaching.

Logic, then, is the science which studies the general principles in
accordance with which we think about things, whatever things they
may be ; and so it presupposes that we have thought about things.
Now our thought about them is expressed partly in the daily con-
versation of life or musings of our minds ; partly and most sys-
tematically in the various sciences. Those sciences are the best
examples of human thinking about things, the most careful, clear,
and coherent, that exist. In them, therefore, the logician can best
study the laws of men's thinking ; and it is in this sense that we
may accept the old definition of Logic, scientia scientiarum.2 What
* the courses of the stars ' are to astronomy, what figures, lines, and

1 Essay, Bk. IV. c. xvii. § 4. .

* Joannes Philoponus cites it ad Ar. Anal. Post. a. is. 76a 15.

62

4 AN INTRODUCTION TO LOGIC [chap.

surfaces are to geometry, what plants are to botany, or the calendar
of Newgate to the crimi aologist, that the other sciences are to the
logician : they are the material which he has to investigate, the
particular facts which are given him, in order that he may discover
the principles displayed in them. He has to ask what knowledge
is as knowledge, apart — so far as possible — from the question, what
it is about ; and he must therefore examine divers ' knowledges ',
and see in what they are alike ; and among the best pieces of know-
ledge that exist, the best ' knowledges ', are the various sciences.
But he is not concerned with the detail of any particular science ;
only with those kinds (or forms) of thinking which are exemplified
in all our thinkings — though not necessarily the same in all — but
best exemplified in the sciences.

It is important to understand what is meant by saying that
Logic is concerned with forms of thinking ; for many logicians wTho
have laid stress on this, and pointed out that Logic is a formal
science, have understood by that expression more than seems to be
true. There is a sense in which Logic is undoubtedly formal. By
form we mean what is the same in many individuals called materially
different — the device, for example, on different coins struck from the
same die, or the anatomical structure of different mammalian verte-
brates, or the identical mode in which the law requires the different
Colleges of the University of Oxford to publish their accounts. And
all science is formal, in the sense that it deals with what is common
to different instances. A scientific man has no interest in a specimen
exactly similar to one which he has already examined ; he wants
new types, or fresh details, but the mere multiplication of specimens
all alike does not affect him.1 So the logician studies the forms of
thinking, such as that involved in referring a quality to a substance
possessing it ; but when he has once grasped the nature of this act
of thought, he is quite uninterested in the thousand different such
acts which he performs during the day ; they differ only materially,
as to what quality is referred to what substance ; formally, so far
as the notion of a quality as existing in a substance is concerned,
they are the same ; and the forms that run through all our thinking
about different matters are what he studies.

But those who have insisted most that Logic is a formal science,
or the science of the formal laws of thought, have not merely

1 Unless indeed he is collecting statistics as to the comparative frequency
of different types.

I] GENERAL CHARACTER OF THE ENQUIRY 5

meant that Logic is in this like other sciences, which all deal with
what is formal or universal in their subject-matter.1 They have
meant to exclude from Logic any consideration of forms or modes
of thinking which are not alike exemplified in thinking about
absolutely every subject. It is as if the botanist were to regard
only those laws which are exemplified in every plant, or the geo-
meter were to consider no properties of figures, except what are
common to all figures. They have thought that one might abstract
entirely from and disregard all question as to what he thinks about,
and still find that there are certain principles in accordance with
which, if he is to think about anything, he will think. But the
truth is, that we think in different ways about subjects of different
kinds, and therefore we must, if we wish to study the principles
that pervade our thinking, consider to some extent the differences
in our thought arising from differences in that about which we
think. The distinction between form and matter may as it were
be taken at different levels. This is plain in a science that
deals with some order of sensible things, like zoology. We may
say of all men and all horses that they have severally a common
form, that as compared to a man a horse is formally different, but as
compared to one another all horses are formally the same, though
each horse in his body is materially different from every other.
Or we may consider not the form of horse common to Black Bess

1 It is important to realize that the subject-matter of Logic is our thinking
about divers things, not the things which that thinking is about ; they are
the subject-matter of that thinking. Just as the form and matter of a coin
are both in the coin, so the form and matter of thought (if we are to keep
the meaning of the antithesis) must be both in the thought ; we must not
suppose that the formal identity is in the thought, the material differences
in the things thought about. An analogy may help to make this point clearer,
on which I confess that there was some confusion in the hist edition of this
book. Hunger and thirst are formally the same as being both appetites,
materially different as being the one for food, the other for drink ; but the
material difference is not the difference between food and drink, nor the
matter of the appetites food and drink respectively ; the matter is rather
the special character which the appetites have through being for these objects.
So the matter of a thought is the special character which it has through
being about a certain subject, not the subject which it is about. It may
be added, as a caution to the unwary, that the antithesis of form and
matter is used in various ways by no means all of them analogous ; and its
application to thought is not really the same as its application to coins or
animals. What is different in particular thoughts is not related to their
common form as the gold or silver of two coins to their common device, or
the flesh and bones of two animals to their common structure, but rather as
the specialty of their structures to the generic identity, or as particular in-
stances to the common nature of which they are instances. Cf. infra, pp. 75-77.

6 AN INTRODUCTION TO LOGIC [chap

and Bucephalus and Rosinante, but the form of vertebrate common
to man, horse, eagle, crocodile, &c. ; and now man and horse (as
compared with oysters for example) are formally alike. Or we
may take the four orders in Cuvier's now obsolete division of the
animal kingdom, vertebrata, coelenterata, radiata, and annulosa,
and regard them as only different examples of the common form of
animal ; and from this point of view a horse and an oyster differ
materially, but not formally. When however we have reached this
stage, and achieved the conception of animal, as something exem-
plified equally in kinds of animal so different, it is clear that we can
only understand what animal structure is by seeing it as it exists
in all the different orders of animals ; whereas we can understand
fairly the structure of a mammalian vertebrate without seeing it
as it exists in every genus of mammals ; still more can we under-
stand the structure of a horse without familiarity with all horses.
The higher the level therefore at which in Zoology the distinction
between form and matter is taken, the less can we study the form in
isolation from variety of matter ; no example taken from one order
of animals, say the starfish, will enable us to realize what animality
is. It is the same in studying the forms of thought. The most
general forms of thought exist diversely modified in thinking about
different subjects ; and they can no more be fully known without
attending to the different matters in which they appear differently,
than animal nature can be fully known without attending to the
different orders of animal in which it appears differently. Thus we
may take the Proposition, and point out that in every affirmative
categorical proposition there is a subject about which something is
said, and a predicate, or something which is said about it. This is
true equally of the propositions, ' A horse is an animal,' ' First-
class railway tickets are white,' and ' Londres is London * . We may
if we like, because in all propositions there is formally the same dis-
tinction of subject and predicate, take symbols which shall stand
for subject and predicate, whatever they are, and say that all affir-
mative categorical propositions are of the form ' S is P ' . But when
we ask for the meaning of this form, and in what sense S is P, it is
clear that the meaning varies in different propositions.1 Londres
is just the same as London ; but a horse is not just the same as an
animal ; it may be said that ' animal ' is an attribute of horse, and

1 Professor Cook Wilson has called attention in his lectures to the dangerous
ambiguity of this symbolization. Cf. infra, pp. 22-24.

I] GENERAL CHARACTER OF THE ENQUIRY 7

' white ' of first-class railway tickets, but animal is an attribute
belonging to horses in quite a different way from that in which white
belongs to first-class railway tickets ; these might as well be of any
other colour, and still entitle the holder to travel first-class by the
railway ; a horse could not cease to be an animal and still continue
to be a horse.1 The meaning of the formula 'S is P ' cannot pos-
sibly be fully known merely by understanding that 8 and P are some
subject and predicate ; it is necessary to understand what kind of
subject and predicate they are, what the relation is between them, and
in what sense one is the other ; and if this sense is different in different
cases, just as animal is something different in a dog and a starfish,
then the thorough study of the form of thought involves the con-
sideration of material differences in the thoughts also. But logicians
who emphasize the purely formal character of Logic maintain that
it can exhaust the form of thought in treating that as one and the
same in every possible instance of thinking ; an impracticable task,
because the form itself (as in the above example of a form of
thought which we call a proposition) is modified according to the
instance in which it appears. On the other hand, and even although
the forms of our thought cannot be studied apart from the differences
connected with the particular sort of subject about which we may
think, yet Logic is not interested in these differences for their own
sake, but only for the sake of the divers forms of thinking involved
in them ; and so far as the same form is exemplified over and over
again in different particular ' bits ' of thinking, the study of the
common form alone belongs to Logic.

[The truth that form cannot be studied apart from matter might
be otherwise expressed by saying, that the general form can only
be studied in one or other of the special forms in which it is mani-
fested ; and these special forms can only be illustrated in examples
that are materially different from one another. The proposition
' Londres is London ' is a special form of proposition equally well
exemplified in ' Koln is Cologne ' ; as Bucephalus is an animal of
a special form equally well exemplified in Black Bess. What is
important to realize is the need of following the common form out
into the differences which it displays in different matter.]

The foregoing discussion will probably become plainer if it be
read again at a later stage, when the reader is more practised in
reflecting on his thoughts. A distinction which is readily seen in

1 In strictness, the generic nature of a subject should not be called an
attribute of it. Cf. infra, pp. 82-83.

8 AN INTRODUCTION TO LOGIC [chap.

regard to material subjects, like animals or plants, is not so easily
seen in immaterial subjects, like our thoughts. The natural man
thinks much about things, and asks and answers questions
about them ; but it is by an effort that he comes to see how
these things are only known to him in his perceptions of them and
his thoughts about them, and so comes to turn his attention inward
upon the nature of the acts of perceiving or of thinking. Nor can
these new objects of his study be preserved and dissected like a
material thing ; a man cannot catch a thought and bottle it ; he
must create it by thinking it, if he wishes to think about it ; and the
task will be found difficult while it is strange, and not altogether easy
when familiar.

[Mediaeval logicians sometimes say that Logic deals with second
intentions ; by this is meant what has been pointed out in the last
paragraph. The mind intends or directs itself at first upon material
things and their qualities or relations ; and these are its first inten-
tions ; it may afterwards intend or direct itself upon its own modes
of thinking as exhibited in its first intentions ; and what it then
discovers are its second intentions. Thus we observe animals, and
give them names according to their kind, calling them stag and ox,
worm and lobster ; and again we observe how these kinds agree and
differ, and call some vertebrate, and some invertebrate, but all
animals ; and all these names, which are names we give to objects,
are names of the first intention. But we may also observe how we
have been thinking about these animals, as having some properties
common to all, and some peculiar to the members of each kind ;
and we may call the members of each kind, or their common nature,
a species, and the members of the several kinds together, or the
wider common nature, a genus ; and genus and species are names
of the second intention. The unity on the strength of which we call
them of one species or of one genus will indeed be something in the
animals themselves ; and so our names of second intention will in
this instance signify something real in things. The distinction
therefore presents difficulties which call our attention to the fact,
that we cannot altogether keep reflection upon thought apart from
reflection upon the nature of things thought about.]

If now we ask for a definition of Logic, to keep before our minds
in the following chapters, perhaps it is simplest and least objection-
able to call it the Science, or the Study, of Thought ; for to say of
the Formal Principles of Thought might imply both that there were
sciences which did not seek for principles, and that the form of
thought can be studied without reference to differences in the matter
of it ; neither of which things is true.

I] GENERAL CHARACTER OF THE ENQUIRY 9

It is sometimes held that Logic is rather an art than a science, or
at any rate that it is an art as well. In considering this question,
we must remember that there are two senses of the word art. We
may say that a man understands the art of navigation when he is
skilful in handling a ship, though he may be unable to explain the
principles which he follows ; or we may say that he understands it,
when he is familiar with the principles of navigation, as a piece of
book-work, though he may never have navigated a ship. Thus an
art may either mean practical skill in doing a thing, or theoretical
knowledge of the way in which it is best done. In the latter sense,
art presupposes science ; the rules of navigation are based upon
a knowledge of astronomical, mechanical, meteorological and
physical laws, and presuppose much knowledge of mathematical
and other sciences. It is in this sense that Logic is called an art ;
and hence it is clear that if there is an art of Logic, there must first
be a science, for the study of the nature of sound thinking must
precede the giving of instructions for thinking soundly. And even
granting the existence of such an art, it remains distinct from the
science ; so that the name Logic would be used of the two in dif-
ferent senses, and we ought rather to say that Logic means the
science or the art of thought, than that it is the science and the art
thereof. That there is an art of Logic, based on the science of Logic,
might be urged on the ground that Logic reveals to us what know-
ledge about any subject really is, and certain canons of reasoning
which no argument can violate and be sound. But more than this
would be required in order to constitute an art. There should be
rules prescribing measures by which to bring our thought into the
forms indicated. An artist, as Aristotle says1, initiates change
in something other than himself : a sculptor e.g. in the clay which
he models, a physician in the body of his patient ; and if in his own
body, he treats himself as he would another. The execution of
such changes is indeed different from the rules to be observed in
executing them. But the logician's business is not to give rules
by following which others or he himself may alter their thought
about things, their geometry or chemistry or biology ; he offers
no prescription for coming to know about all subjects ; it is against
such pretensions that a protest like Locke's, quoted above, may
well be made. His business is to become conscious of the nature of

1 Met. A. iii. 1070a 7 ij fuv olv r^vr, iPXtj iv 5XX», f, fie <f>v<ris aoxn f" «y™
( : Art initiates— sc. change— in another thing, the nature <of a thin g) in itself ).

10 AN INTRODUCTION TO LOGIC [chap.

the thinking carried on in those sciences. Logic, as we have said,
6tudies the way in which we already think about things.

Nevertheless, it is not without effect upon our ordinary thinking.
A good deal of our so-called thinking is incoherent, and breaks down
when we criticize it. That we can indeed discover for ourselves
without learning Logic ; an economist can correct his own or his
predecessors' errors in political economy, a mathematician in mathe-
matics ; they could no more wait for the logician to correct than to
construct these sciences.1 Yet the study of the thinking, good and
bad, which has gone to their construction may give us a more lively
consciousness of the difference between what its character should be
and what it sometimes is, or as the Greeks would have said, between
knowledge and opinion. Herein Logic may be compared with
Ethics. Ethics investigates human conduct ; it discusses the
judgements of right and wrong, of good and evil, that we pass upon
men's acts and them ; it tries to determine what we really mean in
calling an act wrong, and what we really require of a man in saying
he should do what is right. All this would be impossible unless men
already acted wrongly and rightly, and made moral judgements ;
Ethics does not teach men to do that. But it does bring into clearer
consciousness the nature of the ideals which we already have, the
grounds of the judgements which we already make, the frequent
discrepancy between what is done and what we recognize should
be done. To this extent Ethics tells us what to do, though it does
not enable us to do it. Similarly Logic helps us to realize what
knowledge of a subject is : but it does not enable us to bring our
opinions on every subject into the form that knowledge requires.
Both Logic and Ethics are thus in some degree practical ; but we
do not call Ethics an art, and it is not desirable any the more to call
Logic so.2

1 The word logic is sometimes used not for the study of thought which
has been described in this chapter, but for the thinking which it studies :
as when we say that some one is a man of powerful logic, or of great logical
acumen. It is important to recognize that this is a different sense of the
word.

2 It must not however be supposed either that Ethics can determine what
ought to be done in every difficult case of conscience, or that Logic determines
exhaustively the forms of reasoning which the sciences must employ. Cf.
F. H. Bradley, Principles of Logic, pp. 247-249. The phrase normative science,
which some writers have of late applied to Logic, Ethics and Aesthetics, has
perhaps been suggested by the character in them to which this paragraph
refers. But it is liable to create misunderstanding, as if it were the business
of these enquiries to prescribe rather than to ascertain the principles which
our rational thinking, or action, or appreciation of beauty exhibits. The

I] GENERAL CHARACTER OF THE ENQUIRY 11

It is perhaps from a desire to show the practical value of the
study of Logic that men have insisted on viewing it as an art. But
it would be a mistake to suppose that it can have no practical value
unless it can furnish rules for ' the conduct of the understanding '.
The direct help that it can give in this way is not very great. Its
practical value in general education is firstly this : that it demands
very careful and exact thinking about its own subject-matter, and
thus tends to produce a habit of similar carefulness in the study of
any other subject. In this it only does for the mind what a thorough
training in any exact science might do. Secondly, it makes us
realize better what the general forms of speech that we habitually
use really mean, and familiarizes us with the task of examining our
reasonings and looking to see whether they are conclusive. In this
it has an effect which the study of some special science like botany
is not equally calculated to produce. Thirdly, it brings more clearly
into consciousness, as aforesaid, what knowing is, and so far furnishes
us with a sort of standard by which to judge what we commonly call
our knowledge of things ; it makes us more alive to shortcomings in
our ordinary opinions. But it does not need for its justification
that we should point to effects which it produces upon our thoughts
about other subjects ; the nature of thought and knowledge is
itself a subject worthy of investigation.1 And, if we are to look
also beyond this, its chief value lies in its bearing upon those ultimate
problems, concerning the nature of reality, and man's place and
destiny in the world, from which at first sight it might seem far

peculiar character of Logic, Ethics, or Aesthetics seems to be this, that we
who, in them, reflect upon thought, conduct, or art, ourselves also in other
moments of our activity create these objects of our reflection ; and because in
our reflection we recognize the failure of many of our attempts to think
soundly, act rightly, or work beautifully, it is supposed to be the business
of reflection, logical, ethical, or aesthetical, to rectify these failures. Such
a supposition is in the main erroneous. It is by becoming better men of
science that we shall correct our scientific blunders, by becoming better men
that we shall correct our moral judgements and choices, by becoming better
artists that we shall correct our aesthetical ; nor does the recognition of
a should-be surpassing what is require that we pursue those reflective dis-
ciplines. But the exercise of intelligence which they require presupposes the
capacity and provokes the activity of that displayed in science, morality, or
art themselves ; and so there is a connection between them and the improve-
ment of our scientific or moral or aesthetic thinking, such as does not exist
between biology and the improvement of species or between dynamics and
the improvement of locomotives.

1 Cf. Bosanquet, Logic2, i. p. 1 : ' I am wholly of Hegel's mind when he says
that the species of syllogism are at least as well worth discovering as those
of parrots or veronicas '.

12 AN INTRODUCTION TO LOGIC [chap.

remote. ' Logic,' says J. S. Mill, in the Introduction to his famous
work \ ' is common ground on which the partisans of Hartley and
of Reid, of Locke and of Kant may meet and join hands.' Conserere
manus — it is only in this sense that rival schools join hands on the
field of Logic. The dream of a Logic that shall be ' neutralized '
like the physical sciences will not be fulfilled. These may move
securely within the limits of certain well-defined assumptions, which
all workers, though they may fight over minor points, agree to
respect. Logic, which studies the principles of our thought about
all things, cannot be content to leave unquestioned the assumptions
within the limits of which thought proceeds : for it is those very
assumptions that it investigates. The history of Mill's own work
disproves his saying, for it is on its metaphysical side that it has been
most vehemently attacked. Into metaphysical controversies, how-
ever, it is not the aim of this book to enter more than is absolutely
necessary. But he would essay a vain task, who should attempt
to expound the rudiments of Logic with no presuppositions about
the nature of things. We may distinguish thought from the things
thought about, but we cannot study it without any reference to
what they are. All thought is thinking ' this about that ' ; and
the general nature of the ' this ' and the ' that ' must be considered,
if we are to consider what thought is ; otherwise, our subject
becomes a blank. The operations of the mind are unintelligible,
if we disregard altogether the nature of their objects. To know
what desire is, we must know what can be desired ; there are some
who hold that desire, by its very nature, is for pleasure ; if so, could
we understand it without considering what pleasure is ? So we
cannot understand thought without considering in general what
thought is of. And consequently Logic, just because it studies our
thought about things, is concerned with questions about the
general nature of things.2 Some would dissent from what in
the following chapters is said on such questions. The controversies
involved are not there pursued as they deserve, for this is not
primarily a work on Metaphysics ; but they have at any rate been
indicated where they arise.

1 System of Logic, In trod. § 7.

2 Thus recent Symbolic Logic is full of discussions about classes and the
relations between classes, because it holds thinking to be fundamentally
thinking about the relations of classes. It seems to me that classing and
class -relations are a very secondary subject of thought, and that for thia
reason Symbolic Logic gives a very distorted theory of thinking.

i] GENERAL CHARACTER OF THE ENQUIRY 13

[The connection between questions about our thinking, and what
we must think things to be, is excellently shown in the so-called
Laws of Thought. These are certain very general principles exem-
plified in all thinking, and some have supposed the task of Logic
to consist merely in developing their implications. They are known
as the Law of Identity, the Law of Contradiction, and the Law of
Excluded Middle. The Law of Identity may be formulated by saying
that ' whatever is, is ' : or symbolically, that ' A is A ' ; the Law
of Contradiction, that ' a thing cannot both be and not be so and so ',
that ' contradictory propositions cannot both be true ', or that ' A
cannot be B and not be B ' ; the Law of Excluded Middle, that ' a
thing either is or is not so and so ', that ' contradictory proposi-
tions cannot both be false ', or that ' A either is or is not B '. In
other words, if we think about anything, then (1) we must think
that it is what it is ; (2) we cannot think that it at once has a char-
acter and has it not ; (3) we must think that it either has it or has
it not. Now though these are called laws of thought, and in fact
we cannot think except in accordance with them, yet they are really
statements which we cannot but hold true about things. We cannot
think contradictory propositions, because we see that a thing cannot
have at once and not have the same character ; and the so-called
necessity of thought is really the apprehension of a necessity in the
being of things. This we may see if we ask what would follow, were
it a necessity of thought only ; for then, while e.g. I could not think
at once that this page is and is not white, the page itself might at
once be white and not be white. But to admit this is to admit that
I can think the page to have and not have the same character, in
the very act of saying that I cannot think it ; and this is self -con-
tradictory. The Law of Contradiction then is metaphysical or
ontological. So also is the Law of Identity. It is because what is
must be determinately what it is, that I must so think. That is why
we find a difficulty in admitting the reality of absolute change,
change when nothing remains the same ; for then we cannot say
what it is which changes ; ' only the permanent ', said Kant, ' can
change '. The Law of Excluded Middle 1 is so far different as
a disjunctive proposition expresses doubt, and doubt belongs to the
mind, not to things But to deny that this page need either be or
not be white is to deny that it need be anything definite ; determin-
ateness involves the mutual exclusiveness of determinate characters,
which is the ground of negation ; and that is a statement about
things. In other words, unless the primary Laws of Thought were
Laws of Things, our thought would be doomed by its very nature to
misapprehend the nature of things.]

1 On this cf. further infra, p. 41, n. 1.

CHAPTER II
OF TERMS, AND THEIR PRINCIPAL DISTINCTIONS

Logic, we have seen, studies our thought about things ; and that
cannot be studied without some consideration of the nature of things ;
but further, it cannot be carried on, nor yet studied, without the
use of signs — generally written or spoken words, which make what
we call language. The relations of thought to things on the one
hand and words on the other are difficult and intricate ; but we
cannot without some regard to them profitably attack the subject
of this chapter.

The true unit of thought, the simplest complete act of thought,
or piece of thinking, is the Judgement, or Proposition : between
which where a distinction is intended, it is that the proposition
is the expression in words of a judgement. The close connection
of language with thought appears already here ; for the utterance of
the words, unless we were at the same time meaning with them, or
judging, would not really be making a proposition ; else the man
who repeated the words of an unknown tongue would be ' propound-
ing '. We may indeed understand a proposition without judging
it, but only by imaginatively putting ourselves in the situation of
a man who is actually expressing his judgement by it.

We may perceive without judging, though our present perception
may be possible only through past judgements ; and here as else-
where the history of how the individual mind has come to be able
to do what it now does is elusive ; but that belongs rather to
Psychology. I may pass a man in the street, and only afterwards
eay to myself ' That must have been so-and-so ' ; I may be walking
along a railway line in the dark, and hear a sound, and then hear
it again, and for the first time think ' That is the noise of a train
approaching '. I perceived the man, or heard the sound, the first
time ; I judged about them after ; and when I judged (we shall
return to this) I distinguished in the ' subject ' I judged about
a character which I ' predicated ' of it.

In judging then I always distinguish a particular element, the

TERMS, AND THEIR PRINCIPAL DISTINCTIONS 15

predicate, in the being of a subject which I could not think of unless
I recognized in it some other than the predicated character.1 I must
think, severally yet together, of both ; and if I want to call attention
to them separately, I must indicate them by different signs ; but
in order to make the judgement, though I need a sign, I do not need
to indicate them by different signs. The child that learns to say
1 Pussy ' when it sees the cat means by the single word what we
should express by the proposition ' There is the cat ' or ' I love the
cat ', or whatever it may be ; and Mr. Alfred Jingle expressed his
judgements with less than the full complement of words.

Whether any thinking can be carried on without some sort of
sensible signs 2 is disputed ; certainly it cannot be carried far. The
signs need not be written or spoken words ; they may be gestures, or
sensations of touch, by means of which Helen Keller was taught to
think. In algebra, though they can be written, they are not words ;
in geometry the figure serves to a great extent, and one may think
out a demonstration by help of drawing the lines of the construction
with less use mentally of words than would be necessary to com-
municate it. Perhaps, with the figure before one, attending suc-
cessively to its parts, one may dispense for a time with other signs
altogether — other signs, because the figure itself is, as Plato noticed,
a sort of sign : our demonstration is not true of it, since it is im-
perfectly drawn, but it helps us to think of the figure whereof it is
true.3 And perhaps when we are perceiving a thing we can make
judgements to ourselves about it, without help of any sign, because
it is itself sensible ; and when we are not perceiving it, some ' mental
image ' may serve instead of language. For the imagery which ac-
companies thinking is not the object of thinking ; I may as a
psychologist make it the object of my thinking, and say that it is

1 Hence a definition is not properly a judgement, as Aristotle saw (v. Met.
0. x. 1051b17 sq.). For when I define anything— e.g. a triangle, and say
that it is a three-sided rectilinear figure — I have not before me a subject
already distinguished by some other character than what I predicate. Even
here however I distinguish elements in an unity which they constitute ; and
hence the definition can be expressed in a proposition. For I give a name
to this unity as an unity, and also to the elements distinguished in it. There
are some objects of thought which have names, and by the help of instances
we come to know them, but because they are simple, or because they are
unique in nature, what they are cannot be expressed in a proposition— e.g.
difference— though judgements may be so expressed which tell us various
things about them : e.g. 'difference is a relation ' or ' attracts attention '.

2 I do not imply that some signs are not sensible, but merely wish to call
attention to the fact that all are so.

* Rep. vi. 510 d, E.

16 AN INTRODUCTION TO LOGIC [chap.

vivid, or evanescent, or what not ; but that is not the thinking in
connection with which it first arises. Its service to thought seems
to be comparable with that of words, so that it has been called the
1 inner speech-form ' ; though it is not articulated as language is.
These considerations seem to point to the conclusion that language
is necessary to thought because so much that we think of in things
is not itself sensible, and we cannot fix our attention on what is not
sensible, without the help of something that is ; but there need be
no correspondence in detail between the sensible sign, and the
structure of our thought and of its object. This has not always
been realized ; and because a child first learns separate words,
and then learns to combine them in sentences, and then to combine
sentences in continuous discourse, it has sometimes been supposed
that thought begins with isolated apprehensions of what it after-
wards makes subjects and predicates in judgement, and then builds
up judgements into reasoning. Such a view is an illusion produced
by language, particularly through the consciousness of the separate-
ness of words which modern writing and reading produces. It is
indeed supposed by many that in early language words had not a
separate existence, but only existed as it were confluently with one
another in sentences.1 Anyhow, there are no ' ideas ' 2 which we put
together in thinking as we do words in speech and writing.

Though the signs by help of which we think are thus various,
words are incomparably the most important ; and they are almost
always 3 the only ones by help of which we express logical doctrine.
Words are signs sometimes of things thought of, sometimes of opera-
tions of thinking, sometimes of both together. The subject-word
in a proposition is a sign of something thought of, for which it is
said to stand, and the proposition is not about it but about what it

1 I have seen a letter written by an Alpine guide in admirable French,
but wildly at fault in its division of words.

2 No word in philosophy has been responsible for more confusion than the
word idea. In Plato it meant what is called in Logic an universal, the common
nature which thought recognizes in different particular things. Nowadays,
it sometimes means an opinion (as when I say that my ideas on a subject
have changed), sometimes ' mental images ', sometimes it is merely an element
in a periphrasis : to ' have an idea of ' is simply to conceive or think of ;
then we are apt to suppose that we think of things by means of ideas of
them, which is no more an explanation of thinking than if I say that I think
of things by means of thinking of them.

3 Most writers make some use of symbols which are not words to represent
objects of thought (e.g. Arabic numerals) ; and in Symbolic Logic they are
extensively used to represent both objects and operations of thought.

ii] TERMS, AND THEIR PRINCIPAL DISTINCTIONS 17

stands for : except when we say something about the word itself ;
an instance of the former is ' Barkis is willin' ', of the latter ' Barkis
is a proper name \ Words like if, because, therefore are signs of the
acts of supposition or inference, and is is the sign of the act of
judgement, though also implying that something exists.2 Other
verbs, and also adjectives, are signs at once of some object of thought
predicated, and of the act of predication ; and the same verb may
be a sign of the subject of predication as well. Thus in the pro-
position ' Dogs bark ', ' dogs ' stands for the things about which the
statement is made, ' bark ' both is the sign of (or expresses) what is
predicated about them, and also of its being predicated ; if I wish
to disentangle, as it were, the sign of what is predicated from the
sign of predication, I must say ' Dogs are barking animals ', or some-
thing of that sort. The word Perii expresses both the subject
about which the statement is made, viz. the speaker (though it
does not stand for it), what is predicated of it, and the act of
predication ; and if subject and predicate are to be disentangled,
one must say ' I am undone '. Even here the disentanglement is not
complete, because ' undone ' does not so stand for what is predicated
of me that I could make it the subject in another proposition about
that ; for this purpose I should have to say ' I am a man undone ' ; I
could then go on and say ' A man undone has no energy ', or what-
ever it may be.3 Words are often made signs of these divers things
at once by means of inflection* To substitute for a proposition
expressing subject or predicate or both by the same word or words
that express also the act of predication another in which distinct
words express each of the three is called putting it into logical form.
Where (as often in Logic) we wish to make subject and predicate
separately subjects of logical discussion, this transformation is
necessary, though it often does violence to the idiom of language.

Now the subject and predicate (Gk. v iroKeCfxevov and naTrj-yopovpevov),
but not the act of predication, are called the terms in a judgement;

1 Cf. infra, p. 19. 2 Cf. infra, pp. 163-166.

8 Neither the words ' a man undone ' nor (in the previous example) ' barking
animals ' stand for the character attributed ; that is ' being undone ' or ' the
habit of barking ' ; and if we use words that stand for it, and not for the
things characterized by it, it cannot be attributed by the verb to be, but by some
verbiikehave — e.g. 'Dogs have the habit of barking'. Cf. infra, pp. 37-38, 157.

4 Even in a comparatively uninflectional language like English, in a suitable
context, a single word may be a proposition : for example, in a telegram, the
word ' coming '.

5 There is no reason why Logic should ' put into logical form ' the examples
in which it studies thinking, where this is not wished.

1779 O

18 AN INTRODUCTION TO LOGIC [chap

and thus every judgement contains two terms, and they may be called
elements in the judgement or the proposition, and it may be said
to be resolved into them.1 This again illustrates how language and
thought are bound up together. A proposition is a sentence, but
not merely a sentence : it is a sentence expressing or meaning a
judgement. Otherwise we could not speak of resolving it into its
terms ; for the subject and predicate words, at which we thus
arrive, need not have been in the unresolved proposition ; and
a mere sentence could not be resolved into words that were
not in it.

It is easy then to see that a term is not the same as a word. In
a judgement there are always two terms, but a single word may
express both ; Caesar's famous message of three words ' Veni, vidi,
vici ' contains as many distinct propositions, each of which may be
resolved into the same subject-term ' I ' and a predicate-term
which is different. Contrariwise many words may make one term ;
and this is the commonest case. Subject and predicate may each
be expressed by a single word, e.g. 'Tastes differ', 'Regret is
foolish ' ; but in ' Dead men tell no tales ', ' The kingdom of heaven
is within you ', each term consists of several words. Again some
words cannot normally be the terms of a proposition at all. They
do not indicate by themselves any object of thought, but are either
used, like an article, in conjunction with some descriptive word, to
designate an object, or, like an adverb, to qualify what another word
expresses, or, like a preposition or conjunction, to mark some
relation between different parts of a complex object of thought, or

1 "Opov <a\S> els ov SiaXverai 17 Tvporams (' I call that a term into which the
proposition is resolved'), Ar. Anal. Pri. a. i. 24b 16. 'Term' is terminus,
a translation of the Greek opos. It is not quite easy to see why the parts
into which the judgement can be broken up were called opoi. The statement
that ' a term is so called because it forms one end of a proposition ' (Jevons)
is clearly wrong ; for that is an accident of language ; even in English ' hungry
I was, and ye fed me ' would not be impossible, instead of ' I was hungry'.
It may be that Aristotle, like the manuscripts of the Organon, symbolized
the proposition in the form ' A — B ' (where we should write ' B is A '), and
that the use of the word comes from the position of the symbols. Bonitz
(Index Arist., s.v. opos, 530a 21) thinks it a metaphor from mathematics,
where if the ratio of two quantities was considered, these were called opm,
being represented by lines, which are the boundaries of a plane ; in the
judgement, there is a relation of subject and predicate, which might therefore
be called opm too. The word is, however, also used like opm p6s, to mean
definition ; and it may be that subject and predicate were called opoi as the
determinate objects of our thought in a particular judgement, or as together
comprising what is propounded, and limiting the judgement in which they
occur to its own field.

ii] TERMS, AND THEIR PRINCIPAL DISTINCTIONS 19

(as we have seen) to express an operation of thought.1 Such
words are called syncategorematic (o-vyKarriyoprifiaTiKa) because only
capable of being used along with others in predication ; while
words which signify what can by itself be a subject or predicate in
thought are called categorematic. These, indeed, while capable of
being used by themselves as terms, may also enter into a term among
the words of which it is composed ; thus man is a term in the
proposition ' Man hath found out many inventions ', but not in the
proposition ' The heart of man is deceitful ' : the sea in the proposi-
tion ' The sea shall give up his dead ', but not in the line ' She left
lonely for ever the kings of the sea '. In this line the words italicized
are syncategorematic ; but sea is not syncategorematic, because it
can stand for a term, though here it does not do so. Terms com-
posed of words of both kinds have been called ' mixed terms \ It is
true that syncategorematic words, though standing for nothing
whereof anything can be asserted, or which can be asserted of any-
thing, can yet as words be made the subject of linguistic or gram-
matical discussion, as when we say 'Of is a preposition ', or ' is the
sign of the genitive case in English \ When words which stand for
no complete object of thought are made objects of our thought
themselves as words, it is said to be by a suppositio materialis.

1 With the articles may be coupled words like some and any ; not, and no
in 'no man ', are also syncategorematic ; so is the copula iff, as the sign of
predication, though not when it means ' exists ' and is itself the predicate.

2 The doctrine of suppositio, as of divers other ' properties of terms ', has
happily fallen into oblivion ; but for the benefit of any one who wishes to
understand the phrase suppositio materialis it may be worth while to add
a note on it. All parts of speech were said to have signification ; then, as
sounds having signification, they acquired properties which did not belong
to them as mere sounds. These properties were not the same for every part
of speech. Suppositio belonged to substantives denoting substances, copulatio
to verbs and adjectives. Substantiality and adjectivality were characters of
the things signified ; the adjective coupled some adjectival with some sub-
stantival thing, the substantive ' put ' the latter ' under ' the former (v. Prantl,
Geschichte der Logik im Abendlande, vol. II. Abschn. xv. Anna, 67 ; vol. III.
xvii. 59). So far, the sense of suppositio seems to be active ; it is defined
as acceptio termini substantivi pro aliquo ; suppositio puts the substantive,
instead of what it stands for, under what is adjectival ; it takes the sub-
stantive term for or as representative of something, and predicates about it.
But since we do thus supponere the substantival term, suppositio was said to
belong to it, in the sense that not the act of ' supposition ' belongs to it, but
being the subject of that act ; and then it was itself said supponere pro aliquo,
i.e. to stand for, or be put for (not to put for), something (cf. Prantl, vol. III.
xvii. 61, 201 : Sanderson's Compendium Logicae Artis, Lib. II. c. 2). The
same term had different kinds of 'supposition ' according to what it 'stood
for'; e.g. in 'Homo est animal', homo stands for all men, and this is the
tuppositio naturalis of a common term; in 'Homo currit', it stands for

02

20 AN INTRODUCTION TO LOGIC [chap.

Some logicians have preferred to speak of names, rather than
terms, or have been ready to apply to a term Hobbes's well-known
definition of a name. 'A name ', he says, ' is a word taken at
pleasure to serve for a mark, which may raise in our minds a thought
like to some thought we had before, and which, being pronounced
to others, may be a sign to them of what thought the speaker had,
or had not, before in his mind . This definition, if we omit the
words ' or had not ', expresses fairly well the function of a name ;
but it is not equally appropriate to define a term ; for not all words
or phrases which can be predicated of anything would be called
names of it, and yet they may all serve as terms. That word is the
name of anything which we might give in answer to the question
' What is it called ? ' — either, if the thing is a concrete individual,
a word used to direct our thought just to that individual, irre
spectively of what it is, or, if our attention is to be directed by a
name that signifies what that is which we are to think of, a word
signifying not some attribute or detail in its being, but its essential
or (if one may so say) most constitutive being.2 Of the first sort

some individual, and this is suppositio personalis. Now as a sound having
signification, the term was distinguished into the sound as matter, and the
signification as form ; and when a predication was true of a term as a sound
or in respect of its matter, as in 'Homo est disyllabum ', it was said to be
by suppositio materialis : when in respect of what it signified, by suppositio
formalis. There can be suppositio materialis of any part of speech, but
formalis only of substantives ; for only a substantive, or substantival phrase
(haec enim significat rem ut subsistentem et ordinabilem sub alio, v. Prantl,
vol. III. xvii. 60) can have suppositio formalis. Cf. p. 157, infra.

1 Computation, or Logic, c. ii. § 4. By the words ' at pleasure ' Hobbea
does not mean that everything about the formation of names is arbitrary,
but that there is nothing in a particular sound making it of itself more suited
than another to suggest what it stands for ; of course this does not apply
to names derived from others already significant, but to the formation
of underived names it does apply, unless they are ' onomatopoeic '. So
Aristotle says that a name is (pa>vfj ar]iu.avTLKq Kara <tvv8i]kt]v, 'an articulate
sound having signification by convention ' (de Interp. ii. 16a 19). The words
' or had not ' should go out : a name cannot be a sign of what I am not
thinking of, and even a negative judgement does not express the thought
I have not in my mind, but the thought which I have, that ' this is not that '.
What does Hobbes mean by a thought ? — thinking, or the thing thought of ?
a name makes one think of a thing, or 'raises in my mind the thought ' of
a thing. My using it is a sign to others that I am thinking of that thing ;
but it itself is rather a sign of the thing ; and when I use names only in my
private thinking, they are not signs of my thinking at all, but rather instru-
ments. A name also may consist of more words than one, e. g. Stoke Poges.

2 The usage of the word ' name ' is somewhat uncertain, and the distinction
not sharp, because it is often difficult to say whether what a word signifies
about that of which it is predicated is its essential being. We should probably
agree that we give a screw-wrench its name when we call it a screw-wrench,

ii] TERMS, AND THEIR PRINCIPAL DISTINCTIONS 21

are proper names, like Caesar, the Thames, Europe ; of the second,
the general names of substances, like man, river, lead, and the names
of the kinds, attributes, and relations of things, like humanity
( = human nature), jealousy, distance. But words used of a subject
to signify its possession of some attribute or relation, or used of these
to signify their presence in a certain subject, or something ' about '
them, are not names ; ' the Great Commoner ' is not a name of Pitt,
' the sin of Adam ' not the name of disobedience, ' the needful ' not
the name of money, nor ' the continuous ' of quantity.1 Amber-
gris is a valuable substance found in the body of some sperm-
whales ; ' ambergris ' is the name of that substance, ' found in the
body of some sperm-whales ' is not ; but both are terms in that
proposition. And there is another reason for distinguishing name
and term. There is always a contrast in our minds between a name
and what it stands for ; but a term is so bound up with its meaning,
that we often mean by ' terms ' the objects of thought which are
subject and predicate, not the words signifying them. Only so could
we speak of resolving into its terms a proposition which does not
contain the words which we get by our resolution of it. We say too
that the subject-term in a proposition is that about which we predi-
cate ; but we seldom predicate about the words ; when the messenger
announced to Macbeth ' The Queen, my lord, is dead ', it was not
of the words that he spoke. To avoid confusion, it is sometimes
necessary to indicate whether by the terms of a proposition we mean
what is thought of, or the words signifying that ; and we might
call the former the terms of thought, the latter, the terms verbal.
We shall have to give different definitions of a term accordingly.
We may define a term of thought as ' whatever can be thought of as
the subject or predicate of a proposition ' : 2 a term verbal as ' a

but not a carpenter, when we call him a carpenter ; because in the being of
a carpenter to be a man is fundamental, to be a carpenter incidental, but
in the being of a screw-wrench to be a screw-wrench is fundamental ; carpentry
however is the name of the carpenter's trade. Aristotle has a formula which
can be adapted here. If you can say of a thing called A that it is not some-
thing else in order to be A, ovk a\\o n 6v tcrnv A, then A is its name. (What
he says is that a predicate belongs to A essentially as A, if it is not something
else than A in order to have the predicate : v. Anal. Post. a. iv. 73b 5-8.)

1 On the function of a name cf. Lotze, Mikrokosmos*, Bk. V. c. iii. § 5,
E. T. vol. I. pp. 627-628.

2 Or ' of a judgement '. It will be noticed that subject and predicate are
equally ambiguous with term ; in the one definition they mean what is thought
of, in the other the signifying words. Nothing is a term except when it i3
thought of as subject or predicate, or used to signify these ; but when we
consider terms in isolation, though there is no given judgement, we consider

22 AN INTRODUCTION TO LOGIC [chap.

word or combination of words capable of standing as the subject or
predicate of a proposition'.

To avoid ambiguity between terms as words and what they stand
for or signify, logicians sometimes give to the latter, when they are
not individuals, the name concepts. The word ' concept ' always
signifies something thought of, never the name of it. Conception is
sometimes used equivalently ; indeed in ordinary speech that is the
word that would be used, and if a man spoke of the Greek conception
of the heavens, he would mean what the Greeks conceived the
heavens to be. But ' conception ' also means the act of conceiving,
as when I say that the conception of an immaterial substance is
known to us first in Plato. The ambiguity is common in English
with words of this formation ; ' narration ' may signify either the
act of narration or the story narrated, ' composition ' either the act
of composing or what is composed ; we may say that a man is
engaged in composition, or that he has sent his composition to the
press. The Greek language distinguished the two meanings by
different verbal terminations, the act by nouns in -cls (like aicrdva-Ls
and v6i](ris, sensatio and intellectio), the object by adjectival words
in -tov (like alo-dnrov and vot]t6v, sensatum and intellectum). As it
is important not to confuse the two, it is best to use the word
'conception ' to signify conceiving, and ' concept ', though it sounds
less familiar, to signify what is conceived.

A concept is not the same as a term of thought, because concrete
individuals, like the Thames, may be terms of thought, as when I
say ' The Thames flows through London ' or ' That ship is the
Victory ' ; but they are not concepts, for we may perceive or
think of, but not conceive them. Nevertheless many terms of
thought are concepts, and it is important to recognize the part they
play. The three following paragraphs may throw some light on
this, though they belong in other respects more properly to the
discussion of the nature of judgement.

It is an old objection to judgement, that since its subject and
predicate are different, it cannot be true ; for according to the Law
of Identity, A is A, and not B.1 But there can be no thinking unless
we allow that the unity of a thing with itself does not preclude
variety in what it is. Still the problem of the One and the Many is

their capacity to be terms. Hence I have said ' can be thought \ or ' is
capable of standing ', not ' is thought ', or ' stands '.

1 Cf. supra, p. 13. This puzzle was started by Antisthenes the Cynic in
the fourth century b. o. Cf . Lotze, Logic2, Bk. I. c. ii. B. §§ 56-60.

ii] TERMS, AND THEIR PRINCIPAL DISTINCTIONS 23

among the chief problems of Logic and Metaphysics ; and if thinking
expresses itself in the form ' A is B ', we must ask what this form
means. Now consider the following examples: (1) 'Barkis is willin",
(2) ' the Emperor is captured ', (3) ' a bacillus is a vegetable', (4) ' to
obey is better than sacrifice ', (5) ' to doubt is to think ' . In the first,
' Barkis is willin' ', the predicate is only one detail in the being of the
subject, but the subject is indicated by a name, which does not
single out anything else in its being : in the second, ' the Emperor is
captured ', the predicate again is only one detail in the being of the
subject, but the subject is indicated by a word which singles out
another detail in its being ; in both there is a predicate-concept, in
both the subject is a concrete individual, but in the second there is
besides the concrete subject a subject-concept; this subject-concept
however is but a detail in the being of the concrete subject. In the
third, the subject is again a concrete thing, and there is a subject-
concept ; but this is not a detail in the thing's being, but is its
essential or constitutive being, neither is the predicate a detail in its
being, but the general being of the subject-concept. Hence while the
first ascribes a character to Barkis, viz. willingness, but does not
mean that being Barkis is willingness, nor the second that being an
Emperor is being taken captive, the third does mean that being a
bacillus is being a vegetable. In the fourth, the subject is not a
concrete thing, but a concept, i. e. something we conceive ; and the
predicate is so too ; but it is not the general being of the subject-
concept, and the proposition does not mean that obeying is supe-
riority to sacrifice. Lastly, in the fifth, as in the fourth, the subject
is a concept, but the predicate-concept is its general being, and the
proposition does mean that doubting is thinking.

Now the points to which these examples should chiefly direct our
attention are these : — (i) concepts are characters (not necessarily
sensible) which we find displayed in individuals ; (ii) they may be
characters which as it were cover the whole being of these individuals
— the phrase is Professor Cook Wilson's — or only details in their
being ; (iii) one character may cover the whole being, or be the
general being, of another ; (iv) where the predicate-character covers
the whole being of the subject, or subject-character, the latter is
the former essentially, and not only may the things denominated
from the subject-character be denominated from the predicate-
character (' a bacillus is a vegetable ', ' a doubter is a thinker '), but
the subject-character itself is the predicate-character (being a bacillus

24 AN INTRODUCTION TO LOGIC [chap

is being a vegetable, doubting is thinking) ; (v) where the predicate-
character is only a detail in the being of the subject, whether indi-
vidual subject or subject-character, the latter is not thus essentially
the former : the predicate-character is incidental to the subject, or
coincidental * with the subject-character in the same individual
subject ; and though the subject, or things denominated from
the subject-character, may be denominated from the predicate-
character, the subject, or the subject-character, is not the pre-
dicate-character (Barkis is not willingness, being an Emperor is not
being taken captive, obeying is not being better than sacrifice).

Thus judgement involves concepts 2 among its terms of thought,
but individuals may be terms of thought also ; but these terms of
thought, whether individuals or concepts, are not in every judge-
ment judged to be related to each other in the same way, though
the forms of language do not always bring out these differences in
the relation between subject and predicate.

It was said that a concept is a character of something, not an
individual thing ; neither is an individual sensible quality con-
ceived— e. g. the black colour of this ink ; but its general or uni-
versal character, that of which it is a particular instance, is conceived.
It is only by an act of thought that I can apprehend that colour
which is the same in black and red and blue. It is also only by an
act of thought that I can apprehend blackness as something the
same in the black of that ink and of this. Concepts therefore are
not sensible. But it would be wrong, because they are not sensible,
to suppose that they are not real independently of the conceiving
mind : that they are products of the activity of conceiving. Unless
what I conceive a thing to be and predicate of it is what the thing is,
my thinking is vain, and doomed eternally to defeat itself. Suppose
that a study of the literary or other evidence leads a man to judge
that Gibraltar belongs to the British Crown. His j udgement concerns
a rock at the entrance of the Mediterranean and a fact in its present
history. The rock exists independently of his thinking about it ; but
not less does belonging to the British Crown,3 or his judgement could
not be true. Yet belonging to the British Crown is not sensible.4

1 Cf. infra, p. 76.

2 Except where both terms are proper names — e.g. 'Eboracum is York',
' Verulamium is not Colchester '.

3 The word exist is sometimes confined to the concrete individual and its
particular sensible qualities, and anything else real is said not to exist but to be.

4 Idealists of the school of Bishop Berkeley would say that Gibraltar does
not exist independently of being perceived or imagined. Most idealists would

n] TERMS, AND THEIR PRINCIPAL DISTINCTIONS 25

[The view that concepts are products of the conceiving mind is
as old as Plato, who rejects it in the Parmenides 132 b, c ; it is ex-
pressed by calling them not vo^rd, things conceived, but vorHxara,
products of conceiving (as a poem or 7rotrj/xa is a product of the poet's
making or ttoit/o-is). Aristotle often countenances it, though perhaps
also holding these mental facts, our concepts, to be in a manner
the same as the intelligible nature of things, the vor]\ia the same as
the vot]t6v. Others, and among English philosophers notably Locke,
have held that the object of conception is altogether mental ; that
concepts are created by the mind in order through their instrumen-
tality to acquire knowledge about real things, but are not real them-
selves. This doctrine is known as Conceptualism. The objection
to it is simple. It holds that concepts render possible a knowledge
of real things when they are so formed as to correspond with the
nature of the things ; but it cannot show how we could be aware of
this correspondence without knowing the nature of the things directly,
as well as the concepts. If we only know the nature of the things
through the concepts, we can no more tell that they correspond,
than we could tell that the existing portraits of a man were like him,
if we only knew his features through the portraits. And indeed it
would be nearer the truth to say that only what is real can be con-
ceived, than that what is conceived is not real. We cannot conceive
a square circle, though we can conceive a square and a circle, just
because, though circle and square are real, their combination in the
same individual figure is unreal and impossible. But there are
difficulties also in the way of saying that all that is conceived is
real. We may ascribe to the same individual subject a number of
attributes, each of which is conceived, and their combination also
conceived, and which yet are not really combined in this subject ;
for example, I might think Gibraltar to be a fortress acquired by
treachery ; to be a fortress is a real attribute of some subjects, to
have been acquired by treachery of others, and their mode of com-
bination is a real mode of combination, exemplified, if not in them,
yet in other attributes : nevertheless such a belief would be erro-
neous. The difficulty here is the difficulty of error. It may be
said that other fortresses have been acquired by treachery, and
therefore what I think Gibraltar to be is what they were ; and so
I am conceiving something real, though ascribing it to the
wrong subject. But— not to mention other difficulties which this
answer does not remove — the elements thought to be combined, or
(as it would be expressed) combined in our concept, may be such as

hold that its existence is at least not dependent on the consciousness of this
or that finite individual, whatever be the relation of things to mind in the
universe as a whole. Without entering upon this question, I am concerned
here to urge that what is apprehended in things by thinking, but is not
sensible, is not les3 really in them nor more dependent on the mind than what
is apprehended by sense-perception.

26 AN INTRODUCTION TO LOGIC [chap.

[have never been combined in any real subject. Our fathers thought
Methuselah to be a man who lived for more than nine hundred
years ; there are things that have done it, such as some of the big
redwoods at Mariposa ; there are things that are men ; but none
that both are men and have done it. Perhaps we ought still to say
that what is conceived is something real, but that in these cases
(where we are dealing with questions of historical fact) the elements
of a complex predicate are conceived, but do not form a real unity,
and are not one concept, because we do not see the necessity of their
combination. Where we suppose ourselves to see a connection
between conceived elements, which nevertheless does not exist — as
Descartes thought that ' Vis Viva ' in a body was as the product
of the mass into the velocity, not into the square of the velocity —
there, when we escape from our error, we realize that we never saw the
connection, because it never existed. We may be inclined to say that
we conceived what was unreal ; but we ought rather to say that we
thought we conceived what we did not conceive.

There remains however a further difficulty about the existence,
or reality, of objects of conception. We predicate what we con-
ceive of individuals ; it was agreed above that a concept must be
other than a mere product of our conceiving because we conceive
the nature of what exists. Yet we can still conceive it when the
individual whose nature we judge it to have been exists no longer.
The whole question of the relation of the ultimate reality to its
appearance in time is involved here.]

[It has been said that concepts x are universal : that what we
conceive is the common nature whereof we find instances in in-
dividual things. But though we do not conceive the particular
instance, our knowledge of it involves conceiving. To hear a sound
is not an act of conception ; but if thought is at work, and I know it
for a sound, I must be aware of what ' this heard ' is. I may be
determined to action by hearing a sound, without thinking : and
hearing words determines me to think of what they signify, without
thinking about the words ; in this case too I hear sounds but do not
think that they are sounds (though of course I do not think that
they are not sounds) and so there is no conception of sound. But
when I think about what I perceive, and apprehend what it is,
the elements of its individual being are known as an instance of
that whereof there may be other instances, and that is universal.
Conception therefore is involved with my perception. This common
nature or universal is not itself perceived, though known in the
perception of its instances. But it is to be noted, that in some cases
the instances can no more be perceived than can their universal
nature. Relations illustrate this. The likeness between my two
hands is not the likeness between your two hands, but each is an

1 On the nature of concepts cf. further, pp. 68-71 infra, especially p. 69, n. 1.

ill TERMS, AND THEIR PRINCIPAL DISTINCTIONS 27

[instance of likeness ; nevertheless though we can see our hands,
we cannot see the likenesses. By and bye it will appear how im-
portant this fact is to the theory of induction. The inductive
sciences seek to discover causal relations. Now causal relations
are displayed in things ; the impact of these stones causes Achan's
death, of those Stephen's. Yet the particular instances of causality
cannot be perceived ; otherwise it would be as easy to perceive
the cause of a flower's drooping as to see it droop.

The existence of universals is often denied ; men are apt to
imagine that if they exist one should be able to find them as one
finds instances of them. Hence the remark of Antisthenes — Xititov
fxev 6p&, linroTrjTa 8e ovx 6p<3, ' I see a horse, but not horseness ' :
to which Plato replied, that it was because, though he had eyes, he
had no intelligence.1 The universal is not one of its own instances,
and cannot be found like them. Nevertheless to deny that there
are universals is to deny all identity between different individuals,
and to do this is to say that we can never, by what we learn of the
connection of characters in one individual, infer one from the
presence of another in a second individual. We may allow that the
relation of an universal to its instances is puzzling ; but the puzzle
comes partly from trying to describe it in terms of some other
relation. The universal is sometimes called a whole, or (for
distinction) a logical whole, and its instances particulars, and we
ask how they partake of or divide the whole among them ; is
there in each a part, or is the whole present in each ? the first is
inconsistent with its unity, the second makes it to be in many places
at once.2 But the question here assumes that the ' logical whole '
is like a physical whole or thing in space : that horses share horseness
as they do a pottle of hay. If we wish to know the relation of its
parts to a physical whole we must consider examples of the quanti-
tative—England and its counties, a day and its hours ; so, if we
wish to know the relation of its parts to a logical whole, we must
consider examples of that in which this relation is exhibited — things
of a kind, different instances of the same quality. We find in re-
flecting on our thoughts about things, that we do think them to be
things of a kind, instances of the same. That is why the present
discussion is logical ; though it is one of the logical problems that
concerns also the being of things.

It has been maintained 3 that there are no instances of relations :
that the likeness between my hands and the likeness between yours
are not two likenesses but the same likeness — not instances of

1 Cf. Ritter and Preller, Historia Philosophiae Graecae9, § 287. In the story
which gives the answer, it is Diogenes who speaks, and a cup and a table
take the place of the horse. 2 Cf. Plato, Parmenides, 131.

3 Cf. B. Russell,TAe Problems of Philosophy, c. ix. (Home University Library),
Principles of Mathematics, § 55.

28 AN INTRODUCTION TO LOGIC [chap

[likeness, but numerically one likeness. Without accepting this,
it may be granted in regard both to relations and attributes that we
are very apt to confuse the instances and the common nature. And
we often denote them by the same name ; ' colour ' means particular
colours when I speak of the colours in last night's sunset ; it also
means colouredness ; every distance is a particular distance, but
their common nature is called distance also. We do not make this
confusion in regard to substances ; men and horses are instances of
their kinds ; and individual men or horses are so much more obvi-
ously different from one another than individual likenesses or dis-
tances or ultramarines that we cannot overlook in them the distinction
between the manifold individuals and the one common nature.1
But perhaps this distinction is more readily seen in substances
because individual substances are something more than instances of
their kind. The true instances of human nature are the human
natures of individual men ; but the human nature of Caesar is what
Caesar is ; and sensible individual substances at any rate we do not
seem to discriminate altogether by what they are.2]

The foregoing consideration of what a term is in general, and of
its relation on the one hand to a word and on the other to an object
of thought, will have helped to familiarize us with some of the facts
determining the main kinds of terms that Logic has to recognize.
The ordinary classifications of terms are classifications of them as
words which signify objects of thought ; but the distinctions are
based on differences in what we think of, or what in general we
think things to be.

In respect of the objects of thought signified, terms are commonly
divided first of all into abstract and concrete : but if we regard
also their character as words, or terms verbal,3 they must be divided
into abstract, concrete, and attributive. A concrete term (verbal)
is the name of a person or thing, an abstract term the name of
a quality or attribute, or relation ; so that the distinction between
the thing and its qualities, between substance and attribute or rela-

1 Yet biologists do not seem always to have asked themselves which they
mean when they write about evolution. Do individual men evolve, or is it
the human nature which is displayed in them all ? and if the latter, and
men are descended from animals whose nature was not human nature, but
has evolved into human nature, what is the relation of the two, or are human
nature and pithecanthropous nature the same common nature ? and if so,
are there many species or only one ?

2 Cf. infra, pp. 54-57. If there are individual substances that are not
sensible but purely intelligible, they must be discriminated by the under-
standing only.

3 i. e. terms as = the word or words signifying an object of thought.

ii] TERMS, AND THEIR PRINCIPAL DISTINCTIONS 29

tion, is the basis of the distinction between concrete and abstract
terms. Attributive terms will be explained later.1

Our notion of a thing, as has been already indicated, involves
two elements, which furnish the basis for a further division of both
concrete and abstract terms into those which are singular and
those which are common or general.2 A thing is, first, an indi-
vidual, having an existence distinct from that of other individuals *.
the page, for example, on which these lines are printed is a different
page from every other in this book. But secondly, a thing has
a character, which may be the same in other things ; just as other
pages in this book, though individually different, are equally pages.
This character, which belongs alike to many individuals, is some-
times called, as we saw, an universal ; and they, as so many different
cases or examples of it, are called particulars : particulars, as we
often say also, of a kind.3

Now the various particulars of a kind, so far as they have the
same character, may be called by the same name : so far as they are
distinct particulars, they will require different names to distinguish
them. Their names as things of a kind are common or general
names : for the name is common to all particulars of the kind, or
applies generally to any ; acorn, squirrel, file, metal, are general
names. Their names as individuals, if they have any, are singular ;
like London, Zoroaster, the Matterhorn ; such names as these we
call proper names. A general term is thus one that is predicable of
any number of individuals in the same sense : a singular term one
that is predicable of one individual only in the same sense : and a
singular term is a proper name if it does not indicate what individual
it stands for by reference to any special element in its being. Smith
for example, as meaning one who works in metal, is a general term,
because I mean the same by calling Dick or Thomas a smith ; if
I use it as a proper name, numerous as are the persons who bear it,
I do not mean the same in each use of it. I may refer to the de-
fender of Acre, or to the witty canon of St. Paul's, or to any of
a hundred and one others, and in each case my meaning is different ;

1 v. infra, p. 36.

2 That this distinction is applicable also to abstract terms is apt to be
overlooked, and I wrongly denied it in the first edition, through not dis-
tinguishing abstract terms and names of universals. I owe the correction
to Mr. H. A. Prichard.

3 Strictly, if what was said on the previous page is right, it is the constitu-
tive nature of each concrete individual that is the instance of the kind.

30 AN INTRODUCTION TO LOGIC [chap.

nor is it through referring to anything in particular of what he was
that I know, when I hear the name, that Sir Sydney Smith is meant,
as it would be if my thought were directed to the same man by calling
him ' the defender of Acre '.

We are seldom at a loss for some general term by which a par-
ticular thing may be denoted ; but comparatively few particulars
have singular terms appropriated to them. Many particulars of
a kind — for example, new pennies — are not distinguishable at all
to our senses, except by each occupying (when we see them together)
a different place ; these will not have each a different name, for we
should never succeed in calling each individual always by its own
proper name. In other cases, though the particulars of a kind
might be tolerably distinguishable — for example, lumps of chalk of
varying shapes and sizes — we have no occasion to refer to them
individually, nor to burden our memory with so many names. We
are content to employ a common or general name, and to specify
the particular object (from among all those that bear the name) to
which we wish to refer, by pointing, or the use of a demonstrative or
possessive pronoun, or some periphrasis. Thus we say ' the picture
there ', and point : or ' this year ', or ' my great-coat ', or ' the bust
of Julius Caesar in the British Museum of which Froude used an
engraving for the frontispiece of his life of Caesar '. Such expres-
sions are indeed in a manner singular terms, for they serve to
designate particular objects ; they are not however proper names ;
they commonly include general terms and are partially descriptive,
and they have been conveniently called designations.

But where particulars of a kind are distinguishable, and we are
interested in them singly and wish to be able to refer individually
to them, we give them ' proper names '. Thus every individual
man has a name of his own, and every field in the country is named,
because the farmer needs to tell his men which particular field to
work in ; and a railway company for a similar reason names or
numbers its various engines and carriages. Though, however, many
particular things have no proper names, all which have proper names
have general names also ; the ' four- acre ' is a field, the ' Cornish-
man ' is a train, William the Silent is a man ; and on the other hand
any particular thing might, if it were worth while, be distinguished
by a proper name. The proper name and the common name thus
recognize respectively the two elements in our notion of a thing
noted above : the proper name recognizes its distinct existence, the

ii] TERMS, AND THEIR PRINCIPAL DISTINCTIONS 3 J

common name its character that it shares with other things : nor
could our thought about things express itself fully without concrete
terms of these two kinds.

[This has not indeed been always admitted. Thus James Mill in his
Analysis of the Phenomena of the Human Mind (vol. i. ch. viii. p. 260,
London, 1869) writes that it is ' obvious, and certain, that men were
led to class solely for the purpose of economizing in the use of names.
Could the purposes of naming and discourse have been as con-
veniently managed by a name for every individual, the names
of classes, and the idea of classification, would never have existed.
But as the limits of the human memory did not enable men to
retain beyond a very limited number of names ; and even if it had,
as it would have required a most inconvenient portion of time,
to run over in discourse as many names of individuals, and of
individual qualities, as there is occasion to refer to in discourse, it
was necessary to have contrivances of abridgement ; that is, to
employ names which marked equally a number of individuals, with
all their separate properties ; and enabled us to speak of multitudes
at once '. The position here taken up by Mill is known technically
as that of nominalism, the doctrine that things called by the same
name have only the name in common ; a doctrine frequently pro-
fessed, but not often stated with such uncompromising clearness as
in this passage. We do not however really call different individuals
by a common name, except because they have or are believed to
have a common nature ; nor is it conceivable that we could name
an individual by a proper name, without at the same time recog-
nizing in it, however vaguely, some character that, as capable of
existing equally in other individuals, might be the ground of a
general or common name. General names then are no mere means of
abbreviating discourse, but their existence is grounded in what we
must think the nature of objects of thought to be. Aristotle's dis-
tinction x between 6/xw^y^a, or things called by the same name
having only the name in common, and <rvv<avviJ.a, or things called by
the same name having also what is meant by the name in common,
may be mentioned here : the distinction is nowadays embodied
from the side of names instead of things in that between equivocal
and univocal terms (v. infra, p. 46). Opposed to nominalism is
the doctrine known as realism, which maintains the reality of
* universals ' or characters the same in more individuals than one —
of squareness as well as squares, justice as well as just men and
actions, man-ness as well as men. If the common nature be held

1 Most clearly stated Cat. i. la 1-12. The Aristotelian authorship of the
Categories is disputed ; but that the doctrine in it is in the main Aristotelian
can be shewn from treatises admittedly his. Cf. for this distinction Top.
C. x. 148a 24 sq.

32 AN INTRODUCTION TO LOGIC [chap.

[only to exist in the various instances, so that there would be no
squareness unless there were squares, nor man-ness unless there
were men, the doctrine is that of universalia in re ; if it be held to
be eternal, so that with the first existence of squares or men began,
and with their disappearance will end, only the manifestation and
not the being of squareness or man-ness, it is that of universalia ante
rem. Conceptualism (v. supra, p. 25) is an attempt to compromise
between the Nominalists and the Realists by saying that different
individuals cannot indeed share a common nature, because no com-
mon natures but only individuals exist, but that nevertheless we
form concepts which somehow correspond with each of a number of
individuals, and by their means we are able to have general know-
ledge, i. e. (on this view) knowledge about an unlimited number of
individuals at once. Conceptualism is the doctrine of universalia
post rem.]

There are thus two kinds of concrete terms, viz. singular terms,
or names of individuals, and common or general terms ; singular
terms can be further distinguished into proper names, i. e. names
permanently assigned to one individual, and designations, i. e.
phrases which by a pronoun or what not serve to indicate an indi-
vidual otherwise than by a name of its own. Now it has not been
stated in the last sentence, what general terms are the names of.
Are they also the names of individuals, or are they names of the
character common to many individuals ? The former view seems
incomplete, for it does not take account of their difference from
singular terms. The latter view is plainly wrong, for man is clearly
predicated of individual men, not of the nature common to them ;
and when I say that man is mortal, I mean that men die, not that
human nature dies ; that is displayed in a succession of individuals
who are born and perish, but is not born and does not perish itself.
We must then accept the former view. General concrete names
are names of individuals, but names of them in respect of their
common nature. Hence they imply the existence of universals,
though they are not the names of these.

Now such universals sometimes have names. It is true their
names are not often used in ordinary talk, for our practical interests
are in individuals, and only in philosophical reflection are we led
to consider the existence of the universal realities whereof they are
instances. Still, the nature of man is so interesting to us that we

1 But what would happen with the death of the last man ? Cf. p. 26
supra, on the existence of concepts.

n] TERMS, AND THEIR PRINCIPAL DISTINCTIONS 33

have the name humanity x ; and we can form names, like ' horse-
ness ' or ' goldness ', when we wish to distinguish the common
nature of horses or parcels of gold from their instances, or we can
use a periphrasis, like ' the nature of gold '. Are we to call such a
name concrete or abstract ? It would commonly be called abstract,
being the name of the common nature of many individuals, con-
sidered apart or in abstraction from them or from what distinguishes
them from one another ; though the substantial nature of a thing
cannot properly be regarded as a mere attribute of it.

The distinction of individual and universal is not confined to
what is concrete. We have seen that attributes and relations also
have their instances. The red of one rose is not numerically the
same as the red of another, however much their being two depends
on their being in different roses, and otherwise they would be in-
distinguishable.2 The distance from London to York, even if equal,
is not the distance from London to Bideford. But as we can only
distinguish the instances of the same attribute or relation by refer-
ence to the substances to which they attach, only the latter
and not the former have proper names. Hence we are apt to over-
look that there are instances of what is abstract. Yet it is clear
that the death of Caesar is one of many instances of death, just as
Caesar is one of many instances of man ; and when it is said that
there are so many births and deaths a year in London, birth and
death are as clearly general terms as house and street in a total of
streets and houses. And that means that they are used in the same
sense of each birth or death, and that ' the birth of X ' or ' the death
of Y ' is a singular term.

So far the case is the same with abstract and with concrete terms.
But men are interested chiefly in the individual instances of what
is concrete, and in the general nature of their attributes or relations ;
and so not only are there no proper names for these, but the general
name, besides being used of them, is used also of their general
nature, or universal. Death, when I speak of Caesar's death or

1 Humanity has of course other meanings, viz. mankind collectively, and
also kindliness ; in the text it means the human nature common to all men.
Cf. also deity.

2 Cf . supra, p. 27 ad fin.

8 Hence it is a mistake to say that the plurals of abstract terms are con-
crete. Deaths, colours, distances are not substances because there are many
of them ; and a concrete term is the name of a substance. But the plurals
of abstract terms often designate not individuals but kinds of attribute or
relation.

1779 D

34 AN INTRODUCTION TO LOGIC [chap.

Alexander's, is a general abstract term, comparable with the general
concrete term man ; when I say that death comes in many forms,
it is the name of an universal, comparable not with man but with
humanity. So colour is a general abstract term, if I speak of the
colours of yesterday's sunset, but the name of an universal — viz.
colouredness — when I say that colour has divers species. The fact
that many words are used both as general abstract terms and as
names of the universals of attributes or relations helps to make
us regard the names of the universals of substances as abstract.
* Colour ', as predicable not of a coloured thing but of its attribute,
is an abstract term ; meaning colouredness it is a word of the same
sort as ' goldness ' ; hence we think ' goldness ' an abstract term
also.1

[It will be seen that there are really two antitheses confused
together when the division of concrete and abstract is offered as
an exhaustive division of all terms of thought, viz. (a) the antithesis
of individual and universal ; (b) that of substance and attribute
or relation. The second member is called abstract in each antithesis
— though what belongs to the first member in (a) may belong to the
second in (b) — because by abstraction two things are meant, viz.
(a) considering the common or universal nature of divers subjects
apart from the particular instances ; (b) considering some particular
element in the nature or being of anything apart from the rest of
its nature. The former is what Locke has most prominently in mind
when he speaks of the formation of those abstract ideas, which exist,
on his view, only in the mind, and do duty instead of any real identity
in the various things called univocally by the same common name.
The latter is what Aristotle meant when he said that the mathema-
tician considers the subjects of his study h a^cupeVet, in abstraction,
i.e. that he demonstrates the properties that belong to what is circular
or triangular merely in virtue of being circular or triangular, neglect-
ing— because they are irrelevant — all other characters of those things
besides their figure. If we are to avoid confusing the two antitheses
we must say that (A) our thought recognizes, and therefore we have
names for (i) individuals, (ii) universals ; the names of individuals
may be either (1) names of them considered as this or that deter-
minate individual, i. e. proper names or designations ; these are the
singular terms of the traditional doctrine ; or (2) names of them
considered as of a certain sort : these are the general terms of the
traditional doctrine. (B) Our thought also recognizes, and therefore

1 Sometimes particular abstracts and their universal nature may be indi-
cated by different words. Act is a general abstract name, action the name of
the common nature of all acts. But ' action ' is also used as equivalent to
' act ', and we speak of an action, and of actions in the plural.

ii] TERMS, AND THEIR PRINCIPAL DISTINCTIONS 35

[we have names for (i) substances or things ; (ii) their attributes
or relations ; and the distinction of singular and general applies
to the names of both these, since both substances and their attributes
and relations are found as instances of a sort ; but singular names
of attributes or relations are all designations, formed by help of
naming the individual substance involved, and not proper names.
The distinction of singular and general does not apply to names of
universals. Now the traditional doctrine ignores the distinction of
individual and universal in regard to attributes and relations, and
calls the names either of the instances or of their common nature
abstract terms ; and when names are coined for the common nature
of substances, which as a rule in common speech have not got names,
it is inclined to count them as abstract also, not having in mind the
distinction of individual and universal.1

These antitheses, though we certainly make them when we reflect
on things, no doubt present difficulties to a closer examination. The
nature of relations, and their difference from the terms related, have
perplexed many, and have led some philosophers, like Mr. F. H.
Bradley, to deny that relations can belong to Reality ; it appears
to us as a system of things in relation, but transcends this in its
own being. And even if we find no difficulty in the existence of
relations, we may be perplexed by the distinction between the two
kinds of related terms, substances and attributes. The individual
substance, we think, exists, and its attributes are elements in its
being existing only in it and not apart from it. But that of which
they are attributes must be something of a determinate kind, not
a mere point of reference for a multitude of attributes. A concrete
name denotes such a determinate thing ; but on the other hand
its concrete nature threatens to break up into a number of dis-
tinguishable factors, each of which by itself would be called an
attribute. Now they cannot be attributes of each other, nor yet
of that which would be left — if anything would be left — if we
abstracted them all, a ' something, we know not what, which we
feign as a support of qualities ', in Locke's phrase. We might
say that each is an attribute of the complete thing, of the individual
in its whole being : that in fact the so-called attribute is rather an
element in the being of that whereof it is called an attribute. But
this still leaves it a question whether in the being of the individual
substance we rightly distinguish its substantial nature, on the
strength of which we call it by a general concrete name, and the
attributes called by abstract names, or whether the substantial
nature is really but a complex of elements or factors in the thing's
being, which, if they were not so numerous, could be named sepa-
rately, and would then be regarded as so many attributes. On this
cf . infra, pp. 53-54.]

1 I owe the outline of this paragraph to Mr. H. A. Prichard.

D2

36 AN INTRODUCTION TO LOGIC [chap

Abstract terms then are the names of attributes or relations ;
but we must understand this definition rather widely. It is not
only sensible qualities, like flavours or odours, whose names are
abstract terms ; each element in the being of the individual concrete
thing, considered singly and in distinction from, although as quali-
fying, the thing,1 is abstract, and its name (where it has any) an
abstract term. Moreover, the thing in question need not be a single
thing (or person) such as a stone or an elephant ; it may be an
assemblage of what we regard as distinct things (or persons), like
a forest, or an army ; but if there are features belonging to this
assemblage, though they are not qualities of any one thing in it (as
a forest may be extensive and an army skilfully or unskilfully
disposed), these features considered in themselves are abstract, and
their names, ' extent ' or ' disposition ', abstract also. Hence dis-
cipline, civilization, paternity, are all abstract terms, though it is
only by a doubtful extension of language that we could call any of
them a quality, like fragrance or sweetness. And we have seen
that commonly, though confusedly, terms like ' animality ' and
' triangularity ' are also called abstract, names, that is, not of the
distinguishable individual elements in the being of the individual
concrete thing, but of the universals whereof either individual
concrete things, or the various distinguishable individual elements
in their being, are instances.

Besides abstract and concrete terms verbal, there is a kind of
terms verbal which cannot well be classed with either — viz. adjectives
and adjectival terms. These are called attributive terms, e.g. red,
beaten, insolvent. They are not the names of qualities, like redness,
defeat, insolvency ; on the other hand, it is those qualities which
furnish their meaning, not the nature of the various kinds of object
to which the qualities may belong. Thus cloth may be red and so
may silk, but we should not explain what is meant by calling them
red if we were to explain the nature either of silk or cloth ; and a
man may be insolvent and so may a company, but to explain what

1 It may be objected that whether a colour is abstract cannot depend on
our considering it in a certain way ; if it is not abstract, we are wrong so
to consider it ; if it is, it is so however we consider it. But if a substance
is an unity into whose being various elements enter and combine not in the
way in which material things combine into an aggregate, but in the way in
which attributes combine into the being of a concrete thing, then to say
that these elements considered singly are abstract merely means that they
are several and can be distinguished, though only existing in the concrete
unities which they form.

ii] TERMS, AND THEIR PRINCIPAL DISTINCTIONS 37

is meant by calling them insolvent we must explain the nature not
of man, nor of a company, but of insolvency.1

J. S. Mill held that adjectives are really concrete, on the ground
that ' white ' is predicated, or is the name, of snow, milk, or linen,
and not of their colour ; that it is an army and not a defeat that is
beaten.2 But it is clear that the subjects of which an adjective
may be predicated can as well be abstract as concrete ; and if the
adjective is concrete because it is predicated of a thing, it should
equally be abstract because it is predicated of an attribute ; so that
if we say that cabbages are common, common will be concrete ; while
if we say that indolence is common, it will be abstract. The fact is
that the distinction of attributive terms from abstract and concrete
corresponds to no further distinction in terms of thought. There
are substances, and there are attributes or relations, and the latter
qualify the former ; but their qualifying them is not a third co-
ordinate sort of reality. It is the nature of an attribute to be of
a subject, as of a relation to be of its terms3 ; and when we recognize
this in instances, we are said to attribute them to their subjects.
But that is an act of judgement, not a term ; there is an attributive
act, but no third kind of object of thought which we can call attri-
butive. In language however there are words which, though they
can be used as predicates, and therefore satisfy the definition of
a term verbal, are not properly names either of a substance or of
an attribute. Adjectives are such words ; but so also are verbs.
Verbs however were overlooked by those who placed adjectives

1 The meaning of attributives may, however, be incapable of explanation
without reference to that, in the nature of the subjects whereto the qualities
belong, which makes them susceptible of these qualities. Thus neither silk
nor cloth could be red unless they had a surface ; neither a man nor a company
could be insolvent unless capable of having debts. Cf. p. 112,n. 1, infra. It
may be added that terms like father or musician are adjectival in sense, and
would by some be classed as attributive ; for though they are substantives,
and are predicated of concrete things, they do not primarily signify the
concrete things of which they are predicated ; a father must be somewhat
else, to be a father. Cf. p. 20, n. 2, supra, and pp. 156-158, infra. Sometimes
indeed an attributive term may signify more of the nature of the subject
than the subject term does, e.g. if I say 'the obstacle was human ', meaning
* a human being ' ; for to be a human being is more of the nature of the subject
than to be an obstacle.

2 System of Logic, I. ii. 4.

3 Mr. F. H. Bradley however holds that a relation between two terms must
be related to them by a second relation, and so ad infinitum, and the impossi
bility of this infinite process is one reason why he holds that Reality cannot
be, though it may appear as, a system of terms in relation. Cf. Appearance
and Reality, Bk. I. c. ii. The view in the text has the support of Professor
Cook Wilson.

38 AN INTRODUCTION TO LOGIC [chap

among terms. For the terms are the parts into which a proposition
is resolved ; in them, taken singly, the act of predication is not seen ;
they are as it were dead members, which could only have been
taken apart because the life of judgement had fled and no longer
bound them together. But in the verb this life lingers, even if
a verb be taken without its subject. Hence logicians, anxious to
express a judgement in a way to facilitate its resolution into its
terms, have often preferred to sunder, even in language, the word
which expresses the predicate from that which expresses its predi-
cation : to take the term as it were out of the verb, and say of
Lear not, with the doctor,1 that he ' sleeps still ', but that he
' is still sleeping '. Now in such a case the predicate is often
adjectival in form ; although not always, for the proposition ' He
plays cricket ' would become, if it were meant that he played
habitually, not ' He is playing cricket ' but ' He is a cricketer '.
Such an adjectival predicate is one of the parts into which the pro-
position is resolved,2 whereas the verb belongs rather to the un-
resolved proposition. The whole question of the separate character
of the adjective, or adjectival word, belongs indeed rather to
grammar than to logic. But when ' term ' means name, or term
verbal, as these are either substantival or adjectival, and the con-
crete and abstract are both substantival, some place is wanted for
the adjectival, and so they are classed separately as attributive
terms. If their form were to be ignored, and they were to be
referred either to concrete or to abstract, they should rather be
considered abstract than (as J. S. Mill would have it) concrete ;
for their invention implies the consideration of some quality or
character in the thing in abstraction from the rest of the thing's
nature.

A special class of terms is constituted by those which are called
collective. Like most other distinctions of terms recognized in
Logic, this is based on a distinction in things. Individual things
or persons may be considered singly : they may also, since there
are many of them, be considered in groups ; and the names of such
groups are collective terms. Thus a group or collection of books
forms a library ; a group of human beings related in certain ways

1 King Lear, Act iv. 7. 1. 13. Cf. p. 17, supra.

2 Adjectives can indeed be used as subjects, e.g. Beati immaculati in via,
where it is possible to take either term as predicate. In many languages
an article is generally necessary in order to make an adjective do duty as
a substantive.

ii] TERMS, AND THEIR PRINCIPAL DISTINCTIONS 39

forms a family ; related in rather different ways, a tribe ; in other
ways yet, an army or a club. Any term that denotes a collection
of objects, with certain resemblances or relations among them, is col-
lective. Collective terms may be either singular or general ; for we
may wish to refer to a group composed of determinate individuals
(as when we say ' the family of King Henry VIII ') or simply to
a group of individuals, no matter who or what, that is composed in
a certain way, such as a family or a regiment : but they are the
names of the individuals taken together, and not of the mode of
organization among them.1 A general collective term is said to be
used distributively of the different groups that it can severally
denote, and collectively of the individuals in any one group ; thus if
we speak of British regiments the term is used distributively of the
Coldstream Guards, the 60th Rifles, the Argyll and Sutherland High-
landers, &c, and collectively of the men in each several regiment.2

We may sum up what has been so far said of the kinds of terms
as follows : — Terms as individual objects of thought are either con-
crete or abstract ; as names or terms verbal, concrete abstract or
attributive ; there are also names of universals, which are commonly
classed as abstract : concrete terms (verbal) are either singular, and
then either proper names or designations, or else general : abstract
terms can only be made singular by help of a singular concrete term,
and without this are general ; some concrete (and a few abstract)
terms are collective, and some abstract terms denote attributes of
a group or aggregate, not of its members. It may be added that
attributive terms are obviously general.

We pass now to a fresh division of terms, made from another point
of view. As we may give a name to a group of things taken
together, which would apply to none of them by itself, so we may
give to a thing or quality, when we regard it in its relation to some
other thing or quality, a name which would not apply to it con-
sidered in itself. Such terms, attributing to one thing or quality
some definite relation to another, are called relative terms : and
in contrast with them, terms that indicate a thing or quality

1 We may speak collectively of a group of abstracts, as when we say that
thenaturalare more numerous than the theological virtues, or that the Triviuro
and Quadrivium may be traced back to the fourth century b. c. My attention
was called to this by Miss Augusta Klein. But there are no names for groups
of instances of attributes or relations ; terms indicating them must do so by
reference to the individual subjects in which they are displayed.

2 The frequent division of terms into abstract, concrete, and collective, aa
if the third were co-alternative with the other two, is therefore a mere blunder.

40 AN INTRODUCTION TO LOGIC [chap.

considered in itself are called absolute. It is clear that if one thing
or quality stands in relation to another, the latter must also stand
in relation to the first ; and the name applied to the latter to indicate
this reverse relation is ' correlative ' ; or, since each is correlative to
the other, the two together are called correlatives. Instances of
relative terms are equal, greater, subject, parent : with their correla-
tives equal, less, ruler, child ; apple, sound, man are absolute terms.

Relative terms are necessarily general,1 like attributive terms ;
for the same relation may be exemplified in many particular in-
stances, and therefore many subjects may stand in that relation
which the relative term is used of them to indicate. They have this
further resemblance to attributive terms, that though meaning
a relation, they are applied to a subject standing in that relation :
as attributive terms are to a subject possessing the attribute which
constitutes their meaning.2 The existence of attributive terms is
grounded in the fact that the various objects of our thought do
possess distinguishable attributes ; and that of relative terms in
the fact that they do stand in distinguishable relations one to
another. It has been contended that all terms are really relative,
because every object of thought stands in relation to other
objects ; at least only the totality of existence can be absolute,
beyond which there is nothing for it to stand in relation to. But
though it is true that everything stands in relation to other things,
things are sometimes considered rather in themselves, and receive
names accordingly ; and sometimes they are considered in definite
relations to another thing, and receive names that indicate that
particular relation. And this is sufficient ground for the distinction
between absolute and relative terms, though there are cases in which
it is hard to say whether a given term is one or the other. Man
is clearly absolute, and father relative, though mountain might be
disputed ; for a mountain is so only by its elevation above the plain,
and yet in calling it a mountain we have in mind many features
besides this relation.

Terms have been further divided into positive, negative, and priva-
tive. A positive term is said to imply the presence of a quality (or
qualities), e. g. greed, greedy : a negative term to imply the absence
of a quality, e. g. colourless, unfit, unfitness : a privative term to

1 Except when a relative word is combined with others into a term whose
whole meaning is singular: e. g. first is general, but the first Pharaohis singular.
8 Cf. supra, p. 37, n. 1.

ii] TERMS, AND THEIR PRINCIPAL DISTINCTIONS 41

imply the absence of a quality where it has been or might be expected
to be present, e.g. deaf, deafness, desiccated.

There is a certain difficulty in the notion of a negative term, and
in the account of it just given ; for no term can be purely negative,
and imply merely the absence of a quality. The Irishman's receipt
for making a gun, to take a hole and pour iron round it, is not more
difficult to execute, than it would be to frame a term whose mean-
ing consisted simply in the fact that a particular quality was not
meant. A term must have some positive meaning, in order to be
a term at all.

It is indeed sometimes said that a negative term includes in its
meaning whatever is not meant by the corresponding positive term.
According to this view, there is no positive term to which we may
not frame a corresponding negative ; to man there corresponds not-
man, to book not-book, to square not-square, to colour not-colour ; notr
man is everything which is not man, and includes therefore not only
the other animal species, but plants and minerals, books and insti-
tutions, birth and immortality ; not-book includes all these but
books, and man besides ; and so forth. The two ' contradictory '
terms (as they are called) comprise between them all that is ;
nothing can be conceived, of which one or the other is not predi-
cate ; and they divide the universe between them. What the
positive term is, does not matter ; for whatever it be, the negative
term covers everything else ; and therefore it may be expressed by
a symbol ; let A represent any term, and not- A its contradictory ;
we may then say that A and not-A between them make up all
that is, or that there is nothing of which one or other may not be
predicated. ' Everything is either A or not-A' 1

1 This formula, ' Everything is either A or not-^4,' is sometimes given as
the ' Law of Excluded Middle. '. The ' Law of Excluded Middle ' (cf. supra,
p. 13) is that of two contradictory propositions one or other must be true ;
they cannot both be false, and therefore any third or middle course between
accepting one and accepting the other is excluded. It has been asked whether
either of such contradictory propositions as Virtue is triangular and Virtue is
not triangular need be accepted ; the former is clearly false, but the latter
does not seem true. The answer is that if any one were to assert that virtue
is triangular (as the Pythagoreans held justice to have the nature of a square)
we should be right to contradict him ; but that no one who realizes virtue
to be incapable of any spatial character at all would ever put to himself the
alternatives, ' is virtue triangular or is it not ? ' and that to one who, not
realizing this, asserted it to be triangular, the proper contradiction is that it
has no figure. The case therefore furnishes no exception to the truth of the
Law of Excluded Middle, provided the alternatives are not at the outset
realized as nonsense ; but no one to whom they are nonsense would expect

42 AN INTRODUCTION TO LOGIC [chap.

Such negative terms as these do not really figure in our thought ;
they are ' mere figments of logic ' x ; Aristotle long ago pointed
out that ovK-avdpu>nos was not properly a name at all ; and he
perhaps extended his countenance too much to it, when he said that,
if we were to call it anything, we must call it a ' name indeter-
minate ' (ovo/jia aoptcxTov) because, being the name of nothing posi-
tive and in particular, it had a purely indeterminate signification;
it was applicable equally to things existent and non-existent.2

The invention of such terms however is explained when we re-
member the relation of a term to judgement. The latter, as we have
seen, is the primitive and remains the complete act of thought, and
terms are got by abstraction from it. Now the affirmative judge-
ment ' All flesh is grass ' ma}'- be resolved into the terms flesh (the
subject) and grass (the predicate affirmed of it) ; and the negative
judgement ' Man is not a fly ' 3 into the terms man (the subject)
and fly (the predicate denied of it). But since we do therein affirm
that man is not a fly, it seems possible to say that the predicate,
not a fly, is affirmed of man, as well as that the predicate fly is
denied of him. This attempt to reduce negative and affirmative
judgements to a common affirmative type, by throwing the negative
into the predicate, is not really defensible, for the negative term
not a fly does not signify the nature of anything, and so is not really
a term ; it should, if it were a general term covering everything
except the corresponding positive, be predicable of all subjects
except flies in the same sense ; but there is no common character
in all these which it is intended to signify. Hence, as we should not
take the trouble to affirm of man nothing in particular, the only
point of the judgement must lie in denying of him something in
particular ; so that the meaning of the ' infinite ' judgement (as it

to test by them the validity of the laws of thought ; for talking nonsense
is not thinking. The objection to stating the Law of Excluded Middle in
the form ' Everything is either A or not-^4 ' is this, that it seems to sanction
the formation of nonsensical contradictories, such as we have examined, no
less than of contradictories that are rational. Cf. also Bradley, Principles
of Logic, I. v. §§ 23, 24.

1 Stock, Deductive Logic, § 133.

2 de Interpr. ii. 16a 30-33 : the technical term in Latin is nomen infinitum,
whence the English phrase ' infinite term ' is derived : but infinite means in
this context indeterminate ; and for the sake of perspicuity, the latter word
has been used in the text.

8 Why hath not man a microscopic eye ?

For this plain reason, man is not a fly.

— Pope, Essay on Man, i. 193.

ii] TERMS, AND THEIR PRINCIPAL DISTINCTIONS 43

is called) ' Man is not-a-fly ' lies in the negative judgement ' Man
is-not a fly ', and it is clear that we have not resolved the negative
into the affirmative form, when such affirmative can only be under-
stood by restoration to the negative.1 But it is out of such attempts
that so-called purely negative terms like ' not-fly ' have arisen ;
and it is only by understanding that the term A has been the pre-
dicate of a negative judgement, that we can understand how the term
not-A should ever have been formed.

There are however certain negative terms which are not such
mere figments of logic as the ' infinite terms ' considered above.
Where the positive is not a general concrete term but is attributive,
there the corresponding negative may be quite legitimate ; indeed
the distinctions of positive, negative, and privative most properly
apply not to all, but only to attributive terms, or to abstract terms
founded upon these.2 For all attributive terms imply a subject of
which they may be predicated, and to which they refer that attribute
which constitutes their meaning. Therefore even if the term be
negative, it still suggests a subject which, lacking the attribute which
the negative term excludes, is conceived as having some character
instead. And here we have a basis of positive meaning to the nega-
tive term ; for let A be a positive term ; then not-A will signify
what a subject, which might be A, will be if it is not A. Thus intem-
perate signifies what a man, who might be temperate, will be if he
is not that ; uneven suggests what a line or surface, such as the sur-
face of a road, will be if it is not even ; not-blue suggests what a
thing which might be blue (that is, an object having some colour)
will be if it has not that colour. The definiteness of the positive
meaning which a negative term thus conveys will vary greatly, ac-
cording to the range of alternative attributes which we conceive
possible to a subject that might conceivably have possessed the
attribute denied of it ; thus intemperate has a more definite meaning
than not-blue, because when temperance is excluded, though there
are many degrees of intemperance, yet they have more affinity with
one another as opposed to temperance than have the remaining
coiours as opposed to blue ; unruffled has a more definite meaning

1 Cf. Arist. Metaph. A. vii. 1017a 18 ovrat 8e Xeyerat Kai to fif] XevKov tlvai, on
<5 o-vfx$e(iT]Ktv, eKflvo io-nv ('And in this sense the not-white is said to be,
because that is which is not white') — i.e. to be not white cannot itself con-
stitute the being of anything, but that may have a positive being of which
we can deny that it is white.

8 Cf. infra, p. 45.

44 AN INTRODUCTION TO LOGIC [chap.

still, for a surface which is not in any way ruffled can only be
smooth.1

It has been alleged that ' not-blue ' does not necessarily imply
' coloured in some other way than blue ', nor ' not-even ' a surface
of another kind than even ; that it is as true to say of banter that it
is not blue as of a buttercup, and that larceny is as much not-even as
Lombard Street. But such a contention misinterprets our thought.
Just as privative terms imply the absence of an attribute from
a subject that possessed or should have possessed it, and therefore
must convey a notion of what the subject consequently is without
that attribute, so negative terms (at any rate when they are not
mere figments of logic) imply the absence of an attribute from
a subject that might conceivably have possessed it, and therefore
convey a notion of what the subject is instead. The attribute
which a negative term excludes belongs to a genus of attributes
(as blue belongs to the genus colour, or prudence to the genus
feature of human character, or square to the genus figure) ; and
if a subject is unsusceptible of any attribute within that genus, we
should not be at pains to deny of it some particular attribute
therein ; since the soul for example has no figure, we should not
say that it is not-square ; since furniture has no feature of human
character, we should not call a towel-horse imprudent. The nega-
tive term is only used of what must have some attribute within its
genus ; and this genus furnishes a substratum of positive meaning
to the negative term ; not-blue does mean ' coloured not with blue '
and not-even ' having a surface which is uneven ' .2

1 The old Greek proverb will illustrate the point here — e'a-dXoi pev yap
dTrAcof, navTo8(ma>s 8e kcikoL (' Men are good in one way, but bad in many ').

2 The genus within which any attribute falls, or the subjects susceptible
of some attribute within that genus, may be called with de Morgan (Formal
Logic, p. 41) a 'limited universe' ; thus blue is a predicate in the universe
of colour, or of coloured objects : prudent in the universe of human character.
A positive term and its corresponding negative (e.g. blue and not-blue) may
then be said to divide between them not indeed the whole universe, but the
limited universe or whole of things to which they belong ; the members of
this limited universe have a positive common character, which gives the
negative term a positive meaning; whereas if we consider the whole universe,
there is no positive character common to all things included in it, except
the character of being — which, as Aristotle pointed out, considered in itself
and not as realized in some special mode of being, is not a significant term :
cf. p. 50 infra, and de Interp. iii. 16b 22. Such a ' limited universe ' is some-
times called an ' universe of discourse ' ; but this only means the limited
whole which is the subject of discourse, and its limits — e.g. those of the
whole within which blue and non-blue fall — are determined by the nature
of things, not by our discoursing of it.

n] TERMS, AND THEIR PRINCIPAL DISTINCTIONS 45

Many negative terms indeed are not themselves attributives, but
are abstracts which presuppose an attributive ; and what has been
said of negative attributives is confirmed by the fact that these
abstracts — such as injustice, inequality, non-intervention — are very
positive in their meaning. ' Injustice ' does not mean whatever is
not justice (such as ' accidence and adjectives and names of Jewish
kings '), but the quality of being unjust ; ' inequality ' means the
relation of being unequal ; ' non-intervention ' the conduct of the non-
intervening. Abstract negative terms like not-equality or not-colour
are as unreal as concrete negative terms like not-Socrates or not-book.

It may be asked, if all negative terms (and the same is true of
privative) have a positive meaning, what is the use of the distinction
between them ? The answer is as follows. First, with regard to
the distinction of positive and privative terms ; there are some states
which can only be understood as the privation of a positive state :
deafness would have no meaning, but for our knowing what it is to
hear ; we cannot think of a body as desiccated, except we think
of it as having first contained moisture.1

Secondly, with regard to the distinction between positive and
negative terms : there is a real difference between a term which
signifies one definite attribute, and a term which signifies any attri-
bute within a genus except one; the latter is in most cases" compara-
tively indeterminate and uninstructive ; e. g. vertebrate signifies a

1 These two examples are not quite parallel. A notion of deafness can be
had by any one who knows what hearing is. A notion of ' desiccated ' cannot
be had by any one who knows what moisture is, but he must also know
what dryness is. ' Desiccated ' is a privative term, because it means a dryness
due to the withdrawal of moisture previously present ; but ' dry ' is just as
positive a term as ' moist '. It sometimes happens, with two mutually
exclusive alternatives like dry and moist, that men dispute whether or not
both are positive. Some philosophers have maintained that pain is merely
the privation of pleasure, and evil the privation of good; others, that pain
and evil are just as positive as good and pleasure. In these cases, it will be
also in dispute, whether or not pain and evil are privative terms. But the
dispute arises from our uncertainty how to think about the things ; and so
furnishes another illustration of what has been pointed out in the text, that
logical distinctions of terms reflect and are based upon distinctions in the
things thought about.

2 Sometimes, as Miss Augusta Klein has pointed out to me, the positive
term may be less determinate in meaning ; there are more ways of being
coloured than colourless, of being fed than unfed. Here obviously the negative
term has a positive meaning ; we know the look of a colourless fluid^ and
an unfed animal is in a very positive state. If it be said that ' unfed ' has
a meaning also for those who do not know what state an animal is in which
has not been fed, we may reply that for them it means ' which has not been
fed ', and so mere negation is shown to belong to judgement, as stated above.

46 AN INTRODUCTION TO LOGIC [chap.

definite anatomical structure ; invertebrate signifies an animal struc-
ture which is not vertebrate, but fails to characterize it further.
Positive terms are positive directly and precisely, negative terms
indirectly and for the most part vaguely. This distinction is impor-
tant, and we are therefore justified in calling attention to it ;
it will be seen for example presently * to be one of the rules of
definition to state what a thing is, not what it is not ; this is best
expressed by the injunction to avoid, as far as possible, negative
terms ; and there is no way in which the point of this instruction
could be so well conveyed as by the help of the distinction of negative
and positive terms.

[The doctrine about negative terms impugned in the foregoing
paragraphs furnishes a good example of the dangers that beset
a purely formal logic. If we regard only the form of a proposition,
' A is not B ' (in which the terms are A and B), we may ' permute ' it
to the form ' A is not-Z? ' (in which the terms are A and not-B) ; and
we may formally regard A, B and not-B all equally as terms. But
whether not-B is a genuine predicate, and the proposition ' A is
not-B ' really affirms anything, will depend upon the matter of the
proposition — upon what kind of a term B stands for. In respect of
form, B has a corresponding negative not-B ; but we cannot tell by
considering the form alone whether any thought or notion of not-B
is possible. It may be noted also that the Law of Contradiction
should not be formulated symbolically as ' A cannot be both B and
not-B ', or ' A cannot be not- A ', but rather as ' A cannot both be and
not be B ', or ' A cannot not be A'. For if not-B is something positive
other than B, or not- A than A, what is B or A may have such other
positive character besides. If ' to be not-B ' is necessarily incon-
sistent with being B, it is neither more nor less than ' not to be B '.]

We have still to notice the distinction of univocal, equivocal, and
analogous terms. Univocal terms are terms with only one meaning,
so that they are used in the same sense of every subject of which
they are used at all : equivocal (or ambiguous) terms are terms
with more than one meaning, so that they may be used of different
subjects in different senses — e.g. fair, as used of a complexion and
of a bargain : analogous terms are terms which have more than
one meaning, but the meanings have a certain degree of identity
or correspondence — e. g. we speak of the foot of a man and the
foot of a mountain, meaning different things, but in both cases
that on which something stands. We ought in strictness to regard

1 Cf. infra, p. 98.

ii] TERMS, AND THEIR PRINCIPAL DISTINCTIONS 41

this distinction as one not in terms but in the use of terms ; for
fair is used univocally of all fair complexions, and is only equivocal
when we use it at once in different senses. All proper names be-
longing to more than one individual are used equivocally of such
different individuals.

[The history of the words univocal, equivocal, and analogous
will illustrate the tendency to treat Logic from the standpoint of
an affair of names. The Aristotelian distinction already alluded
to (p. 31) between avvutw^a and oixcawfjia was one of things. Uni-
vocum and equivocum are merely translations of (twwvv^ov and
6fxu)vv[xov, and they were defined in the same way (cf. Cracken-
thorpe's Logic, Bk. II. c. i. ' Aequivoca ita describuntur : aequi-
voca sunt quorum nomen solum est commune, ratio vero illius
nominis est alia atque alia.' c. ii. ' Univoca describuntur in hunc
modum : univoca sunt res vel entia quorum nomen est commune,
et ratio illius nominis est una et eadem in omnibus quibus nomen
convenit '). Similarly, it would have been not the word ' foot ', but
the man's and the mountain's foot that would have been called
analogous. In the sense in which terms are not words, but the objects
of thought intended by the words, we might still say that equivocal
terms are different objects of thought with the same name, rather
than the same name with different meanings. But in English usage
the distinction of names has really displaced that of things : we do
not even (except for the word analogous) retain both, like the Latin,
when it was said that ' aequivoca ' were either ' aequivocantia,
ipsae voces aequivocae ', or ' aequivocata, res ipsae per illam vocem
significatae '. And even in Aristotle, Rheb. y. ii. 2. 1405a 1, we find
an example of the use which calls words synonymous. Cf. also
Journal of Hellenic Studies, vol. xxix. pp. 28 and 32, where avvwvvfxov
ovojxa (= synonymous noun) is reported from a school tablet found
in Egypt and belonging apparently to the third century a. d.]

CHAPTER III
OF THE CATEGORIES

The distinctions between terms discussed in the last chapter are
not primarily grammatical, like the distinction between substantive
and adjective (though here and there, as we saw, the forms of
language have affected the mode in which they have been drawn) ;
nor do they belong to any special science, like the use in chemistry of
names in -um to signify metals, and names in -ide to signify com-
pounds. They may be illustrated from all sciences, and are based
on certain features that reveal themselves to reflection about any
subject whatever ; and that is why they belong to Logic. But
they involve not only features of thinking, like attribution, affirma-
tion, negation, but also features in what is thought of ; and so far
they belong to Logic only because the thought which Logic studies
is thought about things, and we cannot separate the study of
thought from the study of the most general nature of things thought
about — such nature as they must have, if they are to be objects of
thought at all. It is of special importance to remember this in con-
sidering the Aristotelian doctrine of Categories, out of which some of
the preceding distinctions take their rise. The Categories present
a logical, but they present also a real distinction : i. e. a distinction
in the nature of the reality about which we think, as well as in our
manner of thinking about it.

We saw x that reflection on the form of judgement 'A is B '
leads us to ask in what sense one thing is another ; that sometimes
it is meant that the predicate character, B-ness, is incidental to
the subject A, sometimes that to be A is essentially to be B ; thus
1 The Emperor is captured ' does not mean that to be Emperor is
to be captured, but ' Man is an animal ' does mean that to be a man
is to be an animal. Out of such reflection arose the doctrine of the
Categories.2

The word category, Karriyopia, means predicate 3 ; but its predi-

1 Cf. supra, pp. 22-24. 2 Cf. Arist. Metaph. A. vii.

8 The Latin equivalent is Praedicamentum, and Aristotle occasionally writes
Karriyoprjfia instead of Karriyopia, which means predication as well as pre-
dicate : v. Bonitz, Index Aristot., s. vv. Karrjyopr/pa and Karriyopia.

OF THE CATEGORIES

49

cate is what any subject is ; and the categories may be described as
a list of predicates, one or other of which declares the mode of its
essential being belonging to any subject that exists. In Aristotle's
complete list there are ten, viz.

ovaCa

substantia

substance

irocrov

quantitas

quantity

TTOLOV

qualitas

quality

7rpoy tl

relatio

relation

TTOV

ubi

place

TTore

quando

time

K^lcrdai.

situs

situation

Ixew

habitus

state

7roieu>

actio

action

itaaytiv

passio

passion (being acted on)

These Aristotle calls both ' kinds of predicate ', yivr\ tS>v Kar^yopi&v,
and ' kinds of being ', yevq tG>v ovtcov. We must examine the
latter phrase first, if we wish to understand his doctrine.

In the form of proposition ' A is B ', as just observed, the predi-
cate does not seem equally in all cases to declare what the subject is.
A man is an animal, and a man is in the kitchen ; Tray is a dog, and
Tray is happy now ; a musician is an artist, and a musician is break-
ing my hurdy-gurdy : if we look at these judgements, we shall admit
that the second does not tell us what a man is so much as the first ;
that the third is a fuller answer than the fourth to the question
1 What is Tray ? ' ; and that the fifth is a fuller answer than the
sixth to the question ' What is a musician ? '. In Aristotle's phrase
the first, third, and fifth of them declare what their respective sub-
jects are nad' avro, or per se : the second, fourth, and sixth what
they are Kara o-v/i/3e/3jjKoy, or per accidens. In other words, the
predicate is in the one case of the essence of the subject, and ' covers
its whole being ' l, and the subject could not exist at all without its
being predicable of him ; in the other case it is an accident of the
subject. What is predicated of a subject kclO' avro tells you what
it is necessarily, permanently and constitutively 2 ; what is predi-
cated of it Kara avnficfir]K6$ tells you indeed something about it, but
something less important, and perhaps unnecessary, to its being —

1 Cf. supra, p. 23.

2 This is not a complete statement of the meanings in which, according
to Aristotle, a predicate may be said to belong to a subject ko6' avro ; but
it is, I think, a sufficient account of the sense in which the expression is
used in this connexion.

1779 B

50 AN INTRODUCTION TO LOGIC [chap.

something of which it could be divested, and still remain the thing
it is, at least something not constitutive of it as such a subject.

The ultimate subject of predication is the concrete individual
thing — you, Socrates, Bucephalus, or the stone in your signet-
ring 1 ; and if you ask of this what essentially it is, you will have to
specify in your answer some kind of substance2 ; you are a man,
Bucephalus is a horse, the stone in your signet-ring is an agate. All
these — man, horse, agate — are so many different substances ; in
saying what you, Bucephalus, or the stone in your signet-ring is
essentially, or per se, these are the answers I must give ; their
essential being, therefore, is to be some kind of substance, and the
predicates which give their essential being are in the category of sub-
stance. But if I ask what is a substance, I cannot find any more
general character under which to bring that, as I bring Bucephalus,
in declaring what he is, under horse, and horse, in declaring what it
is, under substance. Of substance I can say that it is a kind of
being ; for substances are one kind of things that are ; but it is of
no use to treat mere being as a genus, of which substances are a
species, for to being considered in itself, and not as a determinate
way of being (e. g. being a substance), I can attach no meaning.

On the other hand, there are a great many subjects, about which,
if asked what essentially they are, I could not possibly say that
they are substances. Large, loud, blue, heavier, here, yesterday,
fever, horizontal, running, defeat, virtue — each of these is some-
thing, or nothing could be said to be it : but what are they ? Directly
or indirectly they all presuppose substances ; if there were no
animals, there would be no fever : if no fighters, no one could be
defeated. But they are something incident to substances, attri-
butes or relations and not things. To say that they are attributes,
however, only declares their relation to something else, their de-
pendence ; it does not declare what they are in themselves. If we

1 This is the true meaning of the statement in Cat. iii. lb 10 orav htpav

k<i#' erepov <ar r]y oprjTni cos xaB' VTTOKeipivov, o<rn Kara tov Kmr}yopnvp,ivov Xeytrai,

TTiivm Ka\ Kara t<<v vnoKfitxevov prjfirjaerai ('When one thing is predicated of
another as of a subject de quo, all that is asserted of the predicate will be
asserted of the subject as well ') — a statement sometimes erroneously quoted
as equivalent to the Dictum de Omni et Nullo. Cf. infra, c. xiv. p. 297 n.

2 But concrete things sometimes receive names implying their possession
of predicates in some other category than that of substance ; e. g. a threshold
is a concrete thing, but in calling it a threshold I do not give its substance :
to do that, I should have to say that it was a stone. It is a threshold because
it is a stone in a certain situation.

in] OF THE CATEGORIES 51

ask that, we shall find ourselves ultimately giving as an answer some
one of the other categories.

Thus I may say that ' yesterday was wet ' : but that does not tell
any one the nature of yesterday in itself. But if I say ' yesterday is
the day before that on which I am now speaking ', I explain what
yesterday in itself is. And if next I am asked ' What is that ? ', I
should reply that it is a certain date or time ; and there I must stop.
The kind of being then which belongs to yesterday is not being a
substance, but being a time. Similarly blue is a colour, and colour
is a quality ; loud also is a quality, and virtue ; so that their being
is being qualities ; that is what essentially they are. Large is a
size, i. e. to be large is to be of a certain quantity ; to be heavier is
to be in a certain relation, here is a place, fever is a state of the body,
horizontal a situation, running an action, defeat a being acted on.

There. is nothing then, according to Aristotle, that exists or can
be thought of, which is not either a substance, or a quality, or a
quantity, or in some other of the categories. One or other of them
is predicable of everything ; and they cannot be further reduced,
or brought under any common head.1 A quality is not a quantity,
a time not a place, to do is not to be done to, nor any of these a
situation : and so forth. It might be thought that state is hardly
distinguishable from quality, nor situation from place. But they
are not really the same. A state is something which characterizes
a whole through the condition of its parts ; thus we call a man
shod, because he has shoes on his feet ; or healthy, because each
part of his body is functioning rightly ; the healthiness of his body
as a whole does not mean that each part of it is qualified alike,
nor his being shod that every part of him has shoes on. A quality,
on the other hand, is comparatively simple, and if it characterizes

1 As a matter of fact, however, the category of relation is not equally
excluded by the others ; and Xenocrates is said to have reduced them all to
Substance and Relation. In doing this he would not have effected a real
simplification, any more than if they were all reduced to Being ; for time,
place, action, &c, involve irreducibly different kinds of relation ; and mere
relation, which is not any definite kind of relation, is almost as barren a con-
cept as mere being. Aristotle probably erected relational predicates into
a separate class because they tell us less than others what a subject is (cf.
Metaph. N. i. 1088a23). 'Six feet high' would be in the category of irna-nv :
'taller than his neighbour' in that of 7t/jo? n ; it gives more information
about what a man is to say that he is six feet high, than that he is taller
than his neighbour. The latter predicate may change when his neighbour
changes ; the former only by a change in the man himself. The former
involves relation also ; but the latter is more plainly and purely relational.

£2

52 AN INTRODUCTION TO LOGIC [chap.

a whole, does so through being present in the same way in its various
parts ; if a whole surface is blue, that is because the various parts
of it exhibit the same colour, and if a trader's stock is sweet, that
is because the things it is composed of are severally sweet. A state,
therefore, is more complex than a quality ; and so it is with situa-
tion and place. 'Upside down', 'horizontal', ' sitting', 'standing',
are in the category of situation — predicates which determine not
where a thing is, but its ' lie ' or position there. Without place
there could be no situation ; but you do not determine a thing's
situation by assigning its place.

The categories, therefore, are a list of predicates, one or other of
which must in the last resort be affirmed of any subject, if we ask
what in itself it is. They are yivr) t5»v Kar^yopiSiv, kinds of predi-
cate, and equally yhr\ tS>v ovtcov — the kinds of being which we
recognize, the kinds (if we may put it so) of what things are.1 These
things, the ultimate subjects of predication, are individual sub-
stances, and the categories do not give a classification of these, as
is given when things are said to be animal, vegetable, or mineral ;
they give a classification of the kinds of being displayed in and
predicable of them. Those predicates express most fully the being
of an individual substance which are in the category of substance,
like man, rose, gold ; they tell us what essentially it is. But every
predicate tells us what in some sense it is, and the kinds of being
displayed in what else it is are the other categories beside substance.
Thus the distinction between substance and the other categories
is a prominent feature of the doctrine ; for all the others presuppose
and are incidental to substance, since predicates belonging to them
are displayed in the being of individual substances. Terms in these
other categories may be subjects of predication, as when we say
that blue is a colour, and that wisdom is rare; but they exist not inde-
pendently but in concrete individuals. There is no blue except the
blue of the sea or the sky, of a larkspur or a gentian, &c. ; no wisdom,
except that of the wise. Concrete individual things are substances
in the strict and fullest sense. But what is predicated of them is

1 Cf. At. Met. A. vii, and Apelt, Beitriige zur Geschichte der griechischen
Philosophie, III. Die Kategorieenlehre des Aristoteles. In the expression yivr)
rav KaT>]yoptr~.>i>, ' kinds of predicate,' Kar^ynpi> refers no doubt to the predi-
cates of things, these predicates falling under the kinds enumerated, not to
the heads or most general predicates under which these fall. Hence the
concrete individual is not in any category, since it is not what any further
subject is (cf. Cat. V. 3a 36 «7ro p-iv yap rrjs TrpoiTrjs ovirias ovdefiia e'crri KiiTrjyopia,
'for 6rst substances furnish no predicates').

in] OF THE CATEGORIES 53

partly in the category of substance, partly in the other categories.
We have here that distinction between first and second substances
which once occupied so much of the attention of philosophers and
theologians.1

First substances are individuals like Socrates or Cicero ; second
substances are the kinds of these, and terms are in the category of
substance which, like man, horse, peppermint, parsley, tell what
kind of thing an individual is. All else that is said of an individual
tells only some quality or state that characterizes him, his activity
or situation, his relation to others, &c, and is therefore a predicate
in one of the remaining categories.

Undoubtedly it is here that the chief difficulty in Aristotle's
conception lies. But the difficulties are not gratuitous ; they
arise naturally in our reflection upon the nature of things.2 We
naturally incline to think, in considering any concrete individual,
that out of all that characterizes it some part is more essential than
another, goes more to make it what it is. This we call its kind,
and Aristotle called it also its substance ; and language contains
names that are evidence of this, kind-names like man, horse, gold.
It is indeed very hard to say exactly what constitutes the kind ;
kind-names, as we shall see later, present special obstacles to
definition ; and a positive account of the substance of an individual
seems beyond us. But negatively there is a great deal which we
should say does not belong to the substance — the place where the
individual is, what it momentarily does or suffers, all in fact that
we can refer to other categories. All these we tend to think of as
attributes which the individual has, but that it can exist irrespec-
tively of them : whereas, irrespectively of its kind, it would no
longer be at all. And yet the kind is universal ; it is predicated of
more things than one ; Socrates, Plato, and millions more are men ;
the lumps of iron in the world are uncountable. Hence follow two
lines of reflection.

First, because the kind, though universal, is at the same time
more substantial than the other predicates of an individual are-
more concrete, in fact, than they— the kind, or ' second substance ',
comes to be thought of as having some special claim to independent
existence. Other modes of being, other predicates, depend on it ;
but it is thought of as depending on nothing else for its existence.

1 This mode of expressing the distinction comes from Cat. v. 2a 11-19.
8 Cf. supra, pp. 28, 35.

54 AN INTRODUCTION TO LOGIC [chap.

True that we only find the kind realized in some concrete indi-
vidual ; nevertheless it is not a mere attribute of the concrete
individual, as are predicates in other categories. And some have
held that these ' second substances ', though displayed in divers
individuals, are each not only genuinely one and single, but
real, whether there be any concrete individual of their kind or
not.1

But secondly, because the kind is universal, it is predicated of
the concrete individual, like predicates in other categories. And
as the individual is something which has them, so it is something
to which its kind is attributed. It cannot be identified with its
kind ; for then there would be nothing to distinguish one indi-
vidual from another. Man is predicated equally of Socrates and
Plato, and if each as an individual substance were just man, Socrates
would be the same as Plato. Therefore we must look elsewhere for
what distinguishes them. If we find it in the other predicates of
the concrete individual, and say that he is the kind plus all his par-
ticular attributes, we resolve the individual into an assemblage of
universal predicates. If we do not do this, but suppose that his
kind and all his particular attributes as well belong to the individual,
the individual, to which they all belong, becomes a mere uncha-
racterized something. For in saying what it is, we should merely
assign to it a fresh predicate ; whereas we want to get not at its
predicates but at that which ' has ' them. Thus we should reach
a new way of considering the subject of predication. Originally
it was the concrete individual, Socrates or Plato ; but of what he is,
one part was distinguished as what he is essentially, and the rest
reduced to be attributes or ' accidents ' of him, not necessary to his
being, and not to be included in an account of his essence. Now,
what he is essentially is also reduced to the position of attribute and
mere predicate, and the subject becomes a mere subject of which as
such nothing more can be said except that it exists and is unique in
each individual. This mere subject of predicates, which cannot in
itself be described as specifically of this kind or of that, Aristotle
called matter.2 We only know matter in conjunction with form ;
bricks and timber are the matter or material of which a house is
built, but a brick is in turn clay to which a certain form has been
given ; clay again is matter of a certain form ; but matter by itself —

1 Cf. supra, p. 32.

2 Cf. Ar. Phys. a. vii. 191» 8-12, Z. iii. 1029a 23.

in] OF THE CATEGORIES 55

that which is found in various forms, but has no form of its own —
is unknowable.1

It may be questioned whether Aristotle was justified in his
use of the conception of matter. He started by thinking of the
material out of which a thing is made. Now the material of any-
thing is always something quite determinate. Economists know
in how many ways the products of one industry are ' raw material '
to another ; but the raw material which is rawest, i. e. which has
itself been least worked up, is still matter of a perfectly definite
kind. Timber is the raw material of the carpenter, but trees of the
lumberman : pig iron of the ironmaster, but iron ore of the smelter ;
and neither trees nor iron ore are any nearer being formless matter
than timber or pig iron. In these cases, the matter (or material)
is a concrete thing, in a different state no doubt from that into which
it is worked up, but perfectly familiar to us as existing in that state ;
but in the philosophical antithesis, the matter is not a concrete thing
at all, is in no state, is quite unfamiliar and indeed incapable of
being known to us as such ; and this relation of matter to form has
no real analogy with the relation of matter to what is made out of it
in the arts.2 It is true that in using the metaphysical analysis of the
concrete individual into matter and form in order to find in different
individuals different subjects of the same form, I may not at first
sight seem to need the conception of a quite indeterminate matter.
The matter of a house, says Aristotle, is stones and timber ; the
form — what makes the stones and timber the matter of a house —
is ' to be a shelter for men and goods '. Stones and timber are
determinate material, as ' to be a shelter for men and goods ' is a
determinate form. But suppose two houses built to the same
specification ; what distinguishes them ? We say, that they are
built of different materials — different stones and timbers. But
what distinguishes these ? Not their form, since ex hypothesi they
are of the same form. We may say that they just are different,
and leave it at that. But if we are going to use the analysis into
matter and form to explain their difference, since they are not dis-
tinguished by what they are, their predicates, we must find the ground
of their difference in the difference of the matter ; and this dis-
tinguishing matter must be taken as something divested of predi-

1 fj v\n ayvoxnos *«#' uvttjv, Met. Z. x. 1036a 8. Cf. supra, p. 35.

2 In the foregoing criticism I am particularly indebted to lectures of
Professor Cook Wilson.

56 AN INTRODUCTION TO LOGIC [chap.

cates, because in respect of predicates they are the same. The
outcome of this line of reflection would seem to be that what makes
possible different individuals of the same kind is the indeterminate
matter of which what they are is predicated ; and this at times
Aristotle says,1 and he admits that in one sense matter is substance.
But the corollary, that the nature of Socrates, as predicated of this
matter, is something that may be common to another, and universal,
he does not draw ; and it would seem to be his considered doctrine
in the Metaphysics (however hard to reooncile with some of his
other statements) that what makes Socrates Socrates is his form,
or what he is, and not the matter in which this form is realized.2
This form is really his substance, or substantial being ; and it is
neither merely the specific form of men, nor does it include all that
can be predicated of him ; but we are not told how to distinguish
it from predicates in the other categories. We need not pursue the
Aristotelian doctrine further ; so much has been said in order to
illustrate the difficulty of determining what is in the category of
Substance. We start with the concrete individual, and draw a dis-
tinction, among all that can be predicated of him, between that
which declares what he is essentially, and is his substance, or in the
category of substance, and that which declares about him some-
thing not essential, and belonging to one of the other categories.
But a predicate in the category of substance seems universal, as in
any other ; and if it belongs to several individuals, these must be
distinguished otherwise than by it ; hence the tendency to say that
what individualizes is material substance, not universal, nor capable
of figuring as predicate. But then the kind, what is predicated of
individuals in the category of substance, ceases to be essential to
them, for they would still be, and be individually different, without it.
Thus the attempt to distinguish what is from what is not essential
to the individual must either be abandoned in a doctrine of indi-
vidual forms — for if we suppose that there is something about
Socrates which makes him Socrates, we have no principle on which
to select this from among the sum total of all his predicates ; or else
it leads us to distinguish the individual both from his essential and
from his non-essential attributes, and then he is individualized by

1 Cf . Met. Z. viii. 1034a 5-8 ; and v. Bonitz, Index Arist. s. v. vXt}, 786a 52-58.
But individuality cannot be explained by difference of mere matter: cf. infra,
p. 90.

2 Cf. Met. Z. x. 1035b 27-1036a 9, xiii. 1038b 8-15; H. i. 1042a 28-9. But
one cannot really support any statement on the point except by reference
to his whole discussion.

in] OF THE CATEGORIES 67

neither, and neither is essential to his being the very individual he is.
The ' first substance ' is at the outset the whole concrete individual.
We try to distinguish within what it is what is essential to it, and
we only really find what is essential to its being of a certain kind.
Taking this as what is essential to it, we regard it as constituting
the individual, and so as possessing a substantiality of its own and
being a sort of ' second substance '. But then we find that a second
substance will not individualize.

We shall be met later with the same difficulty, when we consider
the doctrine of the Predicables, and the problem of definition. The
metaphysical issue raised is fundamental. But for the present it is
enough to have called attention to it. Logical and metaphysical
problems have a common root. We cannot reflect upon the being
which is asserted in all predication, without asking how things can
be conceived to exist. And it may readily be shown, with regard
to the different categories in particular, that we could not use predi-
cates in them, except so far as we conceived subjects to exist in certain
ways. Thus no predicates in the category of quantity can be used
of the mind, because the mind is not extended ; if it were, it might
have a capacity of 3 or 30 cubic feet, and an area and maximum
diameter ; since it is not, we cannot apply such epithets to it at all ;
and it is only because the existence of material things is existence
in space, that we can call them large or small, three feet square or
four feet long. In the same way, if it were not for the fact that
the world is spatial, there could be no predicates in the category of
place ; and space also renders possible predication in the category
of situation ; for it contains the distinctions of up and down, front
and back, right and left ; and it allows the parts of a body to alter
their relations to certain fixed points above and below, behind and
before, to the left and right of them, while the whole body remains
within the same limits. This is what happens when a man lies on
the sofa where he was formerly sitting, or when an hour-glass is
inverted on the table. And a perfectly homogeneous sphere, though
it may change its place, can be situated only in one way ; and if
we are to distinguish a right and wrong way up in it, we must mark
or single out some point in the circumference, whereby it ceases to
be perfectly homogeneous ; and this again illustrates how the dis-
tinction of categories arises out of the distinguishable modes of
being in things. For it is because it is a figure of a certain kind,
that such a sphere does not admit of the same varieties of situation

58 AN INTRODUCTION TO LOGIC [chap.

as a cylinder ; and because it does not admit of these, they cannot
be predicated of it ; and if nothing could be perceived or imagined
to admit of them, predicates in the category of situation, and
therefore the category of situation, would not exist. Again, there
are predicates in iroielv and -nda-xetv because things act one on
another ; and the two categories are distinguishable because there
are two terms, agent and patient, in all causal interaction. And
the different tenses of verbs, which make a difference to a predica-
tion in time, though it remains in the same category of irotelv or
ttcktxciv, e'xei^ or Keladai,1 presuppose that things exist in time ;
otherwise, how could we distinguish the meanings of vytaivti and
vytavev, vapulat and vapulabit, vivit and vixit, sits and sat ? Of
that which had no continuous existence through differences of time,
predication would be possible only for a moment in the present. But
reciprocally, as we could not predicate in these categories unless
things existed in certain ways — as substances, with qualities, ex-
tended in space, persisting in time, &c. — so we cannot predicate
about things except in one or other category ; in other words, if
we think of anything, we must think it to be determined in one or
another of these ways.2 That which was not conceived as a sub-
stance, or a quality, or a state, and so forth, would not be conceived
at all ; and a concrete thing that was no substance, had no quality
or state, and so forth, would be just nothing. And therefore the
consideration of these distinctions belongs to logic, since the thought
of them is involved in our thought about objects in general ; and
though logic is not interested in the indefinite variety of existing
qualities — blue, green, sour, shrill, soft, &c. — (because a substance, in
order to be a substance, need not have any one of these qualities in
particular, but only one or other) yet it is interested in the category
of quality, or in noticing that a substance must have some quality
or other : in the category of relation, or in noticing that it must
stand in relations to other things : and so on.

1 i.e. action or being acted on, state or situation. It is to be observed that
the predicate of the same proposition may determine its subject in more
than one category. In the proposition ' The other disciple did outrun Peter '
the predicate is in the category of time, for the past is a time, and the event
is referred to the past : and of action, for running is an activity : and of
relation, for ' faster than Peter ' is a relation. But of course, if we distinguish
these different elements in the predicate, we can refer them, considered
separately, to different categories.

2 It is not necessary, however, to hold that Aristotle's list of categories is
complete.

in] OF THE CATEGORIES 59

The problem underlying Aristotle's doctrine of Categories may be
expressed thus — to discover the forms of existence which must be
realized in some specific way in the actual existence of anything
whatsoever. His classification may exhibit defects, but the impor-
tance of his undertaking must be admitted. And many of the
distinctions between terms insisted on by those who attach least
importance to the Aristotelian doctrine of Categories express an
attempt to solve part of the problem which he was attacking, and
are derived from his doctrine. Those distinctions, as was pointed
out in the last chapter, rest upon certain fundamental features of
the existence which we conceive the objects of our thought to have.
The distinction between singular and general concrete terms corre-
sponds in the main to that between irpu>Trj ovcrCa, the concrete
individual, and predicates in the category of substance ; for the
most noticeable of general concrete terms are in the category of
substance, as man, stone, or beast, though some (which might be
called substantives oflin attributive kind) are in other categories,
as, for instance, officer and organist. The distinction between con-
crete and abstract terms corresponds roughly to the distinction
between substance and the other categories. That relative terms are
predicates in the category of relation is plain. The attention paid to
collective terms reminds us that we can consider not only things
severally, but what they are in certain groupings or combinations ;
and the distinction between quality and state involves the same fact.1
The logical divisions of terms rest on differences apprehended in the
being of things ; this is apt to be overlooked when the subject is
approached from the side of names ; Aristotle's doctrine of Categories
has this advantage, that throughout it fixes our attention on things.

[The Aristotelian doctrine of Categories bulks large in the history
of Logic ; such conceptions are instruments of thought ; the instru-
ments forged by one generation are handed on to the next, and
affect subsequent thinking. On that account alone therefore it is
fair to give some attention to it ; but it is still valuable as serving
to express and distinguish certain important features recognized by
our thought about things. That a quality is not a quantity is a
truth which those overlook who think that sound can be a wave-
length in the vibration of the air ; they forget that it is not possible
to define terms of one category by another.2 Moreover a conception

1 It is not meant that collective terms are in the category of State.

2 Except as terms in a derivative category involve terms in those from
which it is derived.

60 AN INTRODUCTION TO LOGIC [chap.

[of categories not very far removed from that of Aristotle has,
through Kant and Hegel, become one of the chief doctrines of
modern metaphysics.

These admissions do not bind us to consider Aristotle's list as
perfect. One important remark on it would perhaps hardly have
been regarded by him as a criticism. The different categories are
not all equally distinct or ultimate. Thus the distinction between
Ttov1 and 7rore2 is far more fundamental than that between -noulv 3 and
irdaxzt-v- 4 A thing need not have a place because it has duration,
nor can any one doubt under which category such predicates as ' at
home ' and ' belated ' respectively fall. But to be acted on implies
something acting ; indeed, if action and reaction are equal and
opposite, for a thing to be acted on implies that it acts itself ; and
it is often difficult to say to which of these categories a predicate is
to be referred. A ship travels : are we to attribute the motion to
the ship, and say that she acts, or to the engines, and say that she
is acted on ? or shall we say that the engines in turn are acted on
by steam ? Aristotle in a measure recognized the mutual implication
of these two categories, for in one place he includes them together
under the single term kiVtjo-i?.5 Language bears traces of it also,
in deponent verbs, which have a passive form with an active meaning,
and neuter verbs, which have an active form with sometimes a
passive meaning. We cannot admit, as Trendelenburg and others
have maintained, that the distinctions of categories were derived by
Aristotle from the grammatical distinctions between parts of speech ;
but undoubtedly they are reflected (though in an imperfect way) in
grammatical forms. Again, as we have seen, the notions of ex*^ 6
and Keladat 7 are derivative : state presupposes the distinction of
whole and part, which, in material things at least, implies the
category of iroaur,8 and it presupposes also the categories of 7roiet^3
and -nao-yjtiv*, and of ttolov 9 ; for a whole is in a certain state
through the interaction of parts having certain qualities, as when
the body is well or ill ; or through something done to certain parts of
it, as when the body is shod or clad ; a situation presupposes the
distinction of whole and part also (a point can have place, but no
'situation'), as well as the categories of ttov1 and irpos n10; for
when a thing changes its situation, some part that was formerly
above another comes to be below it, and so on. On these two deri-
vative categories Aristotle lays least stress ; they are only twice

1 Place 2 Time. 3 Action. * Being acted on.

6 Movement, or change : v. Met. Z. iv. 1029b 25. See for a conspectus of
the lists of the categories found in different parts of the Aristotelian corpus
O. Apelt, Beitrdge zur Geschichte der griecJiischen Philosophie, Kategorienlehre,
pp. 140-141.

6 State. 7 Situation. 8 Quantity.

• Quality. 10 Relation.

in] OF THE CATEGORIES 61

[included in his enumeration. But though derivative, they are
peculiar, and contain something not in the notions from which they
are derived ; it is quite impossible to treat a state like health as
being of the same nature with a quality like sweetness, or place with
situation in that place. Kant made it a ground of complaint against
Aristotle that he had included derivative conceptions in his list
along with pure or underivative ; but it would probably be a fairer
criticism, that he had not taken account of all the derivative con-
ceptions which call for recognition.

A word may be added upon Kant's doctrine of Categories, and
its relation to that of Aristotle, though it is very difficult to put the
matter at once briefly and intelligibly in an elementary treatise.
Aristotle had sought to enumerate the kinds of being found in the
different things that are ; Kant was interested rather in the question
how there come to be objects of our experience having these
diverse modes of being. He maintained that in the apprehension of
them we are not merely receptive and passive ; on the contrary, all
apprehension involves that the mind relates to one another in various
ways the elements of what is apprehended ; if the elements were not
so related they would not be elements of one object ; and they
cannot be related except the mind at the same time relates them ;
since relation exists only for a mind. Kant called this work of
relating a function of synthesis ; and he desired to determine what
different functions of synthesis are exhibited in the apprehension,
and equally in the existence, of objects ; for the objects in question
are not Dinge an sich, things by themselves, existing out of relation
to the perceiving and thinking mind ; of these, just because they
are out of relation to it, the mind can know no more than that they
are, not what they are ; the objects in question are objects of ex-
perience, and their being is bound up with the being of experience
of them. He maintained in the first place, that the mere perception
of anything as extended, or as having duration, involved certain
peculiar ways of relating together in one whole the distinguishable
parts of what is extended or has duration. These modes of synthesis
we call space and time. As to time, I know that I am the same in the
succession of past, present, and future ; I could not do this unless
I distinguished as different the moments in which I am (as I realize)
the same ; I could not distinguish them except by the differences
of what I apprehend in them ; but unless these differences were
conceived as differences in the being of something persistent and
identical, I could not hold them together ; hence through my function
of synthesis there come to be objects combining manifold successive
states into the unity of one and the same thing. It is the same
with any spatial whole. I must be aware at once of its parts as
distinct in place, and yet related together in space ; space is a
system of relations in which what is extended stands ; but the
relations are the work of the mind that apprehends that manifold

62 AN INTRODUCTION TO LOGIC [chap.

[together. But these two modes of connecting in an unity the parts
of what is manifold Kant attributed to sense, for reasons which
we need not now consider ; thinking, the use of general conceptions,
did not enter into them ; and therefore he did not include them in
his list of categories, which were to be the most general conceptions
by which in understanding we connect into an unity the manifold
parts of an object, and so make it an object for ourselves. The
perception of an object involved space and time ; but perception
was not enough. We think of it in certain ways, or conceive it, in
apprehending it as an object. Now this conception of an object
involved, according to him, four things : (1) its having quality : and
quality can only exist in degrees, each of which is distinguished
from and related to the other degrees of the same quality ; heat
only exists at a definite temperature and blue must be of a definite
shade and saturation : (2) its having quantity, or being a whole
composed of parts : (3) that it should be a substance having attributes,
one or permanent through its changing and successive states, and
that its changes should be determined according to laws by its
relation to other substances with which it stood in interaction :
(4) that every such object conceived to exist should be conceived as
connected with every other existing object in a way that knowledge
could apprehend, and express in the form of necessary inference.
The various peculiar relations involved in these requirements Kant
called Categories ; and he pointed out that, in all the sensible
diversity of concrete objects as we know them, these categories or
forms of relation exemplify themselves. Let something be pre-
sented to me ; if there is nothing which I can call it, or regard it as
being (for the question is one of thought and not of names), it is
so far nothing for me ; but if I call it sky-blue, I am thinking of
it as qualified ; I am ' taking it in ' by help of that conception of
quality (realized in a specific mode of quality, sky-blue) which is
one of the notions by which I relate together all that is sensible in
what objects are. Of course it might have a colour unlike any
colour I had seen hitherto, which I had no name to indicate ; but
I should still be apprehending it as coloured in a certain way, though
I could not name the colour, and therein I should be using the
conception of quality. If I call it a sky-blue tassel, I am using in
a specific form the notion of a whole of parts ; for to one who
could not connect distinguishable parts in one whole a tassel would
not be apprehensible as one thing ; I am also using the conception
of substance and attribute, when I regard it as a thing, one of whose
qualities it is to be sky-blue. I cannot call it woollen, without
connecting its existence by causality in a definite way with the life
of a sheep ; and so forth : the forms of space and time being
presupposed in my apprehension of it throughout. It is not meant
that these notions or categories are abstractly grasped, and con-
sciously applied as guides in our apprehension and description of

in] OF THE CATEGORIES 63

[objects, as a doctor who had recognized that height, weight, chest
measurement, and state of the teeth were important characters in
determining the health of children at a given age might use these
headings in a statistical description of the health of children in
London schools. We only become aware of the part which these
notions play in our apprehension of objects by reflection upon the
use we have unconsciously made of them ; just as we become aware
in the abstract of using certain forms of inference, by reflecting upon
the inferences we have drawn in divers fields. But as there would be
no men if there were no animals, and no circles if there were no figures,
so we should not judge anything to be coloured if we could not
conceive quality ; we should never think that a horse pulled a cart,
if we could not conceive a substance to have attributes and to
determine changes in another substance ; we should never call the
movement of the cart necessary, if we could not think of the different
real things in the world as so connected that we could infer one thing
from another. And in all these different ways, we are relating,
or distinguishing and connecting, features and parts of what we
apprehend : what is merely sensible is not the work of the mind ;
but the mind effects a synthesis in what would otherwise be a mere
chaos or confusion of manifold sensations or sensibilia.

Now it has been seen that Aristotle also noted that what, by
making them subjects of predication, we recognize as existing are
sometimes substances with attributes, sometimes attributes of
various kinds ; we recognize the existence of qualities ; of quantities
in things that are wholes or parts of such and such a size ; of rela-
tions and positions in place and time ; of what things do and have
done to them ; of their states and situations. Eut Aristotle ap-
proached the matter from the side of the object ; he asked what
modes of being we can distinguish in that which we recognize to be.
Kant approached it from the side of the knowing subject, and
asked what were the modes of synthesis on the part of our mind,
through which objects are apprehensible by us as the sort of objects
they are. If Kant is right in thinking that there could be no objects
known to us, except through the mind's activity in relating according
to certain principles their manifold differences, then we should expect
that when we reflect upon the modes of being which these objects
exhibit, we should find just those which the mind by its synthetic
or relating activity makes possible for them. Hence the two lists
of categories should correspond ; and in the main they do ; and the
differences between them can be readily explained. Aristotle's list
we have seen. Kant recognized four classes of category, those of
Quality, Quantity, Relation and Modality. Now Quality and
Quantity appear in Aristotle's list as well (though in Kant's they
are each analysed into three aspects, or ' moments ', which here
need not concern us). But in Kant the category of Relation covers
the three relations of Substance and Attribute, Cause and Effect,

04 AN INTRODUCTION TO LOGIC [chap.

[and Interaction (which last really involves the other two) ; the dis-
tinction of substance and attribute is present in Aristotle's doctrine,
when he says that the rest presuppose Substance, and in iroulv 1 and
ttaayjeiv 2 we have the recognition of the relation of cause and effect ;
but there is nothing in Kant corresponding to the Aristotelian
category of Trpos n3. The reason of this is that all predicates in
the category of -npos tl 3 really involve some other category as well ;
larger involves ttoo-o'i/4, earlier irore5, slave Trao-^eiv2, farthest ttov6,
and loudest ttolov7 ; reciprocally, all categories involve relation,
and Kant's whole point is that the relational functions involved are
different. For Kant, who was interested in distinguishing these
functions specifically, it would have been absurd to treat predicates in
which relating, no matter how, is especially prominent, as involving
a special kind of relating 10 ; or to suppose that there was any other
kind of relation involved when I say that Socrates was more scrupu-
lous than Crito, or taller than Tom Thumb, than when I say he
was scrupulous or four cubits high. All scrupulousness must be of
some degree, and all height of some quantity, so that as far as
the function of relating in the way of quantity or degree is con-
cerned, it is equally present whether my term is positive or com-
parative. But from the side of the object, there are predicates
which relate it particularly to some definite other object ; and these
Aristotle placed under the category of -npos tl 3. It might perhaps
be objected to him that all predicates in the category of -npos tl 3 were
also in ttov6 or ttot45, ttolov1 or irocrov*, TtoLfLV1 or -nacryjtLV2, Hxeiv*
or KtlaOaL9 ; but he would have replied that they were referred to
the category of relation not because they involved qualitative or
quantitative, spatial, temporal, or causal relations, but because they
determined a thing as standing in some relation (of any one of these
kinds) to some other thing, and they were predicated of it not so
much in itself as in relation to something else n. Again, terms in

I Action. 2 Passion. 3 Relation.
* Quantity. 6 Time. 6 Place.

7 Quality. 8 State. • Situation.

10 The reason why Kant gave the name of Relation to the three syntheses
of Substance and Attribute, Cause and Effect, and Interaction was historical.
He quite recognized that all his categories were really modes of relating
a manifold.

II Ta npos n are defined first in Cat. vii. 6a 36 as ' what are said to be that
which they are of another' — bo-a avra amp (<jt\v iripoiv thai XtytTai, and more
closely later in 8a 32 as that ' for which to be is the same as to be related
in some way to another ' — oh r6 thai ravrov tort tQ> irpos W nms (Xetv' The
implication of n^k n with some other category is recognized in particular
cases, but not stated generally; cf. vii. 6b 11, ix. lla 20-38, and esp. 37-38
fTt el Tvyyuvot to uvto 7rpos ti koi jrowv ov, ouftep aronov tv dfx(poTepois rots ytveaLP
airo KnTapiQptlcrQai ('besides, if the same thing happen to be both related
and of such a quality, there is nothing strange in its being counted in both
kinds '). Cf. Met. N. i. 1088a 21-25, where it is said that relation pre-
supposes quality and quantity.

in] OF THE CATEGORIES 65

[irorrov1, like 'three-foot' or 'year-long', involve space or time as
well as the relation of whole and part ; and Kant thought right to
distinguish the perceptual syntheses of space and time from the con-
ceptual synthesis of whole and part ; hence also he objected to the
presence of ttov 2 and irori 3 in the Aristotelian list at all. But Aristotle
cared only to notice the modes of being that were to be found, the
kinds of predicate that concrete things had, and was not interested
to distinguish the parts which sense and thought respectively
play in rendering the apprehension of them possible. Once more,
Aristotle included the 'derived' notions of e'xetf4 and Keln-dai5
with the rest, because they certainly are different modes of being ;
Kant, who thought them to involve only the co-operation of func-
tions of synthesis already recognized, gave no place to them. The
most considerable difference between the two doctrines is the absence
from Aristotle's of anything at all corresponding to the Kantian
categories of modality, i. e. to the notions of actual, possible, and
necessary as determinations of our thought about things ; but their
absence will not surprise us if we consider that to the question, what
essentially a subject is, no one would ever answer that it was actual,
possible, or necessary. Speaking generally, however, we may put
the relation of the two doctrines in this way, that whereas Aristotle
had classified the products, Kant distinguished the processes of that
synthesis or relating, through which (as he held) objects in all their
manifold variety, however much they may materially or sensibly
differ one from another, are all alike objects of knowledge and so
far formally the same. Merely to be, said Aristotle, is not possible :
ov is not a significant predicate 6 ; what is must be in a particular
way, and its being thereby fall under one or other of the yivt\ tG>v
Ka.Tr]yopi£>v, the kinds of predicate, which he enumerated ; and all
the modes of being characterize in the last resort some concrete
individual thing, which exists in and through them. An object,
said Kant, cannot be an object of experience, and therefore cannot
exist in the world of our experience, except through being perceived
and thought in certain ways : the general ways in which an object is
perceived or thought, the forms of perception and conception in-
volved (one or another of them) in every predicate through which
an object is known, are the ' forms of the sensibility ' — viz. space and
time — and the ' categories of the understanding'.7]

1 Quantity. 2 Place. 8 Time. 4 State. « Situation.

6 Unless in the sense of otaia or Substance ; but that is one of the
categories.

7 Kant may have been wrong (as Mr. H. A. Prichard has powerfully argued
in his Kant's Theory of Knowledge) in supposing that the ' formal ' characters
which belong to all objects of possible experience are not merely apprehended
in them by the mind, but are there to be apprehended through the mind's
activity. Nevertheless what has been said above will still express the relation
which, on his doctrine, subsists between Aristotle's categories and his own.

1779 F

CHAPTER IV

OF THE PREDICABLES

The distinctions to which our attention was directed in the last
chapter are distinctions of terms according to the nature of their
meaning ; and if we understand what a term means, we may know
to what category to refer it, without waiting to learn the subject
of which it is predicated ; large, for example, is in the category of
quantity, whether it be predicated of a triangle or of a gooseberry,
and just in the category of quality, whether it be predicated of
Aristides or his actions. Such difficulty as may exist in determining
the category to which a term is to be referred arises through defect
in the list of categories (i. e. of the conceptions under which we are
to classify all possible predicates), or through the complexity of
meaning in the term itself, whereby it involves more than one
category at once, like a verb with tense ; but not through the fact
that we are considering the term by itself and without reference to
the subject of which in a particular proposition it may be affirmed
or denied. And the treatise called the Categories indicates this when
it puts forward the list of ten categories as a division of terms out
of construction.1

In the present chapter we have to consider another division of
terms, based upon the relation in which a predicate may stand to
the subject of which it is predicated. Aristotle recognizes four such
relations, and one of them he subdivides, obtaining five in all ; later
logicians give five, but their list is in one important respect different.
According to Aristotle, in every judgement the predicate must be
either the definition (opos), the genus (yivos), the differentia (bt-acpopd),
a property (Ibiov), or an accident (o-vpLfiefirjKos) of the subject. The
later list,2 losing sight of the principle on which the division was

Tcov kuto. firjdeniau (TVfXTr\o<rjV Xeyo/ievcov eKnarov t'jrot oiaiav armnivei r) noaov
r) 7TOIOV rj tti>6 ti fj ttov rj nore r) KelaBai t] %XeLV h TOieiv r) ndcr^tiv, Cat. iv. lb 25

( ' what is said out of construction signifies either substance or quantity or
quality or relation or where or when or situation or state or action or being
acted on ').

2 The Aristotelian list is given in the Topics, a. iv. 101b 17-25. At the
outset Aristotle names yivos, Ibiov and <jvpj3epr)K6s ; he then says that Siacpopd

OF THE PREDICABLES 67

made, omits definition, and includes instead species (eI8o?), running
therefore as follows — genus, species, differentia, proprium, accidens.

The distinctions are known as the Five Predicables, or more
strictly as the Five Heads of Predicables. The words have passed
into the language of science and of ordinary conversation ; we ask
how to define virtue, momentum, air, or a triangle ; we say that
the pansy is a species of viola, limited monarchy a species of consti-
tution ; that one genus contains more species than another ; that
the crab and the lobster are generically different ; that man is
differentiated from the lower animals by the possession of reason ;
that quinine is a medicine with many valuable properties ; that the
jury brought in a verdict of accidental death ; and so forth. The
fact that the employment of the words is not confined to any special
science suggests that the consideration of them may belong to Logic,
as expressing something recognized in our thought about all kinds
of subject.

' Predicable ' here means a predicable character, i. e. not an
individual substance, but what it is ; all kinds, qualities, states,
relations, &c. ; and these may be exemplified in and belong to more
than one individual subject, and so we may say that they are
universal.1 All terms, therefore, except proper names may be
brought under one of these five heads of predicables in relation to
the subject of which they are predicated ; but proper names are
not included 2 ; they may indeed be predicates in a proposition (in
Aristotle's view only improperly) ; but they stand for individuals,
and an individual is not the character of anything. The Par-
thenon, for example, is not the genus or species of anything ;
nor is it that which differentiates any species from another species ;
nor is it a property or accident of anything. It is a particular
building ; and the name denotes that building, with all that it is —

may be ranked with y«W, as ov<rav yeviKi)i', i.e. presumably, as being a
modification of that ; and he distinguishes 18101-, as what is common and
peculiar to the subject, into o^os, which gives the essence, and 18101- sensu
strictiore, which does not. In c. viii he offers a proof that the five-fold
division is exhaustive. The later list passed into modern Europe from a little
work by Porphyry (b. A. d. 233), the Elaayayi) or Introduction to the Categories,
through the medium of a Latin version and commentary by Boetius, who
lived in the last quarter of the fifth and first quarter of the sixth century a.d.

1 Except when they are what no second subject can be ; e.g. there can
be only one Omnipotent, and only one superlative in any kind. Professor
Cook Wilson has called attention in an unpublished paper to the fact that
there may be universals with only one instance.

2 Nor designations, though what is general in a designation may be.

F2

68 AN INTRODUCTION TO LOGIC [chap.

a temple, Doric, of Pentelic marble, beautiful by the simplicity of
its lines and the magnificence of its sculptures, the work of Pheidias
and his assistants, the glory of Athens. All these things are pre-
dicable about it, and they are universals ; for might not another
building be a temple, in the same style, of Pentelic marble, and so
forth ? It, however, is not predicable ; nothing else can be the
Parthenon. We may ask what kind of thing is the Parthenon, but
not of what things is it the kind. The distinctions which we have
to consider, therefore, do not afford a classification of things, but of
concepts * : and (unlike the categories) of concepts considered not
in themselves but in their relation one to another.

But things are known to us through these concepts ; and an
enquiry into the relation of concepts is an enquiry into the nature
of things. There is indeed another sense of knowing. It has been
frequently pointed out that the English language uses only the one
verb, ' know,' to represent two different acts, which in some lan-
guages are distinguished by different verbs 2 : the knowledge of
acquaintance with a thing, and the knowledge about it. In Latin,
the former is signified by cognoscere, the latter by scire ; French
uses respectively the cognate words connaitre and savoir ; German
the words kennen and wissen. Knowledge of acquaintance does
not come barely through conceiving ; however much may be
told me about Napoleon, and however clearly I may have
succeeded in conceiving the features of his character, I never
knew him, and never shall know him, in the sense of being
acquainted with him : such knowledge comes only by personal
intercourse, and separate intercourse is needed with each indi-
vidual that is to be known. But knowledge about a thing comes
by concepts ; and this too is necessary to real acquaintance,3
though it does not by itself amount to acquaintance. I may know
a great deal about a man, without having ever met him : but
I may in fact once have met him, without knowing who he was or

1 To use a phrase of Mr. F. H. Bradley's, it is the ' what ' and not the
1 that ' of things which we have to consider.

2 Cf. e. g. J. Grote, Exploratio Philosophica, Pt. I, p. 60 — a work and by an
author less known than they deserve to be ; the expressions ' knowledge of
acquaintance ' and ' knowledge about ' are borrowed thence.

3 Though not to such familiar recognition as a dog may show of its master,
or a baby of its mother. The less developed mind acts in ways very difficult
to describe, because it does not shew completely what mind is ; but it is
wrong in principle to ' interpret the more developed by the less developed ',
as Herbert Spencer would have us do.

iv] OF THE PREDICABLES 69

anything about him ; and I am no more acquainted with him in
the latter case than in the former.

Now most of our knowledge is knowledge about things ; things are
useful and important to us for the most part not because they are
such particular individuals but because of what they are ; this is not
equally the case with persons ; and yet with persons too it is very
largely the case. ' Wanted, a good coat-hand ' : it is not Smith,
who is taken on, that is wanted, but only the coat-hand : the
master-tailor is satisfied to know that he has engaged a coat-hand,
and very often does not desire his acquaintance : if he knows about
Smith, he can regulate his business accordingly, without knowing
Smith.

Through concepts, then, i. e. through what we conceive of their
being, we are not acquainted with things individually, but we know
and think and reason about them thereby. And a concept may be
said to differ from a thing in being universal, not individual : an
object of thought and not of sense : fixed and not changing : com-
pletely knowable and not partially1. Take, for example, the con-
cept of a timepiece : a timepiece is a machine in which the move-
ment of wheels is so stimulated and regulated as to cause a hand or

1 The characters recognized and named in things are often imperfectly
understood ; but they might be understood completely, whereas the individual
thing cannot be. Hence we may say that a concept is completely knowable,
though not completely known. About the unchangeableness of a concept
certain difficulties arise. (1) It is said that men's concepts change as their
knowledge increases, e.g. there are now timepieces indicating the time by
cards on which the hour and minute are printed, and which displace each
other in proper succession ; and therefore we must modify our concept of
a timepiece. But this only means that we must change the meaning of
a name. What was conceived does not alter ; it is still displayed in the
instruments to which the name was hitherto given ; now, when the name
is also given to instruments which effect their purpose in a different way,
something different is conceived when the name is used (cf. infra, c. vi). So,
if we arrange a row of books according to height, we may say that the height
increases along the shelf ; but no book is getting higher. (2) But we may
conceive a changing character ; and here, what is conceived is not unchanging.
In a body moving with an acceleration, the velocity changes. Cannot we
then conceive velocity ? In the growth of an organism, perhaps we ought
to say that the specific form changes ; yet this, one would say, is only known
by conceiving. We must remember here the distinction between an universal
and its instances. The velocity of this bullet may change ; but velocity is
one in all these momentary velocities. When it is said that we know things
through concepts, that means, through what they are ; but what they are
is an instance of an universal nature. Between instances of these universals
relations hold which do not hold between universals ; Juvenal's indignation
may cause his activity in verse-making, but one universal does not cause
another. So in the instances there may be change, but not in the universal.

70 AN INTRODUCTION TO LOGIC [chap.

hands to move at an uniform rate (usually twice in twenty-four hours)
round a dial, and by pointing to the divisions marked upon the dial
to indicate the time of day. That is the concept of a timepiece : it
is clearly universal, for it applies to all timepieces ; it is an object of
thought, and cannot be seen or felt, like the watch in my pocket ;
it is fixed and unchanging, while my watch wears out or gets broken ;
and it is completely knowable or intelligible, whereas there is a great
deal about my watch which I do not know or understand : where the
metals of which it is made were quarried, and by what series of
events they came into the hands of the maker : why it loses 10"
to-day and gains 13" to-morrow, and so forth. No one knows the
whole history and idiosyncrasy of any particular timepiece, but he
may conceive its general nature satisfactorily for all that.

It has been asked, as we noticed above,1 is a concept merely an
object of thought, with no existence in things (as it is put, outside
our minds) ? or does it exist in things 2 ? Much ink, and even much
blood, have been spilt in disputing over this question. An elemen-
tary treatise must be content to be brief and dogmatic. Concepts,
we maintained, have existence in things, as well as in our minds.
The thing which I can pull out of my pocket, and see and feel, and
hear ticking, is itself a machine wherein the movement of wheels
causes hands to tell the time of day as set forth in stating the con-
cept of a timepiece. What I conceive a timepiece to be, that (if my
concept is a right concept) every particular timepiece is ; what
I know about things is the nature of the things ; nor would it
otherwise be they wherewith my knowledge dealt. But though
the features of things exist in the things, besides being conceived by
our minds, the manner of their existence is different in an important
respect from that of our conceiving them. In our minds,3 each is to
some extent isolated ; my knowledge of an individual thing is
expressed piecemeal in many predicates about it ; each predicate
expressing a different concept, or a different feature in the nature
of the thing. But in the thing these features are not isolated.
The individual thing is at once and together all that can be pre-
dicated of it separately and successively (except indeed as far as

1 Supra, pp. 25, 31-32.

2 Or does it (as some have held) exist apart at once from particular things
and from our minds ? Cf. supra, he. cit.

3 What is conceived by the mind is sometimes said to be in the mind. To
be in the mind means to be the object of a conceiving, thinking, remembering,
or imagining mind : not of course to be in the brain, or inside the skull.

iv] OF THE PREDICABLES 71

predicates are true of it successively — a man, e. g., is successively
awake and asleep). Thus in thinking of my watch I may think of it
as a timepiece, as an heirloom, as being two inches in diameter, and
bo on : between these concepts there is no connexion thought of ;
they are as it were separate from one another ; but they and much
besides are united in the thing.1 The individual thing is all that
can be predicated of it (and there is no end to what might be pre-
dicated, if we knew its whole nature and history) ; but one thing
that can be predicated of it is not another.

An object comes into the room, which I call Tray : what is
Tray ? it is a dog, an animal, yelping, at my feet, mine ; Tray is
all these : but is a dog all these ? A dog (that is, any dog) is an
animal, and a dog yelps ; but I cannot say that a dog (meaning
any dog) is mine, or at my feet ; and though a dog is an animal
it is not equally true that an animal is a dog, or that what is at my
feet is mine, or that what is mine is at my feet.

What, then, is the relation of those various concepts to one
another, which can all be predicated of the same individual ? Are
they united in it like stones in a heap, where the stones together
are the heap ? or like almonds in a stewed pippin, where the pippin
is not the almonds ? or like links in a coat of mail, where the links
indeed are the coat, but only because they are peculiarly looped one
into another ? It is easily seen that none of these analogies is
appropriate. According to Aristotle they are related in one of five
ways. Take any proposition, ' A is B,' where the subject A is
not a proper name, but a general concrete term, or an abstract term.
The predicate B must be either definition, genus, differentia,
property or accident 2 of A : one or other of these relations must
subsist between the two concepts A and B, in any individual
characterized by them.

The statement just advanced clearly concerns our thought about
subjects generally : the technical terms have yet to be explained,
but it is the actual procedure of our thought which they profess
to indicate. Logic invented the terms, but it discovered the
relations denoted by them.

1 The word thing here is used first of the concrete subject of predication,
then of the character predicated. It has been used already in both these
senses. The English idiom allows both uses — we may say, for example,
' about that thing I know nothing ' ; and it may be worth while to use the
word closely together in both senses, in order to direct notice to the ambiguity.

2 But cf. pp. 76, n. 1, 104, n. 1, infra. The Porphyrian list of predicates
Will be considered later.

72 AN INTRODUCTION TO LOGIC [chap.

If we take any term that is general, and not singular, and make
it the subject of a proposition, then the predicate must be either
commensurate with the subject, or not. One term is said to be
commensurate with another, when each can be predicated of
everything whereof the other can be predicated x ; equilateral
triangle and equiangular triangle are commensurate terms, because
every equilateral triangle is equiangular, and every equiangular
triangle equilateral ; but the term equiangular is not commen-
surate with equilateral, for there are figures equilateral which are
not equiangular. It may be pointed out (for it is important to
bear in mind that we have to deal now with the relation between
the different ' universals ' predicable of the same individual, and
not the relation between them and the individual of which they
are predicated — with the relation of ' animal ' and ' mine ', &c,
to ' dog ', and not with the relation of these terms to Tray) — it
may be pointed out that when the subject of a proposition is
singular, the predicate is hardly ever commensurate 2 : for the
predicate is an universal, and so commonly predicable of other
subjects besides this individual : mine is predicable, for example,
of other subjects than Tray ; whereas this individual is predicable
of none of those : nothing else that I can call mine is Tray. Now
where the predicate of a proposition is commensurate with the
subject, there it is either the Definition or a Property of it : where
it is not commensurate, there it is either part of the Definition,
i. e. Genus or Differentia 3, or an Accident.

The definition of anything is the statement of its essence 4 :
what makes it that, and not something else. In the following
propositions, the predicate claims to be the definition of the subject :
1 An organism is a material body, of which the parts are reciprocally
ends and means ' ; ' a church is a building devoted to the service

1 And therefore, of course, neither of anything of which the other cannot
be predicated. Here and in some later passages I put triangle as equivalent to
rectilinear triangle. Spherical and other triangles are ignored for the sake of
simplicity.

2 Only if it is a predicate which from its nature can belong to no more
than one individual : cf. supra, p. 67, n. 1.

3 But sometimes a differentia is commensurate : v. p. 74.

* 'Opurpos nev yap tov ri i<TTi Kai ovaias, Ar. Anal. Post. /3. iii. 90b 30. We
may ask the question ri etm ; — what is it ? — of an attribute (like momentum)
as well as a substance (like a man or a lobster) ; and the answer will be
a definition. In strictness we can define the ovaia of an individual, if at
all, only as meaning the kind to which it belongs ; cf. the previous ch.,
pp. 53-57, and also p. 28.

iv] OF THE PREDICABLES 73

of God according to the principles of the Christian religion ' ;
1 momentum is quantity of motion ' ; ' wealth is that which has
value in exchange ' ; 'a triangle is a three-sided rectilinear
figure ' ; ' a line is the limit of a superficies '. The predicate states
what it is that makes anything an organism, a church, a line,
a triangle : what constitutes momentum or wealth, as distin-
guished from everything else, such as apathy or architecture.
In these judgements it is clear that the predicate, in claiming to
be a definition, claims to be commensurate with its subject ; if
an organism is a material body of which the parts are reciprocally
ends and means, then my dog Tray, being an organism, must be
that, and whatever is that must be an organism : for to be such a
body is to be an organism. If wealth is that which has value in ex-
change, then gold, having value in exchange, is wealth, and so forth.
The genus is that part of the essence of anything which is pre-
dicate also of other things * differing from it in kind.2 Each of
the definitions above given begins by declaring the subject some-
thing, which other and different subjects are besides ; an organism
is a material body — so is a machine, or a block of stone ; a church
is a building — so is a stable ; a triangle is a rectilinear figure — ■
so is a square ; a line is a limit — so is a point, but of a line ; wealth
is that which has value — so is honesty, but not in exchange, for
you cannot transfer it 3 ; momentum is quantity — of motion, but
not of matter. These (building, rectilinear figure, limit, &c.) are
the genus, in each case ; and the genus, being predicable of other
subjects, is clearly not commensurate.4 Genus is sometimes
explained as a larger class including the class defined within it ;
figure, for example, as a class including triangles, squares, cones

1 ' Thing ' here again does not mean only a concrete thing.

8 Vivos 8y earl to Kara irXeiovoop Knt duicfjepovTav ra e"8ei ev rep ri icm KnTt}yo-
poifxevou, At. Top. a. v. 102a 31. The notion of a kind is here presupposed.
Some discussion of it will be found below, pp. 91-103. In botanical and
zoological classification, genus is not merely correlative to species, but marks
a certain degree of affinity, lower than specific, higher than that of families,
orders, &c. Hence a genus, and even a family, may contain only one species,
if that diverges as far from the species nearest it as do the species of different
genera or families ; Homo Sapiens is in the zoological genus Homo and family
Hominidae, and is alone in them. (I borrow the latter part of this note from
Miss Augusta Klein.)

3 The honest man, however, commands in many situations a higher price,
and so far some economists would reckon honesty as wealth.

4 This must be received subject to modification from what is said below
as to the genus being in itself indeterminate, and actually different in each
of its species. Cf. pp. 83-88, 138.

74 AN INTRODUCTION TO LOGIC [chap.

and many other subordinate classes besides : building as a class
including churches, stables, barracks, and so forth. This explana-
tion cannot be considered a good one, for reasons to be presently
stated 1 ; but it may put some into the way of grasping a better.

The differentia is that part of the essence of anything — or, as
we may say, of any species — which distinguishes it from other
species in the same genus ; it is the differentia of an organism
that its parts are reciprocally ends and means — in this it differs
from other material bodies ; it is the differentia of a church, to
be for the service of God according to the principles of the Christian
religion — in this it differs from other buildings ; and so forth. The
genus and differentia (or differentiae 2) between them constitute
the species, or make up the essence of that which is defined. The
differentia, like the genus, need not be commensurate with its
subject. The Book of Common Prayer is for the service of God
in accordance with the principles of the Christian religion, but not
being a building, it is not a church. On the other hand the
differentia is commensurate with the subject of which it is pre-
dicated in cases where no genus except that to which the subject
belongs is susceptible of the particular attribute which serves as
differentia ; thus a vertebrate is an animal of a particular structure
which cannot exist except in an animal, so that the differentia of
vertebrate is commensurate with it. And it is only where this is
the case that the ideal of definition is attained, because only there
is it precisely the common genus which is shewn to be realized
in the several species.

Those who speak of the genus as a larger class containing the
species or smaller class within it sometimes explain the differentia
as the attribute, the possession of which marks off the smaller
from the rest of the larger class. If squares and rhomboids,
triangles and pentagons, &c, are all placed in the class of plane
rectilinear figures because they have that character in common,
triangles, on the other hand, are differentiated from the remaining
classes included within that of plane rectilinear figure by possessing
the attribute of being three-sided. Provided it is not supposed
that the differentia is added to the common character of the

1 v. infra, pp. 83-84.

2 In the plural if the genus has divers determinable points, some or all of
which have to be specified differently in the different species. Cf. infra,
pp. 100-101. In the rest of the paragraph, the singular must be taken as
covering a complex of differentiae.

iv] OF THE PREDICABLES 75

' larger class ' in the same extraneous way that sugar is added to
tea, there is no fresh harm in this mode of expressing oneself.

A property is an attribute common and peculiar to a subject1
(and therefore obviously commensurate with it), but not part of
its essence, and so not included in the definition of it. This is
Aristotle's original account of a property, though we shall see that
he also used the term with a less restricted meaning.2 An organism,
for example, is contractile, irritable, assimilates food, reproduces
itself after its kind : these are attributes of every organism, and
of nothing else, and therefore common and peculiar to the subject
organism ; but they are not in its definition. A triangle, again,
has its interior angles equal to two right angles, and its area half
that of the parallelogram on the same base and between the same
parallels ; a line is either straight or curved (here the alternatives
together are common and peculiar) ; and so forth.

All other attributes of any subject are accidents. An accident
is defined as a non-commensurate predicate not included in the
essence : or as an attribute which equally may and may not belong
to a subject. The latter is the better definition, because it tells us
what an accident is, whereas the former only tells us what it is not.3
It is an accident of an organism to be used for food ; for it may
be so used, but need not. It is an accident of a church to be a
cathedral ; some churches are cathedrals, and some are not. It
is an accident that a contractor should be an honest man, and an
accident that he should be a rogue ; for roguery and honesty are
both compatible with being a contractor.

The doctrine just illustrated presents many points for considera-
tion, of which the following are perhaps the most important : —

1. the antithesis between accident on the one hand and all the
other heads of predicables on the other ;

1 The subject being indicated, it must be remembered, by a common, not
a singular term. I cannot speak of yelping as an attribute common to Tray,
but I can speak of it as an attribute common to the dog— i.e. belonging to
every instance of dog. Aristotle sometimes spoke of an attribute peculiar
to an individual, and not to a kind or universal, as a property ; and also of
attributes peculiar to one out of a certain definite number of kinds, and
therefore serving to distinguish it from the rest (though found perhaps again
outside their number) as relatively properties ; thus it is a property of man
relatively to any quadruped to go on two legs ; but so also does a bird. He
recognized that this use of the term 'property' was not the same as that
given in the text, and not (in his view) so proper a use. Cf. Top. c i.

2 Cf. infra, pp. 80-81, 104. , _

8 Cf. At. Top. a. v. 102b 4-14. Cf. Top. e. i. The former also includes
generic properties : cf. infra, p. 104, n. 1.

76 AN INTRODUCTION TO LOGIC [chap.

2. how to understand the analysis of a definition into genus and
differentia ;

3. the ground of the distinction between the essence of anything
and its properties.

(1) When we classify the members of a genus or class, we some-
times, after specifying as many distinct species as we can think of,
add another to include anything that does not fall within any of
these ; I may classify my books, for example, according to subject
into historical, philosophical, philological, scientific, and mis-
cellaneous— the last division being merely added in order to
receive any book which does not fall within the others, though the
miscellaneous books have no common character that distinguishes
them all alike from the rest. Now Accident is a head of predicables
which includes any predicate that is neither definition, genus,
differentia, nor property of its subject1 ; but it is not a heading
like ' miscellaneous ' ; there is a very definite and important differ-
ence between the relation of those predicates to their subject
which are classed as accidents, and that of those which fall under
the other heads ; the latter belong to their subject necessarily and
universally, the former do not.

Of any individual, as we have seen, an infinity of predicates
may be asserted. Some of them are seen to be connected, or (as
we may express it) have a conceptual connexion ; i. e. if we rightly
conceive one predicate, we see how it involves another. Tray, for
example, is a dog and an animal ; and these predicates are con-
ceptually connected, because the concept of a dog involves that of
animal. My watch has hands, and there is a conceptual connexion
between having hands and being a watch, since without hands
a watch could not fulfil the task of telling the time, which is part
of the concept of it as a timepiece. But there are also many
predicates which coincide 2 in one and the same individual, without
being conceptually connected. Besides being a dog, Tray is mine,
and was born at Bishop Auckland ; now there is no reason in the
nature or the concept of a dog, why it should belong to me, nor in
a thing being mine, why it should be born at Bishop Auckland,
nor in being born at Bishop Auckland, why it should be mine, or
be a dog. No doubt in the case of this particular dog Tray, there

'SvfxfiftfrjKos 8- (cftiv o fxrjdev fiev rovrav e'o~ri, p;re opos Ui]TC iSiop prjTf y(POst
Ar. Top. a. v. 102b 4.

2 Cf. supra, p. 24. Coincident is really a better translation of av/i.Se^Kos
than accident.

iv] OF THE PREDICABLES 77

is a reason why he is mine and a reason why he was born at Bishop
Auckland ; but the reason for the first fact (which may be that he
was given me) has nothing to do with the reason for the second
(which is that his mother was there at the time) ; nor has the
reason for either anything to do with his being a dog ; he would
have been a dog still, if he had never been given to me, or if he
had been born at Bishop's Lydeard.

Of course with more knowledge the coincidence of attributes
in an individual may often be explained ; but the explanation
will always be largely historical, connecting the coincident attri-
butes severally according to laws with other facts which are found
conjoined but not seen to be connected. We have here the great
difference between science and history. In science we seek to
ascertain the connexion of universals. Sometimes we can only
do this inductively ; by noticing how attributes are historically
found conjoined or disjoined in divers individuals we determine
which must be supposed to be connected1 ; but having established
these ' laws ', we trace out by mere thinking their consequences in
divers situations of fact. Sometimes, without the appeal to experience
which induction makes, we can, as in geometry, trace necessary
connexions between one character and another in things. But
history is interested in individuals in whose total being we find
characters coincident, the conjunction whereof we can never wholly
see to be necessary. Even where they are so far of a kind that we
know how they must behave in a given situation, yet each situa-
tion presents different conjunctions. No doubt the scientific and
historic interests interpenetrate. Some sciences, like geology, are
largely occupied in applying what they know of the connexion of
universals to the elucidation of the history of individual things,
or aggregates, if we hesitate to call a mountain range or a coal
formation one thing. And the historian attempts to trace con-
nexions among the events that make the history of individuals,
or groups of individuals, and so far to be scientific. Perhaps, even
if we started with complete historical knowledge of the conjunction
of individuals at a given time, the subsequent course of history
could never be wholly explained this way ; it may be that the
nature of individuals cannot be exhaustively given in terms of
universal characters, but that there is in each something unique.

1 The illustration of this forms a considerable part of what is called Inductive
Logic ; we shall find that many connexions are inductively established whose
necessity remains unconceived.

78 AN INTRODUCTION TO LOGIC [chap.

But anyhow there will always be the bare conjunction of facts
in the historic situation, which cannot be deduced except from
the previous conjunction in another historic situation.

That the accidental should be opposed to what is necessary and
universal conforms to the usage of common speech. Sir Robert
Peel was killed by a fall from his horse, and we say his death was
accidental. Why ? he was a man, and for a man it is necessary
to die, and for any one who falls in that particular way it may
be necessary to die ; but it is not necessary that a man should fall
in that way ; that is not predicable universally of man. We
sometimes dispute whether there is such a thing as chance in the
world, or whether everything has a cause, and happens necessarily.
Few people really believe that anything happens without a cause ;
but chance is not the negation of cause ; it is the coincidence of
attributes in one individual, or events in the same moment, when
each has its cause, but not the same cause, and neither helps to
account for the other.

If we bear in mind this fundamental contrast between the
accidental and the necessary, we shall not be inclined to think that
Aristotle was engaged in a trivial pursuit when he attempted to
classify the various relations in which a predicate might stand to its
subject. Discussions as to what we mean by cause occupy much
space in many modern treatises. Now the causal relation is also
grounded in the nature of universals : Tray yelps not because he
is this individual Tray, but because he is a dog, and unless any dog
yelped, it would not be because he is a dog that Tray does so.
But when we call this the cause of that, the relation intended is
not always the same ; just as when we say that A is B, the relation
of B to A is not always the same. It might be supposed that if
one thing X is the cause of another Y, then you could not have
X without Y, nor Y without having had X. And yet we say that
molecular motion is the cause of heat, that the heat of the sun is
the cause of growth, that starvation is sometimes the cause of
death, that jealousy is a frequent cause of crime. We should in
the first case maintain that the cause and effect are reciprocally
necessary ; no heat without molecular motion, and no molecular
motion without heat. In the second, the effect cannot exist
without the cause, but the cause may exist without the effect ;
for the sun shines on the moon, but nothing grows there. In the
third, the cause cannot exist without the effect, for starvation

iv] OF THE PREDICABLES 7B

must produce death, but the effect may exist without the cause,
since death need not have been produced by starvation. In the
fourth case, we can have the cause without the effect, and also the
effect without the cause ; for jealousy may exist without producing
crime, and crime may occur without the motive of jealousy. It is
plain, then, that we do not always mean the same by our words,
when we say that two things are related as cause and effect ; and
any one who would classify and name the various modes in which
two things may be causally related would do a great service to clear
thinking. Now that is the sort of service that Aristotle attempted
in distinguishing the heads of predicables. Many predicates are
asserted of the subject A. Those of them are accidents, whose
cause does not lie in its nature as A, or which, when they belong to
any individual of the kind A, do not belong to it because it is A.
The rest are in some way or another connected causally with A,
and are predicable of any individual because it is A. Whether
Aristotle's account of the different modes of connexion between
a subject and a predicate is satisfactory is another question, in-
volved principally in that of the value of his account of ' property '.
But that the theory of predicables is closely akin to the question
of the various senses in which one thing can be the cause of another
may be seen by this : whenever science tries to find the cause not
of a particular event, such as the French Revolution (whose cause
must be as unique as that event itself is), but of an event of a kind,
such as revolution, or consumption, it looks in the last resort
for a commensurate cause. What is that exact state or condition
of the body, given which it must and without which it cannot be
in a consumption ? What are those conditions in a political
society, given which there must and without which there cannot
be a revolution ?

The kindred nature of the two enquiries will be further seen,
by looking at certain cases where it is disputable whether a pre-
dicate should be called an accident of its subject or not ; for an
exactly parallel difficulty may arise in determining whether one
thing shall be called the cause (or effect) of another or not. An
accident is a predicate of a subject A, the ground for whose exist-
ence in that subject does not lie in its nature as A. Hodge drives
a plough ; and a full knowledge of his history would show me
why he drives a plough, and the ground for it therefore lies in the
history of the subject Hodge ; it is not of him that driving the

80 AN INTRODUCTION TO LOGIC [chap.

plough is predicated as an accident. But a man drives a plough.
That is an accident ; for the subject now is not Hodge wholly,
but a man, and it is not in his nature as a man that the ground
or reason of his driving a plough lies ; else should we all be at the
plough-tail. And yet no animal but man can drive a plough : so
that it is partly because he is a man that Hodge drives it ; and
therefore, when it is said that a man may drive a plough, the
relation of the predicate to the subject seems not completely
accidental. Contrast the statement that a cow may be knocked
down by a locomotive. There the nature of the subject, as a cow,
contributes nothing ; it is in no wise necessary to be a cow, in order
to be knocked down by a locomotive 1 ; and the relation is purely
accidental.

If we consider these two examples, we see that our account of
an accident, just given, may be interpreted in two ways. A pre-
dicate may belong accidentally to the subject of which it is pre-
dicated either

(i) when the ground for its existence in the subject does not lie
completely in the subject-concept,2 or

(ii) when the ground for its existence in the subject does not lie
at all in the subject-concept.2

The first interpretation would rank as accidents of a subject 3 all
predicates that are not either part of its definition, or else common
and peculiar to that subject, i.e. properties in the strictest sense;
and such, if we take him at his word, is Aristotle's view. But we
are then required to say that it is an accident of money to be

1 So far as a cow is a body, and only a body can be knocked down, it must
be allowed that the nature of a cow contributes something to the accident ;
but the second sentence will stand without qualification.

2 When a general term is subject of a proposition, though the proposition
concerns individuals (designated individuals or not, according as the general
term is or is not combined with a demonstrative), yet these are characterized
only by the general term. The character by which they are thus distinguished
is the subject-concept. If I say that a cow was knocked down by a loco-
motive, the subject is an individual cow ; it is distinguished in my proposition
from other obstacles by being a cow; this being a cow, or cowness, is the
subject-concept. What is knocked down is a cow, not cowness ; but being
knocked down is accidental to cowness in the cow ; and I can therefore say that
the relation of accident lies between universals, though exhibited between
the instances of them in this cow. It would of course be absurd to say that
the particular cow contributed nothing to the accident, since it could not
have been knocked down if it bad not been there. Students of Professor
Cook Wilson's lectures will remember this distinction between subject and
subject-concept.

8 i.e. of the subject as distinguished by the subject-concept.

ivl OF THE PREDICABLES 81

valuable, since it would have no value if there were nothing to buy
with it : or of coal to burn, since it would not burn in a vacuum.
The second interpretation would refuse the name of accident to
anything that could be said about a subject, however rare and
unconnected the conjunction of circumstances through which it
came about, where the nature of the subject contributed anything
at all to the result. Thus we could hardly call it an accident that
an animal should die of overeating itself, since it must be an animal
in order to eat. In practice we make a compromise between these
extreme interpretations. We call it a property rather than an
accident of belladonna to dilate the pupil, though the result depends
as much upon the nature of the muscles as on that of belladonna ;
we call it an accident rather than a property of the plough to be
a favourite sign for country inns, though its necessary familiarity
to countrymen accounts for its selection. The further pursuit of
these difficulties does not concern us now ; but it remains to be
shown that they arise also in regard to the relation of cause and
effect. Is the cause of an effect that, given which and without
anything besides, the effect follows ? in other words, must it contain
the whole ground of the effect ? then a spark is never the cause of
an explosion, for it will produce no explosion without powder. Is
the cause anything, however slight, without which the effect could
not have occurred ? in other words, is that the cause which con-
tributes anything whatever to the effect ? then are cooks the cause
of health, since there would be little health without them.

(2) The antithesis between accident and the other heads of pre-
dicates needs perhaps no further illustration. We may pass to
the second of the three points enumerated on pp. 75-76, viz. how
to understand the analysis of definition into genus and differentia.

It should first be noticed that definition is never of an individual,
but always of what is universal, predicable of individuals — whether
it be what we call their ' kind ', or some state or attribute of them,
or relation in which they stand. For what is defined is thereby
marked off and fixed in our thought as a determinate concept ;
but the individual is made the individual he (or it) is by an infinity
of attributes ; he is as it were the perpetual meeting-place of con-
cepts ; we can neither exhaust what is to be said of him, nor make
a selection, and declare that this is essential to him, and that
unessential. Moreover, even if we could, we should still only have
settled what he in fact is, but a second person also might be ; for

1779 a

82 AN INTRODUCTION TO LOGIC [chap.

every concept is universal. What makes him this individual and
not another we should not have denned, nor could we ; for there
is something which makes me me over and above what can be
predicated of me ; else, what makes me me might also make you
you ; for what can be predicated of me might be predicable of
another, you for example ; and then why does the same character
make me me and you you, and not rather make me you and you
me, or each of us both ?

We can only define then what is universal, or a concept. But
we have already said that concepts are the natures of things ;
and therefore in defining concepts, we may define things, so far as
they are of a kind, but not as individuals. It is sometimes main-
tained that definitions are not of things, but only of names x : that
they set forth the meaning (or, as it is also phrased, the connotation ^
of a name, but not the nature of a thing. Yet the names are only
used to convey information about things ; and to explain what the
name means, is to explain what the thing is said to be. Definitions
then are not really of names ; but we shall see later the difficulties
which drove men into saying so.

Now when we define we analyse ; and the elements into which
we analyse that which is defined are called, as we saw, genus and
differentia. These might be called attributes of the subject : it
might be said, for example, that rectilinear figure and three-sided
are attributes of a triangle. But the expression is not quite appro-
priate ; for an attribute implies a subject beyond itself, to which it
belongs ; but the parts of a definition themselves make a whole,
and coalesce into the unity to which they belong. This may be
best explained by a contrast. We may take any attributes we
like — say far, sour, pink, soft and circular — and we may give one
name to the aggregate of these. But they do not form one notion ;
they remain obstinately five. If we took a single name to signify
the possession of these attributes, we could explain the name as
meaning that assemblage, but we should feel that in so doing we
were merely explaining a name, and not defining any unity. But
when we analyse into genus and differentia, this is otherwise ;
then we feel that the two together really make a single notion.
They have such a connexion in their own nature as makes one fit
the other, so that they constitute the essence of one thing, or state,

1 e.g. Mill, System of Logic, I. viii. 5.

2 On ' connotation ' cf. infra, c. vi.

iv] OF THE PREDICABLES 83

or action, or quality, or relation. And the reason for the parts of
a definition being one l is this : that they are not attributes inde-
pendent but coincident, but the genus is the general type or plan,
the differentia the ' specific ' mode in which that is realized or
developed. Take again the definition of a (rectilinear) triangle. It is
a rectilinear figure ; but to be merely that is impossible, because
incomplete. There cannot be a rectilinear figure without a definite
number of sides, though any definite number above two will do ;
and if the number in a triangle is three, then three-sidedness is
the specific mode in which the general plan, or as we may say the
potentialities, of rectilinear figure are realized in the triangle.
We may say that the genus and differentia are one, because they
were never really two. Three-sidedness can only be realized in
a figure, rectilinear figure can only be realized in a definite number
of sides. The genus therefore never could exist independently of
a differentia, as soft may of sour : nor the differentia of the genus.
It may be said perhaps that though three-sidedness can only exist
as the form of a figure, rectilinear figurehood exists independently
of three-sidedness in the square, the pentagon, &c. But it is not
quite the same thing in the square or pentagon as it is in the
triangle. So intimately one are the differentia and the genus, that
though we refer different species to the same genus, yet the genus is
not quite the same in each ; it is only by abstraction, by ignoring
their differences, that we can call it the same. Triangle and square
and pentagon are all rectilinear figures ; but in the sense in which
they actually are such, rectilinear figure is not the same in them
all. Thus the differentia modifies the genus. And the genus also
modifies the differentia. It might be said that three-sidedness is
not confined to the genus figure ; for a triangle is a three-sided
figure, and N is a three-sided letter. And doubtless, so far as the
genus is the same in two species, the differentia may be the same
in the species of two genera. But three-sidedness is plainly
different in the figure, where the sides enclose a space, and in the
letter, where they do not ; and the genus as it were fuses with the
differentia, so that each infects the other through and through.

For this reason the genus is not well described as a larger class
including the smaller class or species within it. For the word class

1 That the parts of a definition are one is a thing on which Aristotle fre-
quently insists, and says that the main problem about definition is to show
how that can be. Cf. e.g. Met. Z. xii, H. vi.

G2

84 AN INTRODUCTION TO LOGIC [chap.

suggests a collection, whereas the genus of any species is not a collec-
tion to which it belongs but a scheme which it realizes, an unity
connecting it with things different from itself. It may seem at
first plain-speaking, without any metaphysical nonsense, to say that
a genus is a class of things that all have certain features in com-
mon ; and that its species is a smaller class composed of some of
those things, which all possess not only the features common to the
whole genus, but others not belonging to the other members of it.
But what is really meant by being included in a class ? The phrase
is sometimes put forward as if it were simple, and presented no
difficulty ; but such is not the case. The words ' to be within ', or
1 to be included in ', have many meanings, and we must know what
meaning they bear in the phrase ' to be included in a class ', before
we can know what that phrase signifies. We may distinguish in
particular two meanings, which are quite inapplicable to the relation
between a genus and its species ; but they are more easy to grasp
than the meaning in which the species can be said to be included in
the genus, because they can be in a manner represented to the
senses ; whereas the relation of genus to species can never be repre-
sented to the senses, but only apprehended by thinking. Because
one of these inapplicable meanings is readily suggested to the mind,
when we are told that the genus of a thing is a class in which it is
included, we fancy that the expression helps us to understand what
a genus is ; for these inapplicable meanings are easily understood.
But as they are inapplicable, they help us not to understand but to
misunderstand the logical relation of genus and species.1

In the first place, one thing may be included in another as a
letter is included or enclosed in an envelope, or
as Mr. Pickwick and the wheelbarrow were en-
closed in the pound. In this case, all that is
included may be removed, yet that in which it
was included will be left. Such is clearly not
the sense in which species are included in a genus ;
for there would be no genus left if the species
vanished. Yet the logical relation is often represented by a diagram,
which inevitably suggests this sense. Two circles are drawn, one

1 Though the relation of a species to individuals is not the same with that
of genus to species in all respects, yet what is said here upon the vice of
calling the genus a class in which species are included applies also to the
habit of calling the species a class including individuals.

iv] OF THE PREDICABLES 85

enclosing the other ; the genus being represented by the outer and
the species by the inner circle. It is not impossible to use such
diagrams without being influenced by their obvious suggestions ;
yet their obvious suggestions are false, and to avoid them is difficult.

Secondly, a thing may be included in an aggregate, which is
constituted by that and all the other things included along with it.
In this sense a cannon-ball is included in a heap, and a particular
letter in the pile on my table. We do actually use the word class
on some occasions to indicate a total formed in this way ; in
a school, for example, a class is a certain number of boys taught
together, and when a boy is moved from one class to another, he is
sent to do his work with a different set of boys. Here we have
a notion which is so far nearer the logical notion,1 as that the class
would disappear upon the disappearance of what is included in it.
But a little reflection will show that the logical relation of genus
to species is no more like that of an aggregate to its members than
it is like that of an envelope to its contents.

If Tom Smith is in the first class in his school, I should look for
him among the boys in a particular class-room ; but if a triangle is
in the class figure, or a Red Admiral in the class lepidoptera, that
does not mean that I should look for either in a collection of figures
or of lepidoptera ; it is true that a collection of these objects would
include specimens of the triangle or the Red Admiral ; but they do
not belong to their respective genera because they are in the collec-
tion ; specimens of them are placed in the collection because they
belong to the genera. Were it otherwise, I could not say that
a triangle is a figure, or that a Red Admiral is a lepidopteron, any
more than I can say that Tom Smith is the first class ; I could
only say that as Tom Smith is in the first class, so a triangle is in
the class figure, and a Red Admiral in the class lepidoptera ;
whereas it is characteristic of this to be a lepidopteron, and of that
to be a figure.

The ' class ' to which species (or individuals) are referred is apt
not to be thought of as something realized in its various members
in a particular way ; but the genus is something realized in every
species (or, if it is preferred, in the individuals of every species)

1 i.e. the notion which the phrase 'to be included in a class' must bear
in logic, if it is to be used in any applicable sense at all. And note that even
a class at school is not a chance collection, but a collection of boys supposed
to share the same level of attainments.

86 AN INTRODUCTION TO LOGIC [chap,

belonging to them, only realized in each in a special way. The diffe-
rentia carries out as it were and completes the genus. Individuals
are not included in one genus because agreeing in certain attributes,
and then in one species within the genus because agreeing in certain
other attributes that have no connexion with the first ; as you
might include in one island all men who had red hair, and then
rail off separately within it those of them who had wooden legs ;
wooden-legged could not be a differentia of the genus red-haired ;
it must be some modification of red-hairedness itself, and not of the
men having it, which could serve as a differentia to that genus.
It is therefore a phrase that may mislead, to say that the differentia
added to the genus makes the species, or makes up the definition.
For adding suggests the arbitrary juxtaposition of independent
units ; but the differentia is not extraneously attached to the genus ;
it is a particular mode in which the genus may exist. And hence,
when we distinguish the various species of one genus, in what is
called a logical division,1 assigning to every species the differentia
that marks it off from the rest, our several differentiae must be
themselves homogeneous, variations, as it were, upon one theme
and, because each cognate with the same genus, therefore cognate
with one another. If rectilinear triangle, for example, is regarded
as a genus, and one species of it is the equilateral, the others will
be the isosceles and the scalene : where each differentia specifies
certain relations in the length of the sides ; if one species is the
right-angled, the others will be the obtuse- and the acute-angled :
where each differentia specifies certain relations in the magnitude
of the angles. The principle that the differentiae must be thus
cognate is technically expressed by saying that there must be one
fundamentum divisionis ; this, however, has its proper place of
discussion in the next chapter.

To define anything then per genus et differentiam is to put forward
first a relatively vague notion and as it were the rough plan of the
thing, and then to render this definite by stating in what way the
rough plan is realized or worked out. And the differentiae are of
the essence of the things, because they belong to the working out
of this rough plan. In the definition of organic species (inorganic
kinds we will consider later) this is what we aim at doing. We
start with the general notion of a living body, and classify its
various forms in such a manner as to show how this scheme is

1 Cf. infra, c. v. p. 115.

iv] OF THE PREDICABLES 87

realized in successively more complex ways. Our first division is
into unicellular and multicellular organisms (protozoa and metazoa) :
the former obviously admit of no composite cellular structure ; in
a multicellular organism there must be a method of constructing
the system of parts. Hence we proceed to differentiate these
according to the principal modes of structure which they exhibit ;
on this basis is founded for example the division of the metazoa in
the animal kingdom into coelentera and coelomata ; of coelomata
into a number of ' phyla ' (cpvXa, tribes), the platyhelmia or flat-
worms, annelida or worms, arthropoda, mollusca, echinodermata
and chordata ; of chordata, according to the form which the noto-
chord assumes, into hemichorda, urochorda, cephalochorda and
craniata ; and of craniates, according to the different forms
which the general principle of craniate structure may assume, into
fish, dipnoi, amphibia, reptiles, birds and mammals.1 When it is
said that we start with the general notion of an animal body, it
is not of course meant that historically we conceive that, before
becoming acquainted with individuals. We first become acquainted
with individual plants and animals. But the use of general names
shews that some apprehension of their common nature comes
to us from the beginning along with our experience of individuals ;
only we may long remain unable, or not endeavour, to formulate
it. This also applies, at a higher level, to the common nature of
various species — horse, dog and fox, oak, elm and apple — with
which we have become familiar ; we may detect that there is
such an identity, before we know what it is, and call them all
by a generic name, like animal or tree. The genus is that with
which, when we have acquired an insight into the nature of these
various kinds, we then start ; it is first in the order of our thought

1 The extent to which, in subordinating species and genera to a superior
genus, a common type or plan can be definitely traced through them all,
may vary at different stages of a classification. The same functions of animal
life are diversely provided for in protozoa and metazoa ; and within the
comparative complexity of metazoa, in coelentera and coelomata ; but it
would be impossible to give any one diagrammatic representation of the
structure of all these, or even of all metazoa. Such representations are given
for coelentera in general, and coelomata in general ; yet they are a mere
outline, in which even the principal organs of many important types are
sacrificed. On the other hand, for each separate phylum among the coelo-
mata zoologists can give a representation, in which a place is found for every
principal organ that all the species of that phylum, though with manifold
variation of development, at some stage of life or other alike exhibit ; and
for the subdivisions of the craniata this can be done more adequately than
for the subdivisions of the chordata.

88 AN INTRODUCTION TO LOGIC [chap.

about them when we understand them, not in the order of our
acquaintance with them when we perceive them. According to
the Aristotelian formula, it is (pvuei irporepov, or Aoyw Trporepov, not
7]p2v -nporepov : first or fundamental in the nature of the thing, or
in an account of it, but not what strikes us first. And Aristotle
also expressed its function by saying that the genus is, as it were,
the matter, vk-q, of the species or kind.

In saying that a genus is related to its species as matter to form,
the relation of matter to form is conceived as that of the less
developed to the more developed, the potential to the actual.
A word of caution is necessary here. We often compare two
particular objects, say a ' bone-shaker ' and a modern bicycle, and
observing that one carries out more completely certain features
imperfectly present in the other, call them respectively more and
less developed. The same thing may be observed in the arrange-
ment of a picture gallery, where the pictures are placed in such an
order as will exhibit the gradual development of an artist's style,
or of the style of some school of artists : and in a museum, where
the development of the art of making flint implements is illustrated
by a succession of specimens each more perfect than the last. Now
in all these cases, the more and the less developed specimens are all
of them concrete individuals : each has an actual existence in space
and time. But with genus and species it is otherwise. They are
not individuals, but universals ; the genus does not exist side by
side with the species, as the bone-shaker exists side by side with
the best bicycle of the present day ; and you cannot exhibit genus
and species separately to the senses. It is our thought which
identifies and apprehends the generic type, say of mammal, in the
different species, man and horse and ox ; and in thinking of them,
we may say that the single type is developed in so many divers
ways ; but genus and species do not exist in local or temporal
succession, the less developed first, and the more developed later,
like the specimens which illustrate the development of a type or
style. Obvious as these remarks may seem, they are not super-
fluous, if they help to guard against the idea that a genus is some-
thing independent of its species.

[It would be travelling too far beyond the limits of an elementary
work to enquire into the meaning of arranging individuals in an
order of development : whether (like plants and animals) they
proceed one from another in a true genealogical series, or are manu-

iv] OF THE PREDICABLES 89

[factured independently, like bicycles or arrowheads. A criticism
of the conception of development is however of great importance ;
for the complacent application of the notion to disparate subjects,
under the influence of the biological theory of evolution, by writers
like Herbert Spencer has diffused many fallacies. Perhaps it may
be suggested that, if we wish to know what we mean when we
apply the conception of greater and less development to the relation
between individual things, we should first examine what we mean
by the conception in the relation of genus and species. We cannot
throw any light on the relation of genus and species by comparing
it with what subsists between individuals at different stages of
' evolution ' ; but we may get some light upon the conception
of evolution from reflection on our conception of the relation of
genus to species. For the ' evolution of species ' is generally
supposed to be not mere change, but development ; yet it is often
supposed also to involve nothing of the nature of purpose, or design.
Now unless we find, in considering individual things, that there
is a character or form suggested to us in what we call the less
developed, but not adequately exhibited there as we conceive it, and
that this same character or form is more adequately exhibited
in what we call the more developed thing, we have no right to call
them more and less developed at all. The relation therefore is not
between the things as individual, but between their characters ;
we cannot identify with the less developed individual the character
or form which is less developed in it ; there is the same at
different levels of development in each individual ; and the evolu-
tionary history of the series of individuals must be a manifestation
of such a character or form in them, unless we are to say that
there is no real development, but only change, and that to call this
change development is to read into things a fancy of our own. The
example of such development best known to us is in the activity of
the intelligence.]

[In the first chapter, the antithesis of form and matter was
employed in explaining how a common character might belong to
divers things. Two shillings, we saw, may be said to be of the
same form, while the matter in them is different : and two proposi-
tions to be of the same form, so far as each asserts a predicate of
a subject, while their matter varies with the difference of subject
and predicate. But in saying that genus is related to species as
matter to form, it is implied, as between two species, that their
common genus, the ' matter ', is that in which they agree : while
the specific form assumed by this matter in either is the basis of the
distinction between them. Indeed, the phrase ' specific differences '
implies that their differences constitute their form. It may seem
strange that whereas in one sense matter is that which is different

90 AN INTRODUCTION TO LOGIC [chap.

[in things of the same form, in another it is that which is the same
in things of different form.

A little consideration will show that the common notion in both
these uses of the term matter is the notion of something undeveloped.
With regard to the phrase that calls the genus the matter of the
species, this point has already been illustrated. And when we
contrast, in a shilling, the matter (silver) with the form, it is still
so. We regard a shilling as an object having a certain form (that
might also be stamped in gold or copper) impressed upon a certain
matter, silver : and say that both are necessary to its being a
shilling. But the material which the minter takes has a shape as
much as a shilling has, though one geometrically less simple ;
whereas the matter which the metaphysician contrasts with form is
really silver as of no shape, or without regard to shape (cf. pp. 55-56
supra). Now in thinking of silver in abstraction from any shape,
our thought of it is incomplete. As the genus only exists in the
species, so the matter, silver, only exists in some form. It is
however true that there is no special relevance between the nature
of silver and the shape of a shilling, whereas the specific form of
man can only be realized in the genus mammal ; and hence the
conception of development applies more closely to the relation of
genus and species, than to the relation of matter and form in
a concrete thing.

Many controversies have been waged over what is called the
principium individuationis. What is it that makes one individual
distinct from another individual of the same species ? Some of the
schoolmen held that, being of the same species or form, they were
distinct in virtue of their matter ; and it followed, since angels have
no matter, that every angel is of a different species : except their
species, there is nothing by which they can be distinguished from
each other. We may be less ready to dogmatize with confidence
about angels than were the schoolmen ; but the fashion of deriding
their speculations because they were exercised in solving that kind
of questions is fortunately in diminished vogue. The problem of
the principium individuationis is a serious philosophical problem.

It may throw some further fight on what has been said of the
antithesis between matter and form, to point out that matter cannot
really be the principium individuationis. Two shillings which have
the same form are said to be of different matter. Now their matter
is silver : but it is not because it is made of silver that one shilling
is different from another shilling. In that respect all shillings
agree ; it is because they are made of different masses or pieces of
silver that they are different shillings. But if so, it follows that to
be of silver is a character common to both pieces (quite apart from
their being from the same die) ; and though we say they differ in
matter, we mean that though of the same matter, they are different

iv] OF THE PREDICABLES 91

[pieces of it. The problem of the principium individuationis is not
therefore solved by the distinction of matter and form ; the shillings
are different, though of the same form, because in each that form
is stamped upon a different piece of silver ; but the pieces of silver
themselves present the same problem, of a common form (the
nature of silver) in different individuals. Matter is indeed, strictly
speaking, not a particular thing or an aggregate of particular
things, but a generic concept. We recognize various species of
it, which we call elements : the elements are different forms of
matter ; and in calling them so, we imply something common to
them all, as we imply something common to man and ox in calling
them both animals ; though we are less able in the former case
than in the latter to conceive the common or generic character in
abstraction from its specific differences.]

It hardly needs now to be pointed out, that where the predicate
of a proposition defines the subject, it is related to its subject far
otherwise than where it is an accident. We realize (or we should
realize, if our definitions were what we aim to make them) that the
genus, modified or developed in the way conceived, is the subject ;
the definition and that which is defined are not two but one. Of
course, when a green thing is square, the same particular thing is
both square and green ; the green thing and the square thing are
one thing ; but here the subject is not an universal, and we have
only to recognize the coincidence of attributes in the same indi-
vidual. Being green and being square are not one, as being a square
and being a four-sided rectangular and rectilinear figure are x ; there
is a conceptual unity between these ; between those only an
accidental.

It follows that there is a conceptual connexion between any
subject and its genus or differentia ; he who understands the
nature of the subject sees that it must be what is predicated of it
as its genus or its differentia. What belongs to the essence of any-
thing must belong to it ; for else it would not be that kind of thing,
but something different.

(3) We may now take up the last of the points raised on pp. 75-76,
viz. the ground of the distinction between essence and property ;
since the last paragraph suggests the question, What do we mean
by the essence ? If the essence of anything be what makes it what

1 Aristotle would express this by saying that to xXopoVmay be r? rpayavov,
but to x^wpw rival is not to TiTpayiovai elvai — the green is square, but green-
ness is not ' squareness ; whereas triangularity is three-sided-rectilinear-
figurehood. Cf. supra, pp. 15, n. 1, and 22-23.

92 AN INTRODUCTION TO LOGIC chap.

it is, of course it would be something different, were any element
in its essence wanting ; but what makes it what it is ?

Those who hold the view, already mentioned, that definition is of
names only and not of things, have an answer ready here, agreeable
to that view. They say that we cannot tell what makes anything
what it is, but only what makes it to be called by a certain name ;
and that the world might have been spared much useless con-
troversy, if men had realized that by the essence of anything they
meant no more than the attributes which they agreed should be
signified by a general name : or, as Locke called it 1, the nominal
essence. The essence is on this view determined arbitrarily, i.e.
by human convention, though doubtless not without regard to
human convenience — in particular, the convenience of conforming
our nomenclature to what experience shews us of the grouping
of qualities in things. The view is readily suggested by a con-
sideration of material things. If we were to regard only the
definitions of geometry, it would appear paradoxical to maintain,
that men determined arbitrarily what to include in the definition
of circle or rectilinear triangle, and what to omit. Manifestly
you declare better what a rectilinear triangle is by saying that it
is a three-sided rectilinear figure than by saying it is a rectilinear
figure whose angles are equal to two right angles ; or a circle, by
saying that it is the figure generated by the revolution of a straight
line in one plane round one of its extremities remaining fixed, than
by saying that it is a plane figure having a larger area than any
other of equal perimeter. What has led men to suppose that
definition is a matter of fixing the meaning of names is chiefly the
difficulty found in defining natural kinds, i.e. the various species
of animal, plant, or inorganic element; in despair they have looked
to the signification of the name for the only meaning of the essence
of the thing. Our procedure with abstract notions like wealth
or crime or liberty has lent support to the same view. In these
cases, the subject defined cannot be presented to the senses in an
example, as can gold, or the holm-oak, or the buffalo ; we cannot
be sure therefore that different men intend to define the same
thing, when they offer definitions of such notions ; and instead of
settling first by its appearance that a given act is a crime, or an
object wealth, or a state one of liberty, and then arguing to its
nature from our definition, we have rather to determine whether
1 v. Essay concerning Human Understanding, Bk. III. c. iii. § 15.

iv] OF THE PREDICABLES 93

it is to be called a crime, or wealth, or a state of liberty by con-
sidering whether its nature is such as mankind, or particular
writers, have agreed to signify by those names. Hence it might
appear that in the case of abstract terms x at any rate, convention
settles what the essence of them shall be ; in the main it is not
really so, even with them ; for the understanding of facts would
not then be facilitated as it is by the substitution of ' better ' for
' worse ' definitions of abstract terms ; but the plausibility of the
view here adds weight to the arguments which are drawn, in the
manner we must now proceed to show, from the definition of
natural kinds.

Suppose that we wish to define the natural substance dog, or
gold. The forms of language recognize a difference between a sub-
stance and its attributes ; for we say that Gelert is a dog, but not
that he is a faithful ; and speak of a piece of gold, but not of a piece
of heavy. Yet when we define a substance we can only enumerate
its qualities or attributes,2 and leave out of account what it is that
has them. What attributes of Gelert then are we to enumerate,
to explain what we mean by calling him a dog ? or what attributes
of a wedding-ring, to explain what we mean by calling it gold ?
In each case a certain fixed nucleus, as it were, of attributes, holding
together in repeated instances and through great varieties of cir-
cumstance, is included in our concept of a thing called by such
a general concrete name. But which attributes are to form this
nucleus, and on what principle are we to make our selection ? If
it be said that we are to include every attribute common to all
dogs, or all gold, two difficulties arise. The first is, that we should
include in our notion of dog or of gold all the properties, as well as the
attributes that are to constitute the definition : for the properties

1 Such complex abstract notions were called by Locke ' mixed modes ' ;
which he said we could define, because we had first made them by putting
together simple notions (or in his language, simple ideas) with which we
were perfectly acquainted. The expression ' mixed mode ' has not estab-
lished itself ; perhaps because the words are not well adapted to convey the
meaning which Locke intended by their combination ; but it would be useful
to have an appropriate expression to indicate what he meant. Cf. Essay,
Bk. II. c. xxii.

2 We have, however, seen, in discussing genus and differentia, that these
cannot well be called attributes. But it might be urged, that although they
cannot be attributed to any other 'universal ' as qualifying it, they must
be attributed to something which in any individual is what has the sub-
stantial character, in virtue of which we call it a dog or gold, as well as
having such other attributes as mangy or fine-drawn ; cf., however, pp. 54-57,
supra.

94 AN INTRODUCTION TO LOGIC [chap

of a kind are the predicates common and peculiar to all the indi-
viduals of that kind ; and hence we should still lack a principle
upon which to discriminate between property and essence. The
second difficulty is more serious. We are to include in our defini-
tion of a kind every attribute common to all individuals of that
kind ; but until we have defined the kind, how can we tell whether
a particular individual belongs to this kind or another ? Let the
definition of gold be framed by collecting and examining every
piece of gold, and noting down the attributes common to them all ;
the task is impossible in practice, but that might be overlooked ;
it is, however, vicious in theory ; for it implies that we already
know what gold is, or what makes a particular thing a piece of
gold, and can by that knowledge select the things which are to be
examined, as specimens of gold, in order to determine the nature
of that substance. Thus we seem to be moving in a circle ; what
is gold we are to settle by an examination of the things that are
gold ; what things are of gold, by knowing what gold is.

Hence our selection must be arbitrary ; for we have no principle
on which to make it. We may take a particular atomic weight, the
power to resist corrosion by air, ductility, malleability, and solu-
bility in aqua regia ; and say these constitute gold, and are its
essence. And in that case its colour is a property, or for all we can
tell, an accident ; for we can see no necessary connexion between
a yellow colour and all or any of those attributes, and if we found
a white metal with those five attributes we should have to call it
gold. But if we chose to include yellow colour with them in our
definition, then nothing could be gold that was not yellow ; yellow
would be of the essence of gold ; but only because we had decided
to give the name to no metal of another colour ; it would be the
meaning of the name that fixed the essence, and the essence would
be only ' nominal '.

It has been assumed in the above that the attributes included in
the definition may be not only arbitrarily selected, but without any
perceivable connexion among themselves ; so that any attribute
omitted from the definition should drop at once into the rank of
accident ; the essence is only a collection of attributes comprised in
the signification of the same name, and there are no properties at
all. And some logicians have maintained that we can never see any
necessary connexion between different attributes ; and that when
we speak of them as universally connected, we really mean no more

iv] OF THE PREDICABLES 95

than that they have been very frequently found accompanying one
another. Without for a moment agreeing with this opinion (which
denies any sense in the distinction between a connexion that is
necessary and universal, and a conjunction that is accidental) it
may be admitted that we often regard attributes as necessarily and
universally connected, because we believe that with fuller know-
ledge we might see into the necessity of the connexion, when as yet
we cannot actually do so. This is markedly the case with the
various properties of an inorganic substance ; and the kinds of plant
and animal also present us with many instances where different
peculiarities in a species are inferred to be ' correlated ', because
the same conditions seem to affect them both, or because within
our experience they are uniformly present and absent together,
without our being able to understand the connexion between
them.

The difficulty of determining what attributes are essential to
a substance, and therefore of discriminating between essence and
property, does not however arise entirely from the seeming discon-
nexion among the attributes of a kind. It arises also, at least in
the organic, from the great variation to which a species is liable
in divers individuals. Extreme instances of such variation are
sometimes known as border varieties, or border specimens ; and
these border varieties give great trouble to naturalists, when they
endeavour to arrange all individuals in a number of mutually
exclusive species. For a long time the doctrine of the fixity of
species, supported as well by the authority of Aristotle and of
Genesis, as by the lack of evidence for any other theory, encouraged
men to hope that there was a stable character common to all
members of a species, and untouched by variation ; and the
strangest deviations from the type, excluded under the title of
monstrosities or sports or unnatural births, were not allowed to
disturb the symmetry of theory. Moreover, a working test by
which to determine whether individuals were of different species,
or only of different varieties within the same species, was furnished,
as is well known, by the fertility of offspring ; it being assumed
that a cross between different species would always be infertile,
like the mule, and that when the cross was uniformly infertile, the
species of the parents were different. But now that the theory
of organic evolution has reduced the distinction between varietal
and specific difference to one of degree, the task of settling what is

96 AN INTRODUCTION TO LOGIC [chap.

the essence of a species becomes theoretically impossible. It is
possible to describe a type ; but there will be hundreds of charac-
teristics typical of every species. Who is to determine what degree
of deviation in how many of these characteristics will make a
specimen essentially or specifically different ? Will it not have
to be decided arbitrarily at the last ? so that here again our use
of names will settle what is essential to the species. Everything
will be essential that we require in a specimen in order to call it by
a certain specific name.

Such are the reasons for saying that the essence of anything is
settled by the meaning that we give to names, and if the essence
is thus arbitrary or fixed by convention, the distinction between
essence and property is similarly infected. But that distinction is
obnoxious to another objection, already noticed on p. 93 : that if
the property is common and peculiar to the kind, it ought to be
included in the essence, because connected with it universally and
necessarily. It is as little possible for a rectilinear triangle not to
contain angles equal to two right angles, as not to have three sides ;
as little possible for a line not to be straight or curved, as not to be
the limit of a superficies. If the property of a subject is grounded
in the nature of that subject alone, why is it not regarded as a part
of its nature ? if it is grounded partly in the nature of the subject,
partly in conditions extraneous to the subject, then the subject
only possesses it in a certain conjunction, and it ought to be
called an accident.1

Having thus presented our difficulties, we must endeavour their
solution.

The inexpugnable basis of truth in the theory of the predicables
lies first in the distinction between the necessary and the acci-
dental : secondly, in the analysis of definition into genus and
differentia. The first underlies all inference ; the second, all classi-
fication. But the notion of essence, and the distinction between
essence and property, are not applicable in the same way to every
subject.

They present at first sight no difficulty in geometry. The
essence of any species of figure includes so much as need be stated
in order to set the figure as it were before us : whatever can be
proved of such a figure universally is a property. Thus the
definition is assumed, the properties are demonstrated ; and

1 Cf. supra, pp. 80-81.

iv] OF THE PREDICABLES 97

that is the true Aristotelian distinction between essence and
property.

But how are the properties demonstrated ? Only by assuming
a great deal else besides the definition of the figure of which they
are demonstrated. We assume, for example, the postulates ; and
that means that we see that we always can produce a straight line
indefinitely in either direction, or join any two points, or rotate
a straight line in one plane about its extremity. We assume the
axioms ; and that means that we see, e. g., that any two right angles
must be equal ; and that if a straight line
AB falling on two other straight lines CD,
EF makes the sum of the angles CAB,
EBA equal to the sum of the angles DAB,
FBA, CD and EF must be parallel, and
if not, not ; we assume also in one proposition all that we have
already proved in others. It is seldom from considering merely the
definition of the figure which we contemplate that the perception
of its properties follows ; we must set the figure into space-relations
with other lines and figures, by an act of construction ; and the truth
of our conclusion involves not solely the essence of the figure as set
out in its definition, but that taken together with the nature of
space ; for it is really the nature of space which we apprehend when
we realize that the sum of the interior angles made by two particular
parallel straight lines with a line that cuts them is the same on either
side of it, or that a given straight line can be produced to meet
another with which it is not parallel. Another point must be noticed.
It was said that whereas the properties are demonstrated, the
definitions are assumed ; but that does not mean that they are
arbitrarily taken for granted. They are assumed, because they are
what we start with. But they are not arbitrarily taken for granted,
because it is self-evident to us that the existence of a figure as defined
is possible ; and this is self-evident, because in the process of defining
we realize in an actual or imaginary example that such a figure can
be constructed. We know that three straight lines are enough to
make a figure, because we make it of them in imagination ; we
know that a figure may have five sides, because we see the pentagon
before us. It is this power which geometry possesses of creating in-
stances of the objects of its own study that distinguishes it from the
non-mathematical sciences. And it creates its objects by construct-
ing them — i. e. by drawing lines and surfaces ; and in this possesses

177» H

98 AN INTRODUCTION TO LOGIC [chap.

a natural principle upon which to distinguish between property
and essence. For though commonly, in geometry, properties are
commensurate with their subjects, and may be reciprocally demon-
strated, yet everything depends upon the power mentally to see the
lines and surfaces ; thus the angles of a triangle determine the position
of its lines as much as the position of the lines determines its angles ;
but it is only through dividing space by lines that the angles can be
realized. The visible figure is therefore our necessary starting-
point. A definition which fails to determine that waits for applica-
tion until the figure can be pictured. Let a circle be a plane figure
having a larger area than any other of equal perimeter ; that does
not set a circle before us ; an infinity of figures can, we see, be made
by a line that returns upon itself and is flexible at will ; and the
property specified will not, previously to demonstration, afford us
any means of selecting the figure intended. But say that a circle is
the plane figure generated by the revolution of a straight line about
one of its extremities remaining fixed, and then we have it before
us ; then we understand what it is about which the property of
having a larger area than any other figure of equal perimeter is
affirmed. Once again, in geometry there are no happenings, no
conjunctures. It is true that in order to geometrize we have,
actually or in thought, to draw the figures : but our process of
drawing only renders visible space-relations which we conceive are
eternally present everywhere in space. Therefore the circle or the
triangle is not subject to mutation on different occasions ; there is
nothing to prevent it at one place or time from being the same as at
another ; and the conditions under which it exists do not vary ;
the general nature of the space in which it is is uniform and constant.
Hence the properties of any geometrical figure, though, as we have
seen, we must take the general nature of space into account, as well
as the definition of the figure, in order to realize their necessity, may
yet without risk of any false deduction be regarded as if they were
grounded in the essence of that figure alone. For the general nature
of space is a ' constant ' ; it is everywhere the same, and conditions
every figure alike ; it is not because that ever changes, that different
figures have different properties, but because the figures are different.1
Geometry therefore deals with subjects capable of definition :

1 Some deny that we know Euclid's axioms ; they are only the most
convenient assumptions. Even on this view, though we shall have demon-
strated the properties which in Euclidean geometry are demonstrated of the

iv] OF THE PREDICABLES 99

in which the definition serves to set the subject before us : and
in which the distinction between essence and property, though
from one point of view questionable, is from another sound. It is
questionable, so far as the properties of a figure are as necessary
to it as its definition, and do not really any more depend on the
definition than the definition on them. But it is sound, so far as
the essence is that with which we must start, in order to have the
figure before us, and say anything about it, while the properties
are what we can demonstrate. The process of demonstration may
require that we should make a further construction than what the
figure itself demands ; but this further construction is not necessary
in order that we may apprehend the figure itself ; and hence the
definition, which as it were constructs the figure, gives us what is
essential, the demonstration what is necessarily bound up therewith.1

Now the science of geometry, both in Aristotle's day and since,
has been apt to seem the model of what a science should be ; and
that deservedly, so far as its certainty and self-evidence go. But
though we may desire an equal certainty and self -evidence in other
sciences, we must not ignore the differences between their subject-
matter and that of geometry ; nor must we assume that the dis-
tinction of essence and property will have the same applicability to
concrete bodies as to figures in space. The subjects which we study
in chemistry, in botany, or in zoology, are not constructed by us ;
they are complex, and for all we know may differ much in different
instances ; and they exist under conditions which are not con-
stant (like space) but infinitely various. Hence in them we cannot
expect to find the determination of the essence, and the separation
between that and its properties, as soluble a task as in geometry.

Let us consider first the definition of inorganic kinds. Here,
since a compound may be defined by specifying its composition,
our problem deals with the elements. It will be instructive to look
for a moment at the Greek treatment of this question. There were

figures defined only subject to these assumptions, the definitions of the figures
will have the function stated above. So far as non-Euclidean geometry
deals with what cannot be constructed or imagined, the above statement of
the distinction between essence and property will for it have to be qualified.
In analytical geometry, the distinction between essence and property is
harder to draw ; yet it must be remembered that unless we could envisage
the figure, there would be nothing to analyse. Cf. also infra, p. 333.

1 Yet where there are alternative modes of constructing a figure (e.g. an
ellipse) it will be arbitrary which of them we select to define it by ; we can
only say that the definition must enable us to construct the figure.

H2

100 AN INTRODUCTION TO LOGIC [chap

two main attempts to define the famous four elements of Empedocles,
earth, air, fire, and water. Plato supposed that they differed in the
geometrical construction of their particles, those of earth being
cubic, of air octohedral, of fire tetrahedral, and of water eicosihedral.
If these were their differentiae, what was their genus ? We can only
reply, solid.1 They were something filling space, of different figures.
In assuming the concrete things which he defined to fill space, Plato
did what every one who defines a natural substance does. We do not
always mention this character in our definition ; we might define a
scabius, for example, as a certain kind of composita ; but to be a com-
posita involves it ; and it is necessary if the definition is to furnish
the conception of a material thing at all. In taking geometrical
figures as his differentiae, he attempted to gain in physics the
advantages which geometry derives from our power of constructing
its objects ; but he failed to show how the sensible properties of the
different elements were connected with their respective figures.
Aristotle preferred the method of those who distinguished the
elements not by the figure of their particles, but by the mode in
which they combined certain fundamental sensible qualities, heat,
cold, moisture, and dryness. Fire he thought was the hot and dry
substance, water the cold and moist, earth the cold and dry, air the
hot and moist. These definitions have the disadvantage of using
terms that possess no very precise signification. How hot is un-
mixed fire, and how moist is pure water ?

Modern science recognizes in each element a whole legion of
common and peculiar attributes. Some of these, such as its atomic
weight, are conceived to be constant or to characterize the element
in all conjunctures ; others it only exhibits upon occasion ; this is
the case, for example, with its reactions towards other bodies. We
have very little insight into the inter-connexion of the various
attributes thus characterizing each element ; but unless we are to
regard everything in nature as accidental, we are bound to believe
them interconnected.2 It is impossible to include in its definition
all that is known to be characteristic of an element ; and for the
mere purpose of identification, many of the attributes of an element
would serve equally well. But we prefer to select as differentiae,
and include in the definition, such attributes as appear, in some

1 Or perhaps, regular solid.

a On what kind of evidence particular attributes are held to be connected,
it is the business of the theory of the inductive sciences to show.

iv] OF THE PREDICABLES 101

form or another, in all or a large number of elements ; because we
are thus able to exhibit the divers elements as related to one another
upon a scheme, or in other words to classify them. Thus the atomic
weight of a substance is more suitable for defining it than some
peculiar reaction which it exhibits, although perhaps less useful for
identifying it ; because all elements must have some atomic weight,
but no other need exhibit the same sort of reaction. If, however,
a reaction is common to a number of substances, it may serve as
a ground for collecting those into one class, like the acids : the
common reaction being a generic character ; especially when for
any reason, such as the number of attributes that are commensurate
with it (i. e. are found where it is found, and not where it is absent),
such reaction seems to count for much in the being of the substances
to which it belongs.

Such considerations may guide us in choosing what to include
in our definition ; and we shall also ceteris paribus prefer for diffe-
rentiae those attributes that are continuously exhibited to those
that an element only exhibits in a rare conjuncture. Nevertheless
it is plain that our procedure is in great measure arbitrary ; and
the distinction between essence and property is not applicable as it
was in geometry. For among the constant attributes of an element
we cannot start with some and demonstrate the remainder ; and
those which it exhibits only in particular circumstances are not pro-
perties in the full sense. We may indeed call it the property of an
element to exhibit a certain reaction in certain circumstances 1 ; but
whereas the ' circumstances ' under which geometrical figures exist
and possess their properties are in every case the same (being their
existence in space), the circumstances relevant to the manifestation
of the several properties of an element are different ; hence we
cannot afford to omit the statement of them in stating its properties ;
and since they are often very numerous and complex, and involve
many other substances, it may be more natural to refer the property
to a compound, than to one element. Nevertheless, since causal
connexion is fundamental in the notion property, we rightly regard
these attributes as properties rather than accidents. For although
the subjection of an element to any particular conditions rather
than others is strictly speaking accidental, since it depends upon

1 Cf. Ar. Top. e. i. 128b 16 oVoSi'Sornt te to t'^iov fj k«0' avTo Kai afi rj npos
trepov Kal nore (' a property is ascribed to a subject either per se and always
or in a particular relation and time ').

102 AN INTRODUCTION TO LOGIC [chap.

historical causes that are independent of the nature of that element,
yet its behaviour when subject to those conditions is not accidental :
so that it is fairly called a property of gold to be soluble in aqua
regia, though very little gold be so dissolved. On the other hand,
we call it an accident of gold to lie in the cellars of the Bank of
England ; for though it is not accidental that it should lie where it
is placed, but its doing so is connected with other features in the
nature of gold, yet that the particular place should be the cellars
of the Bank of England no more illustrates a general principle,
than that the aqua regia in which it is dissolved should have been
bought in Cheapside. No reasonings that apply to gold universally,
but only historical reasons, will show that certain parcels of gold
must be lying there.

The use of the singular without the article (as in a proper name)
when we say that gold is malleable, or iron rusts, or silver tarnishes,
is worth remark. It implies that we think of gold, or silver, or
iron as one and the same thing always : that we are looking to the
unity of kind, and not the particular specimens. Different parcels
of the same element may be found in divers states, solid, liquid or
gaseous, crystallized or uncrystallized, in molecules of different
numbers of atoms, and so forth. But we conceive that any one
sample is capable of all states whereof any other sample is capable ;
they have no ' individuality \ Even when we investigate the
properties of a compound, so far as the composition is really known
with accuracy, we have the same confidence in attributing to that
compound universally the properties discovered in a particular
sample. But in organic kinds, though we may know the chemical
composition of the parts, we cannot know with the same accuracy
the composition of the heterogeneous parts into the whole. Hence
we do not know how far different individuals are capable of the
same behaviour. And if an organism has a real unity, the differences
between one and another individual of the same kind will never be
fully explicable from their composition.

Indeed the problem of distinguishing between essence and property
in regard to organic kinds may be declared insoluble. If species
were fixed : if there were in each a certain nucleus of characters,
that must belong to the members of any species either not at all or
all in all : if it were only upon condition of exhibiting at least such
a specific nucleus of characters that the functions of life could go on
in the individual at all ; then this nucleus would form the essence

iv] OF THE PREDICABLES 103

of the kind. But such is not the case. The conformity of an
individual to the type of a particular species depends on the fulfil-
ment of an infinity of conditions, and implies the exhibition of an
infinity of correlated peculiarities, structural and functional, many
of which, so far as we can see (like keenness of scent and the property
of perspiring through the tongue in dogs), have no connexion one
with another. There may be deviation from the type, to a greater
or less degree, in endless directions ; and we cannot fix by any hard-
and-fast rule the amount of deviation consistent with being of the
species, nor can we enumerate all the points, of function or structure,
that in reality enter into the determination of a thing's kind. Hence
for definition, such as we have it in geometry, we must substitute
classification ; and for the demonstration of properties, the discovery
of laws. A classification attempts to establish types ; it selects some
particular characteristics as determining the type of any species ;
these characteristics should be (a) of the same general kind for each
type within one genus, or, as it was expressed on p. 86, variations
upon the same theme, in order to exhibit the mutual relations of
agreement and divergence among the various types : (b) important,
or, as one might say, pervasive : that is, they should connect them-
selves in as many ways as possible with the other characters of the
species. It will be the description of the type, drawn up on such
principles as these, that will serve for definition. It is avowedly
a mere extract from all that would need to be said, if we were to
define (upon the supposition that we could define) any species of
plant or animal completely.

The full nature of an organic species is so complex, and subject
to so much variation in different individuals, that even if it could
be comprised in a definition, the task of science would hardly con-
sist in demonstrating its properties. To discover the properties
of kinds belongs to the empirical rather than to the scientific stage
of botany or zoology. Science proceeds to ask what it is in any
kind on which a particular property belonging to it depends.
Herein we break up or analyse the complex character of the kind,
in order to determine what we call the laws of organic life. If a
species, for example, is keen-scented, that must depend upon
conditions that are but a small part of what would be included in
a complete account of its nature. In order to find the commen-
surate subject of which a property is predicable, we must abstract
from all in the species which is not relevant to that one property ;

104 AN INTRODUCTION TO LOGIC [chap.

and our subject will not be the concrete kind, but one determined
by a set of conditions in the abstract. The property whose conditions
we have found is of course the property not of those conditions, but
of anything that fulfils those conditions ; keen-scentedness, for ex-
ample, is not a property of a particular construction of the olfactory
organ (though we should call it an effect of this), but of an animal in
whom the olfactory organ is thus constructed ; the laws of organio
life suppose of course that there exist organisms in which they
are exhibited. We may still speak therefore of properties of kinds ;
but the demonstration of them considers the nature of the kind
only so far forth as it concerns the property in question. The
property is not common and peculiar to the kind, if other kinds, as
may well be the case, agree with it in those respects on which the
property depends ; or if it depends on conditions which cannot
be fulfilled except in an individual of that kind, but are not fulfilled
in every individual thereof.

Such reflections led the schoolmen to distinguish four senses of
the term property —

1. id quod pertinet omni sed non soli : thus it is a property of the
cow to give milk ; but other animals do the same ; and to give milk
is the commensurate property not of a cow but of a mammal ; being
causally connected with a feature which though present in a cow is
present in other species besides.1

2. id quod pertinet soli sed non omni : thus it is a property of
man to write poetry, but not universally ; for the writing of poetry
requires powers which no creature but man possesses, but which
also one may not possess and yet be a man.

3. id quod pertinet omni et soli, sed non semper : in this sense it is
a property of the male egret to grow a certain kind of feather, much
used by ladies in their hats ; but only to grow it at the pairing season.

4. id quod pertinet omni et soli et semper : in this sense it is a pro-
perty of a rectilinear triangle to have its angles equal to two right
angles ; but it is difficult to find an example of such a property
among organic kinds, for a feature so constant and universal would
be regarded as part of the essence : unless like the schoolmen we
call it a property in this sense to be capable of exhibiting a property

1 If all the subjects possessing the property are in one genus, it is called
a generic property. Aristotle's definition of property as a commensurate
predicate not included in the essence places a generic property under the
head of accident. Cf. p. 126, infra.

iv] OF THE PREDICABLES 105

in sense 3 ; they often gave it as an illustration of property in the
third sense that man laughs ; and in the fourth sense, that he is
capable of laughter ; for the capacity is permanent, but the exer-
cise of it occasional.

In all these uses of the term property the notion of a necessary
or causal connexion is retained ; but commensurateness with the
subject is not insisted on in all. No doubt a commensurate subject
for every predicate is to be found ; but only by specifying the
precise conditions (in an organism or in whatever it may be) on
which the property depends ; but the concrete thing is the subject
about which we naturally make propositions, naming it after its
kind ; and kinds being complex may agree together in some points
while differing in others with intricate variety ; so that when we
have distinguished the species to which things conform, and the
attributes which they possess, we cannot divide the latter among
the former without overlapping.

Many general and abstract terms, which form the subjects of
propositions, designate neither natural substances nor mathematical
entities. There are names of qualities and states of things, like
softness or putrefaction : or psychical states and processes, like
pleasure, anger, volition : of the material products of human or
animal skill, like pump, umbrella, bridge or nest : of natural features
of the earth's surface, like beach or valley : of determinate parts of
an organism, like cell or sympathetic nerve : of forms of human
association, like army, university, democracy, bank. It would be
tedious to proceed further with such an enumeration. About all
of these terms it is to be observed that the notion of them involves
a certain abstraction. Bridge and pump are concrete terms, but
they are names given to material things because they serve a cer-
tain purpose, or exhibit a certain structure ; and all else in the
nature of the thing is disregarded, in considering whether it is
a bridge, or whether it is a pump. In attempting to define an
element on the other hand, or an organic species, we have to wait
upon discovery, in order to know the nature that a thing must
possess as gold, or as a crab ; the whole nature of the concrete
thing forms the subject of our enquiry. It is the abstract character
of the terms which we are now considering, or the limited extent
of their signification, that renders them more capable of satis-
factory definition ; they are least definable, where that which
they denote is most complex ; thus it is easier to define army

106 AN INTRODUCTION TO LOGIC [chap.

than democracy, and rigidity than putrefaction. The more complex
any subject, the less is it possible to exhaust its nature in any
brief compendium of words, and the greater also are its capacities
of various behaviour under varying conditions ; all these are part
of the notion of it, and no definition will really be worth much to
any one who cannot realize how different the thing defined would be
in different circumstances. Thus a definition of democracy means
most to him whose mind is most fully stored with a knowledge of
history and of institutions and of human life ; he can realize what
government of the people by the people for the people (if that were
our definition) really involves. But comparatively little knowledge
is needed in order that the definition of a bridge may be fully under-
stood. It will be readily seen, that what has been said of the diffi-
culty of determining either property or essence in regard to natural
kinds applies also to such terms as we are now considering in pro-
portion to the complexity of the notion to be defined ; the more
complex the subject, and the greater the range and variation of the
modes in which it manifests itself, according to the conditions under
which it exists, the more arbitrary becomes our choice of characters
to be included in the definition, and the less can properties be com-
mensurate attributes.

We have now reviewed the theory of predicables as it was first
propounded ; we have seen that the scheme of knowledge which it
implies cannot be realized upon all subjects ; that it is best exem-
plified in mathematics, and in other sciences which deal with
abstractions. But we have also seen that it contains distinctions
of great value and importance. These are

1. the antithesis between an accidental conjunction (or coinci-
dence) and a necessary or conceptual connexion ;

2. the conception of the relation of genus and differentia, and of
the unity of genus and differentia in a single notion ;

3. the resting the distinction of essence and property upon the
distinction between that which we start with and that which we
demonstrate therefrom ; though this use of the term property
cannot always be adhered to in practice.

It remains to say a few words upon the Porphyrian doctrine.

It differs to appearance in one point alone ; the Porphyrian list
of predicables substitutes Species for Definition. But that difference
implies a change in the point of view. It implies that we are to
find the meaning of these five terms — Genus, Species, Differentia,

iv] OF THE PREDICABLES 107

Property, Accident — in the relations which its predicates bear to
an individual subject not as an individual of a certain sort, but barely
as that individual ; for it is of individuals as individuals, not as of
a certain sort, that their species (such as man, or horse, or parrot-
tulip) are predicated.1 And various inconveniences arise from this
change. First and foremost we have to determine what is a true
species, and what only a genus within a wider genus.2 Do I pre-
dicate his species of Cetewayo when I call him a man, or when I call
him a Zulu ? if Zulu be a species, man is a genus, though included
within the wider genus of mammal, craniate, or animal ; but if
man is the species, Zulu is an accident. The question thus raised
is really insoluble ; for species, as is now believed, arise gradually
out of varieties. It gave rise to many great controversies, as to
whether a species were something one and eternal, independent of
individuals, or on the other hand no more than a name. These
opposite views were indeed older than Porphyry or the mediaeval
thinkers who discussed them so earnestly ; nor can any philosophy
refuse to face the controversy between them. But it was a mis-
fortune that the theory of predicables should have got involved in
the controversy ; partly because it led to a mode of stating the
fundamental issue which is not the best : partly because the true
value of the theory of predicables, as a classification of the relations
between universals predicated one of another, was lost sight of in
the dust of the dispute between the realists and the nominalists.

A second inconvenience in the Porphyrian doctrine is that while
beginning by distinguishing the relations of its predicates to an
individual, it cannot continue true to this standpoint. Species is
properly predicated of an individual ; we ask what is the species not
of man, but of Cetewayo ; and if the species can be analysed into
genus and differentia, it is possible to regard these as predicated of

1 There is a suggestion in Aristotle's Topics of this point of view, for he
allows that tBiov may mean a peculiarity that distinguishes an individual
from others ; cf. the passage quoted, p. 101, n. 1, supra, and e. i. 129a 3-5.
But his doctrine as a whole implies that the subject term is general.

2 In technical language, what is an infima species and what a species
8ubalterna ; it was said that a species subalterna ' praedicatur de differentibua
specie ', an infima species ' de differentibus numero tantum '. But it is clear
that this does not help us to solve the problem : how are we to determine
whether men differ in number only and not in kind ? It is no easier than to
determine whether man or Zulu is the infima species ; being in fact the samo
problem restated. Looked at from the other side, the species subalterna
can of course be called the genus subalternum: of. Crackenthorpe's Logic,
Bk. I. c. iv.

108 AN INTRODUCTION TO LOGIC [chap.

the individual belonging to the species. Nevertheless they are his
genus and differentia not as this individual, but as an individual of
this species. And similarly with property and accident : a property
is necessary to its subject, either absolutely or under definite condi-
tions, i. e. it belongs to a subject of a certain sort because it is of
this sort, or of this sort under these conditions ; an accident is not
thus necessary ; it belongs in a given instance to a subject of this
sort, but not because it is of this sort, and so need not belong in
a second instance. But of a subject indicated by a proper name x—
of an individual as this individual — we cannot thus distinguish the
predicates. A predicate which is connected with one character in
the being of an individual is merely coincident with another ; but
a proper name does not signify one character to the exclusion of
the rest. Without such selection, we cannot say whether a predicate
is property or accident. If it is asked whether it is a property of
Cetewayo to talk, or fight, or be remembered, we must demand, of
Cetewayo considered as what 1 Considered as a man, it is a property
of him to talk ; considered as an animal perhaps it is a property of
him to fight ; but considered as a man, or as an animal, it is an
accident that he should be remembered, though perhaps a property
considered as a barbarian who destroyed a British force. So long
as we consider him as Cetewayo, we can only say that all these
attributes are predicable of him. They all help to constitute
his being as Cetewayo, though not all as a barbarian who destroyed
a British force.

Thirdly, the Porphyrian doctrine gave rise to a division of acci-
dents into separable and inseparable which, if a singular term be
the subject, is confused, if a general, self-contradictory.2 An in-
separable accident of an individual is an accident of the species

1 Or by a designation, unless we regard only the general terms in the
designation, and not the demonstrative which makes it singular. ' The king *
is a designation ; if I say that it is a property of the king to be exempt from
prosecution, I mean of a king, and therefore of George V.

Ifiuoy fie dictffiepeiv Ae-yerai ertpov crepov, orav dj^apiaroa crvp^f^rjKOTi to erepov
tov trtpov 8ia<ftepei. dx<x>piCTOV fie av^fif^iTjKos olov yXavKorrjs r) ypvnoTijs fj oiXf)
(k rpavfiaros (vo-Kippa>6(7cra, Porph. Isag. c. iii, init. ('One thing is said to differ
peculiarly from another when it differs by an inseparable accident. And an
inseparable accident is such as greyness of the eye, hook-nosedness, or the
scar of a wound.') Porphyry indeed says that accidents in general subsist
primarily in individuals — kcu to. pev <rvpfi(^rjK(Wa ini tg>v cnoputv npnijyovpevuis
vcpiararai, ib. c. x ; and also that they are predicated primarily of individuals —

aXXa irp<>T]yovptva)S pfv tcov nropwv (sc. Kar^yopetrat, from the context) and

secondarily of the species containing these, Kara dfCrepov fie Xoyov /cat twi/

iv] OF THE PREDICABLES 109

under which he is considered, but inseparable in fact from him.
Thus it is an inseparable accident of a man to be born in England,
but a separable accident to wear long hair ; because he can cut his
hair short, but cannot alter his birthplace. Now this notion of an
inseparable accident is confused, because the attribute is called an
accident in relation to the species, but inseparable in relation to the
individual ; the whole phrase therefore involves two standpoints at
once. And the distinction between separable and inseparable acci-
dents thus understood has really nothing to do with the doctrine of
the predicables as a classification of conceptual relations between
a subject and its predicates. There are, properly speaking, no
accidents of an individual as the complete concrete individual. The
Old Pretender might have been born elsewhere than in England,
and might have cut his hair shorter : regarding him as the son of
James II, each of these things is an accident ; but regarding him
completely as the man he was, there was reason for each, and
neither could have been otherwise without certain historical cir-
cumstances being different, though history does not usually concern
itself with tonsorial incidents in the lives even of princes. That
one thing was alterable while he lived and the other unalterable
leaves them equally accidents from one standpoint, and equally little
accidents from the other. If however the subject of which a pre-
dicate is said to be an inseparable accident be a general term,
then the expression is self-contradictory. Porphyry said that
blackness is an inseparable accident of the crow. But if it is an
accident at all, then it is a mere coincidence that all crows are black,
and there is nothing in the fact that a bird is a crow requiring it to

7reptex<n'T(ov ra (irofia, ib. c. vi. But he does not seem to see that it is not
from their relation to the individual that they are called accidents. For
his account of the distinction between separable and inseparable accidents,
cf. C. V crvfifiejiriKos 8e ecrriv 6 yivtrai Kal anoyivtTai X^P15 T*ls T0^ vnaKtinevov
(pdopas. Siaipetrat 8e (Is 8vo' to fiev yap avTov xutpio-rop eort, to 8e a^coptarof.
to fiep ovv Ka8ev8eiv x^P^tov o~vpl3ej3r]K6s, to 8e fie\av elvai axa)P'a"ra,s T<P "opaKi
Kai to) kldiotri o-vuftefirjKe, SiWrai 8e iirivor)8r)vai Kal *opa£ \tv<6s Kal AWio)\}s
drro^aXoiv rrjv xpotav ^copir (f>8opas tov vTroKeipevov. ('Accident is what comes
and goes without the destruction of the subject. It is of two kinds, separable
and inseparable. To sleep is a separable accident, to be black is an inseparable
accident of a crow or an Ethiopian ; a crow can be conceived to be white
or an Ethiopian to have lost his colour without the destruction of the sub-
ject.') That he regarded inseparable accidents as predicated both of species
and of individuals as subject is clear from c. vi to 8e peXav tov ts (18ovs to>u
Kopcixaiv Kai tocv Kara /xepos (sc. KaTrjyopeiTai), o-vfxfieftrjKos ov a^copccrroi/, Kai to
KiviicrOai dvdpinrov re Kal 7mrovf xa>Pl0~T0V °v o~vp@e(3r)K6s. ( To be black is pre-
dicated both of the species of crows and of crows severally, being an inseparable
accident, and to move of man and horse, being a separable accident )

110 AN INTRODUCTION TO LOGIC

be black ; it cannot therefore be inseparable, however constant in
our experiences the conjunction may have been. Per contra, if it
is inseparable, that must be because the nature of a crow as such
requires it, and then it cannot be an accident. The so-called in-
separable accident of a species is really an attribute which we rind
to characterize a species so far as our experience extends, without
knowing whether its presence depends on conditions necessary to
the being of the species, or partly on conditions in the absence of
which the species may still exist. That amounts to saying that we
do not know whether it is an accident or a property ; and so a phrase
is adopted which implies that it is both.

It would be well therefore to abandon the division of accidents
into separable and inseparable ; and it would be well to abandon
the Porphyrian list of predicables in favour of the Aristotelian.1
Either list raises very difficult questions ; but those which have
been discussed in this chapter are questions that must be raised,
whether we attach little value or much to the use of the terms
Genus, Species, Differentia, Property, and Accident. The attempt
to think out the connexions between one thing and another is so
vital a feature of our thought about the world, that Logic may not
ignore the consideration of it. Abstract terms, and general con-
crete terms, signify not individuals as such, but as of a kind. We
do regard attributes as connected with one another, and with the
kind of a thing, sometimes necessarily and universally, sometimes
through a conjuncture of circumstances in the history of an in-
dividual. We need a terminology in which to express these differ-
ences. We do conceive substances, attributes and states, that cannot
be anatysed into mere assemblages of simple qualities, but only
per genus et differentiam. These are the facts which justify this
somewhat difficult part of logical theory.

1 Mr. C. C. J. Webb has called my attention to the following interesting
passage in John of Salisbury, Metalogicon, iii. 5 ' Proinde quid genus aut
diffinitio, quid accidens sit aut proprium, docet [Aristoteles] longe commodius
his qui in Porphirio aut Categoriis explanandis singuli volumina multa et
magna conscribunt. In consilium illorum non veniat anima mea, nee aliquia
emicorum meorum praeceptoribus his utatur.'

CHAPTER V

THE RULES OF DEFINITION AND DIVISION :
CLASSIFICATION AND DICHOTOMY

In the last chapter the nature of Definition was discussed at some
length ; but nothing was said of the technical rules in which the
requirements of a good definition have been embodied. The process
of dividing a genus into species was also mentioned, but neither
were the rules given which should be observed in that. It seemed
better to defer to a separate discussion these and one or two cognate
matters. Treated first, they would have been less intelligible.
But what has been said about the relation of genus and differentia,
the practical difficulties that lie in the way of adequately defining
certain — indeed most — terms, and the homogeneity which ought to
characterize the differentiae of the several species in one genus,
should serve to render the present chapter easily intelligible.

The rules of definition are as follows : —

1. A definition must give the essence of that which is to be defined.

The essence of anything is that in virtue of which it is such
a thing. It is in virtue of being a three-sided rectilinear figure that
anything is a rectilinear triangle : in virtue of being an institution
for the education of the young, that anything is a school : in virtue
of having value in exchange, that anything is wealth. We have
seen, however, that in the case of natural kinds, and in some degree
of highly complex abstract notions, the essence cannot be comprised
in the compass of a definition, or distinguished very sharply from
the properties of the subject. In these cases one must be content
to do the best he can : remembering —

(a) That the attributes included in the definition should always
be such as are the ground of others rather than the consequences.
Thus the higher species of mammal are better defined by the char-
acter of their dentition than of their habitual food ; since the kind
of food that an animal can eat depends on the formation of its
teeth, and not vice versa.

(6) That we must not give only some comparatively isolated

112 AN INTRODUCTION TO LOGIC [chap.

attributes of the subject, but also indicate the kind of subject which
these attributes qualify. This is done by giving its genus,1 and
hence our second rule is :

2. A definition must be per genus 2 el differentiam (sive differentias).
The better the definition, the more completely will the differentia

be something that can only be conceived as a modification of the
genus : and the less appropriately therefore will it be called a mere
attribute of the subject defined. Thus a lintel is a bar placed to
form the top of a doorway ; it can hardly be called an attribute
of a lintel that it forms the top of a doorway, for that implies that
having already conceived a lintel, I notice this further as a charac-
teristic of it ; whereas really, until I have taken this into account,
I have not conceived a lintel. On the other hand, if sodium
be defined as an element exhibiting line D in the spectrum, the
differentia here may fairly be called an attribute. For one may
have a pretty definite notion of sodium without knowing that it
exhibits this line in the spectrum. The complexity of the subject
under definition is in this case such that whatever be taken to serve
as differentia can be only a small part of its whole nature ; we have
a pretty substantive concept (if the phrase may be allowed) without
the differentia ; and therefore this appears as a further charac-
teristic, which is really selected because it is diagnostic, i.e. it is a
feature by which instances of the subject can be readily identified.

3. A definition must be commensurate with that which is to be
defined: i.e. be applicable to everything included in the species
defined, and to nothing else.

4. A definition must not, directly or indirectly, define the subject by
itself.

A subject is defined by itself directly, if the term itself or some
synonym of it enters into the definition. The sun might, for
example, be thus defined as a star emitting sunlight ; or a bishop
as a member of the episcopate. Such error is a little gross ; but
in the indirect form it is not uncommon. It arises with correlative
terms, and with counter-alternatives,3 where one is used to define

1 Cf. Ar. Top. (. v. 142b 22-29. But properties, according to Aristotle
(An. Post. £. x), are defined per causam et subjectum, i.e. by specifying the
subjects in which they inhere, and the cause of their inherence in their subjects.

2 Where there is a series of terms in subordination, per proximum genus.

3 Where a subject occurs in two forms, and every instance must exhibit
either one or other, then these forms may be called counter-alternatives.
Thus in number, the counter-alternatives are odd and even ; in a line,
straight and curved ; in sex, male and female ; in property, real and per-

v] RULES OF DEFINITION AND DIVISION 113

the other. A cause, for example, is ill defined as that which pro-
duces an effect, or an effect as the product of a cause ; for correla-
tives must be defined together, and it is the relation between them
that really needs to be defined ; this is the ground of applying both
the correlative terms, and in defining this, we define them. The
objection to defining a term by help of its counter-alternative is
that the latter may with equal right be defined by it. If an odd
number is a number one more than an even number, the even is
similarly that which is one more than the odd. It sometimes
happens, however, that counter-alternatives cannot reallylae denned
at all ; if a man does not immediately understand from examples
that a categorical proposition either affirms or denies, there is no
other knowledge to which we can appeal in order to explain to him
the nature of the distinction, for it is unique ; and in the same way
there is no defining the difference between straight and curved.
In such cases, to explain one counter-alternative by the other,
though not definition, is sometimes the best course we can adopt ;
for their mutual contrast may help a man to apprehend them both,
and he may be more familiar with one than with the other.

There are subtler modes of defining a thing indirectly by itself.
We may use a term into whose definition that which we profess
to be defining enters. Aristotle illustrates this by a definition
of the sun, as a star that shines by day ; for day is the period during
which the sun is shining.1 J. S. Mill's2 definition of a cause as the
invariable and unconditional antecedent of a phenomenon errs in
this particular ; for unconditional cannot really be explained without
presupposing the conception of cause.

It should be noticed that where the thing defined is designated by
a compound word, it may be legitimate to employ in its definition
the words that form parts of the compound. Thus a ball-race is
the hollow way between the axle and the wheel in which the balls
run that are used to take the thrust of one against the other. The
term ball, used in this definition, is not of course what had to be
defined.

5. A definition must not be in negative where it can be in positive
terms.

The propriety of this rule is obvious. A definition should tell us

sonal, &c. Contraries, and opposites generally, may be wrongly used to define
one another in the same way.

1 Top. (. iv. 142* 34. * System of Logic, III. v. § 6; cf. infra, p. 405.

1779 I

114 AN INTRODUCTION TO LOGIC [chap.

what the thing defined is, not what it is not. This it must do up
to a point in naming the genus ; but unless the species is distin-
guished by lacking altogether some character which, in one form
or another, other species possess, it should continue doing so in
naming the differentia. An acute-angled triangle, for example,
should be defined, not as one containing neither a right angle nor
an obtuse angle, but as one containing three acute angles. In this
case it is true that a very little knowledge of geometry would
enable any one to extract from the negative information of the
former definition the positive characterization of the latter. But
the negative differentia is in itself inadequate, and such would
in most cases leave us quite uncertain what the subject positively
is. If real property were defined as property that cannot be trans-
ferred from place to place, we should not necessarily gather that it
was property in land. If anger be defined as an impulse not
directed to obtaining for oneself a pleasure, who is to understand
that it is an impulse to repay a hurt ? But for the reason indicated
in the exception above, a definition with negative differentia is
not always faulty. In defining a privative or negative concept
it is inevitable. A bachelor is an unmarried man ; and the very
meaning of the term is to deny the married state. Injustice,
said Hobbes, is the not keeping of covenant. A stool is a seat
for one without a back to it.1 And short of this, definition by
a negative differentia is justifiable, in defining a species which
is distinguished from other species in its genus by lacking what they
possess.2 Thus Amoeba proteus is an amoeba without a nucleus ;
the melancholy thistle (Carduus heterophyllus) is differentiated by
the absence of prickles. But it must not be assumed that because
a term is negative in form it need be negatively defined ; intem-
perance is the excessive indulgence in strong drink.

6. A definition should not be expressed in obscure or figurative
language.

The use of obscure words where plain and familiar words are
available is a fault in definition, because it militates against the

1 From Watts's Logic. In the definition of injustice, the genus, conduct,
is not stated.

2 My attention has been called to this class of cases by Miss Augusta Klein,
from whom I borrow the illustrations ; such definitions are diagnostic. The
subject so defined exhibits the generic character as determinately as other
species. But the definition, instead of stating in what ways that character
is positively determined, names a part or feature whose absence makes a
notable difference. For positive and negative terms cf . supra, c. ii, pp. 40-46.

vj RULES OF DEFINITION AND DIVISION 115

object of definition — viz. that one may understand the nature of the
thing defined. The use of figurative, or metaphorical, language is
a graver fault, because metaphors, where they are intended to do
more than merely to embellish speech, may suggest or lead up to a
right understanding of a subject, but do not directly express it.
Memory, for example, is ill defined as the tablet of the mind ; for
though knowledge is preserved in memory, so that we can recover it
again, and writing is preserved in tablets for future reference, yet
the two things are very different, and the actual nature of what we
call memory is as little like that of a tablet as possible.

It must be remembered that language is not necessarily obscure
because it is technical. Every science is bound to use ' terms of art '
which will be obscure to the layman, but may express the matters
belonging to that science clearly and precisely. The obscurity
forbidden is that which would be acknowledged by those acquainted
with the field of study to which the definition belongs.

In the process of Definition, we take some species, or other
concept, and distinguish in it its genus and differentia. Thus
wealth is that which has value in exchange. There may be things
which have value, but not in exchange — the air, for example, which
has value in use ; these are not wealth, and with them, in defining
wealth, we are not concerned ; though they belong to the same
genus. But we might be interested in distinguishing the different
species which all belong to one genus ; and this process of dis-
tinguishing, or of breaking up a genus into, the species that belong to
it is called Logical Division.

Logical Division is a process of great importance in science.
Things belonging to one genus will be studied together ; and the
aim of our study will be to discover all the general propositions
that can be made about them. But though there may be some
statements that will apply to everything contained within the
genus, others will only be true of a portion. And the better our
division of the genus into its species, the larger will be the number of
general propositions that can be made about its species or parts.

Division * is closely allied to Classification ; and both to Defini-
tion. The difference between Division and Classification seems to

1 In Logic, if Division is spoken of without any qualification, Logical
Division is meant ; though there are other operations of thought, to be
mentioned later (pp. 132-133), to which the name Division is also applied.

12

116 AN INTRODUCTION TO LOGIC [chap.

be principally this, that we divide the genus, but classify the parti-
culars belonging to it. In other words, Division moves downwards
from the more general to the more special, Classification upwards
from the particulars through the more special to the more general.
This, at least, is the difference which one would intend to indicate
if he contrasted the two operations ; but in actual practice our
thought moves in both directions at once ; and the process of
dividing a genus is at the same time one of classifying the things in
the genus. If, for example, one were asked to divide the genus
novel, he might suggest a division into the novel of adventure, of
character, and of plot ; but he would at the same time run over in
thought the novels that he had read, and ask himself if they could
be classed satisfactorily under these three heads.

The close connexion between Division or Classification and
Definition is obvious. If we divide a genus into species, it must be
by the help of differentiae, which serve to define the species we are
forming. If the genus rectilinear figure, for example, be divided
according to the number of a figure's sides into those with three,
with four, and with more than four sides, we obtain the definitions
of triangle, quadrilateral, and polygon. In a classification also, the
classes established must be distinguished by characters that will
serve to define them.

A division may be carried through several stages, i.e. the species
into which a genus is first of all divided may themselves be sub-
divided into species ; and this may be continued until the species
reached no longer require subdivision. The species with which a
division stops are called infimae species ; the genus with which it
starts, the summum genus ; and the intermediate species, subaltern
genera, i. e. genera (for they are genera in respect of the species
next below them) subordinated to another genus.1 The proximum
genus of any species is that next above it in the series ; and the
words superordinate, subordinate, and co-ordinate are used to indicate
respectively the relation of any genus to those below it, above it, or
standing on the same level with it (i. e. having the same proximum
genus). These terms are also used in reference to a classification ;
for a classification when completed may be regarded as a division
and vice versa. The co-ordinate species into which a genus is

1 Cf. p. 107, n. 2, supra. According to one doctrine, nature has determined
where division should stop, and infimae species are fixed by nature ; according
to the other, they are fixed by us with reference to our purpose or con-
venience. Cf. p. 95, supra.

v] RULES OF DEFINITION AND DIVISION 117

divided are sometimes called its constituent species,1 as together com-
posing or making up the genus.

A division, or a classification, may be set out in a scheme, some-
what after the manner of a genealogical tree. The following is an

example : —

Nebula

I I

Irresolvable Resolvable

(i.e. clusters of stars)

Spiral Lenticular Irregular Containing variables Not known to con-
tain variables

The following are the rules which should be observed in a logical
division : —

1. A division must be exhaustive : i. e. there must be a place for
everything belonging to the genus in one or other of the constituent
species into which it is divided. This rule may also be expressed
by saying that the constituent species must be together equal to the
' totum divisum '.

The necessity of this rule hardly needs indicating. The aim of
division is to set out in orderly relation whatever is included within
a certain genus ; and if the division is not exhaustive, this is not
done. Suppose that an income-tax is introduced ; it is necessary
that the Act imposing it should state what forms of wealth are to be
regarded as income, and taxed accordingly. The rent of land and
houses is clearly a form of income, and would be included in the divi-
sion of that genus ; but if the owner of a house lives in it instead of
letting it, he receives no rent. Nevertheless, he enjoys an income, in
the shape of the annual value of the house he lives in, just as truly as if
he had let that house, and received for it a sum of money sufficient to
hire himself another ; and he ought to be taxed if he lives in his own
house as much as if he lets it. But if the income-tax Act omitted to in-
clude among the species of income the annual value of houses occupied
by their owners, he would escape payment on that head altogether.
Such is the practical importance of making a division exhaustive.

2. The constituent species of the genus must exclude each other.
Unless Ave secure this, we do not properly divide ; for the parts

of that which one divides must be separate from each other.

1 In Latin, membra dividentia, as the species are conceived to share the
genus amongst them.

118 AN INTRODUCTION TO LOGIC [chap.

There are two ways in which a breach of this rule may come
about. We may co-ordinate with a species another which ought
properly to be subordinated to it ; as Dr. Johnson is said to have
divided the inhabitants of the country north of the Tweed into
Scotchmen and Damned Scotchmen ; or as the proverb distin-
guishes ' fish, flesh, fowl and good red herring '. In these instances
the logical error points a sarcasm ; but in itself it is comparable to
the procedure of the philosopher, who cut two holes in his door,
a large one for the cat and a small one for the kitten.

The second mode in which this rule is broken is by a cross-
division ; the nature of this will be explained in connexion with the
rule now following.

3. A division must proceed at every stage, and so far as possible
through all its stages,1 upon one principle, or fundamentum divisionis.

The fundamentum divisionis, the principle or basis of a division,
is that character of the genus, in respect of which the species are
differentiated.2 Let the genus be soldier ; in a soldier we may
look to the mode in which he fights, the military rank which he
holds, or the conditions of service by which he is bound. Proceeding
upon the first basis, we should divide into artillery, cavalry, infantry,
engineers, and flying corps; perhaps staff and commissariat ought
to be added. Proceeding upon the second, we should divide into
officer and private, officer being again divided into commissioned
officer and non-commissioned. Proceeding upon the third, into
regulars, reserve, and territorials. When the division is carried
further than one stage, the same fundamentum divisionis should
be retained in the later stages which was used in the first. If the
division of soldier into artillery, cavalry, infantry, engineers, and
flying corps be prolonged, we might divide artillery into horse-
artillery, field-artillery, garrison-artillery, and mountain-battery ;
cavalry into light and heavy dragoons, lancers, and hussars ;
infantry into mounted and unmounted. But it would not be
proper, unless we wish to distinguish our species by combinations
of differentiae, after beginning with the mode of fighting as our
fundamentum divisionis, to proceed with that of military rank, and
divide artillery into officers and privates ; for that is a division of
soldier generally, and not of artillery any more than of cavalry,
infantry, or engineers ; so that if it is applied to one of these species
it must equally be applied to the others.

1 Of. infra, p. 131. * Cf. supra, c. iv. pp. 86, 10L

V] RULES OF DEFINITION AND DIVISION 119

A division which proceeds on more than one fundamentum
divisionis at once is called a cross-division ; as if one were to divide
soldier into artillery, cavalry, privates, and territorials. It is called
a cross-division, because the grouping required by one basis cuts
across that required by another ; in distinguishing privates, for
example, from other soldiers, we disregard the distinction of cavalry
and artillery, taking all members of both those arms who are not
officers. A cross-division is worse* than useless; for instead of
assisting to an orderly arrangement of things in thought, it intro-
duces confusion.

It is plain that in a cross-division the constituent species will
not exclude each other. The only security for their being mutually
exclusive lies in their being formed upon one basis ; for then they
are distinguished by the different modes in which they exhibit the
same general character. But if different characters A and B are
taken, both of them belonging to the genus, everything within the
genus will exhibit some mode of both these characters ; and the same
individuals which are included in a species that is constituted by the
particular mode a' in which it exhibits the character A may also be
included in a species constituted by the particular mode b' in which it
exhibits the character B ; hence a' and b' may not exclude each other.

There are two apparent exceptions to be considered to the state-
ment that, where more than one fundamentum divisionis is employed,
the resulting species do not exclude each other.

The ancient division of matter into the four elements, already
alluded to as having been adopted by Aristotle,1 proceeds (or
appears to proceed) upon a double basis, of temperature and of
humidity. Matter is either hot or cold ; matter is either moist or
dry ; and hence four species were established, the hot and dry (fire),
the hot and moist (air), the cold and dry (earth), the cold and moist
(water). But there is not really a cross-division here. We do not,
while professing to divide upon the basis of temperature, at the same
time introduce species founded upon the basis of humidity (as if
we were to distinguish the hot, cold, and moist elements) ; our real
basis is neither humidity nor temperature, but the combination of
the modes of temperature with the modes of humidity. And such
a basis offers a peculiarly favourable opportunity for a good division.
For given a certain number of characters in a genus, each found in so
many different modes, and granted that every member of the genus

1 Cf. supra, c. iv. p. 100.

120 AN INTRODUCTION TO LOGIC [chap.

must exhibit each character in some mode, and no character in more
modes than one, then the possible alternative combinations are
discoverable with mathematical precision. But it is only where the
combination of certain characters happens to be of primary impor-
tance, that such a basis of division can be profitably adopted. There
would be no advantage in applying the method in such a case as
the division of the genus soldier, where, if we took the three bases
of mode of fighting, military rank, and conditions of service together,
assuming five alternatives under the first head, three under the
second, and three under the third, we should obtain a division into
forty-five members. These would be mutually exclusive ; yet
such a result would for most purposes be valueless ; for the three
bases of division are not such as it is useful to attend to together ;
though in a particular connexion, as, for example, in drawing up
a scale of rates of pay, it might be advisable to proceed thus.1

In the above case, a cross-division seemed to be employed when
it was not ; in the next it might seem not to be employed when it is.
It may happen that in respect of the individuals belonging to them,
the constituent species into which a genus is divided upon one basis
coincide respectively with those into which it is divided upon another.
Thus angiosperms, or plants whose seed is contained in a pericarp,
may be divided according to the method in which they form new
wood into exogenous and endogenous ; and according to their mode
of germination in the seed into dicotyledonous and monocotyle-
donous. It happens that all the exogena are dicotyledonous, and
all the endogena monocotyledonous ; so that if the genus were
divided into exogena and monocotyledons, there would not in fact
be any plant that fell within both members. Nevertheless, the
division is logically a cross-division, for there is nothing that we can
see to prevent the existence of such a plant, and we can imagine
endogena which are dicotyledonous ; and therefore that our con-

1 Dr. Venn, Empirical Logic, c. xiii. pp. 318-321, points out the part played
by this method in Symbolic Logic. Suppose a class S, whose members are
characterized by the presence or absence of each of the attributes X, Y, Z;
but not all combinations are found. Then we may work out mathematically
the class-compartments determined by the different possible combinations of
differentiae ; and if we symbolize the absence of X by X', there will be
XYZ, X'YZ, XY'Z, XYZ', and so on. Then the statement that whatever
is X and Y is Z is equivalent to saying that the class-compartment XYZ*
is not 'occupied', and can be written symbolically ' XYZ' = 0 '. Such
methods of symbolization may facilitate the working out of the implications
of a number of propositions relating to the same genus. But they do not
express the common character of all reasoning.

v] RULES OF DEFINITION AND DIVISION 121

stituent species do not overlap must be regarded as our good fortune,
whereas it ought to arise out of the necessity of the method on
which our division proceeds. And even if we came to understand
the connexion between these differences in mode of wood-formation
and of germination, such a division would still be vicious ; for it
would not exhibit our species as necessarily excluding each other ;
and this because (what is more important) it would not exhibit
them as alternative developments of a single, or common, notion.1

There is a form of division called Dichotomy, which is of necessity
exhaustive, and the species yielded by it of necessity exclude each
other ; for it divides the genus at every stage into two members
(as the name implies), which respectively do and do not possess
the same differentia ; everything in the genus must therefore belong
to one side of the division or the other, and nothing can possibly
fall into both. Animal, for example, may be divided into vertebrate
and invertebrate, body into animate and inanimate, substance into
corporeal and incorporeal ; each of these divisions is exhaustive,
and its members mutually exclusive.

Some logicians have held that in order to secure these advan-
tages all divisions ought to proceed by dichotomy. But the truth
seems rather, that when a division is undertaken with the view of
classifying or arranging all that is contained in the genus, dicho-
tomy should not be used. Its use is in analysing or defining some
one subordinate species. It may, however, sometimes be used to

1 A cross-division is in fact a defect of principle, which is not removed
because practical inconveniences are avoided. H. Sidgwick, in his Methods
of Ethics, holds that it is reasonable for a man to seek his greatest private
happiness, and also to seek the greatest happiness of the greatest number;
and he admits that, so far as happiness in this life is concerned, these principles
would conflict in their application to many situations. He thinks however
(v. Concluding Chapter) that this ' fundamental contradiction ' would be
removed, if the Deity by a system of rewards and punishments hereafter
made it for the greatest happiness of the individual to promote the greatest
happiness of the greatest number. But the theoretical difficulty, that reason-
able action is conceived in two ways, between which we see not only no
necessary connexion, but possible collision, would still remain. So in the
division of angiosperms into endogenous and dicotyledonous, the specification
proceeds disparately, and the absence of collision is an ' uncovenanted mercy \
If a genus were merely 'items of connotation', to which differentiae were
added as further items (cf. Venn, op. cit., c. xii. p. 310), such procedure in
dividing it would have no impropriety : angiosperms X ( = abc) could be
divided into Xd and Xe. Thus we see the impropriety is evidence that we
do regard the relation of genus and differentia in the way described in the
previous chapter : that the alternative species of a genus are bo many ways
in which the same nature is realized or carried out.

122 AN INTRODUCTION TO LOGIC [chap.

show that a division which is not dichotomous is nevertheless exhaus-
tive, and the constituent species exclusive of each other.

The reason why dichotomy is out of place as the principle of
a classificatory division is that we desire in a division to exhibit our
various species as alternative developments of a common notion ;
at every stage the genus is further particularized by the differentiae
which we introduce in constituting its species ; thus the division of
the genus soldier, according to mode of fighting, into artillery,
infantry, cavalry, engineers, and flying corps, was carried further
by particularizing the way in which the artillery may be con-
stituted for different fighting purposes, or the cavalry armed, &c.
But one side of a dichotomy is always characterized negatively, by
the non -possession of the attribute which characterizes the other
side ; and there is therefore no positive notion, except the original
genus, which we can develop in the subdivision of this side. Now
it may be sometimes convenient to use negative differentiae in the
course of a classification, when one species or subaltern genus is dis-
tinguished from the rest by lacking a character which they exhibit.1
But this is not done upon any principle of dichotomy ; for there
might be several co-ordinate species or subaltern genera distinguished
by different forms of that character which the one lacked ; and then
the division would not be dichotomous, but as manifold as the facts
required. Thus albinism might be co-ordinated with several varieties
of pigmentation. And the further differentiation of the subaltern
genus differentiated negatively would be made by means of some
fresh generic character ; whereas when dichotomy is adopted as
a principle, the negative differentia is introduced before exhausting
the co-ordinate forms of the generic character first used as a basis ;
so that at each stage the remainder of these appear as variations of
the lack of the last form taken as a positive differentia. Thus the
land of a country may be divided, according to the use to which it is
put, into building-land, farm-land, forest, means of communication,
pleasure-ground, and waste ; each of these ' subaltern genera '
may be subdivided, farm-land for example into arable, pasture, and
orchard : orchard again according as bush-fruit, tree-fruit, or hops
are cultivated. But if we were to proceed by dichotomy, we should
divide land into building-land and land not used for building : the
latter into farm-land and non-farm-land : non-farm-land into forest
and not forest, and so forth. Now such a division would not only be

1 Cf. supra, p. 114, n. 2.

v]

RULES OF DEFINITION AND DIVISION

123

far more cumbrous than one unhampered by the method of
dichotomy, as may be seen by setting both out in scheme as
follows : —

1.

Land

Building-land Farm-land Forest Means of com-
munication

2.

Arable Pasture Orchard

Of bush-fruit Of tree-fruit Of hops

Land

T1 ri

Pleasure- Waste
ground

Building-land Land not used for building

Farm-land
I

Non-farm-land

Arable Not arable Forest Not forest

Pasture Not pasture Means of communication Not means of communication

Orchard Not Orchard

Pleasure-ground Not pleasure-ground

Of bush-fruit Not of bush-fruit

of bi

Waste Not-waste

Of tree-fruit Not of tree-fruit

1 I

Of hops Not of hops

but it fails entirely to exhibit its species as alternative developments
of a common notion, or (as it was put in the last chapter) variations
on a common theme. To build on it, to farm it, to let it grow
timber, &c, are so many ways of using land ; to plough, to graze,
and to raise fruit from permanent stocks on it are three ways of
farming, and therefore of using it ; to grow bush-fruit, tree-fruit,
and hops on it are three ways of raising fruit on it from permanent
Btocks, and therefore of farming and therefore of using it.1 But

1 Perhaps orchards (if they may be held to include all ground used for
raising fruit from permanent stocks) should be divided according as they

124 AN INTRODUCTION TO LOGIC [chap.

to farm land is not a way of not building on it ; a forest is not a
form of not being a farm ; roads and railways, which occupy land
that is used as a means of communication, are not modes of not
being a forest ; to use land as pleasure-ground is not a particular
way of not making a road or a railway along it ; to leave it waste
is not a particular way of not using it as pleasure-ground. Neither
again is grazing a particular way of not ploughing land, nor growing
tree-fruit a particular way of not growing bush-fruit on it. The
positive differentia of any subaltern genus negatively characterized
is therefore really a differentia of the nearest positive genus :
forest-land and farm-land, e.g., are species of land, not of non-
farm-land and land-not-used-for-building. A negative concept
affords no basis for further subdivision, and in a division which
attempts to classify by dichotomy half the differentiae are useless
for the development of the generic notion.

[This is the main objection to a classificatory division by dicho-
tomy ; which is strangely defended by Jevons, Principles of Science,
2nd ed., c. xxx, pp. 694-698, and Elementary Lessons in Logic,
Lesson XII. Other objections, which it seemed unnecessary to add
in the main text, since the first is fatal, may nevertheless be pointed
out. The proper division co-ordinates concepts of the same degree
of speciality ; but the division by dichotomy subordinates them in
several stages ; so that waste-land is placed level with orchards of
bush-fruit. The serial order in which the subaltern genera are placed
(except where a positive concept is divided) is also quite arbitrary ;
building on land might as reasonably be called a way of not farming
it, as farming it a way of not building on it. Lastly, it is claimed
for division by dichotomy that it is the only method which secures us
from possible oversight of a species : if man be divided into Aryan,
Semitic, and Turanian, a race may turn up that is none of these ;
whereas if it be divided into Aryan and non- Aryan, non- Aryan into
Semitic and non-Semitic, and non-Semitic into Turanian and non-
Turanian, we have a class ready (non-Turanian) for any new race
that may turn up. But it must be observed that to say that a race
is non-Turanian does not characterize it ; that the Aryan and
Semitic races are also non-Turanian (so that the constituent species
are not mutually exclusive) ; and that if the last objection is con-
grow bush-fruit, tree-fruit, or bines ; and bine-orchards might be subdivided
into hop-yards and vineyards. Even then it is not clear where strawberry-
gardens would come. Such are the practical difficulties of making a perfect
division. In the text something has been sacrificed to compendiousness, else
nursery-grounds, brick-fields, and other varieties of land distinguished
according to use would need to be included.

v] RULES OF DEFINITION AND DIVISION 125

[sidered captious, because the non-Turanian is expressly made a
branch of the non-Semitic, and that in turn of the non-Aryan, then
it means what is neither Aryan, Semitic, nor Turanian ; now if we
are uncertain that our division is exhaustive, and wish to reserve
a place for things that may fall within none of the species we set
up, it is easy to do that without the pains of all this dichotomy ; we
may divide man into Aryan, Semitic, Turanian, and anything that
is none of these ; this last heading expresses what non-Turanian
means in the dichotomy, and stands, as it should, upon a level with
the rest.]

For this reason, a classificatory division should never use dicho-
tomy as a principle ; the numbers of species into which a summum
or subaltern genus is to be divided can be determined not on any
general logical grounds, but solely with reference to the nature of the
genus in question. Even where, as in the case of the four elements,
the basis of division is the combination of attributes, the number of
possible species that can be formed by different combinations is
determined, under the restriction that contraries cannot be combined
together, not by logic but by mathematics. Of course, if a genus
falls naturally into two species, it ought to be divided in two ; as
number is divided into odd and even, and line into straight and
curved. But this is not mere dichotomy ; for it is not the same
to divide number into odd and even as to divide it into odd and not
odd. The claim made for dichotomy is that its branches exhaust
the genus and exclude each other in virtue of the mere form of the
division * ; since everything in a genus must either be or not be,
and cannot at once be and not be, characterized by any differentia
that can be taken. And this is true ; and we need realize no more
than this, in order to see that number is either odd or not odd ; but
in order to see that it is either odd or even we need to understand the

1 Cf. S. H. Mellone, Introductory Text-book of Logic, c. v. § 10, who points
out that although division by dichotomy ' has been adopted by the mediaeval
and formal logicians because it appears to provide a theory of division which
does not make the process depend entirely on the matter of our knowledge,
as classification does ', yet this appearance is illusory. I know on formal
grounds that of any genus x the species either are or are not characterized
by any attribute a ; but I cannot therefore divide x into the two species
a and not-a, since in fact a may be an attribute never found in the genus at
all. Every circle must be either rectilinear or not ; but there are not two
species of circle, the rectilinear and the non-rectilinear. For this reason, in
Symbolic Logic (cf. supra, p. 120, n. 1), XYZ, X'YZ, &c, represent not classes
but class-compartments, which may be necessarily empty; and some writers,
like Mr. Bertrand Russell, recognize by the name of null-class a class which
has no members.

126 AN INTRODUCTION TO LOGIC [chap.

peculiar nature of number, and not merely the general ' laws of
thought ', as they are called, that hold of every subject. The com-
pleteness of the division of number into odd or even is not therefore
vouched by logic, any more than the completeness of the division of
rectilinear triangle into equilateral isosceles and scalene ; nor in the
fact that it is twofold does the first possess any guarantee which the
second lacks in being threefold. And if a genus is seen to fall into
thirteen species instead of three, it should be divided into thirteen ;
just as rectilinear triangle should be divided into three and not two.
Unfortunately there are few subjects where we can see at once that
a genus contains necessarily so many species and no more ; and
that makes our divisions precarious, but there is no remedy in the
use of dichotomy.

It may, however, occasionally be possible to show by dichotomy
that a division which is not dichotomous is exhaustive or its species
mutually exclusive. Aristotle thus supported his list of predicables.

Predicable

Commensurate Not commensurate

I I

Essence Not essence Part of essence Not part of essence 1

(Definition) (Property) (Genus or Differentia) (Accident)

But there is no particular logical interest attaching to this mode
of establishing a division ; it is in principle the same as where our
basis is the combination of certain attributes, and we show the
division to be exhaustive by showing that no other possible com-
binations remain, as in the case of the four elements already given.

Element

hot cold

moist dry moist dry

(Air) (Fire) (Water) (Earth)

Dichotomy is really appropriate when we are seeking not to divide
a genus but to define a species. There are two contrasting ways in

1 But generic properties would have to be ranked in this division as acci-
dents. Cf. p. 104, n. 1, supra.

v3 RULES OF DEFINITION AND DIVISION 127

which we may set about to seek a definition. We may take instances
of that which is to be defined, and try to detect what they have in
common, which makes them instances of one kind, and on the
strength of which we call them by the same name. This is the
' inductive ' method. We might thus define ' snob ', comparing
those of our acquaintance to whom we could apply the name, or
those whom Thackeray has drawn for us ; and if we thought that
among all their differences they agreed in prizing rank or wealth
above character, we might accept that as our definition. The other
method is that of dichotomy, and in this we try to reach our defini-
tion rather by working downwards from a genus, than upwards
from examples. Some genus is taken, to which the subject we
wish to define belongs. This genus we divide into what possesses
and what does not possess a certain differentia. The differentia
taken must be something predicable of the subject to be defined ;
and if genus and differentia together are already commensurate
with that subject, the definition is reached ; if they form only
a subaltern genus predicable of it, this subaltern genus must be
again divided in the same way : until we reach a commensurate
notion. At every stage of our division, the differentia taken must
either be a modification of the differentia next before it, or at least
be capable of combining with those that have preceded it in the
construction of one concept in such a way that we are throughout
specifying the general notion with which we started ; and there
should be so many steps of division as there are stages which our
thought recognizes as important in the specification of this concept.
At every stage also we proceed by dichotomy because we are only
interested in the line that leads to the subject we are defining ; all
else contained within the genus we thrust aside together, as what
does not exhibit the differentia characterizing that subject. Had
we further to consider and subdivide it, we could not be satisfied
with characterizing it only negatively ; for a negative notion
furnishes, as we have seen, no basis for any further specification.
But we may disregard, or cut it off : a step to which the technical
name abscissio inflniti has been given, i. e. the cutting off of the
indeterminate.

The following example of definition by dichotomy will illustrate
what has been said. The term to be defined is tuber ; the genus
to which it is to be referred is stem.

1 Cf. infra, pp. 130-131, 133-134.

128 AN INTRODUCTION TO LOGIC [chap.

Stem

creeping not creeping
underground not underground

/\

much thickened not much thickened

possessing buds in the not possessing buds in the
form of ' eyes ' form of ' eyes '

In this process, we reach as our definition of a tuber ' a stem
creeping underground, much thickened, and possessing buds in the
form of eyes '. At every stage by an abscissio infiniti we rejected
from further consideration a large part of the genus we had so
far reached : first all stems not creeping, then all creeping stems
not underground, then all underground creeping stems not much
thickened, &c. ; and at every stage we subdivided that part of
the genus which we had retained by a differentia that specified
further the form to which we had so far brought it.

It might have happened, that creeping stems had a name to

denote them, say Chthamala l ; and that underground Chthamala

had a special name, say Hypochthamala ; that these when much

thickened had again a different name, say Pachysmata ; and that

tubers were pachysmata that possessed buds in the form of eyes.

In this case, the matter would be set out in somewhat different

form, as follows —

Stem

creeping not creeping
Chthamalon

underground not underground

Hypochthamalon

/\

much thickened not much thickened

\

Pachvsma

possessing buds in the not possessing buds in th«
form of eyes form of eyes

\

xSanaXa. Tuber

v] RULES OF DEFINITION AND DIVISION 129

This mode of setting out the definition of anything implies a
classification, in which names have been given to every wider and
narrower genus, and the differentia which distinguishes each within
its proximum genus has been settled. It may indeed be regarded
as an extract from a classification, made for the purpose of exhibit-
ing the nature of a single species. And this is more or less the
character of all definition by dichotomy ; though the classification
may be only in the making, in the very process by which we seek
for our definition. It is only after considerable study of the parts
of flowering plants, enabling us to group them by their less super-
ficial characters, that a tuber would be referred to the genus stem
at all, instead of root ; by that time, the distinction between
creeping and other stems, between those that creep above and
those that creep below the ground, would have been already made ;
so that the method of dichotomy does not so much help us to
discover, as to set out and arrange what we know of, the definition
of a tuber. There may, however, be cases where the method will
guide us in the construction of a definition of that whose nature
has not yet been carefully investigated ; the genus to which a term
is to be referred may be clear, but the appropriate differentiae
unconsidered ; snob, for example, belongs clearly to the genus man ;
but even here, the process of finding a differentia, by which to
distinguish snobs from other men, is classification in the making.
Let us take the prizing of rank or wealth ; if that by itself does
not constitute a snob, we need some further differentia, to
distinguish snobs from other men who prize rank or wealth ;
say they are distinguished by prizing these beyond character ;
we then have a definition of a snob, but in getting it, we
have taken note of a wider class of men within which they are
included.

There are three things which Aristotle 1 says that we must look
to, in reaching definitions by the division of a genus. All the terms
(the summum genus and the successive differentiae) must be of the
essence of the subject defined, they must be placed in their right
order, and none must be omitted. These are requirements also of
an ideal classification, though in the practice of classification, as
of definition, many compromises are necessary ; but just as a study
of the general form of classification does not enable us to classify
any particular set of things, so we are not enabled to define any

x Anal. Post. 0. xiii. 97a 23 sq.
177» K

130 AN INTRODUCTION TO LOGIC [chat.

particular subject, merely by familiarizing ourselves with the scheme
of definition by dichotomy.

[A definition of man, displaying the series of subaltern genera to
which he may be assigned below the summum genus substance, and
the differentia by which each subaltern genus is successively dis-
tinguished within the genus next above it, was long known in
logical textbooks by the name of Arbor Porphyriana. It may be
transcribed here. That of tuber given above on p. 128 is in the same
form.

Substantia

Corporea Incorporea

\

Corpus

Animatum Inanimatum

Vivens

Sensibile Insensibile

\

Animal
Rationale Irrationale

\

Animal Rationale
Mortale Immortale

\

Homo

/l\

Socrates, Plato, &o.

The material for the scheme is to be found in Porphyry's Isagoge,
c. iii ; where the writer points out that the same differentia which is
divisive (SicupeTiK?/) of one genus is constitutive (avarariK-q) of that
immediately below it. The scheme has the advantage of exhibiting
the series of differentiae by which the definition of the species is
reached from the summum genus. Aristotle in Met. Z. xii. discusses
how many differentiae there really are constitutive of the species ;
and decides that if each differentia is itself a true differentia of the
one before it, then the species has only one differentia, namely the
last. For example, if animal is divided into footed and footless
(vttottovi and avow) and if the footed are divided into biped and quad-
ruped, the latter differentia biped is a differentia of footed as such ;
for to be a biped is a particular way of having feet. In the species

v] RULES OF DEFINITION AND DIVISION 131

[animal bipes therefore, the correct analysis is into animal and biped,
and not into footed animal and biped, and though we may proceed
through successive stages to biped, there is nothing in the thing
corresponding to the serial order. If, on the other hand, at any
stage we introduce a differentia which is not merely a further
specification of that which we have used before (as e.g. if we were
to divide biped into feathered and featherless, or rational and irra-
tional), then the species is constituted by more differentiae than
one ; e. g. if we take animal again as the genus, the species man,
defined as a featherless or rational biped, would really be constituted
by two differentiae. We might endeavour to avoid this conclusion
by calling biped the genus and featherless or rational the differentia ;
but that ignores the fact that biped is obviously not summum genus
of man. And if we select a fresh basis of differentiation at more
than one stage, we are each time adding to the number of differ-
entiae that must be recognized in the species. In doing so we
ignore the precept, to proceed throughout any division upon one
basis ; and Aristotle certainly speaks of the introduction of a differ-
entia which is not continuous with that before it as dividing Kara
to <rvixj3e/3 x\ko ? and not Kara to opOov. We may notice too, that
where a differentia which is a continuation of that before it would
be inapplicable to the other member of the preceding genus (e. g.
biped is not applicable to footless, the other member along with
footed of the genus animal), a differentia which is not of that nature
might, for all that we can tell a priori, be applicable to both members
(e. g. feathered and featherless might be applicable to footless no less
than to footed animals) ; hence we shall characterize our species by
the combinations presented in them of the various alternative modi-
fications of several generic attributes.1 The fullness and complexity
of natural kinds constantly leads to the introduction of funda-
mentally new differentiae, especially where, as in the classificatory
sciences often happens, our differentiae are intended as much to be
diagnostic — i. e. features by which a species can be identified — as to
declare the essential nature of the species. Cf. pp. 133-135.]

Before distinguishing Logical Division from the other processes
to which the name Division is applied, it may be well to emphasize
that it deals entirely (like the doctrine of Predicables) with concepts
or universals. The genus which we divide is divided into kinds ;
itself a universal, the specification of it by various differentiae can
only give rise to more determinate universals. The division of it

1 Some of these may be attributes not of the summum genus but only of
some subaltern genus ; and in some combinations, a particular generic
attribute may be altogether absent ; hence the occurrence of negative dif-
ferentiae in scientific classifications. Cf. supra, p. 114, n. 2.

K2

132 AN INTRODUCTION TO LOGIC [chap.

stops therefore with infimae species, and never proceeds to the
enumeration of individuals. For if the infima species could be
logically divided into individuals, we must apply some fundamentum
divisionis ; and that means, that we should have to distinguish
individuals according to the different modes in which the common
character of the species appeared in them ; and to do that would be
to distinguish these modes themselves, which are not individual bub
universal, for many individuals might exhibit the same mode. But
individuals of any species are in fact distinguished from each other
by the coincidence of innumerable attributes ; it is not any attri-
bute singly, but the particular combination of them, that is unique
in each instance ; and whether or not they are sufficient to constitute
individuality, unique combinations of innumerable attributes cannot
be exhibited in a logical division as differentiae of one species.1

There are two processes which have been called division, besides
the division of a genus into its species. They are known as physical
and metaphysical division. In Physical Division, we distinguish
the parts of which an individual thing or aggregate is composed :
as in a man head, limbs and trunk : in a flower bract, sepal, petal,
stamen and pistil. This process is also called Partition. It is still
a process of thought that is meant — not the actual tearing of
a flower to pieces, or quartering and beheading of a man ; it may
be applied to the distinction of the parts composing either a deter-
minate individual, or any individual of a kind : as Great Britain
on the one hand can be divided into England, Scotland, and Wales,
a tree on the other into root, stem, branch, leaf, and flower, or
a forest into its component trees.

In Metaphysical Division, we distinguish in a species its genus
and differentia, in a substance its different attributes, in a quality
its different ' variables ' or ' dimensions ' ; thus we may distinguish
in man animality and rationality, in sugar its colour, texture,
solubility, taste and so forth, in a sound its pitch, timbre, and
loudness. This is obviously a division that can be carried out in

1 Thus in the Arbor Porphyriana the enumeration of the Srofia Socrates,
Plato, &c., in the infima species man is no part of the logical division. Cf.
Porph. Isag. C. ii anpa de Xeyerat ra roiavra, on e£ tStor»;TO)i' <rvvi<jTi]Ktv eicatrrov,
L>v to ti6))m(Tfia ovk av en aXXou TWOS nnre to avTo yevoiro twv Kara ^epos" at yap
~2(CKpnTO\ii ihiuTT^Tit ovk av eV tiXXov rtvbs tg>v Kara, peput yivot"T av at avrai.
('Such things are called individuals because each is constituted by peculia-
rities, the precise collection of which would never be the same in any other
particular instance ; for the peculiarities of Socrates would never occur identi-
cally in any other particular.')

vj RULES OF DEFINITION AND DIVISION 133

thought alone. In Physical Division, the parts of an individual
man or plant may be physically separated ; and in Logical Division,
when the genus is concrete, individual specimens of the infimae
species may be exhibited in different cases in a museum. But in
Metaphysical Division, the ' parts ' cannot be exhibited separately ;
though the colour of sugar may be exhibited without its taste in
a thing of another kind — e. g. in a sample of salt — it can never be
exhibited by itself.

It should be further observed, for the better distinguishing of
these different kinds or senses of division, that in Logical Division
the whole which is divided can be predicated of its parts — animal,
e. g. of man, ox, &c. — and indeed unless it is so predicable of all its
parts, the division is at fault ; in Metaphysical Division the parts
can be predicated (paronymously, to use the Aristotelian expres-
sion,1 or attributively) of the whole — e. g. whiteness, sweetness, &c,
can each be predicated of sugar, in saying that sugar is white, is
sweet, &c. ; in Physical Division, the parts can neither be predi-
cated of the whole nor the whole of the parts — we cannot either
say that a leaf or stem is a tree, or that a tree is a leaf or stem.

[A few words may be added on the relation of Logical Division,
and its rules, to the practical work of Classification. Just as the
theory of Definition, with its sharp distinction of essence and pro-
perty, breaks down amidst the complexity and variety of concrete
things, so it is with the theory of Division. Ideally when a genua
is divided into species, whether once or through several stages, we
ought at each stage to see that just such and so many species are
possible in that genus ; we do see this in geometry, in the division
for example of conic sections into hyperbola, parabola, ellipse, and
circle ; but in other sciences for the most part we must wait upon
experience. Now we do not in experience find that things fall into
kinds which fit into any perfect scheme of logical division. Any
actual division that can be made therefore of animals, or plants,
or forms of government, would exhibit many logical defects ; every
classification involves compromise ; the things, which it puts into
the same class from one point of view, from another claim to be
placed in different classes ; all that was said in the last chapter

1 napowvpa be Xiyerat ocra airo nvos 8ia(pe povra rfj nroxrei rrjv Kara rovvopa
Trpoo-rjyopiav «X«, olov anb TVS ypapp-aTUcrj? 6 ypap-p-ariKot Ka\ atro ttjs avbpeias
6 dvSpe'ios, Cat. i. la 12. ('That is paronymous which receives its designa-
tion from something with a difference in inflexion, as a grammarian from
grammar and a courageous man from courage.') The Latin for n-apairv/oov
is denominatum or denominativum, according as the subject or its attribute is
meant.

134 AN INTRODUCTION TO LOGIC [chap.

[about the difficulty of defining concrete natural kinds might be
repeated to show the difficulty of classifying them ; and the same
reasons which prevent our satisfactorily continuing a division down
to a point at which it would find a separate specific concept for
every individual prevent our satisfactorily classifying them at all.
Classification is, as Jevons called it,1 a tentative operation ; its
results are provisional ; discovery may reveal new species, and
show that characters which have been supposed always to go
together may be separated, or those hitherto considered incom-
patible combined in the same individual : there are limits indeed
to this, for there are ' laws of nature ' with which all particulars
must be consistent ; but many so-called ' laws of nature ' them-
selves rest on the same evidence on which our classifications are
constructed.

Thus the ideal which Logical Division sets before us is very
different from anything which Classification achieves. The first
is or would be an a priori process ; by which is meant that it would
fain develop specific from generic concepts not indeed prior to
any experience of that which belongs to the various species of
the genus divided, but with a perception that the species revealed
in experience are such as must necessarily have existed in that genus.
Classification is an a posteriori process ; it appeals for support to
the facts which we are classifying, and argues that they reveal such
connexions of attributes as we take to mark the classes proposed ;
it does not attempt to show that attributes could be connected in
individuals of the genus in nO other ways than these. Logical
Division again would fain be exhaustive, and establish constituent
species which do not overlap ; but a classification may have to
acknowledge that there are individuals or whole classes which
might with equal right be referred to either of two co-ordinate
genera, or seem to fall between them, or outside them all. For
these reasons, Division, as treated in a textbook of Logic, is apt
to seem unreal and fanciful to any one familiar with the work of
scientific classification ; its rules seem framed to suit not the world
he has to deal with but a fictitious world of the logician's imagina-
tion ; the consideration of a process which, outside geometry, can
scarcely be illustrated by examples except by mutilating facts, is
denounced as a barren pastime. And there is justice in the denuncia-
tion, when Division, or Definition, is studied without reference to
the recalcitrant facts, and on its formal side alone. But if we
realize with what great abatements the rules of Definition and
Division can be fulfilled in the actual classification of concrete facts,
we may yet profitably study these rules, as counsels and not pre-
cepts. That is the best classification which conforms to them most
closely. The case of the logician may be compared with that of the

1 Principles of Science, c. xxx. p. 689, 2nd ed.

v] RULES OF DEFINITION AND DIVISION 135

[geometer. The geometer studies such figures as he conceives, and
he believes that his conclusions are true of the squares or triangles
that exist eternally in space, bounded by the distances between
points therein ; but he does not imagine they would apply without
qualification to a square table, or a triangular lawn. The figures
of these concrete things are much more complex than a simple
square or triangle. So (though the cases are not identical) the
logician studies the problem of classification as it presents itself to
thought ; but is prepared to expect that real things are cross -
related to each other in far too complicated a manner for any single
and simple scheme of classification to embrace them as they stand.
We must consider aspects of them, and attempt to ascertain what
various forms some particular property may assume, and under
what conditions. In tracing a property through all the phases in
which it appears in different instances, we are in a sense pursuing
a genus into its species ; we are realizing its generic identity under
divers forms, and this is part of the business of a logical division.
The things themselves which we have to classify, if we take them
in their completeness, cannot be caged in a neat logical arrange-
ment ; yet even so, the ranking of them in genera and species at all,
which is not the work of logic, but the natural bias of our thought
(for the distinction of man and animal is older than that of species
and genus), implies an effort at such arrangement ; the logician
does no more than render explicit the aims which underlie all
classification : except that the form of his theory takes too little
account of the modifications which are imposed by the particular
nature of the subject-matter with which we may have to deal.1]

1 Some useful remarks on Classification, on the difference between so-
called natural and artificial classifications, and on the relation of different
classifications of the same set of facts to our different purposes, will be found
in J. Venn's Empirical Logic, c. xiii.

CHAPTER VI

OF THE INTENSION AND EXTENSION OF TERMS AND
OF THEIR DENOTATION AND CONNOTATION

We are now in a position to consider certain distinctions in
regard to terms which, owing to the erroneous identification of
them, have become involved in much confusion. These are the
distinctions (1) between Extension and Intension, (2) between
Denotation and Connotation. It was observed by Aristotle,1 that
in one sense the genus is in the species, in another sense the
species is in the genus. ' Animal ' is in ' man ', in the sense that
you cannot be a man without being an animal, so that being
animal is included in being man. ' Man ' is in ' animal ', in the
sense that among the forms of animal nature, man is included.

In the technical language of later Logic, this distinction may be
expressed by saying that in intension the species includes the genus,
in extension is included in it.

The intension of a term is what we intend by it, or what we
mean by it when predicating it of any subject 2 : the extension
is all that stands subordinated to it as to a genus, the variety of
kinds over which the predication of the term may extend.3 Or, if
by term we mean purely the concept, we may say that the extension
is the variety of species in which a common character is exhibited,
the intension the common character exhibited in this variety. The
distinction may be more readily apprehended, if it is noticed that
we analyse the intension of a term in denning it, and its extension
in dividing it.

It is clear that as between two terms subordinated one to the

1 Phys. S. iii. 210a 17-19. Cf. p. 133, supra.

* I do not wish to imply that we may not ' intend ' the same by a term
when it is subject of a proposition, as when it is predicate. But as in the
subject the extension may be more prominent than the intension, while
the predicate is always understood primarily in intension, the expression in the
text is less ambiguous than if I said ' What we mean by it in a proposition '.
Cf. infra, c. ix.

8 For another use cf. p. 143 sq., infra.

INTENSION AND EXTENSION OF TERMS 137

other in a classification, the higher, or superordinate, will normally1
have the greater extension ; animal, for example, is a term of
wider extension than man, and conic section than ellipse ; for the
concept ' animal ' extends or applies to much besides man, and
that of ' conic section ' to hyperbola and parabola, as well as to
ellipse and circle.2 Many hold also, that the superordinate term,
as it is of greater extension, so is of less intension ; less being
meant by calling anything an animal than by calling it a man ;
or by the term ' conic section ', than by the term ' ellipse '. Hence
it has been said that the extension and intension of terms vary
inversely : ' when the intent of meaning of a term is increased, the
extent is decreased ; and vice versa, when the extent is increased,
the intent is decreased. In short as one is increased, the other is
decreased.' 3

This inverse relation of intension and extension in terms may be
illustrated not only by reference to classification, but in another
way. We may take any term, such as Christian, and qualify it by
an adjective or adjectival phrase : as if we were to say ' Armenian
Christian ' or ' Christian of Caesar's household ' ; by the qualifica-
tion we clearly make a term of narrower extension than ' Christian '
simply, for we conceive that there may be Christians not Armenians,
or not of Caesar's household ; and at the same time we add to
the intension, for it is no part of the concept of a Christian to be
an Armenian, or of the household of Caesar.

Still, when we thus qualify a general or an abstract term, we are
instituting a sort of classification ; we make an Armenian species
within the genus Christian, or a class, say, of bright colours within
the genus colour. Therefore we may say generally that it is only
to terms in a classification, and in one ' series of subordination ' in

1 Occasionally, as we have seen (supra, p. 73, n. 2), we find in a classification
Bpecies whose members differ from their nearest kindred as widely as members
assigned to different genera in it differ, so that they are referred to a distinct
genus, although no other species is found belonging thereto ; as in zoology
men are placed in the species Homo Sapiens, which is the only species of the
genus Homo and of the class Hominidae. But that means that we think there
might be other genera of Hominidae, and species of Homo : and if there were,
the relation stated in the text would hold.

2 Porph. Isag. c. viii Ire ra fieu yevr] nXeovafci rrj rfov wr' nvra elftav nepioxTJ,
ra 8e tlbr) rav yevuv 7r\(ova(d rais olKeiais Sta^opals-. ('Further, genera exceed
species in the compass of the species under them, species genera in the
differentiae belonging to them.')

3 Jevons, Principles of Science, 2nd ed., c. ii. p. 26. Of. Sir W. Hamilton,
Lectures on Logic, viii. U xxv ; Thomson, Laws of Thought, § 28 ; Bain,
Logic, Deductive, p. 51 (' the greater the one the less the other ').

138 AN INTRODUCTION TO LOGIC chap.

it, that the doctrine of the inverse relation of intension and extension
applies. It would be ridiculous to compare in this respect such
different concepts as democracy and steam-engine ; it is even un-
meaning to compare terms belonging to the same classification but
to different lines, or ' series of subordination ', in it ; bird and
reptile, for example, both belong to a classification of animals, but
are not subordinate one to the other, and nobody can well tell
which has the greater intension, nor if that were decided would he
be able to infer from the decision, which had the greater extension,
or comprised the larger number of subordinate species.

Applying only to terms subordinated one to another in a classi-
fication, the doctrine is an attempt to explain the nature of
classification, as a series of terms so related that each is of wider
extension and narrower intension than the next below it.

Now it may be questioned whether the doctrine is just. The
generic term undoubtedly exceeds the specific in extension, but does
it fall short in intension ? This question may be put in another
form : is the process of classification one of mere abstraction 1 do
I reach a generic concept from specific concepts merely by leaving
out part of the latter, and attending only to the remainder ? If
our concepts of species and genus were constituted by sets of
attributes disconnected but coincident, then this would be the case.
The generic concept would be formed by picking out from several
sets those attributes, or marks, which occur in them all ; it would
contain fewer marks, or be of less intension, in the same sort of
way as one man may have fewer decorations than another. On
these principles the nature of a classification might be satisfactorily
expressed by the following symbols : —

I

ab

ac ad

t

abe abf abg ach aci adj adk adl

But we have seen 1 that the genus is not something which can be
got by any process of subtraction from the species ; it is not the
same in all its species, and does not enter unchanged into them all
as water into every pipe that leads from a common cistern. You

1 Cf. p. 83, supra.

vi] INTENSION AND EXTENSION OF TERMS 139

cannot form a concept of it apart from all the species, as a can
be read and written apart from other letters with which it may be
combined. Attributes that are really independent, such as blue,
and sweet, and heavy, can be thus conceived apart ; but they
cannot stand to each other in the relation of genus and species.1

If we look at terms which are really in a relation of genus and
species, it is not clear that the wider term has the less meaning.
Take animal and man ; if I say of anything that it is an animal,
I certainly convey less information about it than if I say it is
a man ; but it does not follow that the concept animal is of less
intension than man. For it must be noted, that I should not say
of anything that it is animal, but an animal ; which implies that
I am aware of other animals, and that the concept animal includes
alternatives, among which I cannot or do not at present choose.
But if so, the generic concept would seem to exceed the specific
in intension ; ' animal ' means ' man, or horse, or crab, or jellyfish,
or some other form in which the general nature of an animal may
manifest itself '. As we become familiar with the infinite variety
of animal life, the term comes to mean not less to us, but more.

Or take another illustration. Say that a boy first makes acquain-
tance with the steam-engine in the form of railway locomotives.
For a long time the term means that to him ; but by and by he
meets in his experience with traction-engines, ship's -engines, and
the stationary engines of a factory. His earlier concept of a steam-
engine — the earlier intension of the term for him — will alter ; much
which he included at first in it, because he found it in all railway
locomotives, he will learn to be unessential — first running on rails,
then the familiar shape, then the moving from place to place. And
according to the doctrine before us, he will leave out from the
concept one point after another, and at the end his notion of
a steam-engine will be the unexcised residuum. But surely his
notion of a steam-engine will have become richer and not poorer
in the process ; it is not that he finds that a steam-engine need
not run on rails, so much as that it may run on the roads, nor

1 And therefore the introduction of differentiae into a division which are
not differentiae of those before them is not Kara to 6p6nv (cf. supra, p. 131),
though they may still be such of which only the genus from which we started
is susceptible ; and the introduction of them may be justified as well by
considerations of practical convenience as on the ground that species are
distinguished by variously combining the variations of many generic charac-
ters, or characters not pervading the whole genus.

140 AN INTRODUCTION TO LOGIC [chap.

that its familiar shape is unessential, so much as that it may be
built in quite a different manner ; nor that it need not move from
place to place, so much as that it may work as a stationary engine.
It becomes a genus to him, because it becomes a thing of alter-
native possibilities ; and the experience which leads him to extend
the term to new kinds of subjects leads him to use it with a wider
range of meaning. It is true that in becoming generic, the term
comes to have a less definite meaning, when applied to any subject ;
but it does not therefore come to have less meaning.

The doctrine of the inverse relation of extension and intension
in terms may seem therefore to misrepresent the nature of a classi-
fication. But a doctrine which has been accepted so widely,1 and
is at least at first sight so plausible, must have some degree of
justification. Its justification, or excuse, seems fourfold.

1. The thought which general terms suggest to the mind is
often vague, and the more so in proportion as they less suggest
a definite sensible object. We do not realize all the alternative
possibilities involved in animal nature each time that we use the
term 'animal '. So, because in the term of wider, as compared with
that of narrower, extension there is often little definite, we are apt
to suppose instead that there is a definite little. This error is
encouraged by mistaking for thought the imagery that accompanies
thinking. The nature of this imagery differs with different people,
and any illustration can be only arbitrary. But it might well be
that when one thought of man or horse, he pictured to himself the
look of either with fair completeness ; but that with the notion of
animal there went the kind of image which a child would draw of
a quadruped — four lines sticking out of an elongated trapezium,
with a few more for the head and tail. There is less detail in such
an image than in that of a horse or a man ; and it is not impossible
that one might hence be led to suppose there was less intension in
the term.

2. Our actual classifications, as we have seen, fall short of per-
fection in many respects ; we often do not understand the inter-
dependence of the various characteristics of an organic kind, or of
the various properties of an elementary substance. In these circum-
stances, we are compelled at times to fix on certain characters as

1 There are, however, eminent names on the other side, e. g. Mr. F. H.
Bradley, Professor Bosanquet, and R. L. Nettleship. Cf . especially section xi
of the ' Lectures on Logic ' in The Philosophical Remains of R. L. Nettleship.

vi] INTENSION AND EXTENSION OF TERMS 141

constituting a genus, and then distribute into species the subjects
in which they are found by means of attributes whose connexion
with these characters we cannot conceive. For example, there is
a far-reaching division of angiosperms (already referred to) into
monocotyledons and dicotyledons, based on the number of the seed-
leaves ; but in these two classes the sub-classes are distinguished
by various characteristics of the calyx and corolla, of the mode in
which the stamens are inserted, &c. Now we are ignorant why
a plant with two seed-leaves should be capable of one series of
flower-developments, and a plant with one seed-leaf of another
series ; the number of seed-leaves is, for all we can see, an irrelevant
character, though it cannot really be so ; and the concept of
dicotyledon or monocotyledon is complete, without reference to
the character of the flower. Here therefore the intension of the
wider term is less than that of the narrower. To the botanist
the term Dichlamydeae, whose extension is less than that of Dicoty-
ledon, means plants which in the first place have two seed-leaves,
and over and above that have both calyx and corolla ; the term
Dicotyledon means merely a plant with two seed-leaves. Such
cases give colour to the doctrine, that where terms are subordinated
one to the other, the intension varies inversely with the extension ;
but they do not embody the true spirit of a classification.

3. We have seen that a term may be qualified by an adjective
which is really an accident of it : by which is meant that the ad-
jectival concept is an addition to the original concept, rather than
a further determination of it ; as when we qualify the term Christian
(which implies a certain religious belief) with the adjective Armenian
(which implies a certain nationality) — there being no necessary
connexion between creed and race, but any variety of one being
capable of coinciding in individuals with any variety of the other.
These cases (to which those considered in the last paragraph ap-
proximate) bear out the doctrine of inverse relation, so far as they
go. But it may be observed that they only bear it out, because
they have been as it were constructed to do so. We take a term,
and qualify it by an adjective which in the first place is known
not to be applicable to all instances (and therefore narrows the
extension), and in the second place is not implied by the term in any
way as a possible development of the genus : so that it is a sheer
addition to whatever intension the original term possessed. Then
we call attention to the fact that in the original term, and the term

142 AN INTRODUCTION TO LOGIC [chap.

composed of it and of an adjective, extension and intension vary
inversely. Of course they do, because we have carefully arranged
it, by so qualifying the original term that they must. But it is
ridiculous to infer from this, that in all terms, where one is of wider
extension than the other, its intension is less. Because this holds
where the terms are not related as genus and species should be,
it must not be concluded to hold where they are so related.

4. It may still be felt that there is more truth in the doctrine
than has been conceded. Take the most unimpeachable examples
of genus and species, such as rectilinear triangle, with its species
equilateral, isosceles and scalene. Can we not and do we not con-
ceive a rectilinear triangle with regard to those points in which
equilateral, isosceles, and scalene agree, and without regard to
those in which they differ? and may not this notion be perfectly
precise and definite ? and if such be the intension of the genus -
term, is it not less than that of the species-term ? We must admit
that this is possible. In the words of R. L. Nettleship,1 * we may,
for convenience' sake, mentally hold apart a certain fraction of the
fact ; for instance, the minimum of meaning which justifies us in
using the word " triangularity ". We may call this the generic
triangle, and distinguish it from particular forms of triangle.' But
the true intension of the term is not the ' minimum of meaning '
with which we can use it, but its ' full meaning \

What has been so far said with regard to the relation of intension
and extension in terms may perhaps be rendered clearer to some
as follows. Wherever we have species of a genus, or distinguishable
varieties of a common nature, we may contrast the unity which they
present with the variety. To attend to the intension is to attend
to the element of unity : to attend to the extension is to attend
to the element of variety. Sometimes we are more interested in
one, and sometimes in the other. When Socrates in the Meno
asks what is virtue, and Meno begins describing the virtue of a man,
the virtue of a woman, and so forth, Socrates explains that he wants
to know what virtue is as one in all these, and not what the divers
virtues are ; in later language, he wished for the intension and not
the extension of the term. Aristotle remarks 2 that an enumeration
of these different virtues and a description of them severally are
more valuable than a vague statement of their common nature :

1 Philosophical Remains, i. p. 220. The italics are mine.

2 Plat. Men. 71 D-72 D ; Ar. Pol. a. xiii. 1260a 20-28.

vil INTENSION AND EXTENSION OF TERMS 143

i. e. that here at any rate the element of variety is more worth
consideration than the element of unity, if either is to be neglected.
But if the two are realized together, the unity of the superordinate
whole must be seen as the more comprehensive unity, not as the
more jejune extract. So far however as we cannot realize them
together, and see their necessary connexion, it will have the character
of the jejune extract and be a whole of less meaning, even although
we know that the variety of species into which it enters is great ;
and in these conditions, it may be said to be of less intension.

It follows that the infima species (or the term denoting it), in the
unity of whose being we recognize no variety, has properly speaking
no extension. Equilateral triangles may differ in the length of
their sides, and we may if we like regard this difference as con-
stituting a variety in their common nature. But if we do not —
if we conceive the particular length of the sides to constitute no
difference in equilateral triangularity — then we recognize no such
variety in the unity as makes it possible to distinguish from the
intension the extension through which it ranges. The term equi-
lateral triangularity will denote to us a certain unitary nature, but
no varieties of such.

Logicians have been withheld from acknowledging these terms
to have no extension by two reasons, by one justifiably, by the
other through a confusion. Justifiably by this, that the point at
which logical division stops is generally arbitrary, and what are
treated as infimae species are capable of subdivision into lower
species, which would be their extension ; ellipses may vary in their
ellipticity according to focal length, Christians in their Christianity
according to faith as well as practice. The consciousness of the
variability of the specific nature which forms the intension of the
term makes us regard it as still having extension, though less than
its superordinate terms. Terms within whose intension there is
no variety, like point, or none recognized, like equilateral triangle,
are rare.

The other reason is this, that even where there is no variety
within the intension of a term, there is multiplicity of instances.
Though no species of equilateral triangle are distinguished, innu-
merable equilateral triangles are. Two such triangles interlaced
are a favourite symbol in the decoration of churches ; and the
number of them delineated on church-walls and windows must be
past counting. If the individual instances make the extension,

144 AN INTRODUCTION TO LOGIC [chap.

the infima species will have plenty, though still less than its super-
ordinate terms, because there are more instances of the genus than
of any one species l — more triangles, for example, than equilateral
triangles.

It is plain that this reason involves a confusion between two
different things, between the variety of kinds over which the pre-
dication of a term may extend — the variety of which we conceive an
unity to be susceptible, and the various individual instances in
which a common nature is manifested. On the former view, the
extension of man is Aryan and Semitic, Negro and Berber, &c. : of
triangle equilateral, isosceles and scalene ; on the latter, that of man
is Socrates and Plato, Alexander and Caesar, you and I, &c, that
of triangle every triangle on a church wall or on a page of a copy of
Euclid's Elements. But the relation of genus to species is not the
same as that of universal to individual, of a kind to its instances,
and the antithesis of intension and extension ought therefore not
to be used indifferently in respect of both. We might perfectly
well understand by the extension of a term either the various forma
or the various instances in which the common nature that is its
intension is manifested ; but we ought not to understand both
indifferently.

It is easy to see how the confusion arises. Though the antithesis
between the intension and the extension of terms is based on that
between the unity in wJiat different individuals are, and the variety
in which that unity is displayed, most of the terms in which this anti-
thesis is illustrated are general terms predicated of individuals, like
man or ox and animal, gold or silver and metal, axe or hammer and tool,
musician or painter and artist, triangle or square and figure.2 They
are predicable of individuals, but in respect of their common nature.
The superordinate term — animal, metal, &c. — is predicable of more
individuals, the subordinate — man or ox, gold or silver, &c. — of
fewer. Sometimes there are also proper names predicable of the
individuals singly, but all alike are names of individuals. The
distinction in the meaning of a general name between the individuals
whereof it is predicable and the common nature in respect of which
it is predicable of them is important and obvious. Language
allows us to say that Caesar is a man, and that a man is an animal,

1 Except in a species which is sui generis : cf. p. 137, n. 1, supra.

2 In the last two instances the terms though substantival are attributive
in meaning : cf. supra, p. 37, n. 1.

vi] INTENSION AND EXTENSION OF TERMS 145

that Beethoven is a musician and that a musician is an artist, that
this is gold, an axe, a triangle, and that gold is a metal, an axe a tool,
a triangle a figure. Hence it is supposed that the relation of man
to Caesar, or musician to Beethoven is the same as that of animal
to man, or artist to musician ; the relation of axe or triangle to
' this ' the same as that of tool to axe, or figure to triangle. For
we are misled by the common form of the proposition, A is B, and
do not reflect sufficiently on the different senses in which one thing
is said to be another.1 When I say that a man is an animal or
a triangle a figure, I mean that being a man is a way of being an
animal, to be a triangle is to be a figure ; and I could say instead
that humanity is animality or triangularity figurateness. But
when I say that Caesar is a man, or this a triangle, I do not mean that
Caesarness is a way of being a man, or that thisness is triangularity ;
the concrete individual is something more than can be comprised in
any concept.

With abstract terms and names of universals we are not
tempted to make this confusion. We should not feel the same
hesitation in allowing that ' equilateral triangularity ' as that
' equilateral triangle ' has no extension ; and if we hesitated to
deny extension to humanity or democracy, it would be only
because we are conscious that these concepts are capable of
further specification, that humanity is something different in
different men, democracy in France and in the United States. No
doubt attributes and relations have their instances, and abstract
terms are names of attributes and relations ; they are predicable
of the several instances, and as such are general. But the instances
can only be distinguished by referring to the particular subjects 2
in which the attributes inhere or between which the relations
hold ; and in abstraction we commonly ignore these, and consider
the attribute or relation by itself ; we may be interested in the divers
forms that it may take, and have separate names for these, for the
diversities of colour or constitution, consanguinity or proportion ; but
to be interested in the instances would be to be interested in the con-
crete individuals that display them, and from these we are abstract-
ing. Hence it is that the abstract term becomes the name of the
attribute or relation, of whose instances it is predicable as a general

1 Cf. supra, pp. 23-24.

2 Generally concrete individuals, but not always ; I might e. g. direct
attention to instances of degree by mentioning colour and heat, without
reference to particular coloured or hot things.

177» L

146 AN INTRODUCTION TO LOGIC [chap.

term, and that even when we use it as a general term, e. g. when
we speak of so many deaths, in the plural, we are still apt to think
of the attribute or relation as identical in all its instances ; indeed,
as we saw, it has been denied that there are instances of relations.1

It is plain then that by the extension of a term we should not mean
indifferently species and individuals ; to be specified in divers ways
is not the same as to be found in many instances. And there is the
less necessity for using the word extension thus confusingly, that
another word, denotation, will serve where the instances are meant.
A word denotes anything of which it can be predicated as a name ;
man denotes Socrates and Caesar, artist Beethoven and Giotto,
triangle this and that triangular figure. It is true that universals
are denoted also by their names ; animality, triangularity, proportion,
each denote something; and abstract terms denote not only instances
of attributes and relations, but the attributes or relations con-
sidered each as one in its several instances.1 But this fact need not
disturb us. We use denote in the same sense, in each case.

It will be observed also that the inverse relation of extension
and intension does not hold equally when by the extension of a term
we mean the forms in which the intension is displayed, and when we
mean the instances. We saw how the intension of the term animal
might from one point of view be said to increase, as one becomes
acquainted with fresh forms of animal life ; and how from another
point of view, because what at first one might have regarded as
essential to an animal turns out not to be indispensable, it might
be said to diminish, shrinking to a jejune residuum. But M'hichever
way we look at it, it is only acquaintance with fresh forms of animal
that produces this result ; a mere increase in the number within one's
acquaintance would not produce it. It is said that you cannot
widen or narrow the extension of a term without restricting or
enlarging its intension, and vice versa. But change in the meaning
of a term comes by extending its application to new kinds of subject,
or confining it to some kinds only of those to which it was before
applied. The intension of the term baby does not increase and
decrease with the fluctuations of the birth-rate.2 A change in the
intension of a term will indeed commonly affect its denotation as
well as its extension, just as the superordinate term in a classification
commonly denotes more individuals than the subordinate, besides
having a wider extension ; but only a change in the extension, that is,
1 Cf. supra, pp. 27, n. 3, 33-35. 2 F. H. Bradley, Principles of Logic, p. 158.

vi] INTENSION AND EXTENSION OF TERMS 147

in the kinds of individual denoted, not in the mere denotation, will
affect the intension.1

In place of the terms Extension and Intension, various writers
have used others to mark either what is, or what they wrongly
thought to be, the same distinction ; and in particular, since the
publication of Mill's System of Logic? the antithesis of Denotation
and Connotation has come into favour. Mill regarded this antithesis
as identical with that of Extension and Intension ; but he claimed
for his expressions that they possess an advantage lacking to others,
in the existence of the corresponding verbs, to denote and to connote ;
we may speak of a term denoting or connoting this or that, but with
other expressions we must use a periphrasis and say, e. g., that so and
so is included in the extension, or constitutes the intension, of a term.
This advantage and the jingle of the antithesis have combined with
Mill's authority to bring the word connote into common use ; for
we do require at times, as the passage above referred to in the
Meno shows, a word that will distinguish a term's meaning in inten-
sion from its meaning in extension. In other respects Mill's ex-
pressions are less appropriate ; for extension suggests, and denotation
does not, the range through which the intension is manifested ;
intension suggests, and connotation does not, what we intend by a
term; and connotation contains a suggestion, inappropriate in many
cases, of additional meaning. But the trouble is that the two antitheses
are not really equivalent. A term may denote, which has no exten-
sion; and may have intension, which, in the prevalent meaning
of the word, has no connotation. Mill drew his distinction with his
eye mainly on two classes of terms, attributives and general concrete
names. The functions of denoting and connoting which he found in
these he thought to be the only functions of any term. Then,
because certain terms do not connote like them, viz. proper names
and the names of infimae species of attributes or relations 3 (like
length and whiteness), he thought they only denoted ; and he made
a division of ' names ' into connotative and non-connotative (by which
he understood unmeaning), which he described as ' one of the most
important distinctions which we shall have occasion to point out,
and one of those which go deepest into the nature of language '.

1 Of course, when the term denotes kinds, its intension will be affected by
a change in the denotation. 2 v. Bk. I. ii. § 5.

8 Mill does not mention relations, but the argument applies equally in
their case ; and if they are not always mentioned in the following discussion,
that is only for brevity's sake.

L2

148 AN INTRODUCTION TO LOGIC [chap.

As he expounded it, however, it has been a source of little but error
and confusion. He confounded different distinctions, and raised
a controversy about the connotation of proper names, to which
there has been no satisfactory issue, because he never clearly realized
to himself what he meant by connotation, nor that it was something
different from intension ; and so the word has been used in the
controversy in different senses.

In order to clear up the ambiguities of the word, we must examine
the passage in which Mill expounds his doctrine. It runs as follows.
' A non-connotative term is one which signifies a subject only,
or an attribute only. A connotative term is one which denotes
a subject, and implies an attribute. By a subject is here meant
anything that possesses attributes. Thus John, or London, or
England, are names which signify a subject only. Whiteness,
length, virtue, signify an attribute only. None of these names,
therefore, are connotative. But white, long, virtuous, are connotative.
The word white, denotes all white things, as snow, paper, the foam
of the sea, &c, and implies, or in the language of the schoolmen,1
connotes, the attribute whiteness. The word white is not pre-
dicated of the attribute, but of the subjects, snow, &c; but when
we predicate it of them, we convey the meaning that the attribute
whiteness belongs to them . . . All concrete general names are con-
notative. The word man, for example, denotes Peter, Jane, John,
and an indefinite number of other individuals, of whom, taken as
a class, it is the name. But it is applied to them, because they
possess, and to signify that they possess, certain attributes. . . . The
word man, therefore, signifies all these attributes, and all subjects
which possess these attributes. . . . Even abstract names, though
the names only of attributes, may in some instances be justly con-
sidered as connotative ; for attributes themselves may have attributes
ascribed to them ; and a word which denotes attributes may connote
an attribute of those attributes. Of this description, for example,
is such a word as fault ; equivalent to bad or hurtful quality. This
word is a name common to many attributes, and connotes hurtful-
ness, an attribute of those various attributes.2 . . . Proper names are

1 Mill means that in the case of such terms as these, the schoolmen spoke
of attributes being connoted ; but not that his use of the word connote
conforms generally with that of the schoolmen : cf. infra, pp. 156-158.

2 Mill instances ' slowness in a horse ' as an attribute denoted by the word
1 fault '. It is clear that if ' fault ' is connotative, ' virtue ' should not lmve
been given as an example of a non-connotative name. The italics in this
quotation are his.

vij INTENSION AND EXTENSION OF TERMS 149

not connotative : they denote the individuals who are called by
them ; but they do not indicate or imply any attributes as belonging
to those individuals.'

Thus Mill considers three classes of terms to be connotative —

(a) attributive terms, like white, long, virtuous, &c. ;

(b) general concrete names, like man, snow, &c. ;

(c) abstract terms, if they are names of a genus of attributes, like
fault ;

and two classes to be non -connotative —

(a) proper names ;

(6) abstract terms, if they are names of infimae species of attributes,
like length, whiteness, &c. Designations, i. e. phrases indicating an
individual that contain connotative terms, he regards as connotative.

Now it is true of all his ' connotative ' terms, that they signify
more or less of what that is, whereof they are predicated ; and they
are therefore said to denote the subjects of which they are predicable,
and to connote whatever character x they indicate these subjects to
possess. But further, they are used of these subjects because of their
possessing such character. Mill means then by the denotation of
a term the subjects of which it can be predicated, by the connotation
that character, to indicate the possession of which we use the term
of any subject.

It might seem that we could say simply, that the connotation of
a term is its meaning. But there are two reasons why this is not so.
In the first place, terms have two functions, both of which may be
called meaning. They direct our thought to some subject, and they
suggest what that subject is, to which our thought is directed. I
may be said e. g. to mean by tools either spades, hammers, axes, &c,
or ' things made in order that we may by their means in handling
them do what we could not do, or do so well, with our unaided
hands '. Mill would say that the former is what the word denotes,
the latter what it connotes. In the second place, a term may dis-
charge the function of signifying what the subject is, to which it
directs our thought, in two ways. It may signify the subject in its
entirety, or some character in the subject, with which the subject
is not identical. It is only the latter function which Mill calls
connoting, as in the example tool just given. Terms which mean
what the subject is in its entirety he calls non-connotative, and he

1 Mill says attributes, because he regards e.g. being gold as an attribute
or aggregate of attributes in any piece of gold.

150 AN INTRODUCTION TO LOGIC [chap.

does not think that they signify what the subject is at all. And there
are further differences within both kinds of terms, in their way of
discharging the function of signifying what the subject denoted by
them is, which Mill ignores.

The most important class of his ' non-connotative ' terms is proper
names. If connotation in a term is signifying some character in
a subject, to indicate its possession of which we use the term of any
subject, proper names certainly do not connote. But besides this
signification in a term Mill recognized no other function, except
denoting. Hence he thought that proper names only denoted, and
were ' unmeaning marks '. ' A proper name ', he says, ' is but
an unmeaning mark which we connect in our mind with the idea
of the object, in order that whenever this mark meets our eyes or
occurs to our thoughts, we may think of that individual object ' l ;
and he contrasts connotative names as ' not mere marks, but more,
that is to say significant 2 marks.' Now in thinking that a proper
name merely denotes, and signifies nothing, Mill was wholly wrong.
It is the sense of this error which has led critics to say that proper
names have connotation 3 ; and if we had to make the antithesis of
denoting and connoting cover the ground in regard to the functions of
every kind of term, that would certainly be the less misleading
doctrine. But Mill was calling attention to a real difference distin-
guishing his ' connotative ' from proper names, which may be well
expressed by saying that proper names have no connotation, if we
accept the sense of ' connotation ' which may be extracted by con-
sidering the classes of term to which he ascribes it, and reject his
identification of it with ' signification ' generally. We may the more

1 This account of a proper name closely resembles Hobbes's definition of
a name generally (quoted p. 20, supra), which in the first section of the same
chapter Mill approved. Hobbes says that a name is ' a word taken at pleasure
to serve for a mark which may raise in our mind a thought like to some
thought we had before '. To say that it is taken at pleasure means that
it is not taken on account of any pre-existing signification. This is true at
the outset of all names, proper and general alike, except derivatives. A
general name was unmeaning before it was given to anything ; so also is a proper
name. But a proper name, like a general name, has a meaning after it is
given.

2 Mill obviously means by signifying being the sign of what a thing is ;
else he could not distinguish ' mere marks ' from ' significant marks ' ; for
a mere mark denotes. It is possible to use the word ' signify ' in the sense
of ' denote '. But throughout the following discussion it will be used as
Mill uses it.

3 e.g. Jevons, Elementary Lessons in Logic, Lesson V : Bosanquet, Essentials
of Logic, Lect. V. § 6 ; also the first edition of this book.

vi] INTENSION AND EXTENSION OF TERMS 151

conveniently do this, because the signification which proper names
do possess is perfectly well indicated by the word ' intension '.

A proper name certainly has intension as well as denotation. It is
a mark directing our thought to an individual ; but that which is to
be a mark must have meaning. A scratch may be a mark on a coin
which I am looking at ; it is not a mark of the coin that I am looking
at, but of its being the same coin which I had put in the way of
a suspected thief. I may of course be ignorant of the meaning of
a mark. The broad arrow T which is occasionally seen on gate-
posts, milestones, &c, is a mark; a traveller might not know what it
meant ; but he would not call it a mark, unless he guessed that it
meant something. By enquiry he might learn that it meant that
the spot where it was placed was the precise spot whose height
was recorded in that portion of the ordnance survey. Here the mark
is general. But the mark by which his nurse recognized Odysseus
was equally significant. In its own nature it was a scar, the conse-
quence of a wound, and not (like a brand) intended as a mark. Yet
this scar (its precise form and position being taken into account) to
those who had observed it in Odysseus became a mark by which to
know him. He had been absent twenty years, and was changed
otherwise beyond recognition ; he was supposed to be dead ; but his
nurse, seeing the mark, knew the man before her to be him — knew
that about the man before her which otherwise she would not have
known. How can it be said that it was an unmeaning mark for her ?
And suppose that instead he had at once told her that he was
Odysseus ; the name would have given her precisely the same
information ; how then could the name be unmeaning ? The
doctrine that proper names have no intension is refuted by every
criminal who assumes an alias.

And not only, to any one who knows of what individual it is the
name, has a proper name meaning, but it has more meaning than
a general term. The cry ' man overboard ' would have conveyed to
Aeneas and his companions not more but less information than the
cry ' Palinurus overboard '. It cannot indeed convey to any one,
for he cannot know, the whole character of the individual denoted ;
but it excludes from its meaning designedly nothing of that character ;
whereas another term, if it is not the name of an infima species of
attributes,1 is designedly confined to signifying only some deter-

1 This is intended to signify the whole character of what it denotes : cf.
infra, p. 154.

152 AN INTRODUCTION TO LOGIC [chap.

minate character in what it denotes. Mill speaks as if, were I to
point to some individual person or thing, and to ask who or what is
that, and were another to reply by a proper name, I should only
learn what it is called, not what it is. And if I now heard the name
for the first time, that is true. But it is equally true of a general
name, when I hear it for the first time. If I point in a foreign
country to an unfamiliar object, and ask what is that, though I am
answered with a general name, I shall only learn what in that
language it is called. On the other hand, if a proper name is, and
I know it to be, the name of something with which I am already
familiar, either personally or by report, it may be very instructive.
What would not a man have given to be once truly told, in reply to
the question ' Who is that ? ' ' Napoleon ' ? Or if I cross a country
road, and am told ' That is Watling Street ', do I not learn much
more about it than what it is called, and more than the word
' road ' conveys ?

What then is the important difference between a proper name and
other classes of term, which Mill wishes to indicate by saying that
proper names have no connotation ? It is that they cannot be used
to convey information about an otherwise unknown individual.
A general term, used of any subject, is instructive to those not
acquainted with the subject. If I ask ' What startled you ? ' and
you tell me a karait, I shall know that it was a very venomous snake.
That is because the term ' karait ' is used of a subject merely to
indicate that it has a certain character, of any subject possessing
which it might be used equally. It has a signification on the ground
of which it may be predicated of one fresh individual after another.
But a proper name is not used of any individual for the first time on
the ground of a signification which it already possesses ; the son of
James I and VI was not called Charles on account of his Carolinity.1
It acquires its signification from the individual to which it is given.
Hence it is uninstructive to any one ignorant of the individual
denoted. If I ask ' What startled you ? ' and you answer ' Glamby ',
I shall not know in the least what it was, unless I know already
what that word denotes. On the other hand if I do already know
that — if I know that Glamby is the name of your dog or your baby
or the ghost that haunts your house — I shall learn not only what

1 Hence, as was pointed out p. 47, supra, the name Charles is used equi-
vocally of him, of his son, of the son of Pepin, &c. But an equivocal term
is not a term with no meaning ; it is a term with more than one meaning.

vi] INTENSION AND EXTENSION OF TERMS 153

kind of individual it was that startled you, but what individual of
that kind. For this is a further peculiarity distinguishing a proper
name from terms of any class that Mill calls connotative : it is part
of the meaning of a proper name that the subject denoted is precisely
this or that individual. That is why a proper name can be the
predicate of a proposition ; we make it a predicate when we wish to
say not of what kind something is, about which information is
offered, but what individual of some kind. If I were wandering for
the first time in a country known to me by history, and, coming to
a village, asked its name, the answer Quatre Bras would not tell me
that it was a village, but which village it was. And since I may
point to this village without knowing which it is, I can distinguish
in a proper name the function of designating or denoting an individual
from that of signifying which individual, with all its being and history,
is denoted ; and so I must say that, besides denoting, it has intension ;
only, part of its intension, concerning what it denotes, is that this is
such precise individual.1 Did it signify nothing concerning that
which it denoted, it would not even have denotation. If you say
that you have been reading about Quatre Bras, and I do not know
whether that is a village or a general or a poem or a star, it denotes
nothing to me. A name could only denote and have no signification
if that could be discriminated which had no character.2

All this indeed only amounts to saying that a proper name has
not general meaning. Mill really intended by connotation general
meaning, but thought that to lack it was to lack meaning altogether.
' General meaning ' is not however a complete account of what he
intended by the word. Connotation is signifying some character in
a subject, which can be distinguished from that subject. ' A con-
notative term is one which denotes a subject and implies an attribute.*
When a term signifies a subject in its entirety, what it signifies i3
not an attribute of what it denotes. Hence Mill denied connotation
to another class of terms, the names of infimae species of attributes.
He was assisted to do so by confusing the relation of species to genua
with that of individuals to their kind. Man denotes ' Peter, Jane,

1 If we had to make the words denotation and connotation express all the
functions of all kinds of terms, we might say, though a trifle loosely, that
the denotation of a proper name is part of its connotation.

2 Mill's confusion between an individual's substantial nature and an attri-
bute perhaps helped to mislead him here. Because, if I think away its being
white, snow still remains something that I can discriminate, therefore he
seems to suppose that, if I think away his being man, John still remains
something which I can discriminate.

154 AN INTRODUCTION TO LOGIC [chap.

John,' and connotes their common character ; fault denotes slow-
ness, stupidity, &c, and connotes their common character. Man is
connotative, Peter or Jane or John is not ; and similarly, he thinks,
fault is connotative, slowness or stupidity not. But this result,
unlike his similar view about proper names, is both devoid of
plausibility and in contradiction with his teaching elsewhere. So
unplausible is it, that some, being unable to bring these terms under
the formula just quoted, have preferred to deny to them denotation.
And it flatly contradicts Mill's doctrine, that definition declares the
connotation of a name. For the name of a species of attribute may
be definable 1 by giving its genus and differentia, even where that
of the genus is not ; and yet according to Mill the latter has con-
notation, and the former none.

That these terms are unmeaning is clearly absurd. When Wolsey
says to Thomas Cromwell, ' Cromwell, I charge thee, fling away
ambition ; By that sin fell the angels ', the word ambition does not
denote an object of thought, without signifying what that is which
it denotes, but signifies the nature of the sin which Cromwell is
warned to avoid. Yet it also denotes it. Such terms are names
of universals, of the common character in many instances of an
attribute or relation.2 But of what they denote they signify the
entire being ; and what they denote is general. Hence, if to be
connotative is to have general meaning — and that is why Mill denies
connotation to proper names — they are connotative. If it is to
signify something general in a subject of which that is not the entire
being, they are not. This is the ambiguity which is the source
of Mill's vacillating language.

Proper names, then, and the names of infimae species of attributes
and relations both signify the entire being of what they denote, but
with a difference, because what the latter signify is general, and may
be definable. Those are the classes of term which Mill calls non-
connotative. But the classes which he calls connotative, though he
offers a single account of them all, are not really alike. An attribu-
tive term, like long or white,3 denotes that which, being constitutively
something else, is long or white also 4 ; and it is connotative because

1 It need not be ; e. g. blue is a species of colour, but can no more be
defined than colour, because to know the specific difference presupposes that
I know the species ; I could only say that it is a blue colour.

2 Or names of the instances considered merely in their common character.

3 These terms were called connotative by the schoolmen : v. infra, p. 157.

4 Hence they are commonly combined with a general term, and we speak

vi] INTENSION AND EXTENSION OF TERMS 155

it notes a character found along with the constitutive being of the
subject which it denotes.1 Its ' connotation ' is not the constitutive
or general being of that subject, but some detail in its being, which
would be denoted by the corresponding abstract term length or
whiteness.2 But a general concrete name, like man or snow, does not
denote that which, being something else, or having some other con-
stitutive being, is man or snow also ; and it is connotative because,
besides denoting a subject, it signifies not some detail in its being,1 but
its constitutive or general being ; the prefix, if it has any force, has
not the same force in this case. And those abstract terms which
Mill calls connotative he calls so because, besides denoting species of
relation or attribute, they ' connote ' their generic nature.2

We may now sum up the results of our investigation into the
antitheses Intension and Extension, Connotation and Denotation.
All terms may be said to denote the subjects of which they can be
predicated, but those most directly which are names of, or can
stand for, those subjects ; hence adjectives, when used to denote the
subject of a proposition, are often combined with a demonstrative
word, such as an article. All terms have intension, or meaning ;
that is, they signify all or something of what that is which they
denote. When the intension, or what is thus ' intended ', is some-
thing displayed in divers forms or species, these are said to be the
extension of the term ; and sometimes the individuals in which the
common nature, which is the intension of a concrete general term,
is found are called its extension ; the latter usage is not extended to
general abstract terms, for in abstraction the instances are not dis-
criminated. Either way, proper names have no extension ; names
of infimae species of substances can only be said to have extension,
if the individuals are taken as the extension ; names of infimae
species of attributes can only be said to have it on the same condition,
that we consider their individual instances. What is commonly

of long days or long shadows, white sails or white complexions. The fact
that they may also be combined with proper names, so that we can say
' the envious Casca ' or ' the melancholy Jacques ', shows that proper names
have intension. No one would say ' the envious X ' if he did not know in
the least what that was, which X denoted.

1 A word like traitor or artist might be said to do this ; but we have seen
(p. 37, n. 1, supra) that these, though substantives grammatically, are attri-
butive in function. They also were called connotative by the schoolmen.

2 Attributive terms also may be predicated of attributes and relations, as
when we say that impartiality is rare ; here rare connotes an 'accident ' in
the attribute which it denotes, and not, as virtue would, its generic nature.
Mill takes no account of this difference.

156 AN INTRODUCTION TO LOGIC [chap.

said abont the inverse relation of intension and extension in terms
refers only to terms subordinated one to another in a classification,
and does not regard individuals as the extension. Lastly, terms
have connotation which have general meaning ; the connotation
of a term is that character through signifying which it denotes the
subjects of which it can be predicated.

[It may be added that the instance which Mill takes, on which
to argue that proper names have no connotation, viz. Dartmouth,
confuses the issue. He urges that the town would still have the
same name if the river changed its course, though the name would
not then connote the town's position ; therefore it connotes nothing
now. The argument is not good. If a town is called Dartmouth
because it stands at the mouth of the Dart, so far the name is
a designation. But meaning or intension in a proper name is not
dependent on connotation belonging to a connotative word in it.
Mill should have taken the river's name Dart, or Dartmouth in
New Hampshire. The latter illustrates yet another point. Most
proper names are chosen for a reason. A mountain may be named
after its discoverer or first climber, a town or college after its
founder, a child after its grandparent or godparent, a society after
some one of whom its members wish to be considered the disciples.
But this does not become part of the meaning of the name, which
is derived from that to which the name is given. A similar remark
applies to those names in which, as often happens, something may
give a clue to the nature or nationality or sex of the subject denoted ;
the guess may be wrong ; but even if it is right, the feature which
gave the clue functioned as having general meaning ; and the
meaning which it is important to vindicate for proper names is
not general meaning. It is however relevant to this vindication,
that proper names often come to acquire general meaning ; Caesar
is a familiar instance, and we have all heard of a Daniel come to
judgement, and that Capuam Hannibali Carinas fuisse. For this
acquisition comes about through extending to another subject some
part of the signification which the name derived from the subject
to which it was originally ' proper '.]

[For the sake of the curious, a few words may be added on the
history of the term ' connotative '. In William of Occam a dis-
tinction is found between absolute and connotative terms. Absolute
terms have not different primary and secondary significations ;
* nomen autem connotativum est illud, quod significat aliquid pri-
mario et aliquid secundaria' He gives as instances relative names
(for father signifies a man, and a certain relation between him and
another) : names expressing quantity (since there must be some-
thing which has the quantity) : and certain other words : v. Prantl,

vi] INTENSION AND EXTENSION OF TERMS 157

[GeschicMe der Logih im Abendlande, Abs. xix. Anm. 831, vol. iii.
p. 364. Johannes Buridanus said that some terms connote nothing
beyond what they stand for (' nihil connotantes ultra ea, pro quibus
supponunt'); but ' omnis terminus connotans aliud ab eo, pro quo
supponit, dicitur appellativus et appellat illud quod connotat per
modum adiacentis ei, pro quo supponit '.* Thus meus and tuus
stand for something which is mine or yours ; but they connote or
signify further and ' appellant me et te tanquam adiacentes ' (id. ib.
xx. Ill, vol. iv. p. 30). Elsewhere we are told that ' rationale '
' connotat formam substantialem hominis ' (xx. 232, vol. iv. p. G3 :
cf. Anm. 459, p. 109). Elsewhere again album and agens are given
by Occam (ib. xix. 917, vol. iii. p. 386) as examples respectively
of connotative and relative terms ; and it is explained (ib. Anm.
918) that a connotative or a relative term is one which cannot be
defined without reference to one thing primarily and secondarily
another ; thus the meaning of album is expressed by ' aliquid habens
albedinem ' ; and when by any term anything ' connotatur vel
consignificatur, pro quo tamen talis terminus supponere non potest,
quia de tali non verificatur ' 2, such a term is connotative or relative.
Thus a term was called connotative if it stood for (' supponit pro ')
one thing, but signified as well (' connotat ') something else about
it ; as Archbishop Whately says (Logic, II. c. v. § 1, ed. 9, p. 122),
'it "connotes", i.e. "notes along with" the object [or implies],
something considered as inherent therein.' The Archbishop sug-
gests the term attributive as its equivalent ; and though connotative
terms were not all of them adjectives, since relative terms also
connote, and so do terms like ' mischief-maker ' or ' pedant ', which
though adjectival in meaning are substantival in form, yet adjectives
are the principal class of connotative terms, in the original sense
of that word.

Connotation and denotation were thus originally by no means
equivalent (as they have come to be treated as being) to intension
and extension. Connotative terms were contrasted with absolute,
and their function of connoting distinguished from that of standing
for something. James Mill, who probably by his remarks upon
the word connote had some influence in directing his son's attention
to it, says that ' white, in the phrase white horse, denotes two things,
the colour, and the horse ; but it denotes the colour primarily, the
horse secondarily. We shall find it very convenient to say, there-
fore, that it notes the primary, connotes the secondary, signification '
(Analysis of the Phenomena of the Human Mind, vol. i. p. 34, ed.

1 i.e. to use J. S. Mill's terms, it denotes 'id pro quo supponit', and
connotes ' id quod appellat '. For appellatio cf . Prantl, vol. III. xvii. 59
(' proprietas secundum quam significatum termini potest dici de aliquo
mediante hoc verbo " est " '). Cf. also ib. xix. 875.

2 Occam means that, e.g., snow can be referred to as album, but albedo not.

158 AN INTRODUCTION TO LOGIC

[1869). By the schoolmen it would commonly have been said to
connote the colour, and the primary signification was that ' pro
quo supponit '. J. S. Mill, in a note to p. 299 of the same volume,
objects to his father's inversion of the usage. But he himself, by
extending the term connotative to cover what the schoolmen called
absolute, and opposed to connotative, names, introduced a complete
alteration into its meaning.

John and man are both absolute names in Occam's sense. Man,
no doubt, according to some (though not according to a nominalist
like Occam) signifies in John, or anything else ' pro quo supponit ',
an universal nature ; but John and this are not two things, of
which it denotes one primarily and the other secondarily, or for
one of which it ' supponit ', and ' appellat ' the other ; for John
is a man, and without what the word man signifies would be nothing
for which that word could stand or by which it could ' call ' him.
With white it is different ; I have a notion of paper, and a notion
of whiteness, and whiteness is no necessary part of my notion of
paper ; and so with any other subject of which whiteness is only
an attribute and not the essence. Hence the name white may be
said to signify (or in James Mill's usage to denote) two things, the
colour, and that which is so coloured ; for these can be conceived
each without the other, as John and man cannot ; or, if we prefer,
it may be said to denote or stand for one, and to connote the other.

(Cf. also on the history of the word Connotative a note in Minto'a
Logic, Inductive and Deductive, p. 46.)]

CHAPTER VII
OF THE PROPOSITION OR JUDGEMENT

A general acquaintance with the nature of the judgement or
pioposition has been hitherto assumed. It would be impossible for
Logic to be written, or if written to be understood, unless the acts
of thought which it investigates were already in a way familiar ;
for Logic arises by reflection upon an already existent thought
of things. Now judgement is the form in which our thought of
things is realized, and it is primarily in judgement that we use terms.
Their use in question, command, exclamation or wish presupposes
earlier judgement. The varieties of terms, the different relations
of one to another which form the basis of the distinction of pre-
dicables, would be unintelligible, unless it were realized that, in the
first instance, terms come before us only as elements in a judgement.
They live, as it were, in a medium of continuous judging and think-
ing ; it is by an effort that we isolate them, and considering subject
and predicate severally by themselves ask in what relation one stands
to the other, whether they are positive or negative, abstract or con-
crete, singular or general, and so forth. Without presuming some
knowledge of this medium in which they live it would be of as
little use to discuss terms, as to discuss the styles of Gothic
architecture without presuming some knowledge of the nature of
space.

We must now consider more closely what judgement is, and what
varieties of judgement there are that concern Logic.

A discussion of judgement raises many metaphysical problems,
into which such a work as this cannot enter fully. But a few things
may be pointed out about it.

To judge, in the logical sense of the word, is not to acquit or
condemn, but to affirm or deny a predicate of a subject. There is
however a connexion between the logical and judicial uses of the
word. Judgement, in the logical sense, is often preceded by what
must indeed be called thinking, but is not judging, viz. questioning

1G0 AN INTRODUCTION TO LOGIC [chap.

or ' wondering ' ; but this process, if we do not give it up, is ended
or decided by a judgement, as the judge by his judgement after
considering decides the case. It is true that as the judge may be
mistaken in the opinion which he reaches on the facts, so we commonly
in our judgements form fallible opinions only ; and Logic can render
no greater service than to make us more alive to the distinction,
which the grammatical form of the proposition fails to reflect,
between opinion and knowledge. We shall meet it in discussing
what is called the modality of judgements. So important is it,
that some would hesitate to bring knowledge and opinion under
one genus, judgement. But there is much which may be said about
them in common.

Every judgement makes an assertion, which must be either true or
false. Its propositional form claims truth : i. e. I ought not to make
a statement, such as that the earth is round, unless I think that it is
so, and mean that it is so ; although in fact we often express in this
form opinions which we hold doubtfully. This capacity of truth or
falsehood is the peculiar distinction of judgement, expressed gram-
matically in a proposition by the indicative mood. Imperatives,
optatives, exclamations, and interrogations are not propositions as
they stand, though they imply the power of judging. ' I say unto
this man " Come ", and he cometh.' Here the indicative sentence
' I say unto this man " Come " ' may be true or false, the indicative
sentence ' He cometh ' may be true or false, and both these are
propositions, and express judgements ; but we cannot ask of the
imperative ' Come ', is it false or true ? — it is not a proposition.
Again the question ' Art thou he that troubleth Israel ? ' is not
a proposition ; it is not itself true or false, but enquires whether
the judgement implied is true or false. An optative, as in the line
' Mine be a cot beside the rill ', is not as it stands a proposition ; it
could hardly be met with the rejoinder ' That 's true ', or ' That 's
a lie ' ; if it were, and we were to ask ' What is true ? ' or ' What is
a lie ? ' the answer would be ' That you really wish to live in a cot
beside the rill ' ; so that, although an assertion is implied about the
wishes of the person speaking, it is not so expressed in the optative.
Exclamations may in like manner imply an assertion which they
do not express, as when we say ' Strange ! ' or ' Incredible ! ' They
may also be mere modes of expressing feeling, like an action and
a gesture ; and in such cases, though something doubtless ' passes in
the mind ', the exclamation can hardly be regarded as an attempt

vn] OF THE PROPOSITION OR JUDGEMENT 161

at asserting 1 anything. It is not, however, necessary to go into any
subtleties ; the same grammatical form may indicate different acts
of mind, and the same act of mind be indicated by different gram
matical forms ; ' Let the king live for ever ' may be called imperative
or optative : ' Angels and ministers of grace, defend us,' imperative,
optative, or exclamatory : ' I would that I were dead,' optative or
indicative. It is enough for us to realize that a judgement being
an assertion, capable of truth and falsehood, the full and proper
expression of it is in the indicative mood.

In judging, I affirm or I deny ; in either case, I assert. I can
express doubt — ' matter may be eternal ' ; and herein I neither
assert that it is nor that it is not eternal ; still, I assert something,
though it is not so easy to say what.2 Propositions of the simple
form ' S is P ', or ' S is not P ', are called categorical, but in all there is
a categorical element. We can best elucidate the general character
of judgement by considering examples of this form in the first place.

A proposition makes one assertion 3 ; an assertion is one, when
there is one thing said of one thing — ev *a0' kvos, i. e. when the
subject is one, and the predicate one ; though the subject and
predicate may be complex to any degree. Thus it is one proposition
that ' The last rose of summer is over and fled ' ; but two that ' Jack
and Jill are male and female ' ; for the latter is equivalent to ' Jack
is male and Jill is female ' ; one thing is asserted of Jack and another
of Jill ; one grammatical sentence expresses two judgements.

Subject and predicate are terms which have already been explained,
as that about which something is asserted, and that which is asserted
about it. A proposition — at least a categorical proposition — is often
said to be composed of three parts, subject, predicate, and copula ; the
copula being the verb substantive, is, l<rTiv,est, ist(or is not, ovk €<ttivs
non est, ist nicht), sometimes, though mischievously, represented in
Logic books by the mathematical sign of equation, = (or not — ). We
may consider at this point the nature and function of the copula, and
the propriety of thus reckoning it as a third member of a proposition.

Common speech does not always employ the copula. Take the

1 The reasoning which would make all exclamations imply a judgement
was extended to actions by Wollaston, when in his Religion of Nature
Delineated (first published 1724) he regarded all wrongdoing as a particular
mode of telling a lie.

2 Cf . infra, pp. 197 sq.

3 Some difficulties about the singleness of judgement are discussed in
Mr. F. H. Bradley's Essays on Truth and Reality, c. xiii. pp. 393 sq.

1779 M

162 AN INTRODUCTION TO LOGIC [chap.

line ' It comes, it comes ; oh, rest is sweet '. Here in the proposi-
tion ' Rest is sweet ', we have subject (rest), predicate (sweet) and
copula all severally present ; whereas in the proposition ' It comes ',
we have the subject (it, referring to the omnibus), and for copula
and predicate together the one word, comes. But that word contains
what is said about the omnibus (for it is said to be coming, as rest
is said to be sweet) ; and it also contains, in the inflexion, a sign
that this is said about a subject ; and the judgement may, if we like,
be put in a form that exhibits predicate and copula separately, viz.
' it is coming '. It is true that this change of verbal expression may
sometimes change the sense ; it is not the same to say ' he plays the
violin ', and to say ' he is playing the violin ' ; we must say, ' he is
one who plays the violin ', or ' he is a violinist '. But it is clear that
what the copula expresses is present as much in the proposition ' he
plays the violin ' as in the proposition ' he is a violinist ' ; just as it
is present alike, whether I say Beati immaculati in via or Beati sunt
immaculati in via. The inflexion of the predicate verb, or the in-
flexion of the predicate adjective together with the form and balance
of the sentence, replaces or renders superfluous its more precise ex-
hibition by the copula ; which is, however, always understood, and
if we set down the subject and predicate in symbols whose meaning
is helped out by no inflexion, we naturally insert it. We symbolize
the judgement generally by the form ' A is B ' 2 ; we may write it
' A B ', but that is an abbreviation ; to write it ' A =B ' is an error.

If the copula thus expresses something present or implied in
every judgement, what is its function, and can it be regarded as
expressing one of three parts composing a proposition ? Its
function is to express that the subject and predicate are brought
into the unity of a judgement : that the predicate is asserted of the
subject, and that the subject is qualified by the predicate. I may
think of rhetoric and I may think of trickery, but they may
remain apart in my thought — subjects successively contemplated,
like breakfast and a morning's work ; if I say that ' rhetoric is
trickery ', I show that they are not unconnected, to my thinking,
but that one qualifies the other.

Is the copula then a third member in the judgement, distinct from
subject and predicate ? Strictly speaking, no. For two terms are
not subject and predicate, except in the judgement ; and the act

1 C. S. Calverley, Lines on the St. John's Wood Omnibus.

2 Or ' A is not B ', if the judgement is negative ; and so elsewhere, mutatis
mutandis.

vii] OF THE PROPOSITION OR JUDGEMENT 163

of judging, whereby they become subject and predicate, is already
taken into account in calling them subject and predicate ; it ought
not therefore to be reckoned over again in the copula. In the verbal
expression of judgement, which we call a proposition, we may dis-
tinguish as a third member a word showing that other words are
subject and predicate ; but the whole proposition 'A is B' expresses
a single act, in which though we may distinguish subject and predicate
from the predicating, we cannot distinguish them from it as we can
from one another. To think the copula is the synthesis (or linking)
of judgement : it is the form of the act, as distinguished from
thinking the subject and predicate ; this is the matter, for judge-
ment varies materially with variation of the subject and predicate.
The copula is a word used to express the performance of that act.
Is it of any consequence how that act is expressed — (1) whether
by an inflexion or by an independent word ; (2) if the latter, whether
by the verb substantive or some different word or sign (such as the
mathematical sign of equality) ?

(1) Every categorical judgement is analysable into subject and
predicate ; in the act of judgement we affirm or deny their unity ;
but, whether in affirming or denying it, they are distinguished;
and the predicate may in its turn become a subject of thought. The
separation of the sign of predication from the predicate (as in the
proposition ' He is a violinist ', compared with ' He plays the violin ')
frees the predicate, as it were, from its immersion in the present
judgement. If therefore we wish to set out a judgement in a form
that shows clearly what is the subject, and what the predicate, each
separately considered, an independent word is better, as a sign of
predication, than an inflexion. For the purposes of a logical example,
we should prefer to express a judgement in a form that shows this ;
but it would be pedantry to do it, where, owing to the idiom of the
language, it perverts the sense ; and we do not need to do it at all
when we have no such need to extricate the predicate.

(2) Different languages agree to use the verb substantive, or
verb of existence, as the sign of predication : Homo sum, I am
a man : Cogito, ergo sum, I think, therefore I am.1 The use of the
verb of existence as copula suggests that every judgement predicates

1 Propositions in which the verb of existence was predicate used to be
called propositions secundi adiacentis ; and those which had some other
predicate, where the verb to be was present or implied as copula only, were
called propositions tertii adiacentis.

M2

164 AN INTRODUCTION TO LOGIC [chap

existence, that if I say ' government is a science ', I declare not only
that it is a science, but that it is or exists ; on the other hand, the
content of many judgements seems to negative this ; for in saying
' a griffin is a fabulous monster ', or ' Queen Anne is dead ', I do not
assert that a griffin or that Queen Anne exists. Hence some have
boldly said that the verb ' to be ' is a mere equivocal term employed
sometimes to signify existence, and sometimes to signify predication :
with no more identity of meaning in these two uses than there is
between est = ' is' and est = ' eats \x From this it would follow,
that there is no special appropriateness in using the verb to be as
sign of predication, rather than any other sign.

Yet if there were no special appropriateness in the verb to be, as
the sign of predication, it is strange that so many languages should
have agreed to use it. The case seems to be thus : that every
judgement does imply existence, but not necessarily the existence of
the subject of the sentence. The distinguishing characteristic of
a judgement is, as we have seen, that it is true or false. With the
false we need not here concern ourselves ; for the man who makes
a judgement, unless he says what he does not really think, says
what he thinks to be true, and therefore intends to declare the truth.
All judgements therefore, besides affirming or denying a predicate
of a subject, implicitly affirm themselves as true.2 But a judgement
which affirms itself as true claims to express, so far as it goes, the
nature of things, the facts, or the realit}7" of the universe. In doing
this it maybe said to imply existence, not of its grammatical subject,
but of the whole matter of fact asserted in it.

When I say that a griffin is a fabulous monster, I do not affirm
that griffins exist like pigs and cows. But my judgement implies
the existence of a mass of fable, in which griffins have their place as
fables too. If there were no fables, I could not say that griffins
were fabulous ; but fables are an element in reality — i. e. in the
totality of what is real — no less than pigs and cows. Again, when
I say that Queen Anne is dead, I do not affirm the present existence
of Queen Anne ; I do imply her existence in the past ; and the

1 Cf. James Mill, Analysis of the Phenomena of the Human Mind, vol. L
p. 174 (ed. 1869) ; J. S. Mill, System of Logic, I. iv. 1.

2 Cf. F. H. Bradley, Essays in Truth and Reality, p. 382 : * We cannot,
while making a judgement, entertain the possibility of its error.' It may
be noted that a lie is not a judgement, but rather an action intended, through
the use of words that commonly express a judgement, to influence the action
or opinion of others.

vii] OF THE PROPOSITION OR JUDGEMENT 165

copula therefore still has the meaning of existence. It may be
asked why it should be in the present tense, when the existence
meant is past. The answer is, first, that the predicate corrects this
so far as is necessary ; but secondly, that the past (like fable) has
a kind of existence. If I am the same to-day as I was yesterday,
then I do somehow unite in me at once the present and the past ;
the past has ceased to be present, but it still somehow belongs to
me. What is true of me is true of others, and of reality as a whole.
Its history is in time ; but it is one through that history ; and the
past belongs to it now, as well as the present. Queen Anne, it may
be, does not exist now ; but that exists now in whose past the life
and death of Queen Anne have their place. They belong to the
whole system of things which we call the universe ; therein they
exist, and only in belonging to it can they or anything else exist.
The moon, if it had no place there, would not be ; neither would
justice, nor triangularity ; though these different things play different
parts in the whole.1 When I say what triangularity is, the present
tense is not used because it is contemporary with the time of the
utterance; for it is not temporal at all. Not everything real
belongs to the succession of events in time.

Every judgement then that I make claims to declare some portion
of the whole truth that is to be known about the universe : in what
form (so far as its purview goes) the universe exists. Hence it is no
accident that the verb of existence is employed to express the act
of judgement. There is a kind of thinking called questioning or
wondering, in which we think of various things, and imagine them
connected in various ways, without deciding in our minds whether
they are so connected or not. Thus I may think of Public Schools,
and ask myself whether they are liable to stifle originality in their
pupils ; and I shall be thinking also of that liability, and of the
relation of subject and attribute, and imagining that relation to be
exemplified between these terms. But if I judge one way or the

1 Some writers have used the notion of a ' universe of discourse ' or ' limited
universe ' to express the foregoing contention. In the whole universe fact
and fable, savages and Rousseau's conception of savages alike have their
place ; but I can make statements which are true about Rousseau's concep-
tion which would be false about savages themselves. It is said that these
are different ' limited universes ' ; and that propositions which do not assert
the existence of anything in the material universe may assert it in some
other. ' The royal dragon of China has five claws ' — I do not affirm its
existence in the universe of zoology, but in that of Chinese heraldic design.
Cf. p. 44, n. 2, supra.

166 AN INTRODUCTION TO LOGIC [chap.

other, that public schools are or are not liable to stifle originality
in their pupils, then I believe that this relation really holds, or does
not hold, between these terms, and that what I think of exists
independently of my thinking. And to express that a combination
of which I think is real, I use the verb to be. * Public schools are
liable (or not liable) to stifle originality in their pupils ' ; i.e., the
liability of public schools to do so, or their freedom from such
liability, exists.

[It will be observed that on p. 164 the copula was said to imply,
not to predicate, existence. For existence by itself is not a significant
predicate, as we have already seen,1 and therefore cannot strictly
speaking be predicated. We may ask, for example, whether griffins
exist, as we may ask whether ostriches fly ; but whereas in the
latter case the subject is assumed to exist, and the question is
whether it possesses a certain predicate, in the former case we do
not assume that there are griffins, and enquire whether they possess
the predicate of existence. Their existence would consist in being
griffins, and not merely in being ; and to ask whether griffins exist
is to ask whether anything existing has the character intended by
the term griffin. The existent is thus assumed as the subject of
our judgement, and the judgement claims to declare its nature ;
we do not assume its nature as a subject of which to predicate
existence. Hence it has been said that reality is the ultimate sub-
ject of every judgement ; that, as the distinction of its terms is
not a distinction of two independent things, but of two factors in
the being of one, this whole being, conceived by us in subject and
predicate together, is really one ' content ', and though judgements
differ in their content, these contents are all predicated of the one
reality ; and the contents of all true judgements are factors co-
existing in the being of that reality. To ask ' Is such and such
a proposition true ? ' is to ask whether in its subject and predicate
together I apprehend in part the nature of reality ; and it is because
of this ' reference to reality ' in every judgement that we use in
expressing it the verb to be.

This view that reality is the ultimate subject of every judgement
is wrong if it be understood to mean that it is the logical subject,
or be taken as destroying the force of the logical distinction between
subject and predicate. We may distinguish in fact three subjects,
the logical, the grammatical, and the ultimate or metaphysical.
That the logical subject is not the same as the grammatical subject
of the sentence is readily apprehended. The proposition ' Bella-
donna dilates the pupil ' may be an answer either to the question
1 What dilates the pupil ? ' or ' What do you know of belladonna ? '

1 Cf. supra, p. 65.

vn] OF THE PROPOSITION OR JUDGEMENT 167

[In either case the grammatical subject is belladonna ; but the
logical subject is in the former case ' dilating the pupil ' ; that is
what we are thinking about, and about that the judgement informs
us that belladonna will effect it ; in the latter case, the logical
subject is belladonna, and about that the judgement informs us
that it produces this effect. This distinction of logical subject and
predicate is always present in thought when we judge, though
sometimes the logical subject may be very vague, as when we say
' it rains ' or ' it is hot '. But subject and predicate together may
qualify something further. This is easily seen when the subject is
an abstract term. ' Jealousy is a violent emotion ' : jealousy may
be the logical subject here, but it only exists in those who are
jealous. It is not then the ultimate subject, for it inheres in some-
thing else. Where then do we reach the ultimate subject ? Accord-
ing to our ordinary way of thinking, in concrete individuals ; and
this is the view also of many philosophers, who have thought (and
Aristotle seems to have been among them) that there was no single
metaphysical subject, but as many as there are concrete individuals.
In the Categories 1 the concrete individual is defined as that which
can neither be predicated of nor inhere in anything further.2

But the doctrine which makes Reality the ultimate subject of
every judgement holds that in a sense the metaphysical subject is
always one and the same : i. e. that there can be only one real
system, to which all judgements refer, and which they all contribute
to determine and qualify. That a particular thing should exist or
be real means that it has its place in this system ; and what is
called the existential judgement — the judgement whose predicate
is the verb to be, in the sense of to exist — as in ' Sunt qui non
habeant, est qui non curat habere ', or ' Before Abraham was,
I am '—declares a part of the nature of the one system of reality.
The content of an existential judgement cannot indeed be pre-
dicated of reality as a quality or attribute. When I say that
jealousy is a violent emotion, I think of it as an attribute of jealous

1 ii. lb 3-9, v. 2a 11-14. Cf. supra, pp. 50 sq.

2 It is true that a singular term may appear as predicate of a judgement,
as, for example, if we say ' The greatest epic poet is Homer ' or ' The first
man was Adam '. But in such a case Aristotle regards the predicate as only
accidentally predicate, or Kara avfi^e^KOi (cf. Met. A. vii) : by which he
means that the concrete individual does not really qualify or belong to what
figures as its subject, but that because these two come together, or because
it befalls Homer to be the greatest epic poet, and Adam to have been the
first man, therefore you can say that one is the other, as you can also say
that a grammarian is a musician when the two characters coincide in one
individual, though ' musician ' is not what ' being a grammarian ' is, any
more than Homer is what being the greatest epic poet is, or Adam what
being the first man is. In fact, in making a judgement whose predicate is
a singular term, we cannot help at the same time thinking of the predicate
as qualified by what figures as subject. But cf . supra, p. 153.

168 AN INTRODUCTION TO LOGIC [chap.

[men ; when I say ' Est qui non curat habere ', I do not think of
Horace as an attribute of reality. Nevertheless, his existence is
bound up with the existence of the whole universe ; the universe
of reality is found (when we think the matter out) to be presupposed
by the existential judgement as much as by any other ; and though
in it existence appears to be first affirmed in the predicate, and
therefore not assumed in the subject, yet this cannot represent
the true course of our thought. We could make no judgement at
all, if we did not presume a reality about which it was made.
Even the negative existential — ' Joseph is not, and Simeon is
not ' — implies this ; for not to be means to have no place in that
which is.

We are indeed accustomed to think of things and persons as if
each were complete and independently real ; and in that case, the
metaphysical subject of any judgement would be some concrete
individual or other. The doctrine we are considering carries the
question further, and holds that, since what is predicated of the
concrete individual is not true of him in complete isolation from
all else, therefore he is not, metaphysically speaking, or in the last
resort, the subject of which it is true. There is no desire to deny to
individuals a relative independence, or to pretend that the relation
of attributes or universals to the concrete individual is the same
relation as that of an individual to the system of reality which
includes him. The judgement ' Jealousy is a violent emotion ' can
be so restated as to make the concrete subject man the logical
subject of the judgement ; I may express it, for example, by saying
that jealous men are violent in their jealousy. I cannot so restate
the existential judgement, or any other in which the logical subject
is already a concrete term, as to make Reality the logical subject
instead. But it is the metaphysical subject in the sense that it is
presupposed and referred to even in those judgements. We cannot
maintain the view that the metaphysical subject of every judge-
ment is always in the last resort a particular individual. ' Civiliza-
tion is progressive.' Doubtless civilization is only seen in the fives
of men ; but it is seen in the lives not of this and that man singly
but of the communities to which they belong. We have to think
of men as forming a system and an unity, if we are to give meaning
to a judgement like this. We saw too that the process of biological
evolution, which seems in some way single, yet cannot be exhibited
in any single organism ; nor is it easy to know what is a single
organism. What is contended is, that all judgements involve us in
the thought of one all-embracing system of reality, whose nature
and constitution none can express completely, though each true
judgement declares a part of it. Logic, as has been said before,
cannot be rigidly separated from metaphysics ; indeed, it derives
its chief importance from its connexion therewith. If it had merely

vii] OF THE PROPOSITION OR JUDGEMENT 169

[to work out the scheme of syllogistic inference, and such -like
matters, the problem which the present note has raised would be
superfluous ; but it investigates what is involved in thinking ; and
whether we must think of the universe as a sum of independent
reals or as a system is a fundamental problem.1]

In the act of judgement, the subject 2 with which we start is
thought of as modified or enlarged by the predicate, and in that
form declared to be real. We end with the subject with which we
began, differently conceived.3 The thought of a combination of
elements, and the affirmation of its reality,4 are common features
of every judgement, and the copula expresses them always, and so
far has always the same meaning. Whatever sign be used, whether
an inflexion, or the verb substantive, or the mathematical symbol
for equality, or anything else, this combination, and the affirmation
of its reality, must be meant. The verb to be naturally lends itself
to this meaning. The mathematical symbol of equality has a dif-
ferent meaning ; it is not a sign of predication, but an incomplete
predicate ; it expresses, of one thing, quantitative identity with
some other. If I say A = B, the predicate is not B but ' equal to B ' :
the special force of the sign ' = ' is ' equal to ' ; I must still perform
in thought the act of predication, whether I say ' A is equal to B \
or (A is the first letter of the alphabet ' ; and if = were adopted as
the sign of predication, the equation ' A=B ' (which means ' A is
equal to B ') must be written ' A = = B '.

A judgement then contains subject and predicate ; subject and
predicate in their combination are declared real. To the words
which signify the subject and the predicate separately is added

1 The view that Reality is the ultimate subject of judgement is of course
familiar to all readers of Mr. F. H. Bradley's or Professor Bosanquet's logical
work. Cf. Bradley, Principles of Logic, c. i. pp. 12-14, and Essays on Truth
and Reality, c. ix. pp. 253-254. Mr. Bradley does not distinguish between
logical and metaphysical subject.

2 i. e. the logical subject.

8 Sigwart has pointed out that the movement of thought in a judgement
is different for a speaker communicating information and tor his hearer. The
speaker knows the whole fact, when he starts putting forward one aspect of
it in enunciating the subject, and supplements it with the other by adding
the predicate : if I say ' This book took a long time to write ', the whole
fact is present to my mind in its unity before I begin speaking. To the hearer
I present a subject of thought, ' this book ', which awaits supplementation :
to him the predicate comes as new information, which he has now to combine
with the concept of the subject hitherto formed by him. v. Logic, § 5. 1.

* Even in a negative judgement, subject and predicate are elements thought
of together, as standing in a relation of mutual exclusion.

170 AN INTRODUCTION TO LOGIC

a word which signifies that these are thought to be combined in the
real. This word is called the copula ; it may be omitted in speech
or writing, or be replaced by an inflexion ; but the act of thought
which it indicates cannot be omitted, if there is to be a judgement.
This act, however, is not a part of the judgement in the same way
that subject and predicate are. It is the act or form of judging, and
they determine the matter. Hence it is, at least generically, the
same, while subject and predicate change ; and for this reason the
scheme of a proposition 'A is B ' represents subject and predicate
by symbols, but retains the ' copula ' itself. We write A and B for
subject and predicate,1 because they represent indifferently any
subject and predicate, being themselves none ; we write ' is ', and
not another symbol in its place, because whatever be the subject and
predicate, the act of judgement is, generically, the same.

But judgements are not all so much alike that they can all be
equally well expressed in propositions of the form ' A is B ' ; they
do not differ merely as the places of these symbols are taken by
different terms. For some propositions are of the form ' A is not B ';
and A may be replaced by a singular or by a general term ; and if
by a general, we may judge either that all or some A is (or is not B),
and this difference is one of form, in the sense that it is not a differ-
ence in the terms that replace our general symbols A and B. And
there are other differences in propositions which are not differences
in their terms. Having got some notion of what judgement is in
general, we must now turn to the differences which are expressed in
these differences of propositional form. With differences merely
of the terms, as between ' men are animals ' and ' roses are plants ',
we are not in Logic concerned.

1 Of course any other indifferent symbols will serve, such as X and Y or
S and P.

CHAPTER VIII
OF THE VARIOUS FORMS OF THE JUDGEMENT

Judgements, or the propositions in which they are expressed,
have for long been commonly distinguished according to Quality,
Quantity, Relation, and Modality : — according to Quality, into
affirmative and negative : according to Quantity, into singular,
universal, and particular : according to Relation, into categorical,
hypothetical, and disjunctive : according to Modality, into assertoric,
problematic, and apodeictic. The distinctions in Quality and
Quantity, as the simplest and most familiar, will be discussed first ;
they can only be fully illustrated in categorical judgements or pro-
positions.

In respect then of quality, categorical judgements are distin-
guished as affirmative or negative. An affirmative categorical
judgement assigns a predicate to a subject ; a negative puts it from
it. But the distinction between affirming and denying is too familiar
to need and too simple to admit of being expressed in any other way,
in order to indicate what is meant.

There are certain difficulties connected with negative judgements,
which have already met us in dealing with negative terms. Judge-
ment, as we have seen, refers to the existent, whose manner of being
(so the judgement declares) is as we conceive. But the real is
positive ; it only exists by being something, not by being nothing.
A negative judgement declares what it is not, and how can this
express it as it is ? Dead-nettles don't sting. How does that tell me
anything real in dead-nettles ? You may say that I formed an idea
of a stinging dead-nettle, and in the negative judgement declare it
false, an idea of nothing real. But that only means that I had
thought that, or asked myself whether, dead-nettles sting, and in
correction or reply now judge that they do not. My ' idea ' means
my opinion, or a supposed opinion ; I may reflect on that, and say
that the opinion is false ; but in the example I am judging about
dead-nettles, not about any past opinion about them. And
when I say that they do not sting, what am I saying about them ?

172 AN INTRODUCTION TO LOGIC [chap

in them, what is this property of not stinging ? surely, it may be
urged, just nothing : so that in the negative judgement I assert
nothing real.

These misgivings are sometimes, though unfairly, met by ridicule.
Still, in face of them, we must assert, that everything finite is what
it is, by not being something different : and at the same time, that
it is not something different, in virtue of what it positively is. Hence
we must accept the negative judgement as expressing the real limita-
tion of things ; but we must allow that it rests upon and presupposes
the affirmative. If dead-nettles do not sting, there must be some
characteristic which they do possess, incompatible with stinging.1
There is always a positive character as the ground of a negation.
Snow is not hot, because it is cold ; this is not indeed an explanation
of the temperature of snow ; but it means that a material body
(which must have some temperature) can only not have one degree
of temperature through having another. If snow had no other degree
of temperature, it would have 212° Fahr. ; if it had none but 32°
Fahr., it must have that. And it may be noticed how often in the
building up of knowledge we use negative judgements to reach
affirmative : to know what anything is not is frequently a help to
discovering what it is. In the inductive sciences this procedure
is constant, and we shall find it a fundamental feature of the
induction in them.

To say that negative judgements presuppose affirmative does not
however get rid of the difficulties to which we have referred. If
snow is not hot because it is cold, then the cold is not hot. No one
will deny that ; some people will think it a mere tautological pro-
position. But it is not tautological, though it is superfluous. It is
tautological to say that the cold is cold ; to say that it is not hot
because it is cold informs us that hot and cold are mutually exclusive
attributes. Cold is no more identical with not-hot, than odd with
not-even ; though the numbers which are odd are the same numbers
as are not even. The reciprocal exclusiveness of certain attributes
and modes of being is the real truth underlying negation. But for

1 A critic (Miss Augusta Klein) has objected that this is only a negative
character, viz. the absence of glandular stinging hairs. But the tissues
forming any part of a leaf can only not be glandular stinging hairs if they
ire something else. A body can only not be here if it is elsewhere. However,
a difficulty arises with empty space ; by being what, is it not occupied by
some body ? is emptiness purely negative ? Democritus, and Plato, called
space fi>) ov, not-being. Some have denied that a vacuum can exist.

vin] VARIOUS FORMS OF THE JUDGEMENT 173

that, everything would be everything else ; that is as positive, as
these several modes of being themselves.

Negation, as Plato saw,1 is as necessary as affirmation, if there are
to be any differences or discriminations within reality ; that A is not
B means that it is different from B, and not that it is non-existent.

[The further pursuit of this subject would take us too far into
metaphysics. It may be pointed out in passing that the notion of
an infinite (or, as philosophers sometimes say, an absolute) being
is of a being who is everything that there is to be ; of whom it
cannot be said that he has one attribute by lacking another ;
whereas finiteness comes by limitation and exclusion : whence
Spinoza's Determinatio est negatio. Whether this is a tenable con-
ception is another matter. In particular it raises the problem of
the meaning, and reality, of evil. For if an infinite being is all
things, and evil is something real, he ought inter alia to be evil.
It has been contended therefore that evil is in reality just nothing,
a view against which there are obvious objections on the surface :
or at least that it is a mere appearance incident to limitation, but
in itself no more than limitation ; what is absolute and all-inclusive,
having nothing outside it to limit it, would not be evil, though it
would include what, taken in improper isolation, appears evil.]

It has sometimes been proposed to treat the negative judgement,
A is not B, as an affirmative judgement, A is not-B,2 by combining
the negative with the predicate. But inasmuch as the reciprocal
exclusiveness of certain attributes and modes of being is a positive
fact, it is no use trying to ignore it by a verbal manipulation.
Nothing will make A is not-B an affirmative judgement, unless not-B
is something positive ; and if not-B is something positive, say C, the
judgement is true because B and C are counter-alternatives ; e. g.
the fact that the path of a bullet is not straight may be expressed
by saying that it is a curve, but only because straight and curved
are mutually exclusive and sole alternative determinations of a fine.
It follows that C is not B, and B is not C ; and these negative judge-
ments cannot be evaded by writing ' C is not-2? ', ' B is not-C '.
For if C means the very same as not-B (e. g. curved as not-straight),
then not-C means the very same as not-not-B, and the proposition

Soph. 256 E 7T(p\ €Kci(ttop cipa Tu>v elSa>i> 7roXv pev eori to ov, arretpov 8e nXr/da
to pr/ ov. 257 B onorav to p,f] ov \eywpev, if coiKfv, oIk ivavrinv ti \eyoptv tov

oVto?, a\\' ercpov povov. ('About each Form then there is much that it is,
but an infinite amount that it is not. . . . When we speak of not being,
we speak, it seems, not of what is contrary to being but only of what is
different.')

2 Such judgements, with an infinite term (cf. p. 42, n. 2, supra) for predicate,
have been called infinite judgements.

174 AN INTRODUCTION TO LOGIC [chap.

B is not-C means no more than B is not-not-B (' straight is not-not-
straight '). That however is absurd ; for G is positive, and the
consciousness of the distinction between it and B and of their
reciprocal exclusiveness cannot be reduced to the consciousness that
B cannot be denied of itself. The above argument could equally
be illustrated if we took for B not one of two counter-alternatives,
but a term like dog ; only then not-B would leave us to select in the
dark among a large number of still remaining alternatives.

In respect of quantity, categorical judgements are said to be either
singular, or universal, or 'particular. But the differences at the
bottom of this distinction are not in reality purely quantitative,
though they have sometimes been represented as being so.

The subject of a proposition may be either a singular term like
' Socrates ' or ' Caesar ' or ' the present Cabinet ', or a common
term like ' man ' or ' triangle '. In the former case, the proposition
too is called singular. In the latter, the proposition may affirm
or deny the predicate of the subject either universally, i. e. in every
instance of it, e. g. ' All equilateral triangles are equiangular ',
' Nemo omnibus horis sapit ' : in which case it is called universal ;
or partially, i. e. in particular instances, or of a part of the subject,
only, e. g. ' Some larkspurs are perennial ', ' Some animals cannot
swim ' : in which case it is called particular. The judgements which
these propositions express * are correspondingly distinguished as
singular, universal, or particular.

Now these three kinds of judgement may clearly be represented
as concerned respectively with one individual, with all individuals of
a certain kind or description, or with some part of such aggregate
or class. For though when I say that all acids contain hydrogen, or
that some larkspurs are perennial, I may be thinking primarily
of the kinds or species of acid, or of certain species of larkspur, yet
the statements, if true, are true in every instance of those species.2

1 We judge, commonly, not about words but about what they stand for,
but we express our judgements in words. A common term stands for and is
predicable of not a common nature in things, but things in respect of their
common nature. These things are the subject of the judgement, when
a common term is the subject of a proposition.

2 i. e. of the whole or part of the denotation, as well as of the whole or part
of the extension of the subject-term, if the distinction made on p. 146, supra,
be adopted. It should be remembered that the singular term has no extension ;
and that an individual cannot be called the whole denotation of a singular
term in the same sense in which the divers individuals of a class can be called
the whole denotation of a general or class-term.

vin] VARIOUS FORMS OF THE JUDGEMENT 175

And so they may be represented as concerned with all or part of
what their subject terms denote. And as a singular term denotes
only one individual, the singular proposition is also concerned with
all that its subject-term denotes. Hence it has sometimes been said
that propositions are of two kinds in respect of quantity, universal
when they refer to the whole denotation of the subject-term,
particular when they refer to part of it. We shall see later, when
dealing with syllogism, that in some connexions it is unnecessary to
distinguish between singular and universal judgements or pro-
positions, because they both equally make certain inferences possible.
But at present it is important to realize that what are called differ-
ences of quantity in judgements or propositions, are not primarily
differences in respect of how much of the denotation of the subject
term is the subject of our thought.

The subject of a singular judgement is individual (though it may
be an individual collection) ; that of an universal judgement may
be an universal, or concept, e. g. ' Fear is contagious ' ; or, though
not a concept, it may be determined by a concept,1 e. g. ' Letters
in transit are the property of the Postmaster-General.' The latter
statement, though it concerns individual letters, applies to them not
as this or that individual, but as possessing the character signified
by the words ' letter in transit '. The difference therefore between
it or the former and a singular judgement lies not in the quantity
of the individuals to which they refer (i. e. in the singular referring
to one individual and the universal to all individuals of a certain
collection), but in the logical character of the subject, which in the
singular judgement is a determinate individual, in the universal
judgement a concept or anything characterized and determined by
a certain concept. We may include both these in the expression
1 a conceptual subject '. No doubt an universal judgement has
a quantitative aspect, for it does concern all individuals that share
the subject-concept ; but this aspect is secondary. Primarily, in
making it, we have before us a relation between one character and
another in individuals, not between individuals and a certain char-
acter. Neither therefore is the difference between an universal
and a particular judgement primarily quantitative. A particular
judgement refers to part only of the denotation of some conceptual
subject, an universal to all ; but this is because in the latter the

1 The totality of things exhibiting a certain character is called a class,
and the character which determines membership of the class a class-concept.

176 AN INTRODUCTION TO LOGIC [chap.

relation of concepts is taken to be necessary, and therefore the
subject-concept sufficiently determines the application of the judge-
ment ; in the former it is not, and we indicate by the word some
that the application of the judgement is not completely determined.1
A criticism of the forms in which language expresses judgements
of these different types will throw further light on what has just
been said.

It is common to indicate an universal judgement by the words
all or no (none) prefixed to the subject, according as the judge-
ment is affirmative or negative ; a particular judgement by the
word some, similarly prefixed ; these are called signs or marks of
quantity. The idiom of language will indeed often express a uni-
versal judgement in other ways ; we can say Man is mortal, as well
as All men are mortal : A barometer will not work in a vacuum, as
well as No barometer will work in a vacuum. But in the absence
of a mark of quantity, it is not always clear whether a proposition
is meant to be universal or particular ; if I say Women are jealous,
A flower is a beautiful object, I need not mean all flowers, or all
women. Precision requires the quantity of a judgement to be
expressly indicated : particularly where (as in logical examples) the
proposition is taken out of context and we lack the help which
context often affords us in divining the writer's intention ; and at
least where the subject is in the plural,2 the words all, none, some
are appropriated to that service. A proposition without any mark
of quantity is technically known as an indefinite proposition ;
because it is not clear whether the whole, or only a part, of the
extension or denotation of the subject is referred to, and so the
scope of the proposition is undetermined ; the examples just given,

1 The Aristotelian division of political constitutions (or rather Platonic —
for it occurs in Plato's Politicus) is another example in which differences
not really quantitative have been presented under a quantitative form.
A monarchy, an aristocracy, and a democracy, though said to differ according
as power is in the hands of one man, of the few, or of the many, really differ,
as Aristotle himself pointed out, in quality or kind. It must be added that
Aristotle does not put forward a purely quantitative division of judgements
(cf. de Interpr. vii. 17a 38 eirei 8' tar\ ra fiev kci66\ov to>v 7rpaynora)v to 8e xad
tKatTTov — 'since of things some are universal and some several'), though in
expounding the syllogism in the Prior Analytics he often lays stress on the
quantitative implications of the contrast between universal and particular
judgements.

2 ' Man is mortal ' is clearly universal ; but represented in symbols as
' A is B ' it will not unambiguously show its universality. For ' Iron is
found in Lancashire ' might be represented by the same symbols, but is as
clearly particular.

viii] VARIOUS FORMS OF THE JUDGEMENT 17?

Women are jealous, A flower is a beautiful object, are therefore
indefinite propositions.

At the same time, the words all and none, as signs of the uni-
versality of a judgement, have disadvantages of their own. For
a judgement is really universal, when the subject is conceptual, and
the predicate attaches to the subject (or is excluded from it) neces-
sarily ; but if it is found to attach to the subject (or to be excluded
from it) in every existing instance without any necessity that we
know of, we use the same expressions, all and none. Thus we may
say that No American poet stands in the first rank, or that All the
French ministries are short-lived ; but neither of these is really
an universal proposition. Each expresses a judgement made about
a number of individuals : it states an historical fact, and not a
scientific truth. It would be convenient to call such propositions
collective x or enumerative ; for they really collect in one the state-
ments which may be made about every instance of a certain class,
and make their assertion on the strength not of any conceptual
necessity, but of an enumeration.

We must of course distinguish the question whether a proposition
is meant as universal, in the strict sense, from the question whether
we have a right to enunciate it universally. If instead of saying All
the French ministries are short-lived (where the article the shows that
I am referring to all of a certain number of things), I were to say All
French ministries are short-lived, it might be contended that the
proposition no longer referred primarily to individuals or instances,
but affirmed a necessary character of French ministries as such. In
truth the statement is not clear, and a man would have to ask me,
whether I meant it as an historical summary, or an universal truth ;
but the ambiguity of the statement is the very point to be noticed ;
for the two interpretations indicate the difference between a merely
enumerative, and a true universal, judgement. The difference is
plain in suitable examples : contrast, for instance, ' All, all are gone,
the old familiar faces ', and ' All lovers young, all lovers must, like
chimney-sweepers, come to dust.'

We have seen that there is a marked distinction between a sin-
gular judgement, whose subject is an individual, and an universal
or particular judgement, whose subject is conceptually determined
by a general or abstract term. The enumerative judgement (and

1 Cf. Bradley, Principles of Logic, Bk. I, c. ii. §§ 6 and 45. In the Table
of Contents he speaks of ■ collective ' judgements in this sense.
1779 N

178 AN INTRODUCTION TO LOGIC [chap.

this is true in some degree of the particular also) approximates to the
type of the singular rather than of the universal.1 For though the
subject of the proposition be a general term, and I predicate about
all the members included under that term, yet I do so because I have
examined them severally and found the predicate in them all,
or at least, on good evidence or bad, believe it to attach to them
all, not because of any necessary connexion between the predi-
cate and the common character of these individuals which the
general term signifies. French ministry is a general term ; but (for
all that I see) it is not because being a French ministry involves
being short-lived, that all the French ministries are short-
lived ; I assert it because I have noted each case ; just as it
would be upon the strength of noting the individual case that
I should assert the first ministry of M. Jules Ferry to have been
short-lived. At the same time, the enumerative judgement, though
thus approximating to the type of the singular, gives the hint of
a true universal judgement. It suggests that the ground for the
predicate may lie in the common character signified by the general
term under which all these instances are collected. If I say Luther
was hated, there is nothing to indicate what about him was hateful :
with which of all the coincident attributes in Luther his hatefulness
is universally connected. If I say All reformers have been hated,
though that is as much an historical statement as the first, and there-
fore enumerative only, it suggests that the reason why all those
men have been hated (Luther and Calvin, Cromwell and Gladstone
— the statement implies a possible enumeration) lies in the fact
that they were reformers. Thus from an enumerative judgement
we may pass to an universal ; from a study of individuals to the
assertion of an universal connexion of characters. When we enun-
ciate enumerative judgements, we are on that road : sometimes
farther, and sometimes less far.

The difference between a true universal judgement and one
merely enumerative is exceedingly important. The one belongs
to science, the other to chronicle or history. An universal judge-
ment concerns any and every instance, alike past, present and
future, examined or unexamined. An enumerative judgement
concerns only those instances which have been examined, or have
existed, and which are summed up in the subject. All reformers
are hated : if that is merely enumerative, it does not require me to
1 Cf . Bradley, Principles of Logic, Bk. I. c. ii. § 45.

vin] VARIOUS FORMS OF THE JUDGEMENT 179

anticipate hatred if I undertake reform ; it affords me no explana-
tion of the hatred with which these men have been met. But if it is
a true universal, it explains the past, and predicts the future.
Nevertheless an universal judgement has nothing, as such, to do with
numbers of instances ; if the connexion affirmed in it be necessary,
the judgement is still universal, whether there be a million instances
of its truth, or only one 1 ; so that the form ' All A is B ' hardly does
justice to it. An enumerative judgement contemplates a number
of instances, and refers to all of them ; and the form ' All A is B *
or ' All the A's are B ' expresses it adequately.

The particular proposition may be interpreted as referring either
to individuals not enumerated or to an universal not fully deter-
mined ; and it will approximate more to the enumerative, or more
to the universal, accordingly. If I say Some women have ruled
kingdoms, I mean women whom I could enumerate — Semiramis,
Cleopatra, Zenobia, Elizabeth, Christina, &c. : not women of such
and such a type, but this and that woman. If I say Some pigments
fade, I do not mean pigments that I could enumerate, but any pig-
ments of a certain kind ; and supposing that I could specify or
determine the character of pigment, I could say that all pigments
of that character fade. There is nothing in the verbal form of a
particular proposition to show whether the speaker is thinking
rather of individuals whom he does not name, or of conditions
which he does not specify ; though content and context will often
guide us on this point.

It will be readily seen that there is the same sort of difference
between the particular proposition interpreted of individuals not
enumerated, and the particular proposition interpreted of conditions
not fully specified, as exists between the enumerative and the true
universal proposition. If the women vaguely referred to as some were
enumerated, I could say All the women on my list have ruled kingdoms ;
if the pigments vaguely referred to as some were characterized,
I could say All such pigments fade. The former is the enumerative,
the latter the universal All. And this difference, whether between
the two interpretations of the particular proposition, or between the
enumerative and the universal, may be expressed by saying that in

1 Or, as some logicians would add, none. Such a view makes the universal
judgement, however, purely hypothetical : cf. Leibniz, Nouveaux Essais,
IV. xi. 14 ; Bradley, Principles of Logic, Bk. I. c. ii. §§ 43-6 ; Bosanquet,
Logic2, vol. i. pp. 263-266; v. also Bradley, Appearance and Reality, p. 361.

N2

180 AN INTRODUCTION TO LOGIC [chap.

the one case the subject of the proposition is interpreted in extension
or denotation, in the other case in intension. The subject of a pro-
position is interpreted in extension or denotation, when we are
thinking primarily of the various species or individuals included in
the subject to which the predicate refers ; it is interpreted in intension,
when we are thinking primarily of the subject as of a certain kind,
of the character implied by the subject-term, with which the pre-
dicate is connected. ' Some A is B ' is interpreted in extension or
denotation, if I think of this, that and the other A : in intension, if
I think of ^L's of a certain character. ' All A is B ' is interpreted
in extension or denotation, if I think of every one of the A's : in
intension, if I think of the character of ^4's as such.

What has been said on the quantity of judgements and propositions
may be summed up as follows. Categorical judgements may be
made about either individuals or universals. If about individuals,
these may be indicated either by a proper name or designation —
and then the judgement is called singular — or by a general term.
In the latter case, if the judgement concerns all that is included in
the extension or denotation of its subject-term, it is called universal ;
but a distinction must be made between a true universal judgement,
in which the predicate is affirmed (or denied), without respect of
individuals, of whatever exhibits the subject-concept (or intension
of the subject -term) and one only enumerative or collective, in
which it is affirmed or denied of all of certain species or individuals,
which might be enumerated, but which the subject-term enables us
to indicate collectively. If the judgement concerns an unspecified
part of the extension or denotation of the subject-term, it is called
particular. Judgements about an universal are universal. Pro-
positions are denominated after the character of the judgement
which they express. A true universal judgement can be indicated
by the same words (All and None) as an enumerative, and is often
confused with it. A particular judgement is really incomplete ; it
may be an incomplete enumerative, or an incomplete universal
judgement, according as we think rather of the instances we im-
perfectly denote, or the conditions we imperfectly specify, in the
subject. We make particular judgements chiefly in opposition to
the universal statements of others, to which we ' take exception ',
or in approach towards universal judgements ; and their value
for knowledge is subsidiary and instrumental. The subject of
a categorical proposition may be viewed primarily in intension.

viii] VARIOUS FORMS OF THE JUDGEMENT 181

if the proposition affirms or denies a connexion of characters,
or in extension or denotation, if it affirms or denies a certain
character in individuals.1 The former aspect predominates in
the universal, the latter in the enumerative : in the particular,
sometimes the former and sometimes the latter, according as we
think more of the conditions imperfectly specified, or the instances
imperfectly denoted : the singular proposition merely affirms or
denies in an individual a certain character.1 Sometimes these
distinctions, though we are conscious of them in our thought, are
not expressed in language ; and for certain purposes of inference,
it is enough to consider propositions simply as either universal or
particular : universal, when the whole extension or denotation of
the subject-term or when an individual is referred to, particular
when a part of the extension or denotation is referred to only.

Judgements are distinguished according to relation into categorical,
hypothetical, and disjunctive. We have been considering hitherto
categorical judgements. A categorical judgement merely alfirms
or denies a predicate of a subject : dogs bark, dead men tell no tales.
An hypothetical judgement connects a consequent with a condition
which it does not, however, assert to be fulfilled : if money is scarce,
the rate of discount rises. The condition is called sometimes the
antecedent (in grammar, the protasis), as what is connected with
it is called the consequent (in grammar, the apodosis). A disjunc-
tive judgement affirms alternatives : bees are either male, female, or
neuter.2 The hypothetical judgement is sometimes called con-
junctive, as conjoining the truth of the consequent with that of the
antecedent : while the disjunctive disjoins the truth of one alter-
native from that of the others.3 Both are sometimes called com-
plex judgements, in contrast with the categorical, which is called
simple.

In an hypothetical judgement, the antecedent and consequent
may have the same, or different, subjects or predicates : the scheme
of the proposition may be either ' If A is B, it is C ' (If corn is scarce,
it is dear), or ' If A is B, C is D ' and (// the dead rise not, we are of

1 The singular proposition whose predicate is a proper name does not
assert a character of its subject.

2 For any given bee, these are alternatives : for bees collectively, they are
three forms which are all realized : cf. p. 188.

3 The term hypothetical has also been used by some generically, with con-
junctive and disjunctive to denote the species.

182 AN INTRODUCTION TO LOGIC [chap.

all men most miserable), or ' If A is C, B is C ' (Si tu et Tullia valetis,
ego valeo). Again, antecedent and consequent may be either nega-
tive or affirmative : but these differences make no difference to the
character of the judgement as hypothetical : it still affirms the
dependence of a consequent on a condition : hence the alternative
of affirmative and negative, though applying to the antecedent and
consequent severally, does not apply to the hypothetical judge-
ment as a whole.

It is sometimes said that hypothetical propositions can be reduced
to categorical. So far as the verbal form is concerned, this can often
be done. ' If A is B, it is C ', in which antecedent and consequent
have the same subject, may be written ' A that is B is C ' ; If com
is scarce, it is dear becomes Scarce corn is dear : If that dog is teazed,
he bites becomes That dog bites when teazed. Even where antecedent
and consequent have different subjects, a little ingenuity may
produce from the hypothetical a proposition categorical in verbal
form ; If no war is toward, the temple of Janus is closed might be
written The temple of Janus is closed in peace time : If men are not
free, blame is unjust might be written Men who are not free are not
justly blamed. But whether a judgement is hypothetical or cate-
gorical cannot be determined merely from the verbal form of the
proposition in which it is expressed. The hypothetical judgement
asserts the qualification of the subject by the predicate of the
consequent to be dependent on a condition expressed in the ante-
cedent ; as has been said, it does not assert this condition to be
fulfilled. But where this condition is known to be at times fulfilled,
it may still be expressed by an // (as well as by a When or Whenever) ;
and if it can also be expressed as a qualification of the subject or
predicate of the consequent, then that predicate may be asserted
of the subject so qualified, or the predicate so qualified of that
subject. Now the first three of the above examples are of this sort.
Corn is sometimes scarce, that dog is sometimes teazed, Rome i3
sometimes (though rarely) free from war. And the proposition
// corn is scarce, it is dear, regarded as a statement about scarce corn,
must be ranked as categorical, notwithstanding its form ; while
Scarce corn is dear, regarded as a statement about corn whose scarcity
it leaves in doubt, must be regarded as hypothetical. We are so
well aware that corn is from time to time scarce, that we naturally
interpret categorically in this instance. But in an argument con-
taining the proposition Men who are not free are not justly blamed we

vni] VARIOUS FORMS OF THE JUDGEMENT 183

might see that it was not intended to deny that all men are free,
but only to point out a consequence that would follow from denying
it ; and then the judgement remains hypothetical. Otherwise, and
if taken as implying the existence of men not free, the categorical
proposition is not the equivalent of an hypothetical in which their
existence is not implied.1 The reduction to categorical form is only
justified when the hypothetical proposition is meant merely to
affirm a connexion of one character with another existing in a given
subject or in all subjects of a certain kind.

The difference between the two types of judgement — between
affirming or denying a predicate of a subject, and affirming the de-
pendence of a consequent on a condition not asserted to be fulfilled —
becomes clear where the judgement concerns an individual situation,
and particularly if it contains an unfulfilled condition, in past or future
time, i/ he is insane, he cannot make a will implies, no doubt, a con-
nexion between insanity and testamentary incapacity, but not be-
tween the individual and either. // / had served God as diligently as
I have done the king, He would not have given me over in my grey hairs :
no doubt this implies the categorical judgement God does not forsake
those who serve Him diligently ; but it cannot be reduced to this ; for
it implies also Therefore He would not have forsaken me, if I had served
Him diligently ; and we cannot ehminate this hypothetical judge-
ment. Kpotaos "Akvv 8ta/3as y.eyahrjv apyj]v /caraAucrei, If Croesus
crosses the Halys, he will ruin a great power2 ; here it is not stated
whether Croesus will cross the river or not ; so that, as the fulfilment
of the condition upon which the event in the consequent depends is
left in doubt, there is nothing but a dependence categorically asserted.

It may be urged that, as this at least is asserted categorically, the
hypothetical judgement is categorical after all. And against any
one who attempts to abolish the distinction between the two kinds
of judgement by saying that all judgements are really hypothetical,
it is a good answer to point out that the hypothetical thus involves
the categorical. But that does not invalidate the distinction between
them ; for the distinction rests on the difference between asserting

1 The form ' Men who are not free would not be justly blamed ' retains in
the would the expression of hypothetical judgement, and cannot be regarded
as a categorical proposition.

2 More literally, Croesus by crossing the Halys will ruin a great power, which
might be taken to mean that Croesus will cross the Halys and ruin a great
power. So taken, the oracle is categorical ; and the line well illustrates how
the grammatical form is no sure guide to the logical character.

184 AN INTRODUCTION TO LOGIC [chap.

a dependence of consequent upon condition not asserted to be
realized, and asserting (affirming or denying) a predicate of a sub-
ject. If it be granted that the hypothetical judgement asserts the
former, though it does so categorically, yet it differs from the
categorical.

It has been said x that the very reason just given for maintaining
the essential difference of these two types of judgement excludes the
consideration of that difference from Logic. For both assert ; they
differ in what they assert ; the difference is therefore in the matter
and not the form of judgement. We have the same form, A is B,
whether for A we write Croesus, and for B a king of Lydia, or for A
the destruction of a great power, and for B must follow on Croesus
crossing the Halys. But it will be readily admitted that the dis-
tinction between categorical and hypothetical assertion is formal
in the sense that it is illustrated in our thought about all kinds of
subjects ; and to exclude it from Logic on the ground that, as com-
pared with the common form of assertion in both, it is material, only
shows the impossibility of making Logic a purely formal science.
It is claiming to consider the genus, and refusing to consider the
species : a procedure which would be tolerated in no other subject,
and cannot be tolerated in Logic.

[There is however a difficulty about the meaning of saying that
a consequent depends on a condition, when that condition is unful-
filled. // Hannibal had marched on Rome after Cannae, he would
have taken it. This proposition makes an assertion ; in doing so,
it asserts something about the real, for it claims to be true. But
what does it assert about the real, and what historical fact (as we
may put it in such an instance) does it affirm ? Not that Hannibal
marched on Rome after Cannae, for he did not ; nor that he took
Rome, for he did not ; nor therefore that one event was due to
the other, for neither happened. How then can we say that one
depended on the other ? In the sense, it may be answered, that
if he had marched on Rome at that time, he would have taken it.
But this is the original proposition whose meaning we are trying
to discover. And it does not state a fact in Hannibal's history,
or in the history of Rome, but what is called an unfulfilled con-
tingency ; and how can that be asserted of the real ? Every
hypothetical judgement presents this problem. For its truth does
not require that either condition or consequent be realized, and
yet, if true, it is true of reality ; and reality, we may urge, is
actual. What then does it affirm to be actual in the real ? Mr F. H.

1 Cf. Mansel, Prolegomena Logica, pp. 232, 251.

vm] VARIOUS FORMS OF THE JUDGEMENT 185

[Bradley x replies that it ascribes to reality a character which is the
ground of the connexion stated in the hypothetical judgement.
Rome was in such a state that it could not have resisted Hannibal
after Cannae. This is true ; but it still leaves us with the question,
How can there be the ground in the real universe of something
which nevertheless does not happen ? Or we may put the problem
a little differently by asking how there can be a dependence between
a consequent and a condition that do not exist.

Professor Cook Wilson holds that an hypothetical proposition
affirms the dependence of the solution of one problem upon the
solution of another. ' And they sent the coat of many colours,
and they brought it to their father ; and said, This have we found :
know now whether it be thy son's coat or no. And he knew it,
and said, It is my son's coat ; an evil beast hath devoured him ;
Joseph is without doubt rent in pieces.' 2 Here the hypothetical
proposition is imiDlied, If this is thy son's coat, he has been killed ;
and this means that the determination in the affirmative 3 of the
question whether it is his coat involves the like determination of
the question whether he has been killed. Where the condition is
an unfulfilled condition in past time, this dependence of the solution
of one question on that of another is affirmed not to exist now, but
to have existed. The determination of the question whether Rome
would fall directly after Cannae did depend on that of the question
whether Hannibal would march on Rome. And such a proposition
implies also the assertion that the condition was not fulfilled.

What Professor Cook Wilson points out seems true and important.
An hypothetical proposition does assert the solution of one problem
to be connected with the solution of another. But the particular
difficulty before us is not removed by that doctrine. For neither
problem need be solved or by us soluble, and yet the hypothetical
proposition may be true, as in the instance // the ' Phaedo ' is historical,
Socrates believed in the immortality of the soul. Now our difficulty
concerned the affirmation that a ground exists for what yet does
not happen, or that there is a connexion between terms which yet
do not exist. And there is the same difficulty in asserting one
solution to depend on another, when neither is made. It concerns
the meaning of saying that something is possible, which is not
actual.

It is a partial answer to say that connexions, or principles of
connexion, exist in a different way from the particular things and
events in which they are displayed or illustrated. They are, even
when they are not exemplified. ' If you put a match to that powder,

1 Principles of Logic, Bk. I. c. ii. §§ 50-53. 2 Gen. xxxvii. 32, 33.

8 The determination in the negative of the question in the antecedent
leaves the question in the consequent undetermined, unless the antecedent
is the sole condition of the consequent. Cf. infra, p. 33b.

186 AN INTRODUCTION TO LOGIC [chap.

[it will explode.' Why ? because of a connexion between rise of
temperature and detonation in compounds of a certain kind, which
is not dependent for its being upon the actual process of those
changes. The explosion then is possible, because there are certain
connexions, and some of the things, given which these connexions
are exemplified, exist. These connexions, which somehow are,
even though the conditions for their display do not exist, are the
' ground ' which the hypothetical judgement ' affirms of the
real ' But the connexions whose affirmation is implied need
not be such as are repeatedly illustrated. Sometimes the reason
why the solution of one problem carries the solution of another
lies in principles of connexion displayed in situations that are
repeated ; so it is with the connexion between rise of temperature
and detonation in gunpowder. But sometimes the conditions are
apparently unique, and we cannot resolve them into an assemblage
of repeatable elements ; so it is often with complex historical
situations where all seems to turn on the action of a great per-
sonality. And in some hypothetical propositions the connexion
between the solutions of two problems seems to be the only con-
nexion affirmed, as in the instance already given, ' If the Phaedo
is historical, Socrates believed in the immortality of the soul ' ;
though doubtless this implies an assertion of certain particular facts
about that dialogue of Plato.1]

The disjunctive judgement may be expressed schematically in the
forms ' A is either B or C ' (Every man at forty is either a fool or
a physician), * Either A is B or C is D ' (He either fears his fate too
much, Or his desert is small,2 Who dares not put it to the touch, To
gain or lose it all), ' Either A or B is C ' (Either the Pope or the King
of Italy should retire from Rome). As the hypothetical judgement
always affirms a dependence of consequent on condition, so this

1 The reader must not suppose that these paragraphs deal at all completely
with the problems raised by hypothetical judgement. Nothing, for example,
has been said about distinctions of quantity in them. It has been urged
by some that they are all universal. But though without necessary con-
nexion in the real no hypothetical judgement would be true, such implied
connexion may be remote from the actual hypothetical judgement made.
Again, some hypothetical judgements are concerned with certain individual
consequents and conditions, some with any of a certain kind ; or the
condition may be of the former sort and the consequent of the latter, or
vice versa. These differences however are not of first-rate importance.

2 This might be equally expressed ' He either fears his fate too much, or
deserves little ' : indeed in sense the alternative predicates are predicated of
the same subject, not (as in the proposition Either Tacitus was a slanderer
or Tiberius a villain) of different subjects. This affords another example of
the fact that the logical character of a judgement cannot always be inferred
from the grammatical form of the proposition.

vhi] VARIOUS FORMS OF THE JUDGEMENT 187

always affirms a disjunction, whether the alternatives themselves
be given affirmatively or negatively. So far as the nature of the
disjunction goes, there is no difference between ' A is either B or G ',
and ' A is either not B or not G ' : between ' Either A is B, or C is
D ', and ' Either A is not B, or C is not D ' : between ' Either A or B
is G ', and ' Either A or B is not G \ But it should be noted that
' Neither . . . nor ' is no disjunction at all, but a conjunction of
negations. On St. Paul's voyage to Rome ' neither sun nor stars in
many days appeared. ' ; there is no choice between alternatives here,
but two statements — the sun did not appear, and the stars also
did not.

There may be any number of alternatives in the disjunction ; but
that clearly does not alter the character of the judgement.

It is not always clear in a disjunctive proposition whether the
alternatives offered are meant to be mutually exclusive. If A is
either B or G, then it cannot be neither ; but may it be both ? The
question concerns the right interpretation of a form of speech, rather
than the nature of disjunctive judgement. Sometimes from the
nature of the case we may know that the alternatives exclude each
other : as if we are told that Plato was born either in 429 or 427 b. c.
Where this is not so, it is perhaps safer to assume that they are
intended as mutually exclusive, unless the contrary is stated ; a legal
document is careful so to write it, where ' A or B or both ' is meant,
or to write ' A and/or B ' with that signification.

If has been suggested that the disjunctive judgement is in reality
a combination of hypotheticals ; that ' A is either B or C ' means
' If A is not B, it is G ; if A is not C, it is B ; if A is B, it is not G ;
if A is C, it is not B '. Doubtless these four propositions are in-
volved (supposing B and C to exclude each other) : but we do not
therefore get rid of the peculiar nature of the disjunctive judge-
ment. For they are not four independent hypothetical judge-
ments ; and their force is not appreciated, unless it is seen that
together they make up a disjunction, that they offer us a choice
between alternatives. Thus disjunctive judgement at once includes
and goes beyond hypothetical, in the same sort of way as hypothetical
judgement includes and goes beyond categorical. An hypothetical
proposition makes an assertion, like a categorical ; but what it
asserts is a relation of a consequent to a condition. A disjunctive
proposition involves hypotheticals, which it presents as true together,
but it asserts the truth of one (or, if they are not mutually exclusive,

188 AN INTRODUCTION TO LOGIC [chap.

of at least one), without specifying which one, among alternative
caiegoricals.

The disjunctive judgement also raises a metaphysical problem,
when we ask what real fact corresponds to it. ' Plato was born
either in 429 or 427 b. o.' cannot state the actual fact about Plato :
he was born definitely in one year, not merely in one or other ;
it is because we do not know in which, that we state an alternative,
and there was no alternative in the event. Here, therefore, the
disjunctive proposition seems rather to express the state of our
knowledge, than the state of the facts. On the other hand ' Number
is either odd or even ' seems to express a disjunction in the facts x ;
and the species of the same genus are a kind of real disjunction.
If a colour is to exist, it must be blue, or red, or some other colour,
and if it is one, it can be none of the others. We come back here
upon the same truth which met us in considering negative judge-
ments, that a thing is definitely this or that by not being some-
thing else ; we have to recognize also that there is often a limited
number of possibilities, in the way, for example, of colour, or of
animal species, but why or how there should be a limit to what is
possible in the universe is a hard question.2

We come next to the distinctions of modality in the judgement.
In respect of modality, categorical judgements are distinguished as
assertoric, problematic, and apodeictic (or necessary) ; the first is
sometimes opposed as pure to the other two as modal ; but we shall
find that if judgements are divided into pure and modal, the assertoric
can be regarded as a form of modal judgement. Propositions of
the form ' X is Y ', ' X is not Y ' are assertoric — ' the train is late ',
' the train is not late ' ; of the form ' X may be Y ', ' X may not
be Y ', problematic — ' the train may be late ', ' the train may not
be late ' ; of the form ' X must be Y ', ' X cannot be Y ', apodeictic
— ' the train must be late ', ' the sun cannot be late '. The dis-
tinctions are also expressed by adverbs : X actually, possibly,
necessarily is (or is not) Y.

In the sense of the word to which we have so often called atten-
tion, these distinctions are clearly logical : i. e. they belong to no

1 Of course there is a disjunction in the facts, in the former case as well,
bo far as that the 429th and the 427th years from any point of time whence
we choose to begin our reckoning are distinct years.

2 For the fuller treatment of this form of judgement also the reader is
referred to more advanced works.

vml VARIOUS FORMS OF THE JUDGEMENT 189

special science, but recur in our thought about all kinds of subject.
Whatever X and Y may be1, we may find ourselves asserting that X
is, that it may be, or that it must be Y.2 But their logical character
is specially manifest in this, that they raise a fundamental question
about the nature of the thinking activity, viz. that of the difference
between opinion and knowledge3, just as the distinction of judge-
ments according to quality raises the question of the difference
between affirming and denying. And as the latter difference cannot
be reduced to a difference in the predicate affirmed, by combining
the negative with the predicate, so neither can the former. Still,
we found a ground for the existence of the two ' qualities ' of judge-
ment in a certain fact about the being of things, viz. that each is
positively what it is by exclusion of all else, by difference. It i3
not so easy to find a ground for the existence of the ' modalities '
of judgement in the being of things.

Let us take three judgements differing in modality and expressed
in propositions of the form ' X is Y ', ' X may be Y ', * X must be
Y ' — ' the train is late ', ' the train may be late ', ' the train must be
late \ We can express the same judgements by saying that the
train is actually, or possibly, or necessarily late. But it is clear
that we have not here three judgements with the same subject, the
train, and different predicates, actually late, possibly late, necessarily
late ; for those are not three kinds of lateness. The modality of
a judgement cannot be something qualifying its predicate. ' Nor-
man mouldings were possibly coloured ' : ' Norman mouldings were
actually coloured ' ; the adverbs do not express a mode of colouring,
as if we said that the mouldings were brilliantly coloured, or coloured
blue. ' Water runs down hill ' : ' water must run down hill ' ;
these are not different ways of running, like running fast or run-
ning slowly. Grammarians tell us that adverbs qualify verbs and
adjectives, but adverbs of modality seem to be an exception.4

1 Except so far as in some subjects, like arithmetic, a judgement is nearly
always made with consciousness of its necessity : cf. infra, p. 196. Even
here however I might say, before I had made the calculation, that 37596
may be a square number.

2 For the sake of brevity, I shall not throughout consider negative as well
as affirmative judgements. It should be noted that the problematic affir-
mative ' X may be Y ' is not contradicted by the problematic negative
' X mav not be Y \ but by the apodeictic ' X cannot be Y ' : and similarly
the problematic negative by the apodeictic affirmative.

3 Cf. p. 160, supra.

* Unless indeed they qualify the copula, the verb to be, as some have said.
Cf. next page.

190 AN INTRODUCTION TO LOGIC [chap.

Again, it is not the judgement, in the sense of the act of judging,
that the modal words qualify ; if I judge ' the train may be late ',
my judging is actual ; it is the lateness of the train that is possible.
That, however, as we have just seen, does not mean that its lateness
is a certain sort of lateness, as if we said that the lateness of the
train is scandalous.

Once more, we cannot say that the modal words qualify the matter
judged. I judge that the train may be late, or that the window
may be open ; the judgements have the same form, which I can
express symbolically in the formula ' that X is Y is possible ' ;
the assertoric and apodeictic may be similarly expressed — ' that
X is Y is actual ', or ' necessary ' ; or more compendiously, instead
of the words ' that X is Y ' I can write '17'. But X Y is certainly
not the matter judged ; for when I judge it possible that the train
is late, I do not judge X Y, that the train is late, at all. The matter
judged is that which is judged to be, the subject qualified by the
predicate. X and Y are not the subject and predicate in these
three judgements, as indeed the formulae in which the modal words
are predicates indicate. The affirmative and negative judgements,
1 X is Y ' and ' X is not Y ', can have the same subject and pre-
dicate, but differ in quality ; so we are apt to speak as if the
assertoric, problematic, and apodeictic judgements could have the
same subject and predicate, but differ in modality. The analogy
is false. The true analogy is rather this, that as in a negative
judgement the matter judged is ' that X is not Y ', and is therefore
different from the matter judged in the affirmative, so in the
modal judgements the matter judged is that X is actually, or pos-
sibly, or necessarily Y. But here what actually, possibly, or
necessarily is, as there what is not, is said to be. Hence as Plato
asked what is meant by saying that not being is, so we must ask
what is meant by saying that possible being, or actual or necessary
being, is.

To ask this is the same as to ask whether modality can qualify
the copula. We use the verb to be as the sign of judgement, because
the predicate expresses some further being of the subject than is
expressed already by the subject-term. I look up and say ' the
window is open ', because that is of the being of the window. But
whatever the window is, it is actually, and not possibly ; and
perhaps what it is actually, it is necessarily. If so, what is possible
being, and how can we distinguish actual from necessary being ?

vih] VARIOUS FORMS OF THE JUDGEMENT 191

The modal words cannot indicate different ways in which X is Y,1
any more than differences in Y. What then do we mean by them,
and why do we use them ?

We use them to mark the distinction between knowledge and
opinion, and the differences in the certainty with which we hold an
opinion. This is not a complete answer, because the modal words
are used in divers senses ; but the difference in the modality of
'udgements is the difference between knowledge and opinion, and
between certainty and uncertainty in opining ; and so far as these
words are marks of modality, they mark that. It is no objection
to this view, but rather a confirmation of it, that men often use the
modal forms expressive of knowledge or certainty, when they do not
really know, or are not certain. They may assume a virtue, if they
have it not ; and unless these forms had such meanings, there would
be no motive to use them. But we must turn to a closer examina-
tion of their use.

In the history of thought the assertoric form, 'X is Y\ seems to
come first. Certainty or conviction precedes doubt, and precedes
the reflective consciousness of knowledge. What Bain called
primitive credulity cannot make us know, but it can make us assert.
Our early assertions, however, are made without reflection ; we
do not ask whether they are consistent with others that we have
made, or whether it is possible to doubt them. When we ask such
questions, we may find that different assertions which we have made
are inconsistent, and that they cannot all be true, though we do not
know which are false ; or we realize that we can doubt one, but
not another. Our assertoric thinking is thus displaced by problem-
atic thinking, or by necessary thinking — i. e. the apprehension of
necessity, or knowledge.

But the assertoric proposition itself, ' X is Y ', may express two
different mental attitudes.2 We may hold and express an opinion
without doubt before question has been raised ; after question has
been raised, we may still hold and reassert the opinion as confidently
as before, although we have not been able to prove or see into the
necessity of the fact asserted. There are several kinds of example
that may be given of this. It occurs in regard to sensible facts.

1 Or is not. I have not complicated the discussion by taking also negative
examples.

2 We shall see that it h also often used where the judgement expressed is
apodeictic : v. infra, p. 196.

192 AN INTRODUCTION TO LOGIC [chap.

A man walking up Eskdale in a fog and having lost his way says
that he hears Cam Spout ; if challenged, he may listen again, and
say that he is sure he hears the sound. Or his opinion is asked about
a proposed act, and he condemns it ; another dissents and asks his
reasons ; and he replies that he cannot give any reasons, but is sure
that the act is wrong. Or again (and we shall find this very common
in the inductive sciences) we assert as a fact something which we
cannot explain or understand, because we have had experience of
events that seem only explicable if it is true. Some men detect water
with the divining-rod. That is very extraordinary ; how do you
account for it. I can't, but they detect it. Here the assertoric judge-
ment is challenged ; events are recalled which seem inexplicable
unless there is this power ; and so it is reasserted. On the second
assertion, the word detect would be emphasized in speech ; or
the emphasis could be given in writing by the words ' they do
detect it ', or ' they actually detect it ' ; and language has other
idioms for expressing this assertoric confidence.

The difference between the two mental attitudes just noted lies in
this, that whereas in both we feel confident, in the former this con-
fidence is unreflecting, in the latter it is felt in the face of suggested
doubt, and so is reflective. It might perhaps be best to call a
judgement pure, rather than modal, which is made without any
reflection upon the question of its truth ; and assertoric when, upon
reflection, we can give no proof of it, nor see the necessity of the fact
asserted, but are confident of it. The word actually would mark
a judgement as assertoric, not pure ; but the ordinary categorical
form can also express it ; we are considering the nature of the acts
of judgement, but can only contemplate these by the help of propo-
sitional forms.

A consideration of the problematic and apodeictic judgements
will throw further light upon the assertoric. When an opinion is
challenged, we commonly try to justify it by producing grounds for
it, though we cannot always do this, and our pure judgements,
as just observed, are apt to be displaced by problematic or apo-
deictic. The apodeictic may be taken first ; it is a judgement made
with a consciousness of the necessity of the fact asserted. But we
often use the apodeictic form of proposition, ' X must, or cannot,
be Y* ( ' X necessarily is, or is not, Y '), when we do not apprehend
a necessary connexion between X and Y ; and there are two classes
of case to be distinguished when we do apprehend it, viz. those in

vm] VARIOUS FORMS OF THE JUDGEMENT 193

which we need, and those in which we do not need, in order to see
the connexion, to look beyond the content of the judgement X Y.1
Both are important, because in both we have knowledge.

A boy may believe and assert, because he has been taught it,
or because he remembers to have seen no others, that all lines are
either straight or curved ; if the assertion is questioned — it matters
not whether the question comes from himself or another — and he
asks himself what ground he has for making it, he will realize that
it belongs to the nature of linearity that every line must be straight
or curved. Put symbolically, the ground for the judgement ' X is
T ' is seen to lie within the nature of X.2 We call such a judgement
self-evident. There are self-evidently necessary negative judge-
ments, as well as affirmative, e. g. ' the difference between two
degrees of quality is not a quality '.3 What is self-evident need not
be evident at once, or to everybody ; the intelligible is intelligible
only to the intelligent. In calling anything self-evident we mean not
that it is evident without need for understanding, but that we need
consider nothing but the terms of the judgement, to see its necessity.

[Logicians of two different schools have denied the existence of
the self-evident. The one school are the Empiricists, who, rightly
insisting that there is no knowledge without experience, wrongly
suppose that we cannot by thinking discover the nature of anything
that we have not perceived. The child learns the multiplication-
table by counting marbles, or what not ; but it comes to understand
that the equality of two groups severally of 3 and 4 marbles to
two severally of 5 and 2 marbles is independent of the units being
marbles, or the day Monday, or the place London, or itself the
person counting — that 3+4=5+2 universally; nor does it need,
nor could its judgement be increased in certainty by, experimenta-

1 We may symbolize thus the categorical propositions whose subject and
predicate are X and Y, and which are so far ' materially ' the same, but
whose ' formal ' character — modality, quality, quantity — may differ ; remem-
bering however that in the problematic proposition ' X may be Y ', X and
Y are not the terms of any judgement made, but of a suggested judgement
which is not made. Cf. supra, p. 190, and infra, pp. 196-197.

2 In Aristotle's language, the predicate belongs to the subject icad' alro,
or per se — in virtue of itself.

3 A gallon and a quart are two quantities. I can take a quart from a gallon
of water, and I shall have a certain quantity (three quarts) left. The difference
between two quantities is in this sense a quantity. But suppose two qualities
differing in degree, say a darker and a lighter blue, or a more and a less intense
pain : it is meaningless to say that the quality of lower degree can be takeD
from that of higher, and leave another quality which is the difference of
those degrees. This self-evident fact has an important bearing on the so-called
calculus of pleasures and pains.

1779 O

194 AN INTRODUCTION TO LOGIC [chap

[tion with further particulars ; and from henceforth it sees this
principle to be as true for countable things of which it has not had
experience as for those of which it has. It has thus obtained by
thinking knowledge about things of which it has had no experience
(though it could not have done so without some experience of
countable things). The Empiricist, however, denies this, and holds
that the proposition '3+4=5+2' is a mere generalization from
experience, entertained so confidently not because it is seen to be
necessary, but because it is verified in so many instances. He is
however herein using an argument — ' because this equation holds
good in so large a number of examined instances, therefore it holds
good in the unexamined \ Either the conclusion of this argument
follows necessarily from the premise, or it does not. If it does not
(and in fact it does not), he cannot justify our confidence in any
process of arithmetical thinking ; if we have put 3 shillings into
an empty purse, and then 4, and have taken out 2, we ought not
to say there are 5 left until we look, or to be surprised if we find
more or fewer. If, on the other hand, the conclusion does follow
necessarily from the premise, then here at least is an instance of
our discovering by thinking a fact about things which we have
not learnt by experience. Empiricism breaks down over the validity
of inference ; if it allows that, it gives away its case ; if it dis-
allows it, it cannot argue.

The other objection to self-evident truth is more serious. It is
said that all things are interconnected ; that their relations to each
other are not 'external ', i.e. that relations cannot change without
a change in the nature of the things related, being really an
expression of their nature ; that we cannot know anything in all
its relations, and that a predicate Y which we ascribe to a subject
X must be conditioned by what we do not know of the subject
as well as by that which we have indicated of its nature when we
call it X. Even so simple a fact as that 2+2=4 is part of the
whole system of numerical relations ; it could not still remain, if
per impossibile other numerical relations were different from what
they are, e.g. if 2 +3 =7 and 314 +228 =56. As it is connected
with all these, it can only be fully known through them, and when
seen in its relations to the rest of the system. No particular truth,
on this view, is absolutely true ; only in apprehending everything
could we know anything as it is. This doctrine is maintained
forcibly, and even scornfully, by Mr. F. H. Bradley (e. g. Essays on
Truth and Reality, c. vii) and by Mr. H. H. Joachim in his book On
the Nature of Truth, c. iii ; its most famous advocate in modern times
is Hegel. Something like it was known to Aristotle, and criticized
by him in the Posterior Analytics (/J. xiii. 97a 6-22). l It rests on meta-

1 Professor G. F. Stout has put well the argument against it, in the first
Essay in Personal Idealism (on Error).

vm] VARIOUS FORMS OF THE JUDGEMENT 195

[physical considerations which cannot be lightly dismissed ; and
they might seem to require us to deny that any truth is st7/-evident,
because nothing can be understood except through the whole. But,
even if they are sound, we must still acknowledge that we make
some judgements with a consciousness of their necessity, and others
not ; we cannot abolish the distinction between knowledge and
opinion. Such transformation as complete knowledge would effect
in the thought expressed by a self-evident proposition must be of
the same nature as occurs when a geometrician comes to realize
that a proposition which he has demonstrated about one species
of figure is but a special case of a much wider proposition capable
of a general proof. His original proposition is not thereby shown
to be false, though his insight into the facts was incomplete.]

More often, however, when we use the apodeictic form of pro-
position, the fact asserted in seen to be grounded in some other fact
or facts, not stated in that proposition, which we should assign as
a reason for it. Water must rise in the common pump, when the
piston is raised : why must ? because of the pressure of the atmo-
sphere. Mere observation would lead us to affirm assertorically
that it does rise ; it is the consciousness of the connexion of that
fact with the pressure of the atmosphere in a machine constructed
as a pump is, that makes us affirm it apodeictically. But are we
sure, it may be asked, that the atmosphere must have weight ?
for unless we are, we can only assert that the water must rise if and
when the atmosphere has weight. We cannot here discuss the
grounds on which we regard the general principles of science as
established ; that belongs to a consideration of inductive reasoning.
But two things are clear : first, that if the grounds of the judgement
X Y can only be affirmed assertorically, X Y itself is necessary only
upon the condition that those grounds are as we assert ; secondly,
that, even so, the connexion between those grounds and the fact XY
may be seen to be necessary. We may call the necessity of a judge-
ment, which we see to follow from certain grounds, but whose grounds
we cannot affirm apodeictically, an hypothetical necessity ; when
the grounds can be affirmed necessarily, then perhaps we may
say that the judgement is apodeictically necessary. Thus, if two
straight lines falling on another straight line and making the internal
angles on the same side of it together equal to two right angles can
never meet, the angles of any triangle, however large or small, are
equal to two right angles ; to one who regards the axiom of parallels
as self-evident, the judgement that the angles of a triangle are equal

02

196 AN INTRODUCTION TO LOGIC [chap,

to two right angles will appear apodeictically necessary ; to one
who does not, hypothetically.

It will be seen from this, that there is a close connexion between
the hypothetical and the apodeictic judgement. But we cannot
say that all hypothetical propositions are apodeictic, for we often
use them when we do not see the consequent to be necessarily
involved with the antecedent, e.g. a public speaker says that if
a certain measure is carried, certain results will ensue. This is only
another illustration of the fact that a propositional form which ia
intended to express a certain kind of judgement may be used when
we do not really make the judgement which it should express. Every
deliberate falsehood illustrates this, and every false apodeictic
proposition, whether deliberately intended to deceive or not. And
it often happens that the assertoric form of proposition is used to
express necessary truth, the apodeictic to express doubtful opinion.
In mathematics every step is seen by the mathematician to be
necessary 1> insomuch that it is often summarily said that mathe-
matics deals with ' necessary matter '. There is consequently no
need to distinguish apodeictic from other judgements in mathematics,
and they are all, as a rule, expressed assertorically ; we say '2x2
is 4 ', not '2x2 must be 4 ' : ' the interior angles of a triangle are ' —
not ' must be ' — ' equal to two right angles '. And contrariwise we
use the apodeictic form of proposition to hide our doubts, perhaps
even from ourselves ; we are conscious of grounds for a judgement
and grounds against it, but we look to those only which enforce the
side we wish to take, and in reference to them make our assertion
apodeictic. ' It must be so : Plato, thou reasonest well ' does not
express the same confidence as if the speaker had said ' It is so \
' Methinks the speaker doth protest too much.' The apodeictic
formula, X must be Y, thus covers in use diversities of thinking ;
but it always implies that the speaker has reflected upon the question
of the truth of his judgement.

The problematic judgement, X may be Y, similarly involves re-
flection ; but it does not, like the apodeictic, involve the judgement

1 Many mathematical statements are made without seeing into, or realizing,
their necessity at the time, and the thinking is then assertoric ; but because
their necessity can be seen, we may call them apodeictic. There are a few,
which mathematicians have believed to be true, but found false — e. g. general
formulae for the finding of prime numbers, which have finally broken down.
If it had been seen that the formula must yield a prime for any value, it
could not have broken down.

vin] VARIOUS FORMS OF THE JUDGEMENT 197

that X is Y. He who judges that straight lines making equal angles
with the same straight line cannot meet judges also that they do not ;
he who judges that Mars may be inhabited does not judge that it is.
It involves reflection, therefore, like the apodeictic judgement : but
reflection upon something suggested, as it were in an attempt to
judge, which we cannot find sufficient grounds either to affirm or
to deny. It is an expression of uncertainty.

The problematic is the most difficult of the three modalities of
judgement, and its consideration is complicated by the fact that the
formula X may be Y is sometimes used when there is nothing pro-
blematic in our thinking, because there is no uncertainty. When
a genus G has divers alternative species Sx, S2, S3, we say that a 0
may be either S1, S2, or S3 : a triangle may be equilateral, isosceles, or
scalene — currants may be black, white, or red. So long as such pro-
positions are general, they express knowledge of the divers forms
in which it is seen that a genus must be, or found that it is, realized ;
and they are not problematic. They are only problematic, if the
subject is a definite individual of the genus — 'that currant-bush may
be white, black, or red ' ; for then, they express our ignorance as to
which it is. Again, we use the problematic formula, X may be Y,
when we know that, under certain conditions P, the subject X does,
or must, exhibit the character Y, but we either do not now desire,
or are unable, to state the conditions. In this sense we say ' Water
may boil below 212° Fahrenheit ' — the condition omitted being
a diminution of the ordinary atmospheric pressure ; or ' A man may
die of joy ', the condition here being one which we could not state
precisely, though no doubt it is connected with the condition of his
heart. Disregarding these uses for the present, we must turn to those
which really express problematic thinking.

The plainest examples occur where the judgement concerns
individuals, e.g. 'Rain may fall to-morrow' (this is concerned
not with a particular thing or person, but still with a particular day) ;
or again ' The Sultan may behead his vizier to-morrow '. It is
clear that such judgements imply uncertainty in the speaker. But
is uncertainty only a state of the mind, or is it also a state of the
facts ? A necessary judgement is really an apprehension of necessary
connexion in the facts : is a problematic judgement an appre-
hension of a less than necessary connexion in them ? There is
a sense in which we may intelligibly maintain this. Given that
things of the same kind behave differently in different relations,

198 AN INTRODUCTION TO LOGIC [chap.

and given a complex system containing many things of divers kinds,
whose kaleidoscopic interactions bring different things of the same
kind into different situations, then we can say that there is no
necessary connexion between being of the kind X and behaving
in the way Y. This varying collocation of things is the basis, as we
saw, of the relation of accident or of the ' coincidental ' between
predicate and subject ; and a system of things subject to such
changes may be called not ' necessary matter ' but ' contingent '.*
Yet in a given situation, as we saw in discussing the accidental, we
commonly think that what happens is necessary. Is this opinion
a mistake ? There is one region in which men have been disposed
to think so — that of voluntary action. It has been thought that
the freedom of the will implies that a man's action does not issue
necessarily from his character and circumstances, so that no know-
ledge of these, however complete, would enable one to say that he
must act thus or thus. If this is so, there is a ground for the pro-
blematic character of any judgement about the future actions of a
voluntary agent in the intrinsic uncertainty, or real contingency, of
the event. But this uncertainty can only belong, if at all, to future
actions. If I say ' The Sultan may have beheaded his vizier
yesterday ', I imply no more uncertainty in the facts than if I say
' Rain may have fallen yesterday ' ; the same is true of the judge-
ment ' The Sultan may now be beheading his vizier ', just as much
as of ' Rain may now be falling '. All these alike are problematic
only in virtue of my uncertainty about the facts, not of any uncer-
tainty in the facts themselves. And the same character belongs
to problematic judgements which are not concerned with an in-
dividual but a kind of thing. ' Cancer may be incurable ' means that
though cancer either is incurable or not, and we are aware of certain
facts inclining us to think it is, we have not sufficient grounds for
a decision.

Waiving that case, however, a problematic judgement implies by
the form X may be Y our belief of certain facts which are not suffi-
cient ground for the judgement X is Y, though we believe that along
with other facts they would be. We do not in practice make such
judgements in the absence of all knowledge. ' The grandfather of

1 In Aristotle's phrase, evbtxoficva «\Xeo? ?xeiv> ' things that can be other-
wise ' : but Aristotle does not make it very clear how far he thought
their variability depended on shifting collocations, and how far on a ' real
contingency ', which he did not altogether reject.

viii] VARIOUS FORMS OF THE JUDGEMENT 199

Pocahontas may have died of diabetes ' : that is possible, because any
man may ; but as we do not know that in this particular case any of
those special facts were present which, with others, cause a man to
die of diabetes, we should never so judge. A problematic judgement
is provoked by knowledge ; it is problematic because of ignorance.1
It follows that further knowledge would lead to its supersession
by an apodeictic or assertoric judgement, according as our doubts
were removed by a discovery of conceptual connexions or of historic
facts. A genuine example of cancer being cured would refute the
judgement ' Cancer may be incurable ' ; so also would such an
understanding of the nature of the disease as enabled us to see how it
could be cured. But though further knowledge would lead us to
abandon the problematic judgement, we do not, when we make it,
know whether X is never Y, or alwaj^s Y, or sometimes Y and
sometimes not. In this there is a difference between a genuinely
problematic judgement and those expressed in propositions of the
same form which we noticed and set aside. For in them we imply
that there are conditions, whether we can fully state them or not,
under which X is Y.2 These quasi-problematic propositions have
therefore an affinity with certain particular propositions. In the
particular proposition ' Some X is Y ', we saw that we might either
be thinking of individuals of the kind X, not separately enumerated,
which are Y, or of some general determination of X, not stated,
which would involve its being Y ; the former sort is rather of the
nature of the singular proposition, the latter is on the way to the
universal. In the latter, the conditions, given which any X is Y,
may be either known but not stated, or, though unknown, shown to
exist by examples of X being Y : ' Some triangles have the square
on one side equal to the squares on the other two ' — viz. right-
angled triangles ; ' Some children are taller than their parents '—
doubtless in virtue of certain physiological conditions, but we do not
know them. Particular propositions like these have been called
' modal particulars ', because of their close similarity to the quasi-
problematic propositions just considered. The judgements can
indeed be as easily expressed in the form ' X may be Y ' as in the
form ■ Some X is Y ' ; each form implies that under certain con-
ditions, not specified, though perhaps known, X is Y ; but there
is this difference between them, that the latter implies that the

1 Cf. Bosanquet, Logic2, vol. i, pp. 315-318, on 'real possibility'.

1 Or, if the judgement were ' X may not-be Y ', under which it is not.

200 AN INTRODUCTION TO LOGIC [chap.

conditions are sometimes actually fulfilled, the former does not
do so.1

We may sum up what has been said of the modality of judge-
ment as follows. In every judgement I intend to assert truth,
but not necessarily about the particular reality to which the subject
of my proposition refers ; the truth I assert may be that I am
unable to discover the truth about this reality. I may judge
without pausing to question what I assert ; and in such case my
judgement is called assertoric, and expressed in the form ' X is
(or is not) Y ' ; it can, however, also be called pure, as being pure
or free of any reflection upon the question of its truth. On the
other hand, I may reflect on this question, and if I see the judgement
to be true in virtue of the very nature of its terms, or if I find
that what it asserts is involved in what I already know, or take,
to be fact, my judgement is called apodeictic, and expressed in
the form ' X must (or cannot) be Y '. Those apodeictic judge-
ments which are grounded in facts not forming part of what they
themselves affirm have a different logical character according as
these facts can be affirmed apodeictically or only assertorically ;
if the latter, the judgement resting on them is not strictly apodeictic,
for only the sequence can be affirmed apodeictically. If I find
that what I attempt to assert in a suggested proposition is con-
nected with conditions, some of which I know to exist, while I am
ignorant whether the others required are realized or not, I assert
it to be possible ; such a judgement is called problematic, and
expressed in the form ' X may (or may not) be 7 '. The proble-
matic proposition does not imply that particular events are
unnecessary in their happening, though, when general, it does
often imply that an event of a certain kind depends on a conjunc-
ture, or contingency, which is not universally necessary. It is
possible that when reflecting on the question of a judgement's
truth, we cannot find any ground for asserting what we assert,
except that we perceive or remember the fact stated, or have
had it on good authority ; though this may be reason enough to
convince us of the truth of our assertion ; then the content of
the judgement is affirmed to be actual, and the judgement called
assertoric, and expressed in the form ' X is (or is not) Y ', with an
emphasis perhaps on ' is ', or the addition of the word ' actually '.

1 e. g. ' A man may call at every public-house from John o' Groats to Land's
End.*

vm] VARIOUS FORMS OF THE JUDGEMENT 201

This assertoric judgement, being not a bare unreflective assertion,
but expressing besides our mental attitude towards a suggested
doubt, is different from the assertoric judgement, above called
also pure, that contains no such reflection ; and as involving such
reflection, this is modal.

These distinctions of modality do not then express differences in
the necessity with which elements connected in reality are con-
nected x ; yet they do express this, that whereas some connexions
in reality are seen to be necessary, others, and the existence of
such elements, and their distribution in time and place, are not.
Many philosophers have felt it impossible not to believe that the
existence of all things, and their distribution, and every feature
of their interaction, are as necessary as those matters which are
asserted in our really apodeictic judgements ; and if their belief
could pass into clear vision, judgements at present problematic or
assertoric would be replaced by apodeictic.

[Some further questions connected with modality, and in par-
ticular with the problematic judgement, deserve attention.

In the first place, in a problematic proposition, do we really
judge ? In the assertoric or apodeictic, we judge that X is F,
though there is a great difference between thinking, however con-
fidently, what we do not see to be necessary, and knowing. But
in the problematic, we do not at any rate judge that. Is it more
than an expression of doubt, and of our inability to ' make up our
mind ' ? It certainly is an expression of doubt. But we do not
utter such propositions in vacuo, and out of relation to any question
which we desire to answer ; and if a man were asked whether
there really is telepathic communication, and replied after con-
sideration ' It may be so ', he would mean rather more than that
he did not know. He would mean that there were certain facts
preventing him from denying it, though insufficient to prove it ;
that there was some reason for thinking yes.

If there were no assertion of fact in a problematic proposition,
we should not judge one event to be more probable than another.
The whole mathematical treatment of the probability of events
rests upon the assumption of a limited knowledge of their con-
ditions. If I say that there is more probability of throwing 7 with
the dice than 12, it is because I know that there are six ways of
throwing 7 and only one of throwing 12. In complete ignorance
of a subject I could not say that anything was probable regarding

1 Hence we cannot accept such a definition as Aldrich offers of modality :
' Modalis, quae cum Modo, h. e. vocabulo exprimente quo modo praedicatum
insit subiecto.' Artis Logicae Rudimenta, c. ii. § 2. 1 (Mansel's 4th ed., p. 47).

202 AN INTRODUCTION TO LOGIC [chap.

[it.1 But the attempt to estimate degrees of probability raises
a difficulty which the problematic judgement of itself does not
raise. I say that something may happen on a given occasion,
because I know (or believe) that some of the conditions required
for its happening exist ; but if I say that one event A is more
probable than another B, I do not mean to assert that the con-
ditions necessary for the occurrence of A are more completely
realized than those necessary for the occurrence of B ; for that
implies that the conditions requisite for B are incomplete, and if
I know (or believe) that, I shall call B not less probable, but
impossible. More conditions necessary for A than for B are known
to me ; but as the rest is unknown, it may turn out that the con-
ditions requisite for B are really complete, and those for A incom-
plete. Anyhow, if one or other event must occur, one will occur,
and the other not ; one is necessary, the other impossible ; more
and less probability do not attach to the events. We say therefore
that they attach to our judgements ; the judgement that A will
happen is more probable than the judgement that B will happen.
But one is true, the other false, and we do not know which is which ;
is it not foolish to prefer one, as the more probable ? It is the more
probable judgement, comes the answer, because there are more
grounds for it, i.e. there are more grounds for thinking that A will
occur than that B will. But what does this mean ? the grounds
for thinking that A will occur are the facts, or a knowledge of the
facts, which necessitate A. Less than this is no ground for thinking
that A will occur, but only that it may occur ; and similarly with
B. The real situation then is that there are grounds for thinking
that A or B may occur, but not for thinking that the one will occur
rather than the other. An example will make the point clearer.
Suppose I am to draw from a box, in which there are 5 black balls
and 1 white, and to bet on the result. I shall be told to bet on
drawing a black ball, because it is more probable ; yet all the time
perhaps only the white ball is within reach, and my drawing a black
is impossible. How can that be the more probable judgement
which leads me to act upon the expectation of an event which is
impossible ? The usual answer is, that with the knowledge avail-
able it was more reasonable to bet on black ; but would it not have
been more reasonable not to bet at all ? And indeed, is not that
the only reasonable course ? did I know enough to bet reasonably ?
And if not, can we defend the statement that the one event was
more probable ?

I think we may partially solve this difficulty as follows ; but
not wholly. We must distinguish between what is reasonable if
we are to act many times, and if only once. If I am going to
draw from the box many times, the ball being replaced and the

1 Cf. infra, pp. 423-424.

viii] VARIOUS FORMS OF THE JUDGEMENT 203

[box shaken after each draw, then supposing I always bet on black,
I shall win more frequently than if I always bet on white. In
saying this I know that there are five times as many black balls
as white, and I believe that the movements of shuffling and the
direction of the thrust of my arm are favourable about equally
often to each ball. But what is reasonable to do on each of a number
of occasions is no longer reasonable when there is only one occasion
of acting. The real meaning of the statement that drawing a black
ball is 5 times as probable as drawing a white is that in a large
number of trials black will be drawn about 5 times as often ; but
I cannot transfer the ratio to the event of a single trial. If there
are twice as many boys as girls in a village, it is not because each
child is | boy and § girl ; and it is the same here. Those who call
it reasonable to ' follow the chances ' in an isolated action are like
people who think that an average or percentage is displayed in each
of the items from which it is obtained. An excellent example of
the difference is provided by life insurance. An insurance company,
knowing that, out of a great number of persons who have lived
to 55, so many have died at 56, so many at 57, &c, but that the
average length of life beyond 55 has been (say) 15 years, and
believing that the circumstances favourable or unfavourable to
longevity will continue much as hitherto, offers an insurance policy
to persons of 55 at a premium based on the assumption that they
will live to 70. It does not matter to the company that it loses
on X and gains on Y, provided it makes the calculated profit on
the average. But it matters very much to X whether the company
is going to lose or gain on his insurance. If he dies next year, he
will have made a very good bargain ; if he lives to 90, a very bad
one. What is reasonable for the company to offer is not reasonable
for him to accept, if he regards life insurance as a speculation. If he
insures his life for the sake of the security of his family, the question
is whether this security is worth the price asked for it. The proper
price to ask of him may be settled by applying the theory of proba-
bility ; but you cannot so settle whether a thing is worth its price.

And yet many a man faced with the question, whether it is worth
his while to pay the premium asked, would take into account his
so-called ' expectation of life ' — how long it is ' probable ' that he
will live. If he lived all the lives of all the insurers, this would be
reasonable ; as he lives only his own, is it reasonable ? I think,
applying the same considerations as hitherto, we may justify him a
little further.

We allowed that a man making repeated draws from the box
will draw black more often than white, because black will be more
often under his hand ; and therefore, though he does not know
on any given occasion which will be under his hand, he will act
reasonably if he always bets on black. These occasions of action

204 AN INTRODUCTION TO LOGIC [chap.

[are all of the same kind. But life requires us to act in all sorts
of situations, with very imperfect knowledge of the conditions
affecting the event in each, wherein it is said that we should follow
the more probable judgement, or take the course more likely to
succeed. Our difficulty was to discover what is meant by calling
one judgement (or event) more probable than another ; both seemed
equally problematic. But we were considering an isolated judge-
ment. What is meaningless when our judgement is to guide our
action only once may not be so when very many actions are in
question, and we judge always on one principle. Suppose that the
ratio of the known circumstances favourable to one event to the
known circumstances unfavourable to it, or favourable to another
is roughly the same — or even, more nearly directty than inversely
the same — as the ratio of the existing circumstances favourable to
the one event to the existing circumstances unfavourable to it, or
favourable to another : then if men always acted as if that event
would happen, for which they knew more circumstances favourable,
they would more often succeed than fail. That is what is meant by
adopting the course, or acting on the judgement, for which there are
the better grounds. And the reasonableness of so acting is not dis-
proved by the fact that all men fail sometimes, and some men even
most times, when they act on this principle : any more than the
reasonableness of betting repeatedly on the black ball is disproved
by the fact that every man sometimes loses, or that some men lose
on the whole, within the run of their draws. Indeed the reasonable-
ness is greater in the former case ; for men need not bet, but they
must act ; and if they must act, and act in the absence of the
knowledge which would enable them to secure success in each case
separately, they must act upon a rule which will enable them to
secure the most success upon the whole, and leave its distribution
to fortune.1

Yet I do not think the above a complete solution of the problem.
A rule that can be applied without a fresh exercise of judgement
in each case is only possible in matters like drawing balls from a bag,
or throwing dice, where the factors whose existence we take into
account can be treated quantitatively. We may grant that the
man who acts most prudently on a given occasion may fail on that
occasion, and the imprudent man succeed, and yet that the man
who always acts prudently will succeed more than men who always
act imprudently ; but that does not explain what acting prudently
on a given occasion is. No doubt it requires study of the ascer-
tainable facts ; but it also involves an estimate of their importance ;

1 So, because we do not know enough of the ' merits ' of every case to
decide each ' upon the merits ', we are often compelled in administration to
adopt a rule which suits most cases, and acquiesce in its sometimes failing,
and our not knowing when it will fail.

vin] VARIOUS FORMS OF THE JUDGEMENT 205

[and this is something quite different from the thinking which dis-
covers that out of all possible throws with the dice six give 7 and
only one 12. We speak of a man having a sound judgement, and
the collect prays for ' a right judgement in all things \ The exercise
of such a judgement is not knowledge, and not mere guess-work.
It is better than tossing up, yet it cannot be justified to another.
Though a man is prepared to act on his judgement, he is not
prepared to enunciate it assertorically. It is here, as it seems to
me, that the real puzzle of the problematic judgement lies. We
hold that one man is wiser than another, and that, not only in
reaching opinions on which action is to be based, but also in the
study of matters that do not admit of demonstration, e.g. in
historical inquiry or anthropological. And this wisdom does not
consist in either the advantage resulting from acting as if its opinions
were true, or their confirmation by subsequently discovered fact ;
though these things may be evidence of it. And yet how is one
man wiser than another, when neither knows ?]

[I have spoken frequently of the grounds for a judgement, and
in the previous edition of this book it was said (p. 171) that what
gave modality to a judgement was the presence of the thought of
grounds for what is alleged, though I think it better to say that
a modal judgement expresses reflection upon the question of the
truth of what is judged, or suggested ; for an assertoric, or a self-
evident apodeictic judgement, has not grounds in the same sense
as an apodeictic judgement deduced from others, or a problematic
judgement. By the grounds of a judgement are commonly meant
grounds given in other judgements ; but they are not these other
judgements, i.e. the acts of our minds in judging; they are the
facts which in them we assert. And these alleged facts are only
grounds of our present judgement in the sense that we see a con-
nexion between them and the fact which it asserts. The relation,
however, may be of several kinds. The grounds may be facts
whose existence is seen to account for that of the fact grounded
on them ; this occurs in causal explanations. Or they may be
facts which make intelligible the fact grounded on them, though
there is no causal relation, as in mathematics. Or, thirdly, they
may be facts which do not make intelligible to us what is said to
be grounded on them, but which we think could not exist, but for
that ; this occurs in inductive reasoning, or when we argue from
an effect to the existence of its cause ; if I have known water to
be found by men with apparently no other means of discovering
its presence, I may infer that the divining-rod informs them of it,
though I do not thereby understand its action. There is an old
distinction between ratio essendi and ratio cognoscendi, a reason for
the being of a fact, and a reason for acknowledging its being, which
expresses the difference between grounds of a judgement in the

206 AN INTRODUCTION TO LOGIC [chap.

[first of these three senses, and in the other two. The grounds
which justify an apodeictic judgement must be either rationes
essendi, or such rationes cognoscendi as we get in mathematics ; ior
we can only judge apodeictically if we have insight into the neces-
sity of the fact alleged.]

[There are a few other adverbs (besides possibly, actually, and
necessarily) which may be introduced into a proposition in order
to express that we have reflected upon and made our estimate of
its truth : e. g. probably, truly, falsely, really : although ah1 but the
first of these may also be used merely to qualify some term in the
judgement ; a truly virtuous woman, for example, meaning a woman
virtuous in a particular way, or a falsely delivered message, one
not delivered as it was received, whereas a probably dangerous
undertaking does not mean an undertaking involving a particular
kind of danger. Such adverbs (if used to express our attitude as
to the truth of the proposition reached by omitting them from that
in which they are used) may be called modal, and judgements modal,
in which they are used. But no adverbs of any other kind make
a judgement modal, and no qualification of the content, but only
of the unreflecting directness with which, in a ' pure ' judgement,
the content is affirmed. Differences of tense, for example, must
not be reckoned to affect the modality of a judgement 1 ; they

1 As by J. S. Mill, Logic, I. iv. 2, who rightly rejects the view of those
who would make every adverb the ground of a modal difference in the
proposition where it occurs. The distinctions of modality descend from
Aristotle, de Interp. xii. 1, 21a 34-37, and Anal. Pri. a. ii. 1, 25a 1 sq„ but
the word rponos ( = modus) is said to occur first in the Commentary of
Ammonius ; v. Ainmonius in Ar. de Interp. 172r, (quoted in part Prantl,
vol. i. p. 654) = Berlin ed. p. 214 Tpotvos p.ev ovv cori <ba>vr] o-rjpuivovo-a onus
vnapx*' to Karrjyopovpevov tu vnoKeififvco, oiov to ra^eo)?, orav heympev ij o~ekrjV7)
Taxtcos dnoKa&io-TaTai ", fj to KaXco? ev t<o " 'Saxparris kciXcos biaXeyerai ", fj to ndvv
ev to* " nXaraw Aiuiva trdvv <pi\el ", fj to del ev t<$ " 6 rjXios del Kiveirat ''. dpi6p.os
he avraiv <pvo~ei ptev ovk eo~Ttv aneipos, ov p.r)v 8e ire piXtjirrds ye rjp.lv, uarrep 6 tu>v
K.a8ohov vnoKeip-eveov fj KaTTjyopovpevcov, dvapidprjTwv de avTcav ovtu>v. Ttrrapas oe
pdvovs 6 ' \pio-TOTe\rjs napaXafi^dvei irpbs ttjv Bewpiav to>v p.erd rpojruv npoTdo~eo>v,
tov dvayKciiov tov dvvarov tov ev8tj(6fi€Vov Kal enl tovtois tov aovvarov. . . S Modi
is a word signifying how the predicate belongs to the subject, e. g. " quickly ",
when we say that " The moon waxes quickly ", or " well " in " Socrates
argues well ", or " much " in " Plato loves Dion much ", or " always " in
" The sun always moves ". The number of them is not infinite in the nature
of things, but is beyond our computation, like the number of universals that
can be subjects or predicates, though they cannot be numbered. Aristotle,
however, brings into his consideration of modal propositions four modes
only, the necessary, the possible, the contingent, and further the impos-
sible. . . .' This statement about Aristotle is based on de Interp. xii, and
the modalities were often enumerated as these four, sometimes with the
addition of the true and the false. The same wide definition of rponos is
given by Michael Psellus (v. Prantl, ii. 269), but he singles out for discussion
only those which ' determine the connexion ' of subject and predicate, i. e.
the modalities proper. Cf. Buridanus (Prantl, iv. 22), who explains that the
qualification which is to make the proposition modal must attach to the

vin] VARIOUS FORMS OF THE JUDGEMENT 207

[merely affect the predicate, and not our attitude towards affirming
the predicate of the subject ; and past, present, and future verbs
may all occur (as we have seen) in judgements of any modality.
No doubt differences of tense are a somewhat peculiar affection of
the predicate. If I say Jehu drives furiously, I predicate a different
action from what I predicate if I say that he drives slowly ; but
the action predicated is the same, whether I say that Jehu has
driven, is driving, or will drive, and only the time of the action
differs. This, however, merely amounts to saying that the pre-
dicates of judgements differing in tense differ thereby in the category
of time, and not in another category. Time is a very peculiar
feature in the existence of things, but still it is a feature in their
existence, and gives rise to a great variety of modifications in their
predicates. There is no more reason for reckoning as modal these
differences in time, than there is for so reckoning the differences
in degree, or in place, of which a predicate is susceptible. The
plague raged last year : it is raging now : it is raging here : it is
raging in Calcutta. If the plague can exist in different times, so
also can it exist in different places ; and if judgements do not
differ in modality by connecting its existence with different places,
neither do they differ in modality by connecting its existence with
different times.]

There are a few other distinctions drawn among judgements,
which ought to be noticed. We may deal first with a series of
antitheses whose force is sometimes too readily considered to be
the same : these are analytic and synthetic, essential and accidental,
verbal and real.

' In all judgements,' says Kant,1 ' wherein the relation of
a subject to the predicate is cogitated (I mention affirmative
judgements only here ; the application to negative will be very
easy), this relation is possible in two different ways. Either the
predicate B belongs to the subject A, as somewhat which is con-
tained (though covertly) in the conception A ; or the predicate
B lies completely out of the conception A, although it stands in
connexion with it. In the first instance, I term the judgement
analytical, in the second, synthetical. Analytical judgements

copula, and not to the subject or predicate. The word modus is of course
a term of wide signification, but Logic is concerned with certain modi pro-
positionis ; and it is obviously wrong to suppose that any adverb will make
the proposition in which it occurs modal ; nor can differences of tense do
so, though they express a modification of the predicate.

1 Kritik of Pure Reason, E. T. (Meiklejohn), p. 7. The translator uses
conception as equivalent to concept (cf. supra, p. 22).

208 AN INTRODUCTION TO LOGIC [chap.

(affirmative) are therefore those in which the connexion of the
predicate with the subject is cogitated through identity * ; those
in which this connexion is cogitated without identity, are called
synthetical judgements. The former may be called explicative,
the latter augmentative 2 judgements ; because the former add in
the predicate nothing to the conception of the subject, but only
analyse it into its constituent conceptions, which were thought
already in the subject, although in a confused manner ; the latter
add to our conception of the subject a predicate which was not
contained in it, and which no analysis could ever have discovered
therein.' Kant's example of an analytic judgement is ' all bodies
are extended ' : for our conception of body is extended substance,
and therefore, in order to make the judgement, we need only
analyse the conception. ' All bodies are heavy ', on the other
hand, is a synthetic judgement ; for it is not contained in the
conception of bodies, that they gravitate towards one another.

Kant's statement of the distinction between analytic and syn-
thetic judgements has been much discussed and criticized. He
himself attached to it great importance. For he thought that
analytic judgements could be enunciated universally in advance of
experience under guarantee of the law of contradiction ; because
the predicate was contained in the subject-concept, it could not be
denied of the subject without self-contradiction. Since I mean
by calling anything a body that it is an extended substance, I can
say that all bodies are extended without waiting to examine every-
thing that falls under that denomination. With synthetic judge-
ments it is otherwise. It is no part of what I mean by calling
anything a body, that it is heavy ; and I need experience to assure
me that whatever falls under the denomination body has weight.
But there are some synthetic judgements which we know to be
true universally without appeal to experience ; and how that is
possible Kant conceived to be the fundamental question of meta-
physics.

But we never make judgements analytic in Kant's sense — i.e.
guaranteed by the mere identity of the predicate with an element
in the subject-concept. To do so would be tautology ; and to

1 In speaking of the connexion between the predicate and subject as
cogitated through identity, Kant means that the predicate-concept is identical
with some part of the subject-concept : where it is cogitated without identity
the two concepts are quite distinct.

2 Or ampliative.

viii] VARIOUS FORMS OF THE JUDGEMENT 209

utter a tautology is not to judge, for in all judgement we advance
to the apprehension of a new element in the being of a subject
already partially apprehended. Suppose the constituent elements
of the concept A to be BCD, as those of body are solidity and
extension. Yet the judgement ' A is B ' (all bodies are extended)
is not the equivalent of ' BCD is B ' (all extended solid substances
are extended). This proposition does merely repeat in the predicate
something contained in the subject-concept ; and inasmuch as
the subject is already conceived as uniting in its being elements
whereof the predicate is one, the proposition only goes over old
ground. But that judgement picks out in the unity of what we
call a body an element which it recognizes as combined with others
to constitute a body. And the difference is fundamental. ' A is
B' means 'to the constitution of A, B must go with CD ' ; all
bodies are extended means ' to the constitution of body extension
must go with solidity '. Kant himself tells us that until the judge-
ment is made, the predicate B is only covertly contained in the
subject-concept A ; so that it is really the work of the judgement
to recognize B (as an element along with other elements) in the
nature of A. And it is this recognition of the necessary implica-
tion of different elements in one nature, not the law of Contradic-
tion, which allows us to enunciate the judgement universally.
Suppose that we did not see that a substance could not be solid
without being extended : then (1) if we meant by body merely a solid
substance, we should see no self-contradiction in the statement
that a body need not be extended : while (2) if we meant by the
word a solid extended substance, the statement would indeed be
self -contradictory, as is the statement ' a body need not be a body ' ;
but the so-called analytic judgement all bodies are extended would
be as uninstructive as the tautology bodies are bodies.

In all judgements then — even in those which Kant calls analytic
— we assert a relation of distinguishable elements. Yet his
antithesis of analytic and synthetic judgements is not baseless.
That cats purr is a statement not made on the strength of seeing
that to purr is necessarily connected with other elements in the
being of a cat ; and we may think of a cat without including in
its nature purring. This then he called a synthetic judgement.
But he also called synthetic such judgements as '5 + 7 = 12', or
' Two straight lines cannot enclose a space ' (in which the con-
nexion of the predicate with the subject is seen to be necessary),

1779 P

210 AN INTRODUCTION TO LOGIC [chap.

because in them too the subject can be thought of without the
predicate — whereby is meant not that we can conceive the subject
to lack the predicate, for we cannot conceive what cannot be,1
but that without thinking of the predicate at all, we can still in
a measure conceive the subject. Hence the predicate-concept was
no part of the subject-concept, and, not being included in it, could
be denied of it without self-contradiction ; and so, since we know
the judgements to be true universally, without examining every
instance, we have knowledge of things not guaranteed by the law
of Contradiction before experience of them. This, to Kant's
mind, was the great problem, which he expressed by asking how
synthetic judgements a priori are possible.2

But the difference between the two classes of judgements is
misrepresented when it is said, that in the analytic the predicate
is merely part of the subject-concept, and the necessary truth of
the judgement therefore obvious : in the synthetic the predicate
is no part of the subject-concept, and the necessary truth of a
synthetic judgement therefore a problem. No judgement is
analytic in the sense of asserting of anything in the predicate
what in the subject-concept we have already realized or indicated
it to be. What Kant has really done is to distinguish those judge-
ments in which the predicate is part of the definition of the subject
from those in which it is not. The distinction we may mark by the
antithesis essential and accidental, if accident be taken, as by

1 ayvaxrla e£ dvdyKris eVl nr/ ovn ('Of what cannot be we can only be
ignorant'). Plat. Rep. v. 477 a.

2 To know anything a priori (f*c Trpnrtpov) means to know it by derivation
from something prior ; and a general principle is said to be prior to the
facts, or subordinate principles, that exemplify it ; to know anything a pos-
teriori means to know it by derivation from the particular facts exemplifying
or dependent on it. Thus I know a priori that 5 men in buckram and 7 men
in buckram are 12 men in buckram, for it follows from the general principle
that 7 + 5= 12 : I know a posteriori that cats purr, through observation of
many cats. Analytic judgements might in Kant's view be known a priori,
because their truth followed from the law of Contradiction ; but there was
no principle from which self-evident synthetic judgements could be derived.
Kant spoke of knowing these also a priori in the sense of knowing them not
a posteriori, i.e. not on the evidence of their repeated confirmation in experi-
ence ; properly, they are the priora from which we derive knowledge about
unobserved particulars. Thus to know a priori came to mean to know in
advance of experience ; and his problem comes to this, viz. how, in advance
of experience of that very thing, and therefore merely by thinking, can we
know more about anything than what is guaranteed by the so-called ' laws
of thought ' T Cf . on the meaning of the antithesis a priori and a posteriori,
pp. 436, 437, infra.

vin] VARIOUS FORMS OF THE JUDGEMENT 211

Aristotle in the phrase <ad' avrb o-vfAfieftrjKos 1 ( = essential acci-
dent, or accident per se), to include attributes belonging to any sub-
ject of a certain nature in virtue of that nature, as well as those
coincident in it with that nature.2 Thus the accidental judge-
ment might be in Kant's sense synthetic either a priori or a posteriori.
And we might fairly oppose these, as ' ampliative ' or ' augmenta-
tive ', to essential judgements as ' explicative ', because a subject
and a property or accident of it are not one, as it and the definition
of it are. But the opposition of analytic and synthetic is misleading,
since that insight into the nature of a subject which definition
expresses, though it may be called an analysis, is also an apprehen-
sion of the connexion of elements in an unity, and the necessity
of this connexion cannot be derived from the law of Contradiction.
That law is that contradictory propositions cannot both be true ; but
to know this is not to know which of two given contradictories is true.

Doubtless a man cannot without contradiction deny of a subject
anything which by the subject-term he means that it is. But
how has the subject-term come to have its meaning ? If through
insight into a necessary connexion of elements in the subject,
then the so-called analytic judgement expresses this insight. Only
if definitions were quite arbitrary, mere statements of the mean-
ing of a name, would the truth of Kant's ' analytic ' judgements rest
merely on the law of Contradiction. If I choose to mean by body a solid
extended substance, it is self -contradictory to say that a body is not
extended. But equally, if I choose to mean by body a solid extended
and heavy substance, is it self -contradictory to say that a body is
not heavy. And Kant has forgotten to ask why we regard extension
as belonging to the definition of body rather than weight.

We saw indeed, in discussing Definition, that we often have to
settle arbitrarily what elements shall be included in the intension
of a term, and therefore implied about those subjects to which we
apply the term. Let us take an example of a subject in whose
definition the elements are thus arbitrarily put together.3 In the
Elementary Education Act of 1870, § 3, an elementary school is

1 Cf. e.g. Ar. Post. An. a. vii. 1, 75a 39-b2. o-v/iSepriKos in this sense
includes properties, which are distinguished from accidents in sense of the
Topics by being ko.6' aW6.

2 e. g. two straight lines, in virtue of their straightness, cannot enclose
a space : to be heavy is coincident in bodies (so far as we can see) with their
nature as bodies.

3 Arbitrarily, not because there is no motive, but because the elements,
though compatible, are not necessarily implicated together.

P2

212 AN INTRODUCTION TO LOGIC [chap.

by definition ' a school, or department of a school, at which elemen-
tary education is the principal part of the education there given,
and does not include any school or department of a school at
which the ordinary payments in respect of the instruction, from
each scholar, exceed ninepence a week '. To say therefore that
an elementary school charged less than 10c?. per head per week
in fees was to make an analytic judgement from the standpoint
of the Education Department in 1870 ; but only because it had
been arbitrarily settled that none charging 10c?. or over should
rank as an elementary school, and not because we have such a know-
ledge of what an elementary school must be, as to see that it could
not be elementary and charge a fee so high. The proposition then
is true just because it has been agreed what elementary school
shall mean ; and while that agreement is adhered to, it cannot
be denied without self-contradiction. But if I say that a triangle
has sides, that is true not just because it is agreed to call nothing
a triangle which has not, but because I see that lines can be put
together into the unity of, and are required in, a triangle. Kant's
account of analytic judgements ignores this difference. It implies
that all definition is arbitrary, and that judgements whose predicate
is part of the definition of the subject are necessarily true, only
because what we mean by a name we mean by it.

Some propositions are indeed true universally by mere con-
vention as to the meaning of names, because they give us informa-
tion about the convention. These may be called verbal, and to
them we may oppose as real all which are intended to give informa-
tion about the nature of things. But verbal propositions are
in Kant's sense synthetic. ' Elementary schools charge a fee
below 10c?.' meant that schools called elementary did so ; and to
charge a fee below 10c?. is not part of being called elementary,
but of what was meant by being so called. A proposition about
the meaning of a name is clearly instructive, and ampliative. It
is only inadvertently that we make about things statements, whose
truth rests just on the meaning of words ; and when we discover
that we have done so, we acknowledge that we have really said
nothing. Suppose that some one had argued in 1870 that a par-
ticular school which he knew to give mainly elementary instruc-
tion had a fee below 10c?., because it was an elementary school ;
clearly he would have wasted his breath, unless he knew that
it had a right to be called so within the meaning of the Act ; and

viii] VARIOUS FORMS OF THE JUDGEMENT 213

he could not have known this till he knew that its fee was below
lOd. ; and then the argument would have been superfluous.

There is another objection to Kant's division of analytic and
synthetic judgements. In speaking of analytic judgements, he
had in mind only universal judgements, in which, as he held, we
analyse a concept ; but there are judgements in which we may
be said to analyse the sensible object before us, as when I look
up and say ' the sky is starlit '. These have been called * ' analytic
judgements of sense ' ; they clearly distinguish in a subject an
element which they assert to be combined with others in the unity
of that subject, and so far they are equally analytic with those
which Kant called so ; but yet they differ greatly. They are
singular, not universal ; they rest on perception, not conception ;
and by no possibility could their truth be made to seem dependent
barely on the meaning of names.

Analytic judgements then may be analytic either of a sensible
individual or of a concept : in neither case is their truth guaranteed
by the law of Contradiction, but they rest on our apprehension of
the connexion of elements in the unity of one subject. So far
they do not differ from judgements called by Kant synthetic.
But those analytic of a concept are essential, where without the
predicate the other elements in the subject could not form a
conceivable unity, whereof the predicate could be regarded as
a further attribute. Judgements called by Kant synthetic, whose
subject is something which can be thus conceived before the attribu-
tion of the predicate, may be called accidental (though not in the
sense of that word in the doctrine of Predicables) or ampliative
of their subject. They include both analytic judgements of sense,
and all judgements about the meaning of names. Verbal pro-
positions are therefore not analytic, and real propositions may be
either analytic or synthetic. Essential judgements are true by the
nature of things, not ex vi termini ; or, if we call essential those
judgements whose predicate is part of the arbitrary 2 definition of
their subject, they will be essential in a different sense, and instruc-
tive only as statements about the meaning of a name ; intended

1 F. H. Bradley, Principles of Logic, p. 48 : cf. Sigwart, Logic, § 18. 4
(E. T., Helen Dendy, vol. i. p. 108).

2 Arbitrary (though not therefore settled without good reason) because
what we are defining is something of our own institution, or because our
so-called definition is a compromise of the nature explained pp. 99-102, supra.
In the strict sense of definition, none is arbitrary : things are what they are.

214 AN INTRODUCTION TO LOGIC [chap.

otherwise the propositions are mere tautologies, and not expressive
of any real act of judgement at all. It will be seen therefore that
the three antitheses, of analytic and synthetic, essential and acci-
dental, verbal and real, are by no means equivalent ; they are neither
made on the same fundamentum divisionis, nor do they respectively
bring together and keep apart the same individual judgements.

[Some further points deserve notice in regard to the distinction
of analytic and synthetic judgements.

1. The terms suggest that we in judgement pick to pieces or put
together the object of our thought. And some who use the terms
hold that in the last resort this is true ; that mind by its activity
constitutes its objects, though not perhaps as individual mind,
yours or mine.1 But whatever be the ultimate relation of mind
to its objects, what the individual means to assert in judging is
a relation of elements in the real that holds irrespectively of his
present judgement. A judgement then is analytic in so far as it
recognizes the distinct elements in what the judger starts by envisag-
ing as an unity ; synthetic in so far as it recognizes the union —
whether by way of necessary connexion or of empirical conjunc-
tion— of elements which the judger starts by envisaging as distinct.

2. But hence, because the judger does not lose sight of his start-
ing-point, it has been said 2 that all judgements are at once
analytic and synthetic. In the sense that in all judgements we
assert a diversity in unity, a many in one, this is true. But the
relation of the elements, their mode of combination in the unity,
is not always the same.

3. It has also been said that the same judgement may be analytio
to one person, and synthetic to another : that, e.g., a judgement
analytic to a teacher stating what he already knows is synthetic
to a learner receiving information new to him ; and similarly that
a judgement may be synthetic at one time and analytic at another
to the same person, and that to any one omniscient all judgements
would be analytic. But this is an error. The view rests on the
following consideration, that if, e.g., I learn for the first time that
diamonds are combustible, I make a synthetic judgement, because
to be combustible was no part of what I understood by the word
diamond ; but having learnt it, I include that in what I mean by
the word, and henceforward, when I judge that diamonds are com-
bustible, my judgement is analytic. Now, were this so, it is clear
that the name diamond would have come to be used by me with
a different meaning, i.e. the subject-concept would be different, in
the judgement afterwards expressed by the words ' diamonds are
combustible ', from what it had been in the judgement expressed

1 Cf. e. g. Bosanquet, Logic 2, vol. i. p. 84, vol. ii. p. 237, and Bk. II. c. x.
* e. g. Bosanquet, Logic 2, vol. i. p. 91.

vni] VARIOUS FORMS OF THE JUDGEMENT 215

[by the same words before. The earlier synthetic and the later
analytic would not therefore be the same judgements, though
expressed in the same proposition. Thus at best the view would
involve a confusion between the judgement and the proposition.1
But it is not even true that, when I know that diamonds are
combustible, the meaning of the word diamond must change for
me. The judgement is synthetic because combustibility is not
something without which the nature of a diamond would cease to
be conceivable. That fact is not changed by my learning that
diamonds are combustible. What I know or think once I may
know or think again ; and the nature of a judgement is not
altered by my having made it before. We must, however, acknow-
ledge that there are certain differences in the state of mind of one
who makes a judgement for the first time and one who repeats it ;
there are emotional accompaniments in the former case, or a pre-
ceding attitude of expectation, not present in the latter.]

Two comparatively unimportant classes of proposition, exceptive
and exclusive, may be mentioned before closing this chapter.
An exceptive proposition is one which excepts from its appli-
cation a certain part of the extension of the subject 2 : as
in Clough's satirical version of the Second Commandment — ' No
graven images may be Worshipped, except the currency.' An
exclusive proposition is one which confines the application of the
predicate to the subject of which it predicates it : as in Elijah's
exclamation, ' I, even I only, am left.' Within a given whole,
it clearly makes no difference whether a predicate is affirmed
of one part only, or denied of all but that : Only the brave deserve
the fair would mean the same as the poet's actual line None but
the brave deserve the fair. The scholastic logicians treated these
and some other forms of proposition under the head of Exponibilia,
i.e. statements whose full meaning could only be expounded in
more propositions than one. Thus ' None but the brave deserve
the fan ' or ' Only the brave deserve the fair ' implies two pro-
positions, that the brave (or some of them) deserve the fair, and
that those who are not brave do not. The infinite proposition was
also an exponible ; for if I say that Parliament is not-in-session
I imply that it is not in session, and is in some other state instead.

1 v. L. Nelson, Ueber das sogenannte Erkennlnisproblem, pp. 36-40.

2 In strictness, of what would otherwise be the subject : as the part
excepted cannot be called part of the subject of a judgement which expressly
does not apply to it.

CHAPTER IX

OF THE DISTRIBUTION OF TERMS

IN THE CATEGORICAL JUDGEMENT: AND OF THE

OPPOSITION OF JUDGEMENTS

We saw in the last chapter that all categorical judgements,1 in
respect of their quality, were either affirmative or negative ; and
in respect of quantity, might be treated as either universal or
particular. The latter division indeed strictly applies to those judge-
ments only whose subject is a general term, and therefore not
to singular judgements ; but for the purposes for which these can
be reckoned with universal judgements the division is exhaustive.
These purposes are the determining the distribution of terms,
together with what depends on that. A term is said to be dis-
tributed, when it is used in reference to its whole extension, or to
all that it can denote : undistributed, when not so used.2 Now the
subject of a singular judgement 3 denotes one individual only, and
the judgement 3 refers to that ; the subject of an universal judge-
ment 3 is general, and may denote any number of individuals, but
since the judgement is universal, it applies to them all. Therefore

1 By judgement in this chapter will be meant categorical judgement.

2 We have already seen, in discussing the extension, or denotation, of
terms, that confusion may arise between the relation of a generic concept to
the more specific concepts included under it and the relation of the universal
to the individual, and that, properly speaking, a singular term has no extension,
but only denotes. But in considering the distribution of terms, it is not
always necessary to bear in mind this distinction. I may therefore say
indifferently that a term is used with reference to its whole extension, or to
all that it can denote, even if we reserve the latter expression (denotation)
to signify the individuals of which a term can be predicated.

3 More strictly, of a proposition expressing a singular, or an universal,
judgement. It is terms verbal that are distributed or undistributed, according
as the term of thought, what they cause or help us to think of as subject
or predicate in a judgement, is or is not all that they can denote. For this
reason it might seem more proper to speak only of the distribution of terms
in a proposition. But since it is the act of thought or the judgement that
gives to the terms of the proposition in which it is expressed their distribution,
we may also speak of the distribution of terms in a judgement ; and because
it is important to bear in mind that terms have distribution only through
our use of them in judging, not through their presence in a sentence, I have
spoken thus.

DISTRIBUTION OF TERMS, ETC. 217

in both singular and universal judgements, all that the subject
can denote is referred to, or, in other words, the subject is dis-
tributed ; and, in considering the distribution of terms in a judge-
ment, we may accordingly rank the singular with the universal.

As every judgement has both quantity and quality, and in each
respect there are two alternatives, there are four varieties of
judgement in respect of these two characters combined. An
affirmative judgement may be universal or particular : a negative
judgement may be universal or particular. It is customary in
Logic to indicate these four forms of judgement by the first four
vowels, thus : —

an universal affirmative judgement is indicated by the letter A ;
an universal negative „ „ „ ,, „ ,, E ;

a particular affirmative „ „ „ „ „ „ 7 ;

a particular negative „ ,, ,, ,, „ „ 0.

Thus the affirmative judgements are A (universal) and 7 (particular) :
the negative judgements are E (universal) and 0 (particular) ; and
this may be remembered by noting that A and 7, which indicate
the universal and particular affirmative judgements, are the first
two vowels in the verb ' afKrmo ' : E and 0, which indicate the
universal and particular negative judgements, the vowels in the
verb ' nego '.

All universal judgements (A and E) distribute their subject : all
negative judgements (E and 0) distribute their predicate. No
particular judgements (7 and 0) distribute their subject : no
affirmative judgements {A and 7) distribute their predicate. Thus : —

in A, the subject is distributed, the predicate undistributed ;
in E, „ ,, ,, distributed, ,, „ distributed ;

in 7, „ ,, „ undistributed, ,, „ undistributed;

in O, „ „ „ undistributed, ,, „ distributed.

It is important to understand and become familiar with these
characteristics of a judgement.

A term, as was explained just now, is said to be distributed when
it is used with reference to all that it can denote.1 The term ' book '
is distributed, when used as subject in a proposition that refers to all
books : undistributed, when so used in a proposition that does not

1 i. e. denote univocally : an equivocal term is to be regarded as a different
term in each sense.

218 AN INTRODUCTION TO LOGIC [chap.

refer to all books. It is obvious that an universal proposition about
books (whether affirmative or negative) refers to all; and that a par-
ticular proposition does not : all books are written before being printed :
no book was printed before 1450 1 : some books are published unsewn :
some books are never published. That the subject of universal pro-
positions is distributed, and of particular propositions undistributed,
needs no further illustration. Two cautions, however, may be
offered.

1. The subject of a proposition is the whole subject-term ; if
I say all modern books are printed from movable type, the subject is
not books, but modern books ; it is true that my judgement does not
refer to all books, but it refers to all modern books, and so the
subject is still distributed ; while it is undistributed in the pro-
position some modern books are printed from stereotype plates. But
I may restrict a general term like book not by words which leave it
still general (e. g. modern book, book printed by Elzevir in Leyden),
and therefore capable of being either distributed or undistributed,
but by a demonstrative pronoun, or other words which destroy its
generality (e.g. that book, these books, the first book which I ever
possessed). In the latter case, the term becomes a designation, and
is therefore singular, or (like ' these books ') a singular collective ;
and the proposition should rank with universals. Nevertheless the
general term which is restricted, by a demonstrative or otherwise,
to the designation of a particular individual, is not distributed, since
it does not refer to all that it can denote. ' Book ' therefore is
undistributed, but ' this book ' is distributed, in the proposition
' This book wants rebinding ' ; for ' book ' might be used of other
books, but ' this book ' is already used of the only book of which,
so long as I mean the same by ' this ', it can be used.

2. In speaking of the distribution of terms, we are inevitably
led to view judgements in extension rather than intension : and
indeed as referring (ultimately) to so many individual subjects,
rather than asserting a connexion between universals. Now we
have seen that a judgement may refer to individuals, but need not ;
and that in a judgement properly universal, there is no express
thought of individuals. In saying that a triangle has its angles
equal to two right angles, I am not thinking of all the particular
triangles that have ever existed or may exist ; I am thinking of their

1 The proposition must be taken to refer to European books and movable
type : the nrst dated examples being of 1454.

IX] DISTRIBUTION OF TERMS, ETC. 219

common character as triangles ; this is one and the same in them
all, and so I use the indefinite singular, a or any triangle.1 It may
therefore appear erroneous to say that such a judgement distributes
its subject, if to distribute a term is to use it with reference to all
that it can denote ; for of all the individuals which the term triangle
can denote I am not thinking. But it is true in this sense, that
whatever particular triangle you choose to take, my judgement holds
good of that. We must avoid supposing that in every universal
judgement we are expressly thinking of all the different individuals
of which the subject-term is predicable ; but we must recognize that
our judgement holds of them all.

The distribution of the predicate in a judgement is not generally
so readily understood as that of the subject ; for the extension of
the predicate is not naturally before us. The rule is that negative
propositions distribute their predicate ; affirmative do not : and
this equally whether they are universal or particular.

All preachers praise virtue : some practise it. It is easy to see
here that I refer in one case to all and in the other only to part
of what the term preacher can denote. The subject therefore is
distributed in one case, and not in the other. But what of the
predicate ? That is distributed or undistributed not as it refers to
all or only some preachers ; for a term is distributed or undistributed
when it is used in reference to the whole or to a part only of its own
extension, not of the extension of the subject of which it is pre-
dicated. Now the extension of the terms ' praiser of virtue ' and
* practiser of virtue ' includes everything which can be said to praise
or practise virtue. Preachers may do so, but so may others who
are not preachers ; these also therefore are included in the extension
of the predicate ; but what is thus included is not predicated of
preachers. In the judgement ' X is Y\ I predicate Y of X; but
I might predicate it also of Z ; X and Z are both included in
the extension of Y, or in what Y can denote ; but when I
affirm Y, I do not affirm it in its whole extension ; for then in
saying ' X is Y ', I should mean that it is X and Z, and in saying
' Z is Y ', I should mean that it is Z and X. The predicate
therefore is not used in reference to its whole extension, i. e. is
undistributed.

The predicate of an affirmative judgement in fact cannot be

1 I do not deny that a particular ' representative ' triangle must be con-
sidered in making the judgement.

220 AN INTRODUCTION TO LOGIC [chap.

thought in extension at all. The subject of which it is predicated
forms part of its extension ; but in the predicate, as opposed to the
subject, I am thinking of a character or attribute belonging to that
subject. A great deal of the difficulty which hangs about the
doctrine of the distribution of terms arises from the fact that a term
is said to be undistributed both when it is used with explicit refer-
ence to a part only of its extension, and when it is used without
explicit reference to its extension at all. The subject of a particular
judgement is undistributed in the former sense ; when I say that
Some preachers practise virtue, I am explicitly confining my state-
ment to a part of the extension of the term preacher. The predicate
of an affirmative judgement is undistributed in the latter sense.
When I say that All preachers praise virtue, though it is true that
preachers, even all of them, are only part of the extension of the
predicate, yet I am not thinking in the predicate of its extension
but of its intension. The extension of a term consists of all the
alternative species, or different individuals, in which its intension is
manifested. It is impossible to predicate all the alternative species
of the same subject, or to say of anything that it is so many different
individuals. ' An ellipse is a conic section.' The extension of the
predicate conic section is hyperbola, parabola, ellipse, circle ; I cannot
say that an ellipse is all of these ; I do not want to say that it is
an ellipse ; I am thinking of the common character in them all, i.e.
using the predicate in intension. Still, it is onty part of the extension
of the predicate which is referred to in this judgement, and therefore
the term is said to be undistributed in the judgement, though
in the predicate extension is not considered at all.

In a negative judgement, on the other hand, the predicate is
necessarily denied in its whole extension. Caesar is not ambitious ;
there are a thousand varieties of ambition among mankind ; but if
I deny ambition of Caesar, I deny all these. It is the same whether
the judgement is universal or particular. No Mussulman fears
death. Whether we look to the forms which fearing death may take,
or to the individuals in whom it is exhibited, if I deny the predicate
of Mussulmans, I deny all forms of it, or deny that they are any of
those individuals in whom it is exhibited. But again, Some marine
animals are not vertebrate ; of those animals I do not merely deny
that they are dogs or cats, plaice or salmon, all of which form part
of the extension of vertebrate ; vertebration in every form is denied
of them ; a negative judgement denies its predicate in toto.

ix] DISTRIBUTION OF TERMS, ETC. 221

In an affirmative judgement, the subject is necessarily part of
the extension of the predicate ; in a negative judgement it is as
necessarily no part thereof. And to say that the subject is no part
of the extension of the predicate is to say that the predicate is
denied in its whole extension.

But here again it is primarily the intension of the predicate
which is in my mind. When I say that ' Brutus is an honourable
man ', the only individual referred to is Brutus, though ' they are
all honourable men that have slain Caesar ' ; when I say ' Caesar
was not ambitious ', I need not be thinking of any one who was.
It is an attribute which I affirm in one case and deny in the other.
Nevertheless, whereas if I do attend in affirmative judgements to
the extension of the predicate I cannot affirm the whole, and do not
want to affirm the only part — viz. the subject of the same judgement
— which is referred to, for that would be mere tautology, in a negative
judgement, if I attend to the extension of the subject, I can deny
the whole. ' A cycloid is not a conic section ' ; if I remember that
conic section includes hyperbola, parabola, ellipse, and circle, I can
say that a cycloid is neither an hyperbola nor a parabola nor an
ellipse nor a circle.

We are not thinking primarily of the extension of the predicate
in a negative judgement ; but if we do think of it, we must deny
it in toto, or else our proposition will not mean what we intend it
to mean ; therefore the predicate is distributed. ' The Tenth don't
dance ' ; we are not thinking of those who do ; but bears dance,
and so are part of the extension of the predicate, and if the predicate
were not denied in its whole extension, it would be compatible
with the truth of that proposition to say that the Tenth Lancers
were bears ; or if the predicate were used only in reference to the
ursine portion of its extension, the proposition would mean no more
than that the Tenth were not bears.

[Sometimes the device of circles, representing the extension of
the subject and the predicate, is used in order to explain the
distribution of terms. Collect the mammals in one
circle, and the snakes in another : then if no
snakes are mammals, snakes will lie outside the
whole mammal-area : and if some craniates are
not mammals, some part of the craniate-area will
lie outside the whole mammal-area ; whereas if
some craniates are mammals, some part of the
craniate -area will coincide either with the whole or with a part

222 AN INTRODUCTION TO LOGIC [chap.

[only of the mammal-area ; and if all mammals are craniates, the
mammal -area will fall completely within the craniate-area. But
all the objections which lie against representing in this figurate
way the logical relation of a larger to a smaller class within it lie
equally against so representing the distribution of terms. We may
say that the negative proposition snakes are not mammals excludes
snakes from the whole class of mammals, and not merely from
a portion of it (say men) : but we must not think of the class as
an area cut up into districts called species, or as a collection of
which the species are component groups. And if we ask what is
meant by saying that a larger class craniates is partially coincident
with the whole of a smaller class mammals, we must answer that
the relation is not that of one superficies partially coincident with
another, but of an universal character exhibited in a certain kind
of subjects ; in fact, the logical relation must explain the diagram,
and cannot be explained by it.]

[Any one who realizes that the predicate of a proposition is not
thought in extension will see that there can be no truth in the
doctrine of the Quantification of the Predicate. But the doctrine
has the support of distinguished writers, among others of Sir
William Hamilton, who invented it, and of Stanley Jevons ; and
it ought perhaps to be examined here. It may be easily shown to
be false ; and the conscientious student haply stumbling upon the
mass of intricate technicalities based upon it may be glad to feel
excused from the labour of mastering them by the knowledge that
they are built upon a worthless foundation.

By quantification of the predicate is meant affixing a mark of
quantity to the predicate as well as the subject of a judgement.
Thus instead of the four forms of judgement, A, E, 1, 0, we get
eight, as follows : —

V. All X is all Y. All organisms are all mortals.

A. All X is some Y. All men are some mortals.

Y. Some X is all Y. Some mortals are all men.

/. Some X is some Y. Some men are some (things) fleet of

foot.
E. No X is any Y. No snakes are any mammals.
rj. No X is some Y. No men are some mammals [e.g. not

monkeys] .
0. Some X is no Y. Some mammals are not any quadrupeds.
to. Some X is not some Y. Some quadrupeds are not some

mammals [e.g. not cows].

In defence of this mode of stating propositions it is urged that
as the proposition whose predicate has all before it, and the corre-
sponding proposition whose predicate has some before it, do not

ix] DISTRIBUTION OF TERMS, ETC. 223

[mean the same thing, and we must know which we mean when we
judge, we ought to express it. It is strange, if that is the case,
that no language ever has expressed it ; and it may be confidently
asserted that of these eight forms of proposition only E and 0 express
anything that we ever really mean when we make a judgement
(though others express, in ' portmanteau ' fashion, what we mean
when we make two judgements) ; and that the reason why we
ought not to express in our proposition whether we mean all or
some before the predicate, is that we mean neither.

Let us take an A proposition. It used to be stated ' All X is Y ' ;
we are told to state it ' All X is some Y '. All men are some
mortals : which mortals are they ? the horses ? the grass of the
field ? clearly not, but only the men. Yet it can hardly be meant
by the proposition, that all men are men ; it is something about
men that the proposition tells us. What about them ? that they
die, and not which kind they are among the kinds of things which
die ; we know that they are men already, and that need not be
repeated in the predicate.

But there is a difference between saying that all men are all
mortals, and saying that all men are some mortals ; the first implies
that the terms are commensurate, that there are no mortals but
men : the second that men are mortal, but an undetermined range
of things (cats and dogs and horses and asses and what not) are
so besides. Ought not this difference to be expressed ?

Doubtless, but it requires another proposition ; All men are
mortals — some mortals are not men. In recognizing that men die,
we do not judge that things of any other kind die ; and though
we may be aware of it when we say that men die, it is no part of
the judgement Men die. All men are some mortals is not one judge-
ment, but a ' portmanteau ' proposition — two judgements expressed
in what (in respect of its grammatical form) is one sentence.

It is true that in some judgements we expressly think the
predicate and the subject to be commensurate. In a definition, we
must do this. Momentum is the product of mass into velocity : wealth
is that which has value in exchange ; in these cases, it is included in
our thought that the product of mass into velocity is momentum, or
that which has value in exchange, wealth. But such judgements
are ill expressed in the form ' All X is all Y '. We do not think of
all momenta, all samples of wealth, but of wealth and momentum
each as one thing. Again, the formula ' All X is all Y ' makes us
think of X and Y as different things : whereas the whole force of
a definition is to assert that the subject and predicate, the thing
defined and the definition of it, are the same thing.

There are propositions whose terms are known to be commen-
surate, but which are not definitions, such as all equilateral triangles
are equiangular. These also we are told to represent in the form

224 AN INTRODUCTION TO LOGIC [chap.

[' All X is all Y ', and to say that all equilateral are all equiangular
triangles. But this does not correctly express the true meaning of
the other proposition. For granted that in enunciating it we are
aware that the terms are commensurate : what we wish to assert
is the mutual implication of two attributes in any triangle. It
follows from this that every triangle exhibiting one exhibits the
other ; but those which exhibit one are not a different set of
triangles from those that exhibit the other. By putting a mark
of quantity before the predicate as well as before the subject, we
make it appear as if the extension of one term was affirmed of
the extension of the other, and (if we consider individuals) as if the
individuals denoted by one term were affirmed of the individuals
denoted by another. But that is either impossible, if the individuals
are different, or tautologous, if they are the same.

1 All ' can be no part of any predicate, except where (as in these
are all the apostles) the subject is collective. If the universal judge-
ment ' All living things reproduce their kind ' is true, then it is
true of any living thing and therefore of peas. I may introduce
' perfectly ' into the predicate, and then it will be meant that peas
reproduce their kind perfectly. But I cannot introduce ' all ' into
the predicate. For then, since all living things are all things that
reproduce their kind, peas and even a single pea would be said to
be all things that reproduce their kind ; and that is nonsense. The
predicate of a judgement is affirmed distributively of each that
falls under the subject ; the predicate quantified by all could be
only true of the subject collectively. No equilateral triangle is all
equiangular triangles ; how then can they all be ? The proposition
only means that all equilateral triangles are equiangular and vice
versa. As before, it is a ' portmanteau ' proposition, and not
a single judgement.

The U form of proposition has been considered at some length,
because it is in a way the most plausible member of the series.
Universal judgements whose terms are commensurate do differ
from those whose terms are not, and do form a very important
class of judgements ; and there is no special recognition of them
in the ordinary fourfold classification of judgements (A, E, I, and 0).
It has been wrongly alleged that Aristotle ignored such judgements ;
on the contrary, he recognized their great importance in science.
To remedy this supposed omission the doctrine of the quantification
of the predicate offers us an entirely false analysis of them, and one
which Aristotle himself exposed.1 The analysis overlooks altogether

1 De Interp. vii. 17b 12 eVi Se rov K.aTiiyopovp.ivov KadoXov Kartjyopeiv r6
KaOoXov oi/K 'itJTiv dXrjdts' ov8epia yap KaTa<pao~ts d\rjdr]s coral, ev rj rov Karrjyo-
povfievov Ka66Xov to Ka66\ov KaTrjyope'iTai, olov ((tti nds avoptoiros trav (aov.
(dvdpainos, man, is an universal : when I say ' All men are animals ', I predicate
of an universal universally ; when I say ' Some men are white ', I predicate of

DISTRIBUTION OF TERMS, ETC. 225

[the intension of terms. Professing to complete what is defective
in the current recognition of different kinds of proposition, itself
leaves important differences unrecognized. We have seen that
a proposition of the form ' All I is 7' represents two kinds of
judgement essentially different in thought, according as it is really
universal, meaning ' X as such is Y ', or only enumerative, meaning
' All the X's are Y '. Of this difference, whether in universal judge-
ments whose terms are commensurate (U) or not (A), this doctrine
takes no note ; but sets up instead two kinds which misrepresent
our thought by the sign of quantity prefixed to the predicate.

The particular affirmative propositions may be dismissed briefly.
We are told that ' Some X is Y ' should be written either ' Some
X is some Y ' or ' Some X is all Y '. Take the former, ' Some X is
some Y ' : we ask immediately, which X are which Y ? ; and the
only answer is that the X that are Y are the Y that are X. Some
sowers reap ; if that means some sowers are some reapers, this can
only mean that the sowers who reap are the reapers who sow.
Take the latter, ' Some X are all Y ' ; some animals are all the pigs
(for it does not mean, are all of them pigs : as we might say that
some families all squint, meaning that all the members of some
families squint). Which animals are all the pigs ? surely only the
pigs themselves. If it be said that the proposition means that
there are more animals than pigs, then the real subject of the
judgement is the other animals (which are not pigs), and not (as
this form pretends) the animals which are pigs. If, again, it be
said to mean that all pigs are animals and some animals are not
pigs, we have as before two judgements packed into one sentence.
What is one judgement, and what is the character of a judgement,
are questions to be determined by considering our thought, and not
the verbal devices we adopt to express it. To think that all pigs
are animals, and some animals are not pigs, is to judge not once
but twice, even though we were to write such a pair of judgements
in the form some animals are all pigs.

an universal particularly, or in part. Aristotle goes on to say, in the words
quoted, that the predicate cannot be similarly taken universally [i.e. not
' as an universal ', but ' in its whole extension ']. ' But in the case of the
universal which is predicate, it is not true to predicate universality ; for no
affirmation is true when universality [in extension] is assigned to the pre-
dicated universal, e. g. All men are all animals.' Cf. Ammonius in loc. f. 82,
who points out that then each man would be all animals.) Anal. Pri.
a. XXvii. 43b 17 aiiro 8e to inofifvov ov XrjTTTfov o\ov t'jreadai, Xeya 8' oiov dv8pa>n(&
irav £(5ov fj uovtriicg irdaav eTrtcrTTjfiTju, dWa povov airkcos aKoXovativ, Kaoantp nai
irpoTeivoptdu' Kai yap a^prjaTou ddrepou Kai dhvvarov, oiov ffdvra av8pa>nov tival
ndv (aov, fj hiKaioavvr\v airav dyadov. ('But the attribute must not be taken to
be attributed in toto, I mean for example animal as a whole to man, or
science as a whole to music, but just simply to follow on the subject, as our
premiss says ; for the other is both useless and impossible, e. g. that all men
are all animals, or that justice is all good.')

1779 Q

226 AN INTRODUCTION TO LOGIC [chap.

[To the negative judgement also the quantification of the pre-
dicate does violence. The universal negative is to appear in the
two forms ' No X is any Y * (E) and ' No X is some Y ' (v). The
former may stand ; for as we have seen, if X is not Y, it is not
any case or kind of Y. The latter may well puzzle us. It denies
of X some part of the extension of Y ; pig, for example, is part of
the extension of animal, and sheep are not pigs ; hence sheep are
not some animals ; but this is quite consistent with their being
animals. ' No X is some Y ' is therefore consistent with ' All X
are Y ', and what it means is that ' Some Y are not X ' ; whether
any X are Y or not it leaves doubtful. There remain the particular
negatives, ' Some X is not any Y ', and ' Some X is not some Y '.
Again the former will stand ; but what does the latter mean ? It
does not mean that some X is not Y at all, e.g. that some animals
are not pigs at all, but are something quite different (say sheep or
cows) ; for that is expressed by the form ' Some X are not any Y '.
It can only mean that there are some F's distinct from some X's :
i. e. that though some X may be Y, they are not every Y. ' Some
murderers are not caught ' is sense ; but ' Some murderers are not
some caught ', if different from that, and sense at all, is only true
because fish and cricket-balls are also caught, and some murderers
are not these ; so that if the proposition were to be false, they
would have to be fish and cricket-balls and everything else that is
ever caught ; it is the contradictory of the impossible judgement
that those X are every Y. But as we never make that judgement,
we never want to contradict it ; yet these are forms of judgement
which those who would quantify the predicate condemn Logic for
hitherto ignoring.1

Thus all the eight forms of proposition with quantified predicate
have been found vicious, except E and 0 ; and these are so inter-
preted as to lay undue stress on the aspect of extension in the
predicate. The truth is that if we prefix to the predicate of a pro-
position a mark of quantity, all or some, we are bound to think of
the various individuals (or species) characterized by the predicate,
not merely of the character, or ' universal ' : we are bound to take
the predicate in extension, and that we cannot or do not wish to
do. We cannot affirm of one term the extension of another. If
a set of individuals, or of species, forms the subject of an affirmative

1 We might make them a present of certain forms which they appear to
have overlooked. If the extension of Y be p, q, r, then ' No X is any Y '
means ' No X is either p or q or r '. But the parts of the extension are taken
disjunctively : why should they not be taken together ? Then we should
have the form ' No X is all Y ' — meaning that no X is both p and q and r.
So we might have ' Some X are not all Y '. It is true these forms are
useless ; and in that they resemble the affirmative forms ' All X are all Y '
and ' Some X are all Y '. But they have the advantage over those of being
true.

ix] DISTRIBUTION OF TERMS, ETC. 227

[judgement, another set cannot form the predicate. ' All X is some
F ' is meaningless. ' Some F,' we are told, means ' part of the
class Y ' ; but which part is X ? Let the class Y be divided into
two parts, X and Z ; we do not need to say that X is the former
part ; it is false to say that it is the latter. And in a negative
judgement, unless the predicate is a proper name, which has no
extension, what we wish to deny of the subject is having the pre-
dicate character, not being those individuals which have it.

Still, it is urged, the judgement compares the extension of two
classes. ' All X is all Y ' means that the class X and the class Y are
co-extensive ; ' All X is some Y ' means that the class X is included
in the class F, which extends beyond it. But if the class X and the
class Y are co-extensive, how are they two classes ? Taken strictly
in extension (as the doctrine of the quantification of the predicate
takes its terms) the class X and the class Y are not the common
character X and Y realized in many things, but the set of things
in which this character is realized. If the class X is the things in
which the common character X is realized, and Y is realized in the
same things, then there is only one class or set of things, and not
two co-extensive classes ; so that, after all, we have the class X,
and predicate the character Y of them, i. e. we do not take Y in
extension. And if the class X is included in the class Y, what
does that mean ? Suppose that all F's were collected in one place,
all X's would be found in the crowd ; then, when we said that all X
is some Y, we should mean that all X were included in the crowd of
F's. But now our predicate is no longer Y, and has become ' in-
cluded in the crowd of F's '. We must quantify that if all predicates
are to be quantified, and state whether all or part of what is included
in the crowd of F's be meant. Clearly part ; so that our judgement
will run ' All X are some things included in the class F (or crowd of
F's) '. But which things so included are they ? as before, them-
selves, the X's. If this answer be not accepted, and it be said that
some means ' included in the class of ', then our new judgement must
run ' All X are included in the class of things included in the class
F '. But now the last eleven words become the predicate, and it
must again be quantified ; we must say ' All X are some things
included in the class of things included in the class F '. So the
process goes on ad infinitum. You cannot predicate of one class the
whole or part of another. You may compare the size of two
classes : e. g. when we say that male infants are more numerous than
female ; but then one class is not predicated of another ; female
infants do not include male infants and extend beyond them. You
may predicate a genus of a species, and the genus as compared with
the species has a wider extension ; but it is not the extension of the
genus which you predicate of the species, nor any part of it.

It may be thought that in discussing the quantification of the

Q2

228 AN INTRODUCTION TO LOGIC [chap.

[predicate we have been belabouring errors too trivial for notice. No
one, of course, really supposes that the act of judgement means any
of these absurdities. But many people have supposed that a judge-
ment compares the extension of two terms, or includes a subject in
or excludes it from a class ; and they think of a class as so many
things or kinds of thing. Such views imply the absurdities that
have been dragged to light ; and the custom of elucidating the re-
lation of terms in a judgement by the relative position of circles on
paper, outside each other, one inside the other, or with a common
segment, tends, as has been said before, to make us think wrongly
about a judgement precisely in the direction of these absurdities.
It is of great importance, in speaking of the distribution of terms
(as we shall have to do frequently when examining the syllogism),
not to suppose that the terms of a judgement are all taken in exten-
sion, and that we are always identifying and distinguishing all or
part of what our terms denote. The doctrine of the quantification
of the predicate flourishes upon this mistake, and a thorough
examination of that doctrine is a good prophylactic measure.
Moreover, many of the developments of Symbolic Logic J are based
on the extensional implications of propositions. If I say that all
mammals are craniates, I imply that there are not fewer craniate
animals than mammals ; hence I may write, for ' XisY ', ' X =XY\
and substitute X Y for X elsewhere in my equations. If all organisms
are mortal, and every mortal an organism, I may write ' X = Y ',
and substitute accordingly. When symbols are carefully devised,
we can represent propositions symbolically, and operate with our
symbols without realizing their meaning, and so reach results which
we can retranslate into propositions whose meaning we realize, and
whose truth follows from that of the premisses which we put into
symbols at the outset. But the success of such operations does
not show that we mean by our categorical propositions to assert
numerical equality between classes, but only that, if what we mean
is true, then, whether we determine our class by the subject char-
acter, or by it and the predicate-character together, we shall take
the same things, and so the same number of things. We are not
always thinking of classes and their numerical relations when we
judge. Hence, as it seems to me, the error of representing either all
thinking as a kind of mathematics, or all thinking as class-thinking,
and mathematics as merely a special sort of class-thinking.2]

We may pass now to the opposition of propositions or judge-
ments.

Propositions having the same subject and predicate, but differing
in quantity, or quality, or both, are said to be opposed to one

1 e. g. Jevons's Equational Logic.

2 Cf. Mr. Bertrand Russell's Principles of Mathematics.

ix] DISTRIBUTION OF TERMS, ETC. 229

another. The four forms of proposition A, E, I, 0 admit four kinds
of opposition among them.

1. A — E. Where the propositions differ in quality, and are both
universal, they are called contrary to each other : everything in
Aristotle is true, nothing in Aristotle is true are contrary propositions x

2. / — 0. Where they differ in quality, and both are particular,
they are called sub-contrary : e. g. some things

in Aristotle are true, some things in Aristotle
are not true.

3. A—O, E—I. Where they differ both in
quantity and quality, they are called con-
tradictory : e.g. everything in Aristotle is
true, some things in Aristotle are not true : no
Mussulman fears death, some Mussulmans fear death.

4. A — /, E — 0. Where they differ in quantity but not in quality,
they are called subaltern : e. g. everything in Aristotle is true, some
things in Aristotle are true : no Mussulman fears death, some Mussul-
mans do not fear death.

Contrary and contradictory are terms in common use, though
sometimes treated as equivalent ; the origin of the terms subaltern
and sub-contrary may be seen in the above-given, and ancient, ' dia-
gram of opposition '. / is placed under A, and 0 under E, for the
same reason that in setting out a classification we place the species
under the genus : the wider includes the narrower under it : A and /,
E and 0 are called subaltern, because in each pair one is subordi-
nated to the other : / and 0 are called sub-contrary, because they
are subordinated to the contraries A and E, their respective
universals.

It will be observed that in order to overthrow an universal pro-
position, affirmative or negative, it is only necessary to establish

1 Contraries are what stand furthest apart upon a scale of some kind —
to fidXioTa SiHTTTjKOTa (v ra> avrcp yevci : as white and black on the scale of
illumination, highest and lowest on the scale of elevation, or of pitch, &c.
Contrary propositions are those which stand furthest apart on the scale of
quantity : one asserting that to be true of all which the other asserts to be
true of none. The notion of contradiction belongs properly to judgements
only, and not to terms, though sometimes transferred to the latter, A and
not-^1 (blue and not-blue, &c.) being called contradictory terms. (Cf. Ar.
de Interp. 20a 31-36.) But we have seen that mere not-^4 is no term at all :
there must be some positive meaning. (See however Bradley, Principles
of Logic, p. 119, for the view that all disparate or incompatible terms should
be treated as contraries : e. g. blue and red. ' In logic the contrary should
be simply the disparate.')

230 AN INTRODUCTION TO LOGIC [chap.

the particular negative or affirmative ; that everything in Aristotle
is true is refuted by showing something in his writings false ; that
nothing in Aristotle is true, by showing something true. We con-
tradict the affirmation ' All men are liars ' by saying ' not all ', not
by saying ' all not '. But of course the greater includes the less,
and we refute a proposition by establishing its contrary, as well as
by establishing its contradictory. In common speech therefore we
are said to contradict a proposition when we advance another whose
truth is inconsistent with that of the first, whether it be the con-
trary or the contradictory ; and since the contrary imputes more
error than the contradictory (for if a man tells me that all animals
reason, I impute more error to him by replying that none do, than
that some don't) it may in a sense be said to contradict more fully.
It is, however, convenient to have different words to mark the
relation of A and E to each other, and their relations to 0 and /
respectively ; and Logic confines the title of contradictory opposi-
tion to the latter.

Given the truth or falsity of any proposition, we can see at once
which of the opposed propositions must be true, which false, and
which (upon the information given us) remain doubtful. For
contrary propositions cannot both be true, and therefore if A is
true, E must be false, and vice versa : but they may both be false
(for it is not necessary that either all babies should be disagreeable,
or else none of them), and therefore if one is given as false, the other
remains doubtful. Contradictory propositions cannot both be true,
but neither can they both be false ; and therefore if A, E, I, or 0
is true, 0, /, E, or A must respectively be false, and vice versa.
Subaltern propositions may both be true, or both false, or the
particular may be true while the universal is false ; but the particular
cannot be false while the universal is true, for the greater includes
the less ; hence given the truth of A or E, I or 0 is true, and given
the falsity of / or 0, A or E is false ; but given the falsity of A or E,
I or 0 remains doubtful, and given the truth of / or 0, A or E
remains doubtful. Sub-contrary propositions cannot both be false
(for in that case their respective contradictories; which are contrary
to one another, would both be true) ; but they may both be true,
just as contraries may both be false ; hence given the falsity of /,
0 is true, and vice versa ; but given the truth of /, 0 remains
doubtful, and vice versa.

Of two contrary or of two contradictory propositions one may

ix] DISTRIBUTION OF TERMS, ETC. 231

be advanced against the other, i. e. we may deny one, and advance
the other in its place ; and of two subaltern propositions, the par-
ticular may be advanced against the universal. If any one said
' Some animals reason ', we could not answer ' No, but all do ' ; but
if he said, ' All animals reason ', we could answer, ' No, but some
do '. Sub-contrary propositions, on the other hand, cannot be ad-
vanced one against the other. ' Some animals reason ' : we cannot
retort, ' No, but some don't ' ; ' Some animals don't reason ' : we
cannot retort, ' No (i. e. that is false), but some do '. We may
indeed, to the statement that some animals reason, reply, ' Yes, but
some don't ' ; and to the statement that some animals do not reason,
' Yes, but some do '. In these cases, however, the particular pro-
position ' Some don't reason ', or ' Some do reason ', is advanced
not against its sub-contrary, ' Some do reason ' or ' Some don't
reason ', but against the universal proposition ' All reason ' or ' None
reason ' : which it is feared we might otherwise be supposed to
allow, when we admit that some reason, or that some do not.
Hence it has been urged that we ought not to speak of sub-contrary
propositions as opposed,1 nor include them in a list of the forms of
opposition ; but if they are not opposed, they are anyhow con-
trasted, and that may justify their continued inclusion. Given the
truth or falsity of any proposition, the step by which we pass to
the perception of the truth, falsity or doubtfulness of its several
opposites is in the strictest sense formal. It depends in no way
upon the special content of the proposition, but solely upon the
necessary relations, according to their quantity and quality, in
respect of truth and falsity, between propositions having the same
subject and predicate. And since no other information need be
given, except whether the one proposition is true or false, in order
that we may determine the truth, falsity, or doubtfulness of the
remaining three, the process of inference (if inference it is to be
called) is immediate.

1 Aristotle notices this in Anal. Pri. /3. xv. 63b 27 to yap nvi ra ov nv\
Kara tt]v \i^iv dvTineiTai fiovov (Tor some are is only verbally opposed to some
are not ').

CHAPTER X

OF IMMEDIATE INFERENCES

Inference is a process of thought which, starting with one or
more judgements,1 ends in another judgement whose truth is seen
to be involved in that of the former. This judgement, which, in
relation to the judgement or judgements from which the process
starts, is called a conclusion, must, as compared with them, be a new
judgement ; to repeat in fresh words our original statement is not
inference, any more than translation is inference. For the most
part a new judgement is only got by putting together two judge-
ments, and as it were extracting what they yield. But there are
a few conclusions which we appear to draw not from any ' putting
together ' of two judgements, but simply from the relation to one
another of the terms in one judgement. This is called immediate
inference, etymologically because (in contrast with syllogism 2) it
proceeds without the use of a middle term : but, to put it more
generally, because we seem to proceed from a given judgement to
another, without anything further being required as a means of
passing to the conclusion.3

It was mentioned at the end of the last chapter, that when we
infer, from the truth or falsity of a given proposition, its various
opposites to be true, or false, or doubtful, we perform an act of
immediate inference. We have now to consider other forms ot
immediate inference, of which the fundamental are Conversion
and Permutation (or Obversion).

A proposition is converted, when its subject is made the predicate,

1 Or, more generally, elements, if we allow (with Bradley, Principles of
Logic, pp. 370--373) that, e.g., 2 + 2=4 is inference. But the above is not
intended as a final definition of inference. Cf. infra, p. 244.

2 For the function of the middle term in syllogism, cf. infra, c. xL

8 All inference is immediate in the sense that from the premisses we pass
without the help of anything else to the conclusion ; but this is called
immediate in the sense that from the given relation of two terms in a single
proposition we pass without the help of anything else to a different proposi-
tion. It is doubtful, however, whether, so far as there is any inference in
it at all, it is really in this sense immediate. Cf. the discussion pp. 240 sq.

OF IMMEDIATE INFERENCES 233

and vice versa, its quality (affirmative or negative) remaining
unchanged : as, for example, when from ' No true Mussulman fears
death ' we pass to ' No one who fears death is a true Mussulman '.
The original proposition is called the convertend, and the ne-w
proposition its converse.

Whether, and in what way, a proposition can be converted,
depends on its form, A, E, I, or 0 x : because the process of con-
version is invalid, unless it conforms to the following rule, that no
term may be distributed in the converse, which was not distributed in
the convertend.2 An A proposition is converted by limitation : an
E or an / proposition simply : and an 0 proposition not at all
except through first permuting it.

A proposition is said to be converted simply, when the quantity
of the converse is the same with that of the convertend. In an
universal negative proposition (E) both terms are distributed ; in
a particular affirmative proposition (/) both are undistributed.
Therefore their mutual substitution in the process of simple conver-
sion does not distribute any term that was not distributed before.
Thus E, ' no X is 7 ', becomes E, ' no Y is X ' : e. g. ' no lawyers
are parsons ' — ' no parsons are lawyers ' ; 'no true poet admires
Macaulay's Lays ' — ' no one who admires Macaulay's Lays is
a true poet 3 ' ; 'no snakes suckle their young ' — ' no mammals are
snakes 4 ' ; ' Chatham is not the younger Pitt ' — ' the younger Pitt
is not Chatham \

Again, /, ' some X is Y ', becomes I, * some Y is X ' : e. g. c some
diamonds are black ' — ' some black stones are diamonds ' : ' some
evergreen shrubs flower brilliantly ' — ' some brilliantly flowering

1 The matter of some judgements renders their conversion unnatural, even
where the form allows of it : e. g. ' Civilization spreads by the extermination
of lower races '. Cf. pp. 235-237, infra.

2 Another rule for conversion is sometimes given, to the effect that the
terms (or the subject and predicate) of the converse must be the same as the
terms (or the predicate and subject) of the convertend. But this is not
a rule to observe in converting ; it explains the process of conversion itself.

3 v. M. Arnold, Lectures on Translating Homer, Popular Edition, 1896,
p. 171 : the question before us is not whether the proposition may be rightly
contradicted, but how it may be rightly converted.

4 When the predicate of the convertend is not a substantive or substantival
term, we must either substitute for it in the converse a substantive, if there
be one of equivalent meaning (as in this case), or import some substantival
expression like ' one who ' (as in the previous example) for the original
predicate, now introduced into the subject, to qualify. We often choose the
genus of the subject about which we are speaking, as in the first example
of the conversion of / ; but so far our procedure is not formal.

234 AN INTRODUCTION TO LOGIC [chap.

things are evergreen shrubs ' ; ' some victories are more fatal than
defeat ' — ' some events more fatal than defeat are victories '.

A proposition is said to be converted by limitation, or per accidens,
when, it being universal, its converse is particular. In an universal
affirmative proposition Y is predicated of all X ; but it may attach
to other subjects equally, P, Q, and R ; therefore what is Y need
not be X, and we can only say that some Y is X, not that all Y is X.
To use the language of distribution, the subject is distributed, the
predicate not : if we merely substituted each for the other, the
original predicate, become the subject of an universal proposition,
would be distributed ; for ' all roses are deciduous ' we should have
' everything deciduous is a rose \ We must therefore limit the
extent to which we affirm our original subject rose of our original
predicate deciduous ; and hence such conversion is called ' con-
version by limitation \ So A, ' all X is Y ', becomes /, ' some
Y is X ' : 'all men are mortal ' — ' some mortals are men ' ; 'all
Roman priests are celibate ' — ' some celibates are Roman priests ' ;
' all isosceles triangles have equal angles at the base ' — ' some
triangles with equal angles at the base are isosceles '-1

In the last example, any one who knows geometry will be tempted
to convert simpliciter, and say that all triangles with equal angles
at the base are isosceles. He would not be wrong as a geometrician ;
but he would need a knowledge of geometry, and not merely of logic,
to justify him. In conversion, we look solely to what is justified
by the form of the proposition to be converted, be it A, E, I, or 0 ;
in this respect ' all isosceles triangles have equal angles at the base '
is indistinguishable from ' all isosceles triangles have angles equal
to two right angles ' ; the geometrician knows that it does not
follow from the latter, that all triangles having angles equal to two
right angles are isosceles ; neither therefore does it follow logically
from the former, that all triangles having equal angles at the base
are isosceles. The form of proposition ' all X is Y ' only justifies
a conversion to ' some Y is X ' ; in order to convert to 'all Y is X '
we must know that X and Y necessitate each other, or that there
is nothing accidental in the relation between them ; this is not
implied merely in the one being predicable of the other, because
the relation of a predicate to its subject may be either accidental
or essential. It must at the least be accidental, and therefore
from its bare form, we are entitled to convert an A proposition as
1 With this paragraph, cf. supra, pp. 223-224.

x] OF IMMEDIATE INFERENCES 235

if Y were an accident of X ; but we are not entitled to do more.
For this reason, conversion by limitation is called conversion per
accidens (Kara cry/Ape fir] ko$) ; if Y is an accident of X, i. e. coin-
cides in the same individual subject with X, then X is predicable
of a subject which Y characterizes, and we may say that some
Y is X.1

In a particular negative proposition (0), the subject is undistri-
buted, the predicate distributed ; if here we substituted each for
the other, the original subject, become the predicate of a negative
proposition, would be distributed in the converse. And since the
predicate of a negative judgement cannot, like the subject of a
judgement, be limited by a sign of ' particular ' quantity, an 0 pro-
position is not convertible, except by negation : a process which will
be explained later (p. 238). This is not always realized, when we use
symbols, and forbid the passage from ' some X is not Y ' to ' some Y
is not X ' ; for it is quite possible that both of these propositions
may be true at once : e. g. some freemasons are not freethinkers,2
and some freethinkers are not freemasons. But although ' some
X is not Y ' and ' some Y is not X ' may be true at once, yet we
are not justified by the form of the one in passing to the other ;
and this becomes obvious by comparing such an example as the last
(where both propositions are true) with another, where the converse
is manifestly false : e. g. ' some men are not monks ' — ' some
monks are not men '. In form the two propositions (' some free-
masons are not freethinkers ' and ' some men are not monks ') are
the same ; and therefore formally the conversion must be invalid in
the former case, since it is invalid in the latter.

It is indeed impossible, in converting a proposition, to treat
the terms quite like symbols, and to proceed solely by the con-

1 Even when the predicate is known to be of the essence of the subject,
we must convert per accidens, if the predicate is the genus: e.g. 'all men
are animals ' — ' some animals are men '. We cannot call animal an accident
of man, but we may say that it is an accident that an animal should be
a man, in the sense that an animal may or may not be a man. The term
accident is not wholly suitable, because, though the conditions necessary for
the generation of an animal may exist without those necessary for the genera-
tion of a man, they cannot exist except in a form involving the generation
of an animal of some species, nor can the conditions necessary for the
generation of a man exist without those necessary for the generation of an
animal : there is no coincidence of independent series, as when one series of
events brings a train to a point whither another series has brought a flood
and washed away the metals, and the result is a ' railway accident '. But
the usage is analogous.

a Though certain persons on the Continent seem to believe otherwise.

236 AN INTRODUCTION TO LOGIC [chap.

sideration of the distribution of the terms in the convertend, with-
out considering what the terms are. In an E proposition, for example,
if both terms are proper names, the act of conversion is felt to be
different from what it is where the subject is a general concrete
term and the predicate attributive : in passing from ' no judge
has any right to meddle in politics ' to ' no one who has any right
to meddle in politics is a judge ', the character of the judgement
alters in a way that it does not, when we pass from ' Chatham is
not the younger Pitt ' to ' the younger Pitt is not Chatham '. It
is not natural to say ' no one who has any right to meddle in politics
is a judge ' ; and though it is natural enough to say ' no one who
meddles in politics has any right to be a judge ', this is not the
converse of the proposition with which we started. It is equally
natural to say ' Chatham is not the younger Pitt ' and ' the younger
Pitt is not Chatham ', according as we are discoursing about the
one or the other ; for two individuals stand as it were on the same
level in thought, and each may indifferently be distinguished from
either. But our rights depend upon our position, and not vice
versa ; so that it is natural to deny certain rights to a man filling
a certain position, but not to deny the position to a man possessed
of those rights. Other examples of the same thing might be given.
A proposition both terms of which are singular is called an A pro-
position, but it cannot be converted per accidens : ' Chatham is the
elder Pitt ' can only become ' the elder Pitt is Chatham '. If the
subject is and the predicate is not a singular term, conversion is
a form without meaning ; ' Chatham was eloquent ' becomes ' an
eloquent man was Chatham ', and however we may write it, the
latter means just the same as the former ; we cannot predicate
Chatham of ' an eloquent man ', for this is a general term, and that a
singular.1 Again, 'Demosthenes and Cicero were the greatest orators
of antiquity ' becomes ' the greatest orators of antiquity were
Demosthenes and Cicero ' ; we cannot say ' some greatest orators of
antiquity were Demosthenes and Cicero ' without altering the force
of the term ' greatest orators ' from comparative to positive. 'Some
men are Christians ' is a proper, ' some Christians are men ' an im-
proper mode of speech ; religion can belong only to men, and we
do not predicate of an attribute partially the subject presupposed by
it. A difficulty arises again in a proposition not universal where some

1 We can say ' That eloquent man was Chatham ', but here the subject
is a singular term.

x] OF IMMEDIATE INFERENCES 237

measure is given of the extent to which the predicate characterizes the
subject, e. g. by using such words as ' many ' or ' few ' ; ' most great
men have been of obscure origin ' converts to ' some men of obscure
origin have been most great men ' ; but no one would ever say this,
for the measure ' most ' applies to ' great men ' as taken in extension,
and therefore cannot be predicated of ' men of obscure origin '.

It would be absurd to say that as conversion is a strictly formal
process, we must therefore convert propositions by its rules, accord-
ing to their form as A, E, or I. Logic investigates the actual nature
and procedure of our thought ; and when we find that our thought
is not governed by the bare form of a judgement irrespective of
its content, it is no use to pretend otherwise. The conversion of
propositions may be studied formally, with symbols for terms ;
but when real terms replace the symbols they must affect the
judgement, and our treatment of it in conversion ; for example,
symbols, like X and Y in the proposition ' no X is Y ', are always
regarded as general terms, but the actual terms need not be general.
This is said, not in order to discredit the abstract and formal treat-
ment of conversion, which is sound within its limits ; but in order
to emphasize the fact that the form and matter (or the form and
content) of thought are not capable of separate consideration, like
the mould and the pudding : what from one point of view is form is
from another matter, and the same form in different kinds of con-
tent is not altogether the same, any more than is the same genus
in different species. The importance of this fact must excuse the
reiteration of it ; meanwhile in a text -book of Logic, as of any
other science, we must consider typical cases, with a general caveat
that the subject is thereby artificially simplified.

In conversion, the subject and predicate were transposed, but
otherwise unaltered, and the quality of the proposition remained
the same. In Permutation, or (as it has been also called) Obver-
Sion,1 there is no transposition of terms, but the quality of the pro-
position is changed, and the predicate at the same time replaced
by its contradictory. It consists in fact of substituting for an

1 Jevons, in his Elementary Lessons, calls it Immediate Inference by
Privative Conception. Earlier writers dealt with it under the head of
Equipollency of Propositions : cf. Sanderson, II. 6 ' Aequipollentia com-
muniter sumpta est duarum propositionum, verbo tenus, quoquomodo dis-
crepantium omnimoda in sensu conspiratio '. Aristotle, de Interpr. x. 20a
20-26, notices the equivalence of a proposition and its obverse, but gives no
name to the change.

238 AN INTRODUCTION TO LOGIC [chap.

affirmative or negative proposition an equivalent negative or affirma-
tive of opposite quality, by means of negating the predicate.
Thus—

A, All X is F, becomes E, No X is not- Y : All right angles are
equal, No right angles are unequal ; Barkis is willin', Barkis
is not unwillin'.
E, No X is Y, becomes A, All X is not-F: No dogs allowed, All

dogs forbidden ; Lear is not mad, Lear is not-mad.
7, Some X is Y, becomes 0, Some X is not not- Y : Some stretches
of the road are level, Some stretches of the road have no
gradient.
O, Some X is not Y, becomes I, Some X is not- Y : Some learned
theories are not sense, Some learned theories are nonsense ;
Some swans are not white, Some swans are not-white.
Further transformation of a given proposition may be effected by
a combination of Conversion and Permutation. The process of
permuting and then converting is called Conversion by Negation.
The conclusion so obtained may be permuted again, and this
process of permuting, converting, and permuting is called Contra-
position.

All forms of proposition except I can be converted by negation ;
the process is inapplicable to I, because it becomes 0 by permu-
tation, and a particular negative, as we have seen, cannot be con-
verted. For the same reason I cannot be contraposed.
In conversion by negation —

A becomes E : All X is Y .'. No X is not-F .'. No not-F is X.
All acids turn blue litmus-paper red .'. No acids do not
turn blue litmus-paper red .*. Nothing that does not turn
blue litmus-paper red is an acid.
E becomes / : No X is F .". All X is not-F .*. Some not-F is X.
No stimulant nourishes .". All stimulants are innutritious.
.'. Some things innutritious are stimulants.
O becomes I : Some X is not F .'. Some X is not-F .*. Some
not-F is X. Some sea-animals are not vertebrate .'. Some
sea-animals are invertebrate .". Some invertebrates are
sea-animals. Some things necessary to life have no market-
value .*. Some things that have no market-value are neces-
sary to life.
This is the only way in which a particular negative can be
converted.

x] OF IMMEDIATE INFERENCES 239

In contraposition 1 —

A becomes A : All X is 7 .*. No not- 7 is X .'. All not- 7 is
not-X. All Arabs are hospitable .'. All who are not-hos-
pitable are not-Arabs.

E becomes 0 : No X is 7 .'. Some not- 7 is X .'. Some not- 7 is
not not-X. No unfriendly man is happy .'. Some who are
not happy are not friendly.

0 becomes 0 : Some X is not 7 .'. Some not- 7 is X .'. Some

not- 7 is not not-X. Some reformers are not radicals .*.

Some who are not radicals are not not-reformers (are not

opposed to reform).
The above processes, when worked in symbols, might be supposed
to be equally applicable to all judgements. But when we apply
them to concrete examples, we see at once (as with Conversion) that
it is not so. It is indeed often convenient in discourse to make
what was predicated of a subject itself the subject and starting-
point in our predication, or to lay stress on the affirmative value of
a negative, or the negative value of an affirmative statement. But
the use of these processes is limited in part by the idiom and
vocabulary of the language, in part by the logical character of the
terms in the judgement. The permutation of / to 0 looks almost
ridiculous in symbolic form ; but where there exist two terms, the
affirmation of one of which is equivalent to the denial of the other,
there the process is in practice perfectly natural. No one would
pass from ' Steam is invisible ' to ' Steam is not not-invisible ' ;
but he might naturally pass to ' Steam is not visible '.

Contraposition, as involving the largest number of steps, and
employing permutation twice, may seem to lead to the least
natural modes of expression. For permutation introduces ' infinite '
terms, not- 7 and not-X ; and infinite terms do not ordinarily
figure in speech ; so that unless we can substitute a term that is
not infinite in form, our result seems fantastic. But we may see
that the process of thought involved in contraposition is a common
one (although the mode of expression may be awkward), if we
look at it under the forms of the hypothetical proposition. Given
that all lovers are jealous, it is possible to infer that all the not-

1 What has here been called the converse by negation is by some writers
called the contrapositive (e.g. J. Wallis, Logic, II. 7) ; and what has here been
called the contrapositive, the obverted contrapositive. And the converse of
the obverse of the converse of a proposition has been called its inverse.

240 AN INTRODUCTION TO LOGIC [chap.

jealous are not-lovers. No one would, however, express himself
thus. But the original proposition, if it is a true universal, states
a necessary connexion between the predicate and the subject ; it
involves the proposition that if any one is a lover he is jealous.
Therefore, if any one is not jealous, he is not a lover ; and this is
an inference quite naturally expressed. ' If anything is X, it is
Y .*. if it is not Y, it is not X'; we have here precisely the same
inference as in the contraposition of A, 'All X is Y .'. All not-T
is not-X '. We may interpret in a corresponding way the contra-
position of E and 0, if we bear in mind the modal or problematic
force which may belong to the particular judgement. 'No X is Y *
will mean, ' If a thing is X, it is not Y ' : from this we cannot,
however, infer that if it is not Y it is X ; if a man is insufficiently
fed, he cannot do a proper day's work ; but it does not follow that
if he cannot do a proper day's work, he is insufficiently fed ; this
may or may not be so. Hence we can only infer that ' If a thing
is not Y, it may or may not be X ' : and that is the force of ' Some
not- Y is not-X ', regarded as a modal particular. Similarly with
0 ; ' Some X is not Y ' will mean, ' If a thing is X, it may or may
not be Y ' ; from which it follows that ' If a thing is not Y, it may
or may not be X '.

[The operations whose formal character has been considered in
this chapter are called Immediate Inferences ; but we have seen
that one of them, Permutation, used to be regarded as belonging
to the subject of Equipollency of Propositions, and J. S. Mill x is
not alone in so regarding them all. In his view we have been
dealing merely with equivalent propositional forms ; the processes
are ' inferences improperly so called ' ; and indeed they have once
or twice been called transformations in the course of the text. Thus
conceived, they would belong rather to a study of language than to
Logic. We must therefore consider whether there is really any
inference involved in them or not.2

The question is by no means easy, involving as it does that of
the nature of inference generally. There is no inference where there
is no movement of thought ; but the movement of thought must
spring from a perception of connexion in the objects of thought,
not from subjective conditions in the mind of the thinker ; it must
involve an advance to the apprehension of a fresh object of thought,
and be more than a mere playing as it were upon the same object.
It is not inference, e.g., if the sight of a stormy sea leads one man

1 System of Logic, II. i. 2.

* Cf. Bradley's Principles of Logic, Bk. III. PI. I. c. ii. §§ 30-37.

x] OF IMMEDIATE INFERENCES 241

[to reflect that steam has reduced the terrors of navigation and
another that England owed much to the winds in 1588. Nor, if
a fact involves two terms in a common relation, is it inference to pass
from a statement that makes one term the subject standing in
relation to the other to a statement making the second the subject
standing in relation to the first. For the difference of subject and
predicate, as Professor Cook Wilson insists, is subjective ; it belongs
to the order of our approach to the complete act of judgement, in
which we think the whole fact, and makes no difference to what,
in that act, we think the fact to be. When Achilles was sought,
and found playing with the maidens, the seekers were surprised to
find Achilles their companion, the maidens that their companion
was Achilles ; but both became aware of the same complex fact.
I may live by the Atlantic mouth of the Panama Canal, and learn
one day that it is west of the Pacific mouth, or by the Pacific
mouth, and learn that it is east of the Atlantic mouth ; but in either
judgement I should be aware of the same fact, and there is no in-
ference from one to the other.1 Again, there is no inference from an
universal proposition to its subaltern, though they are not the same,
because what is thought in the latter is only part of what is already
thought in the former ; there is no advance to the thought of some-
thing not thought of, though bound up with what was thought of at
the outset. On the other hand, the obviousness of a transition is no
ground for denying that it is inference, though lack of obviousness
might be taken as a sign that inference is present ; for if in thinking
the premiss we had also thought what is stated in the conclusion,
it could not come to us as a surprise, that we had committed our-
selves to the latter. Neither again is the fact that the conclusion is
implied in the original statement a ground for denying the presence
of inference ; for all premisses imply their conclusion.

We must bear in mind also that the same propositional form may
express different thoughts, and whether there is inference will depend
on the thought which the words express. It is particularly impor-
tant to remember this when working with symbols. Symbolic
notations will often enable us to operate more rapidly than with
words, and without realizing in the process what is meant ; and
when we translate into words the result reached, it is sometimes one
which we should not very readily have seen to be involved in what
we started from, but sometimes also one not warranted thereby.
Thus we may argue in symbols, converting and obverting, ' No
X is Y .-. No Y is X .-. All Y is non-X .-. Some non-X is 7 '. The
original proposition might be ' Things made of asbestos do not burn ',
and the final conclusion ' Some things not made of asbestos burn ' ;
and this arouses no suspicion. But let the original proposition be

1 Such restatements have nevertheless been sometimes called immediate
Inferences.

"779 B

242 AN INTRODUCTION TO LOGIC [chap.

[' No man dies twice ', and we can hardly accept the conclusion
' Some who are not men die twice '. We might hesitate even about
the simple converse, ' Nothing that dies twice is a man ', as implying
an admission that dying twice does occur. Such paradoxes arise
because in working out symbolic sequences we are considering only
what relations of subject and predicate are excluded and what left
possible by the information given ; and the inference to ' Some non-X
is Y ' is intended to mean not that there exist things not X which are
Y, but that the fact that nothing which is X is Y does not exclude
their existence. But propositions in significant terms commonly imply
the existence of instances of their subjects. Not however always ;
and when we pass from a premiss implying the existence of its subject
to a conclusion only asserting compatibility of attributes, or such con-
nexion between them that if there were an instance of one it would
also be an instance of the other, then, and also vice versa, there
is inference which would not equally exist if both propositions
were understood in the same sense. Such inference however may
involve the use of some other premiss besides the convertend ex-
pressed ; and mutatis mutandis the same would be true in obversion.

A categorical proposition commonly implies the existence of
instances of its subject, and therefore, if it is affirmative, of its pre-
dicate also.1 But in making it we may or may not have determinate
instances in mind. We found that the form ' All X is Y ' is some-
times used to state a fact about all members of the group or class X,
sometimes to state a connexion between being X and being Y ; in
the former case, it might be said to be intended historically (e. g.
' all the ruminants part the hoof '), in the latter scientifically (e. g.
1 all rivers run down hill '). But if intended scientifically, the pro-
position need not be intended to assert the existence of instances
of the subject ; e. g. ' a perfect fluid is frictionless ' may be intended
only as a statement of what would be the character of a perfect
fluid if it existed ; and then, though categorical in form, it is intended
only hypoihetically. And a particular categorical might be said to
be intended historically when we make it with instances in mind,
e. g. if we said that ' some garrison towns are important civilly ',
thinking of Winchester, York, and Canterbury : and scientifically
when we wish rather to affirm the compatibility of the subject and
predicate characters (or, if the proposition be negative, the possi-
bility of their disjunction). In the latter case, however, we more
commonly use the modal form ' X may be Y ', than the categorical
particular ' Some X is Y '.

Let us now consider the simple conversion of an / proposition.
Any one starting from the judgement that ' some garrison towns
are important civilly ', whether he has in mind definite instances
or not, must know or believe the fact stated in the converse, that

1 This is sometimes called its existential import.

x] OF IMMEDIATE INFERENCES 243

[' some places civilly important are garrison towns '. The fact, of
which VVmchester, York, and Canterbury are instances, is the same,
whichever way it is put : whether the logical subject be ' some
garrison towns ', or ' some places civilly important '. There is there-
fore here no real inference. There could be inference only if from
a judgement in which we are thinking definitely of certain towns,
though not naming them, we passed to one asserting general com-
patibility. But here in effect we should be passing from the
proposition that Winchester, York, and Canterbury are important
civilly, and to the proposition that some towns civilly important
are garrison towns. This is inference, but syllogistic, not im-
mediate ; and we should not express it by such verbal variation as
is symbolized in passing from ' Some X is Y ' to ' Some Y is X \
The conversion of I then is not a process of inference.

The conversion of the universal affirmative A has more show of
inference, because it proceeds by limitation ; and it might be urged
that there is inference in seeing that I am not entitled to infer that,
since all the ruminants part the hoof, all the cloven-footed animals
ruminate. But surely I know from the outset that in affirming Y
of X, I do not confine the predicate to that subject ; and to realize
that Z also may be Y is to realize that what is Y need not be X.
It can hardly be called inference to realize that information about
X does not extend beyond X, nor to refrain from asserting what I
know that I have no right to assert.1 And I must in the original
proposition, whether understood historically or scientifically, if I im-
plied the existence of instances of the subject at all, have meant that
these were also instances of the predicate ; and therefore I must have
realized that some things exhibiting the predicate character exhibit
also the subject character, which is what is stated in the converse.
So far, therefore, in the conversion of A there is no real inference.

But the universal affirmative, intended scientifically, does not
always imply the existence of instances of its subject. Tout savoir
est tout pardonner ; I might translate this by saying ' Those who
know all pardon all ', not implying that any of us does know every-
thing, but only that, if he did, he would pardon everything. Now
if I convert this and say ' Some who pardon all know all ', I shall
probably mean that there are persons who both pardon and know
everything. Here then there will have been inference ; but again,
it does not lie in the conversion. It lies in combining the thought
of the general connexion with the thought that there are some who
know all about some situations ; and so concluding that there are
some who pardon all in some situations. The inference involves
a premiss not expressed. To pass from the merely hypothetical sense
of an universal affirmative to the categorical involves inference, but

1 Cf. Bradley, loc. cit.
E2

244 AN INTRODUCTION TO LOGIC [chap.

[hypothetical inference \ not conversion. To pass from meaning it
historically to meaning it scientifically is inference, but it is induc-
tion.2 It is more difficult to say whether, if we mean it scientifically,
but categorically, there is inference in passing to a purely hypothetical
meaning : suppose, e. g., that I judge that ' all rivers run down hill ',
meaning that by their nature as running water they must do so,
is it inference to pass to the thought that any other rivers, if they
did exist, would also run down hill ? I think not ; in the necessary
judgement there is really inference from the outset ; it is essentially
inference to see that if a condition X is realized, Y must be realized
too ; I advance herein by mere thinking from X to Y. But if I
have realized this in considering existing instances, there is no further
inference in seeing that it would hold in others.

The last point needed notice in relation to the conversion of the
universal negative, E. ' No X is Y ' converts simply to ' No Y is
X '. The convertend implies commonly the existence of instances
of X, but not necessarily of Y ; the converse however does imply the
existence of instances of Y. Now if in the convertend it be meant
that there are instances of both X and Y, the thought that the latter
are not the former hardly seems separable from the thought that
the former are not the latter ; and there seems to be no inference
from ' No fish are mammals ' to ' No mammals are fish \ If how-
ever this be not meant in the convertend, and in the converse it be
meant that there are instances of Y, then there is inference, but it
involves another premiss. I might judge that ' nothing inductive
is self-evident ', while doubtful whether anything is self-evident ;
if I proceed to judge that ' nothing self-evident is inductive '
meaning that there are self-evident propositions, the judgement that
these are not inductive comes by help of the convertend, but that
they exist at all is independent of it. Still, I cannot reach the
universal ' nothing self-evident is inductive ' without realizing that
if anything were self-evident, it would not be inductive ; and this
connexion of condition and consequent is not the same as what is
realized in the universal negative from which I started ; that was,
that if anything were inductive, it would not be self-evident. From
' if X, then not Y ' to ' if Y, then not X ' does seem to be infer-
ence, the condition being different in the two. It is true that it may
easily be shown that I cannot repudiate this conversion without
self-contradiction ; if a thing might be Y and still be X, then since,
if X, it is not Y, it might be Y and not be Y. But though it is im-
possible to affirm the convertend and deny the converse without
contradiction, inference is involved in realizing this, and the con-
verse is not actually thought in thinking the convertend. Only then
if an E proposition be intended as a statement that two groups of
instances exclude each other (or that the individuals indicated by
two singular terms are different), is its conversion not inference.

1 Cf. infra, o. xv. * Inductive syllogism in Fig. 3. Cf. infra, p. 319.

x) OF IMMEDIATE INFERENCES 245

[As the conversion of 0, the particular negative, is impossible
without first permuting, or obverting, it to /, we must ask next
whether there is inference in Permutation. The process of Permu-
tation involves the use of the infinite or negative term not-F in the
predicate in lieu of F. Now we have seen that an infinite term has
not any meaning at all unless it has some positive meaning ; not- Y
must mean something else than Y.1 We have seen also that the dis-
junctive judgement 'A is either B or C does not always imply that it
cannot be both. But Permutation rests upon disjunction ; Y and not-
Y are alternatives, and it is assumed that if Y is affirmed or denied of
any subject, not-Y can be denied or affirmed accordingly. Bearing
in mind these considerations, we shall find that there is a certain
difference in different cases, in respect of the presence of any real
inference in permutation, according to the meaning attached to the
negative term.

It is unnecessary here to separate universal and particular propo-
sitions. If we are told that X is not Y, and Y and not-Y are
alternatives, one of which must attach to it, then since it does not
exhibit Y, it must exhibit the other, not-Y. We thus reach the
affirmative, ' X is not- Y ' ; and the question is whether that is any
way different from the negative with which we started.

Now we cannot deny that there is any inference in disjunctive
reasoning at all. When I argue that since A is either B or C, and is
not B, therefore it is C, there is clearly inference ; and I could not
argue that, because A is not B, it is C, unless I were given the
disjunctive premiss, A is either B or C, as well. But in permuta-
tion, my alternatives are not two different positive terms, like B
and G, but Y and not-F. Is there any inference in saying that
because X is not Y, it is not- Y ?

It will be allowed that the conclusion would not hold unless X
were either Y or not- Y. But it may be said that this, the ' principle
of Excluded Middle ', like the Principle of Contradiction, though
true, is not a premiss of inference. No one knows what he means in
saying that X is not Y, unless he sees that in that case it is not-F :
any more than he can know what he means in saying that X is F,
unless he sees that in that case it is not not-F. If a proposition is
true, its contradictory is false ; but there is no step from the truth
of the one to the falsity of the other, no movement of thought ;
since the truth of the one is not apprehended without apprehending
the falsity of the other.

If the infinite term not-Y were purely negative, this view of the
matter would demand assent. But F and not-Y are in practice
always alternatives within some definite limits. F may be blue,
and then not- Y will be of some colour not blue : or F may be English-
speaking, and not-Y speaking some language not English. And in

1 Otherwise, the term is Y, and the form not-Y only shows that Y is being
denied of some subject in a judgement.

246 AN INTRODUCTION TO LOGIC [chap.

[passing from one of these predicates to the other, there is inference,
and we do not rely merely on the law of Excluded Middle. ' Noble
blood is not blue .*. it is not-blue ' : if this means ' of a colour not-
blue ', we require the further premiss that it is either blue or of some
other colour. We thus pass from a determinate positive predicate
to another predicate less determinate, but still positive.

If however there is no positive alternative meaning in the
predicate not-Y, then indeed there is no inference, but only equi-
pollency. ' Steam is not visible .-. it is in visible ' seems a mere sub-
stitution of one equivalent expression for another. It follows, that
we cannot tell by the mere symbolic form whether the permutation
of a negative proposition contains any real inference or not, but
must look to the content x ; and if it contains real inference, the
inference is disjunctive.

The permutation of an affirmative proposition may, like this last,
be no real process of inference. We pass here from ' X is Y *
to ' X is not not-Y'. It is not always possible to find in this
any other meaning than that from which we started. We cannot
always interpret not- Y to mean ' possessed of some other of the range
of alternatives to which Y belongs ' ; if a subject must display some
one out of a given range of alternatives, and does not display Y, it
will display one of the others ; but if it does display Y, we cannot
be sure that it may not display one of the others as well. If a
man holds office in the Government, and does not hold an office
that entitles him to Cabinet rank, he must hold an office that does
not entitle him to Cabinet rank ; but if he does hold an office that
so entitles him, he may also hold one that does not. Equally, if
not-Y is quite unlimited in range, and includes everything whatever
except Y, it will not follow that because X is Y, it is not also not- Y ;
because we can predicate of a goose that it hisses, we are not pre-
cluded from applying any predicate but hissing. The only sense,
therefore, in which it is true to say that X is not not-F, is one in
which we deny no alternative, but only deny the denial of Y ;
and that is just equivalent to the affirmation of Y, or at least
can hardly be said to involve any inference from it. If however
we have in mind a range of mutually exclusive alternatives among
which Y is one, then permutation takes us from the affirmation
of Y to the denial of the rest ; and this is again disjunctive reasoning,
wherein the conclusion will be more or less definite according to
the definiteness of our knowledge of the alternatives to Y. But

1 The reader may be reminded, that among the range of alternatives
which the denial of a positive term leaves open, the corresponding negative
term has often come to signify one only. Not-blue may cover all colours but
blue ; but unfriendly does not cover all tho alternatives to friendly ; it
implies a definite degree of hostility which may be absent in those who are
not positively friendly to us. But this is a matter of the interpretation of
language rather than one of Logic.

X] OF IMMEDIATE INFERENCES 247

[so far as there is inference here, there is no use of an infinite term ;
where not-Y is really infinite or unlimited, the only sense in which
the permutation of an affirmative proposition is logically justifiable
is one in which it involves no step of inference.1

If this is a just account of the nature of permutation, then any
inference there may be, apart from disjunctive argument, in con-
verting by negation, must lie in the converting. And the conversion
of 0 by negation will no more be inference, if the permutation of it
is not, than the simple conversion of /. Indeed no one believing
that there existed things which are not F could judge that ' some
X are not Y ', without at the same time thinking that some things
which are not Y are X Similarly the conversion of E by negation —
1 No X is Y .-. some non- Y is X ' — is like the conversion by limitation
of A i H there is anything new in the converse, it is the implied
assertion that there exist instances of what is X, an assertion which
the convertend, if intended hypothetically, did not contain. ' A
perfectly wise man does no wrong, .-. some who do no wrong are
perfectly wise ' ; it is not converting by negation that would justify
us here in passing from a sense of the convertend in which it does
not imply that any one is perfectly wise to a converse that does.
In converting A by negation on the other hand there is inference
to the extent that there is in simply converting E. ' All X is Y .•.
No non- Y is X ' involves the transition from ' If X, then Y ' to ' If
not Y, then not X ', which may be indifferently expressed by
' No not-Y is X ' or ' All not-F is not-X ' — i. e., the inference is in
the conversion, not in the second act of permutation, by which
some distinguish contraposition from conversion by negation.]

The immediate inferences which we have considered so far have
all been of a more or less formal character ; as is shown by the
fact that they have been capable of explanation, up to a point,
by using symbols and not real terms. There are certain kinds of
inference, which have been called immediate, that cannot be
exhibited by symbols at all, but only in concrete. One of these is
known as Immediate Inference by Added Determinants : in which
we add the same qualification to both subject and predicate in
a proposition, and hold the result of our operation to be true, on
the strength of the truth of the original proposition ; e. g. ' A negro

1 This is no doubt why Wallis (cf. p. 239, n. 1, supra) did not distinguish
contraposition from conversion by negation. ' Hanc formulam locum habere
docent in Particulari negativa. Atque huius potissimum causa videtur fuisse
introducta : ut quae per neutram reliquarum converti possit. Puta. Aliquod
animal non est homo : ergo, Aliquod non-homo non est non-animal ; seu
(quod tantundem est) Aliquod non-homo est animal ; seu, Aliquod quod non
est homo, est tamen animal.' loc. cit.

248 AN INTRODUCTION TO LOGIC

is a fellow creature .'.a negro in suffering is a fellow creature in
suffering '.x Another is called Immediate Inference by Complex
Conception : in which the subject and predicate of a given proposi-
tion are used to qualify in some way the same term, and thus
complex concepts are formed, that are made subject and predicate
of a new proposition, e.g. ' Physics is a science .". physical treatises
are scientific treatises '. The following examples, some of them
sound and some unsound, but the sound identical in form with the
unsound, will serve to show that the ground of the soundness of
these arguments does not lie in the form of them : —

The horse is an animal .' . the head of a horse is the head of an animal.

Horses are animals .*. the greater number of horses is the greater
number of animals.

A shark is not a mammal .'. the anatomy of a shark is not the
anatomy of a mammal.

A shark is not a mammal .'. the food of a shark is not the food
of a mammal.

A shark is not a dog . * . the owner of a shark is not the owner of a dog.

It is not worth while multiplying arguments to show how entirely
the validity of such inferences as these involves their content. It
would not be possible to reduce them to a definite number of fixed
types, though in considering generally which are valid, some of
Aristotle's observations in the Sophistici Elenchi, especially those
on what he calls the Fallacy of Accident, would be pertinent. But
their mention here will serve to illustrate, what it is well to realize
early, that inference is not a purely formal process ; that argu-
ments are not all built on the principle of American watches, with
interchangeable parts,2 so that terms from one may be transferred
to another, without interfering with the working of the inference ;
and that the study of inference, like the study of life, is largely
a matter of examining types : though there are a certain number of
common forms, which recur identically in divers contents. One of
the most famous of these common forms is the Syllogism, to which
we must now proceed ; it has often been regarded as the form of all
inference whatever that is not ' immediate ' ; it is indeed highly
general, and found in all kinds of subject-matter ; though the nature
even of it cannot be profitably studied altogether in the abstract, but
is to some extent affected by the concrete character of its terms.

1 Thomson, Laws of Thought, § 55.

2 v. Marshall's Principles of Economics, Bk. IV. c. ix. § 4.

CHAPTER XI

OF SYLLOGISM IN GENERAL

Aristotle, who was the first person to work out the theory of
syllogism, though not, of course (as Locke maliciously suggests
that his followers claimed), the first to reason syllogistically, defines
a syllogism as follows : Ao'yos Zv <o reflevraw tlvu>v erepov rt, t6>v
Kei[A€V(DV i£ avayKrjs <rv/x/3airei tu ravra elvcu * : that is to say, ' dis-
course in which certain things being posited, something else than
what is posited necessarily follows merely from them '.

This definition is too wide. It covers, as the word syllogism in
its etymological signification itself covers, every argument in which
from a consideration of two truths we infer a third — every argument
in which (to use a homely phrase) we ' put two and two together ',
and find a certain conclusion necessarily following.2 But neither
by Aristotle, when he investigated in his Prior Analytics the various
forms of syllogism, nor by the world, which has followed Aristotle,
has the term been actually used so comprehensively. A syllogism
is actually an argument in which, from the given relation of two
terms, in the way of subject and predicate, to the same third term,
there follows necessarily a relation, in the way of subject and predicate,
between those two terms themselves.3

Example will best explain what is here meant by the words
italicized. If A is equal to B, and B is equal to C, then A is equal
to C. If a bullet travels faster than a horse, and a horse travels
faster than a man, then a bullet travels faster than a man. Now
here the terms are A, B, and G : or a bullet, a horse, and a man ; but
the relations between the terms are in the one case relations of
quantity, in the other of velocity. A and B are not related as

1 Anal. Pri. a. i. 24b 18 : cf. Top. a. i. 100a 25, where the same definition
recurs, with the substitution of 8ia rusv Keiyxvoni for rw ravra eivai.

2 ' Putting two and two together ' is often a process which leads people to
conclusions of a highly conjectural character. In such cases, their reasoning
does not come under the Aristotelian definition : for it is expressly stated
by him that the conclusion must be inevitable — e'| dvayicTjs.

3 Bradley's Principles of Logic, Bk. II. Pt. I. c. iv. § 10, et alibi.

250 AN INTRODUCTION TO LOGIC [chap.

subject and predicate, for I do not say of A that it is B, but only
that it is equal (in quantity) to B ; a bullet and a horse are not
related as subject and predicate, for a bullet is not a horse ; its
asserted relation to a horse is in the way of travelling faster, not in
the way of being a subject whereof horse is a predicate. No doubt
it is a predicate of a bullet, that it travels faster than a horse, as it is
a predicate of A to be equal to B ; but then what I proceed in my
argument to compare with C is B itself, and not that which is equal
to it ; what I say travels faster than a man is a horse, and not what
travels faster than a horse. A, B, and C, a bullet, a horse, and a
man, are the terms which I compare, the former in respect of quan-
tity, the latter of velocity ; and from the given relations of A and G
to the common term B, in the way of quantity, I deduce a relation
between A and C themselves in that respect ; or from the given
relations of a bullet and a man to a horse in the way of velocity,
I deduce a relation in the way of velocity between a bullet and a man.
Now the relations between the terms of an argument may be in
the way of subject and predicate ; and then the argument is a syllo-
gism. Let us for the present use the symbols X, Y, and Z to
represent terms related in this way. Suppose that X is predicated
of Y, and Y of Z ; then X must be predicable of Z. For example,
silver prints fade in the sun ; and the photographs which I have
bought are silver prints ; therefore they fade in the sun. Here the
term common to the two premisses (for such the given propositions
are called, from which the conclusion is deduced) is silver prints ( Y) :
that is predicable of the photographs which I have bought (Z), and of
that is predicable to fade in the sun (X) ; hence to fade in the sun (X) is
predicable of the photographs which I have bought (Z). Or again,

Y may be a predicate affirmed or denied both of X and Z ; in the
Dreyfus affair, the French War Office frequently argued that the
man who wrote the famous ' bordereau ' was on the General Staff :
Esterhazy was not on the General Staff, and therefore did not write
it ; here Y (being on the General Staff) is affirmed of X (the man who
wrote the ' bordereau ') and denied of Z (Esterhazy) ; and hence X is
denied of Z — Esterhazy did not write the ' bordereau '. Yet again,

Y may be a subject of which both X and Z are predicates affirmed or
denied ; then X may be predicable of Z, or vice versa. The horse
is strong, and is an animal that lives exclusively upon a vegetable
diet ; therefore an animal that fives exclusively upon a vegetable
diet may be strong. Here we have two terms, strong (X) and being

xi] OF SYLLOGISM IN GENERAL 251

an animal that lives exclusively upon a vegetable diet (Z), affirmed as
predicates of the same term (Y), the horse ; and we hence deduce
that X, strong, is predicable of Z, an animal that lives exclusively upon
a vegetable diet, not indeed necessarily and universally, but as a
possibility in certain cases.

These examples may perhaps explain what is meant by terms
being related in the way of subject and predicate, and how the
relation of two terms in that way to a common third term may
necessitate their relation in the way of subject and predicate to one
another.

What is here called a relation in the way of subject and predicate
may be also called a relation in the way of subject and attribute ;
as it is called, for example, by Mr. Bradley in his Logic, Bk. II.
Pt. I. c. iv. § 10, and elsewhere. If the word attribute is used, it
must be understood generally of anything predicated x ; it is an
attribute of Baal to be a god, to be talking, to pursue his enemies,
to be on a journey, to be asleep, to need awakening, to have 450
prophets in Israel, to be worshipped by the Phoenicians ; whatever
can be affirmed or denied of him is an attribute affirmed or denied ;
the attribute may be in any category, of substance (as when we say
that he is a god), of quality, time, place, state, relation, &c. ; the
only thing necessary is that it should be related to him as what can
be predicated of it to a subject, not (for example) as an uncle to
a nephew, as yesterday to to-day, as cause to effect, as here to there,
as means to end, as more to less, &c. ; all of these are relations in
which terms may stand to one another, if we mean by terms distinct
subjects of thought, and not merely the subject and predicate into
which the judgement which affirms their relation is resoluble. Thus
when I say that the Old Pretender was nephew to Charles II, he and
Charles II may be called the terms placed (in this judgement) in
a relation of consanguinity ; he and ' nephew to Charles II ' are the
terms placed in a relation of subject and attribute. When I say
that Edinburgh is west of Liverpool, Edinburgh and Liverpool are
the terms placed in a space-relation ; but Edinburgh and ' west of
Liverpool ' the terms placed in a relation of subject and attribute.
Understanding the word in this comprehensive sense, we may say
that the theory of syllogism is the theory of inference in the domain 2

1 i. e. in a wider sense than it is used in when the attributes of anything
are distinguished from its substance or kind, and its relations.

2 By a domain here is meant a certain order or system of relations, of

252 AN INTRODUCTION TO LOGIC [chap.

of subject and attribute, just as well as in the domain of subject and
predicate. But it is important to remember that ' attribute ' is
being used in a wider sense than it usually bears ; we should not
ordinarily call it an attribute of Mr. Pickwick to have been once
impounded ; or of Becky Sharp to have thrown Dr. Johnson's
Dictionary out of the carriage window ; the word is not ordinarily
understood to include actions, or the casual relations of one thing to
another ; but in its present use, it includes every predicate. The
advantage of using it is this, that inference depends on perceiving
relations in what is thought of, and in taking the word attribute
instead of predicate, we take a word expressing a real for one express-
ing a logical relation. Blue is an attribute of the star-gentian
really and always : a predicate, only when one judges that the
star-gentian is blue. It is true that in the theory of syllogism we
have to do with attributes only so far as they are predicated ; but
we think of our predicates as attributes.

It has often been held that the syllogism is the type of all reasoning,
except the inferences called immediate.1 No one has done more
to dispel this illusion than Mr. Bradley, in his Principles of Logic ;
though perhaps the zeal of an iconoclast has prevented him from
dwelling enough on the fact that the syllogism formulates reasoning
which is very frequent in occurrence. But our present business is
to become familiar with the theory of syllogism on its formal side.
There is a precision and completeness about this theory, which have

a single kind : as we might call space a domain in which all material things
are related, and time a domain in which all events are related. The domain
of subject and attribute is far less unified than that of space and time.
A thing related to one other thing in space, or an event related to one other
event in time, is necessarily related in those ways to all others. But a term
related to a second term in the domain of subject and attribute is thereby
necessarily related in that way only to those further terms, if any, to which
the second is related in that manner (and not necessarily to all of them).
The domain of subject and attribute is, as it were, a system of relations
embracing group after group of terms, but not necessarily connecting any
of the terms of separate groups ; whereas time and space, which connect
group after group of events or bodies, necessarily connect also any two
members of any two groups. The word category might have been employed
instead of domain, in the Kantian sense of a principle of synthesis or relation.
But it was employed on the last page in the Aristotelian sense of a kind
of predicate (determined indeed, on Kant's view, by the principle, or principles,
of synthesis employed), and has been generally employed in the text in that
sense ; and it would have introduced confusion either to employ it without
notice in a different sense, or to interrupt the present subject in order to
point out the distinction between them.

1 e.g. Hobbes, Art of Rhetoric, Bk. I. c. i, ' all inferences being syllogisms ' :
v. Molesworth's ed., English Works, vi. 423.

xi] OF SYLLOGISM IN GENERAL 253

made logicians dwell on it with something of an artist's concen-
tration ; and the truth of science has sometimes been sacrificed to
neatness of exposition.

The business of syllogism is to establish a relation in the way of
subject and predicate between two terms, by means of their relations
in that way to the same third term. But the proposition which
relates two terms as subject and predicate may be universal or
particular, affirmative or negative.1 Moreover, we have seen that
there are various ways in which the two terms that are to be brought
together in the conclusion may be related to a common third term ;
both may be predicated of it, or it of both, or one of it and it of the
other. Therefore the following general problem presents itself to us,
— Writing S for any subject, P for the predicate which is to be
brought into relation to it, and M for the third or middle term whose
relations with 8 and P are to bring them into relation with each
other, we may ask — What must be the quantity and quality of the
propositions (or premisses) connecting S and P respectively with M ,
and in which relation, viz. subject or predicate, must M stand to S
and P in these premisses, in order to establish in the conclusion
a proposition whose terms are S and P, of the several forms A, E, I,
and 0 ? In other words, what forms of premisses will prove that all
8 is P, no S is P, some S is P, or some S is not P, by means of the
relations, in the way of subject and predicate, of S and P respectively
to M ? Or, yet again, what relations in the way of subject and predicate
between two terms S and P respectively and a common third term M will
establish what relations in the way of subject and predicate between
those two terms themselves ? This is the question, put in its most
abstract form, to which the formal part of the theory of syllogism is
an answer.

1 When it said that a judgement, or proposition, 'relates' terms, 'places'
them in a relation, and so forth, it must not be understood that the terms of
thought come to stand in such relations through that act of judgement. My
judgement is my apprehending, or coming to believe, that they stand in
such relations, and the proposition expresses this apprehension or belief, or
asserts what is apprehended or believed.

CHAPTER XII

OF THE MOODS AND FIGURES OF SYLLOGISM

A. Nomenclature. 1. In any syllogism, there are two proposi-
tions taken as true, and another inferred or following from them.
The latter is called the conclusion (Lat. quaestio or conclusio, Gk.
7rpd/3Ar;jua or avjx-nlpaaixa) : the former the premisses (Lat. praemissa,
Gk. irpoTcicre is).

It was said, that the premisses are taken as true : whether they are
true or false, the conclusion which they yield is the same ; only
that if they are true, it is true, and if they are false, it is probably
false.1 We are not concerned, therefore, in the formal theory of
syllogism, with the truth or falsehood of our premisses or our con-
clusion, but only with the validity of our reasoning : we wish to
know, if the premisses are granted, what must be granted as follow-
ing from them. If our reasoning be correct, a man cannot con-
sistently admit the premisses, and deny the conclusion. Suppose
that a man admits that every restriction upon freedom of contract is
mischievous, and admits that the marriage laws restrict freedom of
contract, then he must admit the marriage laws to be mischievous.

It has been made a reproach to the theory of syllogism, that it
looks only to the cogency of the inference, and not to the truth of
the premisses. We need rules, it is said, by which to determine
whether a proposition is actually true, and not merely whether it
is true, upon the hypothesis that certain other propositions are so.
The theory of syllogism is decried as a Logic of Consistency ; for
the most that it can do is to furnish rules by which to judge whether
different assertions are consistent with one another. In rivalry
with the Logic of Consistency, some writers have projected a Logic
of Truth, and offered it to the world under the name of Induction.2

1 Not necessarily, because a true conclusion may follow from false
premisses (cf. infra, p. 334). But a conclusion correctly drawn from false
premisses implies ignorance in the reasoner, though not ignorance of
reasoning.

■ Cf . Mill, System of Logic, III. iii. 9.

MOODS AND FIGURES OF SYLLOGISM 255

But it has been unfortunately discovered that the ' Inductive
Methods ' that were to test the truth of the premisses, from which
the doctrine of syllogism enquires what may be inferred, suffered
from the same defect as the syllogism itself ; for they also were
processes of inference, in which conclusions were drawn from
premisses ; their conclusions were only true, if the premisses were
true ; they showed themselves quite unable to determine whether
their premisses were true or not, though it was generally just on
that point that disputes were most pronounced.

The fact is, that so far as reasoning can be reduced to fixed
forms at all, and these forms studied in the abstract — whether
or not the forms are syllogistic — we must disregard the truth
of the premisses ; for in expounding an abstract form of reasoning
we may even use symbols for terms,1 i. e. we do not trouble our-
selves to ask what in particular the terms are at all ; and hence
we cannot be asking whether the judgement which connects them
is true.2

Given then the premisses, the conclusion follows necessarily ;
but it may nevertheless be false, if the premisses are false. The
premisses, however, need not in the first place be given, they may be
wanted.

Supposing a man to have admitted that whatever discourages
thrift and independence is evil ; and to have admitted that an
universal system of pensions in old age at the cost of the state
discourages thrift and independence : then he must admit as a con-
clusion that such a system is evil. Here, and to such a man, the
conclusion presents itself in the first place as a consequence of
what is already granted or ' given '. But supposing a man to
be in doubt whether an universal system of pensions in old age
at the cost of the state is evil or not, and to be wanting some
proof, one way or the other ; and that a friend offers him the
above ' premisses ', as showing that it is evil : then, and to him,
the ' conclusion ' presents itself in the first place as a question or
problem, about which he wants to know whether he is to affirm
or deny it ; and syllogism is a process of finding proof, rather than
of drawing consequences.

1 As J. S. Mill does in expounding his Inductive Methods : but his symbols
are very inadequate.

2 Yet inference is at bottom a perception of connexion among facts, and
how can we perceive any in premisses that are not true ? On this difficulty
cf. infra, pp. 331-334.

256 AN INTRODUCTION TO LOGIC [chap.

It makes of course no difference to the form of premisses which
will establish a particular form of conclusion, whether the premisses
be first known, and the conclusion discovered as a consequence : or
the conclusion raised as a problem, and the premisses discovered to
settle it. And in either case alike, the premisses are ' given ' in the
sense of being admitted and not proved in the argument. But they
are not always ' given ' in the sense of being that with which a man
begins : our thought is as often occupied in looking for premisses
to establish what we believe or suspect, as in looking at premisses
to see what follows from them. And that is why Aristotle used
the expressions -np6jiXr]p.a and 7rpora<rei9. For him, the conclusion
was generally regarded as something to be proved x ; the premisses,
as something proffered in proof of it ; and so he asked rather,
1 What kinds of premisses are required to prove various kinds of
conclusion (A, E, I, and 0) ? ' than ' What kinds of conclusion
follow from various combinations of premisses ? ' But so soon as
he had answered his question, and said ' These kinds of premisses
prove the various kinds of conclusion ', then other people could look
at the matter from the side of the premisses first. To them, the
premisses were something which, if given, necessitated a certain
form of conclusion : rather than something which, if a certain form
of conclusion were to be established, must be given.

2. The premisses are called respectively the major and minor
premiss. This nomenclature is adjusted to that of the terms in the
argument. There are, as we have seen, three terms in a syllogism :
two, which form the subject and predicate of the conclusion, and
one with which each of the former is brought into relation (in the
way of subject and predicate) in one of the premisses. The subject
and predicate of the conclusion are called respectively the minor
and the major terms : the term common to the two premisses
is called the middle term.2 The major premiss is the premiss in

1 Or rather, to be proved or disproved : it was a thesis, which might form
the subject of debate between two parties ; one of them, the oppugner, ' held
out ' to the other, the upholder, various propositions, which he asked him to
admit, in hope to obtain admissions wherefrom there followed syllogistically
a conclusion contradictory of the thesis of the upholder.

2 These expressions are based upon what occurs in the first figure, where
the major term is commonly of greater extension than the middle, and the
middle than the minor : and the major premiss, as compared with the minor,
is a more general proposition. But being transferred to the other figures,
in which they cannot any longer be so interpreted, they must be explained
generally as in the text : cf. infra, pp. 259 sq., where this is explained at
length.

xii) MOODS AND FIGURES OF SYLLOGISM 257

which the major term occurs, and the minor premiss that in which
the minor term occurs. Thus in the syllogism

All organisms are mortal
Man is an organism
.'. Man is mortal

the major term is mortal, and the major premiss all organisms are
mortal ; the minor term man, and the minor premiss man is an
organism ; the middle term, organism.

It will be noticed that each term in a syllogism appears twice :
the major and minor terms each in its respective premiss and in the
conclusion, the middle in both premisses but not in the conclusion.

In giving examples of syllogism, it is usual to write down the
major premiss first ; but in ordinary life and conversation, no
particular order is observed ; nor is it necessarily the major premiss
that is written first in a logical example.1 The only mode of deter-
mining the major premiss is to look for the premiss which contains
the predicate of the conclusion.2

3. Syllogisms are said to differ in figure (o-x^a) according to
the position of the middle term in the premisses.3 (i) The middle
term may be subject of the major premiss, and predicate of the
minor : in this case Aristotle called the syllogism of the first (or
perfect) figure. The example just given belongs to the first figure,
as also does the following : —

No insects have eight legs
Wasps are insects
.*. Wasps have not eight legs.

It is convenient to have a conventional symbolism, in which to
represent syllogisms according to their form ; we shall use the
letters P, M, and 8. 8 ( = subject, of the conclusion) will always
indicate the minor term, P ( = predicate, of the conclusion) the
major term, and M the middle. Thus the figure of both these
examples (i. e. their form, so far as it depends merely on the position
of the terms in the premisses) may be written

M P
S M

:. 8 P

1 Of. Locke, Essay, IV. xvii. 8 (fourth or later edition).

2 Except in the ' indirect moods ' of Fig. 1. Cf. infra, pp. 262, 268-269.
■ Cf. c. xi, supra, pp. 250-251.

1779 8

258 AN INTRODUCTION TO LOGIC [chap.

If we wished to indicate in our symbols the character of the pro-
positions which compose the syllogism (i. e. whether universal or
particular, affirmative or negative), we should have to write our two
examples differently. The former is of the type

All M is P
All 8 is M
:. All S is P

the latter of the type

No M is P

All 8 is M

:. No 8 is P.

(ii) The middle term may be predicate in both premisses, the
figure of the syllogism being indicated as follows : —

P M

8 M
.'. S P

e.g. No insects have eight legs

Spiders have eight legs
.*. Spiders are not insects.

Syllogisms in which the middle term is thus placed were called
by Aristotle of the second figure.

(iii) The middle term may be subject in both premisses, the figure
of the syllogism being indicated as follows : —

M P
M S
.'. S P

e.g. The Veddahs of Ceylon show great conjugal fidelity
The Veddahs of Ceylon are savages
.*. Some savages show great conjugal fidelity.

Syllogisms in which the middle term is subject in both premisses
were called by Aristotle of the third figure.

(iv) Aristotle recognized only these three figures. But he pointed
out 1 that the premisses of a syllogism in the first figure would some-
times justify you in concluding to a particular proposition in which
the minor term was predicated of the major, even though no

1 Anal. PH. a. vii. 29a 19-27 (cf. p. 281, n. 2, infra).

xnl MOODS AND FIGURES OF SYLLOGISM 259

conclusion was possible that predicated the major of the minor.
For example, from the premisses

Some parliamentary voters are freeholders
No women are parliamentary voters

it is impossible to determine whether any women are freeholders or
not (for a reason which will be explained later) ; but we can con-
clude that some freeholders are not women.
Again, from the premisses

All persons who have the franchise are eligible to Parliament l
No woman has the franchise

we cannot conclude that women are not eligible to Parliament (for
others might be eligible besides those who have the franchise) ; but
we can conclude that some persons who are eligible are not women.
The famous physician Galen is said by Averroes to have referred
arguments of this kind to a separate and fourth figure (sometimes
called after him the Galenian figure), in which the middle term is
predicate of the major premiss and subject of the minor : the figure
being accordingly symbolized

P M

M 8
:. 8 P.

The theory of syllogism has been much darkened by this addition.3
For in erecting these arguments into a separate figure it is implied
that the distinction between major and minor term depends merely
on their position in the conclusion, and is in no way intrinsic to the
terms themselves. The meaning of that distinction must be con-
sidered next.

4. We have said that the major term is the predicate of the
conclusion, and the minor the subject. But why are they called
major and minor ? Did Aristotle merely want shorter names, to
avoid the constant repetition of such cumbrous expressions as
* subject of the conclusion ' and ' predicate of the conclusion ' ? Are
tne names chosen arbitrary ? And would it have been equally appro-
priate to call the subject of the conclusion the major, and the

1 If the premiss had to be true, the clergy must be excepted.

2 In the second and third figures, where the middle term occupies the
same position in both premisses, either premiss may be regarded as major,
without affecting the situation of the middle term : and hence there is no
possibility of erecting a separate figure bearing the same relation to them
as the fourth does to the first.

32

260 AN INTRODUCTION TO LOGIC [chap.

predicate the minor term ? Or, on the contrary, does the choice of
names indicate a real feature of the relation between subject and
predicate in a judgement ? Is there a reason why the predicate
should be called the major term, and the subject the minor?

Aristotle conceived that there was such a reason, not indeed in
all judgements, but in most and especially in scientific judgements
(i. e. judgements which really express knowledge). We shall do
best to look first at judgements in which the distinction of major and
minor term is arbitrary. ' Some scholars are statesmen ' might be
as well expressed by saying ' Some statesmen are scholars ' ; for
here the two terms or concepts have no necessary relation : it is only
as coincident in the same individual that statesman can be predi-
cated of scholar, or vice versa ; and there is no more reason for
making one term subject than the other. ' Some poulterers are not
fishmongers ' is a judgement of the same kind : the two trades are
frequently conjoined, but merely conjoined, and as there would be
no more reason for making the sale of fish an attribute of a poulterer,
than the sale of poultry an attribute of a fishmonger, so in the
negative judgement, each term is with equal propriety denied of
the other. But where the subject of a judgement is a concrete
thing or person, and the predicate an attribute : or where, though
the subject is an abstract term, yet the predicate belongs to it,
and is not merely coincident with it in the same thing ; there the
two terms cannot equally well be predicated of each other. We
say that Caesar was a great general ; if we said ' a great general
was Caesar ', we should still be understood to make Caesar the
subject, and to have merely inverted the usual order of words in
the sentence. We say that diamonds glitter, rather than that some
glittering things are diamonds ; that blue is a colour, rather than
that a colour is blue.1 To say that a colour may be blue is natural
enough ; just as it is to say that a stone may be a diamond ; but
still we predicate the genus of the species or individual, and not the
species or individual of the genus : it is not the genus colour, but
colour in some particular case, not the genus stone, but some parti-
cular mineral that is blue or that is diamond. Commonly, except
where they are merely coincident attributes,2 the predicate is

1 Unless a definite instance is meant.

2 Terms, though they be general concrete terms, like statesman or fish'
monger, may yet express only a special or ' abstract ' aspect Df the nature
of the thing they denote, if they are not in the category of substance :
cf. tvpta, p. 37, n. 1.

xii] MOODS AND FIGURES OF SYLLOGISM 261

a wider term, or more generic, than the subject in judgement ; it is
something which belongs to this and may belong to other subjects,
not a part of the extension of the subject itself. It is natural to
predicate the genus of the species, the attribute of the concrete
thing. In science especially, whose judgements should be necessary
and universal, the predicate, if not commensurate with the subject,
must be the wider term. We cannot predicate universally of any
term what is only part of its extension. If stone is a wider or more
comprehensive term than diamond, other things besides diamonds
are stones, and therefore that proposition must be particular in
which diamond is predicated of stone. A diamond is a stone,
a stone may be a diamond ; blue is a colour, a colour may be
blue.

In calling the predicate of the conclusion in a syllogism the
major term, then, Aristotle chose a name which was appropriate,
both when the predicate is related to the subject as attribute to
concrete thing, and when it is related to the subject as the more
to the less generic. By the name major he wished to indicate
that the predicate is the more comprehensive term : that it signified
something characterizing the subject, but characterizing, or capable
of characterizing, other subjects also — something therefore which
might be regarded as an attribute of the subject (in a wide sense of
the word attribute), but not as a subject characterized by it.1

1 Cf. infra, pp. 379-380. In Anal. Post. j3. xvii. Aristotle uses the word
iraptKTeivftv, to extend beyond, of the relation of major to middle term. He forgets
however there, and ignores in the Prior Analytics when he adopts the expres-
sions major, middle, and minor terms, what in the Posterior Analytics he
rightly recognizes as characteristic of science (though not of all reasoning),
that it aims at demonstrating commensurate judgements. Still, there are
many scientific judgements which have not that character, and even in
those that have it, the predicate, considered apart from the demonstration,
is conceived as what does belong to this subject, and might belong to others.
It is only in the demonstration by which it is shown to belong to one subject,
that we come to realize it can belong to that subject alone. If we see, for
example, in proving that the angle in a semicircle is a right angle, that the
proof hinges upon a feature which cannot belong to the angle in another
segment (viz. that the subtending chord passes through the centre of the
circle), then we see that the predicate is commensurate with the subject ;
and then also the predicate (if I may so express myself) sinks into the con-
crete nature of the subject, and is conceived as a necessary part thereof.
While a demonstration is still wanted by us, to show us that the angle in
a semicircle is a right angle, we have no ground for supposing that that is
not a property of angles in some other segments as well : so soon as we
realize that it can be the property of none other, we have incorporated the
demonstration with the subject-concept (of the angle in a semicircle) and
major, minor, and middle terms have for us lost their isolation. Demonstration,

262 AN INTRODUCTION TO LOGIC [chap.

The middle term takes its name not simply from being a point
of connexion between the other two, but from being really inter-
mediate in comprehensiveness. This it is, however, only in the first
figure. It is only there that the middle term is predicated of the
minor, and the major predicated of it. In the second, it is predicate
in each premiss ; in the third, the subject, of which both major and
minor terms are predicated. But that which in the first figure is
really a middle term between the major and minor serves equally
in the others to be the means of establishing that relation between
the major and minor which we prove ; and the nomenclature that
is fixed by the first figure is extended to them all.

It follows that Galen was wrong in assigning to a fourth and
separate figure syllogisms in whose conclusion the most compre-
hensive term is subject, and the least comprehensive predicate, as
in the example

What breeds rapidly has a short life
Flies breed rapidly
.*. Some short-lived things are flies.

It is true that in them the middle term is predicate of the premiss
containing the predicate of the conclusion, and subject of the premiss
containing the subject of the conclusion ; but in respect of compre-
hensiveness the predicate of the conclusion is minor, its subject
major ; and therefore such syllogisms are better treated as belonging
to the first figure, but having an inverted or indirect conclusion.
The distinction of major and minor between terms is primarily that
of greater and less comprehensiveness, and this is not altered by
making the more comprehensive the subject, and the less the
predicate, in the conclusion.

But the fourth figure has been taught for so many centuries
among the ' moods and figures ' of the syllogism, that for the sake
of the history of Logic we cannot altogether ignore it, even while
we recognize the error in which it had its birth.1

5. The last paragraph spoke of moods and figures of the syllogism.
The difference of figures has already been explained to depend on
the position of the middle term in the premisses. The difference
of mood depends on the quantity and quality of the propositions

when complete and while completely realized by the mind, may be said to
collapse into a judgement whose terms are interfused. Cf. p. 311, infra.
1 Cf. infra, pp. 280-285.

xii] MOODS AND FIGURES OF SYLLOGISM 263

composing the syllogism. This may be the same in different figures,
or different in the same figure : e. g. in the syllogisms

All organisms are mortal
Man is an organism
.*. Man is mortal :

and No unlicensed body may sell liquor to strangers

A college is unlicensed
.'. A college may not sell liquor to strangers :

the figure is the same (the first), but the component propositions
are in one case of the form A, A, A, and in the other of the form
E, A, E. If the second syllogism be now compared with the
following

No good comrade avoids pleasure
All ascetics avoid pleasure
.*. No ascetic is a good comrade :

it will be seen that the component propositions are of the same
form in both, E, A, E : but the figure is different.

The different moods have received distinct names in the various
figures wherein they occur ; and hence what are called the ' mood-
names ' of the various forms of syllogism indicate both figure and
mood. What moods are possible in what figures — i.e. what com-
binations of premisses, as determined by their quantity and quality,
will yield what form of conclusion (A, E, I, and 0) with each position
of the middle term — is the general problem to which the formal
part of the theory of syllogism has to find an answer. We are now
familiar with the technical terms employed in solving the problem.
We must next consider the solution.

B. The only method of originally determining what combina-
tions of premisses will yield what conclusion is to try them all,
with each position of the middle term, and see. This is what
Aristotle did, in the Prior Analytics. But when it has been done,
it is possible to review the result, and there recognize the nature
of the faults committed in those which are invalid, and the rules
which therefore must be observed (whether in all syllogisms, or in
those of a particular figure) in order to validity. These rules may
then be placed in the forefront of our exposition ; it may be shown,
by the help of an example, that the breach of them brings invalidity ;
and in each figure, out of the whole number of ways in which it is

264 AN INTRODUCTION TO LOGIC [chap.

mathematically possible to combine two premisses, when either
may have any one of four forms, we can show which are conform-
able to the rules that we have found necessary to be observed in
that figure.

The syllogism is now generally taught in the latter manner,
which is the more formal and systematic. But the other is the
more natural, and we shall therefore begin, for the first figure,
with that. Indeed the ' rules of syllogism * could not have been
known first, and then the valid moods determined thence ; their
formulation is the result of an investigation of the valid mooda
conducted without them.

A valid mood of syllogism is immediately seen to be valid by
any one who considers it in a particular example, and though the
example is particular, the form of inference is seen to be valid
universally. The best way, on the other hand, to show that a mood
is invalid, is to produce examples in which the premisses and
conclusion are of the quality and quantity which that mood requires,
and show by them that while the premisses are true, the conclusion
may be indifferently true or false. For if you cannot rely on a form
of argument to produce a true conclusion from true premisses, it
certainly is not a valid form.

Now in the first figure the middle term is subject of the major
premiss and predicate of the minor. Let us take the possibilities
in order.

1. Both premisses universal.

a. both affirmative ; the mood is valid, and the conclusion A :

All organisms are mortal All M is P

Man is an organism All 8 is M

.'. Man is mortal l .*. All S is P

6. both negative ; no conclusion follows :

Sounds have no scent No M is P

Colours are not sounds No S is M

.". Colours have no scent .*,

Sounds are not visible
Colours are not sounds
.*. Colours are not visible *

1 With actual terms, an universal proposition is often more naturally
expressed without the use of the mark of quantity, All men or No colours.
Where this is so, and the content makes it plain that the proposition is

xii] MOODS AND FIGURES OF SYLLOGISM 265

c. one affirmative and the other negative :

i. the major negative ; the mood is valid, and the con-
clusion E :
No Protestant acknowledges the Pope No M is P

Lutherans are Protestants All S is M

.'. No Lutheran acknowledges the Pope .*. No 8 is P
ii. the minor negative ; no conclusion follows :

Lutherans are Protestants All M is P

Calvinists are not Lutherans No 8 is M

.'. Calvinists are not Protestants ,\

Lutherans are Protestants
Romanists are not Lutherans
.'. Romanists are not Protestants
2. One premiss universal, and one 'particular,
a. both affirmative :

i. major universal, minor particular ; the mood is valid
and the conclusion / :
What raises prices injures the consumer All If is P
Some import -duties raise prices Some 8 is M

.'. Some import-duties injure the consumer .'. Some 8 is P
ii. major particular, minor universal ; no conclusion
follows :
Some taxes are levied at death Some M is P

Excise-duties (or Legacy-duties) are taxes All 8 is M

.'. Excise-duties (or Legacy-duties) are levied at death .*.

6. both negative :

i. major universal, minor particular ; no conclusion
follows :
Starches contain no nitrogen No M is P

Some foods (or flesh-foods) are not starches x Some S is not M
.*. Some foods (or flesh-foods) contain no .'.
nitrogen

universal, it has not been thought necessary to mark the quantity in that
way. But with symbols, because there is then no content to guide us, this
is necessary.

1 It is true that no flesh-foods are starches. But if with premisses true
and of the above form the conclusion is to be false, it is impossible to find an
example where it would not be equally true to enunciate the minor premiss
universally. For suppose that only some S is not M : then some 8 is 31,
and with the help of the major premiss, no M is P, it will follow that some
S is not P. But this conclusion was to be false ; therefore no S can be M

266 AN INTRODUCTION TO LOGIC [chap.

ii. major particular, minor universal ; no conclusion
follows :
Some quadrilaterals contain no right angles Some M is not P
The triangle in a semicircle (or The pentagon) No S is M
is not a quadrilateral
.*. The triangle in a semicircle (or The pentagon)
contains no right angle

c. one affirmative, and the other negative :

i. major affirmative and universal, minor negative and
particular ; no conclusion follows :
All living things change (or contain carbon) All M is P
Some compounds are not living Some 8 is not M

.". Some compounds do not change (or do not /.
contain carbon)

ii. major negative and universal, minor affirmative and
particular ; the mood is valid, and the con-
clusion 0 :

No political offence is extraditable No M is P

Some murders are political offences Some S is M

.*. Some murders are not extraditable ,\ Some S is not P

ill. major affirmative and particular, minor negative and
universal ; no conclusion follows :

Some traders are freeholders (or are members

of Parliament) Some M is P

No parson trades No 8 is M

.*. No parson is a freeholder (or is a member of

Parliament)

iv. major negative and particular, minor affirmative and
universal ; no conclusion follows :

Some plants are not edible Some M is not P

Beans (or Monkshoods) are plants All S is M

.'. Beans (or Monkshoods) are not edible

3. Both premisses particular.

a. both affirmative ; no conclusion follows :
Some Germans are Protestants Some M is P

Some Calvinists (or Romanists) are Germans Some S is M
.'. Some Calvinists (or Romanists) are Protestants .*.

xii] MOODS AND FIGURES OF SYLLOGISM 267

b. both negative ; no conclusion follows :

Some things profitable are not pleasant Some M is not P

Some things popular (or pleasant) are not Some 8 is not M

profitable
,*. Some things popular (or pleasant) are not

pleasant

c. major affirmative, minor negative :

Some luxuries are taxed Some M is P

Brandy (or A cart) for some purposes is Some S is not M

not a luxury .'. Some 8 is not P

.*. Brandy (or A cart) for some purposes is

not taxed

d. major negative, minor affirmative :

Some men of science do not study philosophy Some M is not P
Some rich men (or philosophers) are men of Some S is M

science /. Some 8 is not P

,\ Some rich men (or philosophers) do not
study philosophy

This exhausts the possible varieties in form of premisses, so far as
the first figure is concerned ; and we have found only four which
give any conclusion, namely (to represent them by the accepted
symbols, and add the symbol for the conclusion) AAA All

EAE EIO

Since the thirteenth century, logicians have given to each of
these moods, as well as to those in the remaining figures, a separate
name, in which the vowels in order indicate the quality and quantity
of the major and minor premisses and the conclusion. The names
of these moods of the first figure are Barbara, Celarent, Darii
Ferio : and syllogisms of those types are called syllogisms in
Barbara, Celarent, &C1

1 The earliest known work in which these mood-names are found is by
William Shyreswood (born in Durham, student in Oxford, taught at Paris,
died as Chancellor of Lincoln, 1249 ; v. Prantl, iii. 10, Absch. xvii. Anm. 29) :
' Modi autem et eorum reductiones retinentur his versibus — Barbara, &c.'
(ib. Anm. 52). They passed into general currency through the Summula*
Logicales of Petrus Hispanus, afterwards Pope John XXI, who was long
believed to be the author of them (c. 1226-1277), until Prantl found them
in the unpublished MS. of William Shyreswood in the Library of Paris
(vol. ii. p. 264). A somewhat similar memoria technica, but less ingenious,
because it embodies only the form of the moods, and not the rules for the

268 AN INTRODUCTION TO LOGIC [chap.

But an addition has to be made. If the minor premiss is an
universal negative proposition, and the major is affirmative, whether
universal or particular, then though no conclusion can be drawn
in which the major term is denied (or affirmed) of the minor, it is
possible to draw a particular conclusion in which the minor term is
denied of the major. Thus in 1. c. ii. from the premisses

Lutherans are Protestants

Calvinists (or Romanists) are not Lutherans

it was impossible to infer whether Calvinists or Romanists were
Protestants : the former in fact being so, and not the latter. But
it is possible to infer that some Protestants are not Calvinists (or
Romanists). And in 2. c. iii. from the premisses

, (freeholders

Some traders arei , , _, ..

(members of Parliament

No parson trades

it was impossible to infer whether any parson was a freeholder, or
a member of Parliament : none of them, in fact, being eligible to
Parliament, while a rector or vicar is legally a freeholder. But it is
possible again to infer that

„ (freeholders )

Somei , , t. i- ^ r are not parsons,

(members of Parliament)

Doubtless no member of Parliament is a parson, as no Romanist is

reduction of the moods in the second and third figures to the first (v. next
chapter) is found in the margin of the treatise attributed to Michael Psellus
(1018—? 1079), Swo^if cis tt]v 'ApicTTorAotif \oyi<i)v €TTi<TTi']fir]i> (Synopsis of
Aristotle's Logic) (according to Prantl, in the same hand as the text, ii. 275,
Absch. xv. Anm. 46). Prantl believes the work of William Shyreswood to
be borrowed from, and that of Petrus Hispanus to be a mere translation of,
the Synopsis of Psellus. In an article, however, by R. Stapper (Die Sum-
mulae Logicales des Petrus Hispanus und ihr Verhaltniss zu Michael Psellus,
published in the Festschrift zum elfhundertjdhrigen Jubilaum des deutschen
Campo Santo in Rom, Freiburg im Breisgau, 1897, pp. 130 sq. ; cf. also his
Papst Johannes XXI. pp. 16-19, Miinster i. W., 1898), reason is shown for
thinking that the ascription of the Synopsis to Michael Psellus is erroneous,
and that it is really a translation of the Summulae : the Augsburg MS. of
the Synopsis in which the ascription occurs contains also chapters lacking in
the Summulae, and partly identical with other works of Psellus ; these may
have led to his name being placed in the title, which Stapper conceives to
be in a hand fifty years later than the bulk of the MS. No other MS. of
the Synopsis ascribes it to Psellus ; all the rest profess to be translations
from the Latin ; seven give the name of Petrus Hispanus as author, and
four that of Georgius Scholarius (Gennadius) as translator. Cf. also Sir
William Hamilton's Discussions, 2nd ed., pp. 128, 671 sq. : who, however,
wrote before Prantl's work appeared.

xii] MOODS AND FIGURES OF SYLLOGISM 269

a Protestant ; and those who know this would not trouble to
enunciate the subaltern, or particular, propositions ; but our
premisses do not inform us of the universal ; what they do tell us is
the truth, even if not the whole truth.

We have thus two further and indirect moods, i.e. moods in
which the minor term is concluded of the major, not the major of
the minor, viz.

AEO All } ._ . _

IEO Some]

No 8 is M
.'. Some P is not S

And there are other indirect moods also. For in Barbara,
Celarent, and Darii, it is possible, instead of drawing the direct and
natural conclusion, to draw the converse, wherein the major term
will be subject and the minor predicate. Thus in 1. a. we might
have concluded ' Some mortals are men ', in 1. c. i. ' No one who
acknowledges the Pope is a Lutheran ', in 2. a. i. ' Some things that
injure the consumer are import-duties '. There are thus five indirect
moods in all : and the whole nine are given in the first two lines
of the following hexameters (it is to be noted that the extra syllables
after the third, in the fifth and ninth names, are inserted metri gratia,
and have no significance) : —

Barbara Celarent Darii Ferio, Baralipton
Celantes Dabitis Fapesmo Frisesomorum 1 :
Cesare Camestres Festino Baroco : Darapti
Felapton Disamis Datisi Bocardo Ferison.

The first four names in the third line belong to the valid moods in
the second figure : the remainder to those in the third. It would
be possible to show what moods are valid in these figures by experi-
menting with all the combinations of premiss possible in respect
of quality and quantity when the middle term was respectively
predicate or subject in each premiss. But any one who has followed
the process for the first figure can work it out for himself in the
others ; and we may proceed now to the enunciation of the rules
of syllogism, and the briefer deduction of the valid moods from
them.

1 The indirect moods of the first are the same as the moods of the fourth
figure : cf. pp. 280-285, infra.

270 AN INTRODUCTION TO LOGIC [chap.

C. The Syllogistic Rules are eight in number, viz.

1. A syllogism must contain three, and only three terms. The
necessity of this rule is manifest ; for we have seen that a syllogism
is an argument in which a relation (in the way of subject and predi-
cate) is established between two terms, in virtue of their common
relation (in that respect) to a third term. Hence, without a third
term there is no syllogism : and if the terms of the conclusion were
not related to the same third term, there would be no relation estab-
lished between themselves, and so again, no syllogism.

For example, we can draw no conclusion barely from the premisses
What breathes needs oxygen and Fish have gills. Any one who knew
that what has gills breathes might infer that fish need oxygen : but
the inference requires the premiss What has gills breathes no less than
the other two ; and falls really into two syllogisms, each containing
three terms : though four terms occur in the whole argument, viz. :

(i) What breathes needs oxygen

What has gills breathes
.". What has gills needs oxygen

(ii) What has gills needs oxygen

Fish have gills
.*. Fish need oxygen

If the middle term is used equivocally — i.e. in different senses in
the two premisses — there will in reality be four terms, and no con-
clusion is possible ; e. g. it is true that no vegetable has a heart : it
is also true that a good lettuce has a heart : but to have a heart
means something different in these two propositions, and it would
be fallacious to conclude that a good lettuce is not a vegetable.1

A breach of this first rule is technically known as the fallacy of
Quaternio Terminorum or of Four Terms ; and where it arises through
the equivocal use of the middle term, as the fallacy of ambiguous
middle.

2. The middle term must be distributed in one premiss at least.
It will be remembered that a term is distributed, when used in

reference to its whole extension ; and undistributed, when not so
used. Thus in the proposition All jealous men are suspicious, the
term jealous man is distributed (for I expressly refer to all that falls

1 Conversely, the middle term may be really the same, though verbally
different, in the two premisses ; and then there is a syllogism, e.g. Branchiate*
need oxygen, &nd fish have gills .'. Fish need oxygen.

xii] MOODS AND FIGURES OF SYLLOGISM 271

within the range of it) ; but the term suspicious is undistributed,
for I consider it only as characterizing the jealous, and it may very
well have a wider range than that. If again I say that Some jealous
men have killed their wives, in this proposition neither term is dis-
tributed.

Now when the middle term is undistributed in both premisses, it
may refer in each to a different part of its extension ; and then the
major and minor terms are not brought into relation with the same
term in the premisses at all : hence no conclusion can be drawn.1

Examples from the three figures will make plain what is perhaps
hard at first to grasp in an abstract statement. If a Presbyterian is
a Christian, and some Christians think that the order of bishops was
instituted by Christ, it does not follow that a Presbyterian thinks
this. Christian is a term that includes more than Presbyterian ; if
all Christians thought that the order of bishops was instituted by
Christ, then it would follow that Presbyterians thought so ; but if
only some Christians think it, how am I to tell that the Presbyterians
are among these ? Again, in the second figure, from the premisses
Birds fly and Eagles fly, I cannot infer that an eagle is a bird ; for
though birds fly, many creatures may fly which are not birds, and
an eagle might be one of these. If in either premiss the middle
term were used with reference to its whole extension : if nothing flew
but birds, or nothing flew but eagle3, and if my premiss informed
me of this : then I could conclude that all eagles were birds, or
that all birds were eagles ; but as it is, I can make no inference.
Inference is as obviously impossible in the third figure, with the
middle term undistributed. Granted that some working-men are
Tories, and some working-men are tailors : I cannot hence determine
whether or not some tailors are Tories : for the working-men that are
tailors may not be the same working-men as are Tories, and then
the inference would be false. But if in either premiss the middle

1 This is sometimes expressed as follows : though the expression is apt to
be misleading (cf. pp. 272, 273). It is said that the premisses assert agree-
ment (or disagreement, if negative) between the major or minor, and the
middle, terms ; that if the middle term be undistributed in both premisses,
the major and minor may respectively agree (or agree and disagree) with
a different part of its extension ; and therefore we cannot tell that they
agree (or disagree) with one another. The vogue of such language is perhaps
to be traced to Locke : cf . e. g. Essay, IV. xvii. 4 : ' It is by virtue of the
perceived agreement of the intermediate idea with the extremes, that the
extremes are concluded to agree ' ; cf. also Bacon, Nov. Org., Distrib. Operis,
1 tametsi enim nemini dubium esse possit quin, quae in medio termino
oonveniunt, ea et inter se conveniant,' &c.

272 AN INTRODUCTION TO LOGIC [chap.

term were distributed : if working-men were referred to in the whole
extension of the term, and all working-men were spoken of : then a
conclusion would follow. For whether all working-men were tailors,
and some Tories, or vice versa, in either case the some of whom the
one term was predicable would be included among the all of whom the
other term was predicable, and then these two terms (tailor and
Tory) would be predicable — not universally, but in part — one of
the other.

A breach of this rule is technically known as the fallacy of un-
distributed middle.

[It is in the third figure, where the middle term is subject in both
premisses, that the necessity of distributing it once at least is most
obvious. Plainly, there, to say that it is used with reference to
a part of its extension only is to say that only part of what it denotes
is spoken of ; and if this is a different part in the two premisses, there
is not really any middle term. Some animals fly, and some are rodents :
but they are not the same animals ; swallows e. g. fly, and rats are
rodents ; and it is obvious that our premisses do not justify the
inference that the same thing flies and is a rodent. But where the
middle term is not subject, there is a certain awkwardness in talking
of its distribution. This has already been noticed in discussing the
'quantification of the predicate'.1 It was then shown that the
predicate of a proposition is never really thought of in extension.
And yet in explaining the present rule of syllogism, one is tempted
to speak as if it were so thought of. A general demonstration of the
rule is wanted, applicable equally to any figure ; and it is easy to
say that if the middle term is undistributed in both premisses, the
major and minor may be brought into relation only with different
parts of its extension, and therefore not with the same term at all.
Or if we speak of agreement between them and the middle term,
we have a more seductive formula : we can illustrate with circles,
thus :

Fig.1. Fig.2.^- — \ Fig.3.

The inclusion of one area, wholly or partially, within another
symbolizes an affirmative judgement, universal or particular : it is
plain that the area 8 may fall wholly within M, and M partially

1 Cf. c. ix. pp. 222 sq., supra.

xn] MOODS AND FIGURES OF SYLLOGISM 273

[within P, and yet S may lie wholly outside P. This is supposed
to show for Fig. 1, that with an undistributed middle we can draw
no conclusion ; and the other diagrams are as readily interpreted.

Yet a syllogism does not really compare the extension of three
terms, and Euler's diagrams put us upon a wrong train of thought.
It is true, that unless the middle term be distributed once at least,
there is no point of identity in the premisses ; and all mediate in-
ference proceeds in some way by help of an identity. It is not true
that the point of identity need consist in the same subjects being
denoted — in the reference to the same part of the extension of the
middle term in both premisses (for which referring to the whole
extension in one of them would be an obvious security). In the
third figure the inference may no doubt hinge on this ; but not in
the second, or the first. On the contrary, the inconclusiveness
of an argument in the second figure with undistributed middle is
best expressed by saying that it does not follow, because the same
predicate attaches to two subjects, that these can be predicated one
of the other : and in the first figure, that unless P is connected
necessarily and universally with M , it is clear that what is M need
not be P.1

If this discussion of the Undistributed Middle should seem too
lengthy, it must be remembered (1) that for working purposes, in
order to determine the correctness of a syllogism, the main thing to
look to is the distribution of terms : and hence (2) that it is of
great importance, in the theory of syllogistic inference, not to
misunderstand this reference to distribution. In a later chapter
(c. xiv) it will be necessary to consider whether the different figures
of syllogism are really different types of reasoning, or the same ;
and the present discussion will throw light on that enquiry.]

3. From two negative premisses nothing can be inferred. A

negative proposition denies between its terms the relation of
subject and predicate. It is clear that if the major and minor
terms are both denied to stand in that relation to the middle term,
we cannot tell whether or not they are related as subject and
predicate to one another. Ruminant may not be predicable of
rodent, or vice versa : neither carnivorous of ruminant, or vice versa :
we cannot from this infer anything as to the relation of carnivorous
and rodent.

4. If either premiss is negative, the conclusion must be nega-
tive. The same kind of reflection will justify this rule, as the last.
Two terms stand in the relation of subject and predicate ; between

1 The fourth figure has not been considered in this note, but in this matter
it raises no question that is different from those that arise on the other
figures.

1778 T

274 AN INTRODUCTION TO LOGIC [chap.

one of them and a third term the same relation is denied ; if any
inference is possible,1 it can only be to deny the relation also
between the other and the third term.

5. The conclusion cannot be negative, unless one premiss is
negative. This rule is the converse of the last, and equally obvious.
If both premisses are affirmative, and if they justify a conclusion at
all, they must establish and not refute our right to predicate the
major of the minor.

6. No term may be distributed in the conclusion, which was
not distributed in its premiss. For if a term is undistributed in
the premisses, it is there not used with reference to its whole exten-
sion ; and this does not justify us in a conclusion which uses it with
reference to its whole extension.

A breach of this rule is called an illicit process of the major, or
minor, term, as the case may be.

[With an illicit process of the minor term, if (as in the first and
second figures) the minor term is subject in its own premiss, it is
obvious that we are treating information about a part of the ex-
tension of the term as if it were information about the whole. If
all M is P, and some S is M , we can only infer that some 8, and not
all S, is P. Where the minor term is predicate in its own premiss,
or with an illicit process of the major term, the matter requires
a little more reflection. The predicate of a judgement (and the
major term is always predicate in the conclusion, unless the mood
is indirect) not being thought in extension, there is some danger
here again lest we should misunderstand a reference to its distri-
bution. Take the following example of illicit process of the minor
term, where the minor term is predicate in the minor premiss :

To make a corner in wheat produces great misery

To make a corner in wheat is gambling
.-. All gambling produces great misery.
My premisses do not primarily give me information about gambling ;
nevertheless, if there were no gambling except a corner in wheat,
the minor term would be commensurate with the middle, and what

1 It may happen, where the premisses justify no inference, that an affir-
mative conclusion would in fact be true ; e. g. if some M is not P, and all
8 is M, it may be true that all S is P. Here of course the middle term is
undistributed, and therefore there is no real point of identity in the argument.
However, it is worth while noticing that the proof of this rule also is difficult
to express in a quite abstract way. The notion of agreement is employed
here again, but merits the same protest as before : if one term agrees with
a second, and that disagrees with a third, the first will disagree with the
third ; but the relation between subject and predicate is too loosely described
as one of agreement or disagreement.

xn] MOODS AND FIGURES OF SYLLOGISM 275

[is predicated universally of the latter could be predicated universally
of the former. As it is, however, for all the information that is
given me, the minor term may be (and in fact it is) of wider exten-
sion than the middle ; for there are many other modes of gambling
besides making a corner in wheat. It is used therefore with refer-
ence to a part of its extension only, in the minor premiss ; and
it is that part which I am told in the major produces great misery;
I have no right to extend that information to the whole extension
of the term, and say that all gambling produces great misery ; my
only proper conclusion is that some gambling does so. Again, with
regard to the major term : if I argue that productive expenditure
benefits the country, and expenditure on art is not productive ; and
that consequently expenditure on art is of no benefit to the country :
I am guilty of an illicit process of the major term. It may not at
first sight appear that I have treated information given me about
a part of what benefits the country as if it were information about
everything that does so. And indeed expenditure which benefits
the country is not directly the subject of my thought. Yet it is
plain that though productive expenditure may benefit the country,
it need not be the only form of expenditure to do so ; and hence
expenditure on art, though not productive, may be of benefit to
the country for some other reason. Yet my conclusion would only
be justified if I knew every reason why expenditure could benefit
the country, and knew that none of them applied to expenditure
on art : whereas my major premiss mentions one ground, and not
the sole ground, on which expenditure is beneficial. It is therefore
true in effect to say that in the conclusion I treat as referring to
its whole extension information which was confined to a part of the
extension of the major term ; though none the less the extension of
the major term is not the proper subject of my thought.1]

There remain two rules which are corollaries of those already
given, viz.

7. From two particular premisses nothing can be inferred, and

8. If either premiss is particular, the conclusion must be particular.
The truth of these rules is not evident at first sight ; and they

can only be established generally — i. e. without reference to mood
and figure — by considering what combinations of premisses there are,

1 Beginners imagine sometimes that the fallacy of illicit process is com-
mitted, if a term which is distributed in the premiss is undistributed in the
conclusion. This is not so. I must not presume on more information than
is given me, but there is no reason why I should not use less.

It will be noticed, therefore, that no particular conclusion can be vitiated
by an illicit process of the minor term : and no affirmative conclusion by an
illicit process of the major.

T2

276 AN INTRODUCTION TO LOGIC [chap.

both of which, or one of them, is particular ; and it will then be
seen either that there are not enough terms distributed in these
premisses to warrant a conclusion at all ; or not enough to warrant
an universal conclusion, i. e. one that distributes the minor term.

If both premisses are particular, they must either be both affirma-
tive (7 and 7), or both negative (0 and 0), or one affirmative and the
other negative (7 and 0). But in a particular affirmative propo-
sition neither subject nor predicate is distributed ; so that the
combination of premisses 77 contains no distributed term, and
therefore — since the middle term must be distributed if any infer-
ence is to be drawn — will yield no conclusion. From 00, two
negative propositions, a conclusion is impossible. From I and 0, if
there were any conclusion, it would be negative ; but as the predi-
cate of a negative proposition is distributed, the major term (the
predicate of the conclusion) would be distributed in the conclusion ;
therefore the major term should be distributed in its premiss ; and
since the middle term must be distributed in the premisses also, we
require premisses with two terms distributed in them, to obtain
a conclusion ; now the combination of a particular affirmative with
a particular negative provides only one distributed term, viz. the
predicate of the latter (0) ; and therefore from them also a conclu-
sion is impossible.

A similiar line of reasoning will establish rule 8 ; no combina-
tion of premisses, whereof one is particular, contains enough dis-
tributed terms to allow of an universal conclusion. For again,
either both are affirmative (A and 7), or both negative (E and 0), or
one affirmative and the other negative (A and 0 : E and 7). The
two negative premisses may be struck out as before. The combina-
tion of A with 7 contains only one distributed term, the subject of
the universal affirmative (.4) ; and as the middle term must be
distributed if the reasoning is to be valid, the subject of A must be
the middle term ; hence the minor term will be one of those that
are undistributed in the premisses, and therefore also in the conclu-
sion (of which it is the subject) it must be undistributed — i. e. the
conclusion must be particular. The combinations A and 0, E and 7
both contain two distributed terms ; viz. in the former the subject
of the universal affirmative and the predicate of the particular
negative, in the latter the subject and predicate of the universal
negative ; but both of them require negative conclusions, in which
the major term is distributed ; in both therefore the terms distri-

xii] MOODS AND FIGURES OF SYLLOGISM 277

buted in the premisses must be the major and middle, and the
minor term be one of those that are undistributed, so that the
conclusion again will be particular.
The above rules are all contained in four rude hexameter lines :

Distribuas medium, nee quartus terminus adsit ;
Utraque nee praemissa negans, nee particularis ;
Sectetur partem conclusio deteriorem ;
Et non distribuat, nisi cum praemissa, negetve.

The third line (that the conclusion must conform to the inferior
part of the premisses) covers both the fourth and eighth rules ; a
negative being considered inferior to an affirmative, and a par-
ticular to an universal judgement. The fourth line (that the
conclusion must not distribute any term, unless the premiss does
so, nor be negative unless a premiss is so) gives the sixth rule,
and the fifth.

D. Determination of the moods valid in the several figures.

We have seen that syllogisms are distinguished in mood accord-
ing to the quantity and quality of the propositions composing
them ; and in figure according to the position of the middle term
in the premisses. The validity of a syllogism, and the character of
the conclusion that can be drawn, depend very largely on the dis-
tribution of the several terms — middle, major, and minor — in the
premisses ; and this again on the question whether the middle term
is subject, and one of the others predicate, in a premiss, or vice
versa. Hence a combination of premisses which yields a conclusion
in one figure, may yield none in another : e.g. All M is P, All S is
M yields the conclusion All S is P ; but All P is M, All S is M
yields no conclusion, though the quantity and quality of the pre-
misses are unchanged. We shall therefore have to take the possible
combinations of premisses in each figure in turn, strike out those
which yield no conclusion in that figure, and ask what kind of
conclusion — i. e. whether universal or particular l — the others yield
in it.

Now as there are four kinds of proposition, so far as quantity

1 For this depends on the distribution of terms in the premisses, which
varies according to the figure : whether the conclusion is affirmative or
negative depends on whether both premisses are affirmative or not, a point
which can be determined without asking where the middle term stands, i. e.
what the figure is.

278 AN INTRODUCTION TO LOGIC [chap.

and quality are concerned — A, E, I, and 0 — and our premisses must
be two in number, there are sixteen combinations of premisses
mathematically possible.

These combinations are as follows, the premisses being indicated
by the conventional vowels, and the major premiss in all cases by
the vowel which stands first.

AA

EA

IA

OA

AE

EE

IE

OE

AI

EI

II

01

AO

EO

10

00

It is not however necessary to try the validity of all these sixteen
combinations in each figure in turn ; for four can be seen to yield
no conclusion on a ground holding in all figures alike, without
reference to the position of the middle term, viz. those wherein both
premisses are negative, EE, EO, OE, 00. Four more are excluded
by the rules just given, viz. (i) II, 10, 01 (as well as 00 again) on
the ground that both premisses are particular, and (ii) IE, on the
ground that it involves illicit process of the major term ; for since
one premiss is negative, the conclusion would be negative, and so
distribute the major term, while the major premiss, being a particular
affirmative, would not distribute that term, whether it were subject
in the premiss or predicate. But the inconclusiveness of these four
combinations cannot be rightly understood, as that of combinations
of two negative premisses can be, without taking examples in the
several figures : the rules from whose truth it follows being them-
selves a generalization of what we discover in so doing.

There remain eight combinations of premisses, not excluded by
any general rule, on whose validity we cannot pronounce without
reference to the figure and the position of the middle term, viz.

AA AE AI AO EA EI IA OA

It will be found that four of them are valid in the first figure,
four in the second, and six in the third ; there are also five indirect
moods of the first, or moods of the fourth, figure : making in all
nineteen moods.

In the first figure, the middle term is subject of the major premiss
and predicate of the minor : hence in this figure M P

1 . The minor premiss must be affirmative : for if it were S M

negative, the conclusion would be negative, and so distri- 8 P

xnj MOODS AND FIGURES OF SYLLOGISM 279

bute the major term P ; the major term must therefore be distri-
buted in the major premiss ; but as it is there predicate, it cannot
be distributed unless the major premiss is also negative (since
no affirmative proposition distributes its predicate) : we should
thus have two negative premisses, or else an illicit process of the
major term.

2. The major premiss must be universal : for since the minor is
affirmative, its predicate M, the middle term, will be undistributed ;
therefore M must be distributed in the major premiss ; and for this
purpose the major premiss, of which it is the subject, must be
universal. These rules however do not hold for the indirect
moods.

In this figure, therefore, the premisses AE, AO are invalid, by
rule 1 : I A, OA by rule 2 x ; A A, EA, AI, EI are valid. The
conclusions which they yield will be respectively A (universal
affirmative), E (universal negative), / (particular affirmative), and
0 (particular negative) ; and the moods — in which the quantity
and quality of the conclusion are indicated, as well as of the pre-
misses— are AAA, EAE, AH, ElO. Their names are Barbara,
Celarent, Darii, Ferio. But in the first three of these moods, as we
have seen, the converse conclusions can also be drawn ; and with
the premisses AE, IE, a particular conclusion follows denying 8 of
P ; and so we get also the indirect moods AAI, EAE, AH, AEO,
IEO, whose names are Baralipton, Celantes, Dabitis, Fapesmo,
Frisesomorum.

In the second figure the middle term is predicate in both P M
premisses : hence in it 8 M

1. One premiss must be negative, for otherwise the middle 8 P
term would be undistributed.

2. The major premiss must be universal : for since one premiss is
negative, the conclusion will be negative, and so distribute the major
term P : P must therefore be distributed in the major premiss ;
i. e. as it is here the subject thereof, the major premiss must be
universal.

Hence the premisses AA, AI, IA are invalid, by rule 1 : the

1 e. g. from the premisses Contemporary evidence is of great historical value,
Tradition is not (or Some inscriptions are not) contemporary evidence, it cannot
be inferred that Tradition is not (or Some inscriptions are not) of great historical
value (AE, AO) : from the premisses Some pointed arches are (or are not) four-
centred, All Gothic arches are pointed, it cannot be inferred that All Gothic
arches are (or are not) four-centred (I A, OA).

280 AN INTRODUCTION TO LOGIC [chap.

premisses OA (and I A again) by rule 2 x ; EA, AE, EI, AO are valid.
The moods are therefore EAE, AEE, EIO, AOO ; their mood-names
are Cesare, Camestres, Festino, and Baroco.

In the third figure the middle term is subject in both M P
premisses : hence in it MS

1 . The minor premiss must be affirmative, for the same reason 8 P
as in Fig. 1 (the major term, in both figures, being similarly placed
in its premiss).

This rule excludes the premisses AE, AO 2 : the remaining com-
binations, A A, AI, EA, EI, I A, OA, are valid. But because the
minor term in this figure is predicate of the minor premiss, and the
latter is affirmative, the minor term will not be distributed in it ;
hence it must not be distributed in the conclusion ; and therefore in
all cases

2. The conclusion will be particular.

The moods are consequently AAI, All, EAO, EIO, IAI, OAO :
their mood-names are Darapti, Datisi, Felapton, Ferison, Disamis,
Bocardo.

[It is impossible at this point to pass over the so-called fourth
figure. We have seen above (pp. 268-269, 279) that in the first
figure, besides the four direct moods, there are five ' indirect '
moods, i. e. moods in which the conclusion affirms or denies the minor
term of the major. In so describing these moods, we base the dis-
tinction of major and minor terms on their meaning ; the major
term is the more comprehensive ; it signifies the general nature of,
or some element in, the being of that real subject which the minor
term stands for. If we consider ' terms of thought ' we may say
that the minor is characterized by the major, not vice versa. This
relation is natural between two terms, when (as in Fig. 1) we can find
a middle term predicable of one of them, and the other of it. But
two terms of which the same predicate may be respectively affirmed
and denied (as in Fig. 2), or which may be affirmed, or respectively
affirmed and denied, of the same subject (as in Fig. 3), need stand in
no such relation. And if this relation is ignored, and being major
or minor term is made to consist barely in being predicate or subject
of the conclusion, then we cannot describe any mood as affirming or

1 e.g. from Some (or All) daisies have a great number of flowers within
a single calyx, All (or Some) compositae have a great number of flowers within
a single calyx it cannot be inferred that Some, or All, compositae are daisies
{A A, AI, I A) : nor from Some annuals are not (or are) hardy, All poppies are
hardy, that Some poppies are not (or are) annuals {OA, IA).

2 e. g. from the premisses All ostriches have wings, No ostriches can (or
Some ostriches cannot) fly, it cannot be inferred that No creatures that can fly
have wings or that Some creatures that can fly have no wings {AE, OA).

xn] MOODS AND FIGURES OF SYLLOGISM 281

[denying in its conclusion the minor of the major. Instead therefore
of the scheme J

(1) M P What is sensible is in the mind
S M Material things are sensible

.•. P S .'. Some things in the mind are material thinga

we must have the scheme

(2) P M Material things are sensible

M S What is sensible is in the mind
••. S P .*. Some things in the mind are material things.

Aristotle, as already remarked, did not recognize a fourth figure,
but he recognized the possibility of concluding indirectly in the first
figure, though not as a thing peculiar to the first. In one place he
says 2 : ' It is clear that in all the figures, when there is no [direct]
syllogism, if both premisses are affirmative or both negative nothing
at all necessarily follows, but if one is affirmative and one negative,
and the negative is universal, a syllogism always arises with the
minor as predicate to the major : e. g. if all or some B is A, and no
C is B : for, the premisses being converted, it is necessary that some
A is not C. And similarly in the other figures ; for by means of
conversion a syllogism always arises.' This covers the moods
Fapesmo and Frisesomorum in Fig. 1. Elsewhere he points out
that ' whereas some syllogisms are universal [in their conclusion]
and some particular, those which are universal always have more
conclusions than one ; of those which are particular, the affirma-
tive have more conclusions than one, but the negative have
only the [direct] conclusion. For the other propositions convert,
but the particular [negative] does not ' 3. He means that any syllo-
gism concluding to E, No $ is Pt gives also, by conversion of that,

1 It will be noted that the real terms indicated by 8 and P respectively
in (1) are indicated by P and 8 in (2), because in (1) S and P symbolize
minor and major in the sense of less and more comprehensive, in (2) minor
and major in the sense of subject and predicate of the conclusion, and what
is minor in the former sense is major in the latter, and vice versa.

Anal. Pri. a. vii. 29a 19 ArjXov 8( na\ on iv anao~i to'is o~X'IH-a<riv> orap pi) yiprjTai
ovWoyiapos, KaTT]yopiKa>v p.ev r} o~T(pr]Tiica>v dp<portpo>v ovru>v tg>v opcav ovbev o\ws
yiverai avayKalnv, KaTrjyopacov 8e kcl\ are prjriKoii, KadoKov \Tj<pdevros tov o-TfprjriKou
aei yivfrat avWoyiafios tov fXarrovos anpov npos to /uetfov, olov d to pei> A navri
Ta B t] Tivi, to be B fir/oevi tw r* dvTicrrptfpofj.fi'av yap tu>v irpoTa.o~i<&v avaynq to T
Ttvl to) A p.rj vnapx*iv. Sfioicos Be Kan\ Tutv ertpo&v o~xrnjniT<x>v' a*\ yap yiftrat 8ia
rfj? dvTio-Tpo<f)r]i o-v\Xoyio-p.6s. It is plain that otuv pt} yivr)Tai avWoyio-pos
means ' when there is no direct syllogism '.

3 Anal. Pri. j3. i. 53a 3 eVe l 8' ol piv Ka66\ov ratv avWoyio-pcov tlo~\v ol &e Kara
pepos, ol pev KadoXov Tvavra del irXelto o-v\\oyl£ovTai, to>v d' iv pepti ol ptv Karrj-
yoptKol nkfitOf ol 8 djrocpaTiKol to avpnepaapu povov. al pev yap aWai Trpordafis
di/Tio~rpf<j)ovo-iv, rj 8e o-TfptjTiK?) ov< avTio-rpe'cpei. What Aristotle says here
would cover the Subaltern Moods (cf. p. 285, infra) ; but he had not got
them in his mind ; he would not have regarded them as drawing a different,
but part of the same, conclusion.

282 AN INTRODUCTION TO LOGIC [chap.

[the conclusion No P is S, and any concluding to A or I, All S is P
or Some S is P, gives also the conclusion Some P is S. We have
therefore here a recognition of the possibility of the other three
indirect moods of Fig. 1, Baralipton, Celantes, and Dabitis : whose
conclusions are merely the converse of those which follow directly
from the same premisses in Barbara, Celarent, and Darii.

These observations are applied to all three figures, because
Aristotle thought that in the second and third also the major and
minor terms, though not distinguishable, as in the first, by having
different positions in the premisses, could yet be distinguished by
their meaning, so that we could tell whether a syllogism concluded
directly or indirectly, and distinguish, e. g., between concluding
directly in Camestres and indirectly in Cesare, or directly and
indirectly in Darapti. And sometimes this may be done. To
be rock is of the being of all granite, to be granite is not of the
being of all rock ; hence ' rock ' and ' granite ' are by their meaning
relatively major and minor. Now from the premisses

Some rocks are sedimentary
Granites are not sedimentary

I cannot conclude that Granites are not rocks ; but I can conclude,
indirectly, that Some rocks are not granites. Again, in gases
lighter than air, to be lighter than air is a character of the gas, not
vice versa ; hence from the premisses

Steam and hydrogen are lighter than air
Steam and hydrogen are gases

the natural conclusion is that Some gases are lighter than air,
though the converse, that Some things lighter than air are gases,
also follows. But often enough in the second and third figures
there is nothing in their meaning to make us regard one of two
terms as major and the other as minor, rather than vice versa, and
then their position in the conclusion must be taken to decide it. And
if we make that decide it always, there is still no syllogism in Figs. 2
and 3 which cannot formally be referred to one of the direct moods.
The above example in Fig. 2 may be treated as in Festino, thus :

Granites are not sedimentary
Some rocks are sedimentary
.*. Some rocks are not granite

and with premisses A and E in Fig. 2, according as the subject of
A or of E is predicate in the conclusion, the syllogism may be referred
to Camestres or to Cesare ; for each mood yields the converse of the
conclusion of the other. From the premisses

Spiders have eight legs
Insects have not eight legs

xn] MOODS AND FIGURES OF SYLLOGISM 283

[it is more natural to conclude in Cesare that Spiders are not insects
than in Camestres that Insects are not spiders, because ' insect ' is
a more generic term than ' spider '. But whichever conclusion is
drawn, the syllogism can be referred to a mood whose form covers it.
So with Fig. 3, which has no universal conclusions ; an indirect
syllogism with premisses A A still has the form of Darapti, an indirect
syllogism with Al or I A the form of Datisi or Disamis. But in
Fig. 1 it is otherwise. Here, if we draw the indirect, or converse of
the direct, conclusion in Barbara, Celarent, or Darii, we cannot by
transposing the premisses make the premiss containing the predicate
of the indirect conclusion the major premiss, and yet preserve the
scheme of the figure, because in this figure the middle term has not
the same position in both premisses, and their transposition alters
the position of it, and so, if the figure of a syllogism is determined
by the position of the middle term, alters the figure also. And if from
the premisses AE or IE, which have no direct conclusion in this
figure, we draw the indirect conclusion, and therefore treat E as
major premiss because containing the predicate of the indirect
conclusion, again the middle term becomes predicate and the minor
subject of its premiss, and the scheme of the figure is altered. Thus
it is easy to see why the indirect moods of Fig. 1 came to be regarded
as belonging not to Fig. 1 at all, but to a separate fourth figure.
For in many syllogisms of Figs. 2 and 3 there is nothing to settle
which shall be called the major and which the minor term but their
position as predicate or subject of the conclusion. And settling it
thus, we do not need to admit indirect moods in these figures. Only
those syllogisms then remain outstanding which in Fig. 1 from pre-
misses that admit of no direct conclusion draw indirect conclusions,
or from premisses that yield direct conclusions draw the converse
of those. These syllogisms, if the distinction of major and minor
terms is still made to depend on their position in the conclusion, do
not belong to the scheme of Fig. 1, and a fourth figure is therefore
instituted, in which the middle term is predicate in the major
premiss and subject in the minor.

That what Aristotle notices about the three figures generally, in
the passages quoted above, works out rather differently in the
first and in the other two was very early noticed ; and an explicit
recognition of the five indirect moods as supplementary moods of
Fig. 1 is attributed to his pupil and successor in the Lyceum, Theo-
phrastus.1 If Averroes is right in saying that Galen was the first to
regard them as belonging to a distinct figure,2 the view of Theo-
phrastus held the field for some five centuries. Averroes himself

1 v. Prantl, i. 365, Abschn. v. Anm. 46, where the passages from
Alexander, who ascribes the addition of these moods to Theophrastus, are
quoted.

a Prantl, i. 570-574.

284 AN INTRODUCTION TO LOGIC [chap.

[disagreed with Galen, and in this he was followed by Zabarella,1
one of the best of the scholastic commentators on Aristotle, whose
De Quarta Figura Syllogismi Liber is still worth reading on the sub-
ject ; though in the reason he gives for not regarding the Galenian
as really a fourth and independent figure he relies in part upon the
questionable analysis which regards all syllogism as an application
of the principle called the Dictum de omni et nullo (cf . infra, p. 296).
The real objection to Galen's view is that it implies a defective
insight into the character of the thinking in these forms of argument,
and treats the syllogism too much as a matter of verbal manipulation.
In the fourteenth chapter an endeavour is made to explain the
grounds on which this verdict rests. The external and mechanical
way of regarding syllogism, which underlies the reference of these
moods to a fourth and separate figure, finds what is hardly more
than its logical issue in some of the later scholastic writers, who
erect separate moods on no better ground than the order in which
the premisses are enunciated, without there being any actual
difference in the premisses or conclusion.2

Granted, however, that we are to acknowledge a fourth figure,
the following will be the special rules of it : it must be remembered
that as referred to this figure we call that premiss the major which as
referred to the first figure we should call the minor, and vice versa.

1. 7/ either premiss is negative, the major must be universal : for if
either premiss is negative, the conclusion must be negative, and
will distribute the major term ; which in this figure is subject of
the major premiss ; and if it is to be distributed there, the premiss
must be universal (cf. Fig. 2).

2. If the major premiss is affirmative, the minor must be universal :
for the middle term, as predicate of an affirmative proposition, will
not be distributed in the major premiss ; it must therefore be dis-
tributed in the minor premiss, where it is subject ; and therefore
the minor premiss must be universal.

3. If the minor premiss is affirmative, the conclusion will be par-
ticular : for the minor term, as predicate of an affirmative proposi-

1 And by others, e.g. Lambert of Auxerre, thirteenth century med., quoted
Prantl, iii. 30, Abschn. xvii. Anm. 121.

2 e.g. Petrus Mantuanus, quoted Prantl, iv. 178. Petrus, in the edition
of his Logica dated 1492, gives as an example of a syllogism in Cesare,
' Nullus homo est lapis, omne marmor est lapis, igitur nullum marmor est
homo '. If the conclusion drawn from the premisses enunciated in this order
is ' Nullus homo est marmor ', he calls the mood Cesares ; but were they
enunciated in the opposite order, and the latter conclusion drawn, he would
call it Camestres. By such and other even more questionable methods,
Petrus compiles fifteen moods in Fig. 1, sixteen in Fig. 2, eighteen in Fig. 3,
and eleven in Fig. 4. Cf . also Crackenthorpe, Logicae Libri Quinque, Oxoniae,
1670, p. 197, who appears to treat the moods of Fig. 4 and the indirect
moods of Fig. 1 as two different things.

xh] MOODS AND FIGURES OF SYLLOGISM 285

[tion, will not be distributed in the premiss, and so must not be
distributed in the conclusion, which will therefore be particular.

Hence the premisses OA are invalid by the first rule : A I and AO
by the second x ; A A, AE, EA, EI, I A are valid ; but A A will afford
only a particular, instead of an universal, conclusion. The moods
are thus AAI, AEE, EAO, EIO, IAI ; and their mood-names, aa
moods of the fourth figure, are Bramantip, Camenes, Fesapo,
Fresison, Dimaris.

The complete memoria technica, with the fourth figure replacing
the indirect moods of the first, is commonly given in English text-
books nowadays as follows 2 : —

Barbara Celarent Darii Ferioque prioris ;
Cesare Camestres Festino Baroco secundae ;
Tertia Darapti Disamis Datisi Felapton
Bocardo Ferison habet ; quarta insuper addit
Bramantip Camenes Dimaris Fesapo Fresison.
Quinque subalterni, totidem generalibus orti,
No men habent nullum, nee, si bene colligis, usum.

The meaning of the last two lines is explained in the next
paragraph.]

It will be noticed that in five out of these nineteen moods the
conclusion is universal, viz. in Barbara and Celarent in Fig. 1, Cesare
and Camestres in Fig. 2, and Celantes in Fig. 1 ( = Camenes in Fig. 4).
It is, of course, possible a fortiori to draw the particular, or subal-
tern, conclusion in any of these cases ; and the syllogism is then
said to have a weakened conclusion, or to be in a subaltern mood.
Subaltern moods would be used by no one who was asking what
could be inferred from given premisses ; for it is as easy to see that
the universal conclusion, as that the particular, can be drawn from

1 e.g. from the premisses Some change is not motion, All motion is change,
it cannot be inferred that Some cha?ige is not change {OA) : nor from All
great critics are scholars, Some scholars are pedants, that Some pedants are
great critics (AI) : nor from All members of the Government belong to the party
in power, Some of the party in power are not in the Cabinet, that Some of the
Cabinet are not members of the Government (AO).

2 I have not been able to trace this form of the mnemonic verses any
further back than to Aldrich's Artis Logicae Eudimenta. A good many
writers have tried their ingenuity in devising variations upon the original
lines. Watts has a version recognizing only fourteen moods, the indirect
moods of Fig. 1 appearing neither in that capacity nor as moods of Fig. 4.
Sir William Hamilton (Discussions, p. 666) also offers ' an improvement of
the many various casts of the common mnemonic verses '. But the reader
will probably wish for no more. In various modern textbooks, Baroco and
Bocardo are spelt with a k, in order that c medial may not occur with
a different meaning from c initial.

286 AN INTRODUCTION TO LOGIC

them. But in seeking for the proof of some particular proposition,
we might very likely find premisses that would really prove the
universal ; yet, since we are only using them to prove the particular,
our reasoning would fall into one of the subaltern moods. Still, we
should see that our premisses proved more than we had set out to
establish, and substitute at once the wider thesis ; the subaltern
moods are therefore of little importance, and are not included in the
enumeration of valid moods of syllogism.

[It would have been possible to determine what moods are possible
in each figure, without enunciating the special rules (as they are
called) of the different figures. It might merely have been pointed
out, e. g., that in the first figure AA would yield an A conclusion,
AE involve an illicit process of the major term, AI yield an I
conclusion, AO again involve an illicit process of the major, EA
yield an E, and EI an 0 conclusion, I A and OA involve an undistri-
buted middle. And if it were asked why the mood IAI is invalid
in this figure, the proper answer is not because in the first figure
the major premiss must be universal (though that is the second rule
of this figure), but because such a combination of premisses in it
involves an undistributed middle ; the rule being directed to
avoiding this fallacy, and not the fallacy condemned because it
breaks the rule. The rules, however, if the grounds on which they
rest are understood, give in a general form the principles which must
be observed in each particular figure. Therefore the knowledge of
these rules helps us to master the theory of syllogism ; but only if
their grounds are understood. It is better to know what moods are
invalid in each figure, and what fallacy they severally commit, than
to know the special rules and apply them in a mechanical manner,
without being able to justify them.)

CHAPTER XIII

OF THE REDUCTION OF THE IMPERFECT
SYLLOGISTIC FIGURES

Aristotle distinguished between syllogisms which were only
valid (bwaToc) and syllogisms which were perfect (re'Aetot). In the
latter, the necessity of the inference appeared sufficiently from the
premisses as they stand ; in the former, they required to be supple-
mented, in order that it might be seen. The second and third
figures, in his view, were in this plight. Their validity, though
real, needed proving, by means of the first figure. By converting
one of the premisses in the two imperfect figures, he showed that we
might obtain a syllogism in the first or perfect figure, either with
the same conclusion or with one from which that could be recovered
by conversion ; where this direct method of validating an imper-
fect mood fails, we can still validate it indirectly, by proving, in
a syllogism of the first and perfect figure, that the falsity of its
conclusion is inconsistent with the truth of its premisses.1

The process of exhibiting by the help of the first figure the validity
of syllogisms in the other two (or three) is called Reduction. A
knowledge of the method of reducing the imperfect moods to moods
of the first figure belongs to the traditional part of the theory of
syllogism. The present chapter will explain this ; in the next we
must ask whether the process of Reduction, sanctified by the tra-
dition of many centuries, is really necessary, in order to validate
the imperfect figures.

Directions for Reduction are concealed in the mood -names of
4 Barbara Celarent '. Those who have thoroughly mastered the
theory of syllogism will see at a glance how a given imperfect mood
may be reduced ; but the mood-name enables one to do it, as it were,
with a mechanical correctness.

1 This method of establishing the validity of a syllogism per impossibile is
applicable to all the imperfect moods ; but the direct method was preferred
where it is available.

288 AN INTRODUCTION TO LOGIC [chap.

Reduction, as already stated, is either direct or indirect. Direct
Reduction of an imperfect mood to the first figure consists in
showing that from premisses either the same as in the original
syllogism, or inferred immediately by conversion from these, the
original conclusion, or one from which it can be immediately inferred,
follows in the first figure.

As the figures are distinguished from one another by the position
of the middle term in the premisses, it is plain that, to reduce a
figure from one of the imperfect figures to the first, we must alter the
position of the middle term. In the second and third figures, it
occupies the same position in both premisses, being predicate in the
second, and subject in the third, whereas in the first figure it is
subject of the major premiss and predicate of the minor. We must,
therefore, convert one premiss of a syllogism in the second or third,
in order to reduce it to the form of the first. In the second we should
naturally convert the major, for there it is in the major premiss that
the middle term is out of place ; in the third, the minor. But it
may happen that this would give us a combination of premisses
which, in respect of quality and quantity, cannot stand ; e. g. in a
syllogism in Disamis (Fig. 3), by converting the minor premiss A, we
should get the combination II, which yields no conclusion. We
therefore have sometimes to transpose the premisses, making our
original minor premiss the major, and vice versa, and converting in
the second figure that which becomes the major, in the third that
which becomes the minor. Where the premisses are transposed
to make a syllogism in the first figure, they will give a conclusion
in which the terms of the original conclusion have been transposed
likewise ; and it will be necessary to convert this conclusion in order
to recover that of the original ' imperfect ' syllogism.

By way of illustration, we may take the following example in

Camestres, the form of which, as indicated by the vowels of the

mood-name, is

All P is M

No S is M

.'. No S is P

If we were to argue that a spider is not an insect because it has not
six legs, our argument would fall quite naturally into the above form :

Insects have six legs

The spider has not six legs
.*. The spider is not an insect.

xin] REDUCTION OF THE IMPERFECT FIGURES 289

Now if we want to get the same conclusion in the first figure, we
cannot convert the major premiss ; for that would give us a parti-
cular major

Some animals with six legs are insects

and no conclusion as to whether a spider is an insect or not would
follow.1 We must therefore convert the minor premiss, which
being E can be converted without change of quantity : and trans-
posing at the same time, form the syllogism in Celarent :

No animal with six legs is a spider

Insects have six legs
.*. No insect is a spider
From this conclusion we can recover by conversion the original
conclusion

The spider is not an insect

Had our argument run slightly differently, to the effect that the
spider is not an insect because it has eight legs, it would have fallen
into a syllogism in Cesare :

No insect has eight legs No P is M

The spider has eight legs All S is M

.'. The spider is not an insect .'. No 8 is P

Here the major premiss can be converted simply, being E :' ana
transposition is not required. The premisses

No animal with eight legs is an insect
The spider has eight legs
conform to Celarent, and yield at once the original conclusion.

The indirect moods of the first figure (the moods, as others regard
them, of the fourth figure) fall into two groups, when we wish to
show that their conclusions (or others yielding them by conversion)
can be obtained directly in the first figure from the same premisses
(or from premisses which these yield by conversion). Three,
Baralipton, Celantes, and Dabitis, simply draw the converse of the
conclusion which the same premisses yield directly in the first
figure ; all we have to do therefore is to draw the direct conclusion
and convert it. But Fapesmo and Frisesomorum yield no direct
conclusion. From the premisses

Every soldier serves his country
Women are not soldiers

1 Though it would follow by an ' indirect conclusion ' in Frisesomorum
that some insects are not spiders.

1779 U

290 AN INTRODUCTION TO LOGIC [chap

I cannot infer that Women do not serve their country. The only
conclusion is that Some who serve their country are not women.
Now if this is to have the form of a direct syllogism in the first
figure, women must be the major term, soldiers the minor : but if
' Women are not soldiers ' were the major, and ' Every soldier serves
his country ' the minor premiss, the terms would occupy the wrong
positions in the premisses. To obviate this, I must convert both
premisses ; then indeed I shall get the syllogism

No soldier is a woman
Some who serve their country are soldiers
.*. Some who serve their country are not women

which does prove my original conclusion in a direct mood of the
first figure, Ferio ; though whether it is the most natural way of
removing any doubts I may have had about the validity of the
indirect inference in Fapesmo must be considered in the next chapter.

[If these moods, instead of being regarded as belonging to the
first figure, are placed in a fourth, their reduction will be formally
a little different. To reduce the first three, we shall simply have
to draw the conclusion which naturally follows from the same
premisses in the first figure, and then convert it ; but this will now
be said to involve transposition of the premisses ; for what is major
regarded as in the fourth figure is minor regarded as in the first,
and vice versa : thus

Fig. 4. Bramantip. Fig. 1. Baralipton.

Men of stout heart are free The free are happy

The free are happy x Men of stout heart are free

.•. Some who are happy are of stout heart

The premisses in Baralipton are premisses in Barbara ; those in
Bramantip are not so, till they are transposed.

On the other hand, in the last two moods only conversion and not
transposition will now be necessary ; for the fourth figure already
regards the universal negative premiss in Fesapo or Fresison
(= Fapesmo or Frisesomorum) as the major, because it contains
the term which is predicate in the conclusion, though it is subject
in the premiss ; conversion will bring it to the position which the
major term should hold in its premiss in the first figure ; and so
with the minor ; and our original conclusion then follows in Ferio.]

Whether, in reducing a syllogism of any imperfect mood, the
premisses need transposing ; which, if any of them, must be con-

1 to (v8ai{iop to iXtiidepov, to $' theiiOtpov to ev\j/v\ep KpLvavTfs, Thuc. ii. 43.

xm] REDUCTION OF THE IMPERFECT FIGURES 291

verted ; whether we have to convert the conclusion obtained in the
syllogism of reduction, in order to recover the original conclusion ;
and in which mood of the first figure the validating syllogism will
be — all these matters are indicated by the consonants of the mood-
names. The significant consonants 1 are :

1. the initial, always the same as that of the mood in Fig. 1 to
which the imperfect mood must be reduced.

2. m (= muta), which indicates that the premisses must be
transposed.

3. s (= simpliciter), which indicates that the premiss, or con-
clusion 2, signified by the preceding vowel must be converted simply.

4. p ( = per accidens), which indicates that the same must be
converted by limitation.

5. c ( = conversio syllogismi), which, occurring medially, indicates
that we must employ the process of Indirect Reduction, to be
explained immediately.

In order to illustrate the mechanical use of these instructions,
it will be enough to work out in symbols the reduction of a single
mood, Disamis. That, as the mnemonic tells us, is in Fig. 3 ; the
middle term is therefore subject in both premisses. The major,
being indicated by /, is a particular affirmative, and the minor,
being indicated by A, an universal affirmative ; the conclusion
similarly a particular affirmative. Our syllogism is therefore to be
of the type : —

Some M is P /

All M is 8 A

.'. Some 8 is P I

In reducing it, the m of the mood-name indicates that we must
transpose the premisses, and the s that we must convert simply the
premiss indicated by the vowel after which it stands ; the D that
we shall so obtain a syllogism in Darii, thus : —

All M is S
Some P is M
:. Some P is 8

1 Except the initials, these are explained in the old lines —

Simpliciter verti vult 8, P verti per acci,
M vult transponi, C per impossibile duci.

If any one is horrified at the doggerel, he may be assured that much worse
things could have been quoted in earlier chapters.

2 i. e. not the conclusion of the original syllogism (which has to be obtained
again as it stood), but the conclusion of the validating syllogism.

U2

292 AN INTRODUCTION TO LOGIC [chap.

The simple conversion of this conclusion, enjoined by the s after
the third vowel in Disamis, gives us

Some 8 is P

This process of Direct Reduction cannot be applied to the two
moods, Baroco and Bocardo. The reason is obvious. In order that
the middle term may occupy a different position in the two pre-
misses, as the first figure requires, one of the premisses in the second
and third figures must be converted. In these moods, the premisses
are an universal affirmative and a particular negative proposition.
The latter, 0, cannot be converted either simply or per accidens ;
the converse of A is J ; and so by converting that we should obtain
two particular premisses. These syllogisms can, however, be
validated by the process of Indirect Reduction.

Indirect Reduction, or Reduction per impossible, consists in
showing, by a syllogism in the first figure, against which no objection
can be taken, that the falsity of the conclusion in the original
syllogism is inconsistent with the truth of its premisses. This w
done as follows : —

Baroco is of the form

All P is M All negroes have curly hair

Some S is not M Some natives of Africa have not curly

hair
.*. Some S is not P .'. Some natives of Africa are not negroes

Now if this conclusion is false, its contradictory will be true, i. e.
that All natives of Africa are negroes. We can then combine this
with our original major premiss to form a syllogism in Barbara,
thus : —

All P is M All negroes have curly hair

All S is P All natives of Africa are negroes

.*. All S is M .*. All natives of Africa have curly hair

But the conclusion thus obtained contradicts the original minor
premiss ; hence if the original premisses are true, the conclusion we
drew from them cannot be false, and our original syllogism is there-
fore valid.

The method of reducing a syllogism in Bocardo is the same :
except that here by combining the contradictory of the conclusion
with the original minor we reach a result inconsistent with the
original major premiss ; while in the former case, by combining

xiii] REDUCTION OF THE IMPERFECT FIGURES 293

it with the major, we deduced a conclusion contradictory of the
minor. The medial c in the mood -name directs us to substitute for
the premiss indicated by the vowel after which the c is placed the
contradictory of the conclusion.1

[All the imperfect moods could be validated in this indirect
manner 2 : take, e.g., Darapti — All M is P, All M is S .-. Some S is
P; if this is false, then No 8 is P ; and All M is S ; .-. No M is P ;
which is inconsistent with the truth of the original major premiss.
The first figure, on the other hand, cannot be appealed to in order
to confirm itself ; if we suppose its conclusion to be false, and com-
bine the contradictory thereof with one of the premisses, it is only
by a syllogism in the second or third figure that we can deduce
a conclusion inconsistent with the other premiss ; e. g. in Barbara
(All M is P, All 8 is M .-. All 8 is P) ; if the conclusion is false, then
Some 8 is not P ; and All M is P ; .-. Some 8 is not M — which
contradicts the original minor ; and again, Some 8 is not P, and
All S is M .-. Some M is not P — which contradicts the original
major ; but the arguments are in the second and third figures.]

1 It is possible to validate the moods Baroco and Bocardo by the direct
method, if we employ the processes of permutation, and conversion by
negation. From Baroco we obtain a syllogism in Ferio, thus : Baroco, All
P is M, Some 8 is not M .*. Some S is not P : Ferio, No not-M is P, Some S
is not-Jf .". Some S is not P ; from Bocardo we obtain a syllogism in Darii :
Bocardo, Some M is not P, All M is 8 .'. Some S is not P : Darii, All M is
S, Some not-P is M .'. Some not-P is S .'. Some S is not P. Names have
been given to the two moods in place of Baroco and Bocardo, by logicians
who considered these methods of reduction to be preferable, in which the
processes to be followed are indicated. These processes have been relegated
to a note, and the names suppressed, because there is no purpose in burdening
what may be called the mechanical part of the theory of syllogism with any
fresh refinements. ' Barbara Celarent ' may be retained and explained, on
historical grounds ; we need not add to it. On the other hand, the question
as to whether the imperfect moods need validating, and if so, what is the
most proper way of doing it, will be discussed in the next chapter.

2 Though for Fig. 4 the syllogism which employs the contradictory of the
original conclusion as one of its premisses will yield a conclusion contradicting
the converse of one of the original premisses.

CHAPTER XIV

OF THE PRINCIPLES OF SYLLOGISTIC
INFERENCE

When I argue that because A=B and B = C, therefore A=Ct
my reasoning proceeds upon the same principle as when I argue
that because X=Y and Y = Z, therefore X = Z. This principle is
expressed in the familiar axiom that things which are equal to the
same thing are equal to one another. In the particular inference,
A=B, B = G .'. A=C, I do not deduce any conclusion from that
axiom, as from a major premiss. It has indeed sometimes been
contended that the argument is really syllogistic ; that it should
be written

Things equal to the same thing are equal to one another
A and C are things equal to the same thing
.'.A and C are equal to one another.1

But the following considerations will show that this is not the case.
Firstly, we may appeal to an analogous argument, in which a quan-
titative relation is established between A and C on the ground of
the quantitative relations of both to B, although the quantities are
none of them equal. If A is greater than B, and B is greater than
C, A is greater than C. Are we to maintain that this inference
should properly be written

Things of which one is greater and the other less than the same

thing are greater the one than the other
A and C are things of which one is greater and the other less
than the same thing
.'. A and G are greater the one than the other ?

The cumbrousness of this would be no reason for refusing to recog-
nize it, if it were correct ; and if the other is correct, this must be.
Yet where, as in this case, it requires some violence and ingenuity
to give a quantitative inference the appearance of a syllogism, it is

1 Euclid, for example, wrote under the impression that this is the right
way of stating such an argument.

PRINCIPLES OF SYLLOGISTIC INFERENCE 295

not habitually done ; and since men have been content not to force
into the form of syllogism the inference ' A>B, B>G .'. A>G ',
it may be surmised that they would not have so dealt with the
inference ' A = B, B = G .'. A = G ', if it had not been for the apparent
ease of the transformation. But appearances may be deceptive ;
and it must be noticed secondly, that in the syllogism which is
supposed to represent the latter inference, viz.

Things equal to the same thing are equal to one another
A and C are things equal to the same thing
.*. A and G are equal to one another,

our minor premiss and our minor term are both faulty. The minor
premiss is not a correct statement of the grounds of our inference ;
these are, that A and G are both equal to B, and therefore the
major required is ' Things equal to B are equal to one another \
And the minor term ' A and G ' is not really a subject of which we
demonstrate an attribute ; it is two subjects, which are shown to
stand in a certain relation to each other. Thirdly, and chiefly, the
so-called major premiss is itself established through the so-called
minor and its conclusion. It is because I see that if A and C are
both equal to B, they are equal to one another, that I recognize the
truth of the general principle or axiom. If I were incapable of
recognizing the validity of the inference in the case of the three
quantities A, B, and C, or X, Y, and Z, I should not be able to
recognize the truth of the axiom. The axiom, therefore, is not one
of the premisses from which we reason, when we argue that ' A=B
and B = C .'. A = C ' : it is the principle in accordance with which we
reason. If it were denied, the validity of any particular inference
that conforms to it would be denied also ; its truth is therefore
involved in that of the particular inferences. But a man may see
the validity of the particular inference, without formulating the
axiom. This would not be so, if it were really a suppressed major
premiss, and ' A and G ' a true minor term. In the argument that
' Silver is a good conductor because it is a metal ', every one recog-
nizes that it is implied that ' All metals are good conductors ' ; and
without this premiss, the grounds of the inference are not apparent.
But no one requires any further grounds for inferring ' A = G ',
than are contained in the premisses ' A=B and B = G '.

We may therefore dismiss the attempt to reduce this argument
to syllogistic form, and recognize in the axiom not a premiss but

296 AN INTRODUCTION TO LOGIC [chap.

the principle or canon of the argument. But the question then
arises, whether there is similarly a principle or canon of syllogistic
inference. Let us recall what was shown in Chapter XI, of which
what has just been said is only a corollary. We there distinguished
between an argument in which a relation of quantity was estab-
lished between two terms, through their relation in quantity to
a common third term : and an argument in which a relation was
established between two terms in the way of subject and predicate,
through their relation in that respect to a common third term ; the
latter being syllogism. Now the axiom ' Things that are equal to
the same thing are equal to one another ' is a principle of inference
in the domain of quantity. It specifies no particular quantities,
but states that two quantities will stand in a certain relation (of
equality) to one another, if they stand in certain relations (of
equality) to a third. May there not be a corresponding principle
in syllogistic inference — one which specifies no particular terms, but
states that two terms will be related to each other as subject and
predicate in a certain way, if they are so related in certain ways to
a third term ?

Such a principle has been supposed to be furnished in the Dictum
de omni et nullo ; and a consideration of this, and of other canons
which have been proposed in its place, will throw a good deal of
light on the nature of syllogistic inference, and the difference
between its different types or figures.

The phrase ' Dictum de omni et nullo ' is really a short title by
which to refer to a principle too long to enunciate always in full ;
just as we refer to statutes or papal bulls by their first word or two.
The principle may be expressed thus — Quod de aliquo omni praedi-
catur [dicitur, s. negatur], praedicatur [dicitur, s. negatur] etiam
de qualibet eius parte : What is predicated [stated, or denied] about
any whole is predicated [stated, or denied] about any part of that
whole.1

1 I have quoted Zabarella's formulation of the Dictum de Omni, de Quarto
Figura Syllogismi Liber, Opera Logica, Coloniae, 1597, p. 115 A. The words
in square brackets are not his. There are numerous variants of no particular
importance. Crackenthorpe (III. 16, p. 202 in ed. of 1670) gives ' Quidquid
amrmatur (s. negatur) universaliter de aliquo, idem affirmatur (s. negatur)
etiam de omni de quo illud praedicatur'. This form Beems (as Mansel
remarks of Aldrich's) to be more nearly a translation of the passage in
the Categories than of that in the Analytics. The formula 'quod valet
de omnibus valet etiam de singulis ' (the reference for which I cannot now
find) treats the major premiss nakedly as an enumerative judgement ; the

xiv] PRINCIPLES OF SYLLOGISTIC INFERENCE 297

If we take syllogisms in the first figure — and it is enough to
consider Barbara and Celarent — the meaning of the principle will

same view is implied in speaking of the middle term as a class, as e.g.
Whately and Bain do.

The passage in Aristotle from which the Dictum de Omni was primarily
derived is Anal. Pri. a. i. 24b 26-30 to 8e iv 5\a> tlvai trtpov ereptp ku\ to Kara
navroi KaTT)yope't(T@ai darepov Bar tpov raiirov ttrriv. Xeyopev §« to Kara navrbs
Karqyopela&ai, orav prjtiep jj Xaffelv ra>v mv vnoKfiptvov, Ka6 ov Bartpov ov

\fxdt]<reTai' ko\ to Kara p.T]8ev6s ixravToos (' That one term should be contained
in another as in a whole is the same as for one to be predicated of all
another. And it is said to be predicated of all anything, when no part
[ = logical part] of the subject can be found, of which the other term [the
predicate] will not be true ; and to be predicated of none, similarly ').
Aristotle is here explaining the meaning of expressions which he is about
to use in the Analytics ; if mortal is predicated of animal or man Kara rrai>T6s,
it means that there is no animal (e.g. man) or man (e.g. Socrates) who is
not mortal. And no doubt that is involved in the truth of the universal
proposition ; but it does not follow that Aristotle thought of the universal
proposition as no more than an enumerative judgement about every species
(or individual) of which the subject-term can be predicated. He uses the
formula to pio-ov itrriv iv oXa> t<u irpcorco (' the middle is contained in the
major as its whole ') as well as to Trpanov KaTrjyopdrcu koto, navrot tov pto-ov
(' the major is predicated of all the middle ') to indicate the relation of the
major to the middle term in Fig. 1 (and similarly with the relation of the
middle to the minor) ; and SXov means a logical whole or universal, not an
aggregate of individuals. Elsewhere he says of that figure, d yap to A koto
ttclvtos tov B nai to B Kara ttovtos tov T, dfdy<Tj to A Kara iravTos tov T kott)-
yope'io'dat' rrportpov yap elprjrai n&s to koto namros \eyopev (' For if A is predicated
of all B, and B of all C, A must be predicated of all C : for we have already
stated what we mean by predicating of all ') (Anal. Pri. a. iv. 25b 33-4, 37-40).
Doubtless if it is involved in saying ' All B is A ', that every B is A, and in
saying ' All C is B ', that every C is B, then it is involved that every C is A ;
but the universal proposition need still not be viewed as a statement about indi-
viduals. Indeed if it were, each particular C must be already known to be A in
making the judgement ' All C is A \ and therefore the inference that all C is A
would be unnecessary. Aristotle himself points this out in Anal. Post. a. i, and
makes it plain that in his view the universal proposition was not an enumera-
tive judgement about known individuals ; and he hardly ever uses a singular
term to illustrate the minor of a syllogism. And although we must admit
that in regarding Fig. 1 as the only perfect figure, and in exhibiting the
necessity of the inference in Fig. 1 as he does in the words last quoted,
Aristotle lays too much stress on the aspect of extension, and not enough
on that of necessary connexion of characters within the subject, yet he
largely corrects this himself in his account of demonstration, and he did not
think that the essential meaning of the universal proposition, and what
constituted the nerve of the reasoning, lay in the fact that it made an assertion
about every individual falling under it.

There is another passage sometimes quoted as the source of the Dictum,
viz. Cat. iii. lb 10 (e. g. Mansel's Aldrich, p. 85, note a : Baldwin's Dictionary
of Philosophy and Psychology, s. voc. Aristotle's Dictum). The section runs
as follows : orav tTfpov KaO' erepov K(iTT]yoprjTai a>s <a6' liroKfipevov, Saa Kara tov
Karrryopovpfvov XeyeTtu, rrdvTa ko\ Kara tov vnoKciptvov pt]drjo~tTai, otov avdpanos
Kara tov Tivbs dvdpdmov KaTTjyopt'iTai, to 8e (atov Kara tov dvOpdoirov' ovkovv nai
koto, tov tivos dvdpcitnov KaTr)yopr)8rjo~eTai to fojof" 6 yap tis avdpanros kcli ai>6pu>ir6s

tart koA (uov (' When one thing is predicated of another as of a subject de

298 AN INTRODUCTION TO LOGIC [chap

be plain. All (or No) B is A, All C is B .'. AU (or No) C is A. Here
it matters not for what real terms A, B, and C stand, any more than

quo, all that is asserted of the predicate will be asserted of the subject as
well ; e. g. man is predicated of a particular man [as subject de quo], and
animal of man, and therefore animal will be predicated also of the particular
man ; for the particular man is man and animal '). But its context dispels
any presumption that this passage is an enunciation of the Dictum. There
is nothing about syllogism in the Categories at all. In the previous chapter
a distinction was made between different kinds of ovto. — beings, or entities.
Some KuB vTroKeipivnv rivos Xeyerai, iv vnoKeijiivu) he ovbev'i iirnv : they
are said, or predicated, of a subject, but do not inhere in a subject ; man,
e. g., is predicated of Caesar, but not inherent in him. Others iv vnoKeipeva)
fari, nati'1 imoKeifj.evov de ov8ev6s Xtyercu, they inhere in a subject, but are
not predicated of a subject ; as Priscian's grammatical knowledge inhered
in the soul of Priscian, but is not predicable of any subject which could be
said to be Priscian's grammatical knowledge. Others again nad"1 vnoKeip.evov
re heyerai kcu iv vTroKei.jj.evu> eariv, they both are predicated of a subject
and inhere in a subject ; as knowledge is predicated of Priscian's grammatical
knowledge, and inhered in the soul of Priscian. Others, lastly, ovt iv vnoicei-
fxevto iarrlv, ovre Ka8' vrroKeifjievov Xeyerai, they neither inhere in a subject
nor are predicated of any ; such are concrete individuals, like Caesar and
Priscian. Here the opposition between <a6" vnoKeifxevov \eyeo~8m, being
predicated of a subject, and iv vnoKeip.eva> elvm, inhering in a subject, is
parallel to that between essential and accidental predication. If I say of
Caesar that he is a man, of grammar that it is a science, of colour that it is
a quality, those predicates give the general being of their subjects, the sub-
jects are essentially those, or (as it may be put) are their subjects de quo.
But if I say of Priscian that he is a grammarian, or of a map that it is
coloured, grammar and colour are not what the soul of Priscian, and the
map, essentially are ; they inhere in them, and the soul of Priscian and the
map are their subjects in quo. In the language of the Categories, that is in
a subject, which being in anything not as a part of it cannot exist apart
from that wherein it is ; iv inroKeipiva) \eya>, 6 'iv tivi fif] a>? pepos vnap\ov
ahvvarov xaPls 6"/al T°v 6" <? iarlv (ii. la 24-25). Colour cannot exist except
in a body, but is not a part of the body : nor grammar except in a soul,
but it is not part of a soul.

Upon these distinctions succeeds immediately the sentence quoted at the
head of the last paragraph ; and it must clearly be interpreted with reference
to them. The connexion seems to be as follows. There exist (1) individual
substances, like Caesar, which are subjects de quo, and subjects in quo, but
neither predicable of nor inherent in anything else ; (2) universal substances,
like man, predicable of individual substances, and not inherent in anything ;
(3) individual attributes (or accidents), like Priscian's grammatical knowledge,
which is a subject de quo, because the universal attribute is predicable of it,
and it of no attribute else, but is not a subject in quo, being itself inherent
in an individual substance, not that wherein anything inheres ; (4) universal
attributes, like knowledge, predicable of individual attributes and inherent
in individual substances. Man then is not an bnoKeipevov or subject, but
is predicated of a subject ; nevertheless we find terms predicated of man,
in such a proposition as ' Man is an animal '. What then is animal ? for it
is not a subject, and apparently not predicated of a subject, for man is not
really a subject. The answer is, that animal is really predicated of the
subject whereof man is predicated, and therefore, like man, falls into the
Becond of the above classes.

If we consider this doctrine on its own account, it is open to considerable

xiv] PRINCIPLES OF SYLLOGISTIC INFERENCE 299

in the axiom of equals it mattered what real quantities were intended.
Whatever they are, suppose that A can be affirmed or denied of all B,

criticism. There is the same failure to distinguish different senses of imoK*L-
ficvov, subject, as in Mill's discussion of connotative terms (cf. quotation
supra, p. 148). As subject de quo, it is the individual, whether substance
or attribute, the instance of an universal ; as subject in quo, it is the individual
substance. Thus in relation to knowledge, the grammatical knowledge of
Priscian is a subject de quo, as in relation to man, Priscian is ; and language
allows me to say that the former is a knowledge, and the latter is a man.
But again in relation to knowledge, Priscian, or the soul of Priscian, is the
subject in quo ; and language allows me to say not that Priscian is a know-
ledge, but that he is knowing, though both man and knowledge are sometimes
6aid to be ' predicated of ' Priscian : clearly not in the same sense. When the
subject of a predicate is its subject de quo, then if the predicate is in the
category of substance, the subject is a substance ; if the predicate is in some
other category, the subject is in that category ; but when the subject of
a predicate is its subject in quo, the subject is a substance, the predicate in
some other category than substance. Now the language of the Greek,
when it distinguishes <ca#' vnoKeifj.<ii><w XeyeaOm and iv inroKt ineva> tlvm, being
predicated of a subject and inhering in it, does not suggest that the word
inroKfipevov, subject, has any but one sense ; the difference is put as if lying
in the relation of the predicate to it ; but really to be a subject of inherence,
or substance, is not the same as to be a subject de quo, or individual, though
some individuals are individual substances.

There is a further difficulty in the passage. It professes to distinguish
kinds of 6Wa, things or entities. Now when we say that an attribute inheres
in a substance, we mean, according to the teaching of the passage, that an
individual attribute inheres in an individual substance ; and these are rightly
distinguished as things of different kinds. But when we predicate something,
whether substance or attribute, of its subject de quo, we do not mean that
an individual is the universal of which it is an instance. ' His grammatical
knowledge inheres in Priscian ' ; here the words His grammatical knowledge
denote an individual attribute, and Priscian denotes an individual substance.
But when I say that Priscian is a man, or that attribute of his a knowledge,
man does not denote an universal substance, nor knowledge an universal
attribute. Yet these are what are said to be predicated of their subjects
de quo. We have seen (supra, pp. 33-35) that the same abstract term is
commonly used as a general name of attributes, and as name of the universal
whereof they are instances. But here the general concrete term is treated
as the name of the universal whereof individual substances are instances,
and man is said not to be, but to be predicable of, a subject. The word ' man '
is doubtless so predicable ; but what it denotes is some individual subject.
Aristotle however treats general concrete names as names of universals.
In de Interpr. vii. 17a 38-bl we are told that some things are universal, some
individual ; universal, what can be predicated of more than one, individual,
what cannot ; man e. g. is an universal, Callias an individual (enel 8' £<rr\
Ta fieu KadoXov tg>v irpay^iaruiv ra he ko.8' eKaarov' Xeycu de KnOoXov \xev o enl
nXeiovcov nefpvKe Kar-qyopeiaPai, icad' eKacnov de 6 jxrj, olov av&pconos fiev rcov
KadoXov, KaXXlas $e rcov Kciff1 eKaarov' ktX.).

But whatever the scruples which the whole passage raises, the words in
question are far from enunciating the Dictum de ornni et nullo. In the
syllogism ' All men are animals, Socrates is a man .*. Socrates is an animal ' —
if indeed Aristotle would have called that a syllogism (cf. infra, p. 321) —
man is predicated of Socrates cos ku6' imoKemivov, as its subject de quo, and
animal kclt avBpconov Xeyerm, not iv av6pu>n(p e'arip, it is predicable of, not

300 AN INTRODUCTION TO LOGIC [chap.

it can be affirmed or denied of each particular subject, C or any other,
included in B. Here, according to a tradition which has been
strong, is the fundamental principle of syllogistic inference. In
this Dictum is nakedly displayed what is the nerve of our reasoning,
whenever we syllogize in the concrete. It is the assurance that A
is true of all B, which satisfies us that it is true of this B, viz. of
C ; the business of reduction is to bring imperfect syllogisms into
a form, in which we can see at once that the principle applies to
them ; and the title of the first to be the perfect figure lies in its
conforming to the formula of the Dictum de omni et nullo.

There are several objections urged against the claims of this
formula. In the first place, it suggests the ' nominalist ' doctrine
expressed by Hobbes, when he said that reasoning is but the right
ordering of names in our affirmations. It suggests that our ground
for affirming or denying that C is A lies in the fact that A is said of
all, or no, B, and B is said of G. Clearly it is because we believe
that B is A, and G is B — not because B is called A, and G is called
B — that we assert the conclusion. However, this nominalist inter-
pretation of the Dictum is not necessary ; it is not as thus inter-
preted that it will be here discussed ; and therefore this objection
may be dismissed.

It may be said secondly, that if the reduction of the other figures

inherent in man, being in the same category, and its general being ; animal
therefore is predicated of Socrates as its subject de quo, i.e. Socrates is an
animal. The conclusion is justified by the rule in the Categories. But to
most syllogisms it has no application. ' All organisms are mortal, Man is an
organism .*. Man is mortal.' Here the minor term is not an vnoKeifievov,
or subject de quo, in the sense of the passage in the Categories, but some-
thing predicated of a subject ; and though the middle is predicable of the
minor, the major is inherent in the middle. Again if Priscian was a gram-
marian, and a grammarian is scientific, Priscian was scientific ; but here
though the minor term is an hnoKe'ipc-vov, an individual substance, the middle
is predicated of it not as its subject de quo, but as its subject in quo ; it is
not therefore a case orav erc-pov K.a.6' (Ttpov KarrjyoprjTdi cos Kad vTioKcipevov,
where one thing is predicated of another as of a subject de quo, and so does
not fall within the scope of the rule. Once more, if all men are jealous,
and Priscian was a man, Priscian was jealous ; here the middle is predicated
of the minor term as of a subject de quo ; but as in the proposition ' All
organisms are mortal ', so it is in this major premiss ; jealousy is not some-
thing which Kara tov Karriyopovpevov XiytTai ; man is to jealousy not sub-
ject de quo, but subject in quo ; we cannot, according to the language of
the context, say that jealousy kot avdpionov Xiytrai, is predicated of man,
but that it ev iiv$pu>na> toriv, inheres in a man. There is therefore no justifica-
tion for finding in this rule a statement of the Dictum. Whether Aristotle
would have accepted the Dictum as a correct expression of the principle of
syllogistic inference is another question, to which the answer depends very
much on how we interpret the Dictum.

xiv] PRINCIPLES OF SYLLOGISTIC INFERENCE 301

to the first is not necessary, i. e. if the true character of our reason-
ing in them is not more clearly displayed in the first figure, the
Dictum is not the principle of all syllogistic inference. In claiming
to be that, it denies any essential difference between the different
figures ; and those who think them essentially different are so far
bound to question the analysis of syllogistic inference which the
Dictum implies. This is quite true ; but we can hardly discuss
the relation of the different figures, until we have settled whether the
Dictum expresses correctly the nature of our reasoning in the first.
We come therefore to what is the main criticism which has been
urged against the Dictum, and against all syllogistic inference, if
it be supposed that the Dictum is a true analysis of its nature. It
is said that a syllogism would, on this showing, be a petitio principii.
By petitio principii, or begging the question, as it is called in English,
is meant assuming in one of your premisses what you have to prove.
Of course, the premisses must implicitly contain the conclusion ;
otherwise you would have no right to draw it from them, and could
deny it, while admitting them : this much is true of every kind of
cogent inference, whether syllogistic or not, though it has been
sometimes treated as a peculiarity of syllogism by persons who
thought they could find other kinds of inference not obnoxious to it.
But you do not beg the conclusion in the premisses, except where
the conclusion is necessary to establish one or other of the premisses.
For example, I may know that treason is a capital offence ; and the
law might make it treasonable to publish libels against the sovereign ;
and in that case, from the premisses, All treason is a capital offence,
To libel the sovereign is treason, I could infer that To libel the sovereign
is a capital offence. In this argument, there is no petitio principii ;
I can learn the truth of both premisses by consulting the statute-
book, and do not need to be aware that it is a capital offence to libel
the sovereign, in order to know either of the premisses from which
that conclusion is deduced. But the case is different in such a syllo-
gism as that All ruminants part the hoof, and The deer is a ruminant
.'. The deer parts the hoof. I have no means here of ascertaining
the truth of the major premiss, except by an inspection of the various
species of ruminant animals ; and until I know that the deer parts
the hoof, I do not know that all ruminants do so. My belief in the
constancy of structural types in nature may lead me to expect that
a rule of that kind, found to hold good in all the species which I have
examined, holds good universally ; but this presumption, so long

302 AN INTRODUCTION TO LOGIC [chap.

as it rests merely on the examination of instances, is not conclusive ;
I should not accept the conclusion merely on the strength of the
premisses, but should seek to confirm it by an examination of the
hoof of the deer ; the case of the deer therefore is necessary to
establish the rule.

Now it has been alleged that all syllogism is a petitio principii * ;
and the allegation has gained colour from the Dictum de omni et
nullo. * That which is affirmed or denied of any whole may be
affirmed or denied of anything contained within that whole.' What
do we mean by a whole here ? If it is a class or collection, if the
major premiss is to be understood in extension, then it can hardly
be denied that it presupposes a knowledge of the conclusion. If in
the proposition All B is A, 1 mean not that B as such is A, but that
all the B's are A, I must certainly have examined C (if that is one of
them) before making the assertion ; and therefore the major pre-
miss, All B is A, rests {inter alia) on the present conclusion, G is A.
According to this view, the major premiss of a syllogism is (at least
in most cases 2) a statement of fact about the whole of a number of
particulars ; it is really an enumerative, and not a true universal,
judgement.3 We make it, not because of any insight that we have
into the nature of B and A, and into the necessity of their connexion :
but simply because we have examined everything in which B is
found, and satisfied ourselves that A is equally present in all of them.

There is indeed another sense in which the major premiss may be
enunciated without our having insight into the necessary connexion
of characters in things, and in which it no longer makes a collective
assertion about every one of a number of particulars. If I say that
all gold is yellow, I need not mean to assert that every piece of metal,
which by other qualities I should identify as gold, is also yellow
— a statement for which I certainly cannot claim the warrant of
direct experience. I may mean that a yellow colour is one of the
qualities on the ground of which I call a substance gold ; or, in

1 Cf., e.g., Mill, System of Logic, II. c. iii. Mill's own way of avoiding
the charge is not very successful.

2 Where general rules are made by men, as in the case of laws, we can of
course know them, in advance of any knowledge about the particular acts
or events to which they refer. Such syllogisms, therefore, as that about
libelling the sovereign, given in the last paragraph, can in no case be alleged
to beg the question. If any other authority (such as revelation) acquaints us
with general rules, they will serve as major premisses of equally unexcep-
tionable syllogisms. All other general propositions have, by the extremer
cmtics, been interpreted in the way mentioned in the text.

3 For this distinction, cf. supra, p. 177.

xiv] PRINCIPLES OF SYLLOGISTIC INFERENCE 303

Locke's language, that it is included in the nominal essence of gold.
By a nominal essence, Locke means what J. S. Mill called the
connotation of a name — those attributes which, in using a general
name, we imply to belong to the subjects called by it. We may
collect together in our thought any set of attributes we like and
give a name to the assemblage of them ; and then it will, of course,
be true to say that anything called by the name, if rightly called by
it, possesses any of the attributes included in the signification of
the name. The general proposition ceases, in that case, to be
enumerative ; but it does not become really universal. It becomes
a verbal proposition. Gold is yellow, because we do not choose to
call anything gold which is not yellow ; but we are not asserting
that there is any necessary connexion between the other attributes
for which a parcel of matter is judged to be gold, and this of yellow-
ness. Given such and such attributes, we call it gold ; and there-
fore gold has all these. Let any one of them be wanting, and we
should not call it gold ; therefore that is not gold which is not
yellow ; but there may be a parcel of matter, for all that we mean
to affirm, which has all the other qualities of gold, but is of the
colour of silver.1

Locke did not suppose that the ordinary man, who says that
gold is yellow, means only to assert that yellowness is one of the
attributes included by him and others in the nominal essence (or
connotation) of the word gold : but rather that he means, that with
certain other qualities collected in the ' complex idea ' to which the
name gold is attached the quality yellow is constantly conjoined.
This however, on Locke's view, we cannot know ; for knowing is
perceiving a necessary agreement or disagreement between our
1 ideas ' (it would be better to say, connexion or disconnexion
between the characters of things) ; and this in regard to our ' ideas
of substances ' we do not perceive.2 It is not our present business
to discuss this ; we have not to ask how many of the general pro-
positions of the sciences state connexions known to be necessary
(though, if we did, we should find Locke not very far in that matter
from the truth), nor what means there are (if any) of proving uni-
versal propositions about such matters of fact. We are concerned

1 Cf. Locke's Essay, III. vi. §§ 6, 19, and also pp. 92 sq., supra, on Definition.

2 Cf. Locke, Essay, IV. vi, esp. §§ 8, 9. Miss Augusta Klein has justly
objected to me that in the first edition of this book I represented Locke as
holding propositions about ' nominal essences ' to be more verbal than he
really does

304 AN INTRODUCTION TO LOGIC [chap.

with the theory of syllogism, and the allegation that it begs the
question. We found that if the major premiss be interpreted in
extension as an enumerative judgement, the charge is true ; and
that the Dictum de omni et nullo at least lends colour to such an
interpretation. We have now seen that there is another interpreta-
tion, according to which the major premiss may be known to be true
without examination of every individual instance included under the
subject of it, but only by becoming a verbal proposition. On this
interpretation the syllogism will still be a petitio principii, though
not in the way which the Dictum de omni et nullo suggests. For
though the major premiss will no longer presuppose a knowledge of
the conclusion, the minor will do so. If nothing is to be called gold
unless it is yellow, I cannot tell that a parcel of matter is gold, in
which I have found the other qualities which the name implies, unless
I have first seen that it is yellow. Of course, colour being the most
obvious of the properties of a substance, I am not likely ever to be in
the position of inferring the colour of a substance from its name ;
but the argument is the same if I took some unobvious quality,
like solubility in aqua regia. If that is part of the nominal essence
of gold, then I cannot tell that a particular parcel of matter with
the familiar weight and colour of gold is gold, until I know that it is
soluble in aqua regia. I do not therefore infer its solubility from the
knowledge that it is gold, but I call it gold because I know it to be
thus soluble.1

We need not dwell longer on the view that a general proposition
is only warranted by agreement as to the meaning of a name, nor on
the consequences, fatal enough, which this view would entail on the
syllogism. Reasoning is not a mere process of interpreting names ;
and it is not the principle of syllogistic inference, that whatever
a name means may be affirmed of the subjects called by it. In
considering the charge that the syllogism is a petitio principii, it was
necessary to notice the view which makes the petitio lie in the
minor premiss, as well as that which makes it lie in the major. We
must now return to the latter, and to the Dictum which is supposed
to countenance it.

We saw that the crucial question here concerned the nature of
the major premiss ; is it universal, or merely enumerative ? is it

1 It will now be seen why a syllogism was explained to beg the question,
if it presupposed the conclusion not in the premisses together, but in either
of them singly ; all syllogisms in a sense presuppose it in the premisses
taken together (though they do not presuppose a knowledge of it).

xiv] PRINCIPLES OF SYLLOGISTIC INFERENCE 305

based on an enumeration of particulars, or on the connexion of
universals ? If it is enumerative, and rests on a previous review
of all the particulars included in the middle term, the charge of
petitio is sustained. We should then accept the Dictum, de omni et
nullo as the general principle of syllogism, the ' whole ' of which it
speaks being understood as a whole of extension, a collection or
class ; but we should scarcely be able to speak of syllogistic inference.

Now Aristotle, who thought syllogism to be the type of all demon-
stration, could not possibly have understood the major premiss in
this way.1 He thought that, although we might know as a fact
that C is A, yet we did not understand it, without seeing that it
must be so ; and to see that it must be so is to see that in it which
makes it so — to see that it is A in virtue of B. B is a middle term,
because it really mediates between G and A ; it performs for G the
office of making it A, and is the reason why G is A, not merely the
reason why we know G to be A.

We have already, in discussing the modality of judgements, met
with this distinction between the reason for a thing being so and so,
and the reason for our knowing it to be so — between the ratio essendi
and the ratio cognoscendi.2 When I say that wheat is nourishing,
because it contains nitrogen and carbon in certain proportions, I give
the reason for its being nourishing : it is this constitution which
makes it so. When I say that Mellin's Food is nourishing because
Baby grows fat on it, I do not give the reason for its being nourish-
ing, but only the reason for my thinking it to be so : it is not Baby's
condition which makes it nourishing, but its nourishing properties
which produce Baby's condition. The physical sciences always look
for rationes essendi, so far as possible ; though it may be noted that
in what is, in many ways, the most perfect of the sciences, viz.
Mathematics, we reason very largely from rationes cognoscendi.2 If
A =B, and B = C, then A = G ; but it is not because A and G are
both equal to B, that they are equal to one another, though that is
how I may come to know of their equality. The reason why they
are equal is that they contain the same number of identical units.3

The middle term does not in all syllogisms give the reason why
the major belongs to the minor. It does so only in the first figure,

1 The doctrine of the Posterior Analytics must in this respect be taken as
overriding the more formal and external treatment of syllogism in the Prior.

2 v. supra, pp. 205-206.

5 But we cannot give this reason for the equality of the units.

1779 X

306 AN INTRODUCTION TO LOGIC [chap.

and not always there. Because, whenever the middle term really is
a ratio essendi, the syllogism falls into the first figure, Aristotle called
it the scientific figure, o-xwa ^icrr^jxoviKov.1 Why are modest men
grateful ? Because they think lightly of their own deserts. This
implies a syllogism in Barbara. All who think lightly of their own
deserts are grateful, and modest men think lightly of their own
deserts. But if I try to establish the conclusion by an appeal to
instances, pointing out that Simon Lee and Tom Pinch, John Doe
and Richard Roe, were modest, and were grateful, I am giving not
a reason why the modest are grateful, but reasons which lead me to
judge them to be so ; and my syllogism falls into the third figure,
not the first : These men were grateful, and these men were modest,
therefore modest men are (or at least they may be) grateful.

The first figure then is scientific, because a syllogism which makes
you know why G is A falls into that figure ; but the middle term in
the first figure need not be a ratio essendi. ' Parallel rays of light
proceed from objects at a vast distance ; the sun's rays are parallel ;
therefore they proceed from an object at a vast distance.' Here
my syllogism is again in Barbara ; but the distance of the sun is
not due to its rays (at the earth) being (so far as we can detect)
parallel : their being parallel is due to the distance of the sun from
the earth. Nevertheless, the syllogisms in which the middle term
does account for the conclusion are enough to show that syllogism
is not essentially a process of inferring about a particular member
of a class what we have found to be true of every member of it. The
importance of the scientific, or demonstrative, syllogism in this
connexion is, that it most effectually disposes of this analysis of
syllogistic inference. It shows that there are syllogisms which
cannot possibly be brought under the Dictum de omni et nullo, thus
interpreted. We shall, however, find that even where the middle
term is not the cause of the conclusion, in the sense of being a ratio
essendi, the Dictum thus interpreted does not give a true account of
the nerve of our reasoning.

For syllogism really works through the connexion of concepts, or
universals. The major premiss, ' B is A ', is not a collective state-
ment about every B, C included ; if it were, there would certainly
be nothing new in the conclusion ' C is A '. When Jacob lamented
' Me have ye bereaved of my children : Joseph is not, and Simeon

1 Anal. Post. a. xiv. 79a 17. The rest of the chapter is by no means all of
it true. On ' scientific ' and ' dialectical ' syllogisms cf. infra, pp. 398-399.

xiv] PRINCIPLES OF SYLLOGISTIC INFERENCE 307

is not, and ye will take Benjamin away ', he added, speaking
collectively of the loss of two sons, and the threatened loss of a third,
' all these things are against me '.* It would have been no inference
to proceed ' Therefore the loss of Simeon is against me ', because that
was definitely included by the demonstrative these. To be ' one of
these ' is not a common character in each of them, with which a
further character is connected ; it is not therefore a middle term.
Where B is a middle term, the major premiss connects being A with
being B.2 We must not be misled by our use of symbols. In
a syllogism of the form ' All B is A, C is B .'. C is A ', we say that
B is the middle term. The following is a syllogism of that form :

Those who can find things out for themselves are little depen-
dent on education
Men of genius can find things out for themselves
.*. Men of genius are little dependent on education.

Now if we symbolize the major premiss by the formula ' All B is A ',
B represents the words ' those who can find things out for them-
selves '. But if we say that B is the middle term, B really repre-
sents the words ' being able to find things out for oneself \ It is
that which, in men of genius, is either the ground or the sign of
being little dependent on education. The middle term therefore
is not the collection of things called by a general name ; it is the
common nature intended by the name, a ez> eirl ttoW&v, something
one in many subjects. And the same is of course true of the major
term. The minor may indeed be an individual, or a number of
individuals, though it need not be so.

The perception that the middle term is not a class but a character,
universal and not a sum of particulars, has led to the formulation
of a principle intended to express this more satisfactorily than the
Dictum de omni el nullo does ; of which it has already been said
that it at least lends itself to an erroneous view of the major premiss,
as an enumerative proposition, though it was by no means always
so intended. The principle is this — Nota notae est nota rei ipsius
(and for the negative, Repugnans notae repugnat rei ipsi) : i. e. what
qualifies an attribute qualifies the thing possessing it. Certain
objections may be made to this formula also. It suggests that the
minor term is always a concrete individual, and that the syllogism

1 Genesis xlii. 36.

2 Or, if negative, asserts that being B excludes being A. It will be remem-
bered that we are discussing the first figure.

X2

308 AN INTRODUCTION TO LOGIC [chap.

re fere to this (res ipsa) what in the major premiss is stated to char-
acterize its predicates. It speaks also as if one attribute were
conceived to qualify another in the same way as an attribute qualifies
a concrete subject. And the conception of a mark or nota is no
improvement on that of attribute.1 We need not interpret it as
a purely external sign, related to what it signifies as a word to its
meaning or a letter to a sound. The ' notes ' of a thing are its
characteristics, as Cardinal Newman spoke of the notes of the
Church ; they are not the mere indications by which we judge what
thing is present, but themselves contribute to make it the thing
that it is. Yet the nature of a thing is no less ill conceived as an
assemblage of marks than as a bundle of attributes. The notes of
the Church would not exhaust the notion of the Church ; the marks
of a disease, though elements and features of it, would not give a
complete conception of what the disease is. There are predicates
of a thing which include too much of its nature to be called marks
of it. Nevertheless this formula has the great advantage that it
does prevent our regarding the middle term as a class which includes
the minor in its extension.2

But a better formula may be found. Kant said of the syllogism
that it subsumed a cognition (i.e. a subject of knowledge) under the
condition of a rule, and thus determined it by the predicate of the
rule.3 The rule is given in the major premiss, which connects

1 Cf. Hegel's Logic, § 165, E. T. (Wallace), p. 296 : ' There is no more
striking mark of the formalism and decay of Logic than the favourite category
of the " mark ".'

8 J. S. Mill (System of Logic, II. ii. 4 and note) strangely misinterprets
the maxim Nota notae est nota rei ipsius. He understands by res ipsa the
major term, and by nota the minor ; so that the whole, instead of meaning
that what qualifies an attribute qualifies the subject of it, comes to mean
that what indicates the presence of an attribute indicates what the latter
indicates. He naturally gets into great difficulties where the minor term is
singular. We may treat the attributes of man as a mark or indication of
mortality (though this is rather like saying that a Bank of England note
is a mark of the presence of the chief cashier's signature) ; but we cannot
treat Socrates as a mark or indication of the attributes of man. Therefore
in the syllogisms All men are mortal, All kings are men (or Socrates is a man)
.'. All kings are (or Socrates is) mortal, while the minor premiss of the former
is paraphrased The attributes of a king are a mark of the attributes of man,
that of the latter runs Socrates has the attributes of man. This is a rather
desperate shift. But res ipsa never meant the major term, the most general
or abstract term in the syllogism ; and the whole interpretation, which neces-
sitates a measure so violent, is impossible. The formula is really an abridged
equivalent of the passage in Ar. Cat. lb 10-12, quoted p. 297, n., supra.

3 Krit. d. r. Vern., Transcendental Dialectic, Introd. II. B (p. 215, Meikle-
john's Translation).

xiv] PRINCIPLES OF SYLLOGISTIC INFERENCE 309

a predicate (the major) with a condition (the middle term) : the
minor premiss asserts the fulfilment of this condition in its subject ;
and in the conclusion we determine the subject by the predicate
which the rule, in the major premiss, connected with this condition.
This analysis brings out the essential nature of the major premiss,
as a rule connecting a predicate with a condition universally, not an
assertion that the predicate is found in every member of a class.
It also applies equally where the middle term is, and where it is not,
the ratio essendi of the major. And it is free from the objections
just urged against Nota notae.1 If we were to frame from it a ' canon '
parallel to this and to the Dictum de omni et nullo, it would run
somewhat thus : Whatever satisfies the condition of a rule falls under
the rule. In the rule ' Whatever is B, is A ', being B is the condition,
the fulfilment of which involves being A ; and to a given subject C
fulfilling the condition the rule will apply, and it will be A. We
may perhaps accept this as a statement of the nature of the reasoning
employed in syllogisms of the first figure. We need not deny that
the Dictum de omni et nullo, if rightly interpreted, is free from the
offences charged against it. If the omne be understood as an unity
present in many instances — a logical whole or whole of intension,
not an aggregate of individuals — then the principle will serve. But
the other puts more clearly the nerve of the inference. And it
applies to all syllogisms in the first figure, whatever the nature of
the middle term : whether it be a mere sign of the major term, as if
we said that ' All men with large hands and small eyes are choleric '
— where the connexion of the predicate with its condition, though
accepted de facto, is one for which we can see no necessity : or
whether it give, wholly or in part, the reason and the explanation of
the major, e.g. in such premisses as that 'All trees fertilized by the
wind blossom before their leaves are out ', or that, ' Men successful
in a work that gives full play to all their faculties are happy '.
Whatever our particular syllogism is, we shall find it true to say
of it, that it brings a subject under a rule, on the ground that it
satisfies the condition of that rule : that it affirms (or denies) a
predicate of a subject, on the ground that this subject fulfils the

1 Kant himself applied this analysis to hypothetical and disjunctive argu-
ments also. In a later chapter, these are more strongly distinguished from
' categorical ' syllogisms than he allows. But this need not prevent the
acceptance of his analysis. A statement may correctly express the nature
of syllogistic inference, even when some arguments, which are not strictly
syllogistic, are also alleged to fall under it.

310 AN INTRODUCTION TO LOGIC [chap.

condition with which the predicate (or its absence) is universally
conjoined or connected.

This canon is exemplified even where the major premiss rests on
an examination of all the instances included under the middle term ;
so that there is inference there, though not proof of the conclusion.
The major premiss is indeed in such a case a sort of memorandum,
as Mill says of it,1 to which we subsequently refer in order to save
the trouble of repeating our observations ; but a memorandum in
general terms requires inference to make use of it. Suppose a man
intending to dispose of part of his library ; he might look through
his books and put a mark in all those which were not worth keeping ;
if he then forgot what a certain book contained, but finding his mark
in it said that it was not worth keeping, he would be syllogizing.
He would argue ' Books thus and thus marked are not worth keeping,
this book is thus and thus marked, .'. it is not worth keeping'.
There is no real proof here that it is not worth keeping ; that could
only be determined by reading the book ; and his mental note that
no book thus and thus marked is worth keeping requires that he
have read this book and ascertained that it was not worth keeping
before marking it. But he may have forgotten all about it ; and he
now asserts that it is not worth keeping because by containing the
mark it satisfies the condition on which the ascription of that predicate
rests. In applying his rule he trusts of course to his past care in
reading and marking ; and so he may be said to take the major
premiss on trust. But that is common enough. Even when an
universal proposition is capable of proof, many reason from it
syllogistically who never knew the proof, or if they knew it, have
forgotten it. We may go further. Subsumption, or bringing
a subject under the condition of a rule, which is the nature of syllo-
gism in the first figure, always implies that in a measure the rule is
taken on trust. To this extent the major premiss is always a sort
of memorandum. For if we understood at the moment the neces-
sary connexion between the middle term and the major, we should
appeal to no rule, but in considering C, the minor term, itself, pass
from the knowledge that it is B to the further knowledge that it is A.
We should indeed realize herein that the connexion of the character
A with the character B was not limited to the subject G. But

1 System of Logic, II. iii. 4. Mill's mistake lies not in saying that the
major premiss of a syllogism is a memorandum, but in making it a false
memorandum, which records that all B is A when we have only observed
that X, Y and Z (which are B) are A.

xiv] PRINCIPLES OF SYLLOGISTIC INFERENCE 311

we should rather extract the general rule from consideration of the
subject before us than bring it in independently and as it were
ab extra to the consideration of that subject. And that would not
be syllogism. Syllogism does not belong to the level of complete
insight into the connexion of facts. In geometry we never syllogize
except when we rely upon the results of a previous demonstration
whose steps we do not realize in the case before us. ' The triangle
in a semicircle has the square on its hypotenuse equal to the squares
on the other two sides, because it is right-angled ' ; that is a syllo-
gism ; but if we realized at once the construction of Euclid i. 47 for
the figure in iii. 31, the proposition that in a right-angled triangle the
square on the hypotenuse is equal to the squares on the other two
sides would appear rather as generalized from what we saw to be
true in the triangle in a semicircle than as a rule applied to that case.
The subsumption in syllogism belongs therefore to thinking which
has not complete insight into the necessity of all the facts in its
premisses at once. When Aristotle taught that syllogism is the
form of demonstrative thinking, he failed to realize this. Because
C's being A is seen to be involved in its being B, he thought we used
a major premiss, ' All B is A\ He was nearer the truth when he
said that in demonstration our terms are connected per se. The
putting together of, or the appeal to, premisses already known is
not necessary to demonstration. Supposing I already understood
that to be an organism involved being mortal, yet if I discovered
some thing of strange kind to be an organism, I should know that it
was mortal, in virtue of my now understanding the connexion, not
in virtue of having understood it before. But because we have
constantly to appeal to the conclusion of a previous process of
demonstration or other reasoning without re-thinking that process
at the time, we are constantly syllogizing ; and where the premisses
are such of which we remember to have previously satisfied our-
selves by reflection or demonstration or inductive argument, or (if
they concern facts established by authority) by reference to autho-
rity, there syllogism may deserve the name of proof. It is other-
wise where, in order to establish a conclusion, we appeal to a premiss
which itself needs the help of the conclusion to establish it ; this
is not proof ; yet if the premiss has been so established, and is now
appealed to as a record, there is syllogistic inference. Our argument
is one whose general form is given in the canon of syllogism.
That canon, like the axiom of equals, is not itself a premiss but

312 AN INTRODUCTION TO LOGIC [chap.

a principle of reasoning. It is easy to see this. Any one denying it
would as readily deny the validity of any particular syllogistio
argument ; but a man may admit the validity of the inference,
in a particular case, without needing to consider this general prin-
ciple. And, as no one could see that Things equal to the same
thing are equal to one another, who was incapable of seeing the truth
of that principle in a given case, so no one could see the truth of the
principle that What satisfies the condition of a rule falls under the
rule, who failed to recognize that if all organisms are mortal, and
man is an organism, man must be mortal. What then is the use
of the principle, if it is not a premiss of inference ? It might be
used to stop the mouth of a disputant who denied the conclusion
which followed from the premisses he had admitted. We might
ask such a disputant, whether he denied the truth of this principle,
and unless he was prepared to do that, require him to admit the
validity of the syllogism he was disputing. It is true that in con-
sistency he might decline. A man who denies the validity of a given
syllogism in Barbara may with equal reason deny the argument
which attempts to prove its validity. For that argument will
itself take the form of another syllogism in Barbara :

All inferences upon this principle (that what satisfies the condition

of a rule falls under the rule) are valid
The syllogism in question is an inference upon this principle
.*. It is valid

Why should a man admit this reasoning, if he will not admit that
since

All organisms are mortal, and

Man is an organism
.*. Man is mortal ?

The two are of the same form, and this shows that you cannot make
the principle of syllogistic inference into the premiss of a particular
syllogism, without begging the question.1 Yet a man who disputes

1 Cf. an article on 'What the Tortoise said to Achilles', by ' Lewis Carroll ',
in Mind, N. S. iv. 278 (April, 1895). It is obvious that the validity of the
latter of these two syllogisms cannot require to be deduced from the principle
which stands as major premiss in the former. For if until that is done its
validity is doubtful, then the principle by which we are to establish its
validity is equally doubtful. Besides, what proves the validity of the former,
or validating, syllogism ? The validity of a syllogism cannot be deduced
from its own major premiss ; else the fact that all organisms are mortal
would show that the syllogism, of which that is the major premiss, is valid.

xiv] PRINCIPLES OF SYLLOGISTIC INFERENCE 313

in a particular case the conclusion that follows from his premisses
may hesitate to maintain his attitude, if the principle of reasoning
involved is put nakedly before him, and shown to be one which
he daily proceeds upon, and cannot disallow without invalidating
his commonest inferences. For this reason it may cut wrangling
short, if we can confront a man with the principle of the inference
he questions. Show him, for example, that the inference ascribes
to a subject, in which certain conditions are fulfilled, a predicate
connected universally with those conditions, and he cannot longer
refuse his assent. For to do what it does is to be a syllogism x : and
therefore valid.

And there have been writers 2 who thought that the only object of
knowing the theory of syllogism was to cut short wrangling. But
there is another object, connected with a side of logic which the
same writers for the most part ignore. Logic is not an art. Its
business is to know and understand the processes of thought, and
not least the true nature of our processes of inference. To this
business belongs the question, what is the principle of a certain
inference which we make, and recognize to be valid ? To find and
formulate that principle — to extricate it from its concrete setting
in the matter of a particular argument, and set it out in abstract,
— this is the logician's task. Now men may misinterpret the
character of syllogism, and formulate wrongly the principle involved;
yet if their misinterpretation is generally received for true, the
wrong principle will serve in practice to stop dispute as well as
the right principle would have done. Those who are agreed that
syllogism is conclusive, however they define a syllogism, will accept
an argument if it can be shown to accord with their definition ;
and the same misinterpretation which appears in their account of
the general nature of syllogism will appear in their view of particular
syllogisms, from which that account is of course derived. There-
fore, though it be said that a syllogism is an argument which applies
to some one member of a class what is true of every member, yet
even this analysis of it, however faulty, will serve to ' stop wrangling'

If it be said that the validating syllogism needs no proof of its validity, the
same can be said of the syllogism which it validates. But if it needs a proof,
the syllogism which validates it will need validating by another, and so
ad infinitum. No form of inference can have its validity guaranteed by
another inference of the same form with itself ; for we should be involved
at once in an infinite process.

1 Cf. Ar. Post. An. p. vi. 92a 11-16.

8 e. g. Locke, Essay, IV. xvii. 4.

314 AN INTRODUCTION TO LOGIC [chap.

among persons who accept it. For let a particular argument be
exhibited as doing this, and it will be accepted as valid. But the
theoretical objections to this analysis of syllogistic inference are in
no way lessened by its being practically as useful as any other that
men could be brought to accept. The paramount question is,
whether it is true : not whether for any purposes it is useful. And
the present chapter has been quite disinterested ; it has aimed
at throwing light on the question, What is a syllogism ? i. e. What
is the principle of inference which a syllogism exemplifies ?

We have ignored of late the imperfect figures, in seeking an
answer to this question. They furnished a possible objection to the
claims of the Dictum de omni et nullo x ; for if their reduction to the
first figure is unnecessary, then the Dictum, which only contem-
plates the first figure, cannot be the principle of all syllogistic
inference. But this objection was deferred, until the Dictum had
been examined on its own ground. We must now return to the
subject of the imperfect figures.

It may make things clearer, if the view to be taken in the follow-
ing pages is given summarily at the outset. There are difficulties
in any view of the matter ; because the same verbal form may be
used where the thought in the speaker's mind is different. The
true character of an argument depends not on the verbal form, but
on the thought behind it. And therefore sometimes the movement
of a man's thought, though he expresses himself, e. g., in the second
figure, would be more adequately exhibited in the first.2 In such
a case direct reduction may be defensible, though still unnecessary ;
and yet it may be true that, speaking generally, the direct reduction
of the imperfect figures distorts them, and purchases a show of
conformity with the first figure at the expense of concealing the
genuine movement of thought in them.

It would seem then that syllogisms in the second and third figures
do not as a rule merely present under a disguise the reasoning of
the first ; they are independent types. Their validity is confirmed,

1 Cf. supra, p. 301.

2 e. g. in this syllogism in Festino, ' No fragrant flowers are scarlet, Some
geraniums are scarlet .*. Some geraniums are not fragrant ', I think a man
would probably substitute in thought for the major its converse, ' No scarlet
flowers are fragrant', and argue to himself in Ferio. With such a premiss,
where there is no priority as between the two accidents, fragrant and scarlet,
that is the more natural way to argue. But this does not show that all
syllogisms in Festino ought to be thus treated.

xiv] PRINCIPLES OF SYLLOGISTIC INFERENCE 315

in the second figure, by the reductio ad absurdum,1 and in the third,
by the method which Aristotle called eK0e<ris, or exposition. The
fourth figure (or indirect conclusion in the first) is not an independent
type ; its first three moods are merely moods of the first figure, with
the conclusion converted, as the process of reducing them assumes ;
its last two moods draw conclusions which are shown to be valid
most naturally by reduction to the third figure.

Let us begin with the second figure. Take the syllogism : All
true roses bloom in summer : A Christmas rose does not bloom in
summer .'.It is not a true rose. Surely, if a man hesitated for
a moment about the necessity of this consequence, he would reassure
himself, not by transposing the premisses, and converting the
present minor into the statement that No rose which blooms in
summer is a Christmas rose : but by considering, that a Christmas
rose, if it were a true rose, would bloom in summer, whereas it does
not. The same remarks will obviously apply to a syllogism in
Baroco. Nor is it otherwise with the remaining moods. If No
fish has lungs, and Whales (or Some aquatic animals) have lungs, then
Whales (or Some aquatic animals) are not fish. A man sees at once
that if they were, they wrould not have lungs : whereas they have.

It might be said that the last conclusion could be as naturally
reached in the first figure ; that if a man, confronted with the con-
clusion that Whales are not fish, and not feeling that he was clear
about its cogency, were to ask himself ' Why not ? ', he would
answer ' Because they have lungs ' ; and that this implies a syllo-
gism in Celarent, with the major premiss What has lungs is not
a fish. Whether this gives the reason why a whale is not a fish
(in which case Celarent would be a better way of proving it) we
need not dispute ; but there certainly are cases where, in what a
subject is, we can find a reason for its not being something else.
Notes that produce beats are not harmonious : The fourth and fiftf
produce beats ; Therefore they are not harmonious. This argument
might be set forth in the second figure : Harmonious notes do not
produce beats : The fourth and fifth produce beats ; Therefore they are
not harmonious : but here undoubtedly the syllogism in Celarent is
better than the syllogism in Cesare ; and any one who knew that
concord was dependent on regular coincidence in vibrations and
discord on the clashing of them, would extricate from the major
premiss of the latter syllogism the major of the former, and think

1 Called by Aristotle aTrayayfj eh to abvvaTOv.

316 AN INTRODUCTION TO LOGIC [chap.

in Celarent. Nevertheless it is only this knowledge which makes
him do so ; and without it he might perfectly well validate to
himself his conclusion by considering that if those notes were har-
monious, they would not produce the beats they do. If the middle
term gives a ratio essendi, we naturally put our reasoning into the
first figure.1 The Chinese are not admitted into the United States,
for fear lest they should lower the white labourer's standard of
living. The likelihood of their doing this is the cause of their
exclusion. It would be unnatural to express this in Cesare —

None admitted into the United States are likely to lower the

white labourer's standard of living
The Chinese are likely to lower it
.*. The Chinese are not admitted into the United States.

But we are not concerned to prove that no arguments expressed
in the second figure are better expressed in the first ; only that
there are arguments which are more naturally expressed in the
second, and which we should not, if challenged, attempt to validate
by reduction to the first. Thus I may argue that Notes which pro-
duce beats are not harmonious, and A note and its octave are harmonious,
.'. They do not produce beats ; and it is as much a distortion to put
this into the first figure by conversion of the major premiss as to
put the previous example which used that major premiss into the
second figure by the same means. Again, if I give, as a reason
why whales are not fish, that they have not the characteristics of
fish, such as breathing through gills, laying eggs, &e., my syllogism
may very well be in Camestres — All fish breathe through gills, and
Whales do not .'.A whale is not a fish ; if I still ask myself why not,
I should probably answer, ' Because if it were a fish, it would
breathe through gills, which it does not do \ The conclusion states
a fact of difference between two things, which the premisses prove
but do not account for ; and the proof in the second figure may be
said to be here the primary form.2 Moreover, if I were to recur to
the first figure in order to establish this inference, it would naturally
be by contraposing the major premiss

1 It must not be forgotten that most reasoning which explains facts through
their causes is not syllogistic at all ; but if it is syllogistic, it will be in the
first figure. Cf. supra, p. 305.

2 Hence the statement, frequently quoted from Lambert (Neues Organon,
vol. ii. p. 139 ; Dianoiologie, iv. § 229, Leipzig, 1764), that the second figure
points us to the differences between things : ' Die zweite Figur fiihrt auf den
Unterschied der Dinge, und hebt die Verwirrung in den Begriffen auf.'

xiv] PRINCIPLES OF SYLLOGISTIC INFERENCE 317

What does not breathe through gills is not a fish
Whales do not breathe through gills
.*. Whales are not fish

for the absence of a feature essential to any fish may be treated as
explaining why a thing is not a fish. But the syllogism to which
Camestres is supposed to be reduced is not the above ; it is the
following —

What breathes through gills is not a whale
A fish breathes through gills
.*. A fish is not a whale

from which the original conclusion that a whale is not a fish is
recovered by conversion. Now this argument, instead of relying
on something in whales (viz. the absence of gills) to show that they
are not fish, relies on something in fish (viz. the presence of gills)
to show that they are not whales ; whereas whales are really the
subject of my thought. The same line of reflection may be applied
to the argument, Matter containing active bacilli putrefies : Frozen
meat does not putrefy .'.It contains no active bacilli ; where no one
could maintain that non-putrefaction was really the cause of matter
containing no active bacilli.

Thus the second figure is really different in type from the first ;
although reasonings which would naturally fall into the first may be
thrown into the second. And the difference is this, that the second
is fundamentally indirect, the first direct. In the second, we see
the validity of the conclusion through the contradiction that would
be involved in denying it ; in the first (though, of course, it would
be equally self -contradictory to admit the premisses and deny the
conclusion) the perception of this is not a ' moment ' in our thought.
It may fairly be said that the first figure is prior to the second, in
the sense that it is involved in the perception of the contradiction
which would result from admitting the premisses and denying the
conclusion in the second. But that does not justify us in reducing
the second to the first. For it is an essential part of our thought
in the second figure, to see that the conclusion must follow on pain
of contradiction ; and not merely to see the validity of the first-
figure syllogism, by help of which the contradiction is developed.
There is therefore a movement of thought in the second figure which
is absent from the first. This is what makes a new type of it ;
and this is why its direct reduction, representing second-figure

318 AN INTRODUCTION TO LOGIO [chap.

syllogisms as only first-figure syllogisms in disguise, is wrong, and
therefore superfluous.

It may be asked, is even indirect reduction necessary ? Is not the
validity of the argument plain, without our being at pains to show
that, if it were disputed, we should be involved in a contradiction ?
Cannot a man appreciate that if No A is B, and C is B, then G is
not A, without the necessity of pointing out that C would not other-
wise, as it is, be B1 The answer is that a man may certainly not
require this to be pointed out, inasmuch as he sees it at once to be
involved in the premisses. The so-called indirect reduction is really
a part of the thought grasped in the syllogism ; not something
further, by which, when a man has already made his inference, and
realized the act of thought involved in making it, he then proceeds
to justify his act. It rather brings out what is in the inference,
than reduces or resolves it into another. Hence a man may feel it
to be unnecessary, but only because it is a repetition, not because, if
he did not see it, the syllogism would still be seen to hold without it.
Yet it must not be supposed that a form of argument is valid only
because to question it would involve a contradiction. With equal
reason it might be said that unless the argument were valid, there
would be no contradiction in rejecting it. Hence, in the second
figure, the contradiction that would ensue if we denied the con-
clusion, is not the reason for admitting the conclusion, but the
perception of it is involved in realizing its validity. An analogy may

help us. If a straight line, falling on two
other straight lines, makes the exterior and
the interior and opposite angles on the
same side of it equal, the two lines must be
parallel. Strictly speaking, this cannot be
proved by reasoning ; we just see, when we
try to draw the figure otherwise, that it must
be so. But this necessity may be brought out indirectly by the con-
sideration, that if B E F were to be greater than BCD, E F and
C D would cut A B at a different slant, and therefore incline to-
wards one another ; and the perception of this is really part of seeing
the necessity of the original proposition. Nevertheless it cannot be
given as a reason for the truth of that proposition ; for unless the
lines were parallel when the angles B E F, B C D are equal, they
would not necessarily tend to meet when each cuts A B at a different
slant. The confirmation, such as it is, is obtained by looking at the

xiv] PRINCIPLES OF SYLLOGISTIC INFERENCE 319

tame matter from another side ; and so it is in the second figure of
syllogism. The truth of one side cannot really be separated from
the truth of the other, and therefore the one is not dependent on
the other ; but it is not fully appreciated without it. The develop-
ment of the contradiction involved in denying the conclusion in the
second figure is a development of the system of relations between
the terms alleged in the premisses, or of the consequences involved
in these. It is not, like a suppressed premiss, something without
the consideration of which the argument is altogether broken-
backed ; but it is something involved in the full appreciation of the
argument.

If then the second figure is not a mere variation of the first, it
follows that the principle or canon on which the first proceeds is not
that of the second. If the above account of the nature of our
reasoning in the second figure is correct, its principle is this, that no
subject can possess an attribute which either excludes what it
possesses or carries what it excludes.

Of the third figure we must give a different account. Its two most
noticeable features are that the middle term is subject in both pre-
misses, and the conclusion always particular. For this reason it
has been well called the inductive figure ; for induction (whatever
else besides may be involved in it) is the attempt to establish a con-
clusion upon the evidence of instances. The terms of the conclusion
are always general . The conclusion declares two characters to be con-
joined, or (if negative) disjoined : Sailors are handy, The larger carni-
vora do not breed in captivity. In the premisses we allege instances of
which both characters can be affirmed ; or of which one can be affirmed
and the other denied ; and these instances are our evidence for the
conclusion. But the conclusion is not general ; we are never justi-
fied, by a mere citation of instances, in drawing a really universal
conclusion. If All B is A, and All B is C, we cannot say that All G
is A ; in traditional phraseology, G is undistributed in the minor
premiss, and therefore must not be distributed in the conclusion ;
and the thing is obvious, without any such technicalities, in an
example ; if all men have two arms, and all men have two legs, it
does not follow that all animals with two legs have two arms ; for
birds have two legs, besides men, and have not arms at all, but
wings. Yet, though our instances will never justify a really universal
conclusion, they may suggest one ; and they will at any rate over-
throw one. The instances of Queen Elizabeth or Queen Victoria,

320 AN INTRODUCTION TO LOGIC [chap.

of Catherine of Russia or Christina of Sweden, wall disprove the
proposition that No woman can be a statesman ; and truth is often
advanced by establishing the contradictory of some universal
proposition, no less than by establishing universal propositions
themselves.

Now what is the true nerve of our reasoning in such arguments ?
It is the instance, or instances. We prove that some C is A, or some
C is not A, because we can point to a subject which is at once C and
A, or C and not A. Unless we are sure that the same subject is
referred to in both premisses, there can be no inference : Some
animals are quadrupeds, and Some animals are vertebrates ; but they
might be different animals, and then there would be no instance of
a vertebrate that had four legs. But if either premiss is universal —
if e. g., with mammal as our middle term, we take the premisses Some
mammals are quadrupeds, and All mammals are vertebrates — then it
follows that Some vertebrates are quadrupeds ; for the ' some '
mammals of the major premiss are included among the ' all ' of the
minor, and therefore we could pick out, from among the latter,
instances of animals that were both vertebrate and quadruped. The
instances, however, instead of being vaguely indicated as ' some '
of a whole class or kind, may be specified by name ; and then the
nature of our reasoning is unambiguous ; we are manifestly arguing
through instances. In order to show that A woman may be a states-
man, we can appeal to the four queens mentioned above ; these were
statesmen, and these were women ; and therefore some women have
been (or women may be) statesmen. But whether the instances in
which C and A are united, or G is present without A, be cited by
name, or only indicated as ' some ' of a whole class, in both cases
alike it is on them that the reasoning hinges, and it is by producing
them that a sceptic could be confuted, who refused to admit the
conclusion.

Aristotle called this production of the instance by the name
«?K0eo-is, or Exposition. He conceived that the proper mode of
validating a syllogism in the third figure was by direct reduction,1
but added that it was possible to validate it per impossibile or by
* exposition ' : ' if all S is both P and R, we may take some
particular S, say JV ; this will be both P and E, so that there will be

1 Except, of course, where the major premiss is a particular negative and
the minor an universal affirmative proposition (Bocardo), in which case we
can only proceed per impossibile or by exposition. Anal. PH. a. vi. 28b 15-21.

xiv] PRINCIPLES OF SYLLOGISTIC INFERENCE 321

some R which is P ' 1 ; and what is possible where both premisses
are universal and affirmative is equally possible in any other mood.
This seems to exhibit the real movement of thought in the third
figure better than the artificial process of direct reduction. For, in
the first place, if the middle is a singular term, as in this figure it
often is (though Aristotle took little note of such cases), the con-
version of a premiss is forced and unnatural. In words I may say
that since Queen Elizabeth and Queen Victoria were statesmen,
and some women were Queen Elizabeth and Queen Victoria, there-
fore women may be statesmen ; but in thought, Queen Elizabeth
and Queen Victoria will still be subject in the minor premiss. And
secondly, even where the middle is a general term, direct reduction
often conceals, rather than expresses, our thought. No ostrich can
fly, All ostriches have wings .'. Some winged animals cannot fly : here,
though it is possible to substitute for the minor premiss Some winged
animals are ostriches, the other is the form in which we naturally
think ; the more concrete term stands naturally as the subject of
our thought.

It may be admitted that there are cases where direct reduction
is unobjectionable. No clergyman may sit in Parliament, and Some
clergymen are electors to Parliament .". Some electors to Parliament
may not sit in it. Here it would be as natural to say that Some
electors to Parliament are clergymen ; for the franchise, and the
clerical office, are each an ' accident ' of a man, and either, elector to
Parliament or clergyman, can equally well be subject in the pro-
position, and the other predicate. But the character of the argu-
ment seems changed by this alteration. Clergymen are no longer the
instance which shows that a man may be entitled to vote without
being entitled to sit ; the middle term is now a status in virtue of
which certain voters cannot sit. The point contended for is not
that there may not be syllogisms in the third figure, whose conclusion
could be equally well, or even better, obtained with the same middle
term in the first : but that the movement of thought characteristic
of the third figure is not, and cannot be reduced to, that of the first ;
and that reduction, as a general principle, is thereforesuperfluous and
misleading : the true confirmation of the validity of the syllogism
lying in the perception that, if the premisses are true, there actually
are instances of the fact alleged in the conclusion.

One objection to this view of the third figure needs consideration.

1 Anal Pri. a. vi. 28a 24-26.

177» Y

322 AN INTRODUCTION TO LOGIC [chap.

It may be said that the production of a particular instance in support
of the conclusion does not do full justice to the grounds on which
we accept it, in cases where the middle term is general and both
premisses universal. All horned animals ruminate, and they all
part the hoof ; this, it may be urged, is better ground for concluding
that cloven-footed animals may be ruminants, than if I merely
called attention to the cow in my paddock. To settle this, let
us look for a moment at the two meanings, which (as we saw before)
may be intended by a particular proposition.1 If I say that Some
C is A, I may either mean to refer to certain unspecified but definite
members of the class C, and predicate A of them ; or without any
special thought of any particular case, I may mean to declare the
compatibility of the two characters, C and A, in one subject. In
the latter case, I can also express my meaning by the problematic
judgement C may be A ; which contains no doubt the thought of
unknown conditions under which it will be so. Now supposing
I understand the proposition in the latter sense, the cow in my
paddock is as good a middle term as horned animals generally ;
supposing I understand it in the former sense, then my conclusion,
that Some cloven-footed animals ruminate, undoubtedly has more to
rest on, when the premisses speak of all horned animals, than when
for middle term I refer only to a cow or two in a neighbouring
paddock. But it is also really a different conclusion ; the ' some '
intended are a larger number of unspecified animals in the one case
than in the other ; and it is only by the production, or ' exposition ',
of all the instances to which our ' some ' refers, that the reference
to them all, in the conclusion, may be justified.

It may fairly be said that the argument, in this view of it, does
not really amount to a syllogism : it comes to this, that if all horned
animals ruminate, and all part the hoof, then all cloven-footed animals
that are horned ruminate. If the exact sphere of the conclusion is
thus borne in mind when we say that some cloven-footed animals
ruminate, and we mean by ' some ' all that are horned, there is not
really and in thought that elimination of the middle term in the
conclusion which is characteristic of syllogism. It would not be
reckoned a syllogism if we argued that since Wolsey was a cardinal
and Wolsey was chancellor, he was both chancellor and a cardinal 2 ;
neither is it a syllogism (though it is inference) to argue, from the

* Cf. supra, pp. 178-180, 199.

* Cf. Bain's Logic, Deduction, p. 159 (ed. 1870).

xiv] PRINCIPLES OF SYLLOGISTIC INFERENCE 323

premisses above, that all horned animals are both ruminant and
cloven-footed : from which it follows that all cloven-footed animals
that are horned are ruminant.

We may admit the view of the last paragraph to be the right one.
Supposing that when we conclude, in the third figure, that Some
C is (or is not) A, we refer in thought, though not in words, just to
those particular instances, and no others, which in the premisses
were stated to be both B and A (or not A), then we have not got
a proper syllogism. Still our conclusion rests entirely on the pro-
duction of those instances, few or many, beyond which our thought
refuses to travel. The true and characteristic syllogism in the third
figure, however, intends its conclusion in the other sense : as a
statement of the proved compatibility or separability of two attri-
butes. And to establish this too it relies on the production of an
instance ; nor are many instances really more sufficient than one,
to establish mere compatibility, except as minimizing the risk of
mal -observation. The appeal need not indeed be to an individual
it may be to a kind. If we want to prove that an evergreen may
have conspicuous flowers, we can cite the rhododendron ; and we
may mean by that not any particular specimen, but any of the
species generally.1 But very often, and mostly where one premiss
is particular,2 and of course always where the premisses are singular,
it is on determinate individual instances that we rely ; and one
instance or one species is enough. Therefore it is by exposition
— by a production, not of course in bodily form, but in thought, of
one instance or species — that we justify the inference to ourselves ;
we actually make this appeal in our minds, if we realize the ground
of our conclusion. Persons familiar with a type of reasoning may
draw conclusions from premisses as it were by precedent, and
without realizing the evidence on which they act ; but whenever we
are fully conscious of what we are about, there is, in the third figure,

1 It may be urged that the appeal is really to specimens, not species : for
the species does not blossom. The question raised is not peculiar to the
third figure. If I argue that the rhododendron is popular because it flowers
brilliantly, it may equally be said that rhododendrons do so, not ' the rhodo-
dendron '. The relation of an universal truth to its instances is involved
in the question, which is an important one. But it need not complicate the
present discussion.

2 Not always, even there ; I may argue that all breeds of dog are
domesticated, and some are savage, and therefore some domesticated breeds
of animal are savage (Disamis). Here I am speaking, and thinking,
throughout not of individual animals but of their kinds.

Y2

324 AN INTRODUCTION TO LOGIC [chap.

the recognition that the conclusion is proved by its exemplification
in a case cited, or included in what we cite.

Of course there is a way in which the number of instances makes
a real difference to the conclusion which we are inclined to draw.
The case of Prince Bladud is alone enough to show that a man who
washes in the waters of Bath may recover of a disease. The two
events, however, may be accidental and unconnected. But if cases
were multiplied, we should begin to suspect there was a connexion
between the use of these waters and the cure of certain ailments ;
or if the ailments which disappeared after taking the waters were of
all sorts, we might begin to look on Bath waters as a panacea. For
establishing a connexion between two attributes the number and
variety of instances are matters of great importance ; but for estab-
lishing compatibility one instance is enough. Now the third figure
does not prove more than a compatibility ; and never can prove
a connexion, however many the instances are ; and though the
number of instances may make a connexion highly probable, yet we
are influenced in reaching such a conclusion by other considerations
besides the mere number of the instances. For example, a man
who observed in several cows the combination of the cloven foot
with the ruminating stomach would be much less inclined to suppose
that there was any general connexion between these characters in
nature, than if he had observed the same thing in an equal number
of beasts belonging to as many different species. For we are
accustomed to find characters constant throughout one species, and
failing when we go beyond it ; so that the accumulation of instances
would be discounted by the fact that they all belonged to the same
kind. Again, we might meet a Privy Councillor in a light suit, and
yet not be led to regard the next man we met in a light suit as a Privy
Councillor ; but if we met a Guardsman in a breastplate, we should
very likely suppose the next man in a breastplate to be a Guards-
man. The readiness with which we infer connexion is controlled
by our general knowledge of the kinds of attributes that are con-
nected ; such considerations do not appear in our premisses, but
greatly influence our thought. Hence it is, that those who are
thoroughly familiar with the facts of a science, or of some historical
period, can make inferences from isolated facts which to persons
ignorant of the field of investigation, and the controlling principles
applicable to it, appear foolhardy. But all this belongs to rather
a different department of logical theory, the Logic of Induction. It

xiv] PRINCIPLES OF SYLLOGISTIC INFERENCE 325

remains true that so far as we bring no extraneous considerations to
bear, and are guided only by the facts contained in our premisses,
we can infer no more than the compatibility of two characters (or
their separability) from any number of instances ; and we can infer
thus much from a single instance.

It should be noticed, before leaving the consideration of the third
figure, that it always argues from a ratio cognoscendi. It is not
because the rhododendron has brilliant flowers, that this attribute
can be combined with evergreen foliage ; if it were not that there
is no incompatibility between them, the rhododendron could not
exhibit both. Our instance merely teaches us that the two are
compatible ; it is the ground of our assertion, not the ground of
the fact asserted. And this in itself is enough to show that there
is a real difference between the nature of our reasoning in the third
figure, and in the first — at least when our syllogisms in the first
figure are scientific ; and that the attempt to reduce all syllogisms
to one typical form imposes an unreal appearance of conformity
upon arguments which are essentially disparate.

[The fourth figure of syllogism remains for consideration.1 It has
this peculiarity, that its premisses as they stand, if we transpose
them, present the arrangement of terms required by the first figure.
And three of its moods (Bramantip, Camenes, and Dimaris), when
thus regarded as being in the first figure ( =Baralipton, Celantes,
Dabitis), afford conclusions of which those drawn in the fourth
figure are merely the converse ; but the other two moods (Fesapo
and Fresison) yield no conclusion in the first figure, from which the
conclusion in the fourth might be obtained. Are we therefore to
regard this figure as presenting a separate type of inference from the
first, or was Aristotle right in disregarding it ?

Let us look first at the moods which are reduced to the first
figure by a mere transposition, and without any alteration, of the
premisses. In the premisses All nitrogenous foods are flesh-forming ,
All grains are nitrogenous, if we treat flesh-forming as the major
term, we have a syllogism in Barbara ; but if we treat grains as
major term, our syllogism is in Bramantip, and the conclusion
is that Some flesh-forming foods are grains. It is surely true that
the cogency of this inference, as compared with the other, is pecu-
liarly unobvious. The conclusion is not what we should naturally

1 This note may, of course, be equally well regarded as a discussion of the
indirect moods of the first figure. But if a new type of inference were
involved in them, the erection of a fourth figure would be justified. As
that is the question under discussion, it seems fairer to call them moods of
the fourth figure at the outset.

326 AN INTRODUCTION TO LOGIC [chap.

[draw from the premisses ; and we need to look a little closer, in
order to convince ourselves that it necessarily follows. And this
conviction comes to us when we realize either that from the given
premisses it follows that All grains are flesh-forming, and our other
conclusion follows by conversion from that : or else that if no flesh-
forming foods were grains, no nitrogenous foods would be grains ;
and that in that case grains could not all, or any, of them be nitro-
genous. The same remarks would apply mutatis mutandis to syllo-
gisms in Camenes or Dimaris ; and we may therefore conclude that
these moods are not evidently cogent without a further act of
thought than their formulation in the fourth figure displays. Are
we therefore to treat them as belonging to the first figure ? The
reason for doing this is, that the simplest and directest way of
justifying the inference which they contain is by drawing a con-
clusion in the first figure from their premisses, and converting it.

The two remaining moods, Fesapo and Fresison, are less easily
disposed of. As the same considerations apply to both, it will
suffice to take an example of the former. No animals indigenous to
Australia are mammals, All mammals are vertebrate .•. Some vertebrates
are not indigenous to Australia ; if we transpose these premisses,
no direct conclusion follows ; we cannot tell from them whether
any of the animals indigenous to Australia are vertebrate, or not ;
so that if our argument requires validating, we must validate it
either by direct or indirect reduction, or by exposition. That it
does need validating seems to follow from the fact, that in its
present form it is no more obvious than the three preceding moods
of the fourth figure ; no one ever argues in the fourth figure, and
that shows that it does not adequately exhibit the movement
of thought in inference. Aristotle exhibited the validity of this
mood1 by converting both premisses (i.e. by direct reduction):
No mammal is indigenous to Australia, and Some vertebrates are
mammals ; and this is a more natural way of putting the argument.
But there are cases in which conversion would substitute a less
natural mode of expression in the premisses ; e.g. from the pre-
misses No mineral waters are alcoholic and All alcohol is taxed,2 we
can infer that Some things taxed are not mineral waters ; it would be
less natural, although it would yield the same conclusion, and that
in the first figure, to say that Nothing alcoholic is a mineral water,
and Some things taxed are alcoholic. Again we may proceed by
indirect reduction ; we may argue that if all vertebrates were
indigenous to Australia, then since no animals indigenous there
are mammals, no vertebrate would be a mammal ; we thus reach a

1 i.e. of Papesmo and also Fresison = Frisesomorum : v. Anal. PH. a. vii.
29a 24-27.

2 It would complicate the illustration too much to make the exception
required by methylated spirits.

xiv] PRINCIPLES OF SYLLOGISTIC INFERENCE 327

[conclusion inconsistent with the premiss All mammals are vertebrate,
and that shows that our original argument cannot be disputed ; but
we should more naturally say that No mammals are vertebrate than
that No vertebrates are mammals ; and the former contradicts
more directly the premiss that All mammals are vertebrate. Still
more do we feel this, if we apply indirect reduction to our other
example ; there, if Everything that is taxed were a mineral water,
then since No mineral waters are alcoholic, Nothing taxed is alco-
holic ; it is clearly more natural to say that No alcohol is taxed,
and that exhibits better the contradiction with our premiss. If we
employ the method of eK0ecn? or exposition, we must convert the
premiss No animals indigenous to Australia are mammals ; then we
have it given that mammals, in any instance that we like to take,
are not indigenous to Australia, and are vertebrate ; from which it
follows that an animal is sometimes vertebrate, and not indigenous
to Australia. Similarly we may convert No mineral waters are
alcoholic.

Thus we have in this mood an argument undoubtedly valid, yet
lacking something to be obvious ; it is possible to validate it in
several ways, either bringing it into the first figure by conversion
of both premisses, or into the third by conversion of one, or leaving
the premisses and showing, as in the second figure, that the falsity
of the conclusion is inconsistent with their truth. Which of these
methods is preferable ? and to what figure should the mood be
referred ? or is it really of a fourth sort ? That it is not of a fourth
sort is shown by the fact that without one of these methods of
validation its conclusiveness is not apparent, that conversion of both
premisses reduces it to the first figure, and that if reduction ad
impossibile or exposition is to be used, it may as well, and better
(as will be explained two pages below), be regarded as a syllogism
belonging, by the nature of its premisses, to the first figure, but need-
ing validation by this means. Perhaps the first of the above questions
will be best answered, if we ask in what way, by the use of the same
middle term, the conclusion of the given syllogism could most
naturally be reached. How are we to prove that Some vertebrates
are not indigenous to Australia, using mammals as our middle term ?
or that Some things taxed are not mineral waters, using alcohol as
middle term ? In both cases we should appeal to an instance in
point ; the mammals may be cited to show the former, and alcohol
to show the latter. It would seem therefore that exposition is the
natural way of validating the argument ; or in other words, that we
realize its cogency most readily if we realize that in the major premiss
there is involved a converse, from which the conclusion follows at
once in the third figure.

Are we then to reckon the mood to the third figure, and not
(with Aristotle) to the first ? Aristotle would, of course, have said
that since the third figure itself needed validating through the first,

328 AN INTRODUCTION TO LOGIC [chap.

[we had stopped half-way in reducing it to the third ; but if, as
has been held above, the third figure is really a different type of
inference, our question cannot be settled thus. Let us recall the
meaning of the distinction between major and minor terms. The
distinction is not purely formal and external. A term is not really
the major term because it is made the predicate, and minor because
it is made the subject, in a conclusion. It is the meaning of
the terms themselves which determines which ought to be subject,
and which predicate, and therefore which is major and which
minor. Otherwise, Aristotle would have recognized the fourth as
a separate figure. We may take a syllogism in Darii, and by trans-
position of the premisses produce one in Dimaris ; e.g. the pre-
misses White is conspicuous at night, Some flowers are white, whose
natural conclusion is that Some flowers are conspicuous at night,
furnish instead, if we transpose the premisses, the conclusion that
Some things conspicuous at night are flowers. But this is an obvious
inversion, for it is the flower which is conspicuous, and not the
conspicuous, as such, which is a flower. It is true that there are
cases where either conclusion is equally natural, as there are pro-
positions which may be converted without contortion. Those who
are friendless are unhappy, Some rich men are friendless .: Some rich
men are unhappy ; or, in Dimaris, Some unhappy men are rich. Here
the conclusion in Darii is the natural conclusion to draw, because
the premisses give the reason why a rich man is sometimes unhappy,
but not why an unhappy man is sometimes rich ; yet, considered
apart from the premisses, either conclusion is an equally natural
form of judgement. But the reason is, that the concrete subject men
is retained throughout ; in the conversion, the attributes rich and
unhappy change places, but the subject of which they are attributes
is retained in its place. Now these are merely coincident attributes,
and neither is properly the subject of the other ; we feel this in
making the judgement ; and instinctively convert Some rich men
are unhappy not into Some unhappy are rich men (where the concrete
term ' rich men ' could not be predicated of ' unhappy ' as such)
but into Some unhappy men are rich. When, however, this is not
the case — when the subject-concept contains the ground of the
predicate-concept, or is the concrete whole in which the latter inheres
as one feature — then the former is properly the minor and the
latter the major term, and no verbal artifice which inverts them can
alter what the fact is for our thought.

Hence in the first three moods of the fourth figure, reduction to
the first does no more than recognize in outward form as major and
as minor terms what we must acknowledge to be so in our thought.
But in Fesapo and Fresison, the conclusion is the same as what we
should draw in Ferio after their reduction, and not its converse ;
we have therefore no ground so far for giving a preference to the
expression of the argument in the first figure. But the same con-

xiv] PRINCIPLES OF SYLLOGISTIC INFERENCE 329

[siderations which make it not an arbitrary matter, which term is
major and which is minor in the conclusion, will help us to determine
the right position of the middle term in the premisses. If then the
premisses of a syllogism in Fesapo or Fresison were both of them
inversions of what would naturally be expressed in the converse
form, we should instinctively think them back into the form required
by the first figure, in drawing the conclusion. This can hardly
occur with Fesapo ; for bad logic, as well as verbal contortion,
is required in order to express a particular affirmative by an uni-
versal converse ; and therefore the minor premiss A cannot be an
inverted way of stating J : the original of Fesapo cannot be Ferio.
With Fresison it is more possible ; that is to say, a syllogism in
Fresison may be reached by converting both premisses of one in
Ferio (or Celarent) ; and then it is possible that our thought may
validate the conclusion by converting them back again. Gold does
not tarnish, Some ancient ornaments are of gold : we may, however,
say, if we like, that What tarnishes is not gold, and Some things of
gold are ancient ornaments, and from these premisses draw the same
conclusion as from the others, that Some ancient ornaments do not
tarnish ; yet our thought, justifying to itself an inference made by
outward rule, may fly to the other forms of premiss. If so, it is
hard to say that we are not really arguing in the first figure, and in
such a case the syllogism which wears externally the garb of the
fourth belongs really, and is rightly forced by direct reduction
to show that it belongs, to the first. It is however possible even
here to convert only the minor premiss in thought, and reach the
conclusion in the second figure : by realizing that ancient ornaments,
if they tarnished, would not be of gold. But the important cases
are not such as these, where the premisses are palpably in an un-
natural form, and would be restored to natural form by conversion.
They are those in which the position of the middle term, as the pre-
dicate of the major premiss and subject of the minor, is the natural
position. For here conversion to the first figure produces a result
as unnatural as there conversion to the fourth figure produced in the
premisses of an argument naturally belonging to the first ; No
mineral waters are alcoholic and All alcohol is taxed are propositions
put in their natural form ; Nothing alcoholic is a mineral water and
Some taxed things are alcoholic are not.

And if that is so, there is only one ground on which we can justify
Aristotle in reckoning these moods to the first figure. It is, that
what is properly the major term — that is, the most general and
comprehensive — does stand as predicate in its premiss, and what is
properly the minor term — that is, the most concrete and specific
— as subject. Hence looking to the character of the premisses, we
may fairly say that our syllogism is of the first figure. And it
follows that Aristotle is right when he says that we prove the minor,
not universally but partially, of the major ; for major and minor,

330 AN INTRODUCTION TO LOGIC [chap.

[as we have seen, are such intrinsically, and not barely in virtue of
their position in the conclusion ; so that where the two criteria
lead to opposite results, it is right to base our nomenclature on the
former. It was through overlooking this, and taking a purely formal
and external view of the distinction of major and minor terms, that
some of his successors were led to add a fourth figure to the three
of Aristotle. But if we recognize these moods as of the first figure,
we must no less recognize that they need validating ; and the most
natural way of realizing their validity is by the process of exposition
which we found to be the characteristic method for the third figure.
We need not on this account say that the syllogism belongs to the
third figure. The occurrence of a syllogism of the first figure in the
reduction per impossibile by which we validate the second did not
lead us to resolve the second figure into the first. Exposition too,
though the most natural, is not the only way in which we can
realize to ourselves the validity of these arguments ; so that the
third figure could not receive them unchallenged. We must be
guided, therefore, by the character of the premisses, and assign them
to the first : but admit that the conclusion is not really drawn with-
out a further act of inference than appears upon the face of them.]

We may now sum up the results of our enquiry. There are three
figures, each with a distinctive character, and the ' imperfect '
figures are misrepresented by reduction to the first. The first is the
chief, because the demonstrative, but not because the only figure.
Arguments in it need not be demonstrative, but when they are, our
thought is moving on a higher level of intelligence, though not of
cogency, than in the other figures. In realizing the validity of the
second figure, the inconsistency involved in denying the conclusion is
a more prominent ' moment ' in our thought than the necessity of
admitting it. The third figure appeals not to relations of concepts,
but to experience of the conjunction of attributes (or their dis-
junction) in the same subject, and from that argues the general
possibility, under conditions unspecified, of what is exhibited in
a given case. There is no fourth figure ; but in the first three moods
of the first figure we may also argue to the converse of their conclu-
sions ; and two moods may be added, with an universal negative
minor premiss, in which, while the major term cannot be denied of
the minor without fallacy, the minor can be denied of the major ;
though such a conclusion is only particular, and realized by the help
of conversion or of exposition or of reduction per impossibile. It must
always be remembered that the character of an argument is deter-
mined not by the form into which it is thrown in words, but by that
which it assumes in our thought. This is our justification for

xiv] PRINCIPLES OF SYLLOGISTIC INFERENCE 331

recognizing the figures as distinct types. In particular cases,
a syllogism may not belong to the figure into which it has been
verbally compelled ; in others, it may be possible with the same
terms to construct syllogisms in more than one figure ; but then
there must be a real movement of thought in the process of conver-
sion by which the change is effected. The theory of syllogism ought
not to be regarded as a lesson in the manipulation of symbols and
the application of the formulae. What we have to look to is the
character of the thinking involved in it, and to that end we need to
realize our symbols and see how the varying character of our terms,
and of the relations between them in judgement, affects the inference.
If our enquiry has done anything to bring this lesson home, its length
and intricacy will not have been altogether vain.

[One further question about syllogism must be considered. It
was said above (p. 311) of syllogism in the first figure that its use
belongs to the stage of incomplete insight into the nature of facts.
Yet inference depends on seeing the connexion of facts.1 How then
can we infer syllogistically ? The same problem may be reached
another way. We may see that a syllogism is valid, without
knowing whether its premisses are true, or even knowing them to be
false ; or we may follow out syllogistic arguments with symbols,
not knowing what they stand for.2 Now to see the validity of an
argument is a process of inference. How then can inference depend
on seeing connexions of fact ? Again, it is well known that, although
a false conclusion cannot be validly drawn from true premisses,
a true conclusion may be validly drawn from premisses one or both
of which are false ; here then we reach the truth by inference, yet
clearly not by tracing out the connexions of fact.

The problem cannot be solved by distinguishing between the

1 Cf. supra, p. 240.

8 Neither this fact, nor the fact that the validity of an argument may
be considered independently of the truth of the premisses, is confined to
syllogism. Indeed all symbolic logic is an investigation of validity. Mr. Hugh
MacColl (Symbolic Logic, § 52) makes the strange statement that ' it ia
a demonstrable fact that not one syllogism of the traditional logic — neither
Darapti, nor Barbara, nor any other — is valid in the form in which it is
usually presented in our text-books, and in which, I believe, it has always
been presented since the time of Aristotle '. The reason he gives is that it
asserts the premisses and the conclusion, as well as the implication of the
latter in the former ; and he thinks Barbara, invalid as commonly formulated,
is valid in the form ' If every A is B, and every B is C, then every A is G \
But that is exactly what is meant by saying that the traditional form is valid.
To call an argument valid is not to call either its premisses or its conclusion
true, but to say that if the premisses are true, the conclusion is true. The diffi-
culty is, how we can know this, without knowing whether the premisses or
conclusion are true.

332 AN INTRODUCTION TO LOGIC [chap.

[logic of consistency and the logic of truth. Doubtless in the theory
of syllogism we have no more than an account of what conclusions
we must admit, if we are to be consistent, when we have admitted
certain premisses. That indeed is all we have in any attempt to
formulate by the help of symbols types of argument that are found
recurring with various real terms. But what is meant by the con-
dition ' if we are to be consistent ' ? Consistency is not a matter of
arbitrary convention ; it is determined by what is possible in the
nature of things. Inconsistency may be a disregard of the ' laws of
thought ' ; but these, as we know 1, are laws of things. That con-
sistency requires us to admit a certain conclusion if we have admitted
certain premisses means then that the nature of things requires it.
Yet if the premisses and conclusion are false, and if we are working
with symbols, how have we the nature of things before us ? 2

The problem is partly one that arises in regard to all hypothetical
thinking. In the inductive sciences, for example, we are constantly
forming hypotheses, whose consequences we proceed to deduce, only
to reject the hypotheses if their consequences differ from observed
facts. Now here the premisses and the conclusion are both false,
yet the inference from those to this is or should be sound, and
clearly rests on perceiving connexions of fact. What is seen to
involve certain consequences is something in the nature of the facts
supposed.3 Thus in support of the view that so-called acquired
characters are not inherited it has been argued 4 that, if the
conditions of town -life which injure the growth and health of the
individual disposed the individual to produce correspondingly
feebler offspring, and again this offspring to produce a yet feebler,
and so continuously, the later generations of town-dwellers would
be less resistent to the injurious influences of towns than the earlier,
the stock, so far as not replenished from country districts, would
die out, and infants with a long town ancestry, transferred to healthy
surroundings and reared there, would grow up markedly feebler
than infants of country ancestry reared with them. But these

1 Cf. supra, p. 13.

2 In Mind, vol. xix. N. S. 76, pp. 544-546, I drew a distinction between
the (ivdyKr) tlvat, the necessity for certain facts to be thus and thus, which ia
apprehended in demonstrative thinking, and the dvdyicr] Xeyfir, the necessity
to say one thing if we have said another, which was alone considered in the
formal treatment of syllogism. Professor J. A. Smith pointed out to me
the futility of this distinction. What is meant by my being compelled to
say anything ? As far as talking goes, I can say what I please. But the
compulsion here is a logical compulsion, not a moral or physical ; and so
I have not got away from apprehending connexions of fact. By admitting
the premisses I am compelled to admit the conclusion only because, if what
the premisses express exists, what the conclusion expresses exists also. But
how can I see this connexion unless I am considering existents ?

3 I have learnt from Professor Cook Wilson the importance of this.
« By Dr. Archdall Reid, The Principles of Heredity, pp. 335-337.

xiv] PRINCIPLES OF SYLLOGISTIC INFERENCE 333

[things do not happen ; the Jews e.g. have lived mainly in towns
for many centuries, and often under worse conditions than most
town-dwellers ; yet they thrive better, not worse, in towns than
men of other stocks. In this argument, it is because, or so far as,
we understand what are injury, transmission, identity of relation be-
tween every two members in a series of terms, &c, that we can
follow the deduction.1 Whether in the facts of town-life prolonged
over a succession of generations we have an example of all these is
another question, and is the question that has to be settled. So in
syllogism ; we so understand what an universal relation is between
one character M and another P, as to see that it involves the
presence of P in any subject S exhibiting M . The man who sees
that if the premisses of a given syllogism are true, the conclusion is
true, moves by insight into relations which are displayed in some
facts, even if not in the terms of his syllogism. And syllogistic
reasoning can be used in ' indirect reduction ' just as hypothetical
reasoning is used in refuting theories in the inductive sciences.

We may get some further light from our procedure in geometry.
When we wish to follow a geometrical demonstration about a triangle
ot circle, we draw a triangle or circle ; but our power to follow the
demonstration is not dependent on our figure being really triangular
or circular. We are thinking, as Plato says,2 not about the figure
we draw, but about what it represents. So it is when we use
symbols in working out syllogisms, and equally when we use pre-
misses with real terms, that we know to be false. ' Whoever knows
Greek is a compound householder, and all snapdragons know Greek
.•. All snapdragons are compound householders ' is a syllogism whose
validity we can only grasp if we think about relations no more
exemplified in the terms before us than is equality of radii in a badly
drawn circle, but yet understood because exemplified in some real
terms. It is still therefore on insight into connexions of fact that
reasoning rests.

1 If any one likes to say that the consequence depends on the fact that
the relation of parent to offspring is an asymmetrical transitive relation, we
can agree. But if we could think of no example of such relations, we could
not think out their implications. Oar reasoning therefore rests on our
perceiving real connexions. This is so even in non-Euclidean geometries.
We do not really know what would happen if straight lines making equal
angles with the same straight line could meet ; but when it is said that in
consequence a triangle might contain more than two right angles, that ia
because we see that, where x is a positive quantity, l + l + £>2. In other
words, we are guided by our insight into real connexions whenever we follow
out the implications of a false hypothesis, whether the falsity of the hypo-
thesis is only revealed by the discrepancy between its consequences and
observed fact, or is self-evident. From an unintelligible hypothesis we
cannot reason. We must reason therefore from what is intelligible in the
hypotheses of meta-geometry ; and this is relations and analogies exemplified
in fact. Implication is connexion of fact, not of mere thought, or statement.

* Rep. vi. 510 d, e.

334 AN INTRODUCTION TO LOGIC

[As to drawing a true conclusion from false premisses (e. g., if in
the syllogism in the last paragraph we put, for the term compound
householder, the term labiate), if it is simply a question of seeing its
validity, what has been said already applies here equally, and the
truth of the conclusion is irrelevant. But if a man is led by arguing
correctly from false premisses believed true to entertain a true
belief, he has not been thereby enabled to understand the real con-
nexion between the terms in his conclusion. All that he understands
there at all is the subject-attribute relation, which he rightly sup-
poses to be exemplified between the terms of the conclusion, without
seeing the connexion there : as a man might rightly suppose, on
others' testimony, that a portrait was like its original, because he
knows from other examples what likeness is, though he had never
seen the original of this portrait. He does not reach an under-
standing of the connexion which the conclusion states between its
terms. That is impossible apart from the apprehension of the
connexion of its terms with the middle term, and therefore impos-
sible if its terms are not really connected with the middle term
taken. Consequently in drawing a true conclusion from false
premisses, not by way of logical exercise, but in actual life, when we
express in premisses and conclusion what we do think, the inference
does not really make use of the special (or material) nature of the
terms, any more than the inference of the geometer makes any
use of the irregularities or length or thickness or other special
characters of the lines he draws ; and the connexion alleged in the
conclusion between just this nature, say being a snapdragon, and
just that, say being a labiate, is not made manifest, and so, in
strictness, not concluded, but we come to believe it for a reason
independent of the nature of those terms. That reason is, that it
happened to be by help of those terms and the middle term
that we contemplated a certain connexion of relations displayed
frequently in what is real, though not in those terms. We
believe the conclusion, therefore, as Aristotle said, accidentally
(Kara av\xQ(fir)Kos). Only by studying the structure of snapdragons
and other flowers, and detecting in them a common character
variously modified, should we understand that they are labiates. ]

CHAPTER XV
OF HYPOTHETICAL AND DISJUNCTIVE REASONING

The form of argument which we have been examining under the
name of Syllogism has for its premisses only categorical propositions ;
but there are forms of argument to which the name has been
extended, in which this is not so. In what have been called
Hypothetical and Disjunctive Syllogisms, hypothetical and dis-
junctive propositions figure in the premisses. For reasons, how-
ever, to be considered later, it appears better not to call them
syllogisms, but to speak rather of hypothetical and disjunctive
arguments. They are processes of argument that recur with great
frequency both in ordinary thought and in the reasonings of science.

In a hypothetical argument, one premiss is a hypothetical pro-
position, connecting a consequent with a condition or antecedent :
the other is a categorical proposition,1 either affirming the ante-
cedent or denying the consequent. From these follows as con-
clusion a categorical proposition, either affirming the consequent or
denying the antecedent. In the former case, the argument is
said to be in the modus ponens or constructive : in the latter
case, in the modus tollens or destructive. Examples will make
this clear.

1. The modus ponens is of the form : —

If A is B, it is C or If ^ is 5, CisZ> or If A is C, B is G
AisB AisB AisO

.'.AisC :.CisD /.BisC

e. g If the soul is uncreated, it is indestructible

The soul is uncreated
.*. It is indestructible

or If all men are born equal, slavery is unjust

All men are born equal
.*. Slavery is unjust

1 But cf. infra, p 337, iii

336 AN INTRODUCTION TO LOGIC [chap.

or If men have obligations towards their friends, they have

them towards their enemies
Men have obligations towards their friends
.*. They have them towards their enemies.

The following points should be noted further : —
i. The subject of the minor premiss may either, as in the fore-
going examples, be the same as the subject of the antecedent in the
major premiss (if we may retain the name of major for the hypo-
thetical and of minor for the categorical premisses in this form of
argument), or it may be a term that we recognize as included
therein, falling under it. Thus we may argue that

If a beautiful thing is rare, it is costly
Diamonds are rare
.'. They are costly.

Here it is implied and recognized that diamonds are beautiful things.
The argument might of course be expressed

If anything is at once beautiful and rare, it is costly
Diamonds are at once beautiful and rare
.". They are costly.

But diamonds are still ' subsumed ' as a special case under a rule
that applies beyond them ; the condition in the major premiss does
not concern them in particular.

ii. We saw in a previous chapter that the distinction of affirmative
and negative has no application to hypothetical judgements — for
every hypothetical judgement connects a consequent with a condition,
whether that consequent itself be expressed in the form of an
affirmative or of a negative statement : it would be no hypothetical
judgement to say that ' If the weather changes at full moon, it
does not follow that the change will last '} Hence the character
of the modus ponens is unaltered, whether the antecedent or the
consequent (and therefore the conclusion) be affirmative or negative.
I may argue

If the North American colonies were unrepresented in Parlia-
ment, they ought not to have been taxed by Parliament
They were unrepresented in Parliament
.*. They ought not to have been taxed by Parliament.

1 This is the denial of a hypothetical judgement, but not itself hypothetical:
being equivalent to saying ' It is not true that if ', &c.

xv] HYPOTHETICAL REASONING, ETC. 337

Here my conclusion is negative ; but the argument is still in the
modus ponens : for by that is meant not the mood which is affirma-
tive in its conclusion, but the mood which establishes the consequent
set down in the major premiss. The reader will easily see that if
the antecedent were of the form ' If A is not B ', it would still
make no difference to the character of the argument.

iii. It is possible to argue with both premisses and the conclusion
hypothetical, in the form : —

If .4 is C, it is D or If C is D, E is F or H .4 is D, -B is Z)

If A is B, it is C If A is B, G is D If B is D,CisD

.'. li A is B, itisD .' . If A is B, E is F .'. UA is D, GisD

e. g. If the price of an imported article rises, those who manufacture
the same article at home will charge more for it
If a tax is imposed upon the importation of an article, the price
of the imported article rises
.*. If a tax is imposed upon the importation of an article, those who
manufacture the same article at home will charge more for it.

The remarks made in the last paragraph apply, mutatis mutandis,
to this form of the modus ponens also ; and the subject of the
antecedent may be in one premiss the same with that of the con-
sequent, and in the other different. It is unnecessary to illustrate
all these variations.

2. The modus tollens is of the form : —

If A is B, it is G or If ^ is 5, C is D or If^isC,£isC7
A is not G Cis not D £ is not G

:. It is not B .'. A is not B .'. A is not G

e. g. If matter is indestructible, it is uncreated
Matter is not uncreated
.*. It is not indestructible

or If the earth did not rotate, the winds that blow from the poles
to the equator would not be deflected westward
But they are deflected westward
.*. The earth does rotate.

or If any one has a natural right to a vote, every one has
Not every one has
.". No one has.
It is plain that the observations made above with regard to the

1779 z

338 AN INTRODUCTION TO LOGIC [chap.

modus ponens are equally applicable, mutatis mutandis, to the modus
tollens.

Thus, given a hypothetical proposition, we can proceed to draw
an inference whenever we have a further premiss given us, either
affirming the antecedent or denying the consequent. But from the
affirmation of the consequent, or the denial of the antecedent, no
conclusion follows. Arguments of the form

If A is B, it is C
AisC
:. It is B

or A is not B

.' . It is not G

are invalid. It is true that if a member of the Commons House of
Parliament is declared a bankrupt, he loses his seat ; but it is not
true that if he loses his seat, it must be because he has been declared
a bankrupt, or that if he is not declared a bankrupt, he may not
still lose his seat. For the connexion of a consequent with a con-
dition does not preclude the possibility, that there are other conditions
upon which the same consequent may follow ; so that the fact of
the consequent having occurred is no proof of this particular con-
dition ; nor is the fact that this particular condition is not fulfilled
any proof that the consequent has not occurred in virtue of the
fulfilment of some other condition with which it is connected.
Obvious as these considerations are, yet these are among the com-
monest errors to occur in men's reasonings. We are all of us apt
to conclude, that by disproving the allegations advanced in support
of a proposition, we have disproved the proposition itself ; or that
by showing that facts agree with the consequences of some hypothesis
which we have formed, we have established the truth of that hypo-
thesis. We do not realize that it would be necessary to show, not
only that the facts agree with the consequences of our hypothesis,
but that they do not agree with the consequences of any other.
The Teutonic races have during the last three centuries increased
and expanded faster than those which speak languages of Latin
stock ; and some may be inclined to attribute this to the fact that
the former in the main embraced, while the latter rejected, the
principles of the Reformation. Grant that the facts are consistent
with the hypothesis that this difference of growth is due to a differ-
ence of religion ; yet if there are other ways of explaining it, what

xv] HYPOTHETICAL REASONING, ETC. 339

ground has yet been shown for accepting that way ? When facts
are equally consistent with the truth and with the falsity of our
hypothesis, we have so far no reason for believing it true.

It is then fallacious to draw any inference from the affirmation
of the consequent, or the denial of the antecedent, in a hypothetical
argument. It is sometimes said that to do the former is to commit
the fallacy of undistributed middle ; and to do the latter, to commit
the fallacy of illicit process of the major term : for that the argument

If A is B, it is C
AisC
.'. A is B

may

be

exhibited in

the form

ABisO
AisG
.'. AisAB

and the

argument

If A is B, it is G

A is not B

.'. A is not G

may

be exhibited in

the form

ABisG

A is not A B

.'. A is not G

And valid hypothetical arguments, it is said, may be similarly
' reduced ' to categorical syllogisms ; when it will be found, that
the modus ponens is really a syllogism in Barbara, and the modus
tollens one in Camestres.1

It seems to be an error thus to identify hypothetical reasoning
with syllogism. In syllogism, as we have seen, a relation is estab-
lished between two terms in the way of subject and predicate, by
means of their common relation in the way of subject and predicate
to a third or middle term. Hypothetical reasoning rests upon
another relation than that of subject and predicate — the relation
of condition and consequent ; and there is not necessarily any
middle term. Where antecedent and consequent, in the hypothetical

1 A number of modern text-books teach this doctrine. For an older
authority cf. Zabarella, In Lib. Prior. Anal. Tabulae, p. 158, ' syllogismus
hypotheticus an valeat necne cognoscitur per eius reductionem ad categori-
cum.' — Opera Logica, Coloniae, 1597.

Z2

340 AN INTRODUCTION TO LOGIC [chap.

premiss, have the same subject — where that proposition is of the
iorm ' If A is B, it is G ' — a middle term may at times be found,
and the reduction effected ; but where that is not so — where it is of
the form ' If A is B, C is D ' or ' If A is C, B is C '—there a middle
term is wanting, and the violent nature of this process of reduction
becomes manifest.

' If the value of gold is affected by the amount of labour needed
to obtain it, improvements in mining machinery must raise prices.
The value of gold is affected by the amount of labour needed to
obtain it. Therefore improvements in mining machinery raise
prices.' We are not concerned here with the truth of this hypo-
thetical proposition. So many circumstances, many of them varying
independently of one another, combine at any time to affect the
course of prices, that it would be hard to rest on observation the
effect which it is here asserted that improvements in mining
machinery ought to have. Our concern, however, is with the
character of the argument ; it is clearly difficult to reduce it to
a syllogism. There is nothing asserted of improvements in mining
machinery, which in turn is asserted universally to raise prices ;
the connexion between the value of gold and the amount of labour
needed to obtain it is not a predicate of improvements in mining
machinery, nor is raising prices a predicate of that connexion. It
is a consequence of it ; but that is another matter. Attempts have
indeed been made to get round this difficulty. It is said that the
major premiss may be expressed in the form ' The case r of the value
of gold being affected by the amount of labour needed to obtain it
is the case of improvements in mining machinery raising prices.
The existing case is the case of the value of gold being affected by
the amount of labour needed to obtain it. Therefore the existing
case is the case of improvements in mining machinery raising
prices.' But such linguistic tours de force do not alter the nature
of the argument which they conceal. What does that major premiss
mean ? Interpreted literally, it is undoubtedly false. Modification
in the value of gold, because gold has become easier or harder
to obtain, is not a rise in prices due to improvements in mining
machinery. The one fact may be dependent on the other, but the
one is not the other. It is not therefore until we mentally substi-
tute for this premiss the hypothetical proposition it attempts to

1 Had I written, for the case, all cases, the proposition would have been
still more absurd. But the contention should be examined in its strongest form.

xv] HYPOTHETICAL REASONING, ETC. 341

supersede, that we assent to it at all ; the ' reduction ' is purely
verbal ; our meaning remains unchanged, and cannot be put into
the categorical form. Nor does the minor premiss stand criticism
any better. What case is ' the case of the value of gold being
affected by the amount of labour needed to obtain it ' ? To say ' the
existing case ' is useless, unless we are told what the existing case
is a case of. If it is a case of the value of gold being affected by the
amount of labour needed to obtain it, the proposition becomes
tautological, and the conclusion will only repeat the major premiss 1 :
if it is a case of something else, we ought in the first place to have
that something stated, in order that we may know what the pro-
position means ; and in the second place, when it was stated, we
should find the proposition had become false, in the same way as
the major premiss, literally interpreted, was false. It is clear then
that this syllogism is far from exhibiting more correctly the true
character of the hypothetical argument in question ; on the contrary,
the hypothetical form exhibits the true nature of the argument thus
violently forced into a syllogism.

Had we indeed taken an example in which the subject of the
antecedent was the same with the subject of the consequent in
the major premiss — in which, to put it otherwise, the major premiss
was of the form ' If A is B, it is C ' : then the process of reduction
to syllogism would not have appeared to be so difficult or violent.
For then the condition on which it depends that A is G is a condition
fulfilled in A. ' If the moon rotates in the same period as it revolves,
it must present always the same face to the earth. It does rotate
in the same period as it revolves. Therefore it does present always
the same face to the earth.' ' If Christian nations had the spirit of
Christ they would avoid war. They do not avoid war. Therefore
they have not the spirit of Christ.' There is little change made, if
we substitute for these arguments the following syllogisms :

A body rotating in the same period wherein it revolves round

another body presents always the same face to the other
The moon is a body rotating in the same period wherein
it revolves round the earth 2
.'. The moon presents always the same face to the earth

1 The case of A is the case of B : the existing case of A is the case of A :
therefore the existing case of A is the case of B.

2 It will be seen that in this minor premiss not only is the moon ' subsumed '
under the more general notion of a body rotating, &c. : but the earth is also

342 AN INTRODUCTION TO LOGIC [chap.

and Those who have the spirit of Christ avoid war

Christian nations do not avoid war
.*. Christian nations have not the spirit of Christ.

Indeed, if it be granted that the hypothetical premiss is unaltered,
otherwise than in verbal form, by reduction to the form of a cate-
gorical proposition, we must grant that the argument is unaltered
by reduction. And there are logicians who have contended that
all universal judgements are really hypothetical 1 ; from which it
would follow that there is no real difference between a syllogism
in Barbara or Camestres, when it has a genuinely universal (i. e. not
a merely enumerative) major premiss, and a hypothetical argument
in the modus ponens or the modus fattens — though the former rather
than the latter would demand reduction. Yet there do seem to be
some judgements which, in their context, intend to affirm the
existence of the subject about which assertion is made, and not
merely to assert that something would be true about it if it existed.
To say that, if Christian nations had the spirit of Christ, they
would avoid war, leaves it an open question whether any have that
spirit ; to say that those who have the spirit of Christ avoid it,
naturally implies that there are such. The reduction of a hypothe-
tical argument to a syllogism is no merely verbal change, if it
substitutes one of these forms of statement for the other.

Attention ought to be called to one other change incidental to this
reduction in the last two examples. Our hypothetical major con-
cerned the moon and the earth, or Christian nations ; in the syllogism,
the major concerned any two bodies in which certain conditions are
fulfilled, or any in whom the spirit of Christ is found. Thus in
the syllogism, a principle is stated in more general form than in the
hypothetical proposition. Here again, more than a merely formal
change is involved. It is true that no one could assent to the
proposition, that if the moon rotates in the same period wherein it
revolves, it must present always the same face to the earth, without
seeing that its truth has nothing to do with the fact that the bodies

subsumed under the more general notion of the other body. Hence it is
difficult to express the argument completely in symbols. Suppose that we
write ' Any X is Y, the moon is X .". the moon is Y ' : now here, in the
major premiss, X='body rotating in the same period wherein it revolves
round another body ' ; in the minor premiss, X = i body rotating in the same
period wherein it revolves round the earth ' ; and similarly with Y. The
argument is none the less a syllogism ; the difficulty is linguistic ; but we
are really bringing the moon in its relation to the earth under the condition
of a rule. Aristotle recognizes this : cf. Post. An. 0. xi. 94a 36-b7.
1 Cf. pp. 183, 185, n. 1, supra.

xv] HYPOTHETICAL REASONING, ETC. 343

in question are the moon and the earth, but holds equally for any
two bodies ; so that the more general form of the universal cate-
gorical proposition given above is obviously justified. Yet it is
not the mere form of the hypothetical judgement which enables us
to see this ; and it might be contended in the other case that the
more general form of the categorical judgement is not justified, and
that we ought not to have said more than that ' Nations which have
the spirit of Christ avoid war '. It might be said that if a Christian
nation had the spirit of Christ, it would avoid war ; but that an
individual may be morally bound to take part in warfare, though
he has that spirit, when the nation to which he belongs has it not.
Now there is, doubtless, in every true hypothetical judgement of
the form ' If A is B, it is G ', some general principle involved : we
may express this as ' a /3 is y '. But if A is some determinate
individual, or case of a particular kind, and if the condition B is
similarly determinate, we may know that if A is B, it is G, without
knowing generally what conditions /3, occurring in what kind of
subject a, will involve the predicate y. Where this is the case
the hypothetical form is more natural to the expression of our
argument than the syllogistic.1

We find, then, that even when antecedent and consequent have
the same subject in a hypothetical major, reduction of the hypo-
thetical argument to syllogism may mean a real change in the
nature of the argument used ; and that where they have different
subjects, such reduction can only be effected to outward appearance,
and by violent means ; for here the condition on which it depends
that G is D is not a condition asserted to be realized in the nature
of G itself ; in other words, there is no middle term.2 No doubt

1 If the subject of the antecedent in the hypothetical premiss be a singular
term, and we know of no general term under which it falls which can be
substituted as subject in its stead, the impossibility of reducing a hypo-
thetical argument to syllogism is specially obvious ; for we cannot replace
such a hypothetical proposition by any categorical proposition. ' If he
marries her, he will be happy ; he will marry her .\ He will be happy ' — is
an example in point.

2 The inference in a hypothetical argument might hence be called
immediate; but such an expression would readily give rise to misunder-
standing. It is immediate in the sense of having no true middle term : and
in this it differs from syllogism ; it is also immediate in the sense, that given
the premisses, nothing more is needed in order that we may see the necessity
of the conclusion: and in this sense, syllogism, and indeed every step of
valid argument when fully stated, is immediate. But it was in yet another
sense that the processes of conversion, &c, were called immediate, and
distinguished from syllogism : viz. that in them we passed from a single

344 AN INTRODUCTION TO LOGIC [chap.

there is an unity embracing both condition and consequent ; they
belong to a system, of which it might be said that, when affected
by the condition, it exhibits the consequence. Sometimes this
admits of ready expression. ' If the rainfall is deficient, the hay-
crop is light ' : we may express this by saying that ' Grass which is
insufficiently supplied with moisture makes only a small growth
that can be used for hay '. In other cases, the interconnexion
of facts within a whole does not admit of being stated except in
hypothetical form. And anyhow, it must be contended that
hypothetical reasoning is not identical in character with syllogism,
and that we ought not to pretend to validate it by reducing it to
syllogism, nor to identify the fallacies involved in argument from
the denial of the antecedent or the affirmation of the consequent
with the syllogistic fallacies of illicit process of the major term or
undistributed middle.

In a disjunctive argument, one premiss is a disjunctive proposi-
tion ; the other is a categorical proposition, affirming or denying
one of the alternatives in the former. From these follows as con-
clusion a categorical proposition, denying or affirming the other
alternative. In the former case, the argument is said to be in the

proposition to another inferred therefrom, without anything further being
required as a means of reaching the conclusion. (Cf. supra, p. 232, n. 3).
Hypothetical arguments are not immediate in this sense. Given that ' If
A is JB, it is C , I cannot conclude that A is C, unless I also know that
A is B : nor could I conclude that A is C, from the fact that A is B, without
the hypothetical premiss. I can, however, conclude from ' If A is B, it is C
to ' If A is not C, it is not B ', without any further knowledge : and to this
we saw that some forms of so-called immediate inference amounted.

The conditions of valid hypothetical reasoning are of course recognized
by Aristotle (cf. e. g. Top. /3. iv. lllb 17-23 et al.) ; but he does not speak of
hypothetical syllogisms. The term ovWnyta-fxos «£ inroBiaeuis has a different
meaning — viz. a syllogism proving the antecedent of a hypothetical pro-
position, and therefore, by virtue of the acceptance of that hypothesis, proving
the conclusion. Let it be granted that if A is B, C is D : then any syllogism
which proves that A is B will by virtue of this agreement establish also
that 0 is D : but without such agreement, it would not have been shown at
all that G is D : that is therefore said to be proved only ex hypothesi. In
a case between University College, Oxford, and the City of Oxford (v. Times
of July 5, 1904) arising out of a claim by the College to put a bridge between
two blocks of buildings on either side of a narrow street called Logic Lane
without payment of any acknowledgement to the City, it was agreed that
if the soil of Logic Lane were vested in the College, the College was entitled
to do this (subject to any building regulations which the City had power to
make) ; the arguments advanced on behalf of the College (which established
its case) were directed to show that it was owner of the soil ; but, e'| woft'trfs)?,
the College showed by the same arguments that it was entitled to erect the
bridge without acknowledgement.

xv] HYPOTHETICAL REASONING, ETC. 345

modus ponendo tollens : in the latter case, in the modus tollendo
ponens. Examples and observations follow.

1. The modus ponendo tollens is of the form

A is either B or G Either A is B or G is D Either A or B is G
A is B or A is B or A is G

.'. It is not G ,\ G is not D .'. B is not G

e. g. ' Possession by devils ' is either a form of mental derangement,
or supernatural
It is a form of mental derangement
/. It is not supernatural

or Either the interests of religion require the maintenance of the
Temporal Power, or the Popes are actuated by worldly
motives in continuing to claim it
The interests of religion do require its maintenance
.*. The Popes are not actuated by worldly motives in continuing
to claim it

or Either Newton or Leibniz invented the calculus
Newton invented it
.*. Leibniz did not

2. The modus tollendo ponens is of the form

A is either B or G Either A is B or C is D Either A or B is G
A is not B or A is not B or A is not G

.'. ItisC .'. GisD .'.BisG

e. g. The belief in a golden age rests either on history or on hope
It does not rest on history
.'. It rests on hope

or Either God is unjust, or no man is eternally punished
God is not unjust
.*. No man is eternally punished

or Either Aristotle or Eudemus wrote Bks. v, vi, vii of the Nico-
machean Ethics
Eudemus did not write them
.*. Aristotle did write them.

The following points should be noted : —

i. It is sometimes contended that the modus ponendo tollens is

346 AN INTRODUCTION TO LOGIC [chap.

invalid : that the affirmation of one alternative does not justify the
denial of the other. This will depend on the interpretation given
to the disjunctive proposition. If the alternatives therein stated are
mutually exclusive, the argument is valid : if otherwise, it is not.
Whether they are so intended can only be determined in a given
case by reference to the context and the matter of the judgement ;
but mutually exclusive alternatives may exist, and therefore a valid
argument in this mood is possible. Of the examples given above,
the third is clearly the most open to objection ; for Newton and
Leibniz may well have invented the calculus independently, as
they are now believed to have done. In the first, it is implied that
if we can otherwise account for the phenomena of demoniacal
possession, we shall not attribute them to supernatural agency ; and
the argument may be considered valid, provided that we are justified
in that view.1 The second is more doubtful ; men may do from
bad motives what ought anyhow to be done, and the motives of the
Popes in maintaining their claim to temporal power might be
worldly, even though their possession of it were required in the
interests of religion. The premisses do not really prove the un-
worldliness of their motives ; but they show that we need not
assume the contrary, in default of further evidence. The validity
of the present mood of disjunctive argument will, in fact, depend
on what hypothetic als are implied in its disjunctive premiss ; for
we have seen (p. 187, supra) that the disjunctive judgement 'A is
either B or G ' may imply, though it is not reducible to, the hypo-
thetical judgements ' If A is B, it is not G \ ' If A is G, it is not B ',
' If A is not B, it is G ', and ' If A is not G, it is B '. If the alter-
natives are mutually exclusive, all four will be implied, and the
modus ponendo fattens will be valid. If not, we cannot get, out of
the proposition ' A is either B or C ', the propositions ' If A is B,
it is not G '—' If A is G, it is not B '. To say that ' Either the
interests of religion require the maintenance of the Temporal Power,
or the Popes are actuated by worldly motives in continuing to claim
it ' will mean that if the interests of religion do not require it, they
must be so actuated ; but not that if the interests of religion do
require it, they cannot be so actuated ; and therefore to argue
from the premiss that the interests of religion do require it is to
argue from the denial of the antecedent in a hypothetical argument.

1 The argument may be valid even though the conclusion be false : the
truth of the conclusion further presupposes that of the minor premiss.

xv] HYPOTHETICAL REASONING, ETC. 347

Here we might leave this matter, with this as our result — that
the validity of the modus ponendo tollens depends on the alternatives
in the disjunctive premiss being mutually exclusive, and that there
is no way of determining on merely formal considerations whether
they are so * ; that the form of argument is not universally invalid,
because they may be so ; but not universally valid, because they
may not. It is, however, worth noticing that quite independently
of this doubt about the validity of the modus ponendo tollens in
any given case, the modus tollendo ponens is of more impor-
tance on other grounds. We are more often interested in proving
one alternative by disproof of others, than vice versa. A prisoner
indicted on a charge of murder may indeed be content to show that,
whoever committed the crime, he did not ; and his ends may be
satisfied by proving an alibi. But the ends of justice are not satis-
fied except by discovering the murderer. And so it is with disjunc-
tive argument generally ; its use lies more in what it can establish
than in what it can overthrow.

ii. As in hypothetical, so also in disjunctive argument, the major
premiss may make a more general assertion, which in the conclusion
is applied to some special case. Thus a man might argue

Every man at forty is either a fool or a physician
My son at forty is not a physician
.*. He is a fool

or from the premiss ' Either God is unjust, or no man is eternally
punished ', I might have concluded that I shall not be eternally
punished.2

iii. The mood of a disjunctive argument is not affected, any
more than the mood of a hypothetical argument, by the quality —

1 It might be said that we could give an unambiguous form to the
argument by writing it thus : ' A is either B only, or C only, or both B and
C : it is B only .\ it is neither C only, nor both B and C' But here there
seems to be no inference ; for if we already know that it is B only, we must
already know that it is not C. The inference rests upon the knowledge
that A is B, and that B and C are mutually exclusive : if we are doubtful
of the latter point, and only know that A is B, we cannot tell whether it is
C or not : and this information is all that we have ; we must not substitute
for the minor premiss ' A is B ' a different one, ' A is B only '.

a The subsumption involved may be expressed if we like in a separate
and syllogistic argument : thus

Every man at forty is either a fool or a physician
I am forty
.*. I am either a fool or a physician : but I am not a physician, &c.

348 AN INTRODUCTION TO LOGIC [chap.

affirmative or negative — of the minor premiss or the conclusion.
Arguments of the type

A is either B or C
A is not B
.'. It is C

are in the same mood as those of the type

A is either not B or not C
AisB
.'. It is not C

I establish one alternative by way of rejecting the other, equally
whether from the premisses

A diplomatist must either be insincere or fail
Bismarck did not fail

I conclude that he was insincere, or whether I conclude that he was
not honest from the premisses

A diplomatist is either not honest, or not successful
Bismarck was successful

Attempts have been made to reduce disjunctive arguments also
to syllogistic form. We have seen that a disjunctive proposition
implies two or perhaps four hypotheticals ; and every disjunctive
argument can be exhibited as a hypothetical argument using for
major premiss one of these. But as hypothetical argument is not
syllogism, we do not thereby make disjunctive argument into
syllogism ; nor do we really identify it with hypothetical argument ;
for the hypothetical major premiss expresses only a part of the
meaning of the disjunctive proposition, from a perception of the
relations involved in which a disjunctive argument proceeds to
draw its conclusion.1

and having reached the conclusion ' No man is eternally punished ', I can
with the minor premiss ' I am a man ' draw the conclusion that I shall not
be eternally punished. This act of subsumption is a different act of inference
from the disjunctive argument.

1 The term hypothetical was long used (following Boetius) sensu latiore,
to cover both what have in this chapter been called hypothetical and what
have been called disjunctive arguments ; and for hypothetical, in the nar-
rower sense employed above, the term conjunctive. Conditional — originally
equivalent to hypothetical in the wider sense — has by some who retained the
wider sense for the latter been used as equivalent to conjunctive (cf. Sir W.
Hamilton's Discussions, p. 150). A few points may be noted here which did
not seem worth a place in the text.

1. The order in which the alternatives in the disjunction are mentioned

xv] HYPOTHETICAL REASONING, ETC. 349

being irrelevant, it makes no difference to the nature of the argument whether
we proceed from the affirmation of the first to the denial of the second, or
from the affirmation of the second to the denial of the first.

2. A disjunction may contain more than two members : e.g. it may be of
the form A is either B or G or D. In this case, if the minor is categorical,
the conclusion will be disjunctive ; and in the modus ponendo tollens, a dis-
junctive minor will give a categorical conclusion — A is either B or C .'. it
is not D. But the minor ' A is neither B nor C ', which is needed in order
to get a categorical conclusion in the modus tollendo ponens, is not a dis-
junctive proposition. But such details involve no fresh principle of reasoning,
and need not be pursued, any more than it is necessary to work out all the
variations that are possible according as the disjunction is between two
predicates of the same subject, or two subjects of the same predicate, or
two assertions differing both in subject and predicate, when either or both
assertions in each of these cases are affirmative or negative.

3. An argument of the form ' A is either B or C : C is either D or E .". A
is either B or D or E ' is not a disjunctive argument, but the application of
syllogism to one limb of a disjunctive proposition.

CHAPTER XVI

OF ENTHYMEME, SORITES, AND DILEMMA

This chapter deals with certain forms or modes of stating an
argument which introduce no new principle of reasoning beyond
those now already discussed, but for one reason or another deserve
a special name and mention.

An enthymeme indeed is not a particular form of argument, but
a particular way of stating an argument. The name is given to
a syllogism with one premiss — or, it may be, the conclusion —
suppressed.1 Nearly all syllogisms are, as a matter of fact, stated

1 By Aristotle the term ivQv^yLa is defined as avWoyicrpos e£ (Ikotuv f)
a-r)fj.(ia)v, Anal. Pri. i3. xxvii. 70a 10. Its nature is discussed in that chapter
and in various passages of the Rhetoric. Roughly speaking, etKos is a general
proposition true only for the most part, such as that Raw foods are unwhole-
some ; in applying this to prove the unwholesomeness of some particular
article of diet, we are open to the objection that the article in question may
form an exception to the rule ; but in practice we are often compelled to
argue from such probable premisses. A ar)p.iiov is either a particular fact,
to which one can appeal in support of a general proposition, because if the
proposition were true, the fact would follow as a consequence of it : thus
we may argue that ' The wise are just, for Socrates was wise and just ' :
where Socrates is the arj^flov (Rhet. a. i. 1357b 11) ; or it is a particular fact
appealed to as evidence of another particular fact, because the existence of
one such fact implies the previous or subsequent or concurrent existence of
the other : thus ' Pittacus is liberal, because ambitious men are liberal, and
Pittacus is ambitious ' : here his ambition is the orr^xtlov of his liberality
{Anal. Pri. 0. xxvii. 70a 26). In this case, the appeal to a <tt)ph'lov implies
a general principle which, if it is irrefragable, gives to the a-^pdov the nature
of an evidence, or Tenpi'ipiov (Rhet. a. ii. 1357b 3) ; to argue from a TtKp.i)ptov
is not, however, to argue from the true cause of the effect ; for this would
be scientific syllogism, and not iv8vprip.a. It may be added that, where the
general principle implied is not irrefragable, but true for the most part, it
is hard to distinguish the auXXoyio-juoy 4k arjfieiov from a avWoyicrnos ef
flxoTos. It should be noted that Aristotle includes under arjpelov that which,
as a consequence of something else, is assumed, where it exists or occurs,
to presuppose it, whether it could exist or occur without the existence or
occurrence of that other thing or not ; where it could not, we have a reKprjpiov ;
and of this character are what doctors call the symptoms of a disease (and
such reasoning from effect to cause is not 'scientific ') ; where it could, the
argument — as Aristotle recognizes — is not really valid ; it may be true that
persons in a fever breathe rapidly, but I cannot safely infer that a person
who breathes rapidly has fever (ib. 1357b 19) ; there are, of course, symptoms
of disease that are of doubtful interpretation. The evBipqiia is said to be
a rhetorical demonstration, or rhetorical syllogism (Rhet. a. i. 1355a 6, ii.

ENTHYMEME, SORITES, AND DILEMMA 351

as enthyinemes, except in the examples of a logical treatise, or
the conduct of a formal disputation. It must not be supposed,
however, that we are the less arguing in syllogism, because we use
one member of the argument without its being explicitly stated.
Syllogism is an act of thought, and if, in order to perform this act,
we need to recognize in thought all three propositions, we are
arguing syllogistically, whether we enunciate the whole syllogism
or not. That we do recognize a suppressed premiss may be shown
by the fact that, if any one were to deny it, we should feel that
he was attacking our argument, though we had not expressly
asserted it.

The suppressed member may be the major premiss, or the minor,
or — less frequently — the conclusion. Medea, in Ovid's play of that

1356b 4), because public speakers make use of the appeal to such probable
premisses or signs, and do not expect or provide more strictly demonstrative
or scientific arguments. But they also commonly present their thought
without enunciating all three propositions of a syllogism, whereas in a set
debate one endeavours to get both premisses explicitly admitted, and so
establish a conclusion. And Aristotle, in distinguishing the arguments of
the platform from those of such debate, probably had in mind both the
sort of premisses to which orators appeal, and the mode in which they
present their arguments. The chapter (Rhet. ft. xxi) on yv&fiai, sententiae,
or apophthegms, describes a yvu>p.rj as part of an enthymeme, because, when
it is justified by another proposition, we have an enthymeme ; e. g. ' There
is no man free ' is a yva>p,r], but when we add ' For each is a slave to money
or to fortune ', there is an enthymeme (1394b 4-6). So a ypo)p.t) which includes
the ground of its own statement is called enthymematic.e. g. dOdvarov opyfjv p.f)
4>v\aTTe 6vj]t6s a>v (' Nurse not immortal anger, being mortal'). Both the
character of the premisses and its incomplete statement seem then to dis-
tinguish the enthymeme from other syllogisms, according to Aristotle. That,
in spite of the definition quoted from Anal. Pri. ft. xxvii, above, the latter
feature has come to determine the use of the term, may perhaps be due to
a later passage in the same chapter, 70a 24-28 idv fiev ovv 17 pla Xexdfj
TrpoTcuris, arjp.flov ylverai p.6vov, iav be nai 17 iripa Trpoa\ij(f)6rj, <ruXXoyicr/LtoV, oiov
on UtTTaKos e'Xevdepios' 01 yap $tXoVt/xoi (Xtvflepioi, Ulttokos 8e (fyikoriiios
('If the one premiss be stated, there is a sign only, but if the other be
taken also, a syllogism: e.g. Pittacus is generous; for the ambitious are
generous, and Pittacus is ambitious '). This, however, seems merely to mean
that, if I say ' Pittacus is generous, because he is ambitious ', I only state the
sign ; whereas, if I add that the ambitious are generous, I make a syllogism ;
but this syllogism was implied all along, and is an fvdvp.r}p.a, whether I state
it in full or not, because of the character of the premisses. A demonstrative
syllogism could not be so called by Aristotle, even though incompletely
expressed : e. g. ' The moon is liable to eclipse, because it may be hidden
by the earth from the sun '. Cf. on the question Cope, Introduction to
Aristotle's Rhetoric, p. 103, n. The term enthymeme has more commonly been
applied to a syllogism omitting one of the premisses, than to one omitting
the conclusion. Sir W. Hamilton (Discussions on Philosophy, dsC, pp. 153-158)
traces the antiquity of the non- Aristotelian use of the term. It goes back to
the oldest of the commentators.

352 AN INTRODUCTION TO LOGIC [chap.

name, asks Jason — Servare potui, perdere an possim rogas ? here the
major premiss ,Qui servare possunt, perdere possunt, is understood :
Medea supplies only the minor, and — in the form of a rhetorical
question — the conclusion.1 If I argue that ' those cultivate the
land best who have a personal interest in its improvement, and
therefore peasant proprietors are the best cultivators ', I omit — yet
I clearly use, for to deny it would destroy the argument — the minor
premiss, that ' peasant proprietors have a personal interest in the
improvement of the land \2 The conclusion may be omitted from
motives of delicacy, or sometimes for purposes of effect, as in the
Greek couplet

kcu ro'oe <Pa>KvX.Cbov' Aipioi kclkoi, ovx o [iiv, OS o' ov,
ttclvtcs, Tr\r]V Uponkiovs' /cat YlpoKXerjs Aepios.8

It is, of course, possible that an enthymeme may be contained in
what grammatically is only a single sentence ; as in Goneril's address
to King Lear :

You, as you are old and reverend, should be wise,

or in Regan's, later in the play :

I pray you, father, being weak, seem so.

A syllogism, whether expressed in full or as an enthymeme, is
a single act of inference ; it may be analysed into premisses and
conclusion, but not into parts which are themselves acts of infer-
ence. The premisses may, however, be themselves in turn conclu-
sions reached by other acts of inference ; and the conclusion may
itself serve as premiss to a further act of inference. A syllogism
proving one of the premisses of another syllogism is called, in
relation to that, a prosyllogism : and a syllogism using as a premiss
the conclusion of another is called, in relation to it, an episyllogism ;
where the prosyllogism is expressed in the form of an enthymeme,
the whole argument is sometimes called an epieheirema.4 The

1 This example is used in the Port Royal Logic, Pt. III. c. xiv.

8 I am inclined to think it would be found that the major premiss is more
frequently suppressed when the conclusion of the enthymeme is put in the
forefront, the minor when we begin with a reason. If we begin with a reason,
we like to lay down a general principle.

3 ' This too is from Phocylides : The Lerians are bad men, not this one
only and not that, but all of them except Proclees ; and he is a Lerian.'

* v. Mansel's Aldrich, p. 97, note t : and Trendelenburg's Elementa Logices
Aristotelicae, note to § 33, cited by Mansel. The term einx^PW0- was differently
defined by Aristotle (who called it, as well as the epOiprjpa, a dialectical
syllogism, (rvWoyitrnos 8ui\eKTiK6s, Top. 6. xi. 162a 16) : it was an assault
upon a position maintained in disputation by the respondent.

xvi] ENTHYMEME, SORITES, AND DILEMMA 353

following argument contains both a prosyllogism and an episyllo-
gism, and as the former is expressed in abbreviated form, it is also
an epicheirema. ' Those who have no occupation have nothing to
interest themselves in, and therefore are unhappy ; for men with
nothing in which to interest themselves are always unhappy, since
happiness depends on the success with which we advance the objects
in which we are interested ; and so wealth is no guarantee of happi-
ness.' Here the central syllogism is

All who have nothing in which to interest themselves are

unhappy
Those who have no occupation have nothing in which to

interest themselves
.'. Those who have no occupation are unhappy.

The major premiss is proved by a prosyllogism to this effect :

Happy men are those who succeed in advancing objects in

which they are interested
Men who have nothing in which to interest themselves do

not succeed in advancing any object in which they are

interested
.'. Men who have nothing in which to interest themselves are

not happy.

And an episyllogism is added thus :

Those who have no occupation are unhappy
Rich men may have no occupation
.'. Rich men may be unhappy.1

We have in such a case a train of argument, of which the several
steps are not each set out in full, though the premisses necessary to
complete the sequence of thought are readily supplied, as in an
enthymeme. Trains of argument may, of course, be of any length,
and vary indefinitely in composition, according to the nature of the
separate steps into which they can be broken up ; and it would be
useless as well as impracticable to invent names for every variety.
But there is one well-marked variety to which the name of Sorites
has been given by logicians.

1 The schoolmen gave the name of syllogismus crypticus to a syllogism
which lay so concealed in the wording of an argument, that some process
like conversion, or other substitution of equivalent propositions, was necessary
in order to show clearly the terms of the syllogism, and their relation : as,
here, * rich men may be unhappy ' is taken as equivalent to ' wealth is no
guarantee of happiness '.

1779 A a

354 AN INTRODUCTION TO LOGIC |chap.

A Sorites 1 may perhaps be defined as a syllogism in the first figure
with many middle terms ; or if it be thought that nothing should be
called a syllogism that contains more than one act of inference, as
a polysyllogism 2 in the first figure with the intermediate conclusions
suppressed. Schematically, it is of the form

A is B
BisC
CisD
DisE
EisF
.'. AisF

where it will be observed that we start with the minor premiss, and
each subsequent premiss is, in relation to that enunciated before it,
a major.3

There must be, at least, two steps, and therefore three premisses,
in a sorites, else we should have no series or chain of syllogisms ;
and there may be any number of steps more than two ; the premisses
will always be more numerous by one than the steps into which the
argument can be resolved.4 Short sorites are of common occur-
rence. A well-known example occurs in Romans viii. 29, 30, ' For
whom he did foreknow, he also did predestinate to be conformed to

1 The name is derived from o-topoy = heap.

2 A series of syllogisms, one proving a premiss of another, is called a
polysyllogism : while each single step of syllogistic reasoning is called
a monosyllogism.

8 Where the order in which the premisses are enunciated is reversed,
starting with the major and proceeding always to one which in relation to
the preceding is a minor premiss, the sorites is called a Ooclenian Sorites,
after Rodolphus Goclenius, Professor at Marburg at the end of the sixteenth
century, who first called attention to this form of presenting the argument.
But though it is important to notice that the order in which the premisses
are commonly placed in a sorites is the opposite of that which is customary
in a simple syllogism, it must not be supposed that the character of the
argument is affected by reversing the order, or that the Goclenian sorites
is a thing, as such, of any importance. The Goclenian is known also as
a regressive, and the other, or ' Aristotelian ', as a progressive sorites.
Aristotle, however, does not discuss the sorites (though clearly believing it to
occur in science, cf. An. Post. a. xiv. 79a 30, xx-xxiii),so that the progressive
is not entitled to be called Aristotelian. Sir W. Hamilton states that he
could not trace the term back beyond the Dialectica of Laurentius Valla,
published in the middle of the fifteenth century. From the sixteenth century
onward it found a regular place in logical treatises. Cf. his Lectures on Logic,
xix. p. 377.

* ' Sorites est syllogismus multiplex . . . Est enim sorites progressio enthy-
mematica, syllogismos continens propositionibus [ = praemissis] uno tantum
pauciores.' Downam's Commentarii in Petri Rami Dialecticam, 1510, p. 653.

xvi] ENTHYMEME, SORITES, AND DILEMMA 355

the image of his Son. . . Moreover whom he did predestinate,
them he also called : and whom he called, them he also justified :
and whom he justified, them he also glorified.'

But long specimens are less common, not because long trains of
reasoning are rare, but because the successive steps do not generally
continue for long together to be of the same form. Leibniz, in
the second part of his Confessio Naturae contra Atheistas, written in
1668 (and containing doctrines as to the nature of matter which he
subsequently abandoned), offers a proof of the immortality of
the human soul in the form of a continuous sorites ; but even
so, many of the propositions are supported by reasons that do not
enter into the series of premisses constituting his sorites.1 In the
following transcription the premisses that do not belong to the
sorites are placed out of line to the right ; and some of them are
omitted.

The human soul is a thing whose
activity is thinking.

A thing whose activity is thinking
is one whose activity is imme-
diately apprehended, and with-
out any representation of parts
therein.

A thing whose activity is appre-
hended immediately without
any representation of parts
therein is a thing whose activity
does not contain parts.

A thing whose activity does not for all motion is divisible
contain parts is one whose acti- into parts,
vity is not motion :

A thing whose activity is not for the activity of a body is
motion is not a body : always a motion.

What is not a body is not in space : for the definition of body is

to be extended.

What is not in space is insus-
ceptible of motion.

What is insusceptible of motion for dissolution is a movement
is indissoluble : of parts.

1 v. Erdmann's ed., p. 47.
A a 2

356 AN INTRODUCTION TO LOGIC [chap.

What is indissoluble is incorrup- for corruption is dissolution

tible : of the inmost parts.

What is incorruptible is immortal.
.*. The human soul is immortal.

We may pass from examples to a consideration of the form of
the argument, and the rules of its validity. It will be observed
that the predicate of each premiss is the subject of the next, while
the subject and predicate of the first and last premiss are the subject
and predicate respectively of the conclusion. For each premiss is
minor to that which follows, and major to that which precedes it ;
and as we start from the minor premiss of the whole argument, each
middle term is predicate of one premiss and subject of the next.
It follows, that (i) no premiss except the first may be particular,
and (ii) none except the last negative ; for in the first figure, the
major premiss must be universal, and the minor affirmative ; now
each premiss except the last is a minor, in relation to a premiss
following it, and must therefore be affirmative ; and each premiss
except the first is a major, in relation to one preceding it, and there-
fore must be universal. This will be easily seen if we resolve the
sorites into its constituent syllogisms :

1. beginning from the minor

Ai&B

AiaB (i)

BiaC

B is C (ii)

CisD

.*. A is G

DisE

CisD(iii)

EisF

.\ A is D

.'. AiaF

DiaE (iv)

.'. A is E

EisF{v)

.'. AisF

It is clear that if the first premiss were particular, the conclusion
of the first syllogism would be particular ; this stands as minor to
the third premiss in the second syllogism, whose conclusion would
therefore again be particular, and so ultimately would the conclusion
of the whole sorites be ; but if any other premiss were particular,
there would be an undistributed middle in the syllogism into whioh
it entered.

xvi] ENTHYMEME, SORITES, AND DILEMMA 357

2. beginning from the major

EisF (v)

DisE (iv)
.*. D is F

CisD (iii)
.*. C is F

BisC (ii)
.*. B is F

AisB (i)
.'. A is F

Here, if the last premiss (E is .F) were negative, the conclusion of
the syllogism in which it stands as major would be negative : this
as major to the premiss C is D would make the next conclusion
negative, and so ultimately the conclusion of the whole sorites ; but
if any other premiss were negative, there would be an illicit process
of the major term in the syllogism into which it entered. The
rules of a sorites are thus nothing but the special rules of the first
figure.1

A sorites is distinguished from other chains of syllogistic
reasoning by the fact that not only is one of the premisses sup-
pressed, at every step of the argument except one, but the inter-
mediate conclusions, by which the final conclusion is reached, are
all suppressed ; for the conclusion of one argument is the sup-
pressed premiss of the next. This is, perhaps, what has led
logicians to give special attention to it.2

The Dilemma combines into one argument hypothetical and
disjunctive reasoning. Generally it is an argument in which one
premiss is a disjunctive proposition, and the other consists of hypo-
thetical propositions connecting with either alternative in the

1 Either an E or an I proposition may be converted simply. With an
7 premiss for the first, if it be converted, the sorites may be broken up into
a series of syllogisms in the third figure ; with an E premiss for the last,
if it be converted, the sorites may be broken up into a series of syllogisms
in the second figure. Yet, except for the premiss thus converted, the middle
terms stand throughout in the premisses as in the first figure. A series of
premisses in the second or in the third figure will not form a sorites : because
there would be no series of middle terms, but only one middle term through-
out ; hence as soon as we come to combine the conclusion of two premisses
with the next premiss, we should be involved in quaternio terminorum. The
sorites is therefore essentially confined to the first figure, though its resolution
may involve the second or third.

2 It is however only one example of what mathematical logicians like
Mr. Bertrand Russell call a system of asymmetrical transitive relations.

358 AN INTRODUCTION TO LOGIC [chap.

disjunction an unpalatable conclusion. In one form, however, of the
simple destructive dilemma ' the disjunction may be in the conse-
quent of the hypothetical premiss, and the other premiss be cate-
gorical, denying both alternatives in the disjunction.2 We may
therefore define a dilemma, to cover this case, as a hypothetical
argument offering alternatives and proving something against an
opponent in either case. The conclusion may be either the same,
whichever alternative is accepted, or different ; in the former case
the dilemma is called simple, in the latter complex. It is called
constructive, if it proceeds from affirmation of antecedent in the
hypothetical premiss to affirmation of consequent ; destructive, if
it proceeds from denial of consequent to denial of antecedent.

1. Simple Constructive.

If A is B, E is F ; and if C is D, E is F
But either A is B or C is D
.*. EisF3

Troops with a river behind them have sometimes been placed in
a dilemma none the less painful because it is simple. If they
stand their ground they die — by the sword of the enemy : if they
retreat they die — by the flood ; but they must either stand or
retreat ; therefore they must die.

2. Complex Constructive.

HAisB,EisF; and if C is D, G is E

But either A is B or C is D
.'. Either E is F or G is H

Thus we might argue — and this too is unfortunately a dilemma
from which it is not easy to see an escape :

11 there is censorship of the press, abuses which should be
exposed will be hushed up ; and if there is no censorship,
truth will be sacrificed to sensation
But there must either be censorship or not

1 See below, pp. 360-361.

2 The hypothetical premiss is sometimes called the major, in accordance
with the nomenclature used also of hypothetical reasoning : and the other
premiss the minor.

3 Antecedent and consequent may, of course, all have the same subject (if
A is B, it is D ; and if it is C, it is D) : or the same subject in one case and
different subjects in the other ; and the minor premiss will vary accordingly.
It would be tedious to give each time all these varieties, which involve no
difference of principle.

xvi] ENTHYMEME, SORITES, AND DILEMMA 359

.*. Either abuses which should be exposed must be hushed up,
or truth be sacrificed to sensation.

3. Simple Destructive.

If A is B, G is D and E is F
But either C is not D or E is not F
.*. A is not B

Plato, in the Republic,1 urges that children should not learn the

poems of Homer, from which they will derive very false beliefs about

the nature of the gods. One of his arguments might be put thus :

If Homer speaks truth about things divine, the heroes were

sons of gods, and did many wicked deeds
But either they were not sons of gods, or they did not do
wicked deeds
.*. Homer does not speak truth about things divine.

Again, If A is B, either C is D or E is F
But neither is G D, nor is E F

: . A is not B

Of this character was one of the arguments used by Zeno to
disprove the possibility (or perhaps we might say, the intelligibility)
of motion :

If a body moves, it must either move in the place where

it is, or in the place where it is not
But it can neither move in the place where it is, nor in the
place where it is not
.*. It cannot move.

4. Complex Destructive.

ttAisB,EisF; and if Cis D, Gia H
But either E is not F, or G is not H
.'. Either A is not B, or G is not D

A nation having colonies like those of Great Britain might fairly
urge :

If we give our colonies self-government, we shall make
them powerful ; and if we attempt to control their use of
it, we shall make them hostile
But either we ought not to make them powerful, or we
ought not to make them hostile

1 III. 391 G-E.

360 AN INTRODUCTION TO LOGIC [chap.

.'. Either we ought not to give them self-government, or we
ought not to attempt to control their use of it.

[It is sometimes said that a destructive dilemma is always com-
plex, and such arguments as those given under (3) above would not
be allowed to be dilemmas. Mansel's definition (which follows
Whately, and has been adopted by others since) definitely excludes
the simple destructive ; according to him (v. his Aldrich, p. 108,
n. i) a dilemma is ' a syllogism having a conditional major premiss
with more than one antecedent, and a disjunctive minor ' ; as the
destructive dilemma proceeds from denial of consequent to denial
of antecedent, if there is more than one antecedent its conclusion
must be necessarily complex. A number of writers, however, have
admitted the simple destructive dilemma ; and it seems very difficult
to exclude examples of the first form above given, at any rate.
The simple constructive (If A is B, E is F ; and if G is D, E is F)
may be written

If A is B or C is D, E is F

But either A is B or C is D
.-. E is F

The simple destructive runs

If A is B, C is D and E is F
But either G is not D or E is not F
.♦. A is not B

It may be said that there is a disjunction in the hypothetical premiss
of the former, and not of the latter ; but this does not seem to
constitute an essential difference, such as would render one a dilemma
and the other not. In the former, one or other of two alternatives
must be affirmed, and whichever be affirmed, the same conclusion
follows, because it is logically a consequent of affirming either
alternative ; in the latter, one or other of two alternatives must be
denied, and whichever be denied, the same conclusion follows,
because it is logically a consequent of denying either alternative.
The essence of the dilemma seems to lie in the fact of confronting
a man with alternatives at once ineluctable and unpleasant : cf . the
definition quoted by Mansel from Cassiodorus, loc. cit. : Dilemma,
quod fit ex duabus propositionibus pluribusve, ex quibus quidquid
electumfuit, contrarium esse non dubium est. And therefore the other
example given above — Zeno's argument about motion — seems also
to be fairly called a dilemma.1 It is true that its second premiss is
not disjunctive at all, but denies a disjunctive proposition ; it does
not assert the truth of one of two alternatives, but the falsity of both.
But the whole argument is a combination of the hypothetical and

1 So Minto takes it, Logic, Inductive and Deductive, p. 224.

xvi] ENTHYMEME, SORITES, AND DILEMMA 361

[the disjunctive, and drives a man into a corner by way of alterna-
tives between which his choice is alleged to be confined. If we are
to maintain that a body moves, we have to assert one or other of
two propositions which are both self -contradictory ; and that seems
a good example of being placed between the devil and the deep sea.
The simple constructive dilemma is a hypothetical argument in
the modus ponens ; its hypothetical premiss has a disjunctive
antecedent and a simple consequent, and therefore the other premiss
must be disjunctive and the conclusion simple. The simple destruc-
tive dilemma of the second form given above is a hypothetical
argument in the modus tollens ; its hypothetical premiss has a simple
antecedent and a disjunctive consequent ; the other premiss must
therefore be the denial of a disjunctive proposition, and the con-
clusion the denial of a simple one. But the denial of a disjunctive
proposition is a categorical, whereas the affirmation of it is of course
a disjunctive proposition. Hence the difference which has led to
refusing the name of dilemma to this form of argument ; yet its
parallelism with the simple constructive seems correct and clear.
It may be asked why there are two types of simple destructive
dilemma, against one type of simple constructive. The answer
feeems to be this. In the destructive dilemma, I may overthrow
the antecedent, either if its truth involves two consequents, one or
other of which I can deny, or if its truth involves one or other of
two consequents, both of which I can deny ; and each case involves
a disjunction. In the constructive dilemma, I can establish the
consequent, either if two antecedents involve its truth, both of
which I can affirm, or if either of two antecedents involve its truth,
one or other of which I can affirm. But here the former case
does not constitute a dilemma, because no disjunction is involved
anywhere : If A and B are true, C is true ; but A and B are true
.-. C is true. It would appear therefore that so far from there being
no such thing as a simple destructive dilemma, there are two forms
of it, against only one form of simple constructive dilemma.]

A dilemma is sometimes spoken of as if it were a peculiarly
unsound form of argument. It shares with all inference the pro-
perty that it is of no material value unless its premisses are true ;
but formally it is quite sound, and if there is about it any special
weakness, it must lie in some special difficulty in getting true pre-
misses for it. Now it is generally difficult, except where one alter-
native is the bare negation of the other, to get an exhaustive
disjunction ; it is here that any one ' in a dilemma ' would look
for a way out ; and it is this difficulty which inspires mistrust of
the dilemma as a form of argument.

To show that there is some other alternative besides those, on

362 AN INTRODUCTION TO LOGIC [chap.

one or other of which your opponent attempts to drive you, is called
escaping between the horns of a dilemma : the alternatives being the
' horns ' on which you are to be ' impaled '. In reply to Zeno's dilemma
to show the impossibility of motion, it is often said that a body
need not move either in the place where it is or in the place where
it is not ; since it may move between these places. It may be
questioned whether this is a very satisfactory solution of the para-
dox ; for those who offer it might find it hard to say where the body
is when it is between these places ; if it is not in some other place,
the continuity of space seems to suffer disruption. But however
that may be, we have here an attempt to escape between the horns
of Zeno's dilemma.

The other two ways of meeting a dilemma also bear somewhat
picturesque names ; we may rebut it, or we may take it by the horns.
To rebut it is to produce another dilemma with a contradictory
conclusion. The old story of Protagoras and Euathlus, without
which a discussion of Dilemma would hardly be complete, furnishes
a good example of rebutting. Protagoras had agreed with Euathlus
to teach him rhetoric for a fee, of which half was to be paid at the
conclusion of the instruction, and the remainder when Euathlus won
his first suit in court. Observing that the latter delayed to practise,
Protagoras thought he was endeavouring to evade payment, and
therefore himself brought a suit for the recovery of the second half
of his fee. He then argued with the jury that Euathlus ought to
pay him, in the following way :

If, he said, he loses this case, he ought to pay, by the judge-
ment of the court ; and if he wins it, he ought to pay,
by his own agreement

But he must either lose it or win it
.*. He ought to pay.

Euathlus, however, rebutted this dilemma with the following :

If I win this case, I ought not to pay, by the judgement of
the court ; and if I lose it, I ought not to pay, by my own
agreement

But I must either win it or lose it
.*. I ought not to pay.

It will be seen that the rebutting dilemma is produced in this
case by connecting in the hypothetical premiss, with either ante-

xvi] ENTHYMEME, SORITES, AND DILEMMA 363

cedent, the contradictory of the consequent originally connected
with the other. With a destructive dilemma the parallel procedure
would be to comiect with the contradictory of either antecedent
the consequent originally connected with the other. But this is not
the only way of rebutting ; you rebut whenever you produce
a dilemma with contradictory conclusion, even though you do it with
quite different premisses. Nor can every dilemma be rebutted in
this way or in any other way : not in this, for the alternative con-
ditions are not always such with which you can connect the contra-
dictory of each other's consequents. And if a dilemma can be
rebutted, it must be for one of three reasons. Either (1) there must
(as in the last example) be some impossible assumption in the
supposed situation ; and some of the ancients spent much ingenuity
in imagining situations of this kind, in which our reason was
entangled by finding that two contradictory solutions of a problem
could apparently be maintained with equal force.1 Or (2) the
premisses are unsound, and premisses equally or more plausible
can be found for another dilemma proving a contradictory con-
clusion ; in this case, it would be possible to attack the original
dilemma directly, either by showing that you can escape between
the horns of it, if the disjunction is not complete, or in the third of
the ways mentioned above, by ' taking it by the horns '. Or else
(3) as happens, unless there is an impossible assumption in the situa-
tion supposed, when we rebut by transposing and denying the conse-
quents or the antecedents, the conclusions of the two dilemmas are
perfectly consistent, and the second merely shows that you will escape
one or other of the alternatives, of which the first showed that one
or other would be incurred. In a complex dilemma whose alterna-
tives are mutually exclusive this is obviously necessary ; but it is

1 Of this nature are the well-known sophisms of the 'Liar' and the
'Crocodile' ; Epimenides the Cretan said that all Cretans were liars; if they
were, was he lying, or was he speaking the truth ? — a crocodile had stolen
a child, and promised the mother he would restore it, if she could guess
rightly whether he intended to do so or not ; if she said he would not restore
it, she could not claim the child by his promise, because her taking it would
make her guess wrong ; if she said he would restore it, she could not claim
it, for she guessed wrongly ; what was she to say ? (cf . Lucian, Vit. Auct.
§ 22, cited Mansel's Aldrich, p. 151). The solution of the first is easy unless
we suppose that no Cretan ever spoke the truth ; in which case the truth
of the statement attributed to Epimenides is incompatible with his making
it. It may be said generally of both these sophisms, and of the story of
Protagoras and Euathlus, that the difficulty arises from supposing that a
statement or agreement about certain matters can itself be within the scope of
such statement or agreement.

3G4 AN INTRODUCTION TO LOGIC [chap.

small consolation to any one on the horns of a dilemma, to point it
out. When Henry VIII desired to force upon Sir Thomas More the
oath of supremacy, More was asked whether he thought the statute
' giving to the King the title of Supreme Head of the Church under
Christ ' had been ' lawfully made or not. He replied that the act
was like a two-edged sword, for " if he said that it were good, he
would imperil his soul ; and if he said contrary to the statute, it
were death to the body ".' * If a man is threatened with death or
damnation, the threat is not proved empty by showing that he will
escape damnation or death. Sir Thomas More indeed ' declined to
swear at all '. But that also was death to the body.

To take a dilemma by the horns (or by one of them) is to accept
an alternative offered you, but to deny that the consequence, which
the opponent attaches to its acceptance, follows. Perhaps the
following will serve for an example. It is held by many naturalists,
that species are modified in the course of descent only by the accu-
mulation of many slight variations, and not per saltum : variations
not being directly adaptive, but being distributed, in respect of
frequency and degree, in proportions that follow the well-known
' curve of error ', on either side of the standard represented in the
parents. Against this it has been argued, that though the cumula-
tive effect of many slight variations might be useful, it will often
happen that in the incipient stages, while the distance traversed in
the direction of some new peculiarity is still very slight, the variation
would be valueless, and therefore not tend to be perpetuated ;
so that the basis for accumulation would not exist. This line of
objection has been applied to the particular case of protective
colouring in insects in the following argument.2 If, it is said, the
slight variations, with which the process of mimicry in insects must,
as alleged, begin, are of no use in leading birds to mistake the
individuals exhibiting them for members of some protected species,
then they will not be preserved by natural selection, and no accumu-
lation can take place ; while if they are of use, any further and
more exact resemblance to the protected species is unnecessary, and
could not, if it occurred, be preserved by natural selection. Now
against this dilemma we may answer that it does not follow that,
because a slight degree of resemblance is useful, any further degree

1 Political History of England, vol. v, by H. A. L. Fisher, p. 350.
* See an article on The Age of the Inhabited Earth, by Sir Edward Fry, in
the Monthly Review for January, 1903.

xvi] ENTHYMEME, SORITES, AND DILEMMA 365

would be superfluous. On a particular occasion a particular insect
no doubt needs no greater resemblance than what has actually
enabled it to escape ; but with a large number of insects over a long
series of occasions, it may well be that the percentage of escapes
would be higher with those in whom the resemblance was closer.
Thus the dilemma is ' taken by the horns ' ; but that does not settle
the important question at issue as to whether variation ever does
proceed per saltum or not. We saw before that a thesis is not
disproved by the refutation of any particular argument brought
forward in support of it.

CHAPTER XVII

OF THE FORM AND MATTER OF INFERENCE

So far we have considered and examined some of the commonest
forms of inference — syllogism, hypothetical and disjunctive reason-
ing, and certain complications of these. We have not pretended
— what has nevertheless sometimes been maintained — either that
the last two can be reduced to syllogism, or that syllogism, even if the
term be taken to include the three, is the type to which all valid
inference must conform ; though we have maintained, and it will
appear more fully in the sequel, that all are forms of great fre-
quency and importance in our thought. Were Logic a purely formal
science, the analysis of these forms would be, to those who thought
that all reasoning really moved in one or other of them, the end of
the task imposed upon that science ; to those who did not think
them the only forms in which men's reasoning moves, no other task
would be left than to offer a similar analysis of the remainder.
But if it is impossible to understand fully the form of thinking
without reference to differences in it springing from the nature of
that about which we think, then the task of Logic is obviously
harder. It will not suffice to work with symbols. We cannot
make abstraction of the special character of our terms. Already
we have found this to be the case. We saw that what is called
demonstrative syllogism in the first figure rests upon a perception
of the necessary relation between certain notions, or universals ;
while in the third figure such a perception of necessary relation
neither need be given in the premisses of a syllogism, nor can
be reached in the conclusion. We saw too how hypothetical reason-
ing, where it differs most from syllogistic, differs because it establishes
a connexion between subject and predicate in the conclusion by
means of a condition which is apparently extraneous to the nature
of the subject ; and yet how our thought recognizes that there
must be some wider system to which the subject and that condition
both belong, and through which it comes about that the fulfilment
of the latter should affect the predicates of the former. None of
these things could be explained or understood merely through

THE FORM AND MATTER OF INFERENCE 367

symbols : examples were needed not only to show that the argu-
ments symbolized were such as we do actually often use, but because
only in suitable examples could those facts of our thought with
which we were concerned be realized. The symbols, e.g., are the
same, but do not symbolize the same thing, when some terms in our
syllogism are singular, and stand for individual concrete subjects,
whose attributes are set down as we find them, and when they are
all general, and signify universal characters of things, between
which we perceive connexion.

It will be said that if the form of thought be thus bound up with
the matter, and if the matter be different according as we think
about different things, an understanding of the form must wait upon
a knowledge of these, and the task of Logic will not be complete
until we have finished the investigation of what is to be known.
In a sense this is true. It may be illustrated by mathematics ;
no one can understand the nature of mathematical reasoning
except in reflection upon his thinking about number or space or
quantity ; it cannot be seen in application to heterogeneous sub-
jects. And it consists with the position which we have taken up
from the outset, that Logic is the science which brings to clear
consciousness the nature of the processes which our thought per-
forms when we are thinking about other things than Logic. Never-
theless we must bear in mind one or two facts, which may make the
task of Logic seem a little less hopeless than it would appear to be,
if it had to wait altogether upon the completion of knowledge.

In the first place, the dependence of the form of thought upon
the matter is consistent with some degree of independence. It may
be impossible to grasp the nature of mathematical proof except in
application to mathematical subjects ; but an analysis of one or two
examples of geometrical reasoning may serve to show us the nature
of geometrical reasoning in general, and after that the form of it will
not be any better understood for tracking it through all our reason-
ings about every figure and space-relation. So also it may be
impossible except in examples of the relation of subject and predicate
to grasp the distinctive character of syllogistic reasoning ; but we
may grasp it there universally, and realize that it will be the same
for all terms that stand in those relations. If this were not so,
science would be impossible ; for science seeks to reduce a multi-
plicity of facts to unity of principles. Thus our apprehension of
the forms of thought has not to wait upon the completion of our

368 AN INTRODUCTION TO LOGIC [chap.

knowledge so far as that completion means only its extension to fresh
subjects of the same kind. If some branch of our knowledge is
defective in point of extent — as it would appear, for example, that
the science of number must ever continue to be, because the numeri-
cal series is by its nature inexhaustible — yet its further extension
may involve no change in its character ; and so soon as all the main
branches of possible knowledge have been discovered — that is,
knowledge about all the main departments of fact — the forms which
thought assumes in them can be studied even while our knowledge
is incomplete in its extent. The main departments of fact must,
of course, be taken to include not merely those which form the sub-
jects of the physical sciences, but equally those of which philosophy
treats, and not least the relation of the world to the mind that
knows it. It would be rash to assert that this stage has been
reached in the progress of knowledge. The completion of our
knowledge may yet require not only its extension, but in large
degree its transformation. Yet we may assert that a great deal of
our ignorance forms no bar to the completion of the investigations
of Logic.

And in the second place, though Logic is in the main a reflection
upon the nature of knowledge already gained, there is this paradox
about knowledge, that we seem to some extent to know what know-
ledge ought to be, before we know anything as we ought. We
have an ideal, of which we are sufficiently conscious to realize the
imperfections of the actual, though not sufficiently conscious to be
able to put it clearly and fully into words. This paradox is not
confined to knowledge ; it occurs in art and in morality also.1 We
may recognize defect in an aesthetic whole without being able to
rectify it, and yet we may be able to say in what direction its per-
fection must lie ; we may know that ' we have all sinned ', without
having seen ' the glory of God ', and still be able to prescribe some
of the conditions which that must realize. So also we may know
that the form of our thought, even when we think best and most
patiently, often falls short of the full measure of knowledge : that
our way of thinking — our way of looking at things, if one may put
it so — is wrong because it fails to escape contradictions and satisfy
all doubts ; and that there must be some way of thinking (if the
world is as a whole intelligible at all) in which contradiction and
uncertainty will vanish. We may know all this, and know that we

1 Cf. supra, p. 10, n. 2.

xvii] THE FORM AND MATTER OF INFERENCE 369

have not found that better way (for if we had, we should certainly
not remain in the worse) : and still we may be able to say something
about it though we have not found it : to lay down conditions
which our knowledge of any subject must satisfy because it is
knowledge — i. e. to prescribe to some extent the form of knowledge,
not only as a result of reflection upon instances of subjects being
perfectly known or by abstraction from the activity of knowing
perfectly in the concrete, but by way of anticipation, out of reflection
upon instances in which we know subjects less than perfectly, and
know the imperfection of our knowing. The extent to which we can
thus anticipate is not unlimited ; a man must get some way in
science, before he will realize what science should be, and that it is
not what it should be ; just as a man must get some way in virtue
before he will realize how much more it requires of him than he has
achieved. Yet it remains true that thought can in some degree
anticipate a form of knowing a subject which it has not exercised
therein ; and it is the business of Logic to set this form forth.
So far again Logic has not to wait, in order to complete its task,
until our investigation of what is to be known has been completed.

If this is true, we may say on the one hand, that no study of the
nature of inference can be adequate which treats it as an operation
performed with symbols, or one intelligible at any rate when we work
merely with symbols. On the other hand, we may recognize that
there are recurrent forms of inference, whose nature is the same in
their different occurrences,1 and that they occur commonly and are
displayed in regard to subjects in many respects very diverse ; we
may also recognize an ideal of what inference should be if it is to
convey knowledge : if we are to realize in making it not merely that
the conclusion follows from the premisses, but that we are getting
at indubitable truth.

Our discussion of inference up to this point must therefore be
incomplete, in so far as (a) we have failed to deal with all those
distinguishable recurrent forms of inference whose universal nature
can be realized in an example ; (b) we have failed to make plain the
conditions of knowledge as well as the conditions of cogency.

As to the first count, there are certainly forms which have not

1 Some might maintain that it is never quite the same when the matter is
different, any more than the nature of man is quite the same in any two
individuals. I do not wish to subscribe to this view ; but even its upholders
would admit that such differences may be negligible.

1779 B b

370 AN INTRODUCTION TO LOGIC [chap.

been examined. For example, there is the a fortiori argument. 'He
that loveth not his brother whom he hath seen ', asks St. John,
' how can he love God whom he hath not seen ? ' And there is
mathematical reasoning, of which we have only said that it is not
syllogistic ; this from its importance may claim rather fuller con-
sideration. But perhaps more remains to be done in the way of
showing how far inference of these different forms enters into the
building up of our knowledge, and what other operations of thought
enter into it.

As to the second count : it is a charge brought against the analysis
of syllogism, and the other inferential forms considered above, that
such analysis only shows us the conditions of consistency in reason-
ing, and not the conditions of truth. To reason consistently is very
different from discovering truth ; for the consistent reasoner will
reproduce in his conclusion the error there may be in his premisses.1
Those who have brought this charge have sometimes supposed that
what is wanted is other and better forms of inference. It would
be much truer to say that what we want is to realize how much
besides formal validity of inference must be present in an argument
which is to convey knowledge. To realize what is needed is not
indeed the same thing as to supply it ; but Logic cannot help us to
more. The critics of the Logic which was content to analyse the
conditions of validity in some of the common inferential forms (and
which often supposed — it must be admitted — that there were no
other forms of inference) have not always believed this. Many of
them, as has been said in the first chapter, still looked on Logic
mainly as an instrument for the discovery of truth about any
matter on which we might propose to reason, and hoped to find
a new and better instrument than what the Logic which confined
itself to such analysis afforded. This was the object with which
Bacon wrote his ' New Instrument ' or Novum Organum ; and
J. S. Mill, though he calls Logic a science, wrote his famous treatise
in the hope that familiarity with the methods of reasoning used
successfully in the physical sciences would enable men to prosecute
the study of the moral and political sciences with more success.2
Logic is not a short cut to all other branches of knowledge. But
this we may say, that men who know the difference between con-

1 Though formally a true conclusion may be got from false premisses, the
error still infects the mind, and will lead to a false conclusion somewhere.

2 Cf. System of Logic, VI. i., and Autobiography, p. 226.

xvii] THE FORM AND MATTER OF INFERENCE 371

sistency and demonstration, who know what is required before it
can be said that they have knowledge about things, in the full and
proper sense of that term, are less likely to remain content with the
substitutes that commonly pass muster in men's minds for know-
ledge. By a study of the conditions of demonstration we may be
led to see how far from being demonstrated are many of the beliefs
we hold most confidently. To know what we do know, and what
we do not — what, out of the things we suppose ourselves to know,
we really know and are rationally justified in believing : this, as
Plato long ago insisted,1 is neither a small thing, nor an easy ; and
until we understand what knowing a thing means and requires,
we are not likely to achieve it. This is why Logic should do more
than present us with a study of the forms of consistent reasoning,
and should attempt to exhibit the nature of knowledge and demon-
stration : not because such an exposition of the form of knowledge
is itself an instrument for bringing our thoughts upon any subject
into that form, but because it stimulates us to use such powers as we
have, and to appraise the results which we have so far attained.

Now the most obvious criticism that can be made upon a Logic
which confines itself to setting forth the formal conditions of valid
inference is that it ignores the question of the truth of the premisses ;
the validity of the reasoning affords no guarantee that these are
true. It is no doubt possible to direct men's attention so exclusively
to the form of argumentation that they will bestow little upon the
truth of the premisses from which they argue. It has often been
complained that the study of Logic did this — or, as its critics would
say, the study of Deductive Logic.2 The epithet, however, implies
a misunderstanding ; it is a disproportionate attention to validity
of form in general which the critics ought to deprecate. Validity
of form is a thing worth studying, not only for its own sake, but in
some degree lest we infringe it ; yet it is psychologically possible,
by studying it too much and too exclusively, to become distracted
from due care about truth of fact. It is, however, probable that

1 Charmides 171 D.

2 The popular antithesis between Deductive and Inductive Logic has been
so far avoided, and that deliberately ; we shall have to consider presently
what the nature of the difference between deductive and inductive reasoning
is ; but it may be said at once that it does not lie in using the forms of infer-
ence that are commonly expounded under the titles of Deductive and of
Inductive Logic respectively. For inductive reasoning uses forms of inference
with which treatises that would be called Deductive always deal ; and treatises
called Inductive discuss forms of inference which are certainly deductive.

B b 2

372 AN INTRODUCTION TO LOGIC [chap.

in the times when men have been most remiss in the examination
of their premisses, the state of the study of Logic has been as much
a symptom as a cause of this ; and however that may be, so far as
it lies with Logic to provide a corrective, it is very important for
the logician to be clear as to the nature of the corrective he is to
provide. And for that purpose he must distinguish two questions ;
he may try to show what kind of premisses knowledge requires, or
by what process of thought we may hope to get them. In modern
times, the former of these questions has been too much neglected.

These last remarks may be a little expanded. And first as to
the causes which for many centuries made men remiss in the
examination of their premisses ; one sometimes finds the blame for
this thrown upon the futility and misdirection of the scholastic
Logic, which absorbed during the Middle Ages, and even later, so
large a part of the energy of men's minds. It would be hard to
deny that much of it was futile, and that much energy was mis-
directed ; but it is as likely that energy went into this channel
because others were temporarily closed to it, as that others were
robbed of it because it ran in this ; though no doubt there is action
and reaction in such a case, and a habit which certain influences
tend to form may in turn strengthen those influences.

It has been said that the mandate issued to the age of Plato and
Aristotle was Bring your beliefs into harmony with one another ; that
the mandate of the Mediaeval Spirit was Bring your beliefs into
harmony with dogma ; and that the mandate of the new spirit which
rebelled against the authority of the Church was Bring your beliefs
into harmony with fact.1 Such a mode of putting things may suggest
some errors. It is impossible to bring one's beliefs into harmony
with fact, except so far as facts are known to us ; our knowledge of
facts is expressed in propositions which we believe ; and therefore
to bring our beliefs into harmony with fact is to bring them into
harmony with one another (though not conversely). It would be
wrong to suppose that Plato and Aristotle forgot that among the
beliefs they had to harmonize with one another were the beliefs they
held about matters of daily experience, or that they were indifferent
to the necessity of correcting and enlarging those beliefs by more or
less systematic observation ; Aristotle in particular added largely
to men's knowledge of facts. Again, it is clear that to bring one's
beliefs into harmony with dogma is to bring them into harmony
1 Minto, Logic, Inductive and Deductive, p. 243.

xvii] THE FORM AND MATTER OF INFERENCE 373

with other beliefs ; and that those who rated highest the importance
of that task would least have doubted that they were bringing them
into harmony with facts. Propositions do not cease to state facts
because they are presented as dogmas. But it is true, as Minto
wishes to bring out in the passage quoted, that dogma and the spirit
which accepts dogma did during the Dark and the Middle Ages play
a part in the history of thought far greater either than they played
in classical antiquity or than they have come to play since the revival
of learning. And such dogma was not necessarily ecclesiastical
dogma ; it came from the scientific works of Aristotle, or other great
men of old whose works were known, as well as from the Bible and
the Church ; just as to-day there is orthodoxy in science, against
which new scientific doctrines find it at times a little difficult to
battle, as well as in theology.

The schoolmen knew, as well as Bacon or any other of their critics,
that the study of the syllogism was not all-sufficing : that no syllo-
gism could guarantee the truth of its premisses ; and that for a
knowledge of the most general principles to which deductive reason-
ing appeals we must rely on something else than deductive reasoning
itself. Bacon refers to the ' notorious answer ' which was given
to those who questioned the accepted principles of any science —
Cuique in svxi arte credendum.1 And there are seasons in the process
of learning when that is a very proper answer ; men must be content
at many times and in many matters to accept the expert opinion of
their day. But this is only tolerable if in every science there are
experts who are for ever questioning and testing. When tradition
stereotypes doctrine, it is as bad for knowledge as close guilds and
monopolies are bad for the industrial arts ; they shut the door upon
improvement. Authority plays, and must play, a great part in
life — not only in practice, but also in things of the intellect. But
the free spirit is as necessary, which insists on satisfying itself that
what is offered upon authority has claims on its own account upon
our acceptance.

Why was it that for so many centuries so much was accepted
upon authority which afterwards fell to pieces in the fight of inde-
pendent enquiry ? Much knowledge of the human mind, historical
and philosophical, would be needed in order to answer this question
adequately. If a few observations may be made upon it here, it is
with a full consciousness of the inadequate equipment of knowledge

1 Nov. Org. I. 82.

374 AN INTRODUCTION TO LOGIC [chap.

upon which they rest. And it may be doubted whether we can
hope fully to explain why some periods and places are richer
than others in men of fruitful and original thought ; at most we
can hope to show what conditions are favourable to such men's
work when they arise. Now to us, looking backward across the
Middle Ages to the more brilliant days of Athens and of Rome,
and looking also at the great increase of knowledge which the last
three centuries have brought, the stagnation of the sciences in the
period intervening is apt to seem a thing surprising. But how
long was it before ancient science began to appear and to advance?
The power of tradition and authority over the human mind is the
rule rather than the exception.1 And in the break-up of ancient
civilization there perished not only much knowledge, but much
material wealth ; men were of necessity for long absorbed in the
task of restoring this and restoring order ; and it is not wonderful
that they had little time to spend in questioning such scientific
principles as had survived. Moreover, during the darkest times,
the most powerful and the most beneficent institution that stood
erect was the Church ; the most comprehensive and well-reasoned
theory of the world was that which the Church taught ; the strongest
minds, almost the only minds that thought at all, were enlisted in
the ranks of the clergy (which was why independent thought took
so largely the form of heresy), and the interest of men was directed
rather to what concerned the soul than to nature around them. To
this it must be added, that through a series of historical accidents,
a great part of the literature of Graeco -Roman civilization had
perished ; but that of the works of Aristotle some few were known
continuously, and the rest recovered, at least in translations, by
the end of the first quarter of the thirteenth century.2 The works
of Aristotle, by their encyclopaedic range, by the effort after sys-
tematization displayed in them, and by their extraordinary intel-
lectual power, were peculiarly suited to rivet themselves upon the
mind at a time when ability was not wanting, but when detailed
knowledge was slight, and there was little else to serve for an edu-
cational discipline. It is not surprising, if Aristotle and the Church
(especially when the Church pressed Aristotle's philosophy into its
service) acquired a preponderant influence over men's minds.3

1 Cf. Bagehot, Physics and Politics.

2 v. Prantl, Oeschichte der Logik, III. p. 3.

3 Professor W. G. de Burgh calls my attention to the language of Dante,
II Convito, iv. 6, about the authority of the maestro di lor che sanno.

xvn] THE FORM AND MATTER OF INFERENCE 375

Indeed, it is hard for us to imagine what self-confidence and courage
were necessary, in order to question any part of that closely con-
catenated fabric of belief , upon appearing to accept which depended
a man's comfort in society and perhaps his life in this world, and
upon really accepting it — unless he could find for himself something
better — his confidence with regard to the next. It is no small
testimony to the inexpugnable power of reason, that this system
broke down. And it began to break down largely through the
recovery of other monuments of ancient thought and learning
besides the works of Aristotle. This doubtless stimulated, though it
could not produce, the powers of those men by whom the founda-
tions of modern science were laid — men like Copernicus, Galileo,
Harvey, Gassendi, Descartes. It was not the reform of Logic which
liberated the mind, any more than it was Logic which had bound it.

It is, then, rather to the habit of believing on authority, the
strength of which during that period it has been attempted in some
degree to account for, than to the prevalence of an erroneous Logic
(whose errors were not really what the ' inductive ' logicians sup-
posed), that the stagnation of science for so many generations must
be attributed. Given that habit, it was natural that men should
spend time and thought upon a barren elaboration of the more
technical parts of Logic, and leave the traditional assumptions both
of it and of the natural sciences unexamined. When the over-
mastering influence of authority began to decay, the science of
Logic shared with other sciences in the revivification that comes
from thinking out a subject freshly and independently.

But, as was said above, the particular matter which first attracted
the attention of the reforming logician was the barrenness of an
exclusive attention to the forms of valid inference ; and the parti-
cular improvement proposed was the establishment of a Logic that
should do for the discovery and proof of scientific principles what
had already in part been done for the drawing of conclusions from
them. This at least is how Bacon looked at the matter ; and others
have so looked at it after him, in this country more especially.
Now it is a very interesting question, how sciences get their prin-
ciples, and when they may be considered proved ; but it is not quite
the same as the question, what kind of principles knowledge requires.

The works of Aristotle dealing with inference are three — the
Prior Analytics, the Posterior Analytics, and the Topics. Speaking
generally, the first of these deals with syllogism from a formal

376 AN INTRODUCTION TO LOGIC [chap.

point of view — it pays no attention to the nature of the premisses,
but only to the validity of inference ; the second deals with know-
ledge, or demonstration : it asks not when a man is bound by the
acceptance of certain forms of premiss to admit a certain form of
conclusion, but when he can be said really to know a thing abso-
lutely, and not merely on the assumption that certain premisses
are true ; the third asks how positions can be established or over-
thrown, what sort of considerations are useful in weighing their
claims to acceptance, and on what sort of grounds men may be
content to accept their principles in matters where certainty is not
attainable. In the first and in the third of these treatises, Aristotle
was analysing and formulating the actual procedure of his con-
temporaries ; he did not, upon the whole, go ahead of the science,
the disputation, the rhetoric, and the pleadings of his day. In the
second, he was doubtless guided also by a consideration of the
highest types of scientific knowledge then existing ; but he was
guided also by an ideal ; he was trying to express what knowledge
ought to be, not merely what the form of men's reasonings is.

It may be said that in scholastic Logic, the problems of the
Prior Analytics bulked too large ; that those who revolted against
this raised, without realizing it, problems of the same kind as
Aristotle had already discussed in the Topics ; but that for a long
time the questions of the Posterior Analytics received insufficient
attention. It is these last which are the highest, and go deepest
into the philosophy of the subject. The physical sciences employ
many principles of great generality which they try to prove ; but
there are some assumptions about the nature of the world, which
they accept without asking why they accept them. As instances
of these may be mentioned what is called the Law of the Uniformity
of Nature — the principle that every change has a cause upon which
it follows in accordance with a rule, so that it could not recur in the
same form unless the same cause were present, nor fail to recur when
precisely the same cause recurred : or again, the principle that
matter is indestructible : or that the laws of number and space
hold good for everything numerable or extended. There are other
principles less general than these, such for example as the law of
gravitation, of which, as aforesaid, science offers proof ; but
whether the proof of these amounts to complete demonstration, and
whether the assumption of the truth of those is justified — these are
problems with which the special sciences trouble themselves little,

xvii] THE FORM AND MATTER OF INFERENCE 377

and which will not be answered merely by analysing the nature of
the inferential processes that do as a matter of fact lead scientific
men to accept the general propositions which they conceive them-
selves to have proved.

This is only an elementary book, and makes no pretence to give
a complete answer to that most difficult of logical questions, What is
knowledge, in its perfect form ? But from what has been said in the
present chapter, it follows that there are two problems to which
some attention ought to be given. One is the question how, as
a matter of fact, we do get our premisses : the other, what are the
requisites of demonstration.1 The first of these may be called the
problem of Induction.

1 v. infra, p. 524.

CHAPTER XVIII
OF INDUCTION

The history of the word Induction is still to be written ; but it
is certain that it has shifted its meaning in the course of time, and
that much misunderstanding has arisen thereby. The Aristotelian
term e-n-aycoy?/, of which it is the translation, signified generally the
process of establishing a general proposition not by deduction 1 from
a wider principle, but by appeal to the particular instances, or kinds
of instances, in which its truth is shown.2 But if it is to be established
thus, all the instances must be cited ; and induction meant primarily
to Aristotle, proving a proposition to be true universally, by showing
empirically that it was true in each particular case or kind of case :
or, proving something about a logical whole, by appeal to the
experience of its presence in every part of that whole ; as you might
show that all horned animals ruminate, or that whenever the tail of
a fish is unsymmetrical (or heterocercal) it is vertebrated, by a dis-
section of the intestines of every kind of horned beast, or of the tail

1 The history of the term Deduction is also still to be written, cmaymyr)
in Aristotle meant something very different (v. Anal. Pri. /3. xxv : there is also
the use cited p. 315, n. 1, supra), and the nearest Aristotelian equivalent to
Deduction is trvXAo-yioTxor.

2 From what sense of the verb (irdyeiv this use of the word sprang is not
clear; there are two passages (An. Post. a. i. 71a 21, 24: xviii. 81b 5),
where the passive verb, in a logical context which makes it clear that the
process of inaycoyrj is referred to, takes a personal subject ; as if it were meant
that in the process a man is brought face to face with the particulars, or perhaps
brought, and as we could say induced, to admit the general proposition by
their help. In another place (Top. a. xviii. 108b 11 : cf. Soph. El. xv. 174a 34),
it is the universal proposition which is said to be ' induced ' or brought forward
or brought up (whatever the best translation may be) ; and perhaps the not
infrequent antithesis of e'nayayr) and avXXoyicrfios might suggest that the
usual object of the verb is the inductively obtained conclusion ; the conclusion
is certainly what is ' syllogized ', so that the conclusion may also be what is
1 induced '. It has, however, also been thought that the process of bringing
up or citing the instances, by means of which the conclusion is to be estab-
lished, is what the word was primarily intended to signify (Bonitz, Index
Aristotel., s. v. enaywyrj, seems to take this view) ; and anyhow the process
described is one in which a general conclusion is established in that way, by
citing the instances of its truth. Nevertheless, there is no passage where
tjrdyfiu governs an accusative of the instances adduced.

OF INDUCTION 379

of every kind of heterocercal fish. In such a proof, it would be
assumed that the nature of each species of fish or beast might be
judged from the single specimen dissected ; and it is to be noted that
Aristotle thought that the process of induction began here with the
infima species ; the species in his view (as we saw in discussing the
Predicables) being essentially the same in every one of its particulars.1
This form of argument he described in his own technical language as
proving the major term of the middle by means of the minor ; and
he showed how it could be expressed as a syllogism. From the
premisses

The cow, the sheep, the deer, dkc, ruminate
The cow, the sheep, the deer, &c, are horned

I cannot, as they stand, infer that all horned animals ruminate,
because there may be other horned animals besides all that I have
enumerated ; but if I know that this is not the case : if the members
in my enumeration taken together are commensurate or equate
with the term ' horned animals ', then the possibility which forbids
the general conclusion is excluded, and I may infer that all horned
animals ruminate : as is shown by the fact that the minor premiss
may be converted simply ; I may say that all the horned animals are
the cow and sheep and deer, &c. ; and my syllogism becomes formally
correct. In such a syllogism we are said to prove the major of the

1 Induction certainly starts in one sense, according to Aristotle, with
individuals ; for it starts with what we can perceive with the senses, and
only the individual can be perceived : cf . e. g. An. Post. a. xviii. 81b 5-9. But
it may be said that what we apprehend in the individual is its character or
type, and that it is to the individual as such and such an individual that we
appeal : cf. An. Post. a. xxxi. 87b 29. In An. Post. (3. xiii. 97b 7 sq., however,
Aristotle describes a method of searching for definitions — the example which
he uses is fieyaXo^vxia (magnanimity) — in which the instances cited in support
of the definition of fxeya\o\j/vxia are not cited as types at all. This has come
traditionally to be called the method of obtaining definitions by induction ;
and the description of it seems based on those discourses of Socrates to
which Aristotle refers as enciKTiKol \6yoi, inductive discourses ; but the term
inayaiyi) does not occur in the passage. Still in the argument from Example,
or TT<ipa8fiyfia, the instance appealed to is not cited as the specimen of a kind ;
and he calls this the rhetorical form of Induction. Hence, though the state-
ment in the text is true, so far as concerns the proof by induction of the pro-
perties of natural kinds (for in regard to that, Aristotle's particulars are
infimae species), it is difficult to maintain that he never regards induction as
starting with individuals as such. How you are to tell what properties
in a specimen are properties of the species is a question which is discussed
in the Topics ; and certainly he would not have thought of proposing to
prove that by a complete enumeration. The species of a genus are limited
in number, and can all be cited ; not so the individual members of a species.
Cf. infra, p. 384.

380 AN INTRODUCTION TO LOGIC [chat-.

middle by means of the minor, because (as we saw) the minor means
to Aristotle not primarily the subject of the conclusion, but the term
of least generality and nearest to the individual ; it is by the parti-
cular instances that the predicate ruminant is proved of the subject
horned animal. And if we might regard the possession of horns as
the cause of ruminating, then it would be the proper middle term by
which to demonstrate ruminant of cow or sheep or deer ; in Aristotle's
own example, where longevity is proved of gall-less animals by
means of man, horse, mule (and any other particulars that ought to
be mentioned — though for brevity they are not enumerated), it is
supposed that the absence of gall is the cause of longevity.

In symbolic form then we may express Aristotle's Induction
thus : —

ABC D, &c. are P
ABC I), &c. are all the M
.*. All M are P

This, which he calls 6 e£ eiTaycayrjs a-vWoyia-fios, is commonly
called now the Inductive Syllogism. If it is to be valid, our minor
term must, as Aristotle says, comprise all the particulars ; 7/ yap
ciraycoyj] 01a ti&vtmv.1

We have now seen what Induction, as a formal process, meant
in the mouth of the first author who used the term ; and when
Aristotle insisted that it must proceed through all the particulars,
or (as it was afterwards put) by complete enumeration — the require-
ment which, to Bacon and the ' inductive logicians * of modern
times, has given so much offence — he was quite right ; for if you
are going to establish a general proposition that way, you will
clearly not be justified in making it general unless you have made
sure that your enumeration of the particulars is complete ; though,
as has been said, it is not really an universal proposition then, but
only • enumerative ' : a thing which Aristotle fails to point out.
The burden of the charge against Aristotle is, however, not that
he held that, if a general proposition is to be established by enumera-
tion of particulars, the enumeration must be complete : but that
he recognized no other mode of establishing general propositions.
And if this be so, then his Logic falls to pieces. For syllogism needs
a general proposition for its major premiss ; and as Aristotle himself
insists, we cannot be said to know the truth of the conclusion,

1 ' For induction is by means of all ' : Anal Pri. j3. xxiv. 68b 15-29.

xviii] OF INDUCTION 381

unless we know first the truth of the premisses * ; doubt of that will
involve doubt of what is stated in the conclusion, so far as this is
arrived at by inference, and not by direct experience independently
of the inference. Now how can this condition be fulfilled, if our
knowledge of any general principle rests on nothing better than an
enumerative assurance that it holds good in every particular case ?
Let us take the principle that all matter gravitates, and symbolize
it in the form ' All M is O '. If it is possible to know this without
experience of its truth in every parcel of matter, one may use it in
order to prove that this book must gravitate; and therefore may
refrain from adding the book to one's kit in going up a mountain,
or laying it upon a flower that is for show, or on the other hand may
use it to keep one's papers steady in a wind. But if the principle
can really only rest upon a complete enumeration, we must experi-
ment with this book, before we can assert it ; and then we shall know
that this book gravitates by direct experiment, and our deduction
thereof from the general principle will be superfluous, even if the
enumeration be complete — as it would only be, if there were no
other parcels of matter left to be experimented with ; but even so,
the deduction would be but a hollow show, and begging of the
question. For let us symbolize any particular parcel of matter by /z.
We propose to prove that ^ is G, because all M is G, and ^ is M ;
how do we know that all M is G ? Only because fxv fx2, &c. up to fxn
are G, and \xv \x% . . . fxn are all the M, and therefore all M is G.
Hence we use the fact that \x is G to prove the principle by which we
prove that \x is G. And the upshot of this is that we can never prove
anything by reasoning, until we already know it by direct experi-
ence ; so that the use of reasoning, in order to infer that which we
have not learnt by direct experience, must disappear. If we still
try, by appeal to any general principle, to prove anything which
we do not already know, we shall be appealing to a general principle
which we do not know to be true, in order to prove a particular
conclusion which we do not know to be true ; for ex hypothesi our
knowledge of the truth of the general principle depends upon the
knowledge of what occurs in the particular case in question among
others. Such a procedure hardly commends itself to a sane man.
And if again it were said, that however little we may be logically
justified, in advance of experience, in drawing inferences about
some particular from a general principle, yet our experience when it

1 An. Post. a. ii. 72a 25-b4.

382 AN INTRODUCTION TO LOGIC [chap.

comes is constantly confirming the inferences we thus draw, this,
far from being a solution of the logical difficulty in which we have
found ourselves, ought only to be matter of perpetual astonishment,
to a creature that reflects at all upon his experience.

Such is the difficulty that arises, if there is no other means of
proving a general proposition than by enumeration of all the parti-
culars to which it refers * ; and to this criticism Aristotle is obnoxious,
if he recognized no other means. But did he recognize no other ?

Now Aristotle undoubtedly says that we arrive at our first prin-
ciples by a process of Induction.2 He draws a famous distinction
between the order of nature and the order of experience 3 ; in the
order of nature, the general principle is prior to the sensible fact ; in
the order of experience, it is the reverse. To us, the particulars
of sense are known first : the intelligible principles by which these
are explained are known afterwards ; but Nature may be conceived
as starting with principles or laws, and with these in her mind
proceeding to the production of particular objects or events. In-
duction proceeds from what is first in the order of experience to what
is first in the order of nature : from the apprehension of the sensible
facts to the apprehension of the general principles, out of which we
subsequently construct the sciences. Without sense-experience,
there is no knowledge of intelligible principles ; and the process of
obtaining that knowledge out of sense-experience is Induction.

And this, taken together with his analysis of the Inductive
Syllogism, might seem to settle the question of how Aristotle con-
ceived that we come to know general propositions ; if only we could
suppose him capable of overlooking the difficulty in which his
whole system would thereby have been involved. But so far from
overlooking, he shows in one passage that he had considered it, and
uses his distinction between what is prior in nature, and prior in our
experience, in meeting it.4 His view seems to have been this.

The business of any science is to demonstrate the properties of
a kind — such kinds, for example, as geometrical figures, species of
animals or plants, or the heavenly bodies. As we saw in the chapter
on the Predicables, he was influenced much by the fact that geo-
metry and biology were the two most progressive sciences of his
day. Science is concerned with kinds, as what are identical in their

1 Cf. what was said in discussing the Dictum de omni et nullo, pp. 301 sq.

8 See e. g. An. Post. |3. xix. 100b 4.

8 <f>v<Tti nporepov and f]p.lv irportpov : cf. p. 88, supra.

* An. Post. a. iii.

xvin] OF INDUCTION 383

many members, and eternal. In demonstrating their properties, it
starts from a knowledge of their definitions ; such definitions cannot
themselves be demonstrated ; and for them we are dependent on
experience, which familiarizes us with the nature of any kind, or of
its properties, by means of particular cases. But though experience
may thus acquaint us with the definition of anything, yet the essen-
tial nature of a thing (which is what a definition gives) cannot
possibly be an empirical fact. It may be an empirical fact that all
sailors are superstitious ; but how can it be an empirical fact that
a triangle is a three-sided rectilinear figure ? For to say that any-
thing is an empirical fact implies that it might (so far as we can see)
have been otherwise ; and certainly we can conceive that a sailor
may be either superstitious or not superstitious ; but we cannot
conceive that a triangle should not be a three-sided rectilinear figure,
since if that — which is its essence — were removed, there would be no
triangle left to be anything else. It will be asked, how do you
know what constitutes the essence of anything ? The answer is,
that the intellect sees it : sees it, as we might say, intuitively, as
something necessary ; and this is the source of our assurance, in
virtue of which we know the principles from which our demonstra-
tion proceeds more securely even than the conclusions we draw
from them. But the intellect does not perceive it at once ; experi-
ence of things of the kind is necessary before we can define the kind.
The use of these particulars is, not to serve as the proof of a principle,
but to reveal it : as the counters, for example, which a child uses
in learning the multiplication table, though one among innumerable
instances of the fact that three times three is nine, are to be appealed
to not because the general proposition could not be asserted unless
it were tried and found true in the case of these counters as well as
of all other countable things : for had the child learned with nuts,
it would have been quite unnecessary to confirm the generalization
by an examination of the counters ; but because they serve as
a material in which the child can be brought to realize the truth of
a numerical relation, which it apprehends forthwith with a generality
that goes far beyond these particular counters. They are a means
used because some countable material is necessary in order to realize
the general truth ; but the general truth is not accepted simply
because it is confirmed empirically by every instance.

Now we need not ask at the moment whether the sort of intel-
lectual insight with which we do apprehend the necessity of numerical

384 AN INTRODUCTION TO LOGIC [chap.

or spatial relations * can really serve us in determining the essence
of gold or of an elephant or a tortoise ; our present purpose is only
with the nature of Induction, and the different senses in which the
term has been used. And the purpose of the preceding paragraph
is to show that in spite of the analysis which Aristotle gave of
Induction as a logical process, yet when he said that we get our first
principles by induction, he had something else in his mind. Where
your units are species, and you want to prove something about the
genus to which they belong, there you may proceed by appealing
to the fact, that it is found true of every species in the genus ; there
your reasoning may be thrown into the form of the ' inductive
syllogism ', — which is inconclusive unless every species is included
in the premisses. But even there, from the fact that he regarded
the conclusion as an universal and not merely an enumerative pro-
position, we must suppose Aristotle to have thought that the mind
grasped a necessity in that relation between the terms of the con-
clusion, at which it arrived by a process of enumeration ; directly
or indirectly, the connexion of longevity with gall-lessness was to be
seen to be necessary, and freed from the appeal to all the species.
And where your units are individuals, and you want to discover the
essential nature of the species to which they belong, there you do
not work by an inductive syllogism that summons all the instances
to bear witness to the truth of your definition ; for how could you
summon the numberless members of a species ? There is still a use
for experience ; we may still say that we know these things by
induction ; but the induction now is a psychological rather than a
logical process ; we know that our conclusion is true, not in virtue
of the validity of any inductive syllogism, drawing an universal
conclusion in the third figure because the subject of the conclusion
is coextensive with the particulars, taken collectively, by means of

1 There are philosophers who would not agree with what has been said of
the nature and grounds of our assurance of the truth of mathematical prin-
ciples. Some hold that they are only generalizations from experience,
deriving their high degree of certitude from the great number and variety
of the instances in which they have been found to be true. This doctrine
is maintained by J. S. Mill in a well-known passage of his System of Logic,
Bk. II. cc. v-vii, to which he refers in his Autobiography (p. 226) as a crucial
test of his general philosophical position. For a partial examination of the
passage, crushing so far as it goes, see Jevons's Pure Logic and other Minor
Works, pp. 204-221. Others again hold that at any rate geometrical axioms
are only the simplest and most convenient assumptions that fit the facts of
our experience : v. H. Poincare, La Science et VHypothese, c. iii. ad fin. ' Do
la nature des axiomes ', pp. 64-67.

xvin] OF INDUCTION 385

which we prove it : but in virtue of that apprehension of the neces-
sary relation between the two terms, which our familiarity with
particulars makes possible, but which is the work of intellect
or vovs.

Such seems to have been Aristotle's doctrine : and thus he
avoided the bankruptcy that would have ensued, had he taught
that all syllogism rested on universal propositions, and that universal
propositions rested on nothing but showing by enumeration that
they held true in every particular instance that could be brought
under them. But it may be said that thus he only avoids the
Charybdis of moving in a logical circle to be snatched up by the
Scylla of an arbitrary assumption. We are to accept the general
propositions upon which every subsequent step of our inference rests,
because our intellect assures us of their truth. This may satisfy
the man whose intellect gives him the assurance ; but how is he to
communicate that assurance to others ? If a principle is not arrived
at from premisses which another admits, and between which and it
he sees a valid process of inference to lie, why should he accept that
principle ? No evidence is offered, whose sufficiency could be tested.
The ipse dixit of an incommunicable intuition takes the place of any
process of reasoning, as the means whereby we are to establish the
most important of all judgements — the general propositions on
which the sciences rest.

Of this charge Aristotle cannot altogether be acquitted ; yet we
may say this much in his defence. Such an intellectual apprehension
of the necessary truth of the principles from which demonstration
is to start forms part of our ideal of knowledge * ; doubtless it seldom
enough forms part of the actuality. But Aristotle idealized ; he
spoke of what, as he conceived, science in the fullest sense of the
term involved, and forgot to state, or failed to see, that the sciences
did not attain it. And the prominence which he gave to the ques-
tion ' What sort of premisses does knowledge require ? ' led him to

1 With this proviso, that for perfect knowledge all the parts of truth ought
to seem mutually to involve each other. In mathematics, where alone we
seem to achieve this insight into the necessity of the relations between the
parts of a systematic body of truth, we find our theorems reciprocally demon-
strable ; and if twice two could be three, the whole system of numerical
relations would be revolutionized. Yet we do not need to wait till we discover
how all other numerical relations are bound up with the truth that twice two
is four, before we are as fully convinced of this truth as we are capable of
becoming. Whether in every science we should desire that each principle
should thus be apprehended as necessarily true, even when cut off from ita
implications, may be doubted.

1779 0 0

386 AN INTRODUCTION TO LOGIC [chap.

relegate to an inferior position the question ' How can the sciences
as they are validate their premisses ? '

He did not overlook this last question altogether ; indeed he
devotes to it a considerable portion of the longest of his logical
treatises, the Topics ; for when he asks by what sort of considera-
tions you can prove or disprove that a proposition gives in its predi-
cate the definition, or a property, of its subject, he is asking how
you can prove scientific first principles.1 And he knew this ; and
among the uses of Dialectic, or of the disputation whose methods he
elaborates in the Topics, he places as its most proper use the examina-
tion of the truth of scientific principles.2 But he ought to have
seen that, outside mathematics, we seldom have any other means
of establishing general propositions upon the evidence of particular
facts than those of the kind which he discusses in the Topics. For
the rest, his aocount of the logic of the reasoning by which the
sciences do as a matter of fact support the general principles which
they accept contains hints which are in advance of much modern
' inductive logic ' ; though there is much in his conception of the
character of the general principles which science seeks to establish,
that is now antiquated. Science seeks to-day to establish for the
most part what are called ' laws of nature ' 3 ; and these are generally
answers rather to the question ' Under what conditions does such
and such a change take place ? ' or ' What are the most general
principles exemplified in such and such a change ? ' than to the
question ' What is the definition of such and such a subject ? ' or
' What are its essential attributes ? ' 4 It is more in respect of the

1 Though Aristotle does not mention among the premisses of demonstration
propositions giving the properties of kinds, and says that it is the business of
science to prove these, yet he allows incidentally that some such propositions
are indemonstrable (cf. Anal. Post. a. iv. 73a 37-b3) — e. g. that a line is
straight or curved, a number odd or even. In point of fact, as Professor Cook
Wilson has pointed out, the sciences in such cases assume that the genus
displays these alternative properties, but prove which it is that belongs to
some species of the genus.

2 Cf. Top. a. ii. 101 a 34-b4.

3 The term ' Law of nature ' is used especially of these most general prin-
ciples, though sometimes of derivative principles as well. Cf. J. S. Mill,
System of Logic, III. iv. 1, where Laws of Nature in the strict sense are said
to be ' the fewest and simplest assumptions, which being granted, the whole
existing order of nature would result '.

4 I think this contrast is substantially true ; though it is possible to bring
many scientific investigations to-day under one or other of the types of
question which Aristotle says we enquire into, yet looking to his examples,
one must confess that (as is natural) he put the problems of science to himself
in a very different manner from that in which scientific men put them now.
Cf. An. Post. /3. i. S9^ 23 ra (rjTOVfievd iartv Xaa tov apidfxov ovanep eVicrrd/je&j.

xviii] OF INDUCTION 387

problems to be answered, than of the logical character of the
reasoning by which we must prove our answers to them, that
Aristotle's views (as represented in the Topics) are antiquated.

We may briefly indicate the nature of ' dialectical ' reasoning, as
Aristotle conceived it, and of the ' topics ' which it employed. Dia-
lectic is contrasted with science. Every science has its own peculiar
subject-matter : geometry investigates the nature and properties
of lines, surfaces, and figures in space, geology the conditions which
determine the character and distribution of the materials which form
the crust of the earth, physiology the functions of the organs and
tissues of living bodies, &c. Each science, in explaining the facts
of its own department, appeals to special principles, or iSiai apxat>
to the specific nature of its own, and not another, subject-matter — to
laws in accordance with which that particular class of facts is deter-
mined, and not another class. The geometrician makes use of the
axiom of parallels, of the notion of a straight line, of the definition
of a cone or circle ; but the nature of chalk or granite is indifferent
to him. The geologist will use such principles as that stratified
rocks are sedimentary, or that mountains are reduced by denudation ;
but he draws no conclusions from the definition of a cone. The
physiologist in turn has his own problems to explain, and his own
principles to explain them ; that every tissue is composed of cells
which multiply by division is a physiological principle of which we
hear nothing in geology, while the laws of denudation contribute
nothing towards the explanation of the growth of living bodies.1
Dialectic, on the contrary, has no peculiar subject-matter ; all the
sciences submit their principles to its investigation ; the dialectician
may ask whether a geometer would be right in saying that it is
a property of a rectilinear triangle to have its exterior angles equal
to four right angles : whether the geologist has rightly affirmed all

^T]Tovjj.fv be Ttrrapa, to oti, to SuWi, el eort, ri ianv. (' The subjects of
investigation are equal in number to the subjects of knowledge : and we
investigate questions of four sorts — facts, their reasons, whether something
exists, what it is.')

1 One science does often to some extent use the results of another. In
particular, of course, all the other sciences resolve all they can into terms
of chemistry and physics. Yet looking (say) to Physics, Chemistry, Physio-
logy, and Political Economy, no one will deny that they must continue to
rest each in part on different principles, even if the later mentioned may
have to take note of some facts whose explanation involves the principles of
the earlier mentioned. Aristotle noted such partial use by one science of
the results of another ; though the state of the sciences in his day prevented
him from illustrating it as it would be illustrated now, and his remarks on
the subject are open to a good deal of criticism. Cf. An. Post. a. xiii. 78b
32-79a 16.

C02

388 AN INTRODUCTION TO LOGIC [chap.

stratified rocks to be sedimentary : whether the physiologist would
do well to accept Spencer's definition of life, as ' the continuous
adjustment of inner to outer relations '. And in debating such
questions, the dialectician will invoke not special, but common
principles, kolvo.1 apxai1 — i.e. not principles whose application is
confined to the science he happens to be investigating, but principles
of universal application : as, for example, that what is common to
the genus is not a property of the species — whence it follows, that
since all plane rectilinear figures have their exterior angles equal to
four right angles, this is not a property of a rectilinear triangle, or in
other words, that it is because a plane figure is rectilinear, and not
because it is three-sided, that this can be predicated of it ; it is for
the geometer to show that all plane rectilinear figures have their
exterior angles equal to four right angles ; the dialectician's business
is to show that therefore it cannot be called a property of a triangle,
as such. Or again, the dialectician may ask, with regard to Spencer's
definition of life, whether the distinction between ' inner ' and ' outer ',
on which it rests, is clear ; for he knows that the terms of a definition
should be clear, though he does not necessarily know physiology ;
and if Spencer, or his disciples, could not show precisely what it
means, he would say the definition must be faulty ; and if they
replied that ' inner ' meant within the organism, and ' outer ' outside
it, he would ask whether all material systems which change inwardly
in response to changes outside them are living bodies ; for he knows
that a definition should not apply to anything except the species
defined, and if this expression does, it cannot be a definition ; or he
might ask whether many of the peculiar processes of living bodies
are not apparently initiated from within the body ; and if the answer

1 Cf. Anal. Post. a. x. 76b 11-22, xi. 77a 26-34, xxxii. 88a 31-b3, b9-29.
In the second of these passages, Aristotle gives as examples of ' common
principles ' the Law of Contradiction, that the same proposition cannot be
at once true and false, and the mathematical axiom that the differences
between equals are equal. The latter is not really ' common ', but special
to the sciences of quantity ; and if he wished to be consistent with what he
says in 0. xvii. 99a 6-16, Aristotle should have allowed that it means some-
thing a little different in geometry and in arithmetic. By no means all of
the communes loci in the treatise called the Topics are ' common principles '
— e. g. the topics given in y, n(p\ tov alp€Ta)T('pov, which are principles to be
appealed to in determining which of two goods is to be preferred : as, that
the more lasting good is preferable, or the more secure, or the greater, or
the nearer. Most of them however are such, though it must be admitted
that Aristotle does not describe his topics as common principles, or koiv<u
Ap\ai: and I think that the distinction which he intends to convey in the
Posterior Analytics by the antithesis of t8iai and Koivai apxai is really what
has been stated in the text.

xviii] OF INDUCTION 389

was affirmative, he would again object to the definition ; for though
it is not his business to know whether any of the peculiar processes
of living bodies are initiated from within or not (and therefore he has
to ask the physiologist how that matter stands) it is his business to
know that a definition must include everything essential to the thing
defined ; so that if there are such processes, a definition of life which
excludes them must be a wrong one. Or, lastly, the dialectician
might ask the geologist if there are not some igneous rocks that are
stratified : not knowing, as a dialectician, the answer to that question,
but knowing that, since igneous rocks are not sedimentary, the exis-
tence of igneous rocks that are stratified would upset the geologists'
proposition ; while if the geologist were able to answer the question
in the negative, he would so far have come out victorious under
examination.

All these general principles, to which the dialectician appeals,
are called topics 1 : it is a topic, that what belongs to the genus is
not a property of the species ; or that what in some particular
instance is absent from a species is not a property of it ; or that
the terms of a definition must be precise, or that it must be com-
mensurate with what is defined. All these principles hold good in
any science ; it matters nothing what the species may be, or what
the property, or what the definition. A man therefore whose mind
is stocked with principles of this kind has points of vantage, as it
were, from which he may proceed to attack or defend any definition,
any predication of a property ; they are topics in common, ' common-
places', considerations to which you may turn in examining the
statements of any science. Just as a man who knows nothing of
the truth of its premisses may be able to detect a flaw in a syllo-
gism, so the dialectician, without a scientific knowledge of a subject,
may know what sort of questions to ask, if he wishes to test a
scientific man's right to affirm the principles he enunciates.

Aristotle's Topics is written with reference to his doctrine of
Predicables. He regards every proposition as asserting (or denying)
some accident, property, differentia, genus or definition, of its subject ;
and he asks, to what considerations are you to look, if you would
know whether such and such a predicate does stand to such and
such a subject in any one or other of these relations ? Each of these
considerations is a topic. He details an astonishing number of
them. They are of very different degrees of importance and value.
Some are drawn from language. Look, he says, for example, to

1 to770(, loci, communes loci.

390 AN INTRODUCTION TO LOGIC [chap.

conjugate words— the various words, that is, from a common stem ;
if noble is a property of just, then justly is nobly ; perhaps a man
who affirmed generally that justice is noble might admit that it is
possible in some cases to act justly and not nobly.1 Others are
based on the principle that contrary things have contrary properties ;
so that you cannot say that the just is the equal, unless you can
say that the unjust is the unequal. Some aim only at enabling you
to determine whether an expression is elegant, according to accepted
rules. But others are principles of great importance. For instance,
there is what we might call the topic of Concomitant Variation 2 ;
that is not a property of a subject which does not increase or decrease
with an increase or decrease in the subject, and conversely, if you
find two things increasing and decreasing together you may assert
such connexion between them.3 Considerations of this kind enable
you to judge how different concepts are related to one another ; and
relations between concepts furnish the principles with which the
special sciences work.

It may be admitted that this treatise contains much that is
trivial ; that it throws together considerations, or principles, of
great and of little cogency ; that the problems of science assume
other forms than determining the definition of a subject, its pro-
perties, or its accidents (although such problems occur too, and
many problems which we should not express in those forms can be
translated into terms of them). It may also be admitted that
Aristotle had his mind fixed too exclusively upon debate. The
answers to the questions asked were to come from the respondent —
the other disputant ; but in building up the sciences, they must
come from the field and from the laboratory. Aristotle would have
a man test any scientific doctrine that is put forward by interro-
gating its maintainer ; the man of science must test those which he
himself or a fellow worker puts forward by interrogating nature.
It would be easy to do Aristotle an injustice on this head. It may
be assumed after all that the respondent testifies to what he has
seen ; and Aristotle was alive to the importance of collecting and
recording facts.4 But the Topics is a treatise on the art of disputa-
tion ; disputation aims after all more at silencing an opponent
than at establishing truth ; and though we are told that Dialectic
has its use as much in the examination of the principles of the

1 Cf. Top. (. vii. 136b 15. 2 tokos €K rov fiaXkov koi tjttov,

8 e. g. Top. (. viii. * Anal. Pri. a. xxx.

xvin] OF INDUCTION 391

sciences as in the conduct of a disputation, it is in the latter spirit
that it is expounded. Nevertheless, in the distinction drawn
between scientific and dialectical reasoning, as illustrated above,
and in its account of the general nature of the considerations to
which one must appeal in any defence of the principles of a science,
the Topics is a work of great logical value.

What, then, has Aristotle to say about Induction ?

1. He gives the name to a formal process of inference, by which
we conclude a proposition to hold universally of some class, or
logical whole, because an enumeration shows it to hold of every
part of that whole. This is what has been since called Induction by
Complete Enumeration, or Perfect Induction ; and he shows how it
might be thrown into the form of an Inductive Syllogism.

2. He points out that our knowledge of scientific principles
springs historically out of our experience of particular facts ; though
its certainty rests ultimately upon an act of intellectual insight.
And he gives the name of Induction to the process in which the
particulars of our experience suggest to us the principles which they
exemplify. But this is not a formal logical process from premisses
to conclusion ; and it is not the induction (in this sense) which leads
us at the end to accept such principles, but our intellect, or vovs.

3. He shows where (presumably in default of the necessary
insight and assurance from our intellect) we may look for reasons
for accepting or rejecting any principles which a science puts forward.
He does not give to this procedure, which is of a formal logical
kind, the name of Induction, but calls it Dialectic ; nevertheless
what he says on this head is of much the most importance from the
point of view of scientific method, and comes much closer to what
modern writers understand by Induction.

Thus he admitted that our knowledge of general principles comes
from our experience of particular facts, and said that we arrive at
them by Induction ; but the only formal logical process which he
described under the name of Induction was that ' Perfect Induction '
which clearly neither is nor can be the process by which the sciences
establish general propositions ; while the kinds of process which
they really do employ, so far as they appeal merely to the evidence
of our experience, he described under a different name. It is not
surprising that some confusion has resulted.

The critics of whom Bacon is the coryphaeus, recognizing with
Aristotle that we discover universal truths by induction, attacked

392 AN INTRODUCTION TO LOGIC [chap.

him for saying that we only discover them by complete enumeration,
which he had not said ; and finding the name Induction given
to no other formally valid process than this1, supposed he had
nothing else to say of the processes by which such truths are reached.
Bacon himself attempted to systematize the process of discovering
and proving them in a way which undoubtedly possesses value, and
no less undoubtedly owes much to Aristotle ; but as the Aristotelian
doctrines on which it is based do not occur in the Organon in con-
nexion with e-naycoyri, he hardly realized how much he was borrowing.
His analysis is offered in connexion with an unworkable theory of
the nature of the problems which science should set itself to solve.
To put it summarily, he thought that a list of the several sensible
properties of bodies should be drawn up, and that men should then
try to discover on what particular principle of corpuscular structure
in the bodies exhibiting it each property depended. There was
nothing in any particular principle of structure which would lead
you to anticipate that its presence would involve any one sensible
property more than another ; you could not tell, apart from experi-
ence, that a particular motion of the component particles of a body
would exhibit itself to the senses as heat, or that a particular
disposition of its surface particles would show as white, and another
particular disposition as black. Suppose we were to symbolize
the sensible properties of bodies by Greek letters, and the principles
of corpuscular structure in them on which these depend by Roman
letters : how are you to prove whether a property a is connected
with a or 6 or z ? Bacon's answer is as follows. He called the
principles of corpuscular structure Forms : whatever be the Form
of a given property a, it must be so related to a as to be present
in every body in which a is present, to be absent from every body
whence a is absent, and to increase or decrease in any body as a
increases or decreases. Our problem then is, as he says, ut inve-
niatur natura alia (the Form) quae cum natura data (the sensible
property) perpetuo adsit, absit, crescat atque decrescat.2 How are we
to solve it ? No mere enumeration of instances in which a sensible
property a and a Form a are present together will prove that they
are thus related, and that a is the Form of a ; for your enumeration
must be finite, but your conclusion is to be universal. You may

1 It was also given to Induction by simple enumeration — i. e. to any attempt
to prove a general proposition by merely citing a number of instances of its
truth ; but this is not a formally valid process.

2 Nov. Org. II. 4.

xviii] OF INDUCTION 393

find a hundred bodies exhibiting both a and a : yet the presence of
one may be quite unconnected with the presence of the other, and
you may find a body to-morrow exhibiting one without the other.
We must proceed then by exclusions. Where a hundred instances
will not prove an universal connexion, one will disprove it. This
is the corner-stone of his method : maior est vis instantiae negativae.1
If we had drawn up an exhaustive list of the different principles of
corpuscular structure present in bodies in different combinations, all
we should have to do would be to find instances in which any of
these was present in a body that did not exhibit the property a,
or absent in one that did exhibit it, or in which it increased or
decreased without a corresponding variation in the degree of the
property, or vice versa. We could then confidently reject that Form ;
and when we had thus rejected every other Form, then we could con-
fidently affirm that principle of corpuscular structure which alone had
not been rejected to be the Form (or cause of the presence) of a given
sensible property a. Our assurance would rest not on the positive
testimony of its presence along with a in a number of instances,
but upon the fact that we had disproved all possible rival theories.

It will be seen that this procedure presupposes that we know all
the possible Forms, among which that of any particular sensible
property is to be sought ; and Bacon, though he promised to do so,
never showed, and could not have shown, how we were to secure
that. The procedure is formulated too under the belief, that the
immediate task of science is to draw up a complete list of all the
distinct sensible properties found in nature, and then look for what
we should perhaps now call their physical basis. This belief was
mistaken. But the fundamental principle of the method by which
Bacon proposed to ' interpret nature ', the principle on account of
which he gave it the name by which he called it, Exclusiva, is
correct ; it is that where you cannot (as in Mathematics) see that
a proposition must universally be true, but have to rely for the proof
of it on the facts of your experience, there there is no other way of
establishing it than by showing that facts disprove its rivals.2

Bacon called this method inductive ; it may be as well to point

1 Nov. Org. I. 46. Cf. Aristotle, Anal. Pri. a. xxvi. 43a 14 a^a 8e ^\ov on
icat to arao-Ktvd((tv ear) rov KnTuo-Kfva(fti> paov ('And it is plain at the same
time that it is easier to refute than to establish ') : and more fully, Top. rj. v.

2 There are many very valuable remarks in Bacon's account of his ' Ex-
clusiva * about the kinds of instances which are of most evidential value (and
which he therefore calls Prerogative Instances) ; but a discussion of them would
hardly be relevant to the present argument.

394 AN INTRODUCTION TO LOGIC [chap.

out at once that formally the reasoning involved is just that of
a disjunctive argument, with hypothetical argument employed in
the disproof of rejected alternatives. The alternative hypotheses
(with Bacon, the alternative hypotheses as to the Form or physical
basis of a particular sensible property) are so and so : such and such
of them are false, for, if they were true, the facts would be other than
we find them ; therefore the one remaining is true. How we are
to discover what the alternative hypotheses are, he does not explain
to us ; we are to prove that the rest are false by appeal to the facts
of our experience ; these facts he would have men methodically
collect and tabulate, and in making use of them he relies upon the
general principle that nothing can be the Form sought for which is
ever present in the absence of the property whose Form it is alleged
to be, or absent in its presence, or variable when it is constant, or
constant when it varies ; when he has got his premisses, his con-
clusion follows according to the ordinary principles of disjunctive
reasoning.

Bacon wrote in the dawn of modern science, and proclaimed with
splendid confidence its future triumphs. His predictions have been
fulfilled, perhaps to the extent, though not on the lines, that he
anticipated. Spes est una, he wrote, in inductione vera x ; and as
men watched the continuous progress of the inductive sciences, they
came to think that induction was really some new form of reasoning,
ignorantly or perversely rejected by our forefathers in favour of
the deductive reasoning, which they associated with the name of
Aristotle, and now held to be in comparison an idle thing. To
praise induction became a sign of enlightenment ; but the praise of it
ran ahead of the understanding.

Those who did most to advance the sciences had not the need
or inclination to pause and analyse the arguments which they were
so successfully building up ; nor would it imply any disrespect to
add, that many of them probably had not the power of doing so.
It is no more necessary that a great scientific genius should be able
to give a correct account of the methods he uses than that a great
artist should be able to expound the philosophy of art ; those can
often do things best who are quite unable to explain how they do
them. The chief scientific name in the history of speculation upon
the logic of the inductive sciences in this country is that of Sir John
Herschell ; four writers in all, if we exclude those still living, have

1 Nov. Org. I. 14.

xviii] OF INDUCTION 395

made the principal contributions to the subject. David Hume,
in a brief section of his Treatise concerning Human Nature (Of the
Understanding, Part III, Sect, xv), gives ' Rules whereby to judge
of causes and effects ' which contain the pith of much subsequent
writing ; but the work, as he said himself, ' fell stillborn from the
press ' ; this section was not incorporated in the later and more
popular ' Enquiry ' ; and it had no influence on the exposition of
Induction. Sir John Herschell's Discourse concerning the Study of
Natural Philosophy and the various works of Dr. Whewell did, on
the other hand, much to stimulate interest in the subject ; especially
since Whewell propounded an explicit theory of it. The help
which he had derived from both is acknowledged by J. S. Mill,
whose System of Logic for many years held the field as an exposition
of inductive reasoning. To that more than to any other work is to
be traced the prevalence of the opinion, that inductive reasoning,
or Inductive Logic as the theory of it, is a discovery of the moderns
— an opinion which certainly contains less truth than falsehood.
The name induction may be said with him to have stood for more
than a particular form of inference ; it was the battle-cry of a philo-
sophical school, the school, as it is called, of experience. But as a
result of this, and of its previous history, it has become one of the
most confusing terms in Logic. It stands firstly for that induction
by complete enumeration which Mill denies to be properly induction
at all, but from which his influence was unable to withdraw the
name after the prescription of so many centuries. It stands secondly
for the logical processes employed in the inductive sciences, so far
as these infer from particular facts the principles that explain them ;
as to what the nature of these logical processes is, Mill had a theory
different from Whewell's, and others have since had theories different
from Mill's. Thirdly, Mill, who admits that there are certain
common principles assumed as true in the reasonings of the inductive
sciences, gives the name to what he conceives to be the logical
process by which these principles themselves are reached : a process
that, in his view, starts barely from a great number of particular
facts, and without the help of any general principles at all bases upon
these facts the general principles whereon all other inductive infer-
ence rests. Many of Mill's critics have thought, and have thought
rightly, that if the process by which these principles are reached
were as he describes it, it could only be called an illogical process.1
1 The second part of Jevons's Principles of Science ought perhaps to have

396 AN INTRODUCTION TO LOGIC [chap.

It would have been possible to omit the foregoing historical
sketch, and to offer a purely dogmatic account of what Induction is,
and what it is not. But against such a course there were two
reasons. In the first place, a new writer has no right to do such
a thing. It is indeed necessary for him to put forward that account
of the nature of the reasoning of the inductive sciences, which he
believes to be true ; but not as if he was only delivering an accepted
tradition. And in the second place, unless the reader knows some-
thing of its history, he can hardly fail to be confused by the diversity
of senses in which he finds the word Induction used. Men have
rightly felt that an antithesis could be drawn between the inductive
and the deductive sciences ; though they can be classed only
according to their predominant character, since no sciences, except
the mathematical, are exclusively the one or the other. On the
strength of this they have most unfortunately erected an antithesis
between Inductive and Deductive Logic : unfortunately, partly
because Logic is one ; the science which studies the nature of
our thought embraces equally the processes of thought that enter
into the construction of the deductive sciences and of the inductive :
but unfortunately also, because it has led to much misunderstanding
of the nature of inductive reasoning itself. What ' inductive logi-
cians ' have called Deductive Logic, contrasting their own Inductive
Logic with it, expounded forms of argument that belong to their
most typically inductive enquiries. Their ' inductive methods ',
as has been said above, and as will appear more fully by and by,
are, so far as the argumentation is concerned, but a mixture of
hypothetical and disjunctive reasoning ; and these forms were
supposed to be deductive. Nothing but confusion can result from
grouping together all the processes traditionally called deductive,
and opposing to them collectively those of inductive science. If
any clear antithesis can be drawn between Deduction and Induction,
we must not identify them with the forms of argument expounded
respectively by Deductive and Inductive Logic. The names are
perhaps unfortunate. Things may be different without being

been included along with the four works mentioned above (cf. also Lotze's
Logic, E. T., Bk. II. c. 7). Among contributions on the part of living writers to
the criticism of Mill's doctrines (for the great acceptance which his views
obtained has made criticism of him a prominent feature of much subsequent
writing on Induction) may be mentioned Mr. F. H. Bradley's Principles of
Logic, Bk. II. Part ii. cc. 2 and 3, and an excellent discussion in Professor
Welton's Manual of Logic, vol. ii. § 155.

xvin] OF INDUCTION 397

contrary ; but a difference indicated by terms formed from the
same stem with different prefixes is apt to be thought a sort of con-
trariety. Hence we incline to think of Deduction and Induction as
processes moving between the same points, but in opposite direc-
tions ; Deduction, we think, argues from general principles to
particular facts, Induction from particular facts to general principles.
Even if this were true, such a statement tells us nothing of the
difference in the nature of the reasoning between the two cases ;
and in point of fact, though there are arguments of those two kinds,
the distinction is by no means the most important that can be drawn,
does not coincide with the distinction between the arguments
traditionally assigned to Deductive and Inductive Logic respectively,
and leaves out some of the operations of reasoning that best deserve
to be called scientific.

In the inductive sciences we do argue from particular facts to the
principles displayed in them ; in subsumptive syllogism we may
draw conclusions about particular facts from the principles which
we have inductively discovered. Thus a study of certain facts
leads us to conclude that the air exerts a definite pressure upon the
surface of bodies exposed to it, and from this we may deduce that
if we pour mercury into an open glass bulb, it will exert that pres-
sure on the surface of the mercury. This was part of the reasoning
which led Torricelli to the construction of his barometer. So far,
there is a contrariety and an antithesis. But if we look beyond this
simple statement, and compare the structure of a syllogism with the
structure of the reasoning by which the principle that the air exerts
this pressure on the surface of bodies exposed to it is reached, we
shall see a difference, but not a contrariety. Moreover, though
propositions about particular facts are included among the premisses
of inductive argument, it appeals to universal principles as well ;
only these universal principles do not, like those of the syllogistic
reasoning contrasted with it, tell us between what determinate
characters in things universal relations hold, but rather what kinds
of relations between determinate characters of things a causal
system involves. Induction and deduction then cannot be con-
trasted in respect that the premisses of the one are propositions
about particular facts, of the other universal ; for some universal
principles are included among the premisses of both ; the difference
is in the nature of these. Nor again can they be contrasted in
respect that the premisses of the one and the conclusions of the

398 AN INTRODUCTION TO LOGIC [chap.

other are propositions about particular facts ; for though sub-
sumptive syllogism may apply the rule in the major premiss to
determine a conclusion about some individual thing, it may also
(as when used in geometry) apply it to determine an universal
conclusion.

But syllogistic reasoning is not confined to the deductive sciences,
being common in the inductive sciences as well ; nor is subsumptive
syllogism the only reasoning traditionally called deductive. Be-
tween the use of general principles to determine conclusions about
particular things in which they are displayed, and the use of an
examination of particular things to determine conclusions about
the general principles displayed in them, there is indeed a con-
trast. But it is not the most important contrast distinguishing
deductive and inductive reasoning. The most important contrast
is that which Aristotle intended to signalize when he opposed
Demonstration to Dialectic. In demonstrative reasoning we
have a real insight into the connexions of things. Where
this is possible, though Aristotle thought that we used syllo-
gism, yet, as we have seen, there is not really any subsumption.
The conclusion need not be less general than the premisses ;
there need be no application of a rule invoked ab extra ; the
connexions may be traced in an individual subject, though
between characters in it that are universal.1 But we may use
premisses that state connexions which we do not see to be necessary
between certain characters in things ; still, taking these as true,
if combining them together we can see what consequences their
truth involves in some actual or imagined complex of things, we so
far have insight into facts. Explanation, therefore, though the
premisses from which it proceeds are often not seen to be necessary,
is yet to be ranked with demonstration in respect of the nature of
the reasoning. On the other hand dialectical reasoning involves no
such insight. There are, according to Aristotle, dialectical as well
as demonstrative syllogisms ; and the dialectical nature of a syllo-
gism, as we saw,2 turns on the character of the premisses, as well as
the form of the statement. What concerns us here is the syllogism
that argues from signs, not causes — e. g. that the rick is overheated,
because it smells thus. If all ricks that smell thus are overheated,
the argument is sound ; but it explains nothing. Were one however

1 Cf. supra, pp. 310-311 ; infra, pp. 437, n. 1 ; 524, n. 2 ; 545, n. 2.
8 Supra, pp. 305-306.

xvinj OF INDUCTION 399

to argue that it is overheated because being damp it allows the
organisms in the grass to effect too rapid an oxygenation, though
this argument agrees with the former in that it can be stated syllo-
gistically, it differs in that it explains. Now the syllogism whose
middle term is a sign Aristotle called dialectical, because it gave no
understanding of the connexions in the characters of things. And
this is equally true about induction. I establish inductively some
general principle of connexion in nature when I can appeal to the
evidence of particular facts to show that apparently either this
principle holds, or none. But herein I do not make the connexion
intelligible. I should establish it demonstratively, if I could show
that it is involved in the existence of other known connexions.
Thus repeated observations of ice floating on water, in various
times and places, of various sizes and shapes, may lead me to con-
clude that ice is lighter than water ; for as it floats irrespectively
of size or shape, time or place, I can connect its floating with nothing
but a less specific gravity. That it should be lighter, however, remains
a brute fact, nowise apparently necessary. But if I could show that
water expands in becoming ice, then, though this indeed is still
a brute fact, yet, granting this, I see that the ice must float ; so far,
I have explanation, insight into the necessity of the connexion of facts,
demonstrative thinking. And here it will be observed that we are
concerned not with a contrast of opposite directions in the reasoning
process between the same two points (as we saw that the contrast of
deduction and induction was apt to bethought), but with the con-
trast of different ways of reaching the same point * : of establishing
a general proposition either by insight into the necessity connecting
the terms of the system of nature, or by appeal to particular facts
which, though we can find no system of connexion in them unless our
proposition is true, give us no insight into the necessity of it. It is
in this contrast, rather than in the former, that the distinctive
character of inductive reasoning is brought out. But it escapes us
in any simple and collective opposition between the forms of
reasoning traditionally assigned to Deductive Logic and those
which ' inductive logicians ' have called inductive.

1 I owe this remark to Mr H. A. Pricharri.

CHAPTER XIX

OF THE PRESUPPOSITIONS OF INDUCTIVE
REASONING : THE LAW OF CAUSATION

' Why is a single instance, in some cases, sufficient for a complete
induction, while in others myriads of concurring instances, without
a single exception known or presumed, go such a very little way
towards establishing an universal proposition ? Whoever can answer
this question knows more of the philosophy of logic than the wisest
of the ancients, and has solved the problem of Induction.' x How-
ever we may think of the knowledge possessed by the wisest of the
ancients, the question which Mill asks is no doubt an important one.
By what right do we ever generalize from our experience ? and how
can we tell when we have a right to do so ? To these questions we
must now attempt an answer. Afterwards we may note what other
processes of thought besides generalization enter into the sciences ;
and then we shall be able to realize better the true nature of that
antithesis between induction and deduction which was spoken of
at the end of the last chapter.

The present chapter will address itself to the question, by what
right do we ever generalize from experience. This is the primary
question. Syllogism never generalizes. Unless it is provided
with universal propositions for premisses, it cannot arrive at them
in its conclusions, and even so, its conclusion is never more general
than its premisses.2 It is just this fact which raised the difficulty,

1 Mill, System of Logic, III. iii. 3, concluding paragraph. Strictly speaking,
a single instance never is sufficient— if we had really to rely on it alone without
help from conclusions already drawn from other parts of our experience.
Cf . Jevons, Pure Logic and other Minor Works, pp. 295-299 ; and also Lotze,
Logic, §§ 252, 253.

2 The third figure, when both premisses are singular propositions, may
seem to furnish an exception to this statement, and it would hardly be a
sufficient answer to recall the fact that this is the inductive figure ; for the
question is whether a syllogism can generalize, and it is hardly consistent with
Baying no, to add that it can only do so when its character is inductive. But
the statement may stand, because all conclusions in this figure are particulai
or contingent. We may aim at generalizing — at finding a judgement which
is true universally ; but we have failed, with such premisses, to do it.

PRESUPPOSITIONS OF INDUCTION 401

how to get the universal propositions which syllogism needs to
start with. If experience gives us only particular facts, how are
we to get universal conclusions out of them ? A mere enumeration
of particulars will justify a statement about no more than the
particulars which have been enumerated, whereas we claim in any
generalization to go beyond the observed facts on which the general-
ization is based, and to draw a conclusion true in any possible
instance of some sort whatever. By what right do we do this ?

The answer is that all induction assumes the existence of con-
nexions in nature, and that its only object is to determine between
what elements these connexions hold. The events of our experience
are no doubt particular, but we believe the principles which they
exemplify to be universal ; our difficulty lies in discovering what
principles they exemplify ; in that, a close study of particular facts
will help us ; but were we to be in doubt whether there are any such
principles or not, no amount of study of particular facts could resolve
our doubt.

There are many ways in which this assumption may be expressed.
It will be well to consider some of these, and to ask what precisely
it is that we assume. We may then show that (as has just been
said) it is hopeless to attempt to prove the assumption by any
appeal to experience ; and ask ourselves what justification we have
for making it.

The commonest expression for it is the Law of Universal Causation,
or (more briefly) the Law of Causation ; again, we say that we
believe in the Uniformity of Nature ; but the same belief is implied
in the distinction between essential and accidental circumstances, or
in asking what circumstances are relevant to the occurrence of an
event, or what are the material circumstances in the case. For only
that can be called material, or relevant, or essential, without which
the event would not have occurred, or whose non-existence or non-
occurrence would have made some difference to it ; and the existence
or non-existence of any particular circumstances can make no
difference to an event, unless there is some connexion between them
and it. Were everything in nature loose and unconnected, it would
be impossible to say that an event occurred because of any one thing
rather than another. All these phrases therefore imply Causation,
and imply Uniformity.

Both the Law of Causation and the Uniformity of Nature are
phrases open to misunderstanding. There is a sense in which it is

1778 d d

402 AN INTRODUCTION TO LOGIC [chap.

the business of induction to discover laws of causation ; in the
plural, the term refers to the various particular principles of con-
nexion exemplified (whether we detect them or not) in the course of
nature ; it is equivalent to Laws of Nature, or Natural Laws, such
laws, for example, as that matter gravitates, or that organisms
reproduce themselves after their kind. Used absolutely and in the
singular, however, it means the principle that there are such par-
ticular principles, and hence we speak of the Law of Universal Causa-
tion, intending to assert that every event has a cause, and that no
change occurs except under conditions with which its occurrence
is connected universally. And it is because we believe its occurrence
to be connected universally with such conditions, whatever they
are, that we speak of the uniformity of nature. We do not mean
to deny variety, but only to assert the unbroken reign of law. That
which collectively we call nature is a vast assemblage of substances
of divers kinds diversely intermingled : interacting with one another
in ways that depend upon their abiding character and their shifting
situation. Even what we call single things are highly complex,
and their properties and behaviour depend upon their composition,
and upon the situation in which they are placed relatively to other
things ; we may believe that whenever one complex thing of pre-
cisely the same kind is placed in precisely the same situation as
another, it will behave in precisely the same way ; nor is more
required by the principle of the Uniformity of Nature ; and yet we
may doubt whether such precise repetition ever occurs. Watch the
movements of a waterfall, how it breaks into a thousand parts
which seem to shift and hang, and pause and hurry, first one, and
then another, so that the whole never presents quite the same face
twice ; yet there is not a particle of water whose path is not abso-
lutely determined by the forces acting on it in accordance with quite
simple mechanical laws. No one would suppose that because these
mechanical laws are unchanging, the waterfall must wear a mono-
tonous and unchanging face ; and so it is, on a larger scale, with the
course of nature. Nature is uniform in the sense that under like
conditions like events occur ; and in fragments, as it were, she is
ever presenting us with the repetition of conditions that have been
fulfilled before ; so that in fragments there is recurrence of like
events enough. But sooner or later, because the assemblage of
things is not quite the same as before, the likeness in the course of
events is broken ; from the beginning it was probably not complete.

xix] PRESUPPOSITIONS OF INDUCTION 403

Were it indeed possible for the procession of events to bring back
precisely the state of things which had existed at some moment in
the past, then it must follow, from the principle of the Uniformity
of Nature, that the same procession would recur, and terminate
again by reinstating the phase in which it had begun ; so that the
history of the world as a whole would really repeat itself indefinitely,
like a recurring decimal, and to a spectator who could watch it long
enough, might seem as monotonous as the music of a musical box
which, in playing, somehow wound itself up, to pass always from
the conclusion to the recommencement of its stock of tunes. But
nothing of this kind occurs ; the Magnus Annus is but a fancy ;
and the uniformity of nature is consistent, as Mill said, with her
infinite variety.

But it may be said, the Law of Causation is one thing, and the
Uniformity of Nature is another ; every event may have a cause ;
but the same cause need not always produce the same effect, nor the
cause of the same effect be always the same. The human will, for
example, is a cause ; but it does not always act in the same way
under the same circumstances ; to-day in a given situation I may
act meanly ; yet it is possible that in a situation of the same kind
I may act better to-morrow. The Law of Causation, if you will,
is a presupposition of inductive reasoning ; but whether causes act
uniformly, whether the same cause in the same situation always
has the same effect (which is what the Uniformity of Nature means)
can only be determined by experience.

To understand the relation of these two principles, we must enter
a little more fully into the difficulties connected with the notion of
cause. Most people, if asked what they meant by the word, would
probably say, that they meant a thing (or person) producing a
change in something else. But great trouble arises when we ask
what producing is. If I say that a wave, by striking it, produces
motion in a boat, I suppose myself to mean more than that the
motion of the boat ensues immediately upon the contact with it of
the wave. I imagine the wave to exert a force upon, to have a power
to move, another body. But what this power in it is, we do not
understand. Some persons indeed, when challenged on this head,
will answer that they do understand it, because they have experience
of exerting power themselves ; they know themselves as causes in
their voluntary actions. But the answer is unsatisfactory, for the
wave is not intelligent, and we cannot suppose that the action of an

D d 2

404 AN INTRODUCTION TO LOGIC [chap.

intelligent being, and what we call the action of a machine, or other
body whose movements are mechanical, are the same thing. More-
over, as Hume pointed out, the connexion between the movement
of my limbs, and what I regard as the psychical cause thereof, is no
more intelligible to me than that between their movement and the
movement of a body which they strike. He therefore, and many
since, have attempted to eliminate the notions of power, agency, or
force, and to reduce the causal relation to uniformity in succession.1
Words like agency or power, on this view, are voces nihili ; we think
we mean something more by them than habitual sequence, but we
do not. We must say the same of connexion. We observe events
together or in succession ; their conjunction is plain, but not any
connexion ; and uniformity in their succession is all that causation

means.

A little consideration however will show that we do not mean by
the causal relation one of habitual sequence. We take uniformity
in the succession of events — i.e. likeness in the conditions upon
which like changes succeed — to be a sign of causal relation, but not
the same with it. For when I say that a wave striking a boat causes
it to move, I imply that the relation subsists between the blow of
this wave and the ensuing movement of this boat ; whereas uni-
formity can only be exhibited in the sequence of several such
movements of this or other boats upon the blows of several waves.
Connexion is between individuals ; uniformity of succession is in the
sequences of each member of one group of similars upon a member
of another group of similars. We mean by the causal relation some-
thing that might hold between terms that were unique, and does
hold between terms that are individual, even though there are other
individuals of the same nature.

And there is another objection to defining the cause of anything
as its invariable antecedent. Antecedent and consequent are
events. But we cannot treat the world as a mere procession of
events ; there are also things to which the events happen. It is
instructive to observe how Mill is forced to recognize this. In spite
of having denned cause as ' the invariable and unconditional ante-
cedent ', he speaks of ' indestructible natural agents ', such as the
earth, as ' permanent causes ', since the earth affects the movements
of any pendulum upon its surface, and they cannot get out of the

1 Treatise of Human Nature, of the Understanding, Part iii ; and Enquiry
concerning Human Understanding, §§ iv-viii.

xix] PRESUPPOSITIONS OF INDUCTION 405

range of its influence ; he also calls oxygen and hydrogen causes of
water.1 Now the earth is not more antecedent than consequent in
time to the movement of a pendulum which it attracts ; and oxygen
and hydrogen are ingredients necessary to the formation of water,
but they do not happen like their combination. Cause no doubt
implies change and succession. But there can be no change without
something which changes, i.e., which persists through a succession
of states. It would not be change but substitution, if one event
succeeded another, and there were nothing but the events ; just as
a child does not change into the changeling which is substituted
for it. Whatever difficulties there may be in understanding what
a substance is, or the relation of a thing to its attributes, it is
a desperate remedy to offer us instead a ' stream ' of events, loose
and disconnected, in relations of simultaneity and succession.

That causation is more than uniformity in the sequence of events
of one sort upon events of another was felt by Mill when he defined
the cause of an event as its invariable and unconditional antecedent.
' What writers mean ', he tells us, ' when they say that the notion
of cause involves the idea of necessity ' is that the sequence of
phenomena must be not only invariable but unconditional ; and
' we may define the cause of a phenomenon to be the antecedent, or
the concurrence of antecedents, on which it is invariably and uncon-
ditionally consequent \2 If we examine his explanation of this
addition we shall see that without the notion of connexion or
necessity his definition becomes altogether futile. For he distin-
guishes between positive and negative conditions, and the negative
conditions of a phenomenon ' may be all summed up under one head,
namely the absence of preventing or counteracting causes ' 3 ; and
he explains an unconditional sequence to be one ' subject to no other
than negative conditions ' 2. The cause of a phenomenon therefore
is the antecedent, or concurrence of antecedents, on which it is
invariably consequent in the absence of preventing or counteracting
causes. Now what is a counteracting cause ? If the cause of any-
thing is only that whereon it uniformly follows, a counteracting
cause should be that whereon it uniformly does not follow ; and the
invariable and unconditional antecedent of anything should be that
whereon it habitually follows in all cases except those in which it
habitually does not. By such a definition we might call anything

1 System of Logic, III. v. 8, viii. 6 : x. 4.

• lb. III. v. 6. » 76. III. v. 3.

406 AN INTRODUCTION TO LOGIC [chap.

the cause of anything, and say that experience supported us. But
clearly, when he speaks of preventing or counteracting causes, Mill
is forgetting the analysis of the causal relation into uniformity of
sequence, and is thinking of the necessity and the connexion which
he professes to repudiate.

And we do mean something by the words necessity and connexion,
and know what we mean, even though we are unable to see between
what changes in a changing world connexion lies. Within the fields
of geometry and mathematics, and in philosophical enquiries, we see
this necessarily to involve that, and the connexions are apprehended
with their terms. In that which changes we realize that there must
be connexion between successive states, without knowing what is
connected with what ; we understand that a cause produces its
effect necessarily, without understanding that it must produce just
this or that effect.

But need a cause therefore act uniformly? In a sense, yes. If
one thing the same in nature at different times, or two things the
same in nature, are to act in situations the same in their nature,
they must act on both occasions in the same way. This is not
a generalization from experience : it follows from the sameness of
thing and of situation. But to what extent things come again
to the same situation, and whether there exist many things of
one kind, we must learn from experience.

That a cause must act uniformly in the above sense we may the
more readily realize, if we ask what is involved in the supposition
that it does not. It will be found that this is tantamount to denying
the existence of causal connexions altogether. For suppose that
every event had a cause, but that there was no reason why the same
event should have the same cause or the same cause produce the
same effect on different occasions. There need therefore be no
appearance of order in nature at all, but events might happen just
as if all changes were fortuitous. As it is, we believe that plants
produce seed after their kind ; we do not expect to gather grapes of
thorns, or figs of thistles ; where we see garden fruit upon a wild
stock, we look for a graft, convinced that the same stock will only
bear different fruit in virtue of some material difference in the
conditions. If any plant might produce any seed, or any seed any
plant, and it was impossible to discover, in such circumstances as
graft or soil — because no reason of the kind existed — why the same
plant produced now one seed and now another, or the same seed

xix] PRESUPPOSITIONS OF INDUCTION 407

now one and now another plant, then we should just deny that
there was any cause for that which happened. We should not
say that there was always a cause, though the cause need not act
uniformly. If two plants, whose nature is really the same, can
determine the growth of totally different seeds, how can we call
either the seed of that plant at all ? Grant that a seed may some-
times be produced by a plant of its own kind, and sometimes by
a plant of another kind, without any difference of circumstances,
and merely because causes do not act uniformly, and you have
really granted that anything may produce anything ; flint and steel
may produce seed instead of a spark, and oil raise the waves or
quench a conflagration. But to say that anything may produce
anything is to empty the word ' produce ' of all its meaning. For
the causal relation is a necessary relation, such that if you have one
thing you must have another. To add that it does not matter what
the other is, destroys the force of the must. The distinction between
essential and accidental, material and immaterial, relevant and
irrelevant, will vanish. So long as causal connexion is between
determinate terms, there is a meaning in it. That is essential to
health, without which health is impossible, and that is accidental
to it which (though doubtless it has its effects) has no effect upon
health. But if exercise, which is essential to my health to-day,
should suddenly and without any change in my condition give me
epilepsy to-morrow, and the loss of a letter in the post somewhere
in the antipodes on the following day should give to one man epilepsy
and cure another of it, then it would be impossible to say that
anything was accidental, or anything essential, to the same result
for two minutes together. And the discovery of causal connexions
in the succession of events now would certainly be of no use in
enabling any one to forecast the future ; because the connexions
themselves might have altered in the meantime. It is difficult to
see how all this differs from denying that there are any connexions.
Uniformity of action is not indeed the fundamental element in
the causal relation, for it depends on repetition of the action ;
the causal relation has nothing to do with number of instances, so
far as its existence — though much so far as its detection — is concerned ;
it is bound up altogether with the nature or character of things, and
the nature of anything is not a question of the number of such things
that may be or have been fashioned. Yet if a thing is to have any
determinate nature and character at all, there must be uniformity

408 AN INTRODUCTION TO LOGIC [chap.

of action in different things of that character, or of the same thing on
different like occasions. If a thing a under conditions c produces a
change a: in a subject 8 — if, for example, light of certain wave-lengths,
passing through the lens of a camera, produces a certain chemical
change (which we call the taking of a photograph of Mount Everest)
upon a photographic film — the way in which it acts must be regarded
as a partial expression of what it is. It could only act differently,
if it were different. As long therefore as it is a, and stands related
under conditions c to a subject that is s, no other effect than x can
be produced ; and to say that the same thing acting on the same
thing under the same conditions may yet produce a different effect,
is to say that a thing need not be what it is. But this is in flat
conflict with the Law of Identity.1 A thing, to be at all, must be
something, and can only be what it is. To assert a causal connexion
between a and x implies that a acts as it does because it is what it is ;
because, in fact, it is a. So long therefore as it is o, it must act thus ;
and to assert that it may act otherwise on a subsequent occasion is
to assert that what is a is something else than the a which it is
declared to be. It may be replied that no two things ever are the
same, and — what that reply must commit you to — that no one thing
ever is the same for two successive moments. The fact of change is
not disputed, nor the difficulty of finding two things that are quali-
tatively the same. But if the second has a different effect, that
must be because of its qualitative difference from the first, and not
merely because it is a second ; and so far as it is qualitatively the
same, the effect must be the same also : it being understood of
course that to sameness of effect qualitative sameness is equally
necessary in all the material conditions. To deny this is to deny
the possibility of reasoning altogether. If we cannot truly make
the same assertion about a number of things, then, as Aristotle
observes, there will be no universal, and so no middle term, and no
demonstration.2 For an universal judgement connects a certain
attribute with a certain subject in virtue of their nature and without
regard to the frequency of their existence. If we can do this, we can
make the same assertion about all things of such and such a kind ;
if we cannot do it, we are left with nothing but particular things
whose attributes must be ascertained by inspection or experience of
themselves ; and not by transference of what we have in one
instance found true of such a kind of thing to others of the kind.
1 Cf. supra, p. 13. * Anal. Post. a. xi. 77a 5-8.

xix] PRESUPPOSITIONS OF INDUCTION 409

What holds for the relation of subject and attribute holds in this
respect eo ipso for that of cause and effect. To suppose that the
same cause — other things being equal — can have different effects
on two occasions is as much as to suppose that two things can be the
same, and yet so far their attributes different. To reply that two
things cannot be the same, and that the same cause cannot be
repeated, is either to miss the point, or to abandon reasoning. If it
is meant that two complex things cannot be qualitatively the same,
nor can conditions precisely the same in kind ever recur, such an
objection misses the point. One need not maintain that such iden-
tity, or such recurrence, in fact occurs, though it is not perhaps
inconceivable that it should ; all that is maintained is, that so far as
things are qualitatively the same they have the same attributes, and
so far as conditions precisely the same in kind recur, they must, if
there is such a relation as cause and effect at all, have the same effect.
If, on the other hand, it is meant that there can be no qualitative
sameness in what is numerically different, we can only say that if so,
there is no reasoning. For in reasoning we trace connexions between
things in virtue of what they are, not of their being numerically
these individuals or those ; and if we make this distinction, we must
conceive that what one thing is another might also be ; so that to
deny the conceivability of qualitative sameness, or sameness in
what things are, is to deny that we can distinguish what they are
from individual things. But this denial that any identity is con-
ceivable between different things is what is really at the bottom of
the attempt to resolve the causal relation into uniformity of sequence.
For the causal relation which connects a with x connects a cause of
the nature a with an effect of the nature x. The connexion is between
them as a and x, and therefore must hold between any a and any x,
if they really are a and x respectively ; in other words, it must be
uniform. The denial of this is just the denial of universals. How
then could we speak of things of a kind, or hold our sequences
uniform except in the fact that they are sequences, since any other
uniformity must consist in the same antecedent having the same
consequent on different occasions ? x The cause of an event might
then indeed be anything to which it stood in a relation of sequence
at all, and need no more be the same on different occasions than its
antecedent need be ; for we should have agreed that it was impossible

1 Strictly speaking, even sequence could not be a feature common to two
successions.

410 AN INTRODUCTION TO LOGIC [chap.

that the sequence of the same x upon the same a should ever be
repeated.

Must it then be maintained that if man is a cause of any change,
his acts are necessary, and the will not free ? The problem of the
human will is so notoriously difficult, that one might perhaps be
content here to leave it on one side, merely pointing out that what
has been said will still be true of causal relations displayed in things
material, though it were otherwise with an intelligent cause. But
it is worth while to notice that the interpretation of freedom which
could make a man's acts undetermined by his character has been
often rejected for the very reason that they then would not be his,
and that if he might indifferently do or forbear an act, no matter
what his character was, he would be nothing in particular. And
it may be suggested that error lies not in holding a man's actions to
be necessary, but in holding them to be mechanical.

It may throw some further light on the presuppositions of induc-
tion, to consider this distinction. A machine is a collection of
material parts, interacting in various ways, so as to produce in
some part or parts of it a desired movement, such as the regular
revolution of the hands of a watch, the up-and-down or circular
movement of a saw, the lowering and raising of the hook of a crane,
with what is attached to it. But though intelligence has directed
the building it does not direct the action of the machine ; and if
there has been a mistake in the building, whereby it fails to achieve
the result desired, the machine will not correct the mistake, but will
produce an effect rendered necessary by its construction, however
far from that intended. For it moves in virtue of the ' laws ' of
matter. Any two of its parts (and in the last resort the parts which
have to be considered might be molecules or atoms, or even smaller)
interact together in divers ways, according to their kind, proximity,
velocity, and what not ; and every two are in some relation of inter-
action ; and we believe that, if our powers of calculation were
sufficient, we could derive the ' action ' of the machine from the
interactions of all its parts, as their necessary resultant. In this
way would the changes of any closed material system be explained ;
and if no material system that we know is closed — i. e. shut off from
interaction with bodies outside it — yet there are many in the explana-
tion of whose changes most of what is outside them may be treated
as constant.

Now consider what would be involved in the view that a man's

xix] PRESUPPOSITIONS OF INDUCTION 411

actions are mechanically explicable. One form which this view
takes is to suppose that consciousness is an otiose thing, and that
those movements of the body which we commonly think to be
intentional are mere results of the state in which the body and
things surrounding it were at the previous moment. We should
then be, in Huxley's language, conscious automata ; and the laws
of matter and motion would of themselves have sufficed (if we may
borrow an illustration from Professor James x) to produce the manu-
script of Shakespeare's works — and indeed every edition and per-
formance of them — though Shakespeare had been but a lump of
organized matter as devoid of thought and feeling as the pen he
wrote with, or the automaton of Vaucanson. Such a conclusion is
paradoxical, but that does not of itself constitute a refutation. It
is however impossible as a final account of things, because the facts
of consciousness undoubtedly exist, and the theory can give no
account of them. For it demands not only that a physical event
should be physically determined, but that physical conditions should
determine only a physical result. Mass and energy are to remain
constant in amount, but to undergo redistribution in accordance
with certain laws, which can be expressed in mathematical formulae
enabling us to calculate the precise degree of change in one direction
that will be involved in a given degree of change in another.2 In
these redistributions there is no room for knowledge or feeling among
the ' forms of energy ' ; for mechanical conditions are to have their
complete mechanical equivalent, in terms of matter and of motion,
potential or actual. We cannot then bring conscious processes
within the sweep of a mechanical explanation by regarding them as
thrown off in the process of physical change, which they nowhere
influence. But it might still be held that they are themselves
explicable on the same lines as physical processes. On this view
there would be psychic elements, however originated, affecting,
suppressing, intensifying or fusing with one another in such ways
that, from the laws displayed in their several interactions, we
might deduce the resultant complex of our conscious states. Now

1 Principles of Psychology, i. 132.

2 Hence the saying of H. Poincare, that a physical law is a differential
equation, Address on the Principles of Mathematical Physics, St. Louis, U.S.A.,
Sept. 1904 : v. The Monist, Jan. 1905, p. 3. Recent physical research has
suggested doubts about the constancy of mass, or in other words about the
unconditional truth of Newton's Law of Gravitation (cf. H. Poincare, Science
and Method, III. 'The New Mechanics '). But there would be some law of
its variation, if it were not constant. Cf. infra, p. 418.

412 AN INTRODUCTION TO LOGIC [chap.

such a process would be as unintelligent in the soul as it is in a
material system. What we call knowledge and what we call error
would differ merely as two equally necessary results of pre-existing
psychic conditions, like the action of a machine running truly and
of one running awry. But in an intelligent operation of whatever
sort the conclusion wherein it ends is not explicable as a result of the
preceding stages. There is an advance to something new, not the
old in an equivalent form. And this advance is a development,
wherein the soul comes to be what it had it in it to be but was not.
When the development is made, we see it as the realization of what
went before. The man who comes to understand a puzzling set
of facts finds in what he has come to apprehend the solution of his
previous puzzles, the artist finds in his finished work that which he
was feeling after, he who has discovered what he ought to do finds
therein the answer to his questions. At the outset he did not know
what the solution, the finished work, the right course of action
would be ; and he knows it now not as a result of his previous
ignorance, or even as a result barely of his previous knowledge of
other things, but because he is intelligent.

It follows that he, unlike a machine, is an unity, whose later states
and actions are not calculable from the earlier, and involve in their
explanation a soul, or an intelligence, distinct from any particular
act or state. What the powers and resources of this unitary prin-
ciple are is only learnt as they reveal themselves in its activity.
Since it is not resoluble into an aggregate of interacting elements,
its actions are not those of such elements, but are the manifestation
of itself, of its own being, even though in different circumstances it
would have acted in some ways differently. Again, since it is not
compounded out of elements whose modes of interaction with each
other can be exhaustively expressed in the formulae of ' laws ', each
soul may be unique. If indeed we all were purely rational, we
should all think or act in the same way in the same situation ; for
reason is one. And if anything which passes in the soul is mechani-
cally determined, the laws of that may be discoverable. But the
nature of a finite soul is not formed by adding to complete rationality
something not rational but mechanical ; rather it is brokenly and
partially rational, and the broken and partial rationality of each
will develop differently. Moreover we are constantly confronted
with novel situations, in which the rational course required is some-
thing special, not deducible from general laws. To say all this is

xix] PRESUPPOSITIONS OF INDUCTION 413

not to deny necessity in the development of a soul's powers ; but
it is to deny uniformity, so far as no second soul of the same nature
is placed in the same situation.

The amount of success obtained by psychologists and physiologists
in their attempts to discover laws of psychical development or
change is not such as to refute this uniqueness ; and certainly
psychologists have failed to explain such matters as memory, or
space-perception, precisely in this particular, that they have not
made it intelligible why the conditions under which these arise
should lead to them. On the other hand the natures of men are
found to be in large measure the same, and to that extent generaliza-
tion is possible. For generalization, and the use of what we learn
from experience of one thing to anticipate experience of another,
depend on the natures of things being repeated in instance after
instance. So far as things have not the same nature, they will not
act in the same way. This is but the other face to the fact, that so
far as they have the same nature, they will.

If we may accept the foregoing account of the difference between
an intelligent being and a material system \ it follows that the causal
relation is always necessary, and uniform on condition, but only on
condition, that there are more instances than one of the same kind.
Now that there are things of a kind is obvious ; but things of one
kind may and do exhibit individual variations. In the physical
sciences we suppose these variations to be due to diversely combining
elements of various sorts, but qualitatively the same in all instances
of each sort. And the success with which changes in bodies have
been accounted for on the assumption that they are composed of
such elements, whose mutual interactions are statable in quantita-
tive laws, or connected with what may be so stated, leads men to
accept the mechanical view of physical changes as correct. Inani-
mate bodies certainly do not behave as if they were intelligent ;
many living bodies do not. Even for the execution of an intelligent
plan, a material seems needed in which changes are mechanically
determined, and the effects of causes calculable from laws.2 A body
certainly is not a mind, whether a mind can influence its movements

1 Cf. with the foregoing a paper in the Hibbert Journal for April 1914,
vol. xii. No. 3, Mechanism, Intelligence and Life. I have not here, as I have
there, discussed at all the question whether living bodies are mechanical
systems, or more akin to what is intelligent. On the relation of uniqueness
to causality and to freedom, cf. further there, pp. 622-624.

2 Cf. Lotze, Outlines of Practical Philosophy, § 21.

414 AN INTRODUCTION TO LOGIC [chap.

or not. So far as not influenced by a mind, these movements must
exemplify mechanical necessity. They are displayed in things
manifestly of many, but not indefinitely many, sorts. What the
laws are according to which a thing of one sort produces a change
in a thing of another sort, we learn from experience ; but that there
are such laws, that the causality in change involves uniformity, is
evident to reflection.

But if we are to speak of laws exemplified in the changes of things,
it is important to distinguish conditional and unconditional laws.
A law indeed is not a cause, and does not act ; it is a principle of
action displayed in the things that do act, or are causes. The state-
ment of a law is the statement of such principle of action, or of a
connexion between the action or change of action in one thing, and
change in another. The first law of motion is a principle of action ;
a body, left to itself, persists in its state of rest or uniform rectilinear
motion ; but it produces no change in another body, in virtue of
that law, until they collide. The second law of motion is a principle
of the action of one body on another. But it is the body moving
with a certain momentum in a certain direction, and not the second
law of motion, which causes another body to move accordingly.1
Nevertheless, if the causal relation is necessary, a law must be neces-
sary also ; a statement of the way in which a cause does act is a
statement of the way in which it must act. A law then can admit
of no exception ; or, what is the same thing, a law which admits of
exception, and holds good only for the most part, is not the true law,
and indeed as formulated is no law. A true law is true uncondition-
ally ; i.e. there are no conditions, beyond what are included in the
statement of the law itself, variation in which can affect its ex-
emplification. If there are such, the law is only conditionally true,
i. e. exemplified under certain conditions, and not otherwise.

The first law of motion is an example of a natural law which
would perhaps be regarded as unconditionally true — that every body
persists in its state of rest, or uniform rectilinear motion, until it is
interfered with by some other body. The same might be said of
the law of universal gravitation, that all bodies attract one another
with a force that varies directly as the mass, and inversely as the
square of the distance.2 Compare with these the principle that
acquired characters in a plant or animal are not inherited. Supposing

1 Cf. infra, p. 502, n. 1.

2 But cf. supra, p. 411, n. 2.

xix] PRESUPPOSITIONS OF INDUCTION 415

this to be true (for it is still sub iudice), yet it need not be uncon-
ditional. We are not in a position to say that living things could
not be so organized, in respect of their reproductive system, as to
make acquired characters heritable, but only that, with the organi-
zation which we find, they are not heritable. That organization there-
fore may condition the truth of our principle. Just as the prevailing
necessity for sexual union in the reproduction of all multicellular
organisms does not exclude arrangements in some species which
make them parthenogenetic, so there might possibly be conditions
under which the non-heritability of acquired characters held good
no longer. And as conditions may change, those realized at one
time not being realized at another, so the conditional principles
prevailing, or exemplified, may be different at one time from
what they are at another. It appears to be the case that living
matter can only be produced from other living matter ; there is no
spontaneous generation of it from the inorganic ; omne vivum ex vivo.
But many scientific men have supposed that though this is true and
necessary now, yet in an earlier period of the earth's history, under
very different conditions of temperature and so forth, it was not so.
Conditional principles are necessarily derivative : i. e. their truth,
so far as they are true, follows from some unconditional laws, which
under given conditions involve them as their consequence.1 They
therefore admit, theoretically if not as yet actually, of explanation.
But derivative principles, or principles admitting of explanation,
are not necessarily conditional. For when we call a principle
conditional, we mean that the truth of our principle depends upon
conditions which are not stated in it. If we bring the conditions
into the statement, then, though it remains derivative, it is condi-
tional no longer. Supposing that we knew precisely those con-
ditions of organization in animals and plants which made acquired
characters non-heritable ; then the statement that in animals or
plants of that organization acquired characters were not inherited
would be unconditionally true, although no doubt the fact would

1 Not as their effect. The relation of cause and effect is displayed between
things acting together according to their several natures and the complex
change which they produce ; and it involves time. But the fact that the laws
according to which each thing severally acts involve that in a given combina-
tion the things produce this complex change is not itself a change produced
whether by the things or by the laws of their several action ; and there is no
before and after between the simpler and the derivative principles, as there
is between the action of the cause and its effect ; their relation is not a time-
relation. Cf. infra, p. 502.

416 AN INTRODUCTION TO LOGIC [chap.

admit of explanation. It would probably not be called a law of
nature, because derivative ; but it would have all the necessity of
a law of nature.1

The necessity then of the causal relation, and the uniformity that
follows from it in a world containing many things of the same kinds,
involve the truth, without exception or qualification, of all uncon-
ditional laws ; but conditional principles admit of apparent excep-
tions, without derogation to such necessity and uniformity ; and if we
are ignorant of the conditions within which these conditional prin-
ciples hold good, we cannot tell when the exceptions may not occur.
To return to our previous illustration : if we do not know under what
conditions of organization acquired characters are and are not herit-
able, we must be prepared to admit evidence that in some cases they
have been inherited. Where, however, exceptions occur to some con-
ditional principle, they constitute no exception to the Uniformity
of Nature ; but only imply that the conditions, under which that
principle held good, are not fulfilled in the exceptional case. And
the exception leads us, not to deny that ' nature is uniform ', but
to revise or to determine more precisely the particular statement of
principle which we have found invalid. It is only unconditional
laws that can have no exception.

It becomes therefore important to determine, if possible, when
we have discovered an unconditional law. We may disregard here
those derivative laws, which we may be capable of explaining from
others more general than themselves ; for they can only be uncondi-
tional if the more general laws from which they are derived are so.
Now, if we have no better reason for accepting a law as unconditional
than that by assuming it to be true we can account for the facts of our
experience, then, though we might provisionally accept it, we can
hardly be content with our warranty ; for perhaps some other law
might also account for the facts. But if (and this, as we shall see
hereafter, is a distinction of the first importance in inductive theory)
— if, without assuming it to be true, it is impossible to account for the
facts of our experience, we should have to suppose it unconditional ;
though such impossibility may be hard to establish. Still, we
should not be fully satisfied ; for had the facts been otherwise, we
need not have admitted the law ; and we do not see, except on the
hypothesis that the law is true, why the facts might not have been

1 Cf. supra, p. 386, n. 3, and infra, c. xxii. ; the non-reciprocating causal
relations there discussed are all conditional.

xix] PRESUPPOSITIONS OF INDUCTION 417

otherwise. Complete satisfaction would only come, if the law
which the facts had forced us to recognize should, when considered,
appear self-evident.

Are there any unconditional laws known to us ? There is no
doubt that the fundamental principles of physical science are often
so considered. It is often held that we have discovered certain
physical laws prevailing throughout the material universe, in
accordance with which every event in the material order takes place ;
that these laws are mechanical ; and that nature is, in truth, and
in the last resort, a purely mechanical system. And this view is
supposed to be confirmed by the character of the principles with
which physical science works. A great deal is purely mathematical ;
and about mathematical principles at any rate we can say that they
are unconditional because self-evident ; no apparent exception
would make us doubt them or revise them ; we should only doubt
the alleged fact which was supposed to constitute the exception.
And some of the most general physical laws have often been held
to possess the same self-evidence ; the first law of motion, and the
laws of the conservation of energy and the conservation of mass, are
instances. That anything should occur in the material system un-
conf ormably with these principles would then present the same kind
of contradiction as that two and two should make five. The ex-
planations of physical science, at least so far as they rested on laws
of this kind, would be complete and final.

We have however seen that there are grave difficulties in a purely
physical or mechanical theory of the world. Consciousness is to
it unaccountable ; and it cannot therefore be a complete or final
theory. Hence many philosophers have suggested that in the last
resort, instead of vainly attempting to explain consciousness in terms
of physical law, we must find in physical law a manifestation of
intelligence. This view may take the form of saying that an
intelligent Being sustains the physical world, and directs its changes
on mechanical principles, because it is important that men should
be able to calculate and count upon them ; and it is added that
the Being who maintains these principles may depart from them,
if on any occasion any better purpose is to be served by departing
from them than by acting on them.

If, as seems necessary to admit, men, not as physical machines
but as intelligent beings, produce movements in bodies, we cannot
deny the possibility that some intelligence not connected with a

1778 E 9

418 AN INTRODUCTION TO LOGIC [chap.

living body may do the same. And if the so-called physical laws
themselves depend on the will of an intelligent Being, and his plan
involves departure at any time from his own rules of action, what
would appear to us exceptions to these laws would occur. But
such departure cannot be arbitrary. Any one understanding the
plan would see that the exceptions to the law were as necessary as
the illustrations of it. The law therefore would not be unconditional,
and fuller knowledge would introduce into the statement of it the
required qualifications. Only, since intelligent action differs from
mechanical, it would not be possible to express these qualifications in
physical terms, and to substitute for the inaccurately stated law
another really unconditional, and yet connecting changes in a
mechanical way.

Other philosophers have sought to show in a different way that
physical law is a manifestation of intelligence. They have pointed
out that the material order is an object of apprehension, and therein
stands related to the minds that apprehend it ; and they have
urged that the world and minds together form the complete reality,
or res complete/,, and cannot be understood except together. There
is indeed a special difficulty here in the fact that what understands
is itself mind, so that one term in that relation has to understand
both itself and the other term. With the problems of such Idealism
we are not here concerned. But we may point out, in regard to the
unconditionality of physical laws, that if they are known to be un-
conditional, the knowledge of them is not itself a condition of their
truth. It is possible that we may some day know that matter
gravitates as Newton supposed only under the condition that it is
moving with less than a certain velocity, and so not unconditionally.
But matter moving with less than that velocity would gravitate thus
unconditionally. Whatever transformation our view of the material
order may undergo, yet the interconnexions of events within it, the
connexions of cause and effect there traced, will have to be taken over
as it were en bloc, unbroken and undistorted, by any interpretation
of the world which takes knowledge as well its objects into the
account, and holds matter dependent on its relation to mind.
What we call a moving body may be something else at bottom than
a moving body ; but its motion would not because of that any the
less appear determined in accordance with physical laws.

It is different if a body is the subject of the action of a mind
or spirit. That would condition the movement as the knowledge of

xix] PRESUPPOSITIONS OF INDUCTION 419

it would not. And if that is possible, the laws of motion which
physics has formulated cannot be unconditionally true ; for they
make changes in the movements of bodies to be dependent alto-
gether on other bodies, ignoring the influence of anything besides.
Now if we could see the necessity of physical laws, as we can of
mathematical relations, we should have to allow that they are
unconditional. But this we cannot do. Some indeed have thought
the first law of motion self-evident, as only saying that a body
cannot change its state of rest or motion without a cause. But it
says more than this ; viz., that the cause can only be another body ;
and this is not self-evident, for we do not understand how one
body causes a change in the state of rest or motion of another. It
is true that neither do we understand how a mind or spirit does ;
yet we may have to admit it on the evidence of facts. But there
is a little more to be said for the first principles of physical science.
Intelligent action leads to something new, mechanical action does
not ; in a material system there is no development. Therefore the
principles that express the inertia of matter, and the constancy of
the ultimate facts like matter and energy, may be unconditionally
true of a system purely physical. They will not therefore be uncon-
ditionally true of a physical system in relation to intelligent agents.
Nevertheless there is a great difference between what is meant
when in the sciences a physical principle is called conditional, and
what is meant in calling them conditional on something non-
physical. We conceive that for any principle conditional in the
former sense, such as the non-heritability of acquired characters,
the conditions on which it depends might be found, and would be
in eodem genere with the principle itself ; i. e. the principle stated
so as to include these conditions (and in that form called uncon-
ditionally true) would be derivative in an intelligible way from
principles more general, but from principles holding, like itself,
within what is material. But if the ultimate physical principles
are called conditional, it is not because they can be derived from
any physical principles more general than themselves, and the kind
of explanation possible of the other sort of conditional principles,
viz. showing that the facts exemplifying them really only exemplify
simpler principles of the same sort with themselves, is here pre-
cluded. And if there are spiritual conditions upon which the move-
ments of bodies to some extent depend, physical science cannot deal
with these. For as a mind or spirit does not act mechanically, we

E e 2

420 AN INTRODUCTION TO LOGIC [chap.

cannot from observed changes form an hypothesis as to its mode of
action, and thence calculate the effects which it should produce in
another situation.

For this reason, physical science will ignore such conditions. It
is of no use to consider in our calculations a factor which is incal-
culable. The man of science, even if he believes that such conditions
exist, will reasonably consider that he has no means of determining
their influence, and that he can only discover the extent to which
physical principles will account for physical changes by proceeding
as if they would do so altogether. The principle of the Uniformity
of Nature is sometimes understood as claiming that. It need not
be. What has been maintained in the foregoing discussion is that
the Law of Causation is presupposed (not reached) by induction ;
that so far as things and situations are repeated, it carries with it
uniformity ; but that it consists with this uniformity that there
should be unique things, and principles only conditionally true,
and so admitting of exceptions. An unconditional principle admits
of no exceptions ; and a self-evident principle is unconditional.
The fundamental principles of physical sciences are often treated as
unconditional ; but they are not self-evident, and much occurs in
this world which is not explicable from them. If they were self-
evident, what follows from them would have to be retained and not
contradicted in any complete explanation of the world that took
into account what physical science leaves on one side. But if the
first principles of physical science are only conditionally true, yet so
far as the conditions under which they do and do not hold good
are unascertainable, physical science may reasonably push ahead
ignoring such conditions.

We argued indeed that it is no more than a corollary of the Law
of Identity, that the same thing unaltered on different occasions,
or two things of the same nature, should under the same conditions
produce the same effect. But this does not show that anything
remains unaltered, or that any two things have the same nature.
It involves an assumption — if assumption it be — that what is real
is intelligible or rational. Any one who questions this, to the extent
that he does so, despairs of reason and thought ; and his question-
ing cannot be set at rest by reasoning. The assumption however
does not require us to deny uniqueness, and it governs our thinking
in other fields besides that of cause and change. The causal relation
is displayed in change, and involves time ; an effect is always after

xix] PRESUPPOSITIONS OF INDUCTION 421

the action of its cause.1 But the argument from identity of nature
is used in generalizing where time and change do not enter, e. g. in
geometry. And our understanding of connexion between one element
and another in the being of things, where we have such under-
standing, though not the discovery that we must admit connexions
which we do not understand, is independent of their repetition
and consistent with their uniqueness.

With these explanations and qualifications we may say indif-
ferently that the inductive sciences presuppose the Law of Universal
Causation, or the Uniformity of Nature. But as it has been held by
some 2 to be the task of induction to prove this principle about
a world as to whose nature, prior to our observation of what happens
in it, we must presuppose nothing, it may be worth while in con-
clusion to show that this is impossible. It is alleged upon the view
now to be considered that our experience of the great extent to
which like antecedents have like consequents is the ground upon
which we believe that this is universally the case. Against this we
may point out in the first place, that such an inference assumes the
course of events in one time and place to be a guide to their course
in other times and places : which is really the very principle that
is to be proved. As Lotze has urged, if a reason can be given for
the inference, it rests on some previous assumption ; and if no
reason can be given for it, what is its force ? 3 Next it is to be noted
that this view regards as of the same nature two arguments really
very different. It is supposed that to infer an universal relation
between two events a and x from the frequency with which one has
been succeeded by the other, and to infer from the observed succes-
sion of like consequents upon like antecedents in divers pairs, a and x,
b and y, &c, that every event is thus uniformly paired with some other,
are arguments of the same form ; and that since the former is allowed
to have value, so must the latter. This however is not so. We
infer from the frequency of their conjunction in a great variety of

1 This is perfectly consistent with holding that there is no interval of time
between them, just as one body being outside another is perfectly consistent
with their being in contact. It is also consistent with things interacting. If
A and B interact, the initial activity of A produces such a change in B that
B then affects A otherwise than it did initially, and vice versa. The difficulty
of dealing with or understanding the continuity of the operation is not in
principle greater than if the action was not reciprocal. If A produced a
change in B, and B did not react, its next effect on B would still be modified
by the fact that B was no longer quite the same. In the first edition of this
book, note 1 on p. 390 was wrong about this matter.

* Cf . e. g. Mill, System of Logic, III. xxi. 8 Metaphysic, Introd. § v.

422 AN INTRODUCTION TO LOGIC [chap.

circumstances a connexion between a and x, because upon the assump-
tion that there is some set of conditions upon which every change follows
uniformly the view that for x these conditions are a seems alone
consistent with our experience. Now what is thus assumed is just
the uniformity of nature ; and without it the argument from the
frequency of the succession a-x or b-y to their connexion could
not be made. But the argument from the constancy of succession
in divers pairs to the existence for every event of some set of con-
ditions on which it follows uniformly can be made neither with this
assumption nor without it. Not without it, any more than the
other argument ; and not with it, because no assumption can be
used to prove itself. Again, the uniformities which are said to be
the empirical basis of our generalization are not really matter of
direct experience. We have said above, that the particular con-
nexions which we believe to prevail in nature have been inferred
with the help of the assumption that all changes occur in accor-
dance with laws. But if any one likes to question this, he must at
any rate agree that most of the uniformities in which we believe have
been inferred somehow : very little has come directly under our
observation. We believe that winds are caused by differences of
atmospheric pressure : difference of atmospheric pressure is itself
inferred rather than observed ; but waiving that, for what propor-
tion of winds have such differences been noted ? We believe the
sounds of a piano to be caused by the striking of strings : for what
proportion of such sounds which we have heard have we first
seen the strings struck by the hammer ? It is needless to multiply
such examples : but when it is alleged that we are justified in infer-
ring the uniformity of nature to hold good universally because we
have direct experience of it over vastly the larger portion of the
field,1 it is important to point out that our direct experience of it is
singularly small, and that the vastly greater proportion of what we
believe ourselves to have ascertained is matter not of experience but
of inference. Now we may offer the empiricist his choice. If this
inference is made by the help of the assumption of the uniformity
of nature, its results cannot be used to prove that assumption. If it
is made without that help, by his own admission it falls to the ground,
for the inference of any particular uniformity is supposed to need
that assumption ; and so he is not left with experience sufficient to
justify his generalization. We may present the argument against
1 J. S. Mill, System of Logic, III. xxi. 3.

xix] PRESUPPOSITIONS OF INDUCTION 423

his position in yet one more light. The essence of his contention is,
that we must come to the facts of experience without any precon-
ceptions ; we must have no antecedent view of what is conceivable
or possible. For all that we can tell to the contrary until experience
has instructed us, anything whatever is possible ; and if it occurred
with sufficient frequency, anything would be conceivable. Now, it
will be admitted that if there are a number of independent alterna-
tives all equally possible, an event that is inconsistent with only one
of them leaves us quite unable to decide between the rest. But if,
as the empiricist insists, all things are antecedently equally possible,
then all proportions of regularity to irregularity in the world are
equally possible antecedently. All events may occur in accordance
with uniform principles : or there may be no event which ever has
the same consequent twice ; and between these two extremes of
absolute regularity and absolute irregularity an infinity of inter-
mediate alternatives may be conceived, among which we cannot
select except upon the evidence of experience. The extent to which
regularity, or uniformity, prevails may therefore be limited in any
conceivable way, whether as regards place, or time, or subject.
There is no reason why the succession of like consequents upon like
antecedents, while exemplified at other times and places, should not
fail in the hitherto unexplored parts of Central Asia, or on all
Fridays subsequent to the Friday in next week. Nothing less
than this is involved in the refusal to prejudge experience. But if
that is so, past experience itself can never enable us to prejudge
future experience. For why should any degree of uniformity ob-
served till now in the succession of events induce us to expect such
uniformity to continue ? It was antecedently as probable that such
uniformity should continue till to-day, and then terminate, as that
it should continue till to-day and still continue. The fact that it
has continued till to-day has disproved what until to-day was a
possible hypothesis, viz. that it might terminate sooner ; but
between its terminating to-day, and still continuing — two indepen-
dent and antecedently equally probable alternatives with which its
continuing until to-day is equally consistent — it does not in the
least enable us to decide. This argument will hold good, at whatever
point in the series of time to-day may fall ; it will hold good equally
against an inference to the unobserved events of the present or the
past ; so that we never get any nearer being able to infer a degree
of uniformity which goes beyond what has been actually observed.

424 AN INTRODUCTION TO LOGIC [chap.

It seems conclusive therefore against the view that the Uniformity
of Nature can be an induction from experience, if by the term in-
duction any legitimate process of inference is understood.1

With what right then do we assume it ? The answer to this
has been given in discussing what we mean by it. To deny it is
to resolve the universe into items that have no intelligible connexion.
If the universe and the events in it form a systematic whole, then
any change must be determined by something in the nature of that
whole ; and for the same change to occur on different occasions
except under the same conditions is not consistent with its having

1 The last argument may be put in a way that will perhaps to some seem
clearer as follows :

1. An event which is equally consistent with two hypotheses affords no
ground for deciding between them.

e. g. if A and B keep a common stock of boots, and each uses every
pair indifferently, footprints that fit one of these pairs afford no ground
for deciding whether A or B has passed that way.

2. It is admitted by those who regard uniformity in nature as empirical,
that antecedently to experience all issues, so far as regularity and irregu-
larity in the succession of events are concerned, are equally probable.

By an issue is meant a certain course of events, however long.

3. These alternative issues must be regarded as perfectly detached alter-
natives : i. e., antecedently to experience, the rejection of one issue would
not give any ground for or against the rejection of any other. To assume
that it would is to assume, antecedently to experience, the existence of such
degree of uniformity as enables you to say that if one specific issue happens,
another must or cannot.

4. That events should occur with any specified degree of regularity down
to the end of the year 2000 A. D., and with less or no regularity, or in apparent
conformity to different rules, thenceforward, is one such issue ; that they
should occur with the same specific degree of regularity down to the end of the
year 2001 a. d., and thence with less or none or other, is another such issue.
And these issues are perfectly detached alternatives beforehand. Let them
be called X and Y.

5. The empirical observation of that specified degree of regularity down
to the end of 2000 A. D. is equally consistent with the hypothesis that X, or
that Y, expresses the truth. Therefore it affords no ground for deciding
between them.

6. It would therefore be equally likely at the end of 2000 a. d. that the
events should thenceforward exhibit none or less of the regularity that they
had hitherto exhibited, or conform to quite different rules, as that they
should continue to exhibit the same regularity even for a year longer.

7. The dividing date might be taken anywhere ; and one might take
equally a dividing place, or department of fact.

8. Hence the actual issue never affords any ground for preferring the
hypothesis of a continuance of the observed regularities to any hypothesis
of their discontinuance, complete or partial, with or without the substitution
of other regularities, in any period, region, or department of fact, in which
they have not been empirically verified.

xix] PRESUPPOSITIONS OF INDUCTION 425

a determinate nature. It is not, of course, denied that changes
partially the same may occur under conditions partially different ;
and the task of disentangling the identities in what is partially
different is one of the tasks of the inductive sciences ; but ceteris
paribus — a proviso about which it is very difficult for us to know
in individual cases how far it is fulfilled — the same conditions must
produce the same effect, and the same effect must have been due to
the same conditions. A changing universe is otherwise unintelligible
or irrational. If any one likes to accept that alternative, it may be
impossible to reason him out of it ; for he has disallowed at the
outset the appeal to reason. At least let him not maintain that,
while the alternative is conceivable, experience proves that it is
not actual.

CHAPTER XX

OF THE RULES BY WHICH TO JUDGE OF
CAUSES AND EFFECTS

The world, as we have already insisted, is not a mere procession
of events, but the events concern things ; a cause is a thing acting ;
it produces a change in some thing. And the things exist before
and after the action, sometimes apparently unchanged. A wall, for
example, which changes the direction of motion of a ball striking it,
exists before and after producing this effect, and the ball does so also.
And if a bullet struck it, and were scattered in pieces, though we
might say that the bullet no longer existed, the particles would still
exist into which it was broken up, and we should say that the wall
existed, even if scarred or fractured. It may be asked, if the wall
repels a ball striking it, what was it doing until struck ? Can it have
been provoked by being struck, to an action which is momentary, as
a man, we think, may be ? Or is it acting continuously in a way
whose effects vary with varying circumstances ? If we are not to
personify the wall, we must adopt the latter view ; and such terms
as vis inertiae and ' energy of position ' are evidence of the attempt
to reconcile the abrupt occurrence of noticeable changes with the
continuous action of things. In the last resort, we seek to formulate
laws of the action of things, from which we can deduce the changes
that will occur under varying circumstances and in various periods
of time. And having done this, we may disregard the troublesome
questions connected with the nature of action, and treat our laws
as enabling us to determine, from the state of things existing at one
time, what state will exist, or has existed, at another. Even so,
we still find ourselves assuming the existence of things ; for a state
of things is not an event that happens without happening to any-
thing.

These laws, we saw, are principles of connexion between one
change and another in a thing, or between change in one thing and

RULES OF CAUSE AND EFFECT 427

change in another. And all inference from experience rests on
universal connexions in nature. If, for example, there are no
circumstances material to the occurrence of a landslip, it would be
foolish to expect that any examination of the circumstances under
which landslips have been found to occur would enable us to deter-
mine under what circumstances they will occur in the future ; but
to say that certain circumstances are material to its occurrence
means that in a like situation they would always produce such
a landslip. If then we can detect these connexions, we can generalize.
Our problem is, how to detect them.

Now a full account of these connexions requires us to pierce into
the composition of things, and consider the operation even of their
minutest parts. But at the ordinary level of enquiry and for many
practical purposes we trace connexions between the changes that
occur in such aggregates as we can sensibly distinguish and are
interested in, like the land which slips and the rain which loosens it.
And though we know that events must befall things, we take for
granted in our formulation of connexions the things which they
befall, and seek in one change the cause of another — in rain-
fall, for example, the cause of a landslip. The discovery of
causes is indeed popularly regarded as the task of an inductive
science. We may put its questions in the form ' What things
how acting produce what changes in other things ? ' But as
the action of the cause is displayed in a change in itself, or takes
effect upon the occurrence of some other change, the form of question
more commonly put is not that just given, but rather ' What change
in one thing produces what change in another ? ' Whatever the
philosophic imperfection of this formula, yet since such changes
are connected through the causality of things, and our practical
interest lies in discovering connexions of change, we come to speak
of the changes, or events, causally connected as causes one of
another, and of causal relations as lying between them. Often, as
by J. S. Mill, the terms said to be connected as cause and effect are
called phenomena. This word is convenient because it can be used
either of an event, like the fall of a thunderbolt, of a thing, like
the thunderbolt itself, of an attribute, like the velocity of its fall, or
even of a law, like gravitation.1 The difficulty of any description of

1 Mill supposed that by phenomenon he meant not a thing, since he deemed
things themselves unknowable by us, but the appearance or the state of con-
sciousness which that produces in the mind, and whereby we know it. Such

428 AN INTRODUCTION TO LOGIC [chap.

inductive reasoning at once short and accurate arises largely from
the fact that sometimes what we are concerned to discover is the
things, whose agency is involved in the production of an effect,
as in asking for the cause of a disease we might wish to know what
microbe produces it : sometimes it is the attribute in things, which
makes them capable of producing it, as if we asked why blankets
keep us warmer than sheets do : sometimes it is the change in
things, with the occurrence of which the effect is connected, as water
freezes when it falls to a certain temperature : sometimes it is a law
exemplified in the succession of one change upon another. If we
include ah these kinds of question together as enquiries into the
causal connexions of phenomena, we must remember that we are
sacrificing precision to brevity, and that our formula has different
meanings on different occasions.

With these cautions we may proceed to consider how causal con-
nexions are detected. They cannot themselves be perceived.
Events occur and are observed ; the lines of causation that connect
them are not observed. It is here comes in the working importance
of the uniformity which is involved in the conception of a causal
relation. All manner of events are occurring simultaneously at
every moment ; and the events of one moment, taken in the lump,
must be causally connected with those at the next.1 But which
is connected with which, the single experience of their succession
will not determine. A man may run for an hour round his garden
on a frosty night, and when he wakes up next morning may notice
that his legs are stiff, and the dahlias in his garden blackened. If he
had really no other experience of such events than in this succession,
he might equally well conclude that the frost had made him stiff and
his running blackened the dahlias, as vice versa. But it is involved

a meaning cannot be maintained in science or defended in philosophy. Nor
does the word mean, as is sometimes stated, anything that can be perceived
by the senses ; it seems to be used to cover any thing, property, principle or
event that can be a subject of scientific investigation, or used in a scientific
explanation of what is investigated.

1 It may be said that an event of to-day may be due partly to some event
that occurred a long time ago : for example, a man may inherit a fortune on
his twenty-first birthday in virtue of a will made before he was born. We
shall see later that it is by no means always practically convenient to call
the immediately preceding conditions the cause : and others remoter may
without offence usurp the name. But the legatee becomes possessed of hia
fortune because he has just attained the age of twenty-one to-day ; and the
will may be regarded as having initiated a persistent legal position as regards
the money; so that the statement in the text may be deemed sufficiently
accurate in the context which it is intended to elucidate.

xx] RULES OF CAUSE AND EFFECT 429

in the causal relation that if two things are really cause and effect,
the one never exists without the other ; and hence by comparison of
that experience with others, he might conclude that running round
the garden did not blacken dahlias, because at another time they
had not gone black after he had been running round it ; and that
frosty nights did not make his legs stiff in the morning, because he
had waked up after another frosty night without any stiffness in
them. So far he would only have disproved the connexions to
which his mind at first had jumped. To prove that frost does
blacken dahlias, and that it was the running that made his legs
stiff, is a more difficult matter ; for the mere fact that one has
been followed by the other many times constitutes no proof. Yet
the repetition of the same event under different circumstances is
constantly narrowing the field of possibilities ; for no two events
can be precisely cause and effect, of which one in any case occurs
without the other ; so that if we can show that out of all the cir-
cumstances under which the blackening of dahlias has been observed
to occur, a frost is the only one that has not also on another occasion
either occurred without such an effect befalling the dahlias, or
failed to occur when it has befallen them, we may conclude that
there is nothing except the frost to which their blackening can be
attributed.

In this example we find the simple principle upon which the
reasoning of induction rests : though the successful prosecution of
inductive science requires very much besides such reasoning. The
cause of any effect — in the strictest sense of that relation — is so
related to it, as to occur whenever the effect occurs, and never
when it does not ; and to vary or be constant as the effect varies
or is constant, when susceptible of variations in quantity or degree.
From this it does not follow that because in a limited number of
instances some particular two phenomena a and x have been observed
to be present and absent, to vary and be constant together, they are
related as cause and effect ; since there may be another pheno-
menon b which also satisfies the conditions, and it is impossible so
far to tell whether a or 6 or the combination of them is the cause of x.
But it does follow that nothing is the cause of x which fails to satisfy
the conditions ; and it is upon that consideration that all discovery
of causes from experience rests. In saying this we do indeed
but repeat what was said in reference to the ' New Induction ' of
Bacon.

430 AN INTRODUCTION TO LOGIC [chap.

Thus inductive reasoning rests upon understanding what is
involved in the causal relation l; for unless we know this, we cannot
know that certain phenomena do not stand to each other in that
relation. And from the nature of this relation proceed what may be
called Topics of Cause, or rules whereby to judge whether two
phenomena are thus related to each other or not : just as from the
definition of Property proceeded what Aristotle called Topics of
Property, or rules whereby to judge whether a given predicate was
or was not a proprium of a given subject. But you can only
prove that they are related as cause and effect by proving that
there is nothing else with which either of them can be causally
connected.

J. S. Mill formulated four ' Methods of Experimental Enquiry \
or as he also called them, ' Inductive (or ' Experimental ') Methods,'
to which he attached considerable importance in his System of
Logic.2 He called them the Method of Agreement, the Method
of Difference, the Method of Residues, and the Method of Con-
comitant Variations. Among other defects of his exposition, his
treating these as so many separate methods darkens in a special
degree the subject of induction.

We shall be able to appreciate the nature of this defect if we
realize that the essence of inductive reasoning lies in the use of
facts to disprove erroneous theories of causal connexion. It is,
as Mill himself asserts, a process of elimination.3 The facts will
never show directly that a is the cause of x ; you can only draw that
conclusion, if they show that nothing else is. In order to show
that nothing else is, it is of course in the first place necessary to
know what other circumstances there are among which the cause
might be sought ; we cannot ' single out from among the circum-
stances which precede or follow a phenomenon those with which
it is really connected by an invariable law ' (to borrow an instructive
phrase of Mill's 4) unless we have ascertained what circumstances
do precede or follow it on divers occasions. But as to do that is no
part of the inductive reasoning which we are now considering, we
may for the present neglect it, or assume it to have been done.
The important thing to notice here is, that we do not discover what
is the cause, except by eliminating the alternatives. Now it is very
often impossible to do this completely ; nevertheless the nature of

1 Cf. Poste, Sophistici Elenchi, Appendix D, p. 221.

' System of Logic, III. viii. 3 e. g., ib. § 3 init. * lb. § 1 init.

2

xx] RULES OF CAUSE AND EFFECT 431

our reasoning is precisely the same, when we are left with the con-
clusion that the cause is either a or 6 or c, as if we had been able
to eliminate b and c also, and so determine that the cause is o.
Moreover, it makes no difference to the nature of our reasoning, as
a process of advancing to the proof of the cause by the disproof of
the alternatives, what the principle is to which we appeal in order
to disprove them. We know that nothing is the cause of x which
does not satisfy certain conditions — which is not present whenever
x occurs and absent when it does not, which does not vary or remain
constant as x does so. It is sufficient to be able to show that one of
these conditions is not satisfied by a given circumstance p, in order
to conclude that p is not the cause of x ; and which condition it i3
does not matter in the least. It is unlikely that in any particular
investigation every alternative hypothesis which we disprove as
to the cause of the phenomenon that we are studying will be rejected
because it fails to satisfy the same one of these conditions ; the
facts of our experience will probably show us one occurring where
the phenomenon is absent, and the phenomenon occurring in the
absence of another, a third unaffected in quantity or degree through
all the variations of the phenomenon, and so on. All that is essential
to the progress of our enquiry is that we should be able to show some
fact inconsistent with supposing such and such an alternative to be
the cause ; then that alternative is eliminated, and the cause must
lie among the rest.

The essence, then, of these inductive enquiries is the process of
elimination. The reasoning is disjunctive. And the character
of the reasoning is unaffected either by the completeness of the
elimination (i.e. the fact that there are no alternatives left in the
conclusion) or by the ground of elimination used. Yet Mill has
so formulated his ' Methods ' as to make it appear (a) that they
are only used when the elimination is complete ; (b) that they are
different when the ground of elimination is different. From this
it follows that very few inductive reasonings really conform to
any of them ; but the credit which this part of his work has obtained,
and still more the currency given to the names of his ' Methods ',
in which his doctrine is enshrined, threaten us with a repetition of
the same sort of mischief as arose from supposing that every argu-
ment could be put into the form of a syllogism. Just as arguments
not syllogistic at all were forcibly tortured into the appearance of it,
to the destruction of any proper understanding of what syllogism

432 AN INTRODUCTION TO LOGIC [chap.

really is, and how it differs from other forms of reasoning, so in-
ductive arguments are now often forced into a pseudo-conformity
with the canon of one of these ' Methods ', to the utter confusion
of the mind. For in the process, we are made to allege that some
circumstance is (say) the only one in which a number of instances
of a particular effect agree, in order to conclude in accordance with
the canon of the ' Method of Agreement ' that it is therefore its cause,
when we know perfectly well that it is not the only such circum-
stance ; and as we know that it is not by such assumptions that we
really conclude that circumstance to be the cause, we are only con-
fused by a Logic which makes it appear that it is.

There are passages in Mill's work (as is often the case with him)
which implicitly correct his own error. In speaking of what he
calls the ' Method of Agreement ', he writes : ' The mode of dis-
covering and proving laws of nature, which we have now examined,
proceeds on the following axiom. Whatever circumstance can be
excluded, without prejudice to the phenomenon, or can be absent
notwithstanding its presence, is not connected with it in the way
of causation. The casual circumstances being thus eliminated, if
only one remains, that one is the cause which we are in search of :
if more than one, they either are, or contain among them, the
cause ; and so, mutatis mutandis, of the effect.' x It is plain from
this that I am not the less reasoning in accordance with this method,
because I am only able to say in the conclusion that the cause of
the phenomenon is one or other of several alternatives, than if
I were able to offer a definite solution. Yet this is quite ignored in
what immediately follows : ' As this method proceeds by comparing
different instances to ascertain in what they agree, I have termed
it the Method of Agreement ; and we may adopt as its regulating
principle the following canon : —

' // two or more instances of the phenomenon under investigation
have only one circumstance in common, the circumstance in which
alone all the instances agree is the cause (or effect) of the given
phenomenon.'

Every one who has tried knows how difficult it is to find examples
to which this canon can be applied ; for it is seldom that instances
of the subject under investigation have only one circumstance in
common. Where such instances are forthcoming, they are pecu-
liarly instructive to the investigator ; and therefore Bacon placed

1 Logic, III. viii. 1 ad fin.

xxl RULES OF CAUSE AND EFFECT 433

them first in his list of Prerogative Instances (i.e. instances to be
consulted first), under the name of Instantiae Solitariae.1 But
what if your instances have several circumstances in common ?
Are they therefore useless to the investigator ? Throughout the
organic world it is observed that species present a number of adaptive
structures — that is, structures fitting them for the conditions under
which they have to live. To the question how this has come about
several answers have been suggested ; one, the oldest, attributed
them to special design on the part of the Creator : another to the
inherited effects of use and disuse : another to the survival of
those individuals who happened to be born with a body more suited
in any respect than their neighbours' to the conditions of their life,
combined with the elimination of the less fit. Now if it is pointed
out that some adaptive structures, like the horny back of a tortoise
or the shell of a mollusc, cannot be improved by use as a muscle can,
one of these suggestions is overthrown, at least as a complete solution
of the problem ; but it remains doubtful so far whether we are to
refer the structures in question to design or to natural selection :
yet we have certainly made some way in our enquiry, and this
argument is part of our inductive reasoning. Mill's canon, however,
is inapplicable to such a case as that, because the tortoise with his
horny back, and the elephant with his powerful trunk for seizing
branches, though both possessing adaptive structures, which may
in both have been established by natural selection, are not instances
with only one circumstance in common. It is excellent advice to
see in what the instances of your phenomenon agree ; but the
ground of the advice is that you may eliminate the circumstances
in which they differ ; and the principle at the foundation of the
1 Method of Agreement ' is not that ' the sole invariable antecedent
of a phenomenon is probably its cause ' 2, for the ' Method ' is often
employed when we can find no sole invariable antecedent ; it is that
nothing is the cause of the phenomenon in the absence of which it occurs,
The same defect appears in Mill's formulation of the ' Method of
Difference '. In seeking to discover on what conditions an effect
of a certain kind3 depends by way of eliminating all that can be

1 Nov. Org. II. 22, where instances such as are required by Mill's^Method
of Agreement and by his Method of Difference are described under this name.
And this is the proper way to treat them — not as instances the use of which
constitutes a distinct method of inductive reasoning.

2 Jevons, Elementary Lessons in Logic, p. 241 (1880).

■ And mutatis mutandis, if seeking to ascertain the effect of a given cause.

177» F f

434 AN INTRODUCTION TO LOGIC [chap.

shown to be irrelevant, we may rule out not what is common to
different instances in which the effect occurs, but what, among all
the circumstances in which in some instance it occurs, can be shown
equally to exist without the effect occurring. In technical phrase-
ology, instead of comparing various positive instances of the effect,
we may compare a positive and a negative instance : a negative
instance being an instance of circumstances similar to those wherein
the effect occurs, where the effect is nevertheless absent. Mill's
canon for this procedure runs thus : —

' // an instance in which the phenomenon occurs and an instance in
which it does not occur agree in all circumstances but one, the circum-
stance in which alone the two instances differ is the cause, or effect,
or an indispensable part of the cause, of the phenomenon.'
He here implies that the use of this Method depends on finding
a positive and negative instance agreeing in every circumstance but
one. This is not indeed so far beyond achievement as it commonly
is to find a number of positive instances agreeing in only one circum-
stance ; for experiment, when we introduce a factor into or remove
it from an existing situation, provides a positive and a negative
instance ; and we may be able to determine very exactly what factor
we thus vary in a situation maintained otherwise the same. But
even in experimenting the change introduced is often highly complex ;
and situations not artificially produced and maintained are subject
to any number of simultaneous changes. Yet if in the course of
them the effect under investigation should arise or disappear, we
are not precluded by their number from arguing that those elements
which have been the same before and after the emergence or dis-
appearance of the effect do not account for it. We have one positive
and one negative instance. They differ in many more circumstances
than one ; but still the phenomenon must be connected with some-
thing in the total difference, and still the circumstances present alike
in the positive and the negative instance are thereby shown not
fully to account for it.

Again, so obvious is the difficulty of finding such instances as
these canons require, that Mill, having begun by mentioning four
methods (of Agreement, of Difference, of Residues, and of Concomitant
Variations), adds a fifth, which he calls the Joint Method of Agree-
ment and Difference. Such instances as the ' Method of Agree-
ment ' and the ' Method of Difference ' are supposed to require —
positive instances agreeing, or a positive and a negative instance

xx] RULES OF CAUSE AND EFFECT 435

differing, in one circumstance alone — may not be forthcoming ; and
therefore, under the name of the Joint Method, Mill describes the
case in which you look for a circumstance about which it can be
said that it is the only one that is neither absent in any instance
where the phenomenon occurs, nor present in any where it does not.1
Here then both grounds of elimination are employed ; but there is
no reason in the world, as a study of his account of his Methods
would show, why he should not have had other Joint Methods,
of Difference and Concomitant Variations, or of Agreement and
Residues, and so forth. An enquiry into the cause of one pheno-
menon need not confine itself throughout to one ground of elimination.
For the above reasons it would be well to recognize that Mill
has not formulated four (or five) but one ' Method of Experimental
Enquiry ' — as indeed Bacon might have shown him ; of which the
essence is, that you establish a particular hypothesis about the cause
of a phenomenon, by showing that, consistently with the nature
of the relation of cause and effect, the facts do not permit you
to regard it as the effect of anything else (and mutatis mutandis if
you are enquiring into the effect of anything). It is this which
makes the reasoning merely inductive. If you could show in
accordance with known or accepted scientific principles that the
alleged cause was of a nature to produce the effect ascribed to it,
your reasoning would be deductive ; leaving aside the question how
those scientific principles were ascertained, you would be reasoning
from them to a conclusion which you see to be involved in their
truth ; and if we suppose the principles to be of such a nature that
we can see they must be true, then the conclusion will appear

1 Mill's canon for the ' Joint Method ' is by no means carefully worded
{Logic, III. viii. 4). It would be better if for ' the circumstance in which
alone the two sets of instances differ ' we read ' the circumstance in which
alone the second set of instances agrees to differ from the first set '. Note
that Mill represents it as necessary, under the terms of the Joint Method, to
show of every other circumstance than that which is alleged as cause in the
conclusion both that it is absent in some instance where the phenomenon
occurs and that it is present in some instance where it does not. This is
because he develops it as an answer to the objection, that although a circum-
stance b is absent in a particular instance of x there is no reason why it should
not cause x on another occasion. The difficulties created by the so-called
Plurality of Causes will be considered later. The point in the text here is,
that it is quite possible, and very common, to show that one circumstance
is not the cause on one ground — say that the phenomenon occurs without it,
and another on another ground — say that it occurs without the phenomenon,
and a third on a third ground — say that it is variable while the phenomenon
is constant, all in the same investigation.

F f 2

436 AN INTRODUCTION TO LOGIC [chap.

necessary, and a thing that could not conceivably be otherwise.
Take, for example, the maxim that men hate those who have con-
ferred a benefit on them.1 We may regard that as, in the first place,
discovered inductively from the consideration of many instances of
ill will, which are unaccountable otherwise than on that principle ;
yet so far it remains a thing obscure and unintelligible, a relation
which the facts forbid us to dispute, but in which we see no neces-
sity. Now if a man were to say that men hate those who cause
them what they dislike, that they dislike to feel themselves in
a position of inferiority, and that they do feel themselves in a
position of inferiority to those from whom they have received a bene-
fit, the maxim follows deductively ; and these principles are not
only, like the original maxim, capable of being inductively supported
by an appeal to experience, but they are also intelligible to us in
a way in which that was not ; it is mercifully untrue to say that they
appear necessary, but they do appear more or less natural, and we
see that such men must hate their benefactors. Where, however,
we have to rely purely on induction, there is none of this ' natural-
ness ' : I stand on my conclusion because ' I can no other ', and
not because I see any intrinsic necessity in it. Necessity I do see,
if I am right about my facts, and am to reason in this case con-
sistently with what I know to be involved in the causal relation ;
but that necessity is not intrinsic ; had the facts been otherwise,
and for all I can see they might have been, I should have concluded
otherwise ; and then I should have been just as content to accept
that as I now am to accept this conclusion.

There is an enormous number of general propositions, which we
accept for no better reason than that the facts are inconsistent with
our denying them, and not because we see anything in what they
state which could have led us to suppose them true, antecedently
to our experience. When it is said that we ought always to follow
experience, it is meant that we ought not to trust our notions of
what seems antecedently fit to be true, or mere guesses as to the
connexions that subsist in nature, but accept only those connexions
which our experience forces us to accept because it is inconsistent
with any alternative. Such reasoning is called a posteriori, because
it starts from the facts, which are conceived as logically dependent

1 Of course this, like most maxims with regard to human nature, is not
an universal truth : what kind of men hate those who have conferred a benefit
on them would be the next subject for enquiry.

xx] RULES OF CAUSE AND EFFECT 437

on, or posterior to, their principles, and thence infers the principles
on which they are dependent. Conversely, deductive reasoning is
often called a priori, because it starts from general principles, which
are conceived as logically prior to the particular facts that accord with
them.1 When a priori reasoning is condemned, it is not meant that
we are never to reason deductively, but only that we are not to
reason from principles that are not warranted by experience ; at any
rate this is the only sense in which the condemnation can be justified.
But it is an error to suppose that all general principles are arrived
at a posteriori, or by process merely of showing that facts are not
consistent with any other ; the Law of the Uniformity of Nature
itself, as we have seen, is not arrived at in that way, since if we once
doubt it, it is impossible to show that the facts are any more in-
consistent with its falsity than with its truth ; neither are mathe-
matical principles so arrived at : we do not believe that three times
three is nine, because we show successively that it is not five or ten
or any other number except nine. Still it is true that in the in-
ductive sciences the vast majority of our generalizations are reached
either in this a posteriori manner, or by the help of deduction from
other generalizations so reached. And it may be well to show by one
or two examples how generalizations that rest merely on induction
present as it were a blank wall to our intelligence, as something at
which we cannot help arriving, but which we can in no way see through
or find intrinsically plausible. Facts show that the excision of the
thyroid gland dulls the intelligence : could any one see that this
must be so ? Some explanation may be afforded by showing that
on a contribution which the gland, when properly functioning,
makes to the circulating blood depends the health of the brain ;
but that comes later than the discovery of the effects of excision ;
and even so, can we understand the connexion, which facts establish,
between the state of the mind and the health of the brain? Or
take a thing more frequent and familiar. It seems perhaps the most
natural thing in the world, that we should see with our eyes, hear
with our ears, taste with our palate, and so forth. Yet for ail
that we can see a priori, it might just as well have been that we
should see with our ears and hear with our eyes, smell with our

1 Or, in another sense, illustrated in most mathematical reasoning, because
the premisses, without being more general than the conclusion, or giving
the cause why it is true, are not based upon an appeal to facts which might
conceivably have been otherwise : cf. supra, p. 210, n. 2 ; infra, p. 545, n. 2.

438 AN INTRODUCTION TO LOGIC [chap.

palate and taste with our fingers. Doubtless if we tasted with our
fingers, we should not have to eat in order to taste ; there might
be some advantages in that, and at any rate it is not antecedently
inconceivable. It may be said that the mechanism of the eye, by
which light is focused from many points at once upon the extended
surface of the retina, and the eye is readily turned in any direction,
makes it a priori a more suitable organ of sight than the ear could
be ; and it is true that upon the assumptions that light-sensations
are produced by the stimulation of a nerve, that this stimulation
is supplied by wave-motions in the ether, that distinguishable
colours are produced by differences in the wave-length, and that
the arrangement of coloured points in the visual field corresponds
to that of the nerve-fibres appropriately stimulated in the retina,
we can find in the eye an excellent apparatus for securing clear
vision. There is nothing, however, in those assumptions (which
have only been proved inductively) that is any more intelligible
to us than if the wave-motions of the ether stimulated the fibres of
the ear, and those of the air the fibres of the retina ; though doubt-
less our vision would be less serviceable in the latter case. There
is in fact no psycho-physical correspondence that is at present
intelligible to us, although particular correspondences may be in-
telligible in the sense of conforming to the more general principles
which we have found to prevail. The same may be said with regard
to the properties of chemical compounds, which are not for the
most part intelligible from a consideration of the properties of their
elements ; hence in saying that they depend upon the composition
of the substance we rely merely upon this, that no other view con-
sists with the facts which we have observed in our experiments.
The largeness of these two classes of inductive generalizations may
perhaps make it unnecessary to illustrate further what Bacon would
call the ' surd and positive ' * character of conclusions resting only
on induction ; but, as showing how the mind desiderates something
better, we may notice the attempt continually made to conceive
chemical as at bottom only physical processes. In the physical
process, the successive stages do to some extent at least appear to
follow necessarily one out of another ; on their mathematical side,
the principles that connect them are not mere matter of fact, but
matter of necessity which we cannot conceive otherwise. Hence
the attraction of reducing chemical processes to physical terms. It
1 De Principiis atque Originibus, Ellis and Spedding's ed., III. p. 80.

xx] RULES OF CAUSE AND EFFECT 439

is true that the appearance of new sensible properties in bodies in
virtue of their physico-chemical composition is not hereby explained ;
but it is supposed that they only possess these for us : that the
appearance is subjective, or in other words that while the processes
in bodies themselves are purely physical, we are determined to
receive qualitatively different sensations by different physical
stimuli. There is not much prospect at present of rendering psycho-
physical correspondences really intelligible ; thus there is a tempta-
tion to regard the emergence in a chemical compound of properties
which cannot be seen to have any necessary connexion with the
properties of its elements as merely a fresh case of that psycho-
physical correspondence which we already admit that we can ascer-
tain and not understand : in order that we may if possible find in the
principles of chemistry itself something intelligible, and not merely
necessary to be admitted. The gain is more apparent than real ;
but the procedure betrays a sense that though it may lead us far
and win us much, induction turns out at last to be the blind alley
of the reason.

We must return, however, from these general considerations upon
the nature of induction to the particular inductive reasoning which
rests upon our knowledge of the requirements of the causal relation.
By and by we shall find that reasoning which is really inductive
enters into processes of a more complex and partially deductive kind.
What we are at present considering is in principle quite simple.
The cause of a phenomenon x is to be sought among those circum-
stances under which it occurs in the instances that we take. The
causal circumstances are found by a process of exhaustive elimina-
tion. Those which are not causal can be eliminated because the
facts show that in regard to this phenomenon they do not satisfy
the conditions of a cause. Now the grounds on which we may
eliminate are these ; and each points to some particular requirement
of the causal relation, failure to satisfy which disproves that relation
as between two given phenomena :

1. Nothing is the cause of a phenomenon in the absence of which

it nevertheless occurs.

2. Nothing is the cause of a phenomenon in the presence of which

it nevertheless fails to occur.

1 Or mutatis mutandis the effect. I shall not complicate the exposition
by always adding this.

440 AN INTRODUCTION TO LOGIC [chap.

3. Nothing is the cause of a phenomenon which varies when it ia

constant, or is constant when it varies, or varies in no pro-
portionate manner with it.
To these may be added a fourth ground :

4. Nothing is the cause of one phenomenon which is known to be

the cause of a different phenomenon.

This last principle is also, like the others, involved in the general
conception of a reciprocal causal relation ; but in applying it we
appeal not merely to what we observe in the instances of the pheno-
menon under investigation, or in the instances where under more
or less similar circumstances the phenomenon does not occur;
we appeal also to previous generalizations regarding the connexion
of phenomena. These generalizations, however, are used not to
account for the connexion which we are now establishing — it is not
deduced from them ; but merely to exclude alternative explanations
of the present phenomenon, and so force us upon the one which we
finally accept ; and so far the reasoning which appeals to such
a ground of elimination is still inductive.1 But it belongs especially

1 On these grounds of elimination Mill's 'Inductive Methods' severally
repose. The first is the foundation of his ' Method of Agreement ', the second
of his ' Method of Difference ', the first and second jointly of his ' Joint Method
of Agreement and Difference ', the third of his ' Method of Concomitant
Variations ', and the fourth of his ' Method of Residues '. All of them are
quite general, and have been stated above in a way which only holds if in the
cause we include everything necessary and nothing superfluous to the pro-
duction of the phenomenon in question. The illustrations in the present
chapter are not confined to that, the strictest, sense of cause ; but the impor-
tant point involved will be considered later in Chapter xxii, on Non-reciprocat-
ing Causal Relations. Where the cause sought is a non-reciprocating cause,
Other principles call to be applied : e. g. we may wish to ascertain whether
some condition which cannot by itself produce an effect is indispensable to its
production ; and if such sine qua non be called a cause, that is a cause (in this
sense) whose removal from a situation is followed by the cessation of the
effect, though its restoration when the situation is otherwise changed is not
followed by the recurrence of it. Lotze, in Bk. II. c. vii. of his Logic, headed
Universal Inductions from Perception, has paid some attention in § 261 to the
formulation of principles of this kind, stating what degree of connexion between
two elements C and E can be inferred from what kind of observations with
regard to the circumstances of their occurrence. The section is eminently
worth consulting in reference to the nature of inductive reasoning ; and the
principles in question might all be called Topics of Cause, though some of
them are doubtful ; just as Aristotle recognized Topics which hold true in
application only for the most part. Hume too in Part III. § xv. of his Treatise
of Human Nature, Of the Understanding (already, like this chapter in Lotze,
referred to), gives a number of Rules by which to judge of Causes and Effects
which are derivative, but highly important, as for example that ' where several
different objects produce the same effect, it must be by means of some quality,

xx] RULES OF CAUSE AND EFFECT 441

to the later stages of a science, because it presupposes the discovery
of other causal connexions, as a means of prosecuting some present
enquiry.

It is plain that we cannot get to work in the application of these
principles, until we have at least provisionally conceived and learnt
to recognize the phenomenon we are studying, and ascertained and
distinguished the circumstances under which it occurs (or fails to
occur) from one another. And if all this were done, their application
would be an easy matter, as Bacon imagined he could make it. All
symbolic representation of such inductive arguments by letters of
the alphabet, where one letter stands for the phenomenon investi-
gated, and others for the circumstances among which its cause is
sought, presumes these tasks to have been achieved ; and thus it
is apt to convey a totally false impression of the degree of difficulty
attaching to inductive enquiries.1 The truth is, that inductive
reasoning is in form very simple ; but the discovery of the proper
premisses is very hard. As Hume well observes of the rules he

which we discover to be common amongst them \ But those in the text seem
to be really the ultimate principles, if a reciprocating cause is meant.

1 On the artificial simplification which letters of the alphabet also imply,
cf. Venn's Empirical Logic, c. xvii. pp. 406, 407. If they are to be used at
all, to which I see no objection so long as their limitations are understood,
it is important how we use them. In Mill's use of them, which has been
followed by Jevons, Elementary Lessons in Logic, and by Fowler, Inductive
Logic, and I dare say by others, there are two defects. He uses big letters to
symbolize ' antecedents ' or causes, and the corresponding small letters to
symbolize * consequents ' or effects. Now in the first place he has thus
always an equal number of big and small letters ; but when we are looking
for the cause of some phenomenon x, and seek it among a number of alterna-
tives A BC D . . . , we have not also before us effects as many as the alternatives
among which the cause of this phenomenon is sought. Only in symbolizing
his ' Method of Residues ' is this feature appropriate ; there certain circum-
stances collectively are supposed to be known to be the cause of a number of
effects (or of an effect of a certain quantity or degree), and out of these we
reject, as not the cause of one among the effects, those which we know to
produce the others (or if the question is one of quantity or degree, we reject
those whose total effect we know to differ from what we have to account for,
as not accounting for the remaining component). Hence separate symbols
for the effects (or components of the effect) of the various circumstances
among which the cause of one effect (or component) is sought, as well as
separate symbols for the causes, are required. The second objection is, that
he uses corresponding big and small letters (ABC followed by a b c, Sec.).
Now, as Mr. F. H. Bradley points out (Principles of Logic, p. 339, note *), the
letters are intended to symbolize the phenomena as presented to us before we
apply our inductive canons ; and therefore they ought not to imply, as by
this correspondence they do, that the phenomena themselves, as distinct from
the facts of their joint or separate occurrence, have anything about them that
proclaims which is the cause of which. Cf. also Professor Bosanquet's Logic1,
II. iv. vol. ii. p. 122.

442 AN INTRODUCTION TO LOGIC [chap.

gives ' by which to judge of causes and effects ', ' All the rules of this
nature are very easy in their invention, but extremely difficult in
their application.' * It is easy enough to see that if out of so many
alternatives a b c d . . . z, the cause of x is not 6 c d . . . or z, it must be
a ; and it is easy enough to see that if c occurs without x, it is not its
cause. But to show that c occurs without x, and to show some reason
for rejecting b d ... 2, as well, and to discover a h e d ... z, and show
that no other alternatives are possible — all these things are extremely
difficult. Something will be said of these operations in the next
chapter. Here we are concerned with the form of the reasoning,
which is of a disjunctive kind, and may be symbolized thus : —

The cause of x is either a or 6 or c or d ... or z
It is not b or c or d ... or z
.*. It is a.

In this argument the minor premiss is proved piecemeal by hypo-
thetical arguments that rest upon one or other of the above grounds
of elimination, or ' rules by which to judge of causes and effects '.

If 6 were the cause of x, it would be present whenever x is

present
But (in this instance) it is not.
If c were the cause of x, it would be absent whenever x is

absent
But (in that instance) it is not :

and so forth. Or if any one prefers it, he may represent this part
of the argument as a syllogism :

Nothing is the cause of x, in the absence of which x occurs
6 is a thing in the absence of which x occurs .'. &c.
Nothing is the cause of x, which varies without relation to it
d varies without relation to x. .". &c.

It is of course possible that bed . . . z may all be eliminated, or
shown not to be the cause of x, by the application of the same
principle or major premiss ; in this case the minor of the above
disjunctive argument might be proved en bloc, and not piecemeal ;
but this is by no means necessary, and in fact unusual, and does
not affect the nature of the argument. It is, however, the only
case contemplated in Mill's formulation of inductive reasoning. It
is also possible (and this Mill's formulation does not recognize at

1 Treatise of Human Nature, Of the Understanding, loc. cit.

xx] RULES OF CAUSE AND EFFECT 443

all) that we may not be able to prove the whole of the above minor
premiss ; and then our argument will take the form

The cause of x is either a or b or c or d ... or z
It is not c or d ... or z
.*. It is a or 6

or It is not d . . . or z
.'. It is a or b or c

where the degree of uncertainty symbolized as remaining at the
end of our enquiry is greater.

It appears plainly enough in this analysis how all induction rests
on the Uniformity of Nature ; for in proving the minor of the
disjunctive argument a principle is always appealed to, that would
fall to the ground if the Uniformity of Nature were denied. It is
not indeed necessary, in a particular investigation, to assume this
uniformity to extend beyond the department of facts with which
we are dealing ; if I am looking for the cause of cancer, it is enough
that cancer should be subject to uniform conditions in its occur-
rence ; and I should not be impeded in my research by the fact that
thunderstorms occurred quite capriciously. There is, however, no
ground for assuming cancer to be subject to uniform conditions in its
occurrence which does not apply equally to thunderstorms, or to
anything else that could be mentioned ; if I assume the principle
of Uniformity at all, I must logically assume it altogether ; and so,
though I may be said to appeal to it in any particular inductive
argument only so far as concerns the department of nature to
which my investigation belongs, I really assume it universally.1
Nevertheless it is not correct to say that it is the ultimate major
premiss of all inductions 2 ; for that implies that an inductive
argument is, formally considered, a syllogism, and we have seen
that it is not. It is indeed impossible to see how this principle can
be made the major premiss of any inductive argument as a whole,
though its particular applications may afford the major premiss
of an argument by which we prove any part of the minor in our
disjunctive argument. Let us say that ' Nature is uniform ', or
(since we can hardly make a middle term of ' Nature ', which in the
sense of nature as a whole is not predicable of any particular

1 Cf. what Aristotle says of the assumption of the Law of Contradiction
implied in all syllogisms, An. Post. a. xi. 77a 22-24.

2 Mill, System of Logic, III. iii. § 1 med.

444 AN INTRODUCTION TO LOGIC [chap.

subject) that ' All events in nature take place in accordance with
uniform laws ' ; we may then proceed to argue that ' Cancer is an
event in nature ', and therefore that it takes place in accordance
with uniform laws ; but we are thus no further advanced than we were
at the beginning, since so much is assumed in looking for a cause of it
at all. Or if we put our major premiss in the form ' Every relation of
cause and effect that is observed in any instance between one pheno-
menon and another holds good universally ', and then used as our
minor ' The relation between a and z is a relation of cause and effect
between one phenomenon and another observed in certain instances ',
we might indeed take the formal step of concluding that it holds
good universally (though that is already implied in calling it a re-
lation of cause and effect), but the whole question at issue is here
begged in the minor premiss ; for what we want to prove is just that
a is related to a; as a cause, and not in time only and accidentally.
For the formulation of the reasoning by which that is proved —
which is the inductive reasoning — nothing therefore has been done.
And any other attempt to reduce inductive reasoning to syllogism
with the principle of the Uniformity of Nature as ultimate major
premiss will be found equally unsuccessful.

It should be noted that the above account of inductive reasoning
holds good generally, whether by the cause of an event we mean
a thing to whose action it is due, or an attribute which things must
have which are to produce it, or an event with whose occurrence
it is connected by a law, or the law exemplified in its succession
upon another event. We cannot indeed speak of a law being
present or absent in particular instances. But whichever form our
problem takes, we ask ourselves what should happen if the cause
were thus or thus ; and if the effect is absent in any instance where
it should be present, or present where it should be absent, or is
constant or variable as it should not be, or different from what we
know that the cause suggested would involve, the suggestion is
wrong. Nor can any suggestion be inductively established, unless
it successfully, and it alone successfully, runs the gauntlet of such
questions.

It remains to illustrate by a few examples the truth of the con-
tention that inductive conclusions are established disjunctively by
the disproof of alternatives.

1. The power of the chameleon to change colour in accordance
with the colour of its surroundings is well known. But this power

xx] RULES OF CAUSE AND EFFECT 445

is not confined to the chameleon ; it occurs, for examples also in
certain frogs.1 The question raised is as to the cause of this
change. We have first indeed to show that the change is due in
some way to the colour of the surroundings ; that implies a pre-
vious inductive argument ; for so long as it was only noticed that
the frog changed colour from time to time, it would be quite uncer-
tain with what that change was connected. We may disregard such
suggestions as might occur to a collector of portents ; Livy gravely
records as portents of disaster some facts quite on a par with the
statement that ' a frog changed its colour in broad daylight ', but
it would be easy to show that the change had occurred at a time of
no disaster. But of the suggestions that might occur to a biologist
we may conceive the nature of the animal's food to be one : time of
day or season of year to be another : intensity of sunlight to be
a third, and so on ; but when it was shown that the frog might
variously change its diet, and still be of the same colour, and that the
change of colour might take place at any time of the day or year,
and in various degrees of sunlight, these suggestions would be dis-
carded, and so on until the only reasonable suggestion left was that
which connected the change of colour with the colour of the sur-
roundings. Of course this conclusion would acquire great strength
so soon as any one noticed the frog in the process of changing colour
upon removal to new surroundings ; for if the change of colour
is to be connected with some other change that has just occurred,
the range of alternatives is thereby much limited. The preliminary
induction implied in saying that it changes colour according to the
colour of the ground on which it rests need not, however, be further
considered ; we wish to know more precisely what produces the
change. Now differently coloured grounds may vary in tempera-
ture as well as in colour ; but it can be shown experimentally that
the colour-reaction is independent of temperature. Granting then,
in the absence of any other alternative, that it depends on the colour
as such, we may ask in what way the differently coloured rays3
affect the animal. Lord Lister showed that they affected it through
the eyes ; for a specimen of Rana temporaria whose eyes had been
removed was no longer affected by any change in the colour of the

1 This example is taken from Dr. Vernon's Variation in Animals and Plants
(Internat. Scient. Series), pp. 255 seq.

2 To speak strictly, rays are not differently coloured, but of different wave-
lengths.

446 AN INTRODUCTION TO LOGIC [chap.

surroundings in which it was placed ; thus the alternative, other-
wise not unreasonable, is excluded, that the reaction is somehow
determined through the skin, the principle applied being that no
circumstance in the presence of which the phenomenon fails to
occur is its cause. This conclusion is further confirmed by the fact
that in other species that normally exhibit a similar colour-reaction
individuals have been found, in whom the power of adjustment to
the colour of their surroundings is absent, and that these individuals
on examination have been ascertained to be blind ; but it may still
be asked how the stimulation of the eye by different kinds of light
effects the colour-change. Perhaps there are two alternatives here ;
it might be necessary for the frog to be aware of the colour of its
surroundings, or there might be a reflex mechanism. The latter
is supported by the fact that a blinded frog, after a violent struggle
to escape, changed from dark to light, but in half an hour, though
placed in a bright light, became almost coal-black again. Here
it is shown that a colour-reaction can take place without awareness
of colour ; so that awareness of colour is eliminated from among
the conditions necessary to the production of the reaction, on the
principle that a circumstance in the absence of which thephenomenon
nevertheless occurs is not its cause. We must look then for some
circumstance common to a blind frog changing colour after a violent
struggle, and a normal frog changing colour with a change of sur-
roundings ; and we may find this in nervous excitation, for that may
be produced by the action of light upon the eye, and also by the
struggle. Until some other feature common to the two cases was sug-
gested, we should accept this on the principle just cited ; but it is
also supported by the known physiological function of the nervous
system in the building up of reflexes ; it consists too with the fact
that when the excitement subsided the blind frog returned to a
colour not adapted to its environment. Yet how can the animal's
colour be affected by different kinds of nerve-stimulation ? There
have been found in the skin of the frog pigment granules of divers
colours, so arranged that different surface effects can be produced
by different degrees of concentration in the granules. The final
connexion of colour-reaction in the frog with these pigment granules
is indeed rather deductive than inductive ; for the part which
efferent currents from the nerve-centres play in provoking muscular
contractions and relaxations is already known, and so is the fact that
an afferent nerve-current discharges into an efferent nerve ; and we

xx] RULES OF CAUSE AND EFFECT 447

have just shown that the colour-reaction is connected with afferent
nerve-stimulations.

2. Let us take next a simpler example, and one in which there is
little or no generalization : for inductive reasoning may be applied
to discover the cause of a single event, as well as of events of a cer-
tain kind ; and it is not necessary to carry the analysis (of which
more in the next chapter) so far as to make a general conclusion
possible. Let a novice notice that his bicycle makes an unpleasant
noise in running, and try to ascertain the cause. We are to suppose
a novice, because a rider of any experience may be presumed already
to have arrived by induction at the knowledge that one kind of
noise is made in the chain, and another kind in the bearings ; and
the application of this previously acquired knowledge to a particular
case would be deductive. In this problem the determination of the
alternatives among which the cause is to be sought is tolerably
simple ; for the noise must originate in one or other (or it may be
several) of the non-rigid parts. Say that these are, on the machine
in question, the axle-bearings of either wheel and of the cranks, the
bearings of the head, the pedal-bearings, the clutch, the back-
pedalling brake, and the saddle-springs. All that the rider has to
do is to ascertain which of these parts may be at rest while the
noise occurs, and which may be in motion without the noise. If
the noise ceases in free-wheeling, it is not produced in the axle-
bearings of either wheel, for they are still running, and that is not
the cause, in the presence of which the phenomenon fails to occur ;
for the same reason it is not in the bearings of the clutch, which is
now running. If it is not produced in ' wobbling ' the head, or
turning sharp corners, he may acquit the bearings of the head on
the same principle. If it occurs in driving with each pedal singly,
it does not arise in either pedal-bearings, because it occurs with
each pedal in turn undriven, and that is not the cause in the absence
of which the phenomenon occurs. Similarly if it occurs without
putting on the back-pedalling brake, or when he removes his weight
from the saddle, it does not originate in either of those quarters.
Two alternatives remain : it may be in the crank axle-bearings, or
in some looseness of the clutch when that is caught and driving.
Between these alternatives a decision might be made if he dis-
mounted, and listened while he whirled the hind wheel round by the
pedals ; here however he would be reasoning deductively from the
principle that sounds are more distinct when you are nearer to their

448 AN INTRODUCTION TO LOGIC [chap

point of origin. The difficulty of generalizing in such a case arises
from the difficulty of distinguishing the phenomenon investigated
from others that may be like it but have different causes. If the
noise which each part of his bicycle could make were of a distinctive
kind easily recognized, a man might very soon determine that such
and such a noise (at least in his bicycle) only originated in such and
such a part ; and further experience, argued from on similar lines,
might show him that a particular character in a noise was due to
want of oil in a bearing, and another character to a broken ball. But
so long as the phenomenon studied is submitted to no such scrutiny,
it is liable to be confused with others that are not really the same,
and error would obviously arise if we generalized together about
this noise and others like it but differently caused. Hence one may
have to be content with a conclusion that assigns the cause of it in
the particular case. It is, however, instructive to observe that the
same process of elimination among the members of a disjunction is
employed here, as if one were establishing a general conclusion. For
ex hypothesi the novice recognizes in the noise no intrinsic character
which he knows to be connected according to any principle with
a particular origin ; he has therefore to fall back upon ascertaining
its origin by the indirect method of showing that among the possible
origins to which it can be ascribed there is none but one to which
the facts permit him to ascribe it consistently with the principles
of causation.

3. Professor Weismann's theory of the ' Continuity of the Germ-
Plasm ' is well known. The reproductive cells, whether of a plant
or animal, are different in certain important respects from those
which compose other parts and tissues (called somatic or body-cells);
in particular, whereas the latter, in the process of increase and division,
produce only cells of one kind, such as compose the part or tissue
to which they belong, the former produce cells of every kind that
occurs in the organism, and, in fact, are capable of reproducing the
whole organism and not merely a special part of it.1 In so doing
they must, of course, reproduce the reproductive cells also, in order
to provide for the following generation. Now Weismann held that
in the division of the reproductive cell, or germ-plasm, a part is set
aside from the outset to serve the purpose of reproduction once
more, and that this, which is still germ-plasm, remains as it were

1 Sometimes however body-tissue will reproduce a complete organ, or
even (as the leaf of a begonia will) the whole organism.

xx] RULES OF CAUSE AND EFFECT 449

isolated in the developing organism, and unaffected by the other and
heterogeneous parts, or somatoplasm, which also arise in the division
of the reproductive cell ; and as this happens in each generation,
there is an absolute continuity of the germ-plasm ; from which it
follows in his view that no characters acquired by the individual in
the course of its lifetime and not congenital can be transmitted to its
offspring ; for a character so acquired arises in the somatoplasm, and
the germ-plasm is from the first secluded from the possibility of being
affected by the somatoplasm. Influences which reach the germ-
plasm can alone modify subsequent generations ; of which (at least
in metazoa) the most important is the fusion of two reproductive
cells that takes place in sexual propagation ; here the germ-plasm
of the ovum blends with another germ-plasm conveying more or
less different heritable tendencies, and a sort of shuffling takes place,
as a result of which there arises a new individual resembling pre-
cisely neither parent, but exhibiting those ' spontaneous variations ',
as Darwin called them, which form the material for Natural Selection
to work upon. Darwin himself, on the other hand, believed that
' acquired characters ' may in certain cases be inherited, and
that it is very difficult to account entirely for the progressive
modification of species in adaptation to their environment, without
allowing the influence of this so-called ' Lamarckian ' factor.1 The
question has formed a subject of protracted controversy among
biologists, and it is not an easy one to settle conclusively on induc-
tive principles by appeal to evidence, because most facts admit of
being interpreted in either way. One of the most important in-
vestigations into the subject 2 is a series of experiments on guinea-
pigs, conducted during thirty years by Brown-Sequard and extended
by two or three other naturalists ; and it is claimed that in the course
of these experiments certain modifications appeared in some of the
guinea-pigs, the cause of which lay in injuries done to the nervous
system of their parents.

It was found that epilepsy sometimes appeared in animals born
of parents which had been rendered epileptic by an injury to the
spinal cord or a section of the sciatic nerve. Here was a fact to be
accounted for, and the cause must be sought among the circum-

1 Because Lamarck (1744-1829) had propounded a theory which ascribed
the gradual modification of species largely to the inherited and accumulated
effects of use and disuse of organs.

2 The following argument is taken from G. J. Romanes' Darwin and after
Darwin, vol. II. ch. iv.

1779 Q g

450 AN INTRODUCTION TO LOGIC [chap.

stances to which the epileptic offspring were directly or indirectly
exposed. Brown-Sequard attributed it to the injury done to the
parent ; but nobody professes to see how that could produce the
effect, so that one can only be forced to accept that explanation by
default of anything else to which to attribute it. It might be said
that the epilepsy was due to some congenital defect that had no re-
lation to the experiment performed on the parents ; but epilepsy is
not otherwise known to occur spontaneously in guinea-pigs, and
apart from any improbability in the coincidence, we should expect
that if some congenital modification of the germ-plasm produced
epilepsy in these cases, it would have occurred and produced it in
others. Weismann suggested that it was due not to the injury
to the parent, but to ' some unknown microbe ' which, entering at
the incision whereby the injury was made, both produced the
epilepsy in the parent, and by invading the ova or spermatozoa,
produced it also in the offspring. But against this suggestion we
may urge that, though there may be microbes enough unknown
to us, yet if this microbe of epilepsy in guinea-pigs exist, it would
be likely to seize other opportunities of entering ; the disease,
however, as already mentioned, is not otherwise known to attack
them. And it was also found that the epilepsy might be produced
(and apparently transmitted) without incision, by a blow on the
head with a hammer, in circumstances that preclude the entry of
microbes. To this Weismann rejoined that the shock of the blow
might have ' caused morphological and functional changes in the
centre of the pons and medulla oblongata, identical with those
produced by microbes in other cases ', and so set up the epilepsy ;
but these changes would not penetrate, as microbes may be con-
ceived to do, to the ova or spermatozoa, and so the disease in the
offspring occurs without the presence of the cause alleged. More-
over, there are cases (though the facts of them are not so clear or
well confirmed) in which other diseases produced by other traumatic
injuries to the parent have reappeared in the offspring ; these
diseases were not such as could have been produced by microbes ;
and to suppose, with Weismann, that the shock of the injury caused
a general weakness of the nervous system, in consequence of which
the animals would be likely to bear ' weak descendants, and such
as are readily affected by disease ', does not account for the diseases
in the offspring being of the same sort as those respectively pro-
duced in the parents. So far, therefore, the alternative hypotheses

xx] RULES OF CAUSE AND EFFECT 451

to that which attributes the disease in the offspring to the injury
done the parent seem to be excluded ; but Weismann has a final
argument to urge against the ' Lamarckian ' hypothesis. If the
epilepsy was produced in the parent by the injury inflicted, it ought
not to occur in the offspring in the absence of that injury in the
offspring ; and it would therefore be necessary to show that the
nervous lesion which is the alleged cause of the epilepsy, and not
merely the epilepsy itself, is transmitted. To this Romanes replies,
that it very well may be transmitted ; since even if adequate
examination had been made (which is not the case), there may be
structural injuries in a nerve which are not discernible. Never-
theless, he admits that the result of the whole debate is to leave
' the Lamarckian interpretation of Brown-Sequard's results ' rather
unassailed than proved. The facts alleged are ' highly peculiar ',
and hardly sufficient by themselves to furnish ' positive proof of
the transmission of acquired characters '.

This example has been chosen because it illustrates very well how
the inductive proof of a conclusion rests on excluding alternative
explanations. The whole chapter in Romanes' work, from which
it is taken, may be profitably studied from that point of view.1
A further knowledge of facts might enable a biologist to suggest
a cause for the appearance of epilepsy in the second (or later)
generations of guinea-pigs, consistent at once with the facts and
with Weismann's theory of the continuity of the germ-plasm.
But this does not detract from the value of the example as an
illustration of the method of inductive reasoning, which may
be sound, though the conclusion is false, if there is error in the
premisses. Note, however, that in the process of excluding
alternative suggestions as to the cause, it was sometimes necessary
to do more than merely appeal to one of the grounds of elimina-
tion set down earlier in this chapter ; some deduction of the
consequences of accepting such alternative was needed, more
elaborate than is involved in saying that, if such were the
cause, the epilepsy would appear where it did not, or not appear
where it did. Thus it was argued that the epilepsy was not to be
attributed to a microbe, because other diseases equally appeared

1 Cf. Romanes' own words with reference to another experiment on guinea-
pigs : ' Naturally, therefore, the hypothesis of heredity seems less probable
than that of mere coincidence on the one hand, or of transmitted microbes
on the other. But I hope to have fairly excluded both these alternative explana-
tions.' Darwin and after Darwin, p. 119. (The italics are mine.)

Gg2

*52 AN INTRODUCTION TO LOGIC [chap.

bo be transmitted, which a microbe could not have originated ; we
cannot be said to be here applying the simple principle, that that
is not the cause of a phenomenon, in the absence of which it occurs,
for these other diseases are not the same phenomenon as the epilepsy.
To make the evidence of these other diseases serviceable, it had to be
shown that there was no tenable alternative to the Lamarckian
interpretation put forward (in lieu of microbes) in their case ; and
the principle involved in the use of their evidence was this, that if it
is necessary to attribute the reappearance of one kind of disease in
offspring to its production in the parents, it is more reasonable to
attribute the reappearance of another kind of disease (epilepsy) in
offspring to its artificial production in the parents, than to a different
sort of cause of whose presence and operation there is no evidence.
This principle may in turn be said to rest upon the principle that
like effects have causes correspondingly like ; and all rests ultimately
on our understanding of the causal relation ; but in order to see
that facts are inconsistent with the ascription of a given pheno-
menon to some particular cause, a more or less extensive hypo-
thetical deduction of the consequences that ought to follow if that
were the cause is often necessary — more extensive, as Dr. Bosanquet
points out,1 in proportion as with growing knowledge we grasp
more of system in nature. It may be noted, too, in this example,
that some of the steps of the argument are only probable ; if the
entry of a microbe at the incision were the cause of the epilepsy,
it would probably occur in cases of natural injury where, so far as
we can see, the microbe might equally well enter : according to the
principle that that is not likely to be the cause of the phenomenon,
which is probably present on some occasion when the phenomenon
fails to occur.2 And lastly, Romanes cautiously concludes that the
attribution of epilepsy in the offspring to its artificial production in
the parent is not proved, because the cause may lie in something
hitherto undetected ; and this illustrates what was maintained
earlier in the chapter, that the getting of a positive conclusion, but

1 v. his paper ' On a defect in the customary formulation of Inductive
Reasoning ', Proc. London Aristotelian Society, N. S. xi, 1910-11, p. 29.

2 In the Prior Analytics Aristotle discusses at great length modal syllogisms,
i. e. syllogisms where one or both premisses are problematic or apodeictic ;
showing under what conditions the conclusion will be problematic or apo-
deictic. We have here an example of what might be called a modal induction ;
the parallelism may be commended to the notice of any who think, with Mill,
that an inductive argument which can be represented in symbols (like his
' Inductive Methods ') is the less formal because it is inductive.

xx] RULES OF CAUSE AND EFFECT 453

not the inductive character of the argument, depends on the com-
pleteness of the elimination.

4. Adam Smith, in the Wealth of Nations,1 discussing the infer-
ences which can be drawn from the low money prices of goods in
ancient times, and wishing to show that from the low prices of
goods in general nothing can be inferred as to the wealth of a country,
though much can be inferred from the relative prices of different
kinds of goods, such as corn and meat, mentions that it was com-
monly supposed that the said low money prices of goods in ancient
times were a proof of the poverty and barbarism of the countries
where they prevailed. He uses the following argument to show
that this is not so, but that they prove only the barrenness of the
mines which then supplied the commercial world. First, he says
that China is a richer country than any part of Europe, yet the value
(i. e. purchasing power) of the precious metals is higher there than
anywhere in Europe : now on the principle that that is not the
cause of a phenomenon which does not vary proportionately with
it, we cannot attribute low money prices to poverty in the face of
lower prices where poverty is less. Next, he admits that since
the discovery of America the wealth of Europe had increased, and
the value of gold and silver diminished ; but he urges that the
two events have scarcely any connexion ; the first being due to
the fall of the feudal system and the growth of public security, the
second to the discovery of more fertile mines. In support of this
way of connecting the facts he points to Poland. Poland was the
most beggarly country in Europe, as beggarly as before the dis-
covery of America ; yet the money price of corn (the most important
single commodity) had risen equally there : if poverty were the
cause of low money prices, it ought not to be found where prices
were high. On the other hand, Poland was still feudal, so that
her beggarly state was consistent with the connexion of facts alleged
by Adam Smith. Again, Spain and Portugal were the next most
beggarly countries in Europe to Poland, and prices ought there-
fore to be low there, if there were the connexion between low money
prices and poverty that was supposed ; but it was not so ; prices
were high ; as might be expected if they depend on the facility
with which the precious metals are obtained, for, owing to their
control of the American mines, gold and silver were brought more
cheaply to Spain and Portugal than to any other country in Europe.
1 Bk. I. c. xi vol. i. p. 365, 7th ed., 1793.

454 AN INTRODUCTION TO LOGIC [chap.

The cause of low money prices in general, therefore, is not poverty
and barbarism, and may be the barrenness of the mines supplying
the commercial world with gold and silver ; and this has been shown
by inductive reasoning. Adam Smith also offers deductive argu-
ments to show that it is the latter, and is not the former. It is not
the former, because a poor could not afford to pay as much as a rich
country, in labour and means of subsistence, for such comparative
superfluities as gold and silver ; it is the latter, because the pur-
chasing power of gold and silver, or the amount of goods for which
they will exchange, depends on what has to be given in order to
get them ; and where the mines are fertile, a less amount of labour
and means of subsistence needs to be supplied in the work of getting
them, than where they are more barren. The logician may distin-
guish an inductive from a deductive argument ; but investigators will
gladly use arguments of both kinds to support the same conclusion.
5. We may conclude with an example drawn from the Poor Law
Commissioners' Report of 1834, with regard to the cause of the
appalling increase of pauperism in England during the early part
of the last century.1 The Commissioners who were appointed
to find the cause and to suggest a remedy attributed the evil to
one principal fact in the situation, viz. that the condition of those
receiving parochial relief had been allowed to become not less
eligible than the lowest condition of men maintaining themselves
by independent labour. In proof of this finding, they pointed out
in the first place that the cause alleged was present in all instances
of the phenomenon to be accounted for. The great increase of
pauperism had dated from 1796. In that year, an Act of 1723,
providing that no one should be entitled to relief who would not
enter the workhouse, had been repealed ; and it had become
customary for the parish to assure to all labourers, in their own
homes, a certain weekly sum, varying with the numbers in the
family and the price of bread. This sum was made up in various
ways ; sometimes grants were given in supplementation of wages
(which naturally tended to make farmers and other employers give
a lesser wage, and so interested them in the support of a system
from which they saw more clearly their immediately resulting benefit
than the remoter but far greater evils) ; sometimes the parish
found work, generally lighter than what was exacted for the same
price by private employers (and this led men to prefer to work for

1 v. the Blue-book, esp. pp. 18G-216.

xx] RULES OF CAUSE AND EFFECT 455

the parish) ; sometimes a money-grant without any return of labour
was made to men out of work (who were not, therefore, the more
likely to look for work) ; but in any case, it was made possible for
a man to count upon parish pay, sufficient to maintain him as well
as many independent labourers were maintained, whether or not he
endeavoured to support himself.

The cause alleged, then, was present where the pauperism was
present ; but that was not enough to show that it was the cause.
It might indeed be plausibly argued, from familiar principles of
human nature, that such a method of administering poor-relief
would be likely to increase pauperism faster than it relieved it :
but this deductive reasoning was not, and still is not, sufficiently
convincing to men who, from one motive or another, are attached
to such a policy — whether from compassion for the immediate
suffering of those applying for relief, or from desire to get relief
on the easiest terms, or from fear, if relief is less readily given,
that it will become necessary to give higher wages to the labourer.
To bring conviction, it was necessary to show that there was nothing
else to account for the phenomenon. Now several other causes
had been suggested to account for this growth of pauperism. One
was the great rise in the price of corn, which had occurred during,
and partly in consequence of, the French war : another was the
increase of population : and another was the introduction of
machinery — a highly unpopular thing at the time, because its
first and most obvious effect was to displace labour ; and there
had been agricultural riots directed against the use of machinery
in 1830.

It would not be possible to show that none of these causes had
ever made a man a pauper. But it was possible to show that in
the main the pauperism so widely prevailing (which was so great
a national evil because it prevailed so widely) could not be due to
them. The Commissioners were able to point to numerous instances
of three kinds, in which the pauperism so prevalent elsewhere was
absent ; in all of them, the cause they alleged was absent too ; but
the alternatives which they wished to disprove were present.

The first class of instances consisted of certain parishes where
what was called a Select Vestry had adopted the plan (still then
lawful, though not since 1796 compulsory) of refusing relief to any
able-bodied labourer except in a workhouse where a full task of work
was exacted. It was their experience that pauperism immediately

456 AN INTRODUCTION TO LOGIC [chap.

and greatly diminished. And naturally ; for when men who had
hitherto been content to take parish pay found they had to work
as hard all the same, they preferred to work for themselves ; with
a motive for independent industry and thrift, they became more
industrious and thrifty ; becoming more industrious, they were
better worth employing ; and the farmer besides, knowing that
the parish would no longer supplement the inadequate wages by
which he had obtained labourers upon his farm, was compelled, if
he would still have labourers, to give a better wage.

The second class of instances was furnished not by parishes
which, removing the cause alleged, had also removed the pauperism
of which it was alleged to be the cause ; but in the parishes them-
selves where the pauperism existed. It was furnished by the so-
called non-settled labourers, who in all parishes were found to be
more industrious, thrifty, and prosperous, and less pauperized,
than the settled labourers. As the circumstances of two sets of
labourers in one parish are likely to be more similar than those
of labourers in distinct parishes, these constituted what Bacon calls
a prerogative instance ; for all the conditions equally affecting
settled and non-settled labourers may be excluded, in looking for
the cause of this difference between them, on the principle of re-
jecting the circumstances present when the phenomenon is absent.
By a non-settled labourer is meant a labourer living in another
parish than that which is legally bound to support him. If he
becomes a pauper, such a person can be removed to the parish to
which he is legally chargeable ; and to save their own rates, over-
seers were always anxious to remove any one they could. To the
labourer, on the other hand, removal was as a rule by no means
welcome ; such labourers, therefore, found that they had to choose
between removal, which they did not want, and an effort to main-
tain themselves by their own labour ; for if the parish relieved
them at all, they would only get from it — unlike their settled neigh-
bours— little relief on hard terms.

The third class of instances was afforded by parishes which had
never adopted the practice, so common since the Act of 1796, of
relieving able-bodied men out of the workhouse ; i. e. they had
never consented to make the condition of the pauper as eligible
as that of independent labourers ; and in them the same extensive
pauperization and increase in the rates, which had occurred else-
where, had never happened.

xx] RULES OF CAUSE AND EFFECT 457

Now in all these three classes of case, the Commissioners' theory
held good ; for when the effect was absent, so was the cause to
which they attributed it. But the same could not be said for the
alternative theories put forward. If it were alleged that non-
settled labourers had smaller families, which is doubtful, yet the
increase of population was not confined to parishes which had
adopted, or banished from those which had abandoned, the practice
rendered permissive by the Act of 1796. The price of corn had
risen, and the introduction of machinery must have had its effects
— whatever they were — in the parishes which had abandoned or
never adopted that practice as much as in the rest, and among the
non-settled as much as among the settled labourers of any parish.
In short, looking to the mass of pauperism, there was no other
circumstance which might be suggested as its cause, that could
not, upon one or other of the plain grounds of elimination so often
referred to, be rejected ; and the Commissioners' cause was left in
possession of the field ; with the additional support derived from
the deductive reasoning that might not have been thought of —
even if it would have carried conviction — by itself. For it often
happens that we can subsequently show that a cause, to which an
effect has been attributed on the ground that there is nothing
else to which the facts permit us to ascribe it, must, in accordance
with some accepted principles prevailing in the subject-matter to
which the enquiry belongs,1 produce that effect : although, but for
the help which the inductive argument had given us in finding
the cause, the deductive argument would never have occurred to us.

1 i.e. special principles, or "8iat apxaL Cf. supra, p. 387. Cf. the account
of his very successful administration of famine relief in the North-Western,
now the United, Provinces of India in 1896-97, by the Lieutenant-Governor,
Sir A. P. (since Lord) MacDonnell, North-Western Provinces Government Gazette,
Nov. 27, 1897, quoted by Sir Theodore Morison, The Industrial Organisation
of an Indian Province 2, c. xi, pp. 272-283. ' This result was obtained ', says
Sir Theodore Morison, p. 281 n., 'by steadily keeping the pay upon relief
works below the "standard wage " which could be earned in any ordinary
labour market.'

CHAPTER XXI

OF OPERATIONS PRELIMINARY TO THE APPLICATION
OF THE FOREGOING RULES

It was allowed in the last chapter that it is impossible to apply
the kind of reasoning there analysed until a good deal of work has
already been performed upon the material which experience offers
us. That work is really much harder than the reasoning that
succeeds it ; indeed so simple does the reasoning look when thrown
into symbolic form, that it would not be surprising if any one
mistrusted the foregoing account on the mere ground that induction
must be a harder business. A consideration of the present chapter
may reassure him on this point.1

The operations that have to be performed in order that the fore-
going rules, or any other more special rules of the same kind, may
be applied, are difficult to classify in a perfectly satisfactory manner.
Different writers have called attention, and have given different
names, to processes some of which are really more or less the
same. Moreover, we should make our list shorter or longer
according to the extent to which we considered what may be called
the Methodology of the several sciences. By this is meant an
attempt to give special directions, based partly on general logical
considerations and partly on the nature of the facts with which it
deals, for mastering the special difficulties which a particular science
presents ; for example, a mythologist might be enjoined to adopt
the comparative method, and collect, with all the precautions which
the experience of those who know the difficulty of rightly inter-
preting the savage mind can suggest, the myths and customs of
many different lands : in biology again we should probably be told
of the importance of obtaining statistics of a trustworthy kind
regarding the mode in which divergences were distributed on either
side of the average or normal in respect of divers measurable

1 Mill deals with the subject of this chapter for the most part in his Fourth
Book, Of Operations subsidiary to Induction. In the sense that the reasoning
described in the Third Book cannot be profitably performed till they have
taken place, they may be called subsidiary ; but Induction is perhaps rather
the whole process of eliciting from facts the principles that account for them
than merely the form of reasoning involved therein ; and these operations
certainly hold no subordinate place in that process, as indeed Mill recognizes.

PRELIMINARIES OF INDUCTIVE REASONING 459

characters in animals and plants : and so forth. The particular
preliminaries, without which inductive reasoning in each science may
have little prospect of success, could of course only be determined
by some one well acquainted with that science ; though it is quite
possible that a man of logical training, coming fresh to the study of
what others have done, may be the better able for that training to
make contributions to the work of scientific investigation ; still,
here as elsewhere, Logic learns by reflection on the immediate
operations of thought about things. A methodology of the several
sciences lies however beyond the scope of this volume, and would
require far greater knowledge of their detail than the writer possesses.
The list of operations therefore which follows makes no pretence to
go as far as it might, or to embody the only possible division.

First of all may be placed what has been called the Analysis of
the Given l : and this is requisite in two ways,

1 . in determining precisely the phenomenon to be studied ;

2. in distinguishing and detecting the various circumstances under
which it occurs, or under which it fails to occur when perhaps it might
have been expected.

Long before we consciously seek ' rerum cognoscere causas ', a
beginning has been made in the performance of this analysis : and
the results are embodied in the general names by which men group
and distinguish different things, attributes, or events. But there
are many distinctions which ordinary language ignores, and it often
gives different names to things which are in some important respect
identical. For ordinary purposes the identity may be of no account,
and yet in a scientific enquiry it may prove fundamental. For
example, to the lawyer hares and rabbits are vermin, to the sports-
man they are game, and to the zoologist they are rodents ; each
of these men for his own purposes is interested in characters that
unite them respectively with quite a different group of other animals ;
but there is nothing in their specific names to indicate their affinities
with any one of these groups. Or again breathing, burning, and
rusting are three processes occurring in such different connexions
and of importance to us in such very different ways, that they
naturally have obtained distinct names ; yet one of the greatest
steps in the history of chemistry was connected with the discovery
that they are, chemically speaking, all processes of the same kind,
viz. the combination in the first two cases of carbon and in the third
1 Professor Welton's Inductive Logic, c. v.

4C0 AN INTRODUCTION TO LOGIC [chap.

of iron with the oxygen of the air.1 These examples illustrate the
way in which it may be necessary to ignore our customary classifi-
cation of things, and bring together, upon the strength of some
identity which an analysis may have discovered in them, things
that we have habitually kept quite apart in thought. It is equally
necessary at times to distinguish things which we have habitually
classed together, if we are to make any progress in the investigation
of them. Rent furnishes a good instance. The name is given
equally to the sum which a man pays for the occupation of land,
and to that which he pays for the occupation of a building ; as
these are very commonly paid to the same person, as a lump sum
is then charged for the two, and as the ordinary tenant in search of
a dwelling is prepared to pay so much for accommodation, but
indifferent to the question whether the owner considers his charge
to be based on the value of the house or of the site it stands on, it
follows that most of us find no inconvenience in this double use of
the word. The farmer who has to consider separately what the
land he farms is worth to him per acre, and what the value of the
homestead is to him, is more or less aware of the ambiguity ; but
the political economist, when he comes to consider the causes that
determine rents, is bound to distinguish house-rent and ground-rent
by name. Indeed until that is done, his investigation will make
no progress ; for the two depend upon quite different conditions.
The rent of a house, apart from any special history or sentiment,
depends chiefly on the cost of building another like it, and the current
rate of interest on money in the country at the time ; but land
cannot be produced as it is wanted, and this natural limitation of
supply may give to a particular piece of land, in virtue of its fertility
or its situation, a rentable value that depends mainly on its supe-
riority in those respects over other land which cannot be dispensed
with for cultivation or for building, and only very slightly and
remotely, if at all, upon the circumstances which regulate house-rent.
The process of discovering identities between things in which we
commonly ignore them, and that of discovering differences between
things which we commonly take for the same, very generally involve
one another.2 We perform as it were a mental re-grouping ; and in

1 Cf. pp. 471-473, infra. Of course the oxygen need not be atmospheric
oxygen.

* Thus economists have followed up the above distinction between house-
rent and ground-rent (or economic rent) by grouping with the latter, under
the name of quasi-rents, various other differential advantages, not super-
ficially recognizable as of the same kind, such as what the abler entrepreneurs

xxi] PRELIMINARIES OF INDUCTIVE REASONING 461

the act of bringing together what we had hitherto only distinguished
we most probably break up or find distinctions in the groups from
which members are brought together. But in a given case one
aspect may be much more prominent than the other ; and Bacon
has observed * that some men have a ^ /eater capacity for the one
kind of work than for the other, insisting (like Plato before him)
on the necessity of noting, in the investigation of nature, both the
resemblances and the differences that are ordinarily overlooked.
Analysis is at the bottom of each process, for until we have distin-
guished the various characters of things, we have not discovered the
bases on which to compare them. It must be added however that
analysis may be of great importance, yet without leading to any
act of fresh classification, when we want primarily to know the
circumstances under which a phenomenon occurs.

These remarks will indicate generally the nature of the work
involved in the performance of the two tasks above mentioned :
namely, in determining precisely the phenomenon we have to study,
and in distinguishing and detecting the various circumstances under
which it occurs, or under which it fails to occur when perhaps we
should have expected it. It is sufficiently obvious that without
performing them we should hope in vain to discover causal con-
nexions by way of induction. If we have no precise or exact
conception of what is to be studied, or have not (as one might say)
duly determined it, we may examine instances that we ought to
ignore, and ignore instances that we ought to examine. The result
of the former error will be that we shall try to make our theory as
to the cause of x consistent with the facts of the occurrence of a
different phenomenon y : and the result of the latter, that we may be
ignorant of facts which might throw great light upon the cause of x.
The necessity of making a correct enumeration of the circumstances
under which a phenomenon occurs, before asking with which of
them it is causally connected, needs no comment ; nor is it less
plain that, if the question is to be answered, we need equally to
recognize the circumstances, where they occur also in the absence
of the phenomenon.

But though this work is so necessary, it is impossible to give
any rules for the efficient dispatch of it. Familiarity with a science
may help a man to perform it in the investigations of that science,

enjoy over their competitors, or some buyers and sellers over others forced
to do business at the same price. 1 Nov. Org. I. 55.

462 AN INTRODUCTION TO LOGIC [chap.

teaching him the sort of thing to look for, and the sort of way in
which to look for it. Yet the sagacity upon which the discovery
of new truth depends does not come to most men even by such
familiarity. The logician's business at any rate, since he cannot
teach men to do it, is to make them realize the part which it plays ;
and one or two further examples may be given with that object.

A research which has been so frequently cited in works on Induc-
tion as to become almost a stock instance will serve this purpose
— Wells's Theory of Dew. Dew, as is now pretty generally known,
does not rise but falls : the atmosphere can hold water in suspension
in the form of vapour, but the amount depends upon the tempera-
ture of the atmosphere, and increases with it. If the atmosphere
is suddenly chilled, it precipitates such a portion of the moisture
which it holds as exceeds the saturation-point, or maximum it can
hold at the temperature to which it is reduced. It may be chilled
in various ways. One is the contact of a colder surface, on which
the moisture is then precipitated ; another way is by the inrush of
a heavier and colder current : another is by radiation to the sky,
and the degree to which that takes place depends partly on the
amount of cloud about, partly on the substance, surface-form, &c,
of the body itself ; and a sheet or other covering stretched over the
ground acts in the same sort of way over a small area, though with
more effect over that area, as the clouds spread out over the earth.
This precipitation of moisture held in suspension in the air is seen
not only when dew falls ; when warmer weather comes after a frost,
particularly if accompanied by rain, the cold surface of a stone wall,
if painted or otherwise not porous, drips with the water it has
extracted from the air which its contact chills. In the same way
cold spring water poured into a glass in summer will chill the outside
of the glass, so that water is deposited on it from the air without :
and when hot water is poured into a glass without filling it, and
sends its vapour into the air above, some of this vapour bedews the
interior surface of the glass above the water-level, until this portion
of the glass has acquired by conduction the temperature of that
below it. Now our present business is not with the reasoning by
which Wells showed the deposition of dew to depend upon a relation
between the temperature of the atmosphere and of the body on
which the dew falls, taken in conjunction with the degree of satura-
tion of the atmosphere at the time. But it is plain that he could
never have done this, if he had not taken note of all the above

xxi] PRELIMINARIES OF INDUCTIVE REASONING 463

points, the material and texture of bodies, as affecting their surface-
temperature, the clearness or cloudiness of the nights on which he
looked for dew, the conditions of air and wall when the latter drips
with moisture, and so forth. It would have been vain to observe
that one body collected more dew and another less, unless their
roughness and smoothness were noted, as well as their substance :
or that on some nights there was heavy dew and none on others,
unless the saturation of the atmosphere were considered as well as
its temperature. And similarly, it was necessary that he should
get a right conception of the thing called dew that he proposed
investigating. There are days when everything grows damp from
a moist fog hanging in the air. It would not have been unnatural
to take this for a phenomenon of the same nature as dew-fall, and
to overlook such things as dripping walls and moisture-frosted
tumblers. Yet the mistake would have put the enquirer altogether
off the scent.

Curative effects of different kinds are exhibited by certain waters.
To the eye many of the waters are indistinguishable ; and if the
palate detects a difference, yet it would not be found possible to
connect efficacy in particular complaints with particular flavours
according to any explicit and invariable rule. It is plain that no
progress can be made unless the various diseases are described not
merely by their more obvious symptoms but by reference to the
physiological character involved : and the water chemically analysed,
so that one may know each separate ingredient, and the different
proportions in which they are present in different cases. Again, the
bacteriological theory of disease would never have been formulated,
until the bacteria themselves were found — bodies so small that before
the construction of powerful microscopes their presence was of neces-
sity overlooked ; and when one hears of pathologists endeavouring
to isolate the microbe of some particular disease, one realizes how
impossible it is, without the preliminary work of distinguishing the
circumstances, to apply the ' canons of induction ' to any purpose.
Or suppose that an enquiry is undertaken not into the physiological
cause of a disease, but into the causes of its dissemination, either
generally or on some particular occasion : let the disease, for
example, be malaria. Malaria was long supposed to be contracted
from the exhalations of the ground ; and it was true that many
malarious districts were marshy, and that persons who avoided the
swamps at dusk and dawn seemed less liable to be infected ; but it

464 AN INTRODUCTION TO LOGIC [chap.

was not until it was noticed that such districts were infested with
mosquitoes of a particular species, and it occurred to some one to
connect this circumstance with the communication of the disease, that
false beliefs were exposed and the true law of the matter established.
The last remark suggests a transition to the next preliminary
operation that we may notice — the formation of hypotheses.
Much has been written upon the question whether Logic can lay
down any rules by which the formation of hypotheses should be
controlled ; but beyond offering the somewhat obvious and quite
general considerations that an hypothesis must contain nothing
inconsistent with principles which thought finds necessary, and that
it must be such from which we can reason to consequences that
should be found in one set of circumstances or another if it were true,
it does not seem that Logic can be of any more service here than in
the performance of the work of analysis. It would be an illegitimate
hypothesis on the part of a bank clerk confronted with a small
discrepancy in his books, to suppose that on this occasion two and
two made three ; but a petty theft on the part of the Principal
Manager, though very likely a foolish hypothesis, would not be
logically illegitimate. On the other hand, the hypothesis of angelic
intervention, though there is nothing inconceivable in the existence
of angels, would not be a legitimate way of proposing to account for
the event ; for there is no use in attributing phenomena to causes
whose existence, and mode of action if they exist, we have no means
of ascertaining ; since such hypotheses can never be brought to the
test of facts. It is obviously more reasonable to go on trying to
account for them by ascertainable natural causes in the hope of
being able to connect them by general principles with other observ-
able phenomena, than to abandon that hope at the outset and invoke
the agency of beings whose existence cannot be empirically verified ;
so that although we can hardly pronounce it logically inconceivable
(however repugnant to scientific hopes) for the physical order so
to depend on something beyond itself as to make it impossible to
account for a particular natural event by reference solely to other
natural events preceding it, yet we may on logical grounds pronounce
it unscientific 1 : i. e. it is seen to be unscientific 1 not in virtue of any
special knowledge of the particular science to which such hypothesis
belongs, but in virtue of our general appreciation of the aim of any
science, and of the logical conditions under which that aim can be

1 Or at any rate non-scientifio

xxi] PRELIMINARIES OF INDUCTIVE REASONING 4G5

realized. And this is perhaps what Mill really had in his mind when
he said x that ' It appears, then, to be a condition of the most genu-
inely scientific hypothesis, that it be not destined always to remain
an hypothesis, but be of such a nature as to be either proved or
disproved by comparison with observed facts '. It should be of such
a nature that observable facts, if we could find them, might prove
or disprove it 2 : i. e. it should not appeal to the agency of causes
(like the intervention of an angel3, or the influence of the organic
type as a whole upon the growth of the individual organism) of
whose presence we can have no independent evidence, and whose
nature we are not able so to ascertain as to determine deductively
how they must act if they are present ; for with the agency of such
causes as these any facts are equally compatible ; and thus they
furnish no explanation why the facts are so and not otherwise. For
this reason, as Bacon said, in looking for the causes of things in
nature Deum semper excipimus 4 : and Laplace, when Napoleon
observed to him that there was no mention of God in his Mecanique
celeste, replied that he had no need of that hypothesis. But that
an hypothesis should be of such a nature that observed facts will
ultimately either prove or disprove it, and not merely might ulti-
mately do so, seems a condition quite impossible to lay down. We
cannot tell the future in these matters ; how long may an hypothesis
be destined to remain an hypothesis without prejudice to its genu-
inely scientific character ? The ultimate destruction of life on the
earth is assumed by science ; for human minds, an hypothesis which
is not proved or disproved before that date will always remain an
hypothesis. We cannot suppose that its scientific character, when
it is made, is to be estimated by the prospect of its truth being
definitely ascertained a few years, or even a few myriads of years,
earlier or later. Darwin, in the Origin of Species,5 writes as follows :
* As the embryo often shows more or less plainly the structure of
the less modified and ancient progenitor of the group, we can see
why ancient and extinct forms so often resemble in their adult state
the embryos of existing species of the same class. Agassiz believes

1 System of Logic, III. xiv. 4.

2 Facts, as we have seen, cannot prove an hypothesis by their agreement
with it, except so far as at the same time they disprove its rivals by their
disagreement.

3 Cf. Newman's Parochial and Plain Sermons, vol. ii, Sermon xxix, on The
Feast of St. Michael and all Angels.

4 De Principiis atque Originibus, Ellis and Spedding, III. p. 80.
* Origin of Species, c. xiv, 6th ed. p. 396. The italics are mine.

1779 H h

466 AN INTRODUCTION TO LOGIC [chap.

this to be a universal law of nature ; and we may hope hereafter to
see the law proved true. It can, however, be proved true only in
those cases in which the ancient state of the progenitor of the group
has not been wholly obliterated, either by successive variations
having supervened at a very early period of growth, or by such
variations having been inherited at an earlier stage than that at
which they first appeared. It should also be borne in mind, that
the law may be true, but yet, owing to the geological record not
extending far enough back in time, may remain for a long time, or
for ever, incapable of demonstration '. But that the rule in question
is an universal law is a scientific hypothesis.

An hypothesis then must be thinkable1, consistently with the
fundamental assumptions of the science which makes it, and it must
be one whose consequences, if it were true, we can determine by
reasoning : but we cannot restrict, within these limits, the freedom
of scientific hypothesis. What is important is that men should be
cautious not in framing but in testing hypotheses. The publication
of every wild conjecture is undesirable ; but it would be equally
undesirable that a man should never entertain an hypothesis which
contemporary opinion could pronounce wild. Darwin said that he
had framed and abandoned many an hypothesis which he would be
ashamed to avow : he does not imply that he was ashamed to have
framed them. The best control over the licence of the imagination
is exercised by special knowledge. The man who knows most
about any department of nature will see most readily what hypo-
theses are foolish in that department, just as in such practical
matters as legislation the best critics of a bill are those who have
experience of the affairs with which it deals.

1 Lotze would explain this by saying that our hypotheses must conform
to our postulates. He draws a distinction {Logic, § 273) between a postulate
as ' an absolutely necessary assumption, without which the content of the
observation with which we are dealing would contradict the laws of our
thought ', and an hypothesis as ' a conjecture, which seeks to fill up the postu-
late thus abstractly stated by specifying the concrete causes, forces, or pro-
cesses, out of which the given phenomenon really arose in this particular case,
while in other cases maybe the same postulate is to be satisfied by utterly
different though equivalent combinations of forces or active elements '. One
ehould add, that in saying that an hypothesis must be thinkable consistently
with the fundamental assumptions of the science which makes it we are enlarging
as well as restricting the liberty of the mind in framing them. We restrict
it to something which the facts of experience might test ; but the fundamental
assumptions of a science may be metaphysically untenable, and we enlarge it
to extend to all which these assumptions cover, however it may be ultimately
impossible to think the facts in terms of them.

xxi] PRELIMINARIES OF INDUCTIVE REASONING 467

It is clear that every causal connexion presents itself at the out-
set in the light of an hypothesis, to the mind to which it first occurs.
The framing of the hypothesis may sometimes be very simple,
though the proof of it may be very difficult. If we know exactly
what persons were cognizant of a secret which has been betrayed,
it is easy to say that one of them must have betrayed it ; and so far
there is no hypothesis ; hypothesis begins so soon as we ascribe
the offence tentatively to any one of them, and in this there is not
the least difficulty ; but a proper test of it may be impossible.
Whereas here, however, all the alternatives are before us, and in the
abstract any one of them would equally fit the facts, because it is
simply a question of connecting an event x with one of a number of
conditions a b c, about which we do not know enough to say that it
might not be connected with any one of them : yet commonly it
happens that the facts which an hypothesis has to fit are more or
less elaborate ; and then the framing of it is not such a simple matter
as the pairing off of two terms a and x. Take for example the ques-
tion of the authorship of the Acts of the Apostles ; if that book must
have been written as it stands by one of the recorded companions
of St. Paul's journeys, it is a simple thing to say that the author
may be Luke, or may be Silas : although it need be by no means
a simple thing to decide between them. But if that is not necessary,
if the book may be of late date, and contain the work of several
hands, very complicated and elaborate hypotheses will be possible.
We have a large number of facts to co-ordinate ; and the assump-
tions by which we connect them must all be mutually coherent.
Historical criticism presents many problems, where no hypothesis
is free from difficulty ; and though doubtless a problem must have
a solution, yet an ignorance of some details, and very likely the
erroneous accounts that we have received of others, may leave us
permanently unable to find it. And the penetration and ingenuity
of the historian are shown in such cases in devising as well as in
testing hypotheses ; indeed the two operations cannot be kept
altogether distinct : for when our knowledge of the concrete detail
of events is considerable, the process of framing an hypothesis to fit
them all is itself a process of testing. Now what is true in history,
where upon the whole 1 our business is rather to determine events

1 Upon the whole, because the historian has often to rediscover principles —
constitutional, legal, social, or economic ; and history advances by changes
in men's way of conceiving the relations of past facts to one another as well
as by changes in their view of what the facts were. We no longer believe

H h 2

468 AN INTRODUCTION TO LOGIC [chap.

in conformity with acknowledged principles than to determine
principles in accordance with empirically ascertained events, is true
also in science, of whose business the latter would be the more
accurate description. Scientific hypotheses consist for the most
part not in the mere coupling in the mind, as cause and effect, of
two insulated phenomena (if the epithet may be allowed) : but in the
weaving of a large number of phenomena into a coherent system by
means of principles that fit the facts. In the framing of hypotheses
therefore we are called upon to regard facts in new ways : and to
suggest not simply that certain facts are connected, but how. or in
accordance with what principle, they are connected. And this often
involves a radical transformation in our way of looking at the facts
themselves ; for a fact is not such an easily ascertainable thing as
the language we sometimes use might seem to imply. In a sense
facts are stubborn : in another sense they are pliant to our thought.
They are stubborn so far as we have rightly apprehended them ;
but what we call fact is largely matter of inference and interpre-
tation, performed often unconsciously, and often erroneously ; there
is room here for re-interpretation, in accordance with the require-
ments of the rest of our knowledge, and so far as what are called
facts lend themselves to this they may fairly be called pliant. It
would have been called a fact, for example, in the days before
Copernicus (though some of the Greeks had questioned it) that the
sun went round the earth ; but this was only an interpretation of
observations which we now see to be equally compatible with the
earth's revolution round the sun. It would have been called a fact
that species are fixed and immutable ; and it is the case that they
breed so true upon the whole in any one generation as to make that
a fairly accurate statement for practical purposes. Yet we have
learnt to see that this comparative stability is consistent with any
degree of modification over long enough periods of time. These
instances will be enough to show how the familiar facts take on a new
appearance in the light of new theories.

Now some new theories or hypotheses are, as we all know, more
far-reaching in their effects than others ; for some are much more
general, and apply to a much larger number and variety of facts.
Their introduction marks an epoch in the progress of science ; and
Whewell attached more importance to the framing of such hypo-

in William Tell ; but the Patriarchal Theory has also changed our views aa
to the relations between the individual and the State in some ancient societies.

xxi] PRELIMINARIES OF INDUCTIVE REASONING 469

theses than to any other of the operations connected with inductive
reasoning. Indeed he held that this step was the induction ; and
that the history of the inductive sciences could be represented as
the preparation, elaboration, and diffusion of successive hypotheses
each more adequate to all the facts of a science than its predecessors.
He did not use the word hypothesis very prominently in this con-
nexion ; he preferred to speak of conceptions : and what he called
the colligation of facts by means of appropriate conceptions x was in
his view the essence of induction. The new conception, however,
is always put forward at first as an hypothesis, and only accepted as
correct for its superior success in co-ordinating facts. This work of
■ colligation ' therefore must not be regarded as something distinct
in its nature from the framing of hypotheses : it is rather a special
and important case of it, where the hypothesis, instead of merely
connecting facts in a more or less familiar way that leaves our view
of the system to which they belong very much what it was before,
involves a profound and far-reaching change in our view of the
system and so also of the facts themselves. Thus the suggestion
that malaria is communicated by the bite of the Anopheles mosquito
neither altered seriously our notion of the nature of that insect
(though it altered our practical attitude towards it in a way by no
means favourable to the numbers of Anopheles) nor introduced any
new way of conceiving disease ; for the bacteriological theory of
disease had already been applied to many other fevers. But the
first suggestion that a disease depended on or consisted in the pre-
sence and multiplication of some specific noxious bacillus in the
blood altered profoundly men's view of what many a disease is,
of how it was communicable or curable, and of the whole economy
of living nature. In the relation of this ' colligation ' to the general
framing of hypotheses we have an instance of the difficulty of distin-
guishing sharply the different operations of thought which logicians
have enumerated as preliminary (though by no means subordinate)
to such application of the rules on which inductive reasoning rests
as we examined in the last chapter.

A somewhat unprofitable controversy arose between Whewell
and Mill as to the part which the ' colligation of facts ' should be
regarded as playing in induction. While Whewell said it was the
induction, Mill said that it was improperly so called. Mill seems to

1 v. Novum Organum Renovatum, Bk. II. c. iv : Philosophy of Discovery,
c. xxii. §§ 1-37.

470 AN INTRODUCTION TO LOGIC [chap.

have been influenced in part by thinking that an induction must end
in establishing a general proposition, whereas it is possible to bind
facts together by a new conception and so place them in a different
light and reinterpret them, without apparently generalizing ; he
seems too to have considered that nothing in the whole process of
thought, by which general conclusions were reached from the
examination of particular facts, ought to be called induction, except
what could be reduced to the form of inference or reasoning : the
rest was all subsidiary to induction. But the operations of thought
preliminary to the application of such rules as inductive reasoning
rests on are not subsidiary in the sense of being of secondary im-
portance ; and it would perhaps also be better to distinguish induc-
tion as the whole process from the reasoning employed in it. We
might then agree with Whewell that in induction, i. e. the whole
process of the ' interpretation of nature ', what he called the ' colli-
gation of facts ' is an operation of the very first importance, demand-
ing higher and more uncommon powers of mind than inductive
reasoning ; while we agree with Mill that it is not the inferential
operation. But if by induction we mean the inferential operation,
then we shall have to say that this ' colligation of facts ' is more
momentous in the history of science than induction ; for most of us,
as Bacon rightly said,1 would light upon the use of the methods of
inference to which Mill would restrict the name of induction, by our
ordinary intelligence, without their being formulated for us ; but
few can originate the new conceptions that show order and intelli-
gibility in a mass of facts.

The instance which served to illustrate the dispute will help to
show what this ' colligation ' is. The Greek astronomer Eudoxus
supposed the planets to move round the earth as if fixed in concentric
spheres, with the ' fixed stars ' in the outermost sphere. When
further observation showed that this was not so, because the planets
were not always at the same distance from the earth, circles were
substituted for spheres, and the centre of the circle in which a planet
moved was supposed to travel on the circumference of another
circle ; these circles were conceived not as mere imaginary paths,
but as physical entities actually revolving ; and it was possible to
assign such a radius and rate of revolution to them as would account
for the planet fixed upon the outer circle describing the path it does.
This hypothesis had grown more and more complicated, as the mass

1 Nov. Org. I. 130.

xxi] PRELIMINARIES OF INDUCTIVE REASONING 471

of observations upon the movements of the planets had increased ;
and though it was capable of application to the heliocentric no less
than the geocentric theory, Kepler sought for one more satisfactory.
After trying a large number of other curves, and rejecting them on
the ground that they did not agree with the observations, he at last
discovered that the planet Mars — the primary subject of his investi-
gations— moved in an elliptical orbit round the sun, which stood in
one of the foci. Now the ellipse is here the appropriate conception
which binds together into an unity the successive observed positions
of the planet Mars. Any position taken singly must of course
necessarily be on the circumference of that or any other curve ; for
any curve can pass through any point. But he sought for a curve
which would pass through all the positions ; and he found that in an
ellipse. There was indeed nothing disjunctive in his argument.
Other curves were rejected because disproved by the observations ;
but the ellipse was accepted because the observations agreed with
it, and not because no other curve would satisfy them. If it had
suggested itself sooner, the others would not all have been tried.
There are curves, of higher degree, that will equally satisfy a limited
number of observations, and had they occurred to Kepler, he could
perhaps have given no other reason for preferring to accept the ellipse
than an a priori preference for the simplest curve that would do so.
It is to be noted, however, that even here the critical matter was the
thinking of an ellipse, and not the testing its agreement with the
facts : any one with the necessary mathematical training could have
done that, whenever the ellipse had been thought of. And so it
often is, though not always, when the appropriate conception is
a conception of causal relation : not always, because sometimes
there may be as much difficulty or more in testing the conception
than in thinking of it. To test it, we may have to deduce its conse-
quences by some intricate mathematical calculus, as happened with
the Newtonian theory of gravitation ; or to devise an experiment
in which we may see whether the theoretical consequences of our
conception occur. Great mathematical power or great ingenuity
may be wanted here ; but the reasoning will be deductive. Yet
even so, to introduce the appropriate conception is much ; new
theories are scarce ; inductive reasoning, if the material were given
all ready prepared, is easy.

An excellent example of the part which a new hypothesis may
play in inductive enquiry is furnished by the Oxygen theory. It is

472 AN INTRODUCTION TO LOGIC [chap.

borrowed from Whewell,1 whose works afford many more. It
was for a time supposed that combustible bodies were combustible
because of the presence in them of a peculiar substance, that escaped
in the process of burning. This hypothetical substance was called
phlogiston ; and it was very natural to think that one could see it
escaping into the air wherever a fire was burning. When it was
found that there was one air (or, as we should now say, gas) in which
bodies burnt readily, and another in which they would not burn at
all, it was conceived that air could only absorb a limited quantity
of phlogiston in proportion to its volume ; in the former it was
supposed that there was no phlogiston, and it was called dephlogis-
ticated air ; the latter was supposed to be already saturated, and
was called phlogisticated air accordingly. The phlogiston theory
received a shock when it was discovered that if a body were calcined,
or reduced to ashes, in a closed vessel, the weight of the ashes was
greater than that of the body before it was burnt. This, however,
was explained by supposing phlogiston to be a substance naturally
light, whose escape therefore left a body heavier — a view plausible,
perhaps, when we remember how the sparks fly upward, yet really
presenting great difficulties in relation to the theory of gravitation.
The great French chemist Lavoisier, however, conceived the facts
in a new way : he conceived that, when a body burned, what hap-
pened was not that a substance naturally light escaped from it into
the air, and so left it heavier ; but that a substance naturally heavy
was withdrawn from the air and combined with the burning body ;
burning in fact was a process of what we should call chemical com-
bination ; and Lavoisier supported his theory by showing that after
the calcination of a body in a closed vessel the air in the vessel was
lighter by the same amount by which the ashes were heavier ; this
observation perhaps was not conclusive, if the phlogiston had
carried its natural levity into the air ; but the new way of conceiving
the facts accorded far better with the general theory of gravitation.
The substance thus withdrawn from the air in burning he called
oxygen ; and oxygen now took the place of dephlogisticated air ;
while phlogisticated air, instead of being conceived as the same air
saturated with phlogiston, was conceived to be a different substance
from oxygen, incapable of entering into those chemical combinations
which constituted burning. This substance was now named azote,
and afterwards nitrogen. Lavoisier further showed that oxygen
1 Whewell, Hist. Ind. Set., vol. iii. Bk. XIV. cc. 4-7.

xxi] PRELIMINARIES OF INDUCTIVE REASONING 473

was withdrawn from the air and chemically combined with other
substances not only in burning but also in the familiar process of
breathing, and in the rusting or oxidation of iron, which could rust in
water also because oxygen was present there as well ; and thus his
new conception, that burning was really a process of chemical com-
bination between a substance in the atmosphere, which he called
oxygen, and the substance of the body burnt, served to throw light
equally on processes at first sight quite remote from burning. In
this example, therefore, we have as it were a ' colligation ' of two
kinds : primarily, in so far as a large number of facts about burning
were all rendered consistent with one another and bound together
by the help of this new conception of what goes on when a body
burns ; secondarily, in so far as that conception was shown to be
applicable to other phenomena as well as burning, and they are
therefore brought under the same explanation with it. It may be
worth while to give one more example of the transforming and
connecting power exercised by a new and appropriate conception
upon a multitude of facts, in the biological theory of Evolution,
or the modification of species through natural descent. We are
not for the moment concerned with the question whether the only
agency in determining such modification is Natural Selection. The
theory of Natural Selection, as a theory of the way in which modifi-
cations have, not indeed originated, but been established when they
had once arisen, teaches that in each generation individuals vary
more or less in colour, size, structure, &c, from their parents ; that
some of these variations are useful to their possessors under the
circumstances in which they live ; who will therefore, in the con-
stant struggle for existence going on in the world, have an advantage
over less fortunate competitors ; so that those individuals who
happen to possess ' adaptive ' variations will survive and propagate,
while the less well-adapted will perish ; and thus species are brought
into and kept in conformity with the conditions under which they
have to live. Now there is not complete agreement among biolo-
gists either as to the extent to which the peculiarities of different
species of plant or animal are adaptive, or as to the extent to which
those that are adaptive can be accounted for by the theory of Natural
Selection alone ; though there is no doubt that the doctrine of
Evolution won its way through the success of the principle of
Natural Selection in accounting for at any rate a vast number
of adaptive structures, instincts, and colourings. But the doctrine of

474 AN INTRODUCTION TO LOGIC [chap.

the Evolution of Species, or their modification by descent, as opposed
to their special creation in immutable form, does not stand or fall
with the view that Natural Selection is its exclusive modus operandi.
This doctrine has brought into intelligible connexion with one
another whole departments of fact. It explains the various and
intricate relations of likeness and unlikeness between different
species of the same genus, different genera of the same family,
different families of the same order, &c. ; it explains why the same
structural plan is observed in many cases where the function of some
part of the structure has been lost or altogether altered : and why
it is that where their life requires the performance of the same
function in groups otherwise very remote morphologically from one
another, we find the function fulfilled by such very different means
as are, for example, the wing of an insect, of a bird, of a bat, and of
a flying-fish. Again, it explains the divers series of fossil forms : and
accords with the facts of embryology, such as that the embryo of
a given vertebrate only gradually develops the more distinctive
specific features, and at an earlier stage is very little distinguishable
from the embryo belonging to a different genus or family ; for the
characters which appeared later in the course of evolution and
supervened as it were upon a simpler structure appear later in the
growth of each subsequent individual of the same more complex
type, and supervene upon the simpler structure there.1 Again, it
explains the facts of geographical distribution, such as that the
degree of affinity between species is much greater when they inhabit
a continuous area, than on either side of a geographical barrier ; and
that the barriers on either side of which the difference is most
marked are not the same for every kind of organism, but are for
each kind those which would offer the most effective obstacle to the
migration of that kind — high mountain ranges in the case of land
animals or fresh-water fish, wide stretches of open sea in the case of
certain salt-water fish, and so forth : or such facts again as this,
that ' wherever there is evidence of land areas having been for
a long time separated from other land areas, there we meet with
a more or less extraordinary profusion of unique species, often
running up into unique genera '.2 All these facts, and many
others, for which upon the old hypothesis of the special creation

1 Cf. on this the interesting appendix by Professor H. H. Turner in The
Laws of Heredity, by Dr. Archdall Re id.

2 Romanes, Darwin and after Darwin, i. 235 et al.

xxi] PRELIMINARIES OF INDUCTIVE REASONING 475

of immutable species it is impossible to suggest a reason or a
motive, fall into line upon the hypothesis of modification by
descent, and are bound together by that conception as common
consequences.

We have now considered some of the most important operations,
without which inductive reasoning would be powerless to advance
inductive science. One or two others may be noticed. It may
seem unnecessary to mention the observation and registration of facts ;
yet that is no small part of the work that has to be performed
before we are in a position to tell how phenomena may be supposed
to stand related in the way of cause and effect. Every lawyer
knows how hard it is to make an uneducated witness distinguish
rigidly between what he has observed, and what he has been led
thereby to suppose was happening. And scientific observers have
to be trained to be accurate in thus distinguishing, alert in noticing,
quick in selecting what is new and instructive, and, where observa-
tion is of something confused or faint (as often with a microscope)
intelligent in disentangling or interpreting. In such matters
practice and instruction will do little without natural aptitude.
But whatever the aptitude, it is found that there are certain constant
errors of observation to which different men are differently liable.
One man will regard as synchronous two sounds between which
another will detect a slight interval of time ; one man watching to
record the moment when the image of a star touches a line will make
the record just before, another just after contact, and so forth. The
experience of these tendencies to error has led to the establishment
for different observers of what is called their ' personal equation ' ;
i. e. their observations are corrected by a co-efficient which is based
on examination of the direction and amount of error to which they
are severally liable. As no extension of self-recording apparatus
will do away with the necessity for men to observe and record at
certain points, and even the records of the apparatus need observing,
it is clear that in the last resort the average error to which one
man is subject in observing is to be ascertained through the
observations of another man with a personal equation of his own,
and hence the problem is intricate, and the theory of error has
become a difficult branch of largely mathematical reasoning. Again,
the registration of facts, where they are many and their relations
complex, is not a simple matter. We must not ignore the value of
the mechanical aids that can be given in tabulating, cataloguing.

476 AN INTRODUCTION TO LOGIC [chap.

indexing, &c. ; but more important are formulae which enable us
to record in brief collective statements what is relevant to a parti-
cular enquiry in vast numbers of observations. A simple and
familiar instance of this is furnished by an average ; for certain
purposes to know the average of many observations is valuable,
where an enumeration of each detailed observation would only
confuse. But we have to consider carefully where an average is
enough, and where it is not. Thus the average age of women at
marriage is an important figure in relation to a country's birth-rate ;
but the average rainfall of a country is very uninstructive unless we
know how it is distributed in places and years. There are however
more elaborate and difficult devices than averages for reducing to
manageable formulae what is relevant in a mass of observations,
such as the ' co-efficient of correlation ' which attempts to measure
how closely the changes in one variable accompany the changes in
another — e.g. in the corresponding right and left parts of animals
bilaterally symmetrical, in the size of parents and offspring, in the
price of corn and the birth-rate. Here too mathematical problems
arise for solution. And geometrical methods of registration may
also be useful. In many enquiries where the collecting and tabu-
lating of statistics is a necessary preliminary to the application of
the rule that nothing can be the cause of a varying phenomenon
which does not vary proportionately with it, the most helpful way
of exhibiting the facts is by plotting curves or ' graphs ,.1 What
was incidentally referred to on p. 471 2 is also of importance — the
devising of experiments by which to test whether a phenomenon is
present or absent, variable or constant, as it should be if its cause
were what we take it to be. If it be supposed, for example, that
spirit-rapping is really produced by ' cracking ' the joints, it will be
necessary not only to show that a man can produce such noises that
way, but to devise conditions under which one may be certain that
the joints cannot be ' cracked ' without its being detected, and see
whether the ' spirits * still continue to rap.3 This is comparatively
simple ; but all the resources of mathematical reasoning and
mechanical ingenuity are sometimes needed to determine and con-

1 Cf. infra, p. 562.

2 The other process, of mathematical calculation, there referred to, falls
rather to be considered later : as belonging to a stage of science in which
deductive reasoning plays a larger part than in the application of the rules
discussed in the last chapter.

8 v. Podmore's History of Modern Spiritualism, i. 184, 185.

xxi] PRELIMINARIES OF INDUCTIVE REASONING 477

struct the apparatus required for the conduct of an experiment that
shall put a theory to the test.

This is perhaps enough to say upon the present subject. There
are other tasks set to our thought in science, which are of great
importance to its development ; but we have been concerned
especially with those that are presupposed in inductive reasoning.
The help afforded to the ' interpretation of nature ' by a well-chosen
armoury of technical terms, great as it is, is not confined to the use
of inductive reasoning. And account has been taken of abstraction
in what was said of analysis and hypothesis and the formation of
conceptions. By abstraction we mean considering some special
feature of the concrete fact, in mental separation from all with which
it is combined in its existence. It is between feature and feature
that we strive to trace connexion. The tangle of things changes
from moment to moment. Not until we pick it to pieces are we able
to see what in one state of it determines what in another. Every
common term involves some degree of abstraction ; but in science
we have to break up what in daily life we treat as a single matter,
and to consider by itself, or in abstraction, that which had not
hitherto been specially noted and distinguished in the totality of
some comparatively complex nature.

CHAPTER XXII

OF NON-RECIPROCATING CAUSAL RELATIONS

In what has been so far said with regard to the process of induc-
tively determining the cause of a phenomenon, it has been for the
most part assumed that the cause, whatever it is, reciprocates with
the phenomenon studied : i. e. that not only does the phenomenon
occur whenever the cause is present, but that the cause must be
present whenever the phenomenon occurs ; so that you may safely
argue from either to the other, as in geometry you may equally infer
that a triangle is equilateral from the fact that it is equiangular,
and that it is equiangular from the fact that it is equilateral.

But we often speak of one thing as being the cause of another,
where this reciprocal relation by no means obtains. We say that
drunkenness causes crime, although many people get drunk without
committing crime, and many people commit crime without getting
drunk. And in some of the examples of inductive reasoning given
in previous chapters, the cause found was not a reciprocating cause.
The appearance of congenital epilepsy in guinea-pigs was shown to
be due possibly to a wound producing epilepsy in the parent ; yet
it was not alleged that the production of epilepsy by these means
in the parent was always followed by the appearance of epilepsy in
the offspring.

It was said that the inductive proof of the cause of a phenomenon
rested on an understanding of the causal relation ; for nothing that
does not stand to the phenomenon in such relation as a cause should
can be the cause of it ; and it is by eliminating all alternatives that
its cause is inductively established. Our account of cause assumed
that it reciprocated with its effect. But if it does not, we clearly
have no right to eliminate whatever fails to reciprocate. The
admission that there are non-reciprocating causal relations may
seem therefore to invalidate reasoning that starts with the assump-
tion that cause and effect reciprocate.

This difficulty has been postponed till now, partly that the expo-
sition of the subject might not be unduly complicated : but also,

NON-RECIPROCATING CAUSAL RELATIONS 479

because the causal relation is really, and in its strict sense, reciprocal,
and without understanding that first, we could never render non-
reciprocating causal relations intelligible to ourselves. Properly
speaking, to give the cause of anything is to give everything neces-
sary, and nothing superfluous, to its existence. Nevertheless we
should often defeat our ends, if we gave precisely this ; if our object
in seeking the cause of a thing is that we may be able to produce or
prevent it, and if something is necessary to its existence which is
a property of a thing otherwise superfluous, it would be of no use
to specify the property necessary unless we also specified the other-
wise superfluous thing in which it was found.1 Even though we
have no such practical purpose, so long as we do not know what
thing contributes, in the property which it possesses, the factor
necessary to the effect, we can hardly be said to understand com-
pletely the production of the effect. Hearing at a distance, for
example, depends on the transmission of certain vibrations through
an elastic medium ; the necessary elasticity is a property of the air ;
and therefore we can hear at a distance in the air, while if there is
a vacuum interposed between the sounding (i.e. the vibrating)
body and the ear, the transmission of the sound is prevented. It is
true that, except in respect of its elasticity, air is quite superfluous
so far as hearing at a distance is concerned ; not air in the concrete,
but that property in abstraction, is one of the conditions that make
up the reciprocating cause of hearing at a distance. But an elastic
medium cannot be just elastic and nothing else besides.2 We want
to know what possessed of the necessary elasticity is present when
we hear at a distance ; nor could any one, without knowing that,
prevent the transmission of sound by removing the elastic medium ;
for he would not know what to remove.

We may pursue this illustration a little further. It might be
shown inductively that the intervening air was the cause of the
transmission of sound ; indeed it was shown inductively, by the

1 e. g. it may be the texture of pumice-stone that fits it to remove ink-
stains from the skin ; but it would be of more use to tell a man with inky
fingers to get a piece of pumice-stone, than to give him a description of
the fineness of texture which would render a body capable of making his
fingers clean.

2 It is just the fact that we know no more about the ether than its form
of elasticity which makes the conception of it somewhat unsatisfactory ;
and led the late Lord Salisbury, in his Presidential Address to the British
Association at Oxford in 1894, to say of it that it merely 'furnishes a
nominative case to the verb to undulate \

480 AN INTRODUCTION TO LOGIC [chap.

help of a well-known experiment. And speaking loosely, it is true
that from the presence of air it can be inferred that sound will be
transmitted, and reciprocally, from the transmission of sound, that
air intervenes. Yet neither inference is quite safe. The first is
only true with qualifications : the distance must not be too great in
proportion to the loudness of the sound, and so forth. The second
may be altogether false ; for sound can be transmitted through
water, or (with the help of a telephone x) through a vacuum. And
in this case the reason is that the elasticity is provided in some other
way than by means of a continuum of air. We saw that, except
in respect of its elasticity, air was superfluous : but we could not get
the elasticity alone. Now we find that there are other elastic media
which will serve, and the elasticity may be provided by them. An
elastic medium is what is wanted ; but divers things will supply the
want. They are alternatives, and none of them exclusively recipro-
cates with the effect ; for the effect may be produced by the help
of any one of them, so that the occurrence of the effect does not
prove that any one more than another is producing it. But their
common property of providing an elastic medium does reciprocate ;
sound cannot be transmitted without that.

There is, then, always a reciprocating cause ; but it is not always
most instructive to state only that. And very often that is not
what we want to know. There are several reasons for this.

In the first place, though the object of a science is to discover
strictly universal propositions, and though in most sciences 2 these
involve relations of cause and effect, yet as a science advances, its
problems often take a different form than that of an enquiry after
the cause of a given phenomenon. We may start with something
that seems comparatively simple ; and, as we proceed, may find
that it depends upon a number of conditions being combined to-
gether, each of which can be fulfilled in a number of ways, but none of
them without much that is superfluous or irrelevant to the produc-
tion of the precise effect in question ; each is an incident of some
complex event, or involves a property of some concrete thing, like
the elasticity of air in the transmission of sound. To state in abstract
form the conditions that must be satisfied, without indicating

1 The elasticity of the air is employed also in the telephone : but not
continuously. It is hardly necessary for the present purpose to go into the
detail of the apparatus.

1 Not in any branch of purely mathematical study ; nor elsewhere where
we are not concerned with change.

xxn] NON-RECIPROCATING CAUSAL RELATIONS 481

the kind of thing or event in which such conditions can be
realized, is uninstructive ; for it fails to explain by what the pheno-
menon is produced ; yet to mention every thing or event in which
the conditions might be realized would be an endless and unprofit-
able task. Hence we alter the form of our problem. Looking upon
the phenomenon as the complex result of many conditions, we
attempt to determine not what assemblages of things will produce
the result, for many will do so, nor on what properties or incidents
therein it depends ; but what is the principle of action in different
things, in virtue of which any of them will serve equally among the
conditions necessary to the production of the phenomenon. For the
reciprocating cause of a complex phenomenon we substitute as
the object of our search the principle in accordance with which a
certain kind of thing acts. Our problem is better expressed as that
of discovering laws of nature, than causes. For example, we may
ask what is the cause of the monsoons — that is, of the regular and
periodic winds that blow steadily in certain regions for one part of
the year in one and for another in the opposite direction. If we
said that they were due to periodic alternations in the distribution
of atmospheric pressure, it would not be very instructive ; for we
really want to know what events, happening in those regions, pro-
duce these differences. Yet the events which contribute to deter-
mine the deviation and direction of the monsoons are numerous and
variable : the exact combination of them differs from year to year
and from place to place, and produces corresponding differences in
the result. It is better therefore to take the things concerned in
these events, by their kinds, singly : to point out the difference in
power of the sun at any place according to the varying directness
of its rays ; that the sea gives off vapour ; that vapour absorbs part
of the heat of the sun's rays ; how the heated water circulates with
the colder ; that the earth absorbs and retains the heat of the sun ;
that air is expanded by heat ; how the principle of atmospheric
pressure acts under conditions of different expansion ; and so forth.
Then we can see that if a certain combination of events occurs,
a particular complex result must arise ; if the sun travels from over
the sea to over the interior of a continent, we shall find monsoons ;
for the difference between summer and winter temperature will in
the interior be very great, but on the sea, owing to the way in which
the moisture of the air absorbs part of the heat, and the currents
in the water carry away part, it is not so great ; hence as summer

1778 i i

482 AN INTRODUCTION TO LOGIC [chap.

is ending, the air inland will be hotter and have expanded more
than out at sea, as winter is ending it will be colder and have con-
tracted more ; so that at one time the current of air sets inland
in accordance with the laws of atmospheric pressure, and at another
time it sets shoreward. The principles, or ways of acting, on the
part of the sun according to its altitude, of the earth and sea respec-
tively under the influence of heat, of air when unequally expanded,
&c, are not exhibited solely in the phenomena of monsoons ; while
the details of those phenomena display the influence of other prin-
ciples of action on the part of other things (e.g. the action of a
mountain-wall on a moisture-laden wind). To give the cause of
monsoons, without deficiency or superfluity, would mean that we
must not mention the sun (because only the heat of its rays is
material) nor the sea (because only its fluidity and its power of
giving off vapour concern us, and a lake, if it was big enough, would
do as well) nor any other of the concrete things which act in the way
required, but only their requisite actions. If we do not go to this
length of abstraction, we shall have to include in our statement of
the cause elements at least theoretically superfluous ; and even so,
we shall have to choose some particular monsoon, supposing we are
to state everything that goes to produce it. It is clearly simpler to
break up the problem, and look for the principles in accordance with
which things of a certain kind act under certain circumstances ;
then we can show that the monsoon is only the complex result of
the action of a number of things under the particular circumstances
of its occurrence, and in accordance with the principles of action
which our ' laws ' express.

This then is one reason why what we want to know is not by any
means always the reciprocating cause of a determinate phenomenon :
the phenomenon under investigation is often highly complex, and
subject to all sorts of variation on the different occasions of its
occurrence, through variation in the things or events contributing
to its production ; not the whole nature of the things or events
under whose influence it occurs is relevant to its occurrence, but
only certain particular properties or modes of action ; and it is
possible to formulate severally the principles of action involved,
from which the joint result may be seen to follow, where it would
not be possible to assign to the phenomenon any group of concrete
things or events as cause, about which we could say not only that,
given them, the phenomenon must be given, but also that, given

xxii] NON-RECIPROCATING CAUSAL RELATIONS 483

the phenomenon, they must have been given too. These laws or
principles of action may of course be proved inductively in just the
same way as may a causal connexion between two particular
phenomena a and x. Just as we may argue that a cannot be the
cause of x, if it occurs in the absence of x, or is absent when x occurs,
so we may argue that a law or principle of action cannot be rightly
stated, if consequences should follow from it as thus stated which
do not actually arise, or should not follow, which do arise. Here,
as there, we may have no other reason for accepting a theory than
that the facts are inconsistent with any other that we can devise ;
and then our argument is inductive.

Another reason why we do not always look for a recipro-
cating cause is that for practical purposes it is generally more
important to know what means will produce a certain result, than
by what it has been produced. We cannot alter the past ; we may
control the future. The means prescribed for the production of
a certain result may contain much that is not relevant precisely to
the production of that result ; and as this irrelevant matter may be
different on different occasions, there may be a choice of means. To
have a choice of means is undoubtedly useful ; but if any of these
means is called the cause of the result in question, the term ' cause '
is clearly not used in the strict sense ; for we may be able to argue
forward from the means as cause to the result as effect ; but we
cannot argue backward from the result as effect to this particular
means as cause. Yet this will be of comparatively little consequence,
if our interest lies less in being able to determine by which means the
result in question was produced on a past occasion, than whether
certain means will produce it in the future. About a variety of
advertised rat-poisons, all that we should care to know would be
that they would rid us of rats ; and we might endeavour to deter-
mine inductively whether a particular poison was efficacious. But
we should be indifferent to the fact that other poisons might be
equally efficacious, and that rats who died off need not have been
killed by this particular poison ; in other words, we shall not want
to learn the reciprocating cause of the dying off of rats. Indeed as
long as the effect is stated in such a general way, a reciprocating
cause cannot be given. There are, as Mill observed, many causes of
death ; and though he was referring to men, it is also true of rats
But death is not altogether the same thing whenever it occurs ; and
the doctor or the coroner knows this. The many different causes

i i 2

484 AN INTRODUCTION TO LOGIC [chap.

of death do not have altogether the same effects ; if you shoot a man
and if you behead him, the difference in the result is visible ; if you
poleaxe an ox and if you poison him, he is not equally edible. As
soon as we begin to be interested in the particular variety of death
produced, we find the number of causes that produce the result in
which we are interested diminishing rapidly ; if we carried our
interest far enough into detail, we might say that for death of a
particular kind there was only one cause possible. But since much
of this detail is quite unimportant, we treat as instances of the same
kind events which in some respects are different, and then say that
the same event has divers causes : forgetting that the differences
between these several causes consist partly in circumstances irrele-
vant to the kind of the event, which are included in our statement
because indissolubly bound up with what is relevant, but otherwise
superfluous to the production of it : and partly in circumstances that
are represented by differences in the resulting event, but differences
which we ignore. Here then, in the fact that our search is often
for means to the production of a phenomenon of a certain general
character, to the precise form of which we may be indifferent, is
a second reason why the causal relations which we seek to establish
are often non-reciprocating.

On the other hand, thirdly, there are cases where it concerns us
more to be able to argue from one phenomenon to another as its
cause, than from the latter to the presence of the former as effect.
For example, there may be alternative symptoms of the same
disease : for the effects of the disease may differ to some extent in
patients of different age, or sex, or race. Here it may be important
to show, that if a certain symptom occurs, that disease must be
present to produce it ; while the fact that the disease may exist
without giving rise to that symptom is a minor matter, and one which,
if we could be certain that some other equally conspicuous and un-
ambiguous symptom would occur instead, might be called altogether
unimportant. In such a case we shall be anxious to show a causal
connexion between the disease and the symptom in question, though
again the relation will be non-reciprocating ; but it will fail to recipro-
cate this time, because the so-called cause may exist without the so-
called effect, although the so-called effect cannot exist without the
so-called cause ; whereas in such cases as were considered in the last
paragraph, the so-called cause always produced the so-called effect,
but the so-called effect might exist without the so-called cause.

xxii] NON-RECIPROCATING CAUSAL RELATIONS 485

Fourthly, our enquiries are often directed to the discovery of the
cause or effect of some singular event — singular, not in the sense of
unusual, but of individual : we ask, for example, what has been the
effect of the repeal of the corn laws in 1846, or what was the cause
of a particular railway accident, or epidemic. It is plain that the
relation we wish to establish in such cases as these is a non-recipro-
cating relation. The repeal of the corn laws was a measure intro-
duced into a highly complex social and economic state, and whatever
results we can point to depend on much else besides that measure ;
no one would pretend that the same measure would have produced
the same results in other circumstances. It might be possible here
to substitute for the question, What effect has their repeal produced
in the United Kingdom ? the more scientific question, In what way
do corn laws act ? The answer to the latter question might be given
in the form of one or more universal propositions : but the answer
to the former will be a singular judgement. For it is practically
impossible to specify all the conditions which have combined with
repeal to produce the results in which the influence of repeal is
exhibited ; so that we cannot hope to establish an universal pro-
position of the form that repeal of corn laws produces always under
such and such conditions the result which we ascribe to their repeal
in 1846 in the United Kingdom. If a man says therefore that the
repeal of the corn laws has increased the population, or depopulated
the rural districts, or crippled the ancient Universities, or made
inevitable a celibate clergy, he is not to be understood to mean
either that it would always produce any one of these effects, or that
they must always be due to a repeal of corn laws : but only that in
the history of the United Kingdom, had the corn laws remained in
force, other things being equal, these effects would not have occurred
in the same degree. So also when we enquire the cause of a singular
effect : it may be known that the reciprocating cause of small-pox
is the presence of a certain microbe in sufficient strength in the
blood ; but if we ask for the cause of a definite outbreak, something
else than that is wanted. We want to know what particular pre-
caution has been omitted, by taking which this outbreak might have
been prevented ; or in what particular way the infection was con-
veyed to the neighbourhood. Thus we might say that the outbreak
was due to a tramp sleeping in a common lodging-house, or to
insufficient vaccination ; but it is not imagined that a tramp suffer-
ing from small-pox cannot sleep in any common lodging-house

486 AN INTRODUCTION TO LOGIC [chap.

without an outbreak of small-pox following in the place ; or that no
such outbreak ever occurs unless from that reason ; while insufficient
vaccination, even if no serious outbreak ever occurred where it
could not be alleged, may prevail without an outbreak following,
so long as nothing brings the infection. Similarly about a railway
accident the question is, what particular act or omission that some
one is responsible for, or what other preventable event, can be
alleged, without which on this occasion there would have been no
accident : did a signalman give the wrong signal, or pull the wrong
points ? did an engine-driver disregard a signal ? had a flood washed
out the ballast of the line, or an axle failed ? These and many more
are the 'causes' of railway accidents, though railway accidents occur
without them, and they may occur without accidents following.

In previous chapters we have represented the phenomena between
which it is sought to establish causal relations by letters of the
alphabet. Each of these letters is quite distinct from the rest,
insulated as it were, and discontinuous both with those grouped
with it to indicate contemporaneous phenomena, and with those
placed apart to indicate phenomena preceding or succeeding it ;
and the use of them as symbols tends to suggest that the course
of events is a succession of discontinuous phenomena, which pro-
duce each the next in a number of parallel or contemporaneous
series. Nothing could be further from the truth : it is impossible
to conceive the matter thus.1 We have already noted the ambiguity
— the convenient ambiguity — of the term phenomenon ; some ' phe-
nomena ' which we isolate and individualize by a name do succeed
one another ; but others do not precede or succeed at all, but

1 Let nobody object that in such a matter we must ask what experience
teaches, and not what it is possible to conceive. Experience can teach
nothing inconceivable. All thinking is an attempt to make experience more
intelligible, and so far as it is not intelligible, we assume our account of it
to be untrue. It is for this reason that we are always recasting in thought
the account of what appears in our experience. The very search for causal
connexions is an example of this operation. It rests on the principle that
change is only intelligible if it displays necessary principles of change : but
these principles are not presented to our observation. Therefore we believe
that events occurred, which have not fallen within our experience : as
Robinson Crusoe, seeing footprints, concluded that men must have been to the
island whom he had not seen. And if we deny that the events ' experienced '
are all that occur, on the ground that their succession would then be without
principle and unintelligible, we may equally deny that history can consist
of streams of discontinuous events, even though these succeeded one another
according to the most constant rules, on the ground that such a succession
would be unintelligible.

xxn] NON-RECIPROCATING CAUSAL RELATIONS 487

endure or persist. Kant said that ' only the permanent can
change ' * : we look on events as occurring to things ; permanent
things change their states ; and the permanent thing enters into the
earlier and the later state alike, or persists through them. What
that is which remains unchanged, how we are to conceive it, and how
we are to conceive the junction between its abiding nature and its
changing states — these are very difficult questions which do not
belong to Inductive Science. But it is clear that our alphabetic
symbols fail in the first place to represent the persistence of anything
through change : they are discontinuous in their series where they
symbolize a change which is continuous. And secondly they are
discontinuous within the group that represents contemporaneous
phenomena ; whereas the contemporaneous phenomena they
represent are not similarly insulated from one another. What we
commonly speak of as single phenomena are bound together not in
independent series unit to successive unit, but by all sorts of cross
ramifications, so that each is what it is in consequence of conditions
which are at the same time conditioning many others in the most
complicated way. To this complication the letters of the alphabet
do no justice. Doubtless if we carry our analysis far enough, we
may find the a which is the reciprocating cause of x : but a will not
in that case as a rule be anything for which we have any single name ;
a long and carefully guarded statement of conditions will be what
it must symbolize.

The fact is that in most cases the reciprocating cause of anything,
if we push our enquiries far enough, emerges as the conditions that
constitute it, and not those that precede it and bring it about.
The reciprocating cause of small-pox is that activity of a specific
bacillus in the blood in which small-pox consists : the reciprocating
cause of malarial fever is the corresponding activity of another
bacillus. But in the procession of events by which that state is
brought about there may be one, which — for one reason or another
— it concerns us to single out, and call the cause : and that will
often be non-reciprocating. It need not be so ; it is possible to
find an event, whose happening in a given set of conditions or to
a given subject always gives rise to some definite new event or state
of that subject, and without whose happening such new event or
state of that subject never arises. It is supposed for example that
malaria is always communicated to man by the bite of the Anopheles

1 Kritik of Pure Reason, section ' On the First Analogy of Experience*.

488 AN INTRODUCTION TO LOGIC [chap.

mosquito ; there are persons immune to the bacillus, and therefore
the bite of Anopheles is still a non-reciprocating cause ; but if we
knew what state of a subject precluded immunity, then we could
say that the bite of Anopheles caused malarial fever in any man in
that state, and we should have stated a reciprocating relation ; for
no man in that state could be bitten without getting malaria, nor
get malaria without being bitten. If with Aristotle we call the
conditions which constitute anything the formal cause, and that
whose activity brings those conditions into being when they had
previously not all of them existed, the efficient cause,1 we may say
that the formal cause reciprocates or is commensurate with the
phenomenon (as indeed anything must which can in any sense be
called the definition of it : and the conditions into which it can be
analysed may be called its definition) ; while the efficient cause
seldom reciprocates. The concrete thing or complex event which
includes the conditions, or part of the conditions, constituting the
phenomenon, may also be called, in a metaphor of Bacon's using, the
vehicle of the formal cause, or of part of it ; the biting Anopheles is
the vehicle of, or conveys, the bacillus in whose activity malaria
fever consists ; the headsman's axe, or the bullets of the firing party,
convey, or are the vehicle of, that bodily state which we call death.
The expression is not equally metaphorical in both these cases, for
the mosquito really carries the bacillus into the blood of the patient,
as a vehicle carries its occupants, and bullets or axes do not thus
carry death ; but what is meant is that events occur, involving
things, whose existence and activity is irrelevant to the effect in
question except just so far as they contribute to the constitution of
that total state which is the effect.

There are indeed many cases where our ignorance of the con-
ditions constitutive of a certain phenomenon compels us to seek
instead for some event indispensable to its occurrence, even though
our scientific interest would be better satisfied by discovering the
constitutive conditions. And there is one most extensive and
important class of cases where the reciprocating conditions cannot
really be called constitutive of the phenomenon ; it is this class of
cases which made it necessary at the beginning of the last paragraph
to write ' most ' and not 'all'. The former sort may be readily

1 Besides the formal and the efficient, Aristotle distinguished the material
cause, or matter of which a thing is made, and the final cause, or purpose
of its being. These were all causes in the sense of being necessary to the
existence of what they are the cause of. Cf. e. g. Phys. /3. iii. 194b 16-195a 3.

xxii] NON-RECIPROCATING CAUSAL RELATIONS 489

exemplified in the biological sciences. ' That form of barrenness ',
writes an authority quoted by Romanes,1 ' very common in some
districts, which makes heifers become what are called " bullers "
— i. e. irregularly in season, wild, and failing to conceive — is cer-
tainly produced by excess of iron in their drinking water, and I
suspect also by a deficiency of potash in the soil.' Here we have
one and perhaps two causes alleged for an effect, whose nature we
do not understand sufficiently to see how the causes bring it about,
though the facts may prove the connexion. Such a relation may be
called discontinuous — i.e. we do not see how the alleged cause, by
any intelligible procession of events, passes into the effect, or helps
to set up the conditions constitutive of it. We connect one pheno-
menon as cause with another as effect, where from our ignorance of
the intimate nature of the effect, and of the subject in which it is
produced, and from the fact that the intervening process of change
is withdrawn from view, the two seem quite heterogeneous. In
Chicago, one is told, there are machines into which you place a pig
at one end, and receive sausages at the other. The pig and the
sausages, to any one who has no conception of the nature of the
machine and what befalls the pig in it, appear in a relation of
sequence without continuity : first the pig exists, and then instead
of it, the sausages ; but we do not see how the one becomes the
other. This somewhat mythical machine may serve to illustrate
how our ignorance of the nature of the process of change connecting
one event with another may produce apparently discontinuous
causal relations ; and such relations are often all that we can at
present hope to discover ; and they are generally, as may easily be
understood, non-reciprocating relations. This case is different from
that mentioned previously on p. 483 ; for there it was our practical
ends which interested us in causes that were non-reciprocating ;
here it is due to the limitation of our scientific knowledge that we
have to acquiesce in them.

But in the extensive and important class of cases to which atten-
tion must be called next, we find discontinuity even where the causal
relation reciprocates : viz. when the cause is physical and the effect
psychical, or vice versa. It has already been stated that such con-
nexions furnish one of the best kinds of example of purely inductive
reasoning, because there is nothing in the nature of a particular
physical process which would lead us to anticipate the particular

1 J. W. Crompton : v. Darwin and after Darwin, iii. ] 70.

490 AN INTRODUCTION TO LOGIC [chap.

psychical state that we find ourselves led by the facts to connect with
it. What may be the true interpretation of this apparent depen-
dence of psychical states on physical processes, and physical move-
ments on psychical states, is the hardest question in metaphysics.
Meanwhile, at the standpoint at which many sciences and all of us
in our ordinary thought are content to stop, we attribute many
psychical events to physical causes, and vice versa. In science
indeed the attribution of physical effects to psychical causes is less
common than that of psychical effects to physical causes ; just
because between the successive events in the physical order there are
prospects of establishing that continuity, which there seems less
hope of establishing in the psychical series, and none of establishing
between members of one series and members of the other, between
a motion of matter in the brain and a sensation or thought or feeling
or emotion. The series therefore whose members do appear capable
of continuous and coherent connexion is often treated as independent,
and psychical states regarded as by-products of particular terms in
the physical series ; although further reflection can easily show that
such a statement of the case, when thought out into its consequences,
involves us in hopeless contradiction.1 We are however at present
only concerned with the interdependence of physical and psychical
states as it appears to exist, and is at least for many practical
purposes rightly treated as existing.

It is supposed that to every distinct state of consciousness there
corresponds some distinct state of the body ; and this bodily state
is not separated from the state of consciousness by any intervening
process, the discovery of which might help us to see how one gives
rise to the other (as drinking water with an excess of iron in it is
separated from the supervening barrenness in a heifer). There is
perhaps no interval of time between them, but the completion of
the conditions in which the bodily state consists is to ipso the pro-
duction of the corresponding state of consciousness ; so that some
writers have been led to speak as if the state of consciousness could
be analysed into these bodily conditions, and they really constituted
it. That however, when examined, proves to be nonsense.

Yet though in this field we may hope to find relations that recipro-
cate in spite of the discontinuity between the so-called cause and
its effect, there are instances here too where the causal relations
are non-reciprocating ; and of this perhaps the most notable instance

1 Cf. supra, p. 411.

xxii] NON-RECIPROCATING CAUSAL RELATIONS 491

is death. It was explained above, how the many alternative causes
of death are not all of them causes of the same effect ; because
they do not put the body into the same state, although the differ-
ences may not concern us. But if we look not to what befalls the
body, but to the result on consciousness — whether we suppose it
to be that the soul is separated from the body, or that it is destroyed
— we can see no difference in that main result corresponding to
the difference of the means by which it is produced. If th soul,
or individual consciousness, be destroyed at death, there is of
course nothing any longer in which a corresponding difference
can be displayed ; if it be not, we may conceive that as the manner
of a man's death, if it be not absolutely sudden, affects him while he
yet lives — one death being more painful, for example, than another
— so the differences between one death and another are repre-
sented by some difference that persists in the experience of the soul
after death, and therefore the effect is not really the same upon the
soul when the physical ' cause ' is different. But such a suggestion
is quite unverifiable ; and however that may be, it is well to realize
the peculiarity of the relations which we try to establish between
physical causes and psychical effects ; owing to the heterogeneity
of the two terms, we cannot hope to find an intelligible cause of
the psychical state in the conditions constitutive of the physical
state with which it is connected ; at this point there is discon-
tinuity ; and so there may arise an appearance of different causes
producing the same effect which we cannot explain as we explained
it in a purely physical sequence. There we saw that different
series of events having partially the same nature might, in their
course and as a part of their result, agree in establishing the same
complex of conditions constitutive of some particular phenomenon,
although the difference in the events occasioned in the rest of the
result differences which we ignored. Here, inasmuch as we cannot
see that the different causes establish conditions constitutive of the
effect at all, the appearance of the same effect when the causes are
different cannot be exhibited as a case where effects different as a
whole (in a way corresponding to the difference of the causes) agree
so far as concerns the conditions constitutive of the phenomenon
we are investigating.

The term Plurality of Causes 1 has been used to indicate the fact

1 The term was introduced by Mill, who sometimes speaks as if he thought
the Plurality of Causes more than apparent : as if he thought that, in the

492 AN INTRODUCTION TO LOGIC [chap.

that the same phenomenon may have different causes on different
occasions. We have seen that the fact is more apparent than
real : that the alternative ' causes ' of a phenomenon, which make
up the plurality, are none of them causes in the strictest sense,
but rather events which agree so far as the production of the pheno-
menon requires, though taken as a whole they are very different.
It would perhaps be well if there was a term to indicate the corre-
sponding fact, that the same phenomenon may produce different
effects on different occasions : a fact also more apparent than real,
for such phenomenon cannot be the cause, in the strictest sense,
of any of the alternative effects which it produces. We might speak
in this sense of the Diversity of Effects. In neither case do cause
and effect reciprocate.

Where the cause or effect sought is non-reciprocating, it is obvious
that the rules on which the elimination involved in inductive reason-
ing rests are no longer to be trusted. If the same effect may have
divers causes, we cannot say that nothing in the absence of which
a phenomenon occurs can be the cause of it ; it cannot be its cause
in the particular instance in which it is absent ; but it may be on
another occasion. If a small group of plants be geographically
isolated from the main stock, it will diverge, and in course of time
probably give rise to a new species ; but there are other ways in
which a particular group may be prevented from interbreeding with
the main stock (e. g. by flowering at a different season), so that new
species may arise in the absence of geographical isolation ; it would
clearly be unsafe to conclude, from the fact that new species had
arisen without geographical isolation, that geographical isolation
was not a cause of new species arising.

No doubt such an argument would betray insufficient analysis :
it would overlook the fact that geographical isolation was not
a single factor, but highly complex ; and that one feature about it
— viz. that it prevented interbreeding with the rest of the stock —
characterized also such very different phenomena as difference of
flowering-season, or selective sterility.1 However, our analysis is

strictest sense of the term cause, the same phenomenon may have different
causes on different occasions. The Plurality of Causes must be distinguished
from the Composition of Causes : which means that a complex phenomenon,
which we call one, may be due to a number of causes acting together on one
occasion. Clearly none of these is the cause in the full sense, but only part
of the cause.

1 Or ' physiological isolation ' — i. e. that certain members of a species x
which happen to exhibit some modification m are more fertile with one

xxii] NON-RECIPROCATING CAUSAL RELATIONS 493

very commonly incomplete ; and then it is possible, that by applying
the above rule, of eliminating whatever fails to occur in any instance
of the effect, we have eliminated the cause altogether : and that if
some circumstance is left uneliminated, because it occurs in all the
instances of the phenomenon, we take it to be the cause of that with
which it has really nothing to do. If a child were given the same
medicine in a variety of jams, and always had a particular biscuit
afterwards, it might very likely attribute the effects of the medicine
to the biscuit. Suppose my apple-crop fails four years in succession,
and that each year it was * overlooked ' by a woman reputed to have
the evil eye : were I to argue that the failure was not due to insuffi-
cient rain, since in the first year there was plenty — nor to late frosts,
for in the last year there were none — nor to blight, which only
occurred once — nor to high winds, since the third year was singu-
larly quiet, I might at last attribute the failure of the crop to the
' witch-woman ' overlooking it.

In such a situation it is well to test one's results by the second
rule, that nothing is the cause of a phenomenon, in the presence of
which the phenomenon fails to occur. If the child were frequently
given the same biscuit when it had not been dosed, it would learn
to disconnect the biscuit from the effects properly connected with
the medicine ; and if the witch-woman were observed to overlook
my orchard in several years when I subsequently obtained an
excellent crop, I might be cured of my superstition. It is however
possible that I might still hold her responsible for the bad crops,
and apply the doctrine of the Diversity of Effects to explain why
her action had failed of its previous result on other occasions.
Perhaps I might have had the crop blessed by a priest, and attribute
to that an effect counteracting the influence of the evil eye ; or
merely say, that the evil eye cannot be expected always to produce
the same results, when there must be many contributory conditions
that are varying.

There is no remedy against such errors except a wider acquain-
tance with facts, and a closer analysis of them, and a better way of
conceiving them and their connexions. To this end however very
special help is given by experiment. The results of an experiment
are of the same kind with the data of observation — facts, namely,

another than with the rest of the species in which this modification has not
appeared. This would prevent swamping by intercrossing, and so, for
breeding purposes, isolate the new variety.

494 AN INTRODUCTION TO LOGIC [chap.

with which we have to make our theories consistent ; and the
inductive reasoning to which the facts contribute premisses is not
altered in character because the facts are obtained experimentally.
But where we can experiment, we can commonly discover facts
which observation would never reveal to us. We can introduce
a factor into conditions carefully prepared, so that we know more
or less accurately what change we make, and in what we make it ;
and then, when we watch the effect, the work of elimination has
more grounds to proceed on. If we are in doubt whether to refer
some phenomenon to a plurality of causes, or to a single circum-
stance which, as present in all our instances, they have not so far
enabled us to eliminate, we might resolve the doubt by producing
this circumstance experimentally : should the phenomenon not
follow, we have then shown that, at least in the conditions into
which we introduced the factor in question, that factor will not
produce it. We may then try one and another out of the plurality
of alleged alternative causes : and if we find that on the introduction
of each the phenomenon follows, we shall conclude that they are
causes of it. We shall still be far from having discovered its precise
cause, without deficiency or superfluity ; but we shall have advanced
our enquiry. The child who attributed to the biscuit the effects of
the medicine could correct its error by experimenting with the
biscuit separately, and the medicated jams separately. And if 1
could bring myself to experiment with the evil eye, I might convince
myself that it was innocuous to orchards.

It should be noted that though the Plurality of Causes and
the Diversity of Effects render precarious, when our analysis is
imperfect, the application of both the grounds of elimination
just cited — viz. that nothing is the cause of a phenomenon in
the absence of which it occurs, and nothing also, in the presence of
which it fails to occur — yet the amount of error in which we may
be involved is not the same in each case. Should we reject in
turn everything, among the circumstances under which a pheno-
menon occurs, without which also it is found to occur in other
instances, we might reject all its several causes, and fall back on
something whose presence in the instances we have examined is
quite accidental : something altogether immaterial to the pheno-
menon. On the other hand, should we reject everything, among
the circumstances under which it occurs, which at another time is
found without it, though we might be wrong in concluding that

xxn] NON-RECIPROCATING CAUSAL RELATIONS 495

what is left is the whole cause of the phenomenon, or that other
things might not serve as well as it, yet we should be right in con-
cluding that it was not altogether irrelevant to the production of the
phenomenon on this occasion. I give a dog cyanide of potassium,
and it dies ; assuming this to be the only fresh circumstance in the
case, I cannot conclude that dogs do not die without taking cyanide
of potassium ; but I can conclude that taking cyanide of potassium
contributed something to the death of this dog, and that the con-
junction of the two events was not merely accidental, as eating the
biscuit was accidental to the child's subsequent experience, or as
being ' overlooked ' by a witch-woman was accidental to the failure
of my apple-crop. In the former case, where I reject everything
in whose absence the phenomenon occurs, I reject too much : the
essential factor lurks undetected each time in a different ' vehicle ' :
each of these ' vehicles ' is rejected in turn, and the essential facts
rejected with them. In the latter case, where I reject everything
in whose presence the phenomenon fails to occur, I may reject both
too much and too little — perhaps too much, for what I reject,
though insufficient of itself to produce the phenomenon, may con-
tain conditions without which it cannot be produced : perhaps also
too little, for what is left, while I take it all to be essential to the
phenomenon, may be in part superfluous, though containing the
essential factor within it ; so that other things, in which the same
essential factor is contained, may equally serve to produce the
phenomenon ; yet still I retain something essential, and do not
reject everything which I need to retain.

A reciprocating cause would be at once necessary and sufficient
to the production of the effect ascribed to it. What is called the
cause of anything in a looser and commoner sense may fail to
reciprocate with its so-called effect either because it is not sufficient
to its production, although necessary : or because it is not necessary,
although sufficient : or because it is neither sufficient nor necessary.
But what is neither sufficient nor necessary to the production of an
effect x would not be called its cause at all, unless it included some-
thing that was necessary. Now when we seek the cause of an effect
by comparison of instances in which the effect occurs, rejecting those
circumstances which are not common to them all, we proceed on the
principle that what is absent where it occurs is not its cause. But
we mean by this that such circumstances are not necessary to the
production of the effect, since the effect occurs without them ;

496 AN INTRODUCTION TO LOGIC [chap.

we cannot conclude that they are none of them sufficient. There
may be several of them sufficient to produce it ; each might contain
what is really necessary, but none be necessary as a whole ; or if
the effect is one dependent on the maintenance of a complex variety
of conditions, the removal of each might remove some one of those
conditions, and so destroy the effect : in this way asphyxiation,
decapitation, heart failure, are each sufficient to destroy life. The
circumstances which we reject may of course be unnecessary because
wholly irrelevant, and the cause must be sought outside what is
unnecessary in this sense. But if they are severally unnecessary
only because though different they are each sufficient, so that, in
the absence of one, another will serve, we are wrong in looking out-
side them for any part of the cause. In such cases our principle
of rejection misleads us altogether. With the other principle, that
what is present where the effect fails to occur is not its cause, our
risk is less. This principle we use, when seeking to determine the
cause by comparison of an instance in which the effect occurs with
an instance of like circumstances without it. And what we mean
by saying that the circumstances common to both the positive and
negative instances are not the cause of the effect that occurs in the
positive instance is, not that such circumstances are not necessary
to the effect — this they, or some of them, may or may not be —
but that they are not sufficient. And if they are not sufficient,
we must look outside them for something which, though perhaps
also insufficient by itself, is still necessary for the occurrence of the
effect.

What we thus find in the positive instance before us may indeed
be only sufficient for the occurrence of the effect in this situation,
not necessary in all situations ; it may be one of several alternatives,
one or other of which, but none in particular, is necessary. But
though we are liable to error in overlooking this, we are still justified
in the use of our principle to this extent, that what we reject, though
it may contain factors necessary to the occurrence of the effect, does
not contain all the necessary factors, and therefore not what is
sufficient : whereas in using the other principle, we were not justified
correspondingly ; for what we rejected may have contained more
than was sufficient and all that was necessary to the occurrence of
the effect. For the sufficient includes the necessary ; if it recipro-
cates with that to which it is sufficient, it includes no more, and is
the precise aggregate of the factors necessary ; but the necessary

xxii] NON-RECIPROCATING CAUSAL RELATIONS 497

does not include the sufficient ; only the aggregate of the necessary
factors and not each of them is sufficient.

[J. S. Mill, who spoke of what ho called the Plurality of Causes
as the * characteristic imperfection of the Method of Agreement \
said that the Method of Difference was unaffected by it. Clearly
he was wrong. The above argument endeavours to bring out the
truth underlying the exaggeration of his statement. That he was
wrong may be seen further by help of the following considerations.
If x occurs under the circumstances abc, and not under the circum-
stances be, I can infer that be is not sufficient to produce x, and that
a contributed to its production on this occasion ; but I cannot infer
that x could not have been produced without a : pbc might equally
produce it. That a and p can equally produce x (or equally produce
it in be) is an instance of the Plurality of Causes ; and it is the
Plurality of Causes therefore which prevents my inferring univer-
sally that x is produced by a, or requires a for its production, and
limits me to the inference that a produces x, at least in be. It will
be said that a and p must have some common property r, which is
the really essential factor. No doubt : or else they must have this
in common, that each removes one, though perhaps a different one,
of a number of factors collectively necessary to the existence of x.
But, as we have seen, this is equally so in any instance of Plurality of
Causes ; if I refuse to infer, in accordance with the ' Method of
Agreement ', from the fact that x occurs under the circumstances
abc, ode, afg, that a is its cause, urging that for aught I know the
cause may be c in one case, e in the next, and g in the third, I must
believe that c, e, and g contain a common r which is the really
essential factor ; and then a is not the ' only circumstance in
common ', for r is another : just as in the other case a was not the
'only circumstance of difference', where x occurred and where it
did not, but really r contained and overlooked in a was a circum-
stance of difference as well.

The distinction which Mill draws between the two ' Methods ' then
is not altogether sound ; for the appearance of Plurality of Causes
affects the inference which can be drawn in each. But there is this
much truth in it, as was pointed out in the text : that in the ' Method
of Agreement ', where I am eliminating that in the absence of which
the phenomenon occurs, I may unwittingly eliminate the essential
factor : I throw away the baby with the bath, and am left supposing
that a is the cause of x, when a may really have nothing to do with
it, and its presence in each of my instances be a mere accident ; in
the ' Method of Difference ', where I eliminate that in the presence
of which the phenomenon fails to occur, though a large part of
a may be superfluous to the occurrence of x, yet it is not altogether
superfluous ; I do not this time connect x with something that has

1779 k k

498 AN INTRODUCTION TO LOGIC [chap.

[nothing to do with it. But I am unable to infer a reciprocating
relation between a and x for the same reason that in the former
ease I was unable to infer any relation at all — viz. the Plurality of
Causes. And let it not be said that this difficulty would not arise,
if the conditions of the ' Method of Difference ' were fulfilled, and
a were the only circumstance of difference where x occurred and
where it did not. For I should still be unable to infer a reciprocating
relation : I could only conclude that a was necessary to the produc-
tion of x in be : how much of be was also necessary I should not yet
have discovered. In both cases, if the analysis of the circumstances
were complete, the Plurality of Causes would disappear ; in neither,
while it is incomplete, is it without effect on our liberty of con-
clusion.

Mill seems unconsciously to assume that this analysis is more
complete when we employ his ' Method of Difference ' than whea
we employ his ' Method of Agreement '. The reason of his doing so
is probably that the experimenter uses the ' Method of Difference '
(or the principle of elimination which it involves), and a completer
analysis is generally obtainable when we can experiment than when
we are confined to the observation of events as they occur in nature :
the experimenter uses the ' Method of Difference ', because in experi-
menting we introduce or remove some particular factor — and that
under circumstances which we have endeavoured to ascertain as
precisely as possible — and watch the result ; and if we are right in
assuming these circumstances to remain otherwise unchanged, we
do approximate to having only the ' one circumstance of difference '
which Mill's canon requires ; in other words, we are really elimi-
nating at once and by appeal to a single principle all except this
factor removed or introduced by us ; though it must not be forgotten
that what we eliminate is only shown to be insufficient to the produc-
tion of the phenomenon, and may still contain conditions that are
essential though not sufficient. We may note here the reason why
the ' Method of Difference ' seemed to Mill, and in a sense rightly
seemed, to be of superior cogency. The reasoning is clearly no
better in it ; but facts such as are required in order that the reason-
ing may lead to a conclusion of value are far more often available
for this ' Method ', because we can contribute to their production by
experiment, and this ' Method ' is practically a formulation of one
of the commonest ways in which we reason from the results of ex-
periment. We may indeed say that the error into which reasoning
from an incomplete analysis of the facts may lead us is greater when
our ground of elimination is that underlying the ' Method of Agree-
ment ' than when it is that underlying the ' Method of Difference ' :
because in the former case we may reject what is essential, and end
by attributing the phenomenon under investigation to something
whose presence is quite accidental ; while in the latter case, we may

sxii] NON-RECIPROCATING CAUSAL RELATIONS 499

[rather end by supposing that more is essential to it than really is so.
Yet there may be error in both cases, and for the same reason, viz.
our incomplete acquaintance with the facts. What Mill however saw
was, that where you can experiment with precision, your acquain-
tance with the facts is most complete, and hence the conclusions to
be drawn most cogent. It is just in these cases that the ' Method of
Difference ' as he formulates it is specially applicable ; for it requires
instances where the phenomenon occurs and where it does not occur
with ' only one circumstance of difference '. He overlooked the fact
that the Method is just the same, where this condition is not fulfilled,
so long as our ground of elimination is the same — viz. that nothing
in the presence of which the phenomenon fails to occur is its cause ;
and so he attributed universally to the ' Method of Difference '
a superior cogency which really belongs to the ' prerogative '
nature of the instances in connexion with which chiefly he con-
sidered its use.]

It has been the object of the present chapter in the first place to
acknowledge that the ' Rules by which to judge of causes and
effects ', whereon inductive reasoning depends, are not infallible
where we are dealing with non-reciprocating causal relations ; for
they rest on the assumption that one effect has only one cause, and
conversely that the same cause has never any but the same effect ;
and so they furnish no safe guide to the discovery of ' causes ' which
are not the only causes of the effect assigned to them, or of effects
which are not the only effects that the alleged cause may have.
Its second object has been to show that such non-reciprocating
causal relations arise from the fact of our including in the cause
more than is necessary, and perhaps also less than is necessary, to
the production of the effect : or including in the effect less or more
than the cause assigned produces ; i. e. our analysis is not perfect :
we combine with the matters strictly relevant to one another others
irrelevant, but closely bound up in their occurrence with what is
relevant : so that there appears to be a Plurality of Causes for the
same effect, or a Diversity of Effects for the same cause, while really,
if we could ' purify ' our statements of the cause and of the effect
sufficiently, we should see this not to be the case. But we admitted
that for many purposes, practical and even scientific, it is causes
in the looser sense that we need to discover — the sense in which the
cause includes more than is material to the production of the effect
in question, but a more from which what is material cannot be dis-
severed, and so forth. And we saw that science, when pushing its

Kk2

600 AN INTRODUCTION TO LOGIC [chap.

investigation beyond such a level as that, tends to substitute for
the search for the determinate cause of some concrete effect the
search for laws or principles in accordance with which things of
a certain kind act on one another under specified conditions.

In illustrating these points, the rules whose guidance was shown
to become unsafe when non-reciprocating relations were in question
were the first two of the rules laid down in the Twentieth Chapter.
But the last two are also liable to mislead us in such cases. These
are, that nothing which is constant when the phenomenon varies,
or varies when it is constant, or varies independently of it, is
its cause : and that nothing which produces a different effect is its
cause. In particular I cannot, because elimination based upon these
rules reveals that x is not independent of a in the instances before
me, infer that x never occurs without a ; for p might do as well.
If I find that the faster I run, the hotter I get, and if I know that
the temperature of the atmosphere has not altered, and so forth,
I may infer that running makes me hot ; but not that no one gets
hot without running. If I experiment over a series of years with
a particular manure, and take care to ascertain by ' controlling '
experiments the average crop that I might have expected without
its use, I may be led to attribute the excess to the use of the manure ;
but I cannot conclude that a similarly large crop is always due to
the use of it. Errors of that sort would be similar to those which
I might commit in applying the rule that nothing is the cause of
a phenomenon, in the presence of which it fails to occur : then too
I have no right to assume that what I fail to eliminate is altogether
necessary, and that nothing else would serve equally instead of it.
But the danger of eliminating too much, which besets the applica-
tion of the rule that nothing is the cause of a phenomenon, in the
absence of which it occurs, does not equally beset the application
of the two rules we are now considering. It is true that in investi-
gating the cause of a phenomenon that may vary in quantity or
degree, and is due as a whole to a number of contributory factors,
this danger is theoretically possible. The quantity or degree of the
phenomenon might remain constant, owing to divers complementary
variations in the factors, some increasing as others decreased ; and
because the variations masked one another, I might reject each
varying factor in turn, until I had rejected all the contributory
factors, as capable of varying with no corresponding variation in the
phenomenon. But this is not a probable error. And the fact that

xxii] NON-RECIPROCATING CAUSAL RELATIONS 501

the phenomena, to which these rules are applicable, are chiefly
measurable phenomena, is of great importance in the use of them.
Peculiar difficulties no doubt often beset us in tracing the influence
of some particular factor upon a phenomenon which varies in
magnitude dependency upon the joint action of a large number of
conditions independently variable ; it is for example exceedingly
hard to determine inductively whether the corn duty of 1902
influenced the price of bread in Great Britain. But these difficulties
would obviously be altogether insurmountable if no measurement
of the conditions and of their result were possible. The introduction
of the element of quantity enables us to determine laws which
connect a definite amount of change in one phenomenon with some
corresponding amount in another. Where we can do this, we are
already getting clear of the errors lurking in non-reciprocating
causal relations. It still remains true that we cannot, in virtue of
a law which connects with a change in the condition a a corre-
sponding change in the result x, argue backwards from a change in
x to the action of a. But that point has been sufficiently exemplified
already ; and inasmuch as some special attention will have to be
paid in another connexion1, when we are dealing with the importance
of quantitative methods in induction, to the two rules or principles
of elimination last mentioned, it is perhaps unnecessary to say
anything further here upon the care that must be used in arguing
from them when the causal relations which we have it in mind to
establish are non-reciprocating.

1 Cf. infra, c. xxvi, pp. 557-562.

CHAPTER XXIII

OF EXPLANATION

We are said to explain, when a conjunction of elements or features
in the real, whose connexion is not intelligible from a consideration
of themselves, is made clear through connexions shown between
them and others. The connexion explained is said to be the con-
sequence of the connexions which explain it, but this expression
must not be taken as implying sequence in time 1 ; it is also said to
be deduced from them. What is explained may be either a par-
ticular fact, or a general principle ; there is no fundamental differ-
ence between explanation of the one and of the other. If it be a
particular fact, the detail must be accounted for by corresponding
details in the facts referred to in the premisses. If it be a general
principle, we shall omit such detail in the premisses. There are
many conjunctions repeated frequently in our experience with varia-
tions of individualizing detail or concomitant circumstance ; if we
recognize under this their identity of kind, we can explain them
together by reference to relations of elements similarly identical in

1 That which is explained may of course be a sequence in time, and then
time-relations enter into the explanation. Thus if I were to explain the
beneficial effect of root-pruning on fruit-trees, I should point out that roots
whose ends are cut off proceed to throw out fibrous rootlets in greater abun-
dance, and these extract from the soil more of the nourishment which the tree
requires than did the parts removed. But the facts stated in my explanation
do not precede the sequence explained ; I am merely showing what relations
are really involved in the sequence I am explaining. We do indeed speak both
of one event being explained by another which precedes it, and of the sequence
being explained by a principle of sequence displayed in it. Here two cases
need distinguishing, (i) The particular sequence ax-xt may be explained as
an instance of the general principle of sequence a-x, although that principle
is only inductively established and not intelligible ; and to accept this
explanation means simply that we are content with finding in the particular
event an instance of a principle of connexion which we have reason elsewhere
to accept ; the principle stands in no time-relation to its instances. Thus,
granted that belladonna dilates the pupil of the eye, we might be said to
have explained the unusual size of the pupils in X's eyes by the fact that
belladonna had been injected into them, (ii) The sequence al-x1 may be
intelligible from the nature of the terms a and x ; thus the fact that M was
angry with N might be said to be explained, if I learnt that N had insulted
him. Human nature and the nature of an insult are such that when a man
insults another he angers him. This we realize in the case of M and N, and
it again has no time-relation to the sequence of M's anger upon N's insult.

OF EXPLANATION 503

kind, ignoring such detail as does not affect the truth of our general
statement of connexions. But in thus explaining a general principle
we are always explaining at the same time, up to a point, the par-
ticular facts in which it is manifested.

In all scientific explanations, our premisses are ' special ' or
1 proper ' or scientific principles. General logical considerations,
such as direct us in the inductive search for causal relations, account
for nothing in particular * ; every explanation must be consistent
with them, but they will not themselves explain anything. The
explanation of the facts or derivative laws of any science rests there-
fore on a scientific knowledge of the subject-matter of that science.

The first or fundamental principles of science are themselves in-
susceptible of scientific explanation. It does not follow from this that
the principles which at any given time are the most ultimate to which
a science appeals should be insusceptible of explanation ; the Law
of Gravitation, for example, is and has long been a fundamental
physical principle, but various mathematicians have attempted to
show that the behaviour of matter expressed in that law follows
necessarily from some more general principles exhibited also in activi-
ties whose principles we commonly regard as different, like electricity
and light. But the process of explaining must come somewhere to
an end, with principles deducible from nothing prior to themselves.

These principles, as has been pointed out 2, may possibly appear
self-evident when we have reached them ; the First Law of Motion
has often been thought to be a self-evident or necessary truth.
But in most cases, they do not ; and then all that we can say about
them is that nothing so well explains those facts, the study of which
has led us to their enunciation. This however is a pis aller.

It has not infrequently been said that scientific certainty is un-
attainable. Jevons urges that the conclusions of Induction are
only probable at the best. The reason is that the principles which
we arrive at as those which explain things are not — at least as a rule
— seen to be necessary ; and that we cannot absolutely prove that
no other principles will explain the facts : just as in simpler inductive
enquiries our confidence in the cause which we assign to a pheno-
menon is qualified by the difficulty of being sure that we have

1 With the help of these considerations we may be led by the observation
of certain facts to believe some general proposition about their connexion,
but we do not thereby explain the connexion. That we have observed the
facts explains our believing the connexion ; they do not explain it.

2 Supra, pp. 382-386, 414.

504 AN INTRODUCTION TO LOGIC [chap.

overlooked nothing which might equally, upon the facts examined,
be allowed to be the cause.

Jevons indeed suggests l that the true though impracticable road
to certainty would he in Complete Enumeration. ' Perfect In-
duction ' rests on complete enumeration, the ' Imperfect Induction '
of actual scientific procedure does not ; and in this he sees the
source of the ' imperfection ' which conclusions only approximately
certain possess. But though we may agree with him that many of
the conclusions accepted in science fall short of certainty, we
cannot agree that they would rank higher if they were reached by
complete enumeration ; for in that case they would not be universal
truths at all, in the proper sense, but only truths about the whole
of a limited number of particular facts. Indeed the antithesis of
Perfect and Imperfect Induction is an unfortunate one. It belongs
to a different sense of the term Induction from that which, in the
phrase Imperfect Induction, the term now bears. It is drawn from
the completeness and incompleteness of the enumeration of the par-
ticulars on which the Induction rests, and to which its conclusion
refers ; we have seen that if a generalization rests merely on cita-
tion of particular facts, without any attempt to establish connexions
of a causal character by analysis and elimination, the citation should
be complete ; though in such cases, the conclusion has not the true
character of an universal proposition. But the reasoning which
infers general truths from the analysis of a limited number of
particulars does not rely on enumeration, and is not an operation
of the same kind as that which proceeds by complete enumeration.
Though the one therefore may cite every instance, and the other
not, yet they are not to be contrasted as if they were operations of
the same kind differing only in that respect. They are operations
of different kinds ; and their other differences are more fundamental
than the difference in the completeness or incompleteness of the
enumeration they involve. If the one is called perfect because its
enumeration is complete, it must be remembered that it requires
a complete enumeration ; but since the other does not require it, it
is misleading to call it imperfect for not employing it. The im-
perfection attaching to the conclusions of inductive science — con-
clusions which are said to be reached by ' Imperfect Induction ' —
springs from the defective analysis of the instances cited, not from

1 Elementary Lessons in Logic, XXV, ' New Edition ', p. 213 : Principles
of Science, 2nd ed. pp. 146-152.

xxiii] OF EXPLANATION 505

failure to cite every instance ; and it is a mistake to suppose that
' Perfect Induction ', if it could be employed — as it is acknowledged
it cannot — would remove the defect of certainty attaching to scien-
tific generalizations. For science seeks after the necessary and the
universal, not after the merely exceptionless.

However, our present concern is less with the reason for the
want of absolute certainty in the principles of scientific explanation,
than with the fact itself. It cannot be denied that the first prin-
ciples of science rest for the most part on no better foundation than
this, that no others have been suggested which explain the facts
equally well ; and this is not the same as saying that no others can
be suggested which will do so. And even if we were satisfied that
no others could be suggested, i.e. if we could be certain that nothing
so well explains the facts as the principles to which we appeal in our
explanation, yet if we cannot see why these principles need be as we
find them, we are still left with something that at once demands to
be and cannot be accounted for.

We shall be wise therefore to recognize these two things about
scientific explanation at the outset, viz. (i) that it often starts
with principles, or truths, or laws, which are neither accounted for
nor in themselves self-evident, but only warranted by the success
with which they account for the facts of our experience : and
(ii) that these principles are not absolutely and irrefragably proved,
so long as any others which might equally well account for the facts
remain conceivable. But it would be foolish to let these considera-
tions engage us in a general and indiscriminate distrust of scientific
principles. Such principles may lack that demonstrable character
which we should like them to have ; and Logic would abandon its
function, if it hesitated, out of respect for the greatness of scientific
achievement, to point this out. But they hold the field ; we are
not entitled to treat them as dogma, which cannot be questioned ;
but we are entitled to say that so long as they remain unshaken,
they should be treated as true.

It may be objected that they are not unshaken ; that some
of the fundamental assumptions of science are unable to resist meta-
physical criticism : the independent existence of matter, the action
of one thing on another, the production of a conscious state by a pro-
cess in a physical organism, are all unintelligible. And it must be
allowed that the scientific account of reality cannot be the ultimate
truth. But if the provisional nature of certain of its metaphysical

606 AN INTRODUCTION TO LOGIC [chap.

assumptions be borne in mind (for science does not really discard,
though it sometimes professes contempt for, metaphysics), we may
then admit the explanations which it offers within their limits.

If however we are to accept those principles which best explain
the facts of our experience, we must have some antecedent notion
of what a good explanation is. Now it can certainly be required of
an explanation that it should be self -consistent. But we are not
content with this. There are a number of maxims, which do actually
guide us in theorizing about the laws of nature, pointing to some
more positive ideal than self -consistency. The influence of these
maxims shows that there operates upon scientific minds some notion
of what a rational universe should be, as well as a belief that the
universe is rational, not derived from experience, but controlling the
interpretation of experience. ' The common notion that he who
would search out the secrets of nature must humbly wait on experi-
ence, obedient to its slightest hint, is,' it has been said \ ' but partly
true. This may be his ordinary attitude ; but now and again it
happens that observation and experience are not treated as guides
to be meekly followed, but as witnesses to be broken down in cross-
examination. Their plain message is disbelieved, and the investi-
gating judge does not pause until a confession in harmony with his
preconceived idea has, if possible, been wrung from their reluctant
evidence.' What these preconceived ideas are, it would be difficult
to say precisely ; nor is the question of their justification an easy
one. They have formed the subject of considerable discussion on
the part of philosophical writers since the time at least of Leibniz,
who perhaps did most to call attention to them. But one of the
most famous has a much higher antiquity. ' Occam's razor ' 2 —
entia non sunt multiplicanda praeter necessitatem — is a maxim to
which science constantly appeals. It is felt that there is a presump-
tion in favour of theories which require the smallest number of
ultimate principles : that there is a presumption in favour of the
derivation of the chemical elements from some common source,
or of the reduction of the laws of gravitation, electricity, light, and
heat to a common basis. Again, we are inclined to believe that the

1 Presidential Address at the British Association, Cambridge, 1904, by the
Rt. Hon. A. J. Balfour (Times of Aug. 18). He illustrates his statement by
reference to two cases, the persistent belief that the chemical elements will
be found to have a common origin, and the persistent refusal to believe in
action at a distance. It may however be doubted whether this refusal is as
well justified as that belief by the maxims in question.

2 William of Occam, ob. 1347.

xxin] OF EXPLANATION 507

ultimate laws of nature are not only few but simple. The law of
gravitation states that the attraction between any two bodies varies
inversely as the square of the distance. But it is conceivable that
the true relation of the force of attraction to the distance of the bodies
between which it acts is not so simple ; provided it diverged from
the ratio of the inverse square so slightly that the difference would be
less than our observation, with the margin of error to which it is
liable, could detect, such less simple relation would have as much
to be said for it, so far as the facts go, as the simple relation that
Newton established. Yet few would seriously consider its claims.
It may be said, and truly, that there are sound practical reasons
for accepting the simple relation, in preference to any other that has
no better claims, because it renders our calculations much easier ;
yet it may be doubted whether we really regard it as only a more
convenient hypothesis. We are more disposed to think it true
because such a simple relation satisfies better our ideal of explana-
tion. J. S. Mill's definition of Laws of Nature has been already
quoted1 — ' the fewest andsimplest assumptions, which being granted,
the whole existing order of nature would result '. In the words
' fewest and simplest ' are contained perhaps the most important
of the preconceived ideas which we have about the explanation of
the facts of nature.

It is impossible to reduce explanation to any definite formulae.
When nothing but a middle term is wanted, to connect with a sub-
ject a predicate empirically found to characterize it, there it will fall
into the form of syllogism.2 But comparatively few explanations
can be expressed in a single syllogism. Where, as is commonly the
case, they trace the complex result of several principles in some par-
ticular combination of circumstances, the building up of this result
in thought is not a syllogistic process.

As has been said above, there is no fundamental difference be-
tween explanation of a particular fact and of a general principle. In
the latter case, more abstraction has been performed ; we are ex-
plaining something exemplified in facts that constantly occur, which
has been extricated in thought from varying and irrelevant detail.
In the former also, some amount of abstraction must havo taken
place ; but the fact we have thus isolated still retains details that
make it unique. An oculist may explain the common fact that
short-sighted persons grow longer-sighted as they grow older, by

1 Supra, p. 386, n. 3. 2 But cf. infra, p. 524, n. 2.

508 AN INTRODUCTION TO LOGIC [chap.

showing how clear vision depends on focusing all the rays proceeding
to the eye from each several point precisely upon the surface of the
retina ; in short-sighted persons, the curvature of the lens of the eye
is excessive, and therefore objects have to be nearer than would
normally be necessary, in order that the rays proceeding from any
point in them may be focused on the retina and not in front of it ;
but the curvature of the lens is maintained by certain muscles,
which relax with age, and therefore, as years advance, clear vision
of objects is possible at a greater distance. If he were called upon
to explain some unique peculiarity of vision in a particular patient,
the task would still be of the same kind ; but the facts to be taken
into account would partly be facts peculiar to this case, and though
their consequences would be traced according to general principles,
their special combination would make the complex result unique :
unique however not necessarily, for the same combination might
conceivably recur, but only as a fact within medical experience.

Historical explanation is largely concerned with events in this
sense unique. History has generalizations that admit of explanation
also ; but human affairs are so complex, and our interest in them
extends into so much detail, that the unique occupies a quite
peculiar share of attention in its investigations. And its task
consists largely in making facts intelligible by tracing their develop-
ment. For an institution or event, when we come upon it as it
were abruptly, may surprise us : whereas if we know the past, we
may see that its existence or occurrence connects itself with other
facts about the same folk or period in accordance with accepted
principles. The institution of primogeniture for example, according
to which land descends upon the eldest son, is a peculiar institution,
unknown, according to Sir Henry Maine, to the Hellenic, to the
Roman, and apparently to the whole Semitic world ; neither did
the Teutonic races when they spread over Western Europe bring
it with them as their ordinary rule of succession. Whence then
did it originate ? for such institutions do not occur at haphazard.
Maine accounts for it as ' a product of tribal leadership in its decay '.
Chieftaincy is not the same thing as being a landowner ; but some
of the tribal lands were generally the appanage of chieftaincy. So
long as times were warlike, the chieftaincy seems not necessarily to
have gone to the eldest son of the deceased chief ; but ' wherever
some degree of internal peace was maintained during tolerably long
periods of time, wherever an approach was made to the formation

xxm] OF EXPLANATION 509

of societies of the distinctive modern type, wherever military and
civil institutions began to group themselves round the central
authority of a king, the value of strategical capacity in the humbler
chiefs would diminish, and in the smaller brotherhoods the respect
for purity of blood would have unchecked play. The most natural
object of this respect is he who most directly derives his blood from
the last ruler, and thus the eldest son, even though a minor, comes
to be preferred in the succession to his uncle ; and, in default of
sons, the succession may even devolve on a woman. There are not
a few indications that the transformation of ideas was gradual '.
The custom, Maine thinks, was greatly fixed by Edward I's decision
in the controversy between Bruce and Baliol ; where the celebrity
of the dispute gave force to the precedent. The rule of primogeni-
ture was extended from succession to the lord's demesne to succession
to all the estates of the holder of the signory however acquired,
and ultimately applied to all the privileged classes throughout
feudalized Europe.1 In a case like this, a knowledge of past facts
enables us to see how a new custom might emerge conformably to
known principles of human nature. There are motives for allowing
the chieftaincy to devolve upon the eldest son, and motives for con-
ferring it upon the strongest of the near kindred ; when the latter
are weakened by change of circumstance, the former are likely to
prevail. The influence of precedent upon the human mind is also
a familiar principle ; and though it is impossible to show that in
such cases nothing else could at any point have happened (Edward I
for example might have decided differently), yet in the light of what
we know of men's passions and purposes and of the physical con-
ditions under which they live, we are able to understand many of
the connexions which link events together.

Sciences like Geology or Biology set themselves for the most
part to solve more generalized problems of development : though
to them too some particular fact, apparently in conflict with a
theory, may offer occasion for a detailed historical enquiry. But
the explanation of the occurrence of crystallized rock, commonly
as that occurs, is not logically different from what it would be
if it occurred once only ; and if we set about accounting for
that local and temporal affinity of species which is expressed in
Mr. A. R. Wallace's principle that ' Every species has come into

1 v. Maine's Early Institutions, pp. 197-205, from which the above example
is abridged.

510 AN INTRODUCTION TO LOGIC [chap.

existence coincident both in space and time with a pre-existing and
closely allied species ' l, we shall not proceed otherwise than if the
affinities of one particular historical group of species were to be
accounted for.

There are other sciences (e.g. Political Economy or Kinematics)
which do not concern themselves with tracing any particular
historical development, yet have to explain the laws manifested
in a succession of events. Here too it may be of the essence of the
explanation to show how one change determines another, and the
new fact thus introduced determines a third, and so forth. The
laws involved may be various, and the sequence be explained by
resolution into stages, each of which exhibits a general principle,
while the special circumstances in which such a principle is exhibited
furnish the occasion for a further change that exemplifies another.

There are cases where the element of time is one of the most
important of the facts. Many effects depend upon the juxtaposition
of bodies in space, and their juxtaposition depends on time-con-
ditions. The fortune of a campaign may be decided by the rapidity
of a march, bringing troops upon the field at a critical moment ;
the troops may fight upon the same principles and with the same
degrees of courage all through, but the result is determined by their
being there at the time. The working of a machine would be thrown
out by anything that delayed or hastened the movement of a part
with which other moving parts had to engage ; and the same is of
course true as regards the articulated movements of an animal. The
disintegration of mountains is largely produced by frost succeeding
rain ; if rain only came just after frost, it would not take place in
the same way. Professor Marshall has called attention, in his
Principles of Economics, to the great importance of the element of
time in the working of economic laws.2

There are however also many results that are to be accounted for
through the concurrent operation of several principles : or rather
— for principles cannot in strictness be said themselves to operate —
through the concurrent operation of several causes, each according
to its own principle. The path of a projectile at any moment is
determined by its own inertia, the pull of the earth, and the resis-
tance of the atmosphere. It is true that at every moment these
forces are producing a new direction and velocity in the projectile,

1 Quoted Romanes, Darwin and after Darwin, i. 243.

2 e. g. Bk. III. c. iv. § 5, 4th ed. p. 184.

xxm] OF EXPLANATION 511

which forms the basis for an immediate further change ; and that
it is by following the continuous series of these successive changes
that its path is ascertained — a task which the notation of the calculus
alone renders possible. The consideration of any term in the series
of changes as the resultant of simultaneously operating causes ia
however different from the consideration of the succession of one
resultant change upon another in the series. And the explanation
of many problems lies in showing the concurrent operation of diffe-
rent causes, each acting continuously according to its own law ; as
opposed to the case just considered, where one cause may produce
an effect that, by virtue of the conditions with which its production
coincides, then produces a fresh effect in accordance with a different
law. The column of mercury in the barometer is maintained ac-
cording to laws that are all continuously exemplified, and not first
one and then another of them ; the atmosphere is always exerting
pressure, and in the mercury the pressure is always equalized in
virtue of its nature as a fluid. Economists are familiar with
' Gresham's Law ' that bad money drives out good, i. e. that if in
any country the circulating medium is not of uniform quality, the
best is always exported and the worst left behind. By ' best ' is
meant that whose intrinsic value bears the highest proportion to its
nominal value ; a sovereign which contains the proper weight of
fine gold being better than one containing less, and so forth. The
explanation of the Law is simple. Government can make the bad
money legal tender for the payment of debts at home ; it cannot
compel the foreigner to receive it. For discharging debts abroad
the better money is therefore more valuable, for discharging debts
at home it is no more valuable than the worse ; it is therefore more
profitable to export the good, and keep the bad money for home
purposes ; and the desire of wealth being one of the strongest and
most uniform motives in mankind, what is most profitable is natu-
rally done. Nothing turns here upon the resolution of a sequence
into stages exhibiting different laws ; the derivative law is shown
to follow from more general laws, under the special assemblage of
circumstances described in saying that the circulating medium in a
country is not of uniform quality ; but these general laws are
exhibited simultaneously and not successively. That the power of
any government extends to its own subjects only, and that men
desire wealth, are principles more general than Gresham's Law ; and
both apply to money, which is at once, as legal tender, a matter to

512 AN INTRODUCTION TO LOGIC [chap.

which the power of government applies, and, as medium of exchange,
the equivalent of wealth.

No logical importance attaches to the distinction between ex-
planations that derive a complex law from simpler laws exemplified
together, and those that derive it from simpler laws exemplified
successively. Many explanations involve both features. But there
is a difference of more importance between either of these, and that
form of explanation which consists in showing that laws, hitherto
regarded as distinct, are really one and the same. Newton showed
that the familiar fact that heavy bodies fall to the earth, and the
equally familiar fact that the planets are retained in their orbits,
were really instances of the same principle, the general Law of
Attraction. Something of the same sort is done when Romanes
points out that Natural Selection, and Sexual Selection, and Physio-
logical Selection, and Geographical Isolation are in their operation
so many forms of Isolation preventing free intercrossing among all
the members of a species, and thereby leading to modification of
type.1 In cases like these, we do not derive a derivative law from
several more general laws exemplified together or successively in
complex circumstances of a particular kind ; but a single more
general law or kind of process is shown to be exemplified in a diver-
sity of circumstances which have hitherto concealed its identity.
This operation is sometimes called subsumption, as bringing several
concepts under one, in the character of instances, or of subjects of
which it can be predicated in common. Yet even here it is plain
that the operation, of tracing the distinctive peculiarities of the laws
or processes explained or subsumed to the special character of the
circumstances in which the same more general principle is exhibited,
is of the same kind as occurs in all other forms of explanation : only
the further synthesis, in which the complex consequences of the con-
current or successive laws or kinds of process are traced, is lacking.

Explanation, as was said at the beginning of the chapter, is
deductive — deductive, that is, in respect of the reasoning involved
in it. Yet it has a close relation with the work of Induction, and the
consideration of this will form the subject of the remainder of the
chapter.

Explanation starts, as we have seen, from principles already
known, or taken as known ; and it shows that the matter to be
explained follows as consequence from these. But it is clear that

1 Darwin and after Darwin, vol. iii. o. i.

xxin] OF EXPLANATION 513

the reasoning which deduces their consequence from them is un-
affected by the nature of our grounds for taking them as true. If
they were nothing more than hypotheses, we might still argue from
them to their consequence as if they were indubitably certain. Just
as we may syllogize in the same way from true premisses and from
false \ so it is with any other kind of reasoning. Moreover, it was
pointed out that many at least of the most general and fundamental
of our scientific principles are accepted only because they explain
the facts of our experience better than any we can conceive in their
stead ; they are therefore, or were at the outset, hypotheses, used
in explanation of facts, and accepted because of their relative success
in explaining them. We do not see why they are true but only
why we must believe them to be true. They are established induc-
tively, by the facts which they explain, and the failure of any rival
hypothesis ; they are not explained from the facts, but the facts
from them.

It follows that whatever deductive reasoning enters into an ex-
planation enters also into the inductive proof of an hypothesis which
is shown to explain, and to be the only one that will explain 2, the
facts. And many explanations are put forward, which do not appeal
only to principles already known, but have it as their avowed
object to prove one or more of the principles which they employ.
Explanation then figures as an instrument of induction ; and
J. S. Mill spoke accordingly of a ' Deductive Method of Induction ',
and rightly attributed great scientific importance to the process
which he called by that name.

No better instance of this operation can be given than the
familiar instance of the Newtonian theory of gravitation. Sir Isaac
Newton showed that the movements of the heavens could be ex-
plained from two principles or laws — the First Law of Motion, and
the Law of Universal Gravitation. The former is, that every body
preserves its state of rest or uniform rectilinear motion until it is
interfered with by some other body ; according to the latter, every
particle of matter attracts every other particle with a force that
varies directly as the mass and inversely as the square of the

1 On this, cf. supra, pp. 331-334.

2 I add these words, because it is important to realize that an hypothesis
is not really proved by merely explaining the facts : cf. infra, p. 523. But
many hypotheses are provisionally accepted, which are not proved, on the
ground that they explain the facts, and without the performance of what
would often be the impracticable task of showing that no other hypothesis
could equally well do so.

1779 L 1

514 AN INTRODUCTION TO LOGIC [chap.

distance. The former had already been established by Galileo, and
Newton took it for granted ; but the latter he proved for the first
time by his use of it in explanation.

The theory which bears the name of Ptolemy though much older
than he, represented the sun, moon, and stars as moving round
the earth ; and originally it was supposed that they moved in circles
with the earth as centre. While the laws of motion were still undis-
covered, no difficulty was found in their circular motion ; indeed
Aristotle supposed it to be naturally incident to the substance of
which the heavenly bodies were composed, that their motion should
be circular ; for the circle is the perfect figure ; movement in a
circle is therefore perfect motion ; perfect motion belongs naturally
to a perfect body ; and the substance of which the heavens are
composed — the quinta essentia, distinct from the four primary
substances, earth, air, fire, and water, that are found composing
this globe — is perfect.1 The only difficulty arose when it was found
that the orbits of the heavenly bodies, other than the fixed stars,
were not perfectly circular ; and that was met by the hypothesis
of epicycles referred to in an earlier chapter.2 The substitution of
the Copernican for the Ptolemaic hypothesis, though involving a
reconstruction of the geometric plan of the heavens, did not neces-
sarily involve any new dynamics 3 ; Kepler's discovery that the
planetary orbits were elliptical was however a severe blow to the
traditional theory of epicycles, which had already by that time
become highly complicated, in order to make it square with the
observed facts. But when the first law of motion had been grasped,
it was evident that a planet, if left to itself, would not continue
moving in a circle, and returning on its own track, as Aristotle had
thought to be natural to it, and as with more or less approximation
it actually does : but would continue moving for ever forward with
uniform velocity in a straight line. Circular motion, however
uniform, was now seen to involve an uniform change of direction
for which a dynamical reason was required. And as the planets

1 According to Aristotle, every body left to itself had a natural motion,
dependent on its own nature : that of the heavens was round a centre, that
of earth and water to a centre, that of air and fire from a centre. The centre
was the centre of this globe, and so (on his view) of the physical universe.
Bodies need not be left to their own motion ; a stone, for example, may be
thrown towards the sky ; but in such case their motion was not natural,
but violent. 2 Supra, c. xxi, p. 470.

3 The heliocentric hypothesis was put forward in the history of Greek
astronomy by Aristarchus of Samoa.

xxm] OF EXPLANATION 515

were constantly changing direction towards the sun, a force exerted
from or in the direction of the sun seemed necessary.

Now the greatness of Newton's achievement 1 did not lie in the
conception that the orbital motion of the planets was the resultant
of two factors, the inertia of their proper motions which, left to
itself, would carry them forward with constant velocity in a straight
line, and a ' centripetal force ' which, left to itself, would carry them
to the sun. The resolution of curvilinear into rectilinear motions
had been accomplished before him, and the hypothesis of an attrac-
tive force had already been hazarded. It had even been suggested
by others as well that such a force might vary inversely as the
square of the distance ; for the area of the spherical surface over
which it might be conceived as spreading at any distance from the
centre of the sun varies directly as the square of the distance, and its
intensity might be supposed to decrease as the area increased.
Neither was it Newton who ascertained the facts about the move-
ments of the planets — no small or easy contribution to the solution
of the problem. But he did three things. He conceived that the
force which deflected the planets into their orbits was the same as
that which made bodies fall to the earth : or, to put it differently, he
identified celestial attraction with terrestrial gravity, and conceived
the earth as continually falling out of a straight path towards the
sun, and the moon towards the earth ; he conceived that this at-
tractive force was exerted between every two particles in the
universe ; and he invented a mathematical calculus by which
he could work out what were the theoretical consequences of the
principles which he assumed.

All these steps were of the highest importance. The first pro-
vided data to calculate from ; the second made it possible to give
a precise form to the doctrine of attraction ; the third made the
calculation possible. The amount of acceleration produced per
second in near bodies falling to the earth was already known2;

1 It is instructive to note that his law now seems not unconditionally true.

2 Strictly speaking, that acceleration should not be the same at 1,000 feet
from the earth and at 100 feet : and in virtue of atmospheric resistance
a cricket-ball should not fall as far in a given time as a cannon-ball ; but
the theoretical differences would be so small as to escape observation, and
therefore the fact that acceleration is empirically found to be 32 feet per
Becond for all bodies in the neighbourhood of the earth creates no difficulty.
On the other hand, in the oscillations of a pendulum, which vary in the
plains and in the neighbourhood of mountains, we do find evidence agreeable
to the theory, of the same kind as those minute differences would afford if
we could measure them. The logical bearing of these considerations will be

Ll2

516 AN INTRODUCTION TO LOGIC [chap.

Newton proved that the resultant of the attractions of all the par-
ticles of a sphere was as if its mass were concentrated at the centre ;
and that enabled him to show that the same law of attraction would
give the known acceleration of a falling body near the earth and also
that of the moon towards the earth or a planet towards the sun in
their fall from the path of the tangent.

With his proof of this Logic is not concerned. Processes of reason-
ing are too numerous for Logic to study them all, and those of
mathematics are for the mathematician to appraise ; it is enough
if the logician can satisfy himself in general regarding the grounds
of mathematical certainty. But assuming the task of deducing
from his principles their theoretical consequences to have been
performed, we may look at the logical character of the reasoning
in which Newton made use of that deduction.

The principal astronomical facts to be accounted for concerned
the movements of the earth and other planets round the sun, and
the movements of the moon round the earth.1 The former body
of facts had been already generalized by Kepler, in his three laws,
(i) that the planets move in ellipses round the sun, with the sun
in one of the foci ; (ii) that they describe equal areas in equal
times ; (iii) that the cubes of their mean distances vary as the
squares of their periodic times.2 There was also a large body of
recorded observations upon the movements and perturbations of the
moon. But when Newton first worked out his theory he conceived
that a sphere attracted bodies near it as if its mass were concen-

seen if it is remembered that a theory, though not proved by its conformity
with facts, is disproved by any clearly established unconformity.

1 Where the planets are mentioned they may be taken to include the
moon, unless the context expressly forbids.

2 Perhaps it should be explained that as a circle is a curve, every point
on which is equidistant from a point within it called the centre, so an
ellipse is a curve, the sum of the distances of every point on which from
two points within it called the foci is constant ; that the area described by

a planet in moving from a point a to a point b on
its orbit is the area comprised between the arc, and
the lines joining those points to the centre of the
sun : so that if the planet is nearer the sun, it will
move faster, since if ac, be are shorter, ab must be
longer, to make the area abc the same ; that the mean
distance of a planet is its average distance from the sun
during its revolution, and its periodic time the period of its revolution, so
that if the cubes of the mean distance vary as the squares of the periodic
time, it follows that a planet whose mean distance from the sun was twice
that of the earth would have a ' year ' or period of revolution, whose square
was to the square of one (earth's year) as the cube of two to the cube of
one — i. e. that its period of revolution would = V 8 x the earth's year.

xxiii] OF EXPLANATION 517

trated near the surface. On this assumption, the force which gave
an acceleration of 32 feet per second to falling bodies near the earth
would not, if it varied inversely as the square of the distance, account
for the period of the revolution of the moon. It was only after
several years, when his attention had been recalled to the whole
question by Halley, that he demonstrated that a sphere attracted as
if its mass were concentrated at the centre1, and found with this cor-
rection in his premisses that the theoretical results of the law of
Universal Gravitation agreed with the observed facts. But it was
further involved in his demonstration that any other rate of varia-
tion in an attractive force operating between all particles of matter
would give results conflicting with those facts ; and therefore it had
been shown not only that his theory might be true, but that if the
planetary motions were to be accounted for by help of a theory of
universal gravitation at all, the law of that attraction must be as he
formulated it.2

The further confirmations which Newton's Law of Universal
Gravitation has received, from its success in accounting for other
physical phenomena, need not detain us ; we have to look to the
steps involved in its establishment, and they can be sufficiently seen
in what has been detailed already. First, there was the suggestion
that the movements of the planets were to be accounted for by
reference to two factors — the inertia of their proper motions, and
a force of attraction ; this was not due to Newton. Next, it was
necessary to determine or conjecture the way in which these two
factors severally operated ; so far as the inertia was concerned, that
had also been in part already done, and it was expressed in the first
law of motion ; the actual velocity of each planet was ascertained
by calculation from astronomical observations, and the velocity
proper to each planet considered alone was determined by reference
to the actual velocity and the velocity acquired by gravitation.
But the velocity acquired by gravitation, or through the influence
of the attractive force, had to be conjectured ; and though the
law of its variation had been suggested before, unless the amount of
its effect between some given masses at some given distance were
known, the law of its variation left the matter quite indeterminate.

1 Cf. Glaisher's Address in commemoration of the bicentenary of the
publication of Newton's Principia, April 19, 1888, published in the Cambridge
Chronicle and University Journal of April 20, 1888. I owe the reference to
this to the kindness of Professor H. H. Turner.

* i.e. if it was to embody a simple ratio : cf. pp. 470-471, 507, supra.

518 AN INTRODUCTION TO LOGIC [chap.

The identification of the attractive force with terrestrial gravity
and its formulation as a force operating between every two par-
ticles of matter, thus completed the necessary data ; and prin-
ciples and facts were now before Newton, sufficient, if a method
of calculation were devised, to enable him to determine what should
be the consequences of his hypothesis. The next step was the
process of calculation. But he had to show, not barely what the
consequences of his hypothesis would be, but that they would be the
same as the observed facts : and moreover, that his was the only
hypothesis whose consequences would be the same as the observed
facts.1 The comparison therefore of the facts with the theoretical
results of his and of any other hypothesis was the step that succeeded
the calculation ; and having found that they agreed with his, and
with no other, he reasoned thus — Assuming that the continual de-
flexion of the planets from a rectilinear path is due to such an
attractive force, their actual motions, if my statement of the law
of attraction is true, would be thus and thus ; if it is false, they
would be otherwise : but they are thus and thus, and therefore my
statement is true.

Now of the steps in this whole logical process, some are not
processes of reasoning at all — the suggested reference of the resultant
motions to those two factors, the suggested identification of one
of the factors with terrestrial gravity, the suggestion that it operates
between all particles of matter, and the comparison of the theoretical
results with the observed facts. Reasoning may have been em-
ployed in establishing the first law of motion ; but that reasoning
lies outside the present appeal to it. The reasoning involved in
determining the theoretical results of the action of the factors assumed
is deductive. But the final argument, in which the agreement of the
facts with the results of this hypothesis and of no other is shown to
require the acceptance of this hypothesis is inductive. Had the
Law of Gravitation been already proved, we might have said that
Newton was merely explaining certain empirical generalizations
about the movements of the planets ; had it been already proved
and had the attraction of a sphere acted as he at first supposed, the
apparent disagreement of its consequences with the records of the
moon's and planets' movements would have led him not to lay aside

1 Cf. previous page, n. 2. It was possible to show that no other rate of
attraction would give results conformable to the facts, because the problem
was a mathematical one ; and in mathematics it is easier than elsewhere to
prove not only that if a is true, b is true, but also the converse.

xxra] OF EXPLANATION 519

the theory, but to doubt the observations, or to assume (as Adams
and Leverrier afterwards did for the perturbations of Uranus) the
existence of some other factor to account for the discrepancy ; but
inasmuch as it was only now proved by its exclusive success in
explaining the facts, he was arguing inductively to the proof of it.

If we look for a moment at the simpler inductive arguments
which establish the cause of a phenomenon by appeal to ' grounds
of elimination ', we shall find in them too something of this double
character, at once inductive and deductive. The facts appealed to
as showing that a is the cause of x are themselves accounted for by
that hypothesis. If, for example, facts do not allow us to doubt that
malarial fever is conveyed by the bite of the Anopheles mosquito,
then too the power of the Anopheles mosquito to convey malarial
fever accounts for its appearing in persons bitten by that insect. It
is impossible but that, if certain facts are the ratio cognoscendi of
a causal principle, that principle should be the ratio essendi of the
facts. But in these simple arguments there is nothing correspond-
ing to the deductive reasoning which works out the joint conse-
quence, in particular circumstances, of the action of two or more
causes, from a knowledge (or conjecture) of the effect which each of
these causes would produce singly. It is on account of this opera-
tion that J. S. Mill gave to reasoning that involves it, even when its
primary object is the inductive establishment of a general principle,
the name of the ' deductive method of induction '.

Such reasoning can only be used where the joint effect of several
causes is calculable from the laws of their separate effects. Where
the joint or complex effect cannot be determined by thinking, from
a knowledge of what the separate effects would be, we rely entirely
on the inductive method of elimination in order to show that such
complex effect is to be attributed to the action of one particular
conjunction of causes rather than another. But into the investiga-
tion of any complex effect of the other kind, in which the action of
the several causes can be traced as combining to produce it, some
measure of this deductive reasoning will always enter. Most
obviously is this the case in regard to those complex effects which
exemplify what has been called a homogeneous intermixture1 — i.e.

1 J. S. Mill gave the name of ' homogeneous intermixture of effects * to
those cases where the joint effect of several causes acting together is of the
same kind with their separate effects, and differs only in some mathematical
respect from the effects which the same causes would produce singly ; this
happens, e.g., in the mechanical composition of forces — for which reason

520 AN INTRODUCTION TO LOGIC [chap.

where the complex phenomenon is quantitative, and there are many
factors determining its quantity, some by way of increase and some
of decrease. The simpler inductive methods are there quite in-
adequate : for there need be no two instances of the phenomenon in
which its quantity is the same, nor, if there were, need the combina-
tion of factors be the same ; neither can we infer from the non-
occurrence of the phenomenon, or its presence only in an imper-
ceptible degree, where the supposed cause is present, that what we
had been inclined to ascribe it to does not produce it ; since that cause
might be present, but counteracted by another of contrary effect.
Even the rule that cause and effect must vary concomitantly, and
the rule that no such portion of the effect must be attributed to one
among the factors making up the cause of the whole, as is already
accounted for by other factors, are not sufficient to ensure success in
such enquiries. It is necessary to be able to measure more or less pre-
cisely the complex effect, and to know with corresponding precision
the amount of effect that the several supposed causes would pro-
duce alone, in order to prove that any particular one among them
cannot be dispensed with, or rejected from being a part cause.
And into this proof a deductive calculation will obviously enter.
In the fiscal controversy, for example, initiated in Great Britain in
1903, it was alleged that the excess in the value of our imports
over that of our exports was due to the crippling of our production
by free trade ; but this could only be proved by showing that the
difference of value between exports and imports was unaccounted
for, unless we were living on our capital ; and that could not be
shown unless the excess in value of imports were ascertained, which
was attributable to other causes known to assist in producing their
total excess-value — such as the fact that the valuation of our imports
was swollen by the inclusion of the cost of carriage to our own ports
(while our exports, being valued before transport, did not receive this

he spoke also of Composition of Causes in such a case. Where the joint
effect differs in quality from the separate effects (and so cannot be calculated
from a knowledge of them) he called it heterogeneous or heteropathic. He
illustrated this from chemical combination, in which the chemical properties
of the compound (unlike its weight) are not homogeneous with those of its
constituents ; though he quite overlooked the fact that elements were not
the ' cause ' of a compound in his usual sense of that term. But though
homogeneous intermixture of effects allows of deductive reasoning, such
reasoning may also occur where the complex effect is not the sum or difference
or mathematical resultant of the separate effects. And it is the deducibility
of it from a knowledge of the several principles involved that differentiates
this kind of ' intermixture of effects '. Cf. System of Logic, III. vi.

xxin] OF EXPLANATION 621

addition) : and by the value of the goods that paid for ships sold
abroad or for the service which the country performs as ocean-
carrier, although nothing appears in the total for exports on those
heads : and by the value of the goods that represent payment for
the use of British capital invested abroad, or pensions charged on
the Government of India. The difficulty of determining the amount
by which these causes should make our imports exceed our exports
in value rendered it exceedingly hard to prove, at least on this
line of argument, that we could not be paying out of the year's
production for all that we imported in the year.

To sum up — Explanation considered in itself is deductive : it
consists in showing that particular known facts, or laws, or general
causal connexions, follow from principles already established, in the
circumstances of the case ; it discovers therefore nothing new,
except as it makes us understand the reason for that which we had
hitherto only known as a fact. But explanation also enters into
induction, so far as the principles, from which the facts, or laws, or
general causal connexions, are shown to follow, were not previously
established, but are so only now by showing that the actual facts,
laws, or causal connexions would follow from them and not from any
alternative principles. In such induction there are four main steps
distinguishable : (i) conceiving the several agents, or causes, at
work ; (ii) determining or conjecturing how or according to what law
each of them severally would act ; (iii) reasoning from these pre-
misses to the result which they should produce in common, as well
as to the result which would follow on any rival hypothesis as to the
agents at work, and the several laws of their operation; (iv) showing
that the facts are what should follow from these, and not from any
rival premisses.1

Many observations might still be made upon this type of argument
— one of the commonest and most important in the sciences. Its
applications are very various. It may be directed to establish that
a known agent is concerned in the production of a familiar effect
with which it has not hitherto been suspected to be concerned : as
Darwin showed that earthworms play a part in the subsidence of
buildings below the surface level. Or again, it may be used in sup-
porting a theory as to the law or principle displayed in a set of
variable facts : the Mendelian theory, that there are definite alterna-
tive factors, dominant and recessive, determining, according as only
1 This is not always a separate step : v. p. 523, n. 1.

522 AN INTRODUCTION TO LOGIC [chap.

one kind or both are present in the fertilized ovum, various peculia-
rities in individual animals and plants, involves elaborate deduction
of the proportions in which such peculiarities should be found over
a large number of specimens, and of the possibility of establishing
varieties that breed true in this or that respect, and it is recom-
mended by its success in accounting for observed facts of this kind.
Or it may be used to show the existence of an agent, whose mode of
action, if it exists, is known, as Adams and Leverrier argued that there
must bo a planet hitherto unobserved to account completely for the
perturbations of the known planet Uranus.1 The more we can
introduce number and quantity into our statement of the principles
that are to account for facts, and can determine numerically and
quantitatively the facts themselves, the more this type of argument
is available. But we are using it, whenever an explanation of facts
is offered, among the premisses of which is one whose truth is in
question, and is inferred from the success with which by its help the
facts are shown to be explicable. The question may be, what
causes can produce such an effect, or which of the causes that can
produce it are contributing to produce it now ? We may wish to
establish a general principle, or only some special fact as to the
circumstances that are modifying the results of that principle in the
case before us. It is possible too that the laws of the action of the
several agents may some of them have been previously ascertained
and established, while others are only conjecturally formulated ; or,
if the question be as to the agents contributing to the result in a par-
ticular case or class of cases, the laws of the several actions of them
all may have been established previously. But without dwelling on
these points, we may conclude the chapter with four considerations.
First, the inductive arguments of science display in every dif-
ferent degree that combination with deductive reasoning which has
been now analysed. Thus, though we may represent in symbols
the induction whose logical form is a mere disjunctive argument,
and contrast it with this into which the deduction of a complex
result from several premisses so prominently enters, yet in actual
practice the contrast is not so sharp ; in few inductive investigations
is the reasoning merely disjunctive ; but the amount of deductive
reasoning that has to be performed before one is in a position to

1 This celebrated argument is often also used to illustrate Mill's ' Method
of Residues ', as it very well does. For that Method is, as Mill himself
recognized, partly deductive in character. Cf. infra, p. 560.

xxm] OF EXPLANATION 523

apply a disjunction, and to say that this hypothesis is true because
the rest can be proved false, varies very greatly in different inves-
tigations.

Secondly, to show that the facts agree with the consequences of
our hypothesis is not to prove it true. To show that is often
called verification * ; and to mistake verification for proof is to
commit the fallacy of the consequent2, the fallacy of thinking that,
because, if the hypothesis were true, certain facts would follow,
therefore, since those facts are found, the hypothesis is true. It
is the same mistake as that of incomplete elimination, in the estab-
lishment of a simple causal relation : the same as results from over-
looking what is called the Plurality of Causes. A theory whose
consequences conflict with the facts cannot be true ; but so long as
there may be more theories than one giving the same consequences,
the agreement of the facts with one of them furnishes no ground for
choosing between it and the others.3 Nevertheless in practice we
often have to be content with verification ; or to take our inability
to find any other equally satisfactory theory as equivalent to there
being none other. In such matters we must consider what is called
the weight of the evidence for a theory that is not rigorously proved.
But no one has shown how weight of evidence can be mechanically

1 Mill supposes verification, i.e. showing that the facts agree with the
consequences deduced from an hypothesis, to be always a separate stage in
the whole process. But, as Professor Cook Wilson has pointed out, this is
not so, if the only facts appealed to are those which the hypothesis was
framed to explain ; for then its consequences are not deduced first, and the
facts ascertained and compared therewith after. Mill was thinking of what
is very common in this sort of inquiry, viz. that we endeavour to verify
our theory by considering what should happen, if it were true, in circum-
stances which have not been examined, nor perhaps hitherto existed, and
then observing what happens in these circumstances, instituting them if
necessary. Such procedure often involves very delicate and elaborate experi-
ment, as well as very intricate calculation, especially in physics. If the
result which our theory led us to anticipate actually occurs, the theory is
said to have shown a power of successful prediction ; and men are often
more influenced in favour of a theory by its power of successful prediction,
than by its explaining facts already known. This however is unreasonable.
Just as an erroneous theory may successfully explain known facts, i. e. the
facts may be such as would exist if the theory were true, and yet it may
be in some respect false, so it may successfully predict unknown facts. But
what really gives to a theory a greater title to our belief is greater compre-
hensiveness, i. e. power to explain a wider and more varied range of facts ;
for the bigger the system, the harder it is to find several principles that
equally satisfy the facts of it. And when men seek to verify a theory by
way of predicting what will occur in fresh circumstances, they commonly
take circumstances unlike those of what it was first formed to explain.

8 Cf. p. 596, infra. 3 Cf. p. 423, supra.

524 AN INTRODUCTION TO LOGIC [chap.

estimated ; the wisest men, and best acquainted with the matter
in hand, are oftenest right.

Thirdly, there is no logical difference between the reasoning con-
tained in explanation, and the inductive reasoning that involves
explanation, except in one point : that the latter infers the truth of
some premiss assumed in the explanation from its success in explain-
ing the actual facts and the impossibility of explaining them with-
out assuming it. Where this impossibility is not shown, and we
content ourselves with verification — that is, with showing that the
facts consist with the assumption — there the logical difference is
still slighter ; it amounts to this, that in explanation the premisses
are taken as previously known, and in the other case something in
the premisses, not taken as previously known, is accepted on the
strength of its use in the explanation.1

Fourthly, we may answer here the second of the two questions
raised at the end of c. xvii. Demonstration is explanation from
principles that are self-evident, or necessarily true. If it be said that
in that case very little of what we believe is demonstrated, we must
admit it. We can demonstrate little outside mathematics. But we
have an ideal of demonstration, and it seems to be that ; and it is
not syllogistic, as Aristotle thought it to be.2

[Dr. Bosanquet, in a paper already referred to, ' On a Defect in the
customary logical Formulation of Inductive Reasoning ' (Proceedings
of the London Aristotelian Society, N. S. vol. xi, 1910-11, p. 29), has
expressed the opinion that ' the restriction of Inductive proof to

1 J. S. Mill, to whose work the above chapter is not a little indebted
(v. Logic, III. x-xiii), fails to mark sufficiently the difference between showing
that the facts agree with a theory, and showing that the theory is true.
And he does not bring out clearly enough the relation between what he calls
the Deductive Method of Induction (c. xi) and what he calls the Explanation
of Laws of Nature (c. xii). He neither notices how they differ, nor how
closely they agree, though he gives the same investigation (the Newtonian
theory of gravitation) as an example of both of them (xi. 2, xiii. 1). More-
over, in resolving into three steps his ' Deductive Method of Induction ', he
leaves out the first of the four mentioned on p. 521, but cf. xi. 1 and vii.

2 For syllogism, as has been argued above, pp. 308-311, implies the applica-
tion, to a particular case, of a general principle known independently ; with
complete insight, the necessity which connects the different elements in
a complex fact should be manifest in the case before us, and the general
principle or major premiss not brought in ab extra, but rather visible in and
extricable from that case (cf. p. 311, supra). This much however Aristotle
would probably have admitted, and would have called it syllogism, to show
what character it was, in a subject of a given kind S, that involved its
having the predicate P ; but most demonstration cannot even so be put into
the form of syllogism, connecting one term with another through a third
by the relation of subject and attribute.

xxm] OF EXPLANATION 525

[the disqualification of competing hypotheses is a fundamental error
of principle ', and adds that he here finds himself in opposition to
the doctrine of this book. With a great deal of his paper I cordially
agree. It is an attack on the sufficiency of the principle ' Same
cause, same effect ' in inductive enquiry. Though provoked especi-
ally by the writings of M. Bergson, from whose L' Evolution creatrice
he quotes a ' typical passage ' \ it seems to me to hold good equally
against Mill's presentation of induction in his exposition of his
1 Inductive Methods '. Induction is not commonly nor for long so
simple a business as the pairing off of a determinate cause a with a
determinate effect x, each repeated unchangingly amid varying
circumstances from among which it has to pick them out. Our
' causes ' are commonly variables, with whose variations is con-
nected a corresponding variation in the effect, and we seek a principle
from which we may determine what variation in the one is connected
with what variation in the other. And they are commonly co-
operant, so that we need also to trace their consequences through
divers combinations into very divers complex effects. Hence, as
Dr. Bosanquet says, the intelligence will ' bind different to different
in binding same to same ' ; and the universality and generality at
which it aims ' is not measured by millions of repeated instances,
but by depth and complexity of insight into a subsystem of the
world ' (loc. cit. p. 34). ' The value of an Inductive conclusion, as of
any piece of knowledge, lies in the amount of reality which it enables
us to grasp, and this is very slightly tested by the number of cases
in which the nexus is repeated in fact ' (ib. p. 39). This seems to me
very true, and I think the foregoing chapters are in accordance with
it.2 I also agree that what Dr. Bosanquet calls ' this work of the
universal ' is ' the true spirit and mainspring of the inductive ad-
vance of knowledge ' (ib. p. 34). I have indeed said something of
the kind above, pp. 469-470. That the work of the mind in framing
theories cannot be reduced to rule I have pointed out in the same
chapter, when discussing the formation of hypotheses. That
indeed is its most originative work. It may be compared with the
activity of artistic creation. There the mind, ruminating as it were
upon a hint of the beauty which it seeks to articulate, somehow
advances to a fuller apprehension of it ; and in scientific activity,
ruminating upon certain facts, it advances to the thought of a system
in which they might be connected. This is the genuine work of
intelligence 3, but our minds are not fully intelligent. Much suggests
itself to the artist which is not suited to his theme ; sometimes he

1 p. 218 : ' L'intelligence a pour fonction essentielle de Her le meme au
meme, et il n'y a entierement adaptable aux cadres de l'irjtelligence que les
faits qui se repetent.'

2 Cf. e. g. pp. 402, 408-409, 458, n. 1, 483. Cf. also Mill, op. cit., III. xi. 3.

3 Cf. the paper on ' Mechanism, Intelligence and Life ', Hibbert Journal,
xii. 3, April 1914, pp. 626-629.

526 AN INTRODUCTION TO LOGIC [chap.

[rejects it, sometimes he retains it, and others may recognize it as a
defect in the work of his art. There is no general criterion here, any
more than there is a general criterion of scientific truth, as Dr. Bosan-
quet says (ib. p. 38). A theory, as I have urged, may make coherent a
wide range of fact, and yet not be true. But I think that Dr. Bosanquet
underrates at this point the part played by the ' eliminative test '. It
is the function of the intelligence to trace the connexion between one
feature and another in the real, and this it does even if the hypothesis
that these features are exemplified in this or that existent situation
be false.1 But where, as in the inductive sciences, our hypotheses
include so much that, even if true, seems to us mere brute fact, a
connexion of features in the real that is not self-evident, nor what we
have any prospect of coming to find so, there we must, as it seems to
me, rely for our acceptance of such alleged connexions upon the
elimination or the lack of alternative theories. Dr. Bosanquet says
that ' the only criterion of truth is the fuller truth — the science at
a more developed stage ' (ib. p. 38). If by a more developed stage
of a science he means one at which we are aware of fresh facts to be
connected in a system with those previously known, then with our
earlier theory of the principles displayed in them these fresh facts
will either be consistent or not. Supposing that they are not (as
the varying distances of the planets from the earth were not consis-
tent with the theory that they moved round it in concentric spheres),
they lead us to abandon the theory ; and we are applying here
the eliminative test. Supposing that they are (as fresh facts re-
vealed about the apparent movements of the planets by more
accurate observation were consistent with the theory of epicycles),
they do not prove the theory true. By the more developed stage
of a science may be meant a stage at which we have not only ascer-
tained fresh facts, but thought out principles of connexion that will
account for them, whereas those accepted at a less developed stage
will not. But again it seems to me that we now reject those by the
eliminative test, because if they were true, the newly ascertained
facts would not be as we find them ; and we accept the principles
that have displaced those, unless indeed they be self-evident, because
they alone account for all the facts now known to us. Should we
discover further facts inconsistent with them, we should give them
up also ; and therefore they remain subject to the eliminative test.
The mere elaboration of a theory to keep pace with the accumulation
of fresh facts, which might be called a development of the theory,
is no criterion of its truth. It need not be true because it admits of
such elaboration, and the modifications introduced may only make
it more erroneous. If I understand him rightly, Dr. Bosanquet
holds that as our knowledge of the facts belonging to some ' sub-
system of the world ' increases, and our theories of their systematic

1 Cf. supra, pp. 333-334.

xxiii] OF EXPLANATION 527

[connexion are modified and elaborated accordingly, we learn that
we are getting nearer to the truth as to their connexion merely by
comparison of our theories in respect of systematic comprehensive-
ness. I agree that the more systematically comprehensive is pre-
ferred to the less, because it accounts for facts which lead us to reject
the other as not accounting for them. But nothing in its principles
or starting-points which is not self-evident seems to me in the last
resort to have any other warrant than that it alone enables us to find
systematic connexion in the facts. This seems to me to be the only
inductive proof, and if a competing hypothesis enabled us equally
well to find systematic connexion in the same set of facts, I do not
see how we should decide between them, until we discovered a
' crucial instance ' \ a fact which overthrew one of them, because that
could not find a place for it. And even so, the other could not
be proved, unless we could show that all possible competing hypo-
theses had been overthrown. This view of the nature of inductive
proof does not of course involve that such eliminative argument is
the most important part of inductive enquiry, or even the most im-
portant argument in it. There is much work, both of thought and
otherwise, besides argument in inductive enquiry. I have enlarged
on this in Chapter XXI. The activity of the intelligence which
results in the formation of fruitful hypotheses, on whose importance
Dr. Bosanquet so well insists, is not a process of argument. Nor
can rules be given for it ; Bacon promised to show how his method
of ' Exclusions ' could be applied to the formation of bonae ac verae
notiones, as well as to the rejection of notiones that were not bonae
ac verae, but he never showed it, because it cannot be done. And
of argument the deductive processes spoken of in the present chapter
are more difficult, and often bulk much more largely in a scientific
investigation, than the mere eliminative argument, which is charac-
teristically inductive because involved in every attempt to establish
a principle of connexion, neither self-evident nor explicable from
other principles, by appeal to facts that are to show its truth. Such
a principle need not connect ' the same with the same ', with no
provision for variation and diversity. To trace its exemplification
may be, as Dr. Bosanquet says, ' like the continuation of a varying
curve from the datum of a given fragment of it ' (ib. p. 35). 2 But
suppose we were given a series of points lying on a curve, and re-
quired to find others through which it would pass at certain distances
when continued ; a number of curves might satisfy the data, but give
different positions for the quaesita ; how except by fresh data should
we be able to decide between our alternatives, and then how else than
by rejecting those with which these fresh data were inconsistent ?
That, in the last resort, seems to me to typify our case in regard to
those scientific generalizations which rest merely on inductive proof.]

1 Cf. infra, p. 565, n. 1 2 Cf. H. Poincar6, quoted supra, p. 411, n. 2.

CHAPTER XXIV

OF INDUCTION BY SIMPLE ENUMERATION
AND THE ARGUMENT FROM ANALOGY

There are many reasonings which do not prove their conclusion.
It is not merely that we have to use dubitable premisses ; for this,
though it destroys the strictly demonstrative character of our know-
ledge, does not invalidate the reasoning, so long as the conclusions
are what must be drawn, if the premisses are true. It is that we
often draw, and act upon, conclusions, about which we cannot say
even this much, that they must be true if the premisses are. And
in so doing, we often find ourselves right ; nor, if we refused to do
it, could the affairs of life be carried on. Descartes, when he set
himself to examine all which he had hitherto believed, and to doubt
everything which could be doubted, determined with himself that
he would not let this demand for demonstration in things of the
intellect prevent his following the most probable opinion in practical
matters.1 But it is not only in these that we have to hazard an
assent to conclusions which our premisses do not strictly justify.
Many branches of science would not progress at all, unless we did
the same there. In the first place, by committing ourselves to
a conclusion, and working upon the assumption that it is true, we
may be led to results that will help either to confirm or to overthrow
it ; whereas if we had merely withheld our assent from any con-
clusion, because the evidence was inconclusive, we might have
remained indefinitely long possessed only of that inconclusive
evidence. ' Truth ', said Bacon, ' is more readily elicited from error
than from confusion ' 2 ; and perhaps we might add, than from
indecision. Only we must in such cases let our assent be provisional,
and hold our opinion not as demonstrated, but as in default of
a better. The advice of the politician, that a man should make war
with another as with one to whom he may be reconciled, and peace
as with one with whom he may become at variance, may without

1 Discours de la Meihode, Troisieme Partie.
1 Nov. Org. II. 20.

SIMPLE ENUMERATION AND ANALOGY 529

suspicion of cynicism be adapted to the assent or dissent with which
we receive conclusions that are based on insufficient evidence. But
secondly, the sciences differ very much in the amount of evidence
which they can hope to obtain for their conclusions. A fairly
rigorous science may be content to use provisionally principles
which are known to be insufficiently proved (and that means really,
not proved at all) ; but some sciences hardly ever obtain rigorous
proof of their positions, as for example Anthropology ; and yet
much at any rate of their teaching is generally accepted as authori-
tative. Aristotle said that it was ' the business of education to
teach a man to demand rigorous proof of anything according to the
nature of the subject ; for it is as foolish to ask demonstration of
the orator, as to accept plausibilities from the mathematician ' x ;
and he would have allowed that for this purpose education must
include both a training in ' Analytics ' 2 and an acquaintance with
the kinds of subject-matter to which these different attitudes are
appropriate. It is often said that a man whose studies are too ex-
clusively mathematical is at sea when he comes to deal with matters
that do not admit of demonstration ; and that contrariwise, if he is
trained only in sciences where rigorous proof is impossible, he becomes
incompetent to see what is required in matters of a stricter sort.

There are no logical criteria by which to judge the value of such
reasonings, unless what is called the Theory of Probability may
claim to be such a criterion. But the Theory of Probability is
primarily a branch of mathematics ; many of the assumptions
which underlie its applications are open to suspicion on logical
grounds ; and its use is at any rate confined to subjects that admit
of numerical treatment. The object of the present chapter how-
ever is to consider briefly two kinds of argument, which while being
of this inconclusive character are very common, and have attracted
considerable attention from logical writers accordingly.

Induction by Simple Enumeration consists in arguing that
what is true of several instances of a kind is true universally in
that kind. Simple enumeration means mere enumeration ; and

1 Eth. Nic. a. i. 1094b 23 nenatbevfievov yap ianv iiii toctovtov raKpifies
tVifrjreie Kaff" eKaarop yivos, «<£' ocrov f] tov npayfxaros (pvcris eTri8exeTCU' Tapa-
ttXtjctiov yap (paiverai p.adi]p.aTiKOv re nidavoXoyovvTos d7roSe^€cr^ai Kal prjropiKov
u7ro8eit;cis cmaiTUv,

2 Aristotle called by this name his treatises on syllogism and demonstra-
tion, presumably because in them he sought to analyse the argumentation
of ordinary debate and of scientific proof, and so to show what conditions
must be fulfilled in order to justify or compel assent to conclusions in either field.

1779 M m

530 AN INTRODUCTION TO LOGIC [chap.

such an argument differs from scientific induction in the absence of
any attempt to show that the conclusion drawn is the only conclusion
which the facts in the premisses allow, while it differs from in-
duction by complete enumeration in that the conclusion is general,
and refers to more than the instances in the premisses. It should
however be noted here, that induction by complete enumeration, if
the conclusion be understood as a genuinely universal judgement,
and not as an enumerative judgement about all of a limited number
of things, has the character of induction by simple enumeration.
The name of empirical generalization is also given to such argu-
ments by simple enumeration.

Bacon's strictures upon this form of reasoning have been already
referred to.1 Regard it as a form of proof, and they are not unde-
served. Yet it is still in frequent use, in default of anything better.
It has been inferred that all specific characters in plants and animals
are useful, or adaptive, because so many have been found to be so.
So many ' good species ' have become ' bad species ' (i. e. species in-
capable of any strict delimitation) in the light of an increased know-
ledge of intermediate forms, that it has been inferred that all species,
if we knew their whole history, would be found ' bad \2 The familiar
generalization that we are all mortal, though not based solely on
enumeration, draws some of its force thence. Most men's views of
Germans, or Frenchmen, or foreigners generally, rest upon their
observation of a few individuals. The ' four general rules of
geography ', that all rivers are in Thessaly, all mountains in Thrace,
all cities in Asia Minor, and all islands in the Aegaean Sea, are
a caricature of this procedure, drawn from the experience of the
schoolboy beginning Greek History. The history of the theory of
prime numbers furnishes one or two good examples. More than
one formula has been found always to give prime numbers up to
high values, and was assumed to do so universally : x2 + x + 41
worked for every value of x till 40 : 22* + 1 worked for long, but it
broke down ultimately.3 It is needless to multiply illustrations.

What is the assumption which underlies arguments of this kind ?
It is the old assumption that there are universal connexions in
nature ; and the conjunction of attributes which our instances
present is taken as evidence of a connexion. The arguments are

1 Nov. Org. I. 105. Cf. supra, pp. 380, 391-392.

2 Romanes, Darwin and after Darwin, ii. 282.

8 v. Jevons, Elementary Lessons in Logic, pp. 221-222.

xxiv] SIMPLE ENUMERATION AND ANALOGY 531

weak, because the evidence for the connexion is insufficient. If
abed, instances of x, present the property y, it does not follow
that y is connected with those features on account of which they
are classed together as x. Yet a large number of instances furnishes
some presumption. For some reason must exist, why all these
instances exhibit the same property. If it is not in virtue of their
common character x, it must be in virtue of some other common
feature. When the variety of circumstances is great, under which
the instances are found, and the differences many which they pre-
sent along with their identity as x, it is harder to find any other
common features than what are included in classing them as x.
Therefore our confidence in the generalization increases, although
it may still be misplaced. All men are mortal ; for if men need not
die except through the accident of circumstances that are not
involved in being man, is it not strange that no man has avoided
falling in with these circumstances ? There is force in the question.
The number and variety of our observations on the point are such,
that almost everything can be eliminated : almost everything that
has befallen a man, except what is involved in being man, has also
not befallen other men : who therefore ought not to have died, if it
were because of it that men die. Something involved in being
man must therefore surely be the cause of dying.

Induction by Simple Enumeration rests then on an implied
elimination ; but the elimination is half -unconscious, and mostly
incomplete ; and therefore the conclusion is of very problematic
value. But where the instances do serve to eliminate a great
deal, the openings for error are correspondingly reduced in number,
and the conclusion is received with greater confidence accord-
ingly. General considerations of this kind, however, will not
stand against definite opposing facts ; therefore such an empirical
generalization is at once overthrown by a contradictory instance.1
Neither will they overbear more special considerations drawn from
acquaintance with the subject-matter to which the induction be-
longs. Pigmentation is known to be a highly variable property in
many species ; therefore the overwhelming range of instances to
show that all crows are black was felt to be insufficient to give
the conclusion any high degree of value. Again, a difficulty in
conceiving how two properties could be causally connected will
incline us to attach less weight to the fact of their conjunction.
1 "EvcrTacris, instantia, meant originally a contradictory instance.

Bi m 2

532 AN INTRODUCTION TO LOGIC [chap.

And contrariwise, where the connexion to which the conjunction
points is one which seems conformable with other parts of our
knowledge, we are much more ready to generalize from the con-
junction. Many general statements are made about the correlation
of attributes in plants and animals, which rest on simple enumera-
tion ; but the theory of descent suggests an explanation of the
constancy of such a conjunction ; for what was correlated in a
common ancestor might well be correlated universally in the
descendants. We are therefore readier to suppose that attributes
found several times accompanying one another in a species (such
as deafness with white fur and blue eyes in tom-cats, or black
colour with immunity to the evil effects of eating the paint-root in
pigs x) are correlated universally, even though we can see no direct
connexion between them, than we should be if no way of explaining
the constancy of the conjunction presented itself to us.

The Argument from Analogy (at least in the usual sense of the
term) is of the same inconclusive character as Induction by Simple
Enumeration ; and like it, rests on the general belief in universal
connexions, and takes a conjunction of attributes as evidence of
their connexion.

Analogy meant originally identity of relation. Four terms, when
the first stands to the second as the third stands to the fourth,
were said to be analogous, or to exhibit an analogy. If the relation
is really the same in either case, then what follows from the relation
in one case follows from it in the other ; provided that it really
follows from the relation and from nothing else. Where the terms
are quantities, or are considered purely on their quantitative side,
and the relations between them are also quantitative, there the
reasoning is of course mathematical in character : analogy in mathe-
matics being more commonly called proportion. And such reason-
ing is necessary, like any other mathematical reasoning. If in
respect of weight a : b :: c : d, and if a weighs twice as much as b,
then c must weigh twice as much as d. So soon however as we
connect with the relation c : d, on the ground of its identity with the
relation a : 6, a consequence which is not known to depend entirely
on that relation, our reasoning ceases to be demonstrative. Suppose
that by rail the distance from London to Bristol bears the same
relation to the distance from London to Plymouth as the distance
from London to Darlington bears to the distance from London to
1 v. Darwin, Origin of Species, c. i, 6th ed. p. 9.

xxiv] SIMPLE ENUMERATION AND ANALOGY 533

Aberdeen : and that it costs half as much again to send a ton of
timber to Plymouth from London as to Bristol ; we cannot infer that
the rate from London to Aberdeen will be half as much again as it is
to Darlington ; for the rate need not depend entirely on the relative
distance, which is all that is alleged to be the same in the two cases.

There are many relations however between terms which are not
relations of quantity. Here too, four terms may stand in an
analogy : and what follows from the relation of the first to the
second may be inferred to follow from the relation of the third to
the fourth. It might be said that the relation of his patients to
a doctor is the same as that of his customers to a tradesman, and
that therefore as a customer is at liberty to deal at once with rival
tradesmen, so a man may put himself at once in the hands of several
doctors. And if the relations were the same, the argument would
be valid, and indeed in principle syllogistic ; for the common relation
would be a middle term connecting a certain attribute with a man's
position towards his doctor. ' Those who employ the services of
others for pay are at liberty to employ as many in one service
as they pay for ' : such might be the general principle elicited from
our practice in shopping, and proposed for application to our
practice in the care of our health. The case of patient and doctor
is ' subsumed ' under the principle supposed to be exhibited in the
case of customer and tradesman. Even however if it were not
possible to disentangle a general principle, and reason syllogistically
from it, we might use the analogy ; thinking that there was an
identity of relations, and that what was involved in the relation in
the one case must be involved in it in the other.

Unfortunately however the identity of the relations may be
doubted. Relations are not independent of their terms. Quantita-
tive relations are no doubt independent of everything except the
quantitative aspect of their terms, and are on that account usually
stated as between quantities in the abstract. But with other
relations it may be very difficult to abstract, from the concrete nature
of the terms between which they hold, the precise features which
involve the relation. Hence we may say that two relations are
similar, and yet doubt whether they are similar in the way that
would justify the inference. They may be partially the same, but
the difference may just invalidate the consequence x ; and reason-
ing by analogy cannot then possess the character of necessity.

1 Cf. infra, pp. 589-590.

534 AN INTRODUCTION TO LOGIC [chap.

David Hume held that virtue and vice are not attributes of any
act or agent, but only feelings which an act may arouse in a spec-
tator ; so that if nobody approved or disapproved my actions, they
could not be called either virtuous or vicious. And one of the
arguments by which he endeavoured to sustain this opinion was as
follows. A parricide, he said, is in the same relation to his father
as is to the parent tree a young oak, which, springing from an acorn
dropped by the parent, grows up and overturns it ; we may search
as we like, but we shall find no vice in this event ; therefore there
can be none in the other, where the relations involved are just the
same ; so that it is not until we look beyond the event to the feelings
with which other persons regard it, that we can find the ground for
calling it vicious.1 Doubtless there is an analogy here ; but the
relations are not altogether the same ; for the relation of a parent
to a child is spiritual as well as physical, and in the parricide there
is an attitude of the will and the affections which cannot be ascribed
to the oak.

Many arguments from Analogy, in the sense of this loose identity
of relations, have become famous ; and they are a favourite portion
of the orator's resources. How often have not the duties of a colony
to the mother-country been deduced from those which a child owes
to a parent ; the very name of mother-country embodies the ana-
logy. Yet it is by no means easy to find the terms which stand
in the same relation. The soil of Britain did not bear the soil of
Australia ; and the present population of Australia are not the de-
scendants of the present population of Britain, but of their ancestors.
To whom then does the Commonwealth owe this filial regard, and
why ? Doubtless the sentiment has value, and therefore some
justification ; but this argument from analogy will not quite give
account of it. Alexis de Tocqueville again said of colonies, that
they were like fruit which drops off from the tree when it is ripe.
Here is another analogy, and two of the terms are the same as
in the last. The relation of a colony to the mother-country sug-
gests different comparisons to different minds, and very different
consequences : which cannot all of them follow from it. We may
take another instance, where the relations are really closer, and the
argument therefore of more value. To grant that Natural Selection
may be able to do all that is claimed for it, and yet object to it on

1 Treatise of Human Nature: Of Morals, Part I. § 1, Green and Grose's
ed. vol. ii. p. 243.

xxiv] SIMPLE ENUMERATION AND ANALOGY 535

the ground that the facts which are accounted for by it may equally
well be ascribed to intelligent design, is, it has been urged, as if
a man were to admit that the Newtonian theory of the solar system
works, and yet were to continue to suppose with Kepler that each
planet is guided on its way by a presiding angel ; if the latter
therefore be irrational, so must the former be.1 Or consider the
following passage 2 : — ' It has been objected to hedonistic systems
that pleasure is a mere abstraction, that no one could experience
pleasure as such, but only this or that species of pleasure, and that
therefore pleasure is an impossible criterion ' [viz. of good : i. e. it is
impossible to judge what is good by the amount of pleasure which it
affords]. ' It is true that we experience only particular pleasurable
states which are partially heterogeneous with one another. But this
is no reason why we should be unable to classify them by the amount
of a particular abstract element which is in all of them. No ship
contains abstract wealth as a cargo. Some have tea, some have
butter, some have machinery. But we are quite justified in arrang-
ing those ships, should we find it convenient, in an order determined
by the extent to which their concrete cargoes possess the abstract
attribute of being exchangeable for a number of sovereigns.' The
force of this argument will depend on whether the particular con-
crete pleasurable states do stand to their common character as
pleasures in the same relation as the concrete cargoes of ships stand
to the abstract attribute of wealth or exchangeability for sovereigns.
Doubtless the relations are partly the same, for each, pleasure and
wealth, characterizes its concrete subjects. But the concrete sub-
jects, tea, butter, machinery, are measurable in terms of wealth, by
the fact of being exchangeable for a definite number of sovereigns ;
and the question is whether there is anything that renders the others
similarly measurable in terms of pleasure. On the value of this
argument doctors will probably disagree : and this again shows how
arguments from analogy are inconclusive.

There is however another sense in which the terms analogy and
argument from analogy are used. The analogy may be any re-
semblance between two things, and not merely a resemblance of
the relations in which they respectively stand to two other things ;
and the argument from analogy an argument from some degree of
resemblance to a further resemblance, not an argument from the

1 Romanes, Darwin and after Darwin, i. 279.

2 McTaggart, Studies in Hegelian Cosmology, § 113. Cf. supra, p. 193, n. 3.

636 AN INTRODUCTION TO LOGIO [chap.

consequences of a relation in one case to its consequences in another.
Expressed symbolically the argument hitherto was of the following
type : a is related to b as c is to d ; from the relation of a to 6 such
and such a consequence follows, therefore it follows also from the
relation of c to d. The present argument will run thus : a re-
sembles 6 in certain respects x ; a exhibits the character y, therefore
6 will exhibit the character y also. Argument of this type is ex-
ceedingly common.1 ' Just as the flint and bone weapons of rude
races resemble each other much more than they resemble the metal
weapons and the artillery of advanced peoples, so,' says Andrew
Lang, ' the mental products, the fairy tales, and myths of rude
races have everywhere a strong family resemblance.' 2 The fact that
mental products, which resemble certain material products in being
the work of rude races, resemble them in the further point of
exhibiting the strong family likeness that is known to characterize
the latter, is here perhaps suggested to be something more than
a coincidence. Or take this argument from Sir Henry Maine.
He is discussing the various devices by which in different systems
of law the lack of a son to perform for a man the funeral
rites can be supplied. We are familiar with adoption. But
adoption in England does not carry the legal consequences of
legitimate sonship. The Hindu codes recognize adoption and
various expedients besides ; and the son so obtained has the full
status of a real son, can perform satisfactorily the important cere-
monies of the funeral rites, and succeed to property as the real son
would succeed. One of their expedients is known as the Niyoga,
a custom of which the Levirate marriage of the Jews is a particular
case. The widow, or even the wife, of a childless man might bear
a son to him by some other man of the family, and the son became
his son, and not the natural father's. How did Hindu thought rest
content in so fictitious a relation ? ' All ancient opinion,' says
Maine3, 'religious or legal, is strongly influenced by analogies,
and the child born through the Niyoga is very like a real son.
Like a real son, he is born of the wife or the widow ; and though
he has not in him the blood of the husband, he has in him the
blood of the husband's race. The blood of the individual cannot
be continued, but the blood of the household flows on. It seems to
me very natural for an ancient authority on customary law to hold

1 It was called by Aristotle trapabeiyfxa : cf. Anal. Pri. (3. xxiv, Ehet. a. ii.
1357b 25-36, and pp. 540-541, infra.

2 Custom and Myth, p. 125, ed. 1901 (' The Silver Library').
• Early Law and Custom, p. 107.

xxiv] SIMPLE ENUMERATION AND ANALOGY 537

that under such circumstances the family was properly continued,
and for a priest or sacerdotal lawyer to suppose that the funeral
rites would be performed by the son of the widow or of the wife
with a reasonable prospect of ensuring their object.' We may
find in the exacter sciences this sort of argument from analogy
employed. Before it was known that light travelled in waves, it
was known that sound did so. Light and sound were both capable
of being reflected, and the direction of their reflection obeyed the
same law, that the angle of reflection is equal to the angle of inci-
dence. From these facts it was inferred by analogy that light,
like sound, travelled in waves : as it was afterwards shown to do.
Among the properties of gold was long enumerated fixity, i. e. that
it was incapable of volatilization. As one element after another
was successfully volatilized, it might have been inferred by analogy
that gold could be volatilized too.

We may now compare this with the former type of argument
from analogy ; and afterwards consider their logical value, and
their relation to induction by simple enumeration.

Since analogy properly involves four terms, the latter and looser
but commoner sense of the expression argument from analogy seems
at first sight difficult to account for. Why should a resemblance
which is not a resemblance of relations be called an analogy at all ?
Perhaps the answer is that where the relation is no longer a quanti-
tative one, it is apt to be regarded as a property of the subject that
stands in the relation. The quantitative relation of one thing to
another does not affect the intrinsic character of the thing ; but
other relations do. We should not regard it as constituting a
resemblance between a child and a young elephant that one weighed
half a hundredweight, and the other half a ton ; but that they both
had mothers (though that is also a resemblance of relations) would
seem to constitute a resemblance. Such a relation rests on and
involves important characters in the thing related of a less purely
relational character than quantitative predicates are. And in
this way the term analogy may well have come to be extended
to resemblances generally, even where the resemblance is not a
resemblance of relations.1

1 I give in a note another possible explanation of the change that has
taken place in the logical use of the term analogy, but one that seems to me
less likely than the foregoing. The ' rule of three ' is in a sense an argument
from analogy. Starting with the conception of an analogy, in the strict
sense, it supplies from three given terms the fourth term which will complete

538 AN INTRODUCTION TO LOGIC [chap.

Even in the stricter sense then, the argument from analogy does
not commonly mean the mathematical argument from an identity of
ratio : the relations are only similar, and must be conceived to
involve intrinsic attributes of the things related.1 In considering
the value of the argument therefore we may for the future ignore
the distinction pointed out between the two types of inference to
which the name is given, and may take the second (to which the
first tends to approximate) as fundamental. The argument from
analogy is an argument from a certain ascertained resemblance
between one thing and another (or others) to a further resemblance ;
because a and 6 are x, and a is y, .*. b is y. What is the logical
value of this argument ?

It is plainly not proof. As Lotze has pointed out2, there is no
proof by analogy. Many conclusions drawn in this way are after-
wards verified ; many are found to be false. Arguments from
analogy can often be found pointing to opposite conclusions.

The Parmenides of Plato, a dialogue of his later period, discusses
various difficulties with regard to the relation between the universal
and the particular, which many scholars consider to be criticisms
upon his own 4 doctrine of ideas ' as presented in his earlier writings.
One of these is identical with an objection afterwards frequently
urged by Aristotle against the Platonic doctrine as he understood
it.3 It has been suggested that the dialogue incorporates criticisms
which Aristotle may have originated as a young man of about 17,
when a pupil in the Academy. Are the points Plato's own, or are
they borrowed from his pupil ? On the one hand it may be said
that when he wrote the Parmenides Plato was too old to revise his

the analogy. It is therefore an argument from the general conception or
form of analogy to the actual analogy (or complete terms of the analogy)
in a particular case. Now when I argue that because a and b both exhibit
the property x, and a exhibits besides the property y, therefore b will also
exhibit the property y, I may be said to be completing an analogy. The
presence of a; in a is to the presence of y in a, as is the presence of x in b to
that of y in b. In this case, the argument would be from the existence of
an analogy to the fourth term of it. But if the looser usage of the term be
interpreted thus, it bears less resemblance to the earlier usage than upon
the interpretation in the text.

1 Metaphysical criticism could easily raise difficulties against the view that
relations as such are extrinsic and attributes intrinsic to their subject. But
we are concerned here rather with a common way of regarding the matter
than with its ultimate tenability ; and I think we do commonly so regard it.

2 Logic, § 214.

3 Farm. 132 D-133 A. It is possible that the argument was originally
neither Plato's nor Aristotle's.

xxiv] SIMPLE ENUMERATION AND ANALOGY 539

system, as this interpretation of the dialogue conceives that he was
doing ; on the other, that at 17 Aristotle was too young to develop
criticisms so original and profound.

But Kant's chief works, embodying the system which has made
him famous, were written after he was 50 ; and Berkeley at the
age of 20 was entering in his Commonplace-book important and
original criticisms of Locke.1 One analogy supports the attribution
to Plato, the other that to Aristotle.

If it is not proof, has argument from analogy any value ? Can we
give any rules by which to judge its value in a given case ? Here we
must remember that the argument rests altogether on a belief that
the conjunction we observe discovers to us a connexion ; the presence
of both x and y in the subject a points to such a connexion between
them as will justify our inferring from x to y in the subject 6. If
we definitely thought that x and y were irrelevant to one another,
it would be foolish to expect 6 to exhibit one because it exhibited
the other. But though the argument thus presumes a connexion
between x and y, it makes no pretence of showing that y depends
on x rather than on some other property z in a, not shared with
a by b. There is no elimination. If however there were any
implicit, though not formal, elimination : or again, if there were
anything known to us which seemed to support the hypothesis of
a connexion between x and y : we should attach more weight to the
argument. Hence if the ascertained resemblance between a and b
is very great, we may think the argument from analogy stronger.
For there must be something in a to account for the presence of y ;
and if y is not connected with x, we must look for that something
in the remaining nature of a ; but the more we include in x (the
ascertained resemblance), the less there is that falls outside it, and
the fewer therefore the alternatives open to us, to account for the
presence of y in a. Still it must be admitted that so long as we
rely merely on this sort of consideration, it remains to the end
possible that y is unconnected with x, and therefore that y will not
be found in 6. Of much more weight is the consideration, that the
connexion between x and y implied in the argument is one for which
our previous knowledge prepares us. The fact that the angle of
reflection is equal to the angle of incidence might well be supposed
due (as indeed it is) to the propagation of sound in waves ; and if

1 Cf. D. G. Ritchie, Plato, pp. 108, 120. I have not reproduced the exact
use which he makes of the analogies.

640 AN INTRODUCTION TO LOGIO [chap.

so, we should expect the same fact in the case of light to be produced
by the same cause.

It will be seen that the considerations which must influence us in
determining what weight we are to attach to an argument from
analogy are the same as those by which we must estimate the
value of an induction by simple enumeration. Both point to
a general principle, which if it were true would account for the
facts from which we infer it ; neither proves its truth ; and to try
to prove it must be our next business. Mill rightly says that,
however strong an analogy may be, any competent enquirer will
consider it ' as a mere guide-post, pointing out the direction in
which more rigorous investigations should be prosecuted '. And
the same might be said of an empirical generalization. The next
sentences from the same passage of Mill's Logic may well be quoted :
4 It is in this last respect that considerations of analogy have the
highest scientific value. The cases in which analogical evidence
affords in itself any very high degree of probability are, as we have
observed, only those in which the resemblance is very close and
extensive ; but there is no analogy, however faint, which may not
be of the utmost value in suggesting experiments or observations
that may lead to more positive conclusions.' *

How then does argument from analogy differ from induction by
simple enumeration ? In the latter, because a number of instances
of a kind x exhibit the attribute y, we infer that all x are y ; in
the former, because two particulars a and b agree in certain respects
x, we infer that y, which is exhibited by a, will be exhibited by
6 also. In the latter, from the limited extension of an attribute
over a class, we infer to its extension over the whole class ; in the
former, from a partial agreement between two individuals in inten-
sion, we infer to a further agreement in intension. But the one
passes gradually into the other ; for the former may be called the
application to a particular case of a general principle inferred in
the latter from a larger number of instances than in the former.
This is very plain in an illustration which Aristotle gives of the
' Example ' (his name for the argument from analogy). A man
might have inferred that Dionysius of Syracuse designed to make
himself tyrant, when he asked the people for a bodyguard ; for
Peisistratus at Athens asked for a bodyguard, and made himself
tyrant when he got it ; and likewise Theagenes at Megara. Both
* System of Logic, III. xx. 3 med.

xxiv] SIMPLE ENUMERATION AND ANALOGY 541

these fall under the same general principle, that a man who aims
at a tyranny asks for a bodyguard.1 One of the instances of
argument from analogy given above concerned the volatilization of
gold ; and it might perfectly well be said that it would be contrary
to all analogy for gold to be incapable of a gaseous form. But
we might equally well say that our experience of other elements
warranted the empirical generalization that they could all be
volatilized, and therefore gold must be capable of it. This affinity
between the two processes of inference is however often concealed
by the fact that the points of resemblance in two (or more) subjects,
which form the basis of an inference to a further resemblance, have
not given rise to any special denomination ; there is no general
name by which the subjects can be called on the strength of the
resemblance, and the resemblance may even be one that we recognize
but cannot precisely describe. In gold, we might pick out the
fact of its being an element, as justifying the expectation that it
can be volatilized. In Dionysius, his asking for a bodyguard is
the circumstance that classes him with Peisistratus and Theagenes,
and excites our fear that he aims at a tyranny. But a weatherwise
man might be unable to describe what it is in the appearance of
the sky that makes him fear a great storm, though he can say that
it was on just such a night as this that some other storm broke out.
The general proposition (the induction as some would call it),
which mediates his inference from that past occasion to the present,
cannot be formulated ; and so he may appear to work without it,
and the affinity between such a process and induction by simple
enumeration may be unobserved. Yet it exists, and, as has been
said, the one process passes imperceptibly into the other, as the
number of instances increases from which the conclusion is inferred ;
though where we cannot formulate a general principle, we should
certainly speak of the argument rather as one from analogy.

It is of some importance to realize that a general principle is
always involved in such an argument, because it has been contended
that all inference goes really from particulars to particulars.3
There may be psychological processes in which a man's mind passes

1 Rket. a. ii. 1357b 25-36. To make the inference to Dionysius necessary
(it is of course Dionysius I who is meant), the principle would have to be,
that a man who asks for a bodyguard aims at a tyranny ; and that is really
what the suspicious citizen of Syracuse would have had in his mind.

2 Mill, System of Logic, II. iii. 3, and supra, c. xiv, pp. 300-310: of. also
Bradley's criticism, Principles of Logic, Bk. II. Pt. ii. c. ii.

542 AN INTRODUCTION TO LOGIC

direct from a to b, and he predicates of the latter what he was
predicating of the former, without grounding it on anything recog-
nized to belong to them in common ; just as a man who passes
a letter-box in the wall may look round at it to see the time. Psy-
chologists explain such actions as due to the ' Association of Ideas '.
But this has nothing logical about it, and is not inference. Any
one must admit when questioned, that unless he supposed b to share
with a the conditions on which the presence of y depends, he could
not rationally infer it in b because he found it in a ; and a process
which cannot rationally be performed can hardly be called a process
of reasoning. But that supposition is the supposition of a general
connexion ; and therefore inference from particular to particular
works through an implicit universal principle.

CHAPTER XXV

OF MATHEMATICAL REASONING

Mathematics is frequently and rightly called a deductive science.
Yet it has been said to rest on generalizations from experience, and
for this reason to be fundamentally inductive. There are also
certain particular processes of reasoning in mathematics to which
the name ' induction ' is habitually given.

One of these is just induction by complete enumeration, which
does occur sometimes in mathematics. A proposition may be proved
independently of a right-angled, an obtuse-angled, and an acute-
angled triangle, and therefore enunciated of the triangle universally :
or of the hyperbola, the parabola, and the ellipse *, and therefore
enunciated of all conic sections. The formula for the expansion
of a binomial series is proved separately to hold good when the
exponent is a positive integer, negative, and fractional ; and only
therefore asserted to hold good universally. The peculiar nature of
our subject-matter in mathematics enables us to see in each case
that no other alternatives are possible within the genus than those
which we have considered ; and therefore we can be sure that our
induction is ' perfect '. The nature of our subject-matter further
assures us, that it can be by no accident that every species of the
genus exhibits the same property ; and therefore our conclusion
is a genuinely universal judgement about the genus, and not a mere
enumerative judgement about its species. We are sure that
a general ground exists, although we have not found the proof
by it. This kind of mathematical induction needs no further
consideration.

The case is different where some proposition is inferred to hold
good universally because it is proved to hold good in one or two
instances. This sort of inference occurs in geometry, when we
prove something about a particular square, or circle, or triangle,
and conclude that it is true of any square, circle, or triangle ; and
again in algebra, when a formula for the summation or expansion

1 The circle being treated as the limiting case of the ellipse.

544 AN INTRODUCTION TO LOGIC [chap.

of a series, and such-like, being shown to hold good for certain
values of x, is inferred to hold good for any value. The former kind
of procedure is too familiar to need illustration ; of the latter, the
simplest illustration is the proof of the formula for the sum of the
first n odd numbers — i. e. of the odd numbers, beginning with 1,
and taken continuously up to any term that may be chosen. The
sum is always n2 ; and this is shown as follows. It is found by
addition that the sum of the first three, four, or five odd numbers is
32, 42, or 52 ; and then proved that if the sum of the first n— 1 odd
numbers = n — l2, then the sum of the first n odd numbers must = na.
For the n — 1th odd number is 2n — 3. Let

1+3+5+1 + ...+2n-3=n-l*=n2-2n+l.
Add to each side 2 n — 1 (which is the next or nth odd number)

\ 1+3 +5 +7+... +2n-3 +2n-l =n2 -2n +1 +2n -1 =n*.

If the formula holds for n— 1 places therefore, it holds for n places :
that is, it may always be inferred to hold for one place more than
it has been already shown to hold for. But it was found by addition
to hold (say) for 5 places ; therefore it holds for 6 ; therefore again
for 7, and so on ad infinitum ; and therefore universally.

It is instructive to compare this reasoning with the induction
of the inductive sciences. In one respect it presents the same
problem, viz. What is our warrant for generalization ? Yet it
cannot be said that the reasoning is of the same kind.

We saw that in the inductive sciences all generalization rests
on the existence of universal connexions — whether we express that
as the Law of Causation, or the Uniformity of Nature, or in some
other manner. But the particular problem of any inductive enquiry
is to determine what are the conditions with which a determinate
phenomenon x is connected universally ; and that is only to be
done by an exhaustive process of showing with what, upon the
evidence of the facts, it is not connected universally, until there is
only one alternative left unrejected, which we are therefore bound
to accept. Now it is by no such process of elimination as this, that
we demonstrate the properties of a figure, or the sum, for any number
of terms, of a series. We do not conclude that the angles of a
particular (rectilinear) triangle are equal to two right angles, be-
cause we have tried and found that there is nothing else to which
they can be equal ; but we so far understand the nature of space

xxv] OF MATHEMATICAL REASONING 545

as to see, by means of drawing a line through the apex parallel to
the base,1 that the mere three-sidedness of the figure necessarily
involves that equality. The geometrician sometimes appeals to
the conclusion of a previous demonstration, without realizing to
himself the reasons for the necessity of that conclusion ; thus, for
example, in proving that the angle in a semicircle is a right angle,
he appeals to the fact that the three angles
of the triangle in which it is contained are
equal to two right angles, and to the fact that
the angles at the base of an isosceles triangle
are equal to one another, and shows now only
that the angle in the semicircle must there-
fore necessarily be equal to the other two angles in the triangle in
which it is contained. So far as he thus appeals to the conclusion
of a previous demonstration, and applies it to the figure before him,
he syllogizes ; but when he realizes the necessity of that conclusion,
he does not syllogize, but sees immediately that it is involved in the
truth of other space-relations ; and this he finds out by help of
drawing the figure. It is felt that a reductio ad absurdum is a defective
proof in geometry just because we should be able to show that such
and such a proposition is true by direct reference to conditions
which necessitate it, and not indirectly by the refutation of the
contradictory. Thus the reasoning proceeds directly from the
apprehension of certain necessary relations among characters in the
subject of our study to the apprehension of other relations seen to be
bound up with those 2 ; not, as in induction, from the observation
of facts to belief in the only connexions with which they cannot be
shown to be incompatible. In our constructions, we have insight
into the necessary implication of one fact with another within
a system of spatially related points, lines, surfaces, and figures.
Our reasoning therefore is deductive ; and its premisses are proper
premisses, iSiat apyai — geometrical truths which explain other

1 Or, from the intersection of one side with the base, a line parallel to
the other side.

2 We might say that our reasoning proceeds from conditions to their
consequences ; but it must be remembered that in mathematics different
facts in the system of spatial or quantitative relations mutually condition
one another ; and therefore the order of demonstration is often indifferent,
and condition and consequence may change places. Still the reasoning is
deductive, since our premisses display to us the rational necessity of the
conclusion, and do not leave it merely as one which the facts force us to
accept, if there is any principle of connexion in them at all, but which we do
not see from the nature of the terms to be necessary : cf. p, 437, n. 1, supra.

1779 N n

546 AN INTRODUCTION TO LOGIO [chap.

geometrical truths. It is the same with any process of calculation
in arithmetic or algebra. There too we argue deductively ; and
there too our premisses are proper premisses, truths about relations
of quantity which render necessary other relations of quantity.
Nor is there any special difficulty about the ' mathematical induc-
tion ' employed in proving the formula for the summation or
expansion of a series, &c. When we prove that a formula which
holds for n — 1 terms holds for n terms, n represents any number
in just the same way as the circle on a blackboard represents any
circle. Geometrical proofs rest on the intuition of spatial relations,
and algebraic on the intuition of quantitative relations, and so far
the two sciences differ. But that is not more surprising than the
fact that moral philosophy, in which our proofs rest on insight
into relations neither of quantity nor space, differs both from
geometry and from algebra.

Yet we may return to the question, What warrant have we for
generalizing ? We must grant that the reasoning by which I prove
that the angle in this semicircle ABC is a right angle, or that a
formula which holds for the sum of the first n - 1 odd numbers
holds for the sum of the first n odd numbers, is different from that
by which I prove connexions of cause and effect in the inductive
sciences. Yet why do I conclude that the angle in any semicircle
is a right angle, or that the formula for the sum of the odd numbers,
which holds up to the term next to the n — 1th, holds up to any next
term, when I have only proved it about this semicircle, and the series
up to the next to the n — 1th odd number ?

Probably most people's natural impulse would be rather to
express surprise at the question than any sense of difficulty in the
matter. What difference can it make, they would ask, what circle
is taken ? What difference can it make that in proving that what
holds for so many places of odd numbers holds for one place more,
the place you take is represented by n - 1 ? Such counter-questions
would be a very proper rejoinder. But it may be useful to see
whereon our confidence rests, and so where the real difficulty about
generalization lies in the inductive sciences.

Our confidence rests, as has been indicated already, on our power,
in regard to the relations of points, lines, surfaces, and figures in
space, or to the relations of quantities or numbers, to apprehend
what must be : a power lacking in regard to the subject-matter of
the inductive sciences. In geometry and mathematics we have

xxv] OF MATHEMATICAL REASONING 547

a direct insight into the special nature of our facts. This insight
is expressed in a variety of ' necessary judgements ' as to the con-
nexion of one character with another in any particular quantitative
or spatial subject. Some of these judgements have attracted attention
as axioms or postulates. Others, which are implicit in our reasoning,
have often passed unnoticed. Geometrical definitions, for example,
are not made without insight into the possibility of constructing what
is defined. If it is an axiom that two straight lines cannot enclose
a space, the definition of a plane triangle implies the axiom that
three can. Again, we constantly take it as evident, in a geometrical
demonstration, that certain lines must intersect, with no warrant but
our insight into the nature of our subject.1 The very first proposition
of Euclid assumes that the circumferences of two circles described
with the same straight line for radius, and its opposite extremities
for centres, will intersect ; others assume that the diagonals of a
quadrilateral will do the same, either within the figure, if it has no
re-entrant angle, or without it, if it has one. Every simple numerical
equation, such as 2 + 2 = 4, asserts a relation seen to be necessary.
Every application of algebraical induction involves that we see that
one term has necessarily a certain relation to the next term or the
term so many places from it in a series, no matter which term it be.
Such insight we lack in regard to the connexions between one
quality and another in concrete things, or between change in one
thing and change in another or one change and another in the
same thing ; though we believe that there are necessary connexions
here also, and that because what is necessary is universal, we can
use our experience of particular things and events to determine
what the connexions are.

But there is another point to consider. In discussing the prin-
ciple of Universal Causation, and its relation to the so-called Uni-
formity of Nature, we saw that the necessity involved in the causal
relation between a change of one kind and a change of another was
irrespective of the repetition of such changes. An unique cause
would produce its unique effect necessarily. The repetition of like
changes, the multiplicity of like things, is important not because
otherwise there would be no causal connexions, but because other-
wise we could not discover them. And we could not otherwise
discover them just for the reason that we have not that direct

1 My attention was called to this by Professor Cook Wilson, from whom
I borrow the illustrations in the next sentence.

Nn2

648 AN INTRODUCTION TO LOGIC [chap

insight into the connexions of terms here which we have in respect
of geometrical and mathematical terms ; we rely on the repetition
of the like for eliminating the irrelevant, since that is not relevant
which is not repeated as connexion would require.1 But in the
mathematical sciences we have this insight, and hence the repetition
of like instances is superfluous to the process of proof. That the
angles at the base of an isosceles triangle are equal would not only
be as necessary, but would as easily be seen to be necessary, though
there could be only one isosceles triangle ; that 1+2+3+&C. +n

= - — would as easily be seen to be necessary, though (per

impossibile) only one value and application could be given to n.
Only in this case we should not generalize. That we do generalize
depends therefore not merely on our direct insight into necessary
relations, but on our apprehending that the terms between which they
lie are not unique. Indefinite repetition with no qualitative variety
belongs to the nature of space, and also of the numerical series.
Any space is divisible into spaces which are smaller, but not other-
wise different, and is a portion of a space which is larger, and not
otherwise different. Therefore whatever space-relations are exem-
plified in one part of space may be equally exemplified in any
other. This homogeneity or, as we might say, indifference of space is
of course taken for granted in all the physical sciences ; for we never
regard mere difference of position as affecting the state of a body,
but only difference of relation to other bodies involved in difference
of position.2 So with the number-series ; at any point in it there is
the same difference between one number and the next ; a ratio
found in one part of the series can be found in another3, and so on ;
otherwise our x and y and n could not be general symbols.

But this insight into the homogeneity of space, or the uniform
construction of the numerical series is, after all, only of a piece with
the insight into the nature of our subject-matter which we display
when we see the necessary truth of particular mathematical pro-

1 Though the repetition need not be without variation, if we can find
a formula to connect the differences of our variables : cf. supra, p. 525.

8 Some theories of non-Euclidean space, just because they reject this
indifference of it, have to represent the consequences by saying that bodies
would be distorted through translation : i. e. distorted in terms of Euclidean
space. That they have to represent the consequences in terms of Euclidean
space seems to show that we cannot really conceive the possibility of the other.

3 Not of course in any other : e. g. the ratio 5 : 7 can be found beginning
at 10, but not beginning at 8.

xxv] OF MATHEMATICAL REASONING 549

positions. To see what is irrelevant is but the other side of seeing
what is sufficient to a given consequence.1 If we understand that the
equality of the angles at the base of a triangle is necessarily involved
just with the equality of the sides, we also understand that the length
of the sides or the place of the triangle can make no difference. If
we understand that the connexion between the truth of a formula
for n terms of a series and its truth for w + 1 terms depends just on
their being next terms, we also understand that it has nothing to do
with the magnitude of n. We believe indeed that, as in respect of
number and quantity and of space, so in the attributes and changes
of bodies there is a fixed system of necessary relations ; but we
cannot tell what it is by thinking. If we could, so as to see for
example that gold must be heavier than lead, as we can see that the
angle at the centre of a circle must be double that at the circumfer-
ence, the physical sciences would be deductive, where they are now
empirical.

But it has been said that the principles of geometry and of mathe-
matics are themselves generalizations from experience, and these
sciences therefore at bottom empirical and inductive, like the
physical sciences2. This question was referred to at the outset of
the chapter. Now, were it so, it is hard to see why the same should
not as well be said of the inferences in mathematics.3 Their de-
monstrative force arises from the fact that the nature of space or
quantity allows us to see immediately the consequences involved in
certain conditions. But any one who requires repeated experience
to convince him of the truth of a geometrical principle (such as that
two straight lines cannot enclose a space) may just as well require
repeated experience to convince him of the truth of a geometrical
deduction ; we have to do with the mutual implication of spatial
conditions in both cases. And so it is also in the science of pure
quantity. The multiplication table up to 12 x 12 might be said to
contain principles, and the multiplication of 266 x 566 to apply
them ; but whatever reason there is to doubt that 6 x 6= 36, there will
be the same reason to doubt whether it follows that 60 x 60 = 3600,

1 We may indeed prove something of a subject which is not the com-
mensurate subject, as if we proved that the external angles of a square were
equal to four right angles, when it is true for any rectilinear figure ; but
even here we see that nothing relevant is omitted from the conditions of the
property. Cf. Arist. Anal. Post. n. v.

2 Mill, System of Logic, II. v-vii. Cf. Autobiography, p. 226.

* Or of any form of inference : cf. supra, pp. 312-313, infra, p. 567.

550 AN INTRODUCTION TO LOGIC [chap.

However, it will be sufficient if we confine ourselves to the con-
sideration of the alleged inductive character of the process by
which we ascertain mathematical principles, without attempting to
determine how much would have to be regarded as principles, and
how much as valid consequence.

What is really meant by the allegation is, that whereas every
mathematical principle, such as the axiom of parallels, or 2 +2 = 4,
is universal, our reason for accepting it as universally true lies in
the fact that we have always found it to hold good in experience.
Two apples and two apples make four apples ; it is the same with
cows or sovereigns, window-panes or waterpots. And whenever we
have seen a straight line falling on two other straight lines and
making the alternate opposite angles measurably equal, we have
found — if we have tried — that however far we produced the two
other straight lines, so long as they continued apparently straight,
they remained at the same measurable distance from one another.
All experience confirms these principles, and none is contrary to
them ; so we accept them as empirical generalizations, possessing,
on account of the extent and variety of the circumstances under
which they have been found to hold good, the same degree of
certainty as if they had been proved by a rigorous elimination of all
other hypotheses.

It is really sufficient answer to this view, to recur to what was
said upon a similar attempt to treat the Law of Causation as em-
pirically established. If the Law of Causation is true, the facts
of our experience help us to determine what are the particular
causal connexions in nature ; if we start by doubting it, the facts
will never bring us any nearer the proof of it. Similarly, if we start
by doubting whether the relations between the same spatial or
numerical terms are constant, the facts will never begin to prove it.
Grant that the sum of 2 +2 is always the same, and it is worth while
to see what it is ; and whatever countable things we take to reckon
with will make no difference. But question whether it is always the
same, and proof that it is so becomes impossible. For you have no
ground for supposing that if 2 +2 could sometimes make 5, cases of
the occurrence would have occurred in your experience. Everything
becomes problematical ; the frequency of any particular sum of 2 +2
is quite indeterminate, if the sum is indeterminate ; and your
experience may assure you that you have never found them making
anything else than 4, but cannot assure you that you are never

xxv] OF MATHEMATICAL REASONING 551

likely to do so. And so it is with geometrical principles also. If
geometrical relations are not necessary and universal, we have
nothing but a conjunction of facts empirically ascertained. In each
place and time the conjunction may be different ; there is no reason
to suppose that what occurs here and now conveys any instruction
about the occurrences at other times and places. If each place and
time is loose and independent, the next may always contradict even
the uniform results of previous experience.

Other lines of refutation are also possible. It might be pointed
out that in point of fact we do not look for confirmation of our
principles to repeated experience ; but we interpret experience in
the light of our principles. Two drops of quicksilver and two drops
of quicksilver will make one drop of quicksilver ; but we insist that
the four drops are there, in a new figure. The angles between the
end-lines and the side-lines of a tennis-court may seem each to be
a right angle, and the sides to be drawn straight ; but if we find
that one end-line is shorter than the other, we say that we know that
the angles cannot be true. It may be said that by this time our
principles are well established, and facts in apparent conflict with
them are therefore reinterpreted so as to be consistent with them.
But facts in apparent conflict must have been frequent from the
beginning. Again, it is hard to see what meaning can really be
attached to the statement that 2+2 might conceivably make 5,
or that lines making equal angles with a third straight line might
conceivably remain straight and yet converge ; for such a thing
cannot be represented to thought as possible.

It is of course true that in the application of mathematical
reasoning to what is concrete, our conclusions will only be true if
our premisses were so. If a wheel which I assume to be circular
is not circular, conclusions based on the assumption will prove false.
If I am wrong in my linear measurement of a floor, I shall be wrong
as to the number of square feet of floor-cloth required to cover it.
But this does not shake the certainty and universality of mathe-
matics ; indeed nothing else would be consistent therewith.

It is also true that without experience of counting numerable
things, and of constructing figures in space, I should be unable to
apprehend or understand the truth of mathematical principles.
But this does not make their truth empirical, or my mode of ascer-
taining it inductive. For these principles are seen to be intrin-
sically necessary as soon as they are understood ; whereas inductive

552 AN INTRODUCTION TO LOGIC [chap.

conclusions are never seen to be intrinsically necessary, but only to
be unavoidable. Nor does further experience add anything to our
assurance, when we have once made the construction or the calcu-
lation in which their truth becomes manifest to us ; whereas further
experience of the same conjunction amidst variation of circum-
stance is precisely what does add to our assurance of the truth of
an empirical generalization.1

We must conclude that in mathematics there is (or at least should
be 2) no generalization from experience. To suppose mathematical
principles to be such generalizations is like supposing the Law of
Causation to be so. Their universality is the counterpart to the
reign of law in physical nature. But the deductive character of
mathematical science is due to the nature of the subject-matter,
and our peculiar insight into the rational connexion of its parts.
What is implied in our possession of this insight is a metaphysical
question lying beyond our purview.

[The nature of mathematical certainty is a question of far-reaching
metaphysical importance ; and J. S. Mill, in his Autobiography
(loc. cit.), frankly acknowledges that the chief strength of the oppo-
sition to the truth of the Empirical Philosophy had always seemed
to lie here. It was on this account that he sought to show that
mathematical principles in their turn were generalizations from
experience. He held the same with regard to logical principles. It
is logically important to see that there can be no knowledge unless
there are truths not empirical — i. e. not open questions, for a decision
on which we must go to the tribunal of sense-perception or events.
And no one will understand the structure of knowledge, who does
not see that mathematical principles are truths of this kind. But
it may be asked what their relation is to logical principles. There
are some who have represented logic as at bottom a branch of
mathematics ; and others seem inclined to suppose that mathe-
matics can be reduced to formal logic. A non-mathematician is
not well fitted to discuss these matters in print ; and the discussion
belongs in any case to a more advanced stage of logical science than
this book pretends to attain. But I ought perhaps to say that I
do not understand how either theory can be true.3]

1 Cf. p. 531, supra. * Cf. p. 530, supra.

3 On the circularity of attempts to define fundamental mathematical notions
by help of what is regarded as the general logical notion of a class and
its members, cf. H. Poincare, Science and Method (E. T., F. E. Maitland,
pp. 155-157). A similar circle seems to be involved at a critical point in
Dedekind's treatise Was sind und was sollen die Zahlen? He explains (pro-
fessedly without presupposing the thought of number) what a well-ordered

xxv] OF MATHEMATICAL REASONING 553

system is, and what an image of that system is. The alphabet, e. g., ia
a well-ordered system, and a cipher in which other symbols correspond to
the several letters is an image of it. Now the image of a system may lie
within itself. This would be the case with a cipher that put b for a, c for
6, &c, and finally a for z. Here however every term in the original system
occurs also in the image. Suppose this not to happen (e. g. that a did
not occur in the cipher) : and suppose also no term in the image to corre-
spond to more than one in the original system ; then we should say that
a system cannot contain a complete image of itself. Dedekind however
assumes a system containing a complete image of itself, with a separate
term corresponding to each term in the original system, yet not employing
the first term of it ; and he develops certain properties of such a system,
which are those of the number-series. Now unless we thought at the outset
of the number-series, the whole of that procedure would be just words standing
for nothing conceivable. Because the number-series is endless, therefore
however many terms we take in it, starting from 1, we can find in it as
many terms starting from 2, and there will be a ' one-one correspondence '
between the two systems. But Dedekind's professedly logical considerations
do not elucidate the number-series ; on the contrary, it is required to
elucidate them. One who finds paradox in the number-series will not think
that it elucidates them completely ; but they certainly do nothing to resolve
such paradox.

CHAPTER XXVI

OF THE METHODOLOGY OF THE SCIENCES

We have seen that inferences cannot all be reduced to a small
number of fixed types. They are not all syllogistic, not even all
that are deductive. Their form is not altogether independent of
their matter. All inference, according to Mr. F. H. Bradley, is a
construction and an intuition.1 The putting together of the pre-
misses is the construction, but it is the terms which determine how it
can be effected. The perception of something new to us in the whole
which we have constructed is the intuition ; and if we do not see its
necessity, there is no help for us. But within the unity of this
definition, we may examine any particular type of inference which,
for its frequency or importance, seems to demand our special atten-
tion. Syllogism is one of these types ; the disjunctive argument
as applied to establish causal connexion is another. The relation
of subject and attribute is one of the commonest which our thought
uses, and therefore inferences based on it are common. The causal
relation is not less important, and the type of inference used in its
establishment equally deserved our study.

We found that this type of inference rested on some insight into
the causal relation.2 We considered very generally what that
relation involved, and how we could satisfy ourselves that we
were right in bringing any particular relation of facts under it.
We noticed some of the difficulties which the complexity of
nature places in our way ; and some of the cautions which we
must constantly bear in mind in interpreting facts in accordance
with our conception. We found that general truths present
themselves to the mind at first in the form of conjecture or hypo-
thesis, and that often there is no means of testing such hypothesis
except by first deducing — it may be by very elaborate reasonings
— the consequences that should follow in specified circumstances

1 Principles of Logic, p. 235. ' The process is a construction and the result
an intuition, while the union of both is logical demonstration.'

2 Not that all disjunctive argument involves that relation; but only
disjunctive argument applied to the discovery of causes.

OF THE METHODOLOGY OF THE SCIENCES 655

if it were true and if it were not. But all these matters were
discussed and illustrated in a very general way.

Now different enquiries have their own peculiar difficulties,
arising out of the nature of their subject-matter, and of the problems
which it sets. And any rules for dealing with these peculiar diffi-
culties will constitute rules of method, instructing us how to set
about the task of singling out the laws or causal connexions from
amidst the particular tangle in which the facts are presented in
such science. The consideration of such rules, as distinct from the
use of them, is Methodology ; and so far as herein we consider how
certain general logical requirements are to be satisfied in a particular
case, it is sometimes called Applied Logic.1

To this subject belongs Mill's discussion of the proper method
of studying the moral or social sciences.2 He points out how methods
of enquiry appropriate to certain chemical investigations (to which
he therefore gives the name of the Chemical Method) are inapplic-
able in dealing with the sciences of human nature. The chemist,
unable in a great degree to predict from his knowledge of the
properties of elements the properties which will belong to their
compounds, has to proceed by experiment conducted with every
precaution to secure a precise knowledge of the conditions ; and thus
discovers the effect of a new condition or ingredient upon a whole
of a certain kind. But we cannot experiment with society out of
a merely speculative curiosity ; the practical interests involved
are too great ; and were that not so, yet the thing is impossible.
Our material is not under control. It would be most instructive
to prevent the use of alcohol in England for a generation, and watch
the difference in the amount of pauperism and crime ; but there
is no means of performing the experiment, for to pass a law is not
to enforce it. Nor can we ever know precisely into what conditions
we introduce the factor whose effects we wish to study ; nor can
we maintain those conditions unchanged in all but what is due to
the influence of that factor during the course of our experiment.
For these and other reasons, it is hopeless to expect much light to
be thrown upon the laws of social phenomena, merely by watching
what follows in different cases upon the adoption of the same policy,
or by comparing the results of different policies. There are so

1 Cf. Kant, Introduction to Logic, ii. 4 (T. K. Abbott's tr., p. 8), who gives
a different sense to the term, but notices this use of it.

2 System of Logic, VI. vii-x.

556 AN INTRODUCTION TO LOGIC [chap.

many factors which modify one another ; each effect depends on so
many conditions, and each condition by its presence or absence
makes a difference to so many effects by us regarded as distinct, that
it is useless to suppose that the effect of any particular social experi-
ment will stand out sharp and recognizable amidst its surround-
ings, or that we could say — Here is something which could not have
occurred but for the measure we took.

We must have recourse then to deduction. From what we
know of the laws of human nature, we must attempt to determine
the effect which a measure must produce, or the conditions out
of which a given state of society must have arisen. But again the
great complexity of the subject imposes certain restrictions upon us.
We must not expect to be able to trace any pervading feature of
society to a single motive, as political obedience to fear,1 or good
government to a system by which the ruler's private interest is
engaged in governing well. And Mill lays stress on one feature in
particular of the method by which the course of human history is to
be explained. Instead of working out first the theoretical conse-
quences of certain general principles, and then checking ourselves by
comparing our result with the facts, he holds that we should en-
deavour first to ascertain empirically the subordinate principles
that manifest themselves in history, and check our formulation
of them by considering whether they are consistent with the more
ultimate laws of human nature and conduct from which in the
last resort they must be derivable. For the facts of every period
are so diverse and manifold, that the former procedure would
probably be a waste of time. We may know the laws of human
nature, but until we know the circumstances of a given state of
society, we cannot tell what result these laws will produce. We never
know them sufficiently for it to be worth our while to attempt to
develop human history a 'priori, as the astronomer might attempt
to develop a priori the course of a comet or of the tides. We must
be content if such generalizations as we can frame a posteriori are
confirmed by showing that they present nothing surprising when
they have happened, although we might have been unable to
predict them.2

1 Cf. Bryce, Studies in History and Jurisprudence, Essay ix.

2 Mill gives to this order of procedure the name of the ' Inverse Deductive,
or Historical Method ' : by which he means the method appropriate to the
study of history. The Historical Method now however commonly means
interpreting present facts in the light of their past history. The contrast

xxvi] OF THE METHODOLOGY OF THE SCIENCES 557

In the chapter on Non-reciprocating Causal Relations, questions
of methodology were really to some extent discussed. For we were
engaged in considering the difference between the evidence required
to establish a pure causal relation, where nothing irrelevant enters
into the statement either of the cause or of the effect, and a non-
reciprocating relation such as is implied when we speak of a Plurality
of Causes. Now some sciences find it much harder than others to
eliminate the irrelevant ; and to them it is specially important to
remember the sort of tests by which the non-reciprocating character
of a relation may be detected.

In that chapter, two of the ' Rules by which to judge of Causes
and Effects ' which had been previously enunciated were reconsidered
at some length, and it was shown that, although nothing which
failed to satisfy their conditions could be in the strict sense the
cause of any phenomenon, yet if cause were understood in a looser
sense, as non-reciprocating, it was not safe to make the same asser-
tion. But of the precautions to be attended to in the application
of the other two Rules little was said.

These rules were, that nothing which varies when a pheno-
menon is constant, or is constant when it varies, or varies inde-
pendently of it, is its cause ; and that nothing is so of whose
effect account has already been taken in other phenomena. Both
these rules are especially useful where we are dealing with measur-
able effects, the total amount of which is dependent on a large
number of conditions ; and the investigations which employ them
have been called ' Methods of Quantitative Induction '-1 It may
be worth while to consider some of the difficulties which beset the
use of them ; and that will furnish an example of a methodological
problem ; for a science which deals with measurable phenomena,
in spite of the great advantage which their measurability brings,
generally meets also with some special difficulties, which it needs
particular precautionary measures to surmount.

What is measurable must so far be homogeneous. Sometimes it
is for all practical purposes entirely homogeneous. A gas company

intended by the word inverse is that noted on the preceding page ; but it does
not really amount to more than that precise deductions of the consequences
implied in our general principles, and their experimental verification, are
impossible in social and political investigations, for the reasons above given.
Kepler formulated his empirical generalizations about the planetary orbits
before Newton deduced them from the laws of gravitation and inertia.
1 Jevons, Elementary Lessons in Logic, XXIX.

558 AN INTRODUCTION TO LOGIC [chap.

supplies gas by metre; the gas is measured, and one cubic foot
is practically indistinguishable from any other. Sometimes the
homogeneity is less complete, but there can be no measurement
except so far as it is found. It may be important for a general to
know what percentage of men he is likely to lose by casualties other
than in the field ; these casualties may be of various kinds, and to
the individual soldier it may make a great deal of difference whether
he breaks down through dysentery or fatigue ; but they are all alike
in incapacitating men for service ; and the general wants a measure
of the extent to which that occurs. A valuer assesses the value of
the personal property of a man deceased ; it consists of pictures,
plate, furniture, horses, stocks and shares, books, and all kinds of
miscellaneous articles ; but so far as these are all exchangeable for
money they have a common property which can be measured in
terms of money.

Now contributions may be made from many sources to any homo-
geneous quantity, but when you are merely told what the quantity
is, there is nothing to show of how many parcels, so to say, it is
made up. The total quantity is a sort of unity. Had one parcel
been greater, the total would have been greater ; should one parcel
fluctuate in amount, the total fluctuates ; but there is nothing to
show which parcel is fluctuating and which is constant, and the
variation seems to belong to the whole.

It follows that where an effect is quantitative, and there are
a number of contributory factors which, one way or the other,
influence its amount, fluctuations in these do not necessarily
stand out in the result. There is no doubt that overcrowding
affects the death-rate ; yet the death-rate in a town may rise
while overcrowding has diminished, if other causes operate to
increase it faster than the improvement in housing operates to
diminish it.

Hence a hasty application of the rule that nothing is the cause
of a varying phenomenon which does not vary proportionately with
it may lead us into grave mistakes. We might suppose, for instance,
in the last example, that overcrowding had no influence on the
death-rate, because the death-rate seemed to rise and fall inde-
pendently. Doubtless the independence is only seeming ; and if
the other contributory factors could be kept constant, we should
find the rise and fall proportionate. But we cannot keep them
constant.

xxvi] OF THE METHODOLOGY OF THE SCIENCES 559

And even if we could, we should be exposed to other errors of
interpretation. The death-rate, many as are the causes which
contribute to it, is yet measured as a whole, and treated as one
phenomenon. If all the causes which contribute to it were constant
except one, and that one fluctuated, the whole result might be
attributed to the one circumstance which exhibited proportional
fluctuations with it. In this particular matter, indeed, we know too
much to fall into such an error ; we know that overcrowding is not
the only cause of death. But where our previous knowledge is less,
it is very easy to attribute the whole of a varying effect to the factor
which varies in proportion, instead of attributing to it only the
increase or decrease beyond a fixed amount. The influence of
education upon character is great ; and that is shown by the effects
of giving and withholding it. But we cannot thence infer that it
is all-powerful, or that the whole difference between the criminal
and the good citizen and father is due to comparative defects in
the criminal's upbringing.1

It is clear, then, in the case of a fluctuating effect which is the
complex result of several causes, that though there must no doubt
be a proportionate fluctuation in the cause, yet it is unsafe to reject
from being a cause either a factor which fluctuates when the effect
is constant, or one which is constant when the effect fluctuates.
For we see the effect as a whole ; and the whole need exhibit no
fluctuations proportionate to those of any one part. The rule
of elimination is not false ; for if the separate effects of each
factor were not lost and undistinguished in the total, we should
observe the facts conforming to it. But this not being so, the rule
is unsafe.

The best remedy lies in determining the precise amount of effect
which each factor can produce ; and as each factor may perhaps be
liable to fluctuation, what we need is a principle or law connecting
each degree of its activity with a corresponding quantity of the
effect. This is done, for example, in the Law of Gravitation. And
could we thus calculate the amount of effect which the other causes
at work, at the strength at which they were severally present, were
capable of producing, we might then safely attribute any difference
beyond this to some circumstance that fluctuated proportionately
with it.

1 The ' Perfectibilitarians ', like Godwin, at the beginning of the last
century, held very nearly this Cf. Godwin's Political Justice, I. iv.

560 AN INTRODUCTION TO LOGIC [chap.

But in such a procedure we should no longer be appealing merely
to the principle that the cause of a varying phenomenon must be
something that varies in proportion. We should be invoking also
the fourth of our grounds of elimination, that it can be nothing
whose effect is already accounted for. Only because we have
determined the amount of effect which the other factors can produce
are we entitled to say that the residue is in no part due to them.
And unless we know with fair accuracy what amount of effect may be
justly assigned to other factors present, we cannot upon the strength
of this principle attribute any part to some particular further
factor a. The application of this rule therefore is involved in the
same difficulties as that of the former, through the fact that the
effects of many different causes are compounded and lost in one
total amount.

Moreover, so long as all these causes are freely varying, and
masking their separate effects in one total, the determination of the
law of any single cause, much as it would help us to discover the
others, is the very thing that is so difficult. Hence the necessity
for experimenting with each suspected cause singly. It may be
impossible to exclude the influence of any others ; we must then
endeavour to keep it constant ; or we may employ what is called a
controlling experiment at the same time. We may see what happens
both when a certain factor is introduced, and when it is not, under
circumstances which, though we cannot keep them constant, we
have good reason to believe to be varying alike in either case. A
farmer, for example, wishes to know whether some new dressing is
of any use to his grass. He cannot remove the other causes which
promote or hinder the growth of grass, and see how large a crop of
hay this dressing could produce alone ; for alone it would produce
none at all. Neither can he control those other causes, so as upon
the same field to use it one year and not the next, and maintain all
other factors the same. But he can select two plots, or series of plots,
on which he has reason to believe that the other causes all operate
equally, and use the dressing on one and not on the other.

But even so, we have not got a great way towards determining
the law of a cause. To show through all that masks it that some
part of an effect is due to a particular cause is not the same as
showing how much is due to it : still less as finding a mathematical
expression that connects definite fluctuations in the one with defi-
nite fluctuations in the other. There are many cases where this last

xxvi] OF THE METHODOLOGY OF THE SCIENCES 561

achievement is impossible, even though the phenomena we study
be quantitative and to some degree measurable ; indeed it is impos-
sible except in dealing with the physical properties of bodies.
Elsewhere we must be content with a vague much and little. In
time of war, the risk of capture at sea is a great deterrent to neutral
commerce ; but we cannot say precisely how great. The history of
times of plague shows that increased uncertainty of life relaxes the
bonds of custom and morality ; but it would be impossible to give
any measure of the connexion between the two facts, though the
measurability of the facts, in the sense that as the death-rate from
plague rises the frequency of criminal or reckless acts increases, helps
to betray the connexion. The one fact may be, in mathematical
parlance, a function of the other ; but it is not a function of the
other alone ; and we cannot so disentangle the many causes and
their complex result as to give precision to the degree in which one
affects the other. Moreover, where the phenomena are more purely
quantitative, the law of variation that connects them is not always
by any means easy to establish ; for a formula which holds good over
a considerable range of variation may break down beyond those
limits. The coefficient of expansion of a metal, which indicates the
rate at which its bulk increases with successive increments of heat,
no longer applies when the metal vaporizes. There are what have
been called critical points, at which the change in an effect no longer
observes the same proportion as hitherto to the change in the cause.
Great caution must therefore be observed in formulating any law
upon the evidence of concomitant variation between two phenomena,
even where we are satisfied that we have excluded any variation
due to other causes, and can give a precise measure of the phenomena
in question.

The causes whose effects are merged in a total may not only vary
independently of one another ; some may be intermittent in their
operation. And whether they are continuous or intermittent, they
may be periodic ; and one may have a longer period than another.
There may again be causes which are both intermittent and irregular
in their action, recurring at no definite and periodic intervals.
Yet it is possible to cope with many of the difficulties which these
facts present by taking averages. No one would expect the rainfall
of one year to agree closely with that of another in the same locality ;
the circumstances affecting it are too numerous and inconstant. But
we have no reason to expect that the average annual rainfall over

1779 0 0

562 AN INTRODUCTION TO LOGIO [chap.

a considerable period of years should not agree closely for different
periods ; for though in one year there may be more circumstances
than usual that are favourable to rain, in the next there may be
fewer. If, then, the average rainfall for one considerable period of
years were greater than for another, we should look for some definite
reason for the difference : which we might find perhaps in a differ-
ence in the amount of forest standing in the district at the different
dates ; for the intermittent and irregular causes of whose operation
we are aware would have roughly balanced in the two periods, though
not perhaps in any two single years. Another method is to plot curves.
A base line for example is taken, and perpendiculars drawn to it at
equal intervals for the successive years. On each of these a point
is taken whose height above the base is greater or less in proportion
to the number of inches of rainfall in that year ; and a line is drawn
through those points. The line will rise and fall irregularly ; but
it is possible that in spite of these intermediate fluctuations there
may be long-period fluctuations which stand out clearly ; what may
be called the crests and troughs of the curve may be at fairly equal
intervals, though its course is not uniform from trough to crest.
This would indicate the action of some cause having a similar
period ; and if we discovered any factor with a corresponding period
of fluctuation, there would be a strong presumption that it was the
cause.

The profitable use of statistics depends very largely on methods
like these ; but the devices for bringing out their teaching are often
much more elaborate than has been indicated. These belong, how-
ever, to the detail of particular sciences rather than to the general
principles of logical method. Enough perhaps has been said to
indicate the misinterpretations of causal relation to which we might
be led, regarding quantitative phenomena that vary in their amount,
by too hastily applying rules true in themselves to any unanalysed
total effect : as well as the difficulties that beset us in disentangling
the component parts and fluctuations.

A few further and miscellaneous examples of the way in which
precepts for the better prosecution of a particular science may be
drawn from general logical principles will serve to conclude this
chapter. It must not be supposed that the subject is at all ade-
quately treated here ; it is only illustrated.

What is called the historical or comparative method has in the
last few generations revolutionized many branches of enquiry. It

xxvi] OF THE METHODOLOGY OF THE SCIENCES 5G3

is but an application of the general principle of varying the cir-
cumstances in order the better to discover the cause of a phenome-
non. But of old, enquirers into matters of historic growth, such
as language, or myth, or religion, or legal ideas, were content
to attempt an explanation of the facts of some particular age
or country by observations carried on within that age or country
alone, or if beyond it, only in adjacent ages or countries of the
same type. The historic method looks farther afield. It compares
the institutions of widely different ages, or of peoples who though
contemporaneous stand at widely different levels of civilization and
of thought. In the light of such a comparison, facts may take
on quite a new appearance. Legal or other customs for which
a later age has found a reason in some supposed meaning or utility
which they now possess are seen to have had a very different
origin, in conditions no longer existing, and beliefs no longer enter-
tained. Folk-lore is full of such surprises. The custom of throw-
ing rice after a married couple as they drive away is sometimes
explained by saying that rice is a symbol of fertility; Sir J. G.Fraser,
comparing a number of other facts, thinks that the rice was origin-
ally intended to lure back the spirit of the bride or bridegroom
to its body ; it was supposed that at critical times — and every-
thing connected with marriage was critical — the spirit left the body,
in the form of a bird ; the rice would attract it, and if it hovered
about the body it would be more likely to re-enter. Whether
this be the true explanation of the custom or not, only the com-
parative method could have suggested it. It is the same with
myth ; the account of the origin of Greek and Roman mythology
popularized by Max Miiller represented it as, in the language of
Dr. Andrew Lang, a disease of language, the pearl in the oyster.1
Names originally designating the attributes of earth or sun or
moon were confused with words of similar sound but different
meaning, and out of these other meanings myths arose. Apollo
Lykios had no connexion with the wolf ; he was only the Shining
One ; but when that was forgotten, some wolf story would be invented
to account for the name. Such theories are however discredited
when it is found that a myth occurs in forms substantially alike
among widely different peoples, whose languages do not all admit
of supposing it to have originated through confusion between
similarly sounding words of different meanings. There is no new

Custom and Myth, p. 1.
002

564 AN INTRODUCTION TO LOGIC [chap.

principle in the use of such an argument against the ' Sun-myth '
theory of mythology ; we simply say that the theory fails, because
the phenomena it is intended to account for occur where it cannot
be applied. But Aryan mythology is a large subject by itself ; an
enquirer might naturally think that it could be explained without
going to the mythology of African or American savages ; it has
been found that this is not the case ; the long descent of man
connects his present with a past very dissimilar, and connects
thereby with one another contemporary forms of civilization wide
apart. Therefore it is important to insist upon studying the pre-
sent in the light of history and comparing as extensive a range of
facts as can be gathered together.

We hear sometimes of ' methodological assumptions '. By the
term is meant assumptions made for the sake of getting forward
with the investigation of a subject, but not necessarily regarded
as true. For example, there is obviously some connexion
between states of mind and states of body. The psychologist,
seeing quite clearly that to suppose the former to be produced by
the latter soon lands him in the most hopeless contradiction, and
ignorant as to the true way of stating the relation between them,
may think the hypothesis of interaction the most convenient assump-
tion to make, with a view of increasing and systematizing his
knowledge of the laws which determine the development of the
individual mind ; or instead of the hypothesis of interaction (which
conceives mind and body as producing changes in one another) he
may prefer the hypothesis of parallelism, according to which every
mental change has a corresponding bodily change, and vice versa,
but the two series proceed each uninfluenced by the events of the
other. Either hypothesis, if not regarded as true, but only as
facilitating enquiry, would be a methodological assumption.1 Simi-
larly, if he believes in the freedom of the will, the psychologist
may still, as a methodological assumption, accept the doctrine of
determinism ; because so far as actions have not any cause suffi-
ciently accounting for them in some pre-existing state of the agent,
but spring from the activity of a will not acting according to any
laws, it is hopeless to try to explain their occurrence. In his
attempts to do this therefore he will assume what is necessary to

1 But it is strange that a psychologist should think that the truth or falsity
of such an assumption makes no essential difference to psychology. Cf. Pro-
fessor G. F. Stout, Manual of Psychology3, Pref. p. v.

xxvi] OF THE METHODOLOGY OP THE SCIENCES 565

the possibility of doing it, even though he may believe that it
cannot be altogether done.

Lastly, general logical considerations may indicate the weak
places in a particular science at a given time, and thus show what
line of enquiry is logically of most importance to the science in
question. The theory of Natural Selection assumed the existence of
variations, that is, divergences from the parent type in offspring ;
and it assumed these variations to be accidental or non-adaptive.
It concentrated itself at first on the task of showing how great
a degree of adaptation between an organism and its environment
could be brought about, through the operation of the struggle for
existence among individuals varying slightly from type in all
directions ; and how by the accumulation of such small variations
as happened to be favourable in each generation a profound modi-
fication of specific type might ultimately be produced. It was
quite worth while to work this out even upon a basis of assumption
as to certain of the facts. But the pressure of criticism has directed
attention to the question whether variations are all of them non-
adaptive ; and one of the logical requisites of the theory of Natural
Selection is a suitable collection of facts throwing light upon this
point. The facts are not very easy to obtain or estimate ; but
biologists are working at this problem with great assiduity. A
study of the contemporary state of biology from a logical point
of view would have to consider with some care the kind of facts
required on such a point as this, and the sort of instance that would
be crucial^ i.e. decisive against one or other theory.

1 From crux, a sign-post : as directing our choice between two (or more)
theories : v. Bacon, Nov. Org. II. 36. A crucial instance, though it can
disprove, can never prove a theory, except upon the assumption that there
is no other theory with which it agrees. And it is easier to imagine instances
fatal to the view that all variation is non-adaptive than to the view that
adaptive variation sometimes occurs.

CHAPTER XXVII

APPENDIX ON FALLACIES

A fallacy is an argument which appears to be conclusive when
it is not ; and the chief use of studying fallacies must be that we
may learn to avoid them. Regarding Logic as a science, we might
therefore justly say that we are not called upon to discuss them. The
only way in which their study can help us to understand how our
thought works is by the force of contrast. Show a man an argu-
ment which he recognizes to be unsound, show him where the
unsoundness lies, and he may very likely realize more clearly, so
far as they can be formally prescribed, what are the conditions of
valid reasoning. On this account as we went along we contrasted
examples of invalid with examples of valid inference. What more
then is wanted ? for the case is not as it is, for instance, with psy •
chology. To the psychologist few things are more instructive than
the study of marked abnormalities of mental life : just as to the
physiologist diseases reveal much which cannot be seen in health.
For psychology is an empirical science, so far as it is a science at all :
it aims at discovering the principles in accordance with which the
various manifestations of consciousness develop in the life of the
individual ; what these are it is to a large extent unable to anticipate,
although the metaphysician may have his views as to the conditions
under which alone their action — whatever they may be — is possible.
Now insanity is just as much a fact as any normal mental develop-
ment ; the conditions under which it occurs must be equally
ascertainable ; and doubtless the same principles, in accordance with
which this development proceeds under certain conditions normally
and to a sane result, are often exemplified in the mental disturbances
which other conditions evoke. They are exemplified there too in a
more prominent way ; so that such cases furnish what Bacon called
a glaring instance1 to assist us towards their discovery. But it
would be absurd to say that the principles of rational thought are
equally exemplified in fallacy as in sound thinking ; and it would
be absurd to hope to discover, in the procedure of a fallacious mind,

1 Instantiae Oslensivae, or Elucescentiae. Nov. Org. II. 24.

APPENDIX ON FALLACIES 567

the nature of true thinking. We have said once and again that
Logic analyses the operations of thought which the mind has
already performed in thinking about things ; but it must not be
supposed that it is on that account, any more than mathematics, an
empirical science. The mathematician can only recognize the
necessary relations of number or space by the help of some quantities
or figures in which he finds them ; yet he recognizes their necessity
to be absolute and universal, and the fact that his non-mathematical
friends make mistakes in their mathematical thinking is not taken
by him as evidence that there are really two ways of thinking about
the matter ; he merely says that on such subjects they cannot really
think. So also with Logic. Only in some thought in which they
are found can the necessary relations involved in thinking be
recognized ; but their necessity too is recognized to be absolute,
and we say that those who hold otherwise are incapable of thinking
about how they think. If any one is inclined to hold otherwise,
and to suppose that the laws of our thinking are psychological laws,
exemplified no less in fallacy than in its opposite, let him reflect
that even in doing so he is bound to assume the contrary. For he
who in that opinion sets out to ascertain what the principles of
thought, as a matter of empirical fact, are, will be unable by rights
to know that the thought is valid by which he conducts that investi-
gation. How then could he have any confidence in its results ?
Yet the fact that he intends to trust them implies that he assumes
the principles of thought, in accordance with which he conducts the
investigation, to be valid, whatever principles the investigation may
report in favour of ; and herein he takes for granted that he can
recognize immediately what rational thought is, without reference
to empirical facts revealed by psychology.

Nevertheless the insertion of a chapter on Fallacies may be
defended. It has tradition in its favour ; and without it, the
nomenclature of fallacies — a nomenclature by no means fallen out
of common use — would remain unexplained. There are practical
uses in it also ; and it would be ridiculous to say that because Logic
is a science we may not turn the study of it to advantage in practice.
Familiarity with some of the commonest types of fallacy is no
security that we shall never fall into them ourselves ; still less are
we bound to fall into them unless we have acquired that familiarity.
But it may help us to avoid them, by helping us more readily to
perceive them. The overtones which a man has never noticed till

568 AN INTRODUCTION TO LOGIO [chap.

they were pointed out to him he may afterwards detect easily for
himself. A flavour in a dish, a line in a picture, whose presence had
gone unobserved, a man may be unable to ignore, if it has been
singled out and presented to him in isolation. So it may be with
a fallacy. There are many whose perception of the unsoundness
of an argument is not unaffected by their belief in the truth or
falsity of its conclusion : they will detect it where they think that
what it proves is false ; but let that be true — still more, let the sup-
posed truth be precious to them, or familiar — and the same form of
argument in its support may pass unchallenged. Yet if we have
accustomed ourselves to the look, or type, of the fallacy, we are less
likely to be the victims of such an imposition. It is true that, in
the words of Archbishop Whately l, ' After all, indeed, in the prac-
tical detection of each individual Fallacy, much must depend on
natural and acquired acuteness ; nor can any rules be given, the
mere learning of which will enable us to apply them with mechanical
certainty and readiness : but still we shall find that to take correct
general views of the subject, and to be familiarized with scientific
discussions of it, will tend, above all things, to engender such a habit
of mind, as will best fit us for practice.' And, as Aristotle intimates,2
a man who, if you give him time, may be well able to detect a fallacy
by the light of nature, may be placed at a practical disadvantage
by not being able to do it quickly enough : here the systematic study
of fallacies will help him. Nor is it only in arguing with others that
he may reap some benefit from the study ; it will accrue to him also
in the conduct of solitary thinking.3 It was however chiefly with
reference to the conduct of debate that Aristotle discussed the sub-
ject. It was from this point of view that he observed, that a man
might be suspected of incompetence, who only found fault with an
opponent's argument, and could not show in what the fault con-
sisted.4 It may be added, that so far as fallacies are referable to
recognized types, it is a great abridgement of criticism to be able to
name the types, and refer a particular fallacy to one of them.

These are practical considerations ; and it would probably be
found that importance has been attached to the doctrine of fallacies
chiefly by those who have viewed Logic as an instrument for reason-
ing. But an use may be found in the doctrine, of a more theoretical
kind. It is intellectually unsatisfactory to see that an argument

1 Logic, p. 153, 8th ed. 2 Soph. El. xvi. 175a 23.

8 lb. 175a 9. * lb. 175a 14.

xxvii] APPENDIX ON FALLACIES 669

is faulty, and not to see precisely why. We desire for ourselves,
no less than we owe to our opponent, an analysis of the error.
Otherwise, and if we can only see it, and not see through it, the mind,
as Aristotle expresses it, is entangled, and unable to proceed.1 It
is probable that some of the fallacies of which he finds the solution
in different ambiguities of language did once constitute a more
serious entanglement than they do to-day. This is partly because,
as others have pointed out, such fallacies may disappear by trans-
lation into a foreign tongue ; and peoples more familiar than the
Greeks were with a diversity of tongues have an advantage in
detecting such. It is partly also because an analysis new in his
day is common property in ours ; and many of its results are so
incorporated into the currency of common thought and speech, that
a man whose attention is called to them feels as if he was taught
only what he already knew.

If however we are satisfied that Logic should treat of fallacies,
it is very difficult to be satisfied with any treatment of them. Truth
may have its norms, but error is infinite in its aberrations, and they
cannot be digested in any classification.2 The same inconclusive
argument may often be referred at will to this or that head of
fallacies. ' Since, in any Argument ', says Whately, ' one Premiss
is usually suppressed, it frequently happens, in the case of a Fallacy,
that the hearers are left to the alternative of supplying either a
Premiss which is not true, or else, one which does not prove the Con-
clusion. E.g. if a man expatiates on the distress of the country,
and thence argues that the government is tyrannical, we must sup-
pose him to assume either that " every distressed country is under
a tyranny ", which is a manifest falsehood, or, merely that " every
country under a tyranny is distressed ", which, however true, proves
nothing, the Middle-Term being undistributed '.3 The assumption
of a false premiss is not indeed perhaps to be called a fallacy, as we
shall see presently ; it is at any rate different in its nature from
inconclusive argumentation. But the choice may equally well lie
between two modes of inconclusive argumentation, when we have to
classify a fallacy ; a man who attempts to refute by an enumeration
of striking instances the proposition that some specific characters

1 Eth. Nic. ij. iii. 1146a 24.

2 Cf . de Morgan, Formal Logic, p. 237. * There is no such thing as a classifi-
cation of the ways in which men may arrive at an error : it is much to be
doubted whether there ever can be.'

3 Logic, p. 159, 8th ed.

570 AN INTRODUCTION TO LOGIC [chap.

in plants and animals are not adaptive might either be charged
with illicit process of the minor term, in drawing an universal
conclusion where his premisses only entitle him to a particular one,
or with what is called Ignoratio Elenchi, in supposing that a par-
ticular affirmative refutes a particular negative.1 And not only is
it impossible to make such a classification of fallacies as will never
leave it in doubt to which class a particular example is to be referred ;
if that were all, it might be said that the types were distinct, and
the classification so far a good one, although individuals could not
be assigned to their types unambiguously : but it may be doubted
as well, if the types of error can be exhaustively detailed, and the
classification completed.

The reason for this is twofold. In the first place, there may be
arguments so foolish and inconsequent, that they cannot even be
said to simulate cogency ; these cannot be positively characterized,
but must be lumped together by the mere negative mark of incon-
clusiveness. And secondly, there are many fallacies, the detection
of which requires not general logical training, but acquaintance
with a particular scientific subject-matter. The latter point is of
some importance, as connecting with what has been already said
about demonstration.

We have seen that the syllogism cannot sustain the claim once
made in its behalf, to be the type of all valid inference ; but that
— to say nothing of hypothetical and disjunctive argument — there
are deductive reasonings whose validity lies in no conformity
to a scheme exhibitable in the abstract, or symbolically, but rests
for its apprehension upon acquaintance with the nature of the
special subject-matter with which they deal. The readiest illustra-
tion of this, but by no means the only one, is furnished by geometry.
Now what is true of valid is equally true of invalid reasonings.
There are many which are not of a sort that can occur in reasoning
on every subject-matter, but are bound up with misconceptions
of the special subject-matter in which they occur. This too may
be readily illustrated from geometry. ' Lewis Carroll ' devised
a proof that ' a right angle is sometimes equal to an obtuse angle '.
The demonstration was in all other respects unimpeachable, but
vitiated by one — of course intentional — error in the construction of

1 Cf. At., Soph. El. xxiv. 179b 17 ov8ev St KaAvei rbv avrbv \6yov vXtiovs
Ho\8ripias c\*iV ('There is nothing to prevent the same argument having
several faults '), and xxxiii. 182b 10.

xxvn]

APPENDIX ON FALLACIES

571

the figure, in which a line was drawn to one side of a point which
must in fact fall on the other.1 Just as a knowledge of geometry
can alone show where this line must fall, so a knowledge of geometry
can alone expose the inconsequence of the false demonstration.
And similar inconsequences occur in every particular science, which
only an understanding of that science can show to be inconsequences.
Thus if it were argued that because a and b were halves of the same
thing, therefore they were halves of one another, and since a =4,
b must = 2, it is only a perception of the nature of quantity that
reveals (doubtless in this case to the least mathematical of us) the
invalidity of the first step in the argument. It is less obvious that
among a people who acknowledge kinship only through the female,
a man would inherit not from his father but from his brother or
maternal uncle. Yet a little reflection shows this to be the case,
and shows therefore the fallacy of arguing, where female kinship
prevails, that because A is in possession of a property, his son will
possess it after him. Here the detection of the fallacy rests upon
our perception of the system of relationships uniting the members
of a society which takes account only of union by descent through
the female line.

Aristotle, who noticed that every science afforded its own special
opportunities for erroneous inference, gave to those that involved

1 v. the Lewis Carroll Picture Book, edited by S. Dodgson Collingwood,
(London, 1899), pp. 266-267. (GK must really fall to the right of 0.)

' Let ABCD be a square. Bisect AB at E, and through E draw EF at
right angles to AB, and cutting DC at F. Then DF = FC.

' From C draw CG = CB. Join AG, and bisect it at H, and from H draw
HK at right angles to AG.

' Since AB, AG are not parallel, EF, HK are not
parallel. Therefore they will meet if produced. Pro-
duce EF, and let them meet at K. Join KD, KA,
KG and KG.

' The triangles KAH, KGH are equal, because
AH = HG, HK is common, and the angles at H are
right. Therefore KA = KG.

' The triangles KDF, KCF are equal, because DF =
FC, FK is common, and the angles at F are right.
Therefore KD = KC, and angle KDC = angle KCD.

' Also DA = CB = CG.

' Hence the triangles KDA, KCG have all their
sides equal. Therefore the angles KDA, KCG are

equal. From these equals take the equal angles KDC, KCD. Therefore the
remainders are equal : i. e. the angle GCD = the angle ADC. But GCD is
an obtuse angle, and ADC is a right angle.

1 Therefore an obtuse angle is sometimes = a right angle.

4 Q. E. D.'

572 AN INTRODUCTION TO LOGIC [chap.

mistakes in geometry the name of \f/€vboypd<prjfji.a, or false construc-
tion.1 As an example he gives Hippocrates' method of squaring the
circle by lunules. A lunule is a figure enclosed between arcs of two
circles concave in the same direction. Hippocrates found a rectili-
near area equal to a lunule whose upper arc was a semicircle, and its
lower arc the fourth part of the circumference of another circle ; he
then found another rectilinear area equal to the sum of (a) three
equal and similar lunules whose outer arcs were semicircles, and
their inner arcs the sixth part of the circumference of another circle,
and (6) a semicircle of the same diameter as the three lunules
(i. e. of diameter equal to the chord of the arcs enclosing them) ; and
he supposed that it would be possible, by subtracting from this
rectilinear area an area equal to the three lunules, to obtain in the
remainder a rectilinear area equal to the semicircle. He overlooked
the fact that because you can find a rectilinear area equal to a lunule
of the former sort, whose inner arc is a quadrant, it does not follow
that you can find one equal to a lunule of the latter sort, whose
inner arc is a sextant ; and in fact a rectilinear area equal to these
three lunules cannot be obtained.2

Now it will indeed be seen that, in this or any other case of
erroneous reasoning dependent on misconceiving the consequences
which follow from given conditions in a special subject-matter, the
error can be expressed in a false proposition. It is false that because
a rectilinear area can be found equal to one of these lunules, it can
be found equal to the other : it is false that things which are halves
of the same thing are halves of another : it is false that, if we take
account only of kinship through the female line, a man will be in
the same line of descent with his father. But we cannot see that
any of these propositions is false, unless we understand something
of the respective subject-matter. They are as it were false ' special
principles ', or Ihiai apyai? It is not desirable to call every false
proposition a fallacy, as e.g. that snakes eat dust, or that South

1 Soph. El. ix, xi. There is not however any false construction made and
used in this argument, as in that quoted in the last note ; though it is falsely
concluded that a required construction has been shown possible. •$rev8oypd(f>r]ij.a
practically means a fallacy involving special subject- matter : cf. the interesting
example quoted from Proclus by Sir Thomas Heath, Elements of Euclid, i. 206,
to prove that ' if two straight lines falling on another straight line make the
interior angles on the same side of it together less than two right angles,
those two straight lines being produced can never meet '.

2 v. Poste's ed. of Soph EL, App. F, pp. 245-247.

3 Cf. supra, p. 387.

xxvn] APPENDIX ON FALLACIES 573

America is an island ; nor can we extend the name to every valid
argument that uses a false premiss. If the falsity of the premiss can
only be ascertained empirically, there is error, but not fallacy. If
however the falsity of the premiss is to be ascertained by thinking
out the consequences of certain relations, or concepts, in the
circumstances of a given case, then we are guilty of fallacy, or defect
of reasoning, in overlooking it ; and that is what frequently occurs
in the matter of any particular science.

There are indeed general heads, under which many such fallacies
can be brought. In particular, they very often arise from over-
looking some of the special circumstances of the case : from assum-
ing that what is true under certain conditions will still be true when
those conditions are in some way modified. Thus, if two things
a and 6 are equal to the same thing, they are equal to one another ;
from which we may conclude, that if they bear any same quantitative
relation to a third thing, they bear that relation to each other ; and
then it would follow that if they were halves of the same thing they
would be halves of one another. But in fact, it is only when their
same relation to a third is one of equality, not merely when their
relation to it is the same, that they bear to one another the relation
borne to it. We shall meet with this type of fallacy by and by under
the name of Secundum Quid. That heading embraces a great range
of examples. But though we can detect in them a common char-
acter, it is only by understanding something of the special matter
of the argument, that we can see that the fallacy is being committed
in a given case. The type, if one may say so, is fluid ; the instances
are not so far of one form, that we can separate their common form
from the variety of their matter, and exhibit it symbolically ; nor,
though the type admits of all this diversity, can we subdivide it,
and carry our classification down to infimae species. We recognize
that its character differs in different cases ; but the differences
cannot be formulated.

Our task then is one which does not admit of fully satisfactory
performance. Still no doubt it can be better and worse done.
What classification of fallacies are we to adopt ?

The earliest, and for long the accepted, classification is that of
Aristotle, given in the last book of his Topics, called the Sophistici
Elenchi. It is not free from defects ; and others, some of which
will be referred to, have been propounded. But the subject is
emphatically one upon which some consensus is desirable. If it ia

674 AN INTRODUCTION TO LOGIC [chap.

useful to have a nomenclature of fallacies, it is useful to have a
standard nomenclature. And it is remarkable how, even in rival
classifications, many of the Aristotelian species of fallacy still hold
their own. Later writers have given new meanings to certain of
the Aristotelian names ; or have invented new names for special
forms of some of the Aristotelian fallacies ; or have included in
their list what are not forms of erroneous argument, but sources of
error of a different kind x ; yet it is surprising how little there is
which cannot be brought within Aristotle's list. And if we consider
not the enumeration of types of fallacy, but their classification, it
will appear, I think, that there is no such merit in any alternative
scheme as justifies us in sacrificing the advantage of keeping to the
standard and traditional scheme of Aristotle.

Aristotle divided fallacies into two main groups — fallacies in
dictione, -napa r?;y Ae£u-, arising through ambiguity of language,
and fallacies extra dictionem, Ifw rijs Ae'fews, which do not have
their source in such ambiguity. Although one of his species of
fallacies extra dictionem — the fallacy of Many Questions — might
perhaps be referred more naturally to the other group, yet the
division, being dichotomous, is sound. It suffers, however, like all
such divisions, from the defect of not positively characterizing one

1 Thus the fallacy of Accident has practically been identified with Secundum
Quid by many writers : that of Consequent has, e. g. by de Morgan and Jevons,
been explained as ' the simple affirmation of a conclusion which does not
follow from the premisses ' (de Morgan, Formal Logic, p. 267) : divers forms of
Ignoratio Elenchi have received special names : Whately has explicitly included
under fallacies, in defiance of his own definition, ' any false assumption em-
ployed as a Premiss ' {Logic, 8th ed. p. 168 : cf. def. on p. 153) : Mill includes
among fallacies such sources of error as Mal-observation — i. e. mingling
inference with the report of what is perceived (System of Logic, V. iv. 5) ;
and his first great group of fallacies, which he calls A priori Fallacies, or
Fallacies of Simple Inspection, consists of a number of maxims which he con-
siders erroneous (though it is not equally clear that they all are so), such as
that what is inconceivable cannot be true, that effects must resemble their
causes, that motion can only be produced by motion, that the same effect
must always have the same cause (V. iii.) ; in iv. 1, Fallacies of Simple Inspec-
tion are called ' Prejudices, or presumptions antecedent to and superseding
proof ', and in ii. 2 they are called supposed connexions or repugnances between
facts, ' admitted, as the phrase is ', on their own evidence, or as self-evident.
Whately (op. cit. p. 208) speaks of the fallacy of References, i. e. giving refer-
ences in support of a statement to passages which do not really bear it out, in
the trust that readers will not look up the references and discover this. Pro-
fessor William James gives the name of the Psychologist's Fallacy to the
mistake of supposing that a man who has a given psychical experience knowa
it, when he has it, to be all that I as a psychologist know or believe it to be
(Principles of Psychology, vol. i. p. 196). Locke's argumenta ad verecundiam,
ad ignorantiam, ad hominem, which he opposes to an argumentum ad indicium,
might be called heads of fallacies (Essay, IV. xvii. 19-22).

xxvn] APPENDIX ON FALLACIES 575

group.1 Later writers, willing to remedy this defect, called the
fallacies extra dictionem fallacies in re, or material fallacies. But
this introduces a cross-division. For it cannot be said that fallacies
in dictione are independent of the res or matter of the argument.
On the contrary, inasmuch as they arise through giving different
meanings to the same words either in the two premisses, or in
premiss and conclusion, they disappear if we abstract from the
matter of the argument and look only to the form in which it is
cast. The proper antithesis to matter is form ; a fallacy not in the
matter must be in the form : i.e. it must be independent of what
the terms are, and must therefore persist, if symbols be substituted
for the terms, and whatever term be substituted for the symbols.
This cannot be said of the fallacies in dictione.

It is true that Whately gives a somewhat different interpretation
to the expression material fallacy. He divides fallacies into logical
and material. By the former title he means fallacies where the
error lies in the fact that the premisses do not prove the conclusion ;
by the latter, those in which the premisses prove the conclusion, but
either the premisses are false, or such at least as we are not entitled to
assume, or else the conclusion proved is not that which we profess
or are required to establish.2 He then subdivides logical fallacies
into two groups, according as their defect of proof can be seen in the
mere form of the argument (e.g. undistributed middle) or only
if we attend to the ambiguity of the terms employed ; the former
group he calls purely logical, and the latter semi-logical. Though
the nomenclature here is unfortunate (for according to his own
definition of a logical fallacy, those which lie in ambiguity of lan-
guage are altogether and not only half logical), yet the division is
sound. It includes however arguments which have no fault except
that their premisses are false ; and it is true that in this he follows
the words of Aristotle 3 ; but in the body of his treatise Aristotle

1 Cf. supra, pp. 122-124.

2 By the matter of an argument he means the propositions in it, not the
terms of the propositions.

3 Top. a. i. 1001' 23 ei>l(TTLKos §' tCTTi crvWnyicr fJ.os 6 eK (prnvotievwv evhot-cav,
firf ovtg>v 8e, kcl\ 6 e'£ evSo^cDv rj (pnivnfj.evaii' e'v^o^oov (fcaivoiievos (' A contentious

syllogism is one whose conclusion follows from premisses that appear to be
endoxical but are not, or that appears to follow from premisses that are or
appear to be endoxical ') : cf. Soph. El. ii. 165b 7 £pi(n<«n St ( vi-yoi) nl s'k t op
(pnivo^.ei'U)V fi/Sn^uii' [xi) ovtg>v fie irvWoyiaTi^oi r] cjxnvo^i'oi a""V'\' yicr ikoi ( Con-
tentious arguments are those that conclude or appear to conclude from
premisses that appear to be endoxical but are not'). The latter definition
excludes unsound arguments from premisses really endoxical (i. e. probable
or supported by opinion, and allowable in non-scientific discussion) ; but this

676 AN INTRODUCTION TO LOGIC [chap.

proceeds as if he had not included them. And the practice of
Aristotle appears preferable in this respect ; for false premisses are
certainly incapable of any classification, and the consideration of
one does not help us to detect another. That, if the premisses are
false, the conclusion, though valid, need not be true, every one should
certainly realize ; and it is good advice to a disputant to consider
well the truth of the premisses he is asked to grant, or to a solitary
thinker to consider well the truth of what he proposes to assume
and build upon. Nevertheless there seems to be a real difference
between a plausible but inconclusive argument, which we can see
through by clearer and more attentive thinking, and a false pro-
position (whether or not plausible), which cannot be exploded by
any more attentive consideration of itself, though it may by reason-
ings that are within our power. For this reason the extension of
the term fallacy to cover ' any false assumption employed as a
premiss ' seems undesirable ; the only sort of false proposition to
which it ought to be applied is false canons of proof . If this correc-
tion is made, Whately is left with only two kinds of material fallacy
(Petitio Principii and Ignoratio Elenchi), both of which are in
Aristotle's list of fallacies extra diciionem ; and there is no particular
advantage in that regrouping of the species enumerated in both
lists, which the adoption of Whately's principle of division carries
with it. Whately certainly enumerates under the head of purely
logical fallacies those breaches of syllogistic rule with which we
long ago became familiar by the names of undistributed middle,
quaternio terminorum, and illicit process of the major or minor term ;
and Aristotle makes no mention of these. But that is not because
his classification provides no place for them ; they are clearly
fallacies extra dictionem. They were omitted because they did not,
in Aristotle's view, simulate cogency ; no one who could not detect
these ought to undertake a disputation ; and even a sophist, aiming
only at appearing to confute his adversary and not at truth, would
hardly dare to employ such methods as these. And so it was with

can hardly be supposed to be deliberate. The expression twice used in Soph.
Mil. 1. (164a 23 on fxev ovv ol fiev elai crvWoyio-fjioi, ol 6' ovk ovres Sokoikti, (pavepov
—-' It is plain then that there are real syllogisms and what appear to be syllo-
gisms without being so' : 165a 17 8ia pev ovv ravrqv ttjp ahlav koi tos Xe^^(ro-
fiivat errri iea\ <riA\oyi07xo? Kai e'Keyxos <f)atv6fi.cvns fuv ovk tsv hi — ' For this reason
then and those that follow there are both apparent syllogisms and apparent
confutations which are not really such ') might perhaps by itself be more
naturally understood to refer only to fallacious arguments, and not to include
arguments that have no fault except in the falsity of their premisses.

xxvnl APPENDIX ON FALLACIES 577

the writers who for many centuries reproduced — often with increas-
ing divergence — the Aristotelian doctrine. ' The pure syllogism
and its rules were to them as familiar as the alphabet. The idea of
an absolute and glaring offence against the structure of the syllogism
being supported one moment after it was challenged, would no more
suggest itself to a writer on logic than it would now occur to a writer
on astronomy that an accidental error (which might happen to any
one) of affixing four ciphers instead of five when multiplying by
a hundred thousand would be maintained after exposure.' * A
sophism, or sophistical confutation, as Aristotle called a fallacy
(for he had in mind throughout the conduct of a disputation, and
the methods by which one might attempt to confute a thesis main-
tained by an opponent : though these are of course equally methods
of establishing a conclusion that confutes it), must be at least
(fxuvoixevos <rv\Xoyi<rix6s, apparently conclusive ; these he wished
in his treatise to enable the student to expose 2 ; but a plain breach
of syllogistic rule had not any appearance of conclusiveness, and
enough had already been said in the Prior Analytics to enable any
one to expose that.

We may therefore abide by the Aristotelian division into fallacies
in dictione and extra dictionem. In each member of the division
he enumerates a variety of types. The lists are as follows 3 : —

1 de Morgan, Formal Logic, p. 240.

2 Cf. Soph. El. i. 165a 26 tov be -^-evbo^evov e'fi<fiavi£et.v bvvacrdai.

3 Whately, as was observed above, regroups the fallacies here enumerated
to suit his division. It is of course inadmissible to adopt the nomenclature
of his division, and retain Aristotle's grouping, as is done by Jevons in his
Elementary Lessons in Logic, XX and XXI. He treats as purely logical
fallacies the four breaches of syllogistic rule above mentioned ; as semi-
logical, Aristotle's six fallacies in dictione ; and as material, Aristotle's seven
fallacies extra dictionem. He does not therefore understand the distinction
between logical and material as Whately does. ' The logical fallacies ', he
says, ' are those which occur in the mere form of the statement. . . . The
material fallacies, on the contrary, arise outside of the mere verbal statement,
or as it is said, extra dictionem ' (p. 170). This is not of course what Whately
meant. But clearly Jevons means by a logical fallacy one which can be
detected in the form without consideration of the matter ; it should therefore
be capable of illustration in symbols, as his ' purely logical ' fallacies are.
A material fallacy, on the contrary, needs that we should understand the
terms for its detection. From this point of view, it is nonsense to speak of
4 semi-logical ' fallacies ; a fallacy either can be detected in symbols or not :
it must either be ' logical ' or not, and cannot be ' semi-logical '. The fallacies
in dictione, which he ranks as ' semi-logical ', he ought undoubtedly to have
ranked as ' material '. On the other hand, some of those which he ranks as
' material ' — the fallacy of the Consequent certainly (which however he
misunderstands) and one type of Petitio Principii — can be exhibited in sym-
bols, and ought to have been enumerated among the ' purely logical '. The

1779 p p

578 AN INTRODUCTION TO LOGIO [chap.

a. Fallacies in dictione, or irapa rr)v Ac£iv.

1. Equivocation, or irapa rr/y 6p.(t>vvp.(av.

2. Amphiboly, or irapa rr\v a.p.<pi(3oX(av.

3. Composition, or -rrapa rqv arvvQzviv*

4. Division, or irapa ti]V 8iaipe<ru/.

5. Accent, or irapa ttjv irpoo-^bCav.

6. Figure of speech, or irapa to <ryr}p.a rrjs X{£ea>$.

b. Fallacies extra dictionem, or !fa> rrjs Ae£ecos.

1. Accident, or irapa to o-vp.(3€(3r)Kos.

2. Secundum Quid, or irapa to airX&s r/ irfj Xeyecrdai Kal p.))

KVpCoiS.

3. Ignoratio Elenchi, or irapa ttjv tov eXiyxov ayvoiar.

4. Petitio Principii, Begging the Question, or 7rapa to iv

apxy Xap.fia.veiv.

5. Non Causa pro Causa, False Cause, or irapa to pvq afrtov

O)? aiTtov.

6. Consequent, or irapa. to kir6p.evov.

7. Many Questions, or irapa to to. bvo kpoiTT]p,aTa kv iroieiv.

The fallacies in dictione are so many different forms of error that
may arise through the double meanings of language. They differ
according to the character of the ambiguity ; and in a syllogism it
may be any of the three terms which is ambiguous.1 Obviously
such arguments are invalid ; and if the different meanings were
expressed by different terms in each case, we should have a plain

fact is that, if the distinctions of logical and material, and in dictione and extra
dictionem, are to be combined in one classification, they cannot be identified,
as Jevons identifies them. We may either start with the distinction of
fallacies into logical and material, according as they lie in the mere abstract
form of the argument, and can be exhibited in symbols, or not : and then
divide the latter into those in dictione and those extra dictionem, according as
they arise through ambiguity of language, or not ; but of course those fallacies
extra dictionem which are logical in this sense must be removed from Aristotle's
list of fallacies extra dictionem, if that title is used to indicate a subdivision of
material. Or else we may begin by dividing them into fallacies in dictione and
extra dictionem, and treat logical and material as subdivisions of extra dictionem.
In the former case, what Jevons calls semi-logical ( = Aristotle's fallacies in
dictione) will enter by the latter name as a subdivision of material ; in the
latter, what he calls purely logical will enter as a subdivision of extra dictionem.
Cf. the remarks in Mr. St. George Stock's Deductive Logic, c. xxx, who points
all this out very clearly in discussing fallacies. It may be added that there
may be in algebra fallacious arguments which use symbols, but are not on
that account logical in the above sense, because the symbols are not logical
symbols, standing for any term, but specifically symbols of quantity.

1 Many arguments referable to Aristotle's heads of fallacy are not syllogistic.

xxvn] APPENDIX ON FALLACIES 579

quaternio terminorum, which would impose on nobody. As it is, the
shifting of the meaning may sometimes pass unobserved ; or the
identity of the language seem to afford some proof of identity of
meaning ; and even where it is obvious that we are tricked by the
argument, we may wish to be able to show how.

1. Equivocation is the simplest form of ambiguity, where a single
word is used in divers senses. ' The sick man is well ; for men
who have recovered are well, and the sick man has recovered ' x ;
here the equivocation is in the minor term, and arises from the fact
that the expression ' the sick man ' may mean either ' the man who
is sick ' or ' the man who was sick '. The following is an old example :
1 Finis rei est illius perfectio : mors est finis vitae : ergo mors est
perfectio vitae ' ; the equivocation in this case lies in the middle
term. Trivial and punning examples of this fallacy, as of all those
that depend on ambiguity of language, will occur to any one ; but
in many cases it is serious and elusive. ' It is the business of the
State to enforce all rights : a judicious charity is right : therefore
it is the business of the State to enforce a judicious charity.' ' A
mistake in point of law ', says Blackstone, ' which every person of
discretion not only may, but is bound and presumed to know, is in
criminal cases no sort of defence ' 2 ; the State must perhaps presume
a knowledge of the law, and so far we are bound to know it, in the
sense of being required under penalty ; but a criminal action done
in ignorance of the law that a man is legally bound to know is often
considered morally discreditable, as if the knowledge of the law
on the matter were a plain moral duty. How far that is so in a
particular case may be a very doubtful question ; the maxim quoted
tends to confuse the moral with the legal obligation. In a long and
closely reasoned argument, where important terms have been
defined at the outset, it may still be very difficult to hold them
throughout to the precise meaning set forth in the definition ; and
so far as this is not done, the fallacy of Equivocation arises. James
Mill held that a so-called necessity of thought arises from the
constant conjunction in experience of the ' ideas ' between which
a necessary connexion is asserted ; his son, endeavouring to make
this doctrine cover negative propositions which assert a necessity
of thought, says 3 that here one idea is inseparably associated with

1 Ar., Soph. El. iv. 165b 39. 2 Quoted by Austin, Jurisprudence, i. 482,

3 v. Jas. Mill's Analysis of the Phenomena of the Human Mind, i. 97, note 30,
by J. S. Mill.

Pp2

580 AN INTRODUCTION TO LOGIO [chap.

the idea of the absence of the other. But the idea of the absence
of an idea means the opinion that an idea is absent.

2. Amphiboly * is ambiguity in a phrase, in which the words are
used univocally throughout, but the meaning of the phrase as
a whole changes through change of the construction in which the
same words are taken. A traditional example in Latin is ' Quod
tangitur a Socrate, illud sentit : lapis tangitur a Socrate : ergo lapis
sentit ' ; in the major premiss, illud is the object of sentit ; the con-
clusion is drawn as if it had been the subject. So we might say in
English : ' Polyphemus what he best loves doth devour : the ram
that leads the flock he loves the best : therefore the ram devours
him.' Lawyers are well aware of the importance of avoiding
ambiguity in the construction of a legal document (though under
that head they would include the ambiguities which Aristotle
assigned to Division and Composition, as well as Amphiboly and
Equivocation too). Whately cites a good example from the rubric
at the beginning of the Form of Service formerly ordered for use on
Jan. 30, the anniversary of the execution of King Charles I : 'If
this day shall happen to be Sunday, this Form of Prayer shall be
used and the Fast kept the next Day following ' ; is the form of
prayer to be used on Sunday and the Fast kept on Monday, or are
both to be deferred ? Another famous and deliberate example is in
the oracle which Ennius said was delivered by Apollo to Pyrrhus
— ' Aio te, Aeacida, Romanos vincere posse.' 2 Ambiguous words
and constructions are still not unfrequently used to deceive by those
' That palter with us in a double sense ;

That keep the word of promise to our ear,

And break it to our hope '.

1 The Greek word is afi(f)i^o\ia, which is said to be an imarr} ivnpa tov \6yov,
' being misled by a form of words ', as distinct from 6uowfj.ia, where the ambi-
guity is in an ovona or word {Soph. El. vii. 169a 22). Hence arose the compound
an(f)i[3o\o\oyia, which became corrupted into Amphibology, as etfiwXoAuTpeia
became corrupted into Idolatry. There seems to be no reason for not saying
Amphiboly in English ; Amphibolia is frequent in Latin (e.g. Crackenthorpe,
Aldrich). It will be seen that the fallacy of Composition may also turn on
taking words together in different ways. I think that Aristotle (who notices
their affinity Soph. El. xx. 177a 38) would have called a fallacy Amphiboly
where the ambiguity arose through taking variously the government within
the same group of words, and Composition where it arose through taking the
same words now with these and now with those others in a sentence. Whately'a
example would in this case have been referred by him to Composition.

2 Cf. Cic. de Divinatione, ii. 56. Cicero reasonably observes that Apollo
did not speak in Latin. Cf. Augustine, de Civ. Dei, iii. 17 ' Cui sane de rerum
futuro eventu consulenti satis urbane Apollo sic ambiguum oraculum edidit,

xxvn] APPENDIX ON FALLACIES 581

3 and 4. Composition and Division are the converse one of the
other. They consist in taking together in the conclusion (or one
premiss) either words, or objects of thought, which in the premiss
(or the other premiss) were not taken together, or vice versa. Plato
in the Republic1 argues, from the fact that a man can refuse the
thing that he desires, that there must be a principle of reason as
well as of appetite in the soul. For, he says, it is impossible to be
contrarily affected at the same moment towards the same object
in the same part of oneself (one cannot for example at once loathe
and long for the same object) ; yet a man who is thirsty and refuses
to drink is contrarily affected at the same moment towards the same
object ; he does not therefore refuse drink on account of the char-
acter of his appetites, but because of his reason ; he reckons that
to indulge his appetite would interfere with the pursuit of some
other end which he prefers. Now a sophist might attack this
conclusion as follows : ' Are you now drinking ? No. Can you now
drink ? Yes. Therefore when you are not doing a thing, you still
can do it ? Yes. But if you can do a thing when you are not doing
it, you can desire a thing when not desiring it ? Yes. And so you
can be contrarily affected in the same part of yourself (your appeti-
tive nature) towards the same object at the same time.' 2 The
fallacy is one of composition. The admission is that a man can
when not desiring a thing desire it, i. e. that when not desiring it,
he is capable of doing so ; this is used as if it meant that he can
desire when not desiring it, i.e. that he is capable of at once desiring
and not desiring it ; the words ' when not desiring it ' are taken,
or compounded, in one case with ' can ' and in the other with ' desire '.
If a man were to argue that three and two are five, and three and
two are odd and even, therefore five is odd and even, and the same
number may thus be both, he would be committing the same fallacy ;
when it is said that three and two are odd and even, it is true only if
* three and two ' are not taken together, as the subject of which ' odd
and even ' are predicated, but ' three ' is made the subject of ' odd '
and ' two ' of ' even ' ; but the conclusion is drawn as if they were

ut, e duobus quicquid accidisset, ipse divinus haberetur : ait enim, Dico te
Pyrrhe vincere posse Romanos : atque ita sive Pyrrhus a Romanis sive
Romani a Pyrrho vincerentur, securus fatidicus utrumlibet exspectaret
eventum.' Cf. also Henry VI, Part 2, Act i. Sc. 4, 11. 60-65.

1 Rep. iv. 436 A sq.

2 To 8vva(rdai fxf] ypdcpovra ypdcfxiv (' to be able to write without writing ')
is an example of fallacy irapa rr/v avvdeaiv in Soph. El. iv. 166a 24. I do not
know if the principle involved was ever brought against Plato's argument.

582 AN INTRODUCTION TO LOGIC [chap.

taken together. On the other hand, the same argument furnishes
an example of the counter fallacy of taking separately in one premiss
words which were taken together in the other ; for three and two
together are five, but it is separately that they are odd and even,
and separately that in the conclusion each of them is declared to be
both. And the reader will doubtless have observed that the pre-
vious example illustrates no less the division from one another
in the conclusion of words that were combined in the premiss than
the combination in the conclusion of words that in the premiss were
divided.1

It was said above that in these fallacies either words or objects
of thought are taken in one place in the argument together and in
another separately. Of course the combination or separation of
certain words carries with it that we think differently in either case
of the things signified. But sometimes the illicit combination or
division made in thought is not reflected by taking words together
or apart. If any one were, upon the strength of the text in
Gen. i. 27 — ' So God created man in his own image, in the image of
God created he him ; male and female created he them ' — to argue
that man was originally created bisexual,2 and that the present
division into male and female was the result of the Fall, and were
to base on that a condemnation of marriage, he would be guilty of
the fallacy of Composition ; and quite as foolish arguments have
been drawn from the words of Scripture upon such subjects. Now
here the fallacy lies in referring the words ' male ' and ' female '
together to each person signified by ' them ', instead of referring
' male ' to one and ' female ' to another. But the point is the same
in the story of the showman who announced that children of both
sexes were admitted free, and then charged admission to boys and

1 It is difficult to keep Composition and Division apart. Aristotle gives the
last example slightly differently as an example of Division — ' Five is two and
three, and therefore odd and even ' ; five is two and three together, and so
inferred to be what they are separately. He gives also as an example of
Division one which might equally well be called Composition, t7(vtt]kovt dv8pa>i>
(k(it6v XiVe t'tos 'Ax&Xevs, where the sophist charges you with saying that
Achilles left 100 men out of 50 (centum ex hominibus quinquaginta liquit divus
Achilles : the ambiguity cannot be reproduced in English) — v. Soph. El.
iv. 166a 33-38. But one cannot wrongly combine certain words with these
instead of those without also wrongly separating them from those instead of
these. Note that in the numerical example in the text the ambiguity arises
through understanding the same words of things separately or together, but
is not reflected in an ambiguous grouping of words.

* Cf. the fancy in Plato's Symposium, 189 D E.

xxvii] APPENDIX ON FALLACIES 583

girls alike on the plea that neither of them were children of both
sexes. Yet in the latter case there are no words that are wrongly
taken together ; it is the sexes thought of, to which the showman
pleaded that he had only promised to give free admission when
combined. Words like both and all, which may have equally a
distributive and a collective reference to the things signified by the
substantives to which they belong, are specially adapted to facilitate
this fallacy.1 Another and a double example of the fallacy of Com-
position, in a business transaction, is afforded by the tale oi a
railway enterprise in one of the British Islands. A company is
said to have been formed to build a railway, and to have announced
in its prospectus that a guarantee of 3 per cent, on the share
capital had been given by the Government, and a guarantee of
2 per cent, by the local authority ; and later in the same document
to have stated that a guarantee of 5 per cent, had been given by
the Government and by the local authority.

5. The fallacy of Accent meant to Aristotle one arising through
the ambiguity of a word which has different meanings when differ-
ently accented. It was perhaps distinguished from Equivocation,
because words differently accented are not strictly the same word.
The Latin writers illustrate it in words which have different mean-
ings when the quantity of a vowel is different ; e.g. ' omne malum
est fugiendum, pomum est malum : ergo fugiendum '. The am-
biguity is of course one which is more likely to occur in communica-
tion by writing than by speech.2 In English, which does not
distinguish words by tonic accent, the name is generally given to

1 It illustrates again how much akin the different fallacies in dictione are, and
how the same example may from different points of view be regarded as
falling under different heads, that any one who likes can call the showman's
trick, or others where words like all and both figure similarly, fallacies of
Equivocation. Aristotle does not give any such instances under the head of
<rvvde<ris or 8inipt<ris ; it has been however done by divers writers, and if we
look to the nature of the thought involved, justly. And the fallacies in
question might have been defined above as arising, when a conclusion is
reached by taking those things together which we are only entitled to take
separately, or vice versa ; for even where words are taken together or separately
in one part of the argument, which were intended to be taken separately or
together in the other, it is only as this leads to our so taking what they signify
that fallacy results. But as this is reflected often in a definite combination
and division of words, and as that fact probably led to the erection of these as
particular species of fallacy based on ambiguous language, it seemed right to
make express mention of such cases in describing them (cf. Crackenthorpe,
Logic, ed. quart, p. 353, cum quia ab iis coniunctia arguat, quae aeparatim vera
aunt, non coniuncta).

a Ar., Soph. El. iv. 166b 1.

A

584 AN INTRODUCTION TO LOGIO [chap.

arguments that turn on a wrong emphasis of some particular word
in a sentence ; in which if the emphasis were placed differently,
the meaning might be very different. The words of the Catechism
in the ' Duty towards my Neighbour ' — ' to hurt no body by word
nor deed ' — have by laying stress on body been wrested to include
the injunction to be kind to animals.1

6. The fallacy of Figure of Speech arises through the ambiguous
force of some verbal inflexion, which is wrongly alleged to imply in
one case what it really implies in others. If a man were to argue
from the use of such an expression as ' I am resolved what to do ',
that, because the passive signifies not action but being acted on, as
in ' I am beaten ', ' I am praised ', therefore a man's resolution is
not his own free act, but the result of something done to him, he
would be guilty of this fallacy. Arguments from linguistic usage
of that sort are by no means uncommon or necessarily unsound : as
that the object of sight is not a visual sensation, because you say
that you feel a sensation, but no one would say that he felt a
colour. In this case there is no ambiguous inflexion, which is what
was held to constitute the differentia of the fallacy now under con-
sideration. But let a man say that important is a negative notion,
because imperturbable or impenitent is, and we have a case in point.2
J. S. Mill in his Utilitarianism3 affords an excellent example of
this fallacy in a critical point of his argument. He is trying to
prove that the chief good, or one thing desirable, is pleasure. ' The
only proof ', he says, 'capable of being given that an object is visible,
is that people actually see it. The only proof that a sound is audible,
is that people hear it : and so of the other sources of our experience.
In like manner, I apprehend, the sole evidence it is possible to pro-

1 This example was given me from personal recollection. Not unlike this
fallacy, understood as consisting in basing on a wrong emphasis a conclusion
not intended by the speaker or writer, is the error of inferring from the stress
which a man lays on one element of a truth that he necessarily overlooks
another. It might be said to be Hegel's conception of the progress of specu-
lative thought, that it advances by emphasizing first one and then the other
side of a contrast in such a way that the emphasis on one leads to overlooking
the other : until a new conception is reached which unites the two. This
indeed he considers inevitable in the development of philosophy. But many
writers have been erroneously interpreted, because it was thought that when
they insisted upon one aspect of a truth they intended to deny some other
aspect. This error of interpretation however could hardly be classed with
fallacies in dictione, since the misinterpretation does not arise through the
doubtful stress-accentuation of particular words.

2 A lady once observed : ' The question is, is he a postor or an impostor ? '

3 p. 52 (Routlcdge's ed., ' New Universal Library ', p. 66).

xxvn] APPENDIX ON FALLACIES 585

duce that anything is desirable, is that people do actually desire it.'
But visible, audible mean what can be seen or heard ; whereas Mill
is trying to prove that happiness ought to be desired, or is the thing
worth desiring. Yet the termination -able or -ible must be taken to
have the same force in the word desirable as in audible or visible, if
the argument is to have any force at all ; and the only thing shown
is really that men can desire happiness : which was never in question.
To distinguish the different sources of the ambiguity in the
different fallacies enumerated above is not a matter of first-rate
importance ; but to be alive to the errors into which ambiguities of
language may lead us is so. ' Verba plane vim faciunt intellectui,
et omnia turbant,' wrote Bacon.1 Perhaps the disturbance which
they caused was in some respects more serious of old than now. We
do not suffer less from the subtle and unconscious shifting of the
meaning of important terms in a sustained argument ; but some of
the more trivial and (as we should say) obvious ambiguities may
have been a more real puzzle in olden days. ' The genius of un-
cultivated nations ', says de Morgan,2 ' leads them to place undue
force in the verbal meaning of engagements and admissions, inde-
pendently of the understanding with which they are made. Jacob
kept the blessing which he obtained by a trick, though it was in-
tended for Esau : Lycurgus seems to have fairly bound the Spartans
to follow his laws till he returned, though he only intimated a short
absence, and made it eternal : and the Hindoo god who begged for
three steps of land in the shape of a dwarf, and took earth, sea and
sky in that of a giant, seems to have been held as claiming no more
than was granted. The great stress laid by Aristotle on so many

1 Nov. Org. I. 43. The false ideas about nature generated through language
Bacon called idola fori. These false ideas or idola were classified by him
according as they had their sources in universal properties of human nature,
in idiosyncrasies of the individual, in language, or in false theories of science
and philosophy. The division was not logically perfect, and the enumeration
in each group is doubtless not complete. This illustrates in a parallel field
the difficulties above acknowledged to render a perfect classification of fallacies
impracticable. Bacon himself calls attention to the parallel that exists
between his undertaking and a classification of fallacies : ' Doctrina enim de
idolis similiter se habet ad interpretationem naturae, sicut doctrina de sophisticis
elenchis ad dialecticam vulgarem ' (I. 40). The ' interpretation of nature '
involved more than reasoning ; it required the use of the senses in observation,
the recording of facts, the formation of conceptions, or hypothesis, the inven-
tion of a nomenclature, &c. There are obstacles in the way of the successful
performance of these operations, no less than of reasoning. The fallacies of
the common Logic waylay us in the work of reasoning. His idola arise from
circumstances that waylay us in all these tasks.

2 Formal Logic, p. 244.

586 AN INTRODUCTION TO LOGIC [chap.

forms of verbal deception may have arisen from a remaining tendency
among disputants to be very serious about what we should now
call play upon words.' Just as many people tend to think that
in conduct the claims of veracity are satisfied or broken, according as
the facts can or cannot, by some verbal quibble, be brought within
the four corners of what they said or promised, so with argument
men may think that there is something in it, though the conclusion
turns upon an ambiguity of language. Not but what men are often
also too ready to assume that a controversy is merely verbal when
it is not.

In the enumeration of the fallacies which he recognizes, Aristotle
obviously had before him the practices of disputants in his own
day.1 One man, the ' respondent ', undertook to defend a thesis ;
the other, the ' questioner ', attempted to extract admissions from
the respondent which involved the contradiction of his thesis. But
we find that a man might endeavour to discredit his opponent by
confuting him on a side issue ; and that it was a recognized device
to get him to admit something easier to attack than his original
thesis ; though when Aristotle wrote, men had learned to reply
to the entrapping question by asking what it had to do with the
original thesis.2 Similarly we are told that answers in the form
of a plain yes or no were less insisted on when he wrote than for-
merly ; whereby a bountiful source of unfair confutations was cut
off.3 The questioner is advised also not only to endeavour to involve
the respondent in a contradiction of his own thesis, but to bring out
its inconsistency with what is held by those whose authority he or
others may respect, or by mankind at large, or by the majority of
mankind, or by his own school.4 Nowadays formal disputation has
gone out of fashion. Men still harangue ; and we understand by
a debate a series of set speeches, in which a proposal is attacked
and defended. Many of the devices which can be employed to pro-
duce the appearance of confuting an adversary are common to
rhetoric and dialectic — to the harangue and to the interchange of
question and answer. But if we were more familiar with the latter

1 Minto, in the first chapter of his Logic, Inductive and Deductive, speaks
as if Aristotle worked out his system of logic as a whole chiefly with the
conduct of disputation in view. He seems to me to have very much over-
stated his case ; but so far as the treatise on Sophistical Confutations is
concerned, it is true.

2 Soph. El. xii. 172b 16-24.

* lb. 175b 8-10. Cf. on the fallacy of Many Questions, p. 597, infra.

* lb. xv. 174b 19-23.

xxvn] APPENDIX ON FALLACIES 687

mode of trying an issue, we should perhaps understand better the
scope that exists for some of the sophistical confutations that
Aristotle mentions. Such disputation is seen chiefly to-day in
courts of law, when counsel cross-examines a witness ; and an un-
scrupulous counsel can still confuse a timid witness, and discredit
him before the jury, by involving him in contradictions more ap-
parent than real. And there have been times when matters, which
to-day are submitted to the judgement of the public by means of
speeches to and fro, reported in the newspapers, were argued by
chosen disputants according to fixed rules of debate before an
audience whose verdict, as to which side got the best of the discus-
sion, was of high practical importance. Not a few controversies of
that sort were argued during the Reformation, at Leipsic or at
Marburg or at Zurich or elsewhere.

The fallacies in dictione have to some extent become of less im-
portance through the decay of the habit of disputation. The same
cannot be said of those extra dictionem.1 These are not united
by any common character, as the others were by springing from
ambiguity in language.

1. The first in the list is the fallacy of Accident. The following
are some of the examples referred by Aristotle to this head : ' This
dog is yours : this dog is a father : therefore he is your father.'
* Do you know Coriscus ? Yes. Do you know the man approach-
ing you with his face muffled ? No. But he is Coriscus, and you
said you knew him.' ' Six is few : and thirty-six is six times six :
therefore thirty-six is few.' His solution of the error involved
seems to be this. A subject has divers accidental predicates,
i.e. predicates indicating attributes which are not commensurate
with it nor essential to it ; what is predicable of the subject may
or may not be predicable of these accidents, and vice versa.2
Thus the dog is a father, and is yours ; but it does not follow
that the father is yours — that he is yours as a father, as he is
yours as a dog. Coriscus is approaching with his face muffled ;
to be a man approaching with his face muffled is an accident of
Coriscus ; and it does not follow that, because Coriscus is known,
a man approaching with his face muffled is known to you. It is
an accidental way of regarding thirty-six things, that they are six
groups of six things ; and though the groups are few, the thirty-six
are not therefore few. The defect of the solution offered is, that

1 Except perhaps ' Many Questions ' ; but cf . infra, p. 598.

2 Soph. El. v. 1661' 30-32, xxiv. 179a 27-31.

588 AN INTRODUCTION TO LOGIC [chap.

it does not enable us to distinguish between those cases in which
what is predicated of the accidents of a subject may be predicated of
the subject itself, or vice versa, and those in which it may not ' This
dog is yours, and this dog is property (or, a spaniel) : therefore he is
your property (or, your spaniel) ' : why is this argument valid and
the former one not ? If you say that the former is invalid because
it equates subject and accident * when they are incommensurate,
why do you allow the latter, which does so just as much ? A term
and its definition may be equated : they are interchangeable, and
wherever one occurs in a proposition you may substitute the other
without detriment to its truth. But you cannot extend that rule
to terms which have any less close relation ; for these you may
be led into error by such substitution or you may not ; the rule
would not be infallible.

We learn from Aristotle himself that other solutions than what
he formulated were offered for some of the fallacies referred by him
to the head of Accident 2 ; and as Poste says 3, ' the fallacy per
accidens has been generally misunderstood.' It has been very
commonly expounded in a way that does not really distinguish it
from the fallacy next to be considered, Secundum Quid. Indeed
what has happened is that the notion of the former has been dropped,
being somewhat ill defined, and the name of the latter, being
somewhat clumsy ; so that what to-day is commonly called Accident
is what the Aristotelian tradition called Secundum Quid. But be-
cause the tradition recognized them as two, a distinction between
the direct and the converse form of the latter fallacy was drawn,
which is really quite unsubstantial.

1 The phrase is from Poste's ed. of Soph. El. {v. p. 73) : cf. esp. his remarks
on p. 158, from which the above interpretation and criticism are borrowed.
Cf. also H. Maier, Die Syllogistik des Aristoteles, 2. Teil, 2. Halfte, pp. 280,
288-291. It should be observed that the subject and predicate, in whose
equation the error lies, are not terms of thought, but terms verbal (cf. supra,
p. 21). No one, e.g., would suppose that for a substance could be substituted
one of its attributes : that being a father could take the place of a dog, and
wag a tail or bite. But a man might suppose that in a proposition one term
could take the place of another, when each is predicable of the same subject-
term : and so might proceed to enunciate a new proposition which seemed aa
if it ought to follow from the others, though what is meant by it was not
implicated in what they mean. Because it is true to say of a certain dog
that he is your dog, and also that he is a spaniel, or property, or a father,
therefore it might seem equally true to say of him that he is your spaniel, or
your property, and that he is your father. Thus this fallacy, though not
traceable to the ambiguity of definite words, is not independent of the part
winch language plays in the conduct and expression of thought.

a Soph. El. xxiv. 3 0p^ ciL p 15s4

xxvn] APPENDIX ON FALLACIES 589

2. The fallacy of Secundum Quid, or — to give the formula in full —
A dictosimpliciter ad dictum secundum quid (from which the argument
a dicto secundum quid ad dictum simpliciter is sometimes distinguished
as its converse1), is one of the subtlest and commonest sources of
error. It consists in using a principle or proposition without regard
to the circumstances which modify its applicability in the case or
kind of case before us.2 Water boils at a temperature of 212° Fah-
renheit ; therefore boiling water will be hot enough to cook an egg
hard in five minutes : but if we argue thus at an altitude of 5,000 feet,
we shall be disappointed ; for the height, through the difference in
the pressure of the air, qualifies the truth of our general principle.
A proposition may be stated simpliciter, or without qualification,
either because the conditions which restrict its truth are unknown,
or because, though known, they are thought seldom to arise, and so are
neglected ; and we may proceed to apply it where, had it been quali-
fied as the truth required, it would be seen to be inapplicable. Per-
haps it holds good normally, or in any circumstances contemplated
by the speaker ; the unfair confutation lies in taking advantage
of his statement to bring under it a case which, had he thought
of it, would have led him to qualify the statement at the outset.
But it is not only in disputation that the fallacy occurs. We are all
of us at times guilty of it ; we argue from principles that hold good
normally, without even settling what conditions constitute the nor-
mal, or satisfying ourselves that they are present in the case about
which we are arguing. Freedom is good, and therefore it is sup-
posed that every community should have free institutions, though
perhaps there are some races only fit for a very moderate degree of

1 No real distinction can be made here. It is sometimes said that in the
direct fallacy we argue from a general rule to a special case, in the converse
from a special case to a general rule. But the former is not fallacious unless
because the rule is applied in a sense in which it is not strictly true ; and the
latter, if it misleads, is erroneous generalization, which is by no means the
converse of Secundum Quid. We may distinguish between applying a rule
to a case to which it is inapplicable because of the presence of conditions
whereby the rule was not qualified, and applying it to one where it is inapplic-
able because of the absence of conditions whereby it was qualified ; but the
latter error is hardly likely to deceive, and if it did, it would do so rather by
suggesting a false statement of particular fact. To argue that because wine
is pernicious its use should be forbidden may be plausible (cf. de Morgan,
Formal Logic, p. 251) ; but to argue that because wine is pernicious in excess,
X ought not to drink it, is hardly plausible, unless it is taken to be meant
that X cannot drink wine without drinking to excess. If that is true, there
is no fallacy ; if it is false, the fallacy is not antithetical to the other.

2 Cf. Dicey, Law and Opinion in England, p. 487, on the extension of prin-
ciples to fresh cases in ' judge-made law' : also Ar. Eth. Nic. e. x. 4, 1137b 14-19.

590 AN INTRODUCTION TO LOGIC [chap.

1 freedom '. A man should be allowed to do what he will with his
own ; and that is often urged as a conclusive argument against any
interference either with his disposition of his property, or his educa-
tion of his children. Paris did nothing wrong in carrying off Helen,
for her father left her free to choose her husband ; but the freedom
allowed her extended only to her first choice, like the authority of her
father.1 There are trivial examples of this as of any other fallacy, as
that if it be maintained that an Ethiopian is black, it is contradictory
to say he has white teeth.2 But there is no fallacy more insidious
than that of treating a statement which in many connexions is not
misleading as if it were true always and without qualification.

3. Ignoratio Elenchi means proving another conclusion than what
is wanted. The name does not literally mean that, but ' ignorance
of confutation '. But the business of any one undertaking to con-
fute a statement is to prove the contradictory ; and if I prove
anything else, I show that I do not know what confutation requires.
Of course every fallacious confutation shows that I am ignorant of,
or ignore, what is required.3 But other fallacies have other defects ;
in this, the argumentation may be perfectly sound, and the sole
defect lie in the fact that the conclusion proved does not confute the
thesis maintained. Or — since it makes no difference whether we
regard a man as undertaking to confute one thesis or to sustain
another contradictory to it — we may say that the fallacy lies in
proving what is not the precise conclusion which we are called upon
to prove. Against a minister who proposes to put a small duty on
corn to-day it is no sufficient answer to prove that the people are much
more prosperous under free trade than in the days when corn stood
at 60 or 80 shillings a quarter ; against a free-trader it is no suffi-
cient answer to prove that foreign nations injure us by their tariffs.
Subterfuges of that kind are however so frequent a resource of the
orator, that it is hardly necessary to illustrate them. Every reader of
Plato's Apology will remember how Socrates refused to appeal to his
judges with tears and entreaties, or to bring his wife and children into
court to excite their commiseration ; for his part was to persuade
them, if he could do it, of his innocence and not of his sufferings.4

Such appeals as Socrates declined to make are sometimes called
argumenta ad misericordiam, arguments addressed to show that a

1 Ar., Rhet. 0. xxiv. 1401b 34, quoted Poste, op. cit. p. 117.

* Soph. El. v. 167a 11.

3 Cf . Soph. El. vi. 168» 17 sq. « Apol. U C, 35 B a

xxvn] APPENDIX ON FALLACIES 601

man is unfortunate and deserves pity, when it ought to be shown
that he is innocent, or has the law on his side. Other favourite
forms of irrelevant conclusion have also received special names.
The best known is the argumentum ad hominem, in which, being called
upon to confute an allegation, I prove something instead about the
person who maintains it. The politician who attacks an opponent's
measures by showing that they are inconsistent with his former
opinions commits this fallacy ; it is the same if I condemn Home
Rule for Ireland on the ground that Parnell was an adulterer. But
the argumentum ad hominem need not be altogether irrelevant.
A barrister who meets the testimony of a hostile witness by proving
that the witness is a notorious thief, though he does less well than
if he could disprove his evidence directly, may reasonably be con-
sidered to have shaken it ; for a man's character bears on his credi-
bility. And sometimes we may be content to prove against those
who attack us, not that our conduct is right, but that it accords
with the principles which they profess or act upon. Christ replied
to those who censured him for healing on the Sabbath, by asking
which of them, if his ox or his ass had fallen into a ditch, would
not pull it out on the Sabbath day.1 Their practice was sufficient to
justify him to them, whatever were the true theory of our duties
on the Sabbath. And Aristotle ansM'ers the Platonists, who held
all vice to be involuntary, by showing that they could not discrimi-
nate in that respect between vice and virtue ; there was no more
reason for calling one involuntary than the other ; virtue, however,
they called voluntary ; and whatever be the true state of the case,
their position at least was not sustainable.2

4. The nature of Petitio Principii is better expressed in the
English name, Begging the Question.3 It consists in assuming

1 Luke xiv. 1-6.

2 Eth. Nic. y. vii. 11 14a 31-b25. In the game of disputation, we may be
held to score a victory if we force an opponent to an admission inconsistent
with the thesis he propounded. But in the search for truth, to convict any one
of inconsistency is irrelevant ; we have to determine what is true.

3 Gk. to iv apxff Xapftaveiv, to «'£ dpxns alreicrdai, to assume or claim the
admission of the very thing propounded for debate at the outset — the Tvp6^~\r^p.a.
The word petitio belongs to the terminology of disputation, where the ques-
tioner sought his premisses in the admissions of the respondent. He had no
right to ask the respondent to admit the direct contradictory of his thesis ;
let the thesis, for instance, be that the Pope cannot remit the temporal punish-
ment of sin in Purgatory : the opponent may not ask the respondent to admit
that he can. If by some verbal disguise he gets the respondent to admit it,
it is only a sophistical confutation ; the respondent did not see what he waa
granting, and would have refused to grant it if he had seen — not because it

592 AN INTRODUCTION TO LOGIC [chap.

what is to be proved, in order to prove it. To do this within the
compass of a single syllogism — assuming in the premisses the very
thing to be proved, and not merely some thing which depends on
that for its proof — is only possible by the use of synonyms. If
I argue that C is A because B is A and C is B, and if the middle
term B is identical either with the major or the minor, then I use
the proposition to prove itself ; for let B be the same as A : then,
by substituting A for B in the minor premiss, I get ' C is A ' as
a premiss ; or let B be the same as C : then by substituting C for
B in the major premiss, I again get ' G is A ' as a premiss ; and in
either case therefore the conclusion is among the premisses. Thus
let the syllogism be that to give to beggars is right, because it is a
duty to be charitable ; so far as charity is taken to include giving
to beggars, we have no business to assume that it is a duty ; for
the question whether it is a duty and the question whether it ia
right are the same question : to call it a duty is to call it right.
Here the major premiss, that duty is right, is a tautology, and the
minor contains the petitio. On the other hand, if I defend legacy
duties by saying that property passing by will ought to be taxed,
I beg the question in the major ; for a legacy duty is a tax on pro-
perty passing by will, and to say that such property should be taxed
is only to assert in other words the justice of a legacy duty.1

But the fallacy is generally committed less abruptly. The premiss
unduly assumed is generally not the conclusion itself differently
expressed, but something which can only be proved by means of
the conclusion ; and arguing thus is often called arguing in a circle.
If I argued that early Teutonic societies were originally held together
by kinship, because all societies were so held together originally,2

led to the contradictory of his thesis, for a man is often f airty refuted by showing
that he cannot reasonably deny something which does that : but because it
was the contradictory of it. It is quite fair to try to get a man to admit
a general principle, and then to show that his thesis is inconsistent with it,
provided that the general principle does not really require the disproof of his
thesis in order to its own establishment. Hence the term principium is a mis-
translation. The fallacy lies in begging for the admission not of a principle
to be applied to the determination of the matter, but of the very matter, in
question. As occurring in a book or speech, where a man puts forward his
own premisses, and has not to get them by the admission of a respondent, it
consists in assuming among the premisses either the conclusion itself which
a show is made of proving, or something more or less directly depending
thereon. Cf. Mansel's ed. of Aldrich's Artis Logicae Rudimenta, App. E.

1 It is also possible to beg the question when the conclusion is negative,
but then only in the major premiss ; and to beg it in other figures than the
first (for details see Poste, Soph. EL, App. A). Cf. also supra, p. 578, n. 1.

1 For the general statement see Sir Henry Maine, Early Institutions, p. 64.

xxvn] APPENDIX ON FALLACIES 593

I might be accused of arguing in a circle ; for the major premiss,
it might be said, is only arrived at by enumeration ; early Teutonic
societies have to be examined in order to show that it is true. Of
course to show that the generalization was not enumerative would
be to rebut the accusation ; but, as we saw in discussing the view
that all syllogism is petitio principii, every syllogism whose major
premiss is an enumerative judgement is so.1 The circle is fairly
manifest in such cases ; but in others it may often escape the notice
of its author. ' There are certain people ', says Dr. McTaggart,2
1 who look on all punishment as essentially degrading. They do not,
in their saner moods, deny that there may be cases in which it is
necessary. But they think, if any one requires punishment, he
proves himself to be uninfluenced by moral motives, and only to be
governed by fear. . . . They look on all punishment as implying
deep degradation in some one, — if it is justified, the offender must
be little better than a brute ; if it is not justified, the brutality is in
the person who inflicts it. This reasoning appears to travel in a
circle. Punishment, they say, is degrading, therefore it can work no
moral improvement. But this begs the question. For if punish-
ment could work a moral improvement, it would not degrade but
elevate. The humanitarian argument alternately proves that
punishment can only intimidate because it is brutalizing, and that it
is brutalizing because it can only intimidate.' Romanes finds an
example of petitio in an argument of Huxley's, adduced to show that
all specific characters are adaptive.3 ' Every variety which is
selected into a species is favoured and preserved in consequence of
being, in some one or more respects, better adapted to its surround-
ings than its rivals. In other words, every species which exists,
exists in virtue of adaptation, and whatever accounts for that adap-
tation accounts for the existence of the species.' Here the fallacy
lies in substituting, for ' every variety which is selected ', ' every
species which exists ' ; the statement in the first clause is true for
every variety which is selected, since selection means the survival of
those best adapted to the conditions of life. But the question is
whether every species which exists has originated by ' selection \
One more instance may be cited, from a work on the squaring of the

1 pp. 304, 305, supra.

2 Studies in Hegelian Cosmology, § 142. By punishment here is meant ' the
infliction of pain on a person because he has done wrong ' (§ 137). And it ia
of corporal punishment that we most often hear this view expressed.

* Darwin and after Darwin, ii. 307.

1779 Q q

594 AN INTRODUCTION TO LOGIC [chap.

circle, called The Nut to Crack, by James Smith.1 Smith held the
ratio of circumference to diameter to be 3J, and proved it thus :
1 1 think you will not dare to dispute my right to this hypothesis,
when I can prove by means of it that every other value of it will
lead to the grossest absurdities ; unless indeed you are prepared to
dispute the right of Euclid to adopt a false line hypothetically, for
the purpose of a reductio ad absurdum demonstration, in pure
geometry.' That is, he argued first that if 3| be the right ratio,
all other ratios are wrong ; and then, that because all other ratios
are wrong, 3| is the right ratio. And he conceived that he had
established his conclusion by a reductio ad absurdum — by showing that
the denial of his thesis led to absurdity. But the absurdity, in such
an argument, ought to be ascertained independently, whereas here it
rests upon the assumption of the truth of what it is used to prove.

5. The fallacy of False Cause is incident to the reductio ad ab-
surdum. That argument disproves a thesis by showing that the
assumption of its truth leads to absurd or impossible consequences,
or proves one by showing the same for the assumption of its falsity.2
In False Cause, the thesis alleged to be discredited is not really
responsible for the absurd or impossible consequences, which would
follow equally from the other premisses, whether that were affirmed
or denied. ' It is ridiculous'to suppose that the world can be flat ;
for a flat world would be infinite, and an infinite world could not be
circumnavigated, as this has been.' Here the supposition incon-
sistent with the fact of the circumnavigation of the world is not that
the world is flat, but that it is infinite ; it might be flat and still
circumnavigable, if it were finite ; the thesis of its flatness is there-
fore unfairly discredited.

From a passage in the Prior Analytics it would seem that Aris-
totle regarded this fallacy as of frequent occurrence.3 But the fact
that later writers have largely given a different meaning to the name
suggests that it is not really a prominent type. It is often iden-
tified with the fallacy Post hoc, ergo propter hoc: i.e., supposing
that one event is due to another, merely because it occurred after

1 Cf. de Morgan, Budget of Paradoxes, p. 327.

2 James Smith argued, not that ' if A is false, B will be true : but B is
false, /. A is true ' ; but ' if A is true, B will be false— (as to which nothing
was known) — .-. A is true '.

8 Anal.t Pri. 0. xvii. 65a 38 ri 81 fit, napa tovto vvpfrdvuv to ifrtvSos, t
-ro\Xd(<tr iv ro'it \6yoit fla>$(ifj.(v Xtyav, kt\., ' that the false statement does
not arise from the premiss alleged, as we are accustomed often to say in argu-
ment, &c.' Cf. Poste's Soph. EL, App. B, on this passage.

xxvn] APPENDIX ON FALLACIES 595

it ; as the countryman is said to have declared that the building of
Tenterden Steeple was the cause of Goodwin Sands, because the
sands only appeared after the steeple was built. Such, as Bacon
truly says, is the origin of almost every superstition — of men's
astrological fancies, and their fancies about omens or dreams. The
story which he quotes may well be repeated in his own words.
' Itaque recte respondit ille, qui, cum suspensa tabula in templo ei
monstraretur eorum qui vota solverant, quod naufragii periculo
elapsi sint, atque interrogando premeretur, anne turn quidem
deorum numen agnosceret, quaesivit denuo, At ubi sunt Mi depicti
qui post vota nuncupata perierint ? ' *

Inferences of this kind are undoubtedly both frequent and falla-
cious ; and Post hoc, propter hoc is a definite type or locus of fallacy.
That is, it is a general or dialectical principle — a principle applicable
in divers sciences, and not exclusively appropriate in one : and
it is a false principle, the application of which is as likely to lead
to error as to truth. Nor is it peculiar to this fallacy, that it can be
expressed as a false principle. Equivocation proceeds on the false
principle that a word is always used with the same meaning :
Accident, on the principle that a term and its predicate are inter-
changeable : Secundum Quid, on the principle that what is true
with certain qualifications is also true without them. And the fact
that these different types of fallacious inference severally depend on
a false, or misleading, principle is what was meant by calling them
loci of fallacy.2 But the locus Post hoc, propter hoc is not quite the
same as that of Non causa pro causa : in other words, the type is
a little different. In False Cause we are dealing with the logical
sequence of premisses and conclusion ; the fallacy lies in connecting

1 Nov. Org. I. 46. Bacon cites the story in illustration of one of the ' Idola
Tribus ', the tendency to overlook or despise facts which do not agree with an
opinion which we have once adopted. J. S. Mill would call this the fallacy of
Non-observation (System of Logic, V. iv). The meaning Post hoc, propter hoc
does occur in Aristotle, Rhet. 0. xxiv. 1401 b 29-34 a\\os napa to dvalriov <ur
airiov, aiov tg> ap.a rj fxera tovto ytyovtvaC to yap pera tovto ws 8ui tovto
\ap.(Sdvovo~i, Ka\ p.a\io~Ta ol iv rats iroXireiais, oiov las 6 Arjuadrjs T>]V Arjpo-
o~devovs noXiTfinv irnvrcov rav kokcov alriav' per iKeivrjv yap o~vvtf5r) 6 noXffios
(' Another locus of fallacy is through taking what is not the cause as cause,
because one thing has happened together with or after the other ; for what
arises after something is taken as arising through it, especially in political
argument, as Demades for example said that the policy of Demosthenes was
the cause of all their ills ; for after it came the war ').

2 The Sophistici Elenchi is the concluding book of Aristotle's Topics. The
false principle is exemplified in the fallacious argument; it is not one of the
premisses of the argument.

Q q2

596 AN INTRODUCTION TO LOGIC [chap.

the conclusion with a particular premiss which might, so far as
getting the conclusion is concerned, have been equally well included
or omitted ; and because the conclusion is false, we erroneously
infer this premiss to be false also. In Post hoc, ergo propter hoc we
are dealing with the temporal relation of cause and effect ; the fal-
lacy lies in connecting the effect with a particular event which might
equally well have happened or not happened, so far as the effect in
question is concerned ; and we erroneously suppose that the effect,
which did occur, occurred because of that event. But if any one
likes to use the name False Cause as equivalent to Post hoc, propter
hoc, there is not much harm done ; for the fallacy which in the
Sophistici Elenchi Aristotle describes under the name is not one
that we have much occasion to speak of.

6. It is otherwise with the fallacy of the Consequent, which some
modern writers have also misunderstood.1 For this is one of the very
commonest, and we have already had occasion to notice it in dis-
cussing inductive reasoning.2 It consists in supposing that a con-
dition and its consequent are convertible : that you may argue
from the consequent to the condition, no less than vice versa. If
a religion can elevate the soul, it can survive persecution : hence it
is argued that because it has survived persecution, such and such
a religion must elevate the soul ; or perhaps (for we may follow
Aristotle 3 in including under the name both the forms of fallacy

1 e.g. de Morgan, Formal Logic, p. 267; J evons, Elementary Lessons in Logic,
p. 181.

2 p. 523, supra.

8 Cf. Soph. El. xxviii. 181a 27 nap' b Kai 6 tov MeXiWov Xoyos- ft yap to
ytyovbs e^ei dp^rjv, to ayei>t]Tov d£ioi urj e^ftj/, coar el dyevr)TOS 6 ovpavtk, km.
anupos. to 8' ovk sariv' dvdiraXiv yap f) aKoXovdrjais ('with this accords the
argument of Melissus ; for he thinks that if what is generated has a beginning,
what is ungenerated has not ; so that if the heaven is ungenerated, it is also
infinite. But this is not so ; for the sequence is the other way ') ; i. e. from
' A is B ' you cannot infer • not- J. is not-iJ ', but only contrariwise, ' not-ZJ is
pot-%4 '. It appears by the same chapter that Aristotle would bring the
illicit, viz. simple, conversion of an universal affirmative judgement under
the same heading. This illustrates the close parallelism between the modi
ponens and tollens in hypothetical, and Barbara and Camestres in syllogistio
reasoning (cf. pp. 339-342, supra). But that Aristotle did not identify them
might perhaps be inferred from the fact that he does not include Undistributed
Middle and Illicit Process of the Major in his list of sophistical confutations,
while he does include, under the name of the fallacy of the Consequent,
the corresponding though not identical errors which may be committed in
hypothetical reasoning. It may be noted that such inferences would only
not be fallacious where condition and consequent reciprocated — a relation
which corresponds to that of commensurate terms in an universal affirmative
judgement. Hence Aristotle says that the fallacy of the Consequent is

xxvn] APPENDIX ON FALLACIES 597

to which hypothetical reasoning is liable) that because it is in-
capable of elevating the soul, it will succumb to persecution. Such
fallacies are committed whenever a theory is assumed to be true for
no better reason than that the facts exist, which should exist if it
were true — i. e. whenever verification is mistaken for proof x ; and
whenever the refutation of an argument advanced in support of a
theory is supposed by itself to be fatal to the theory. If it can be
shown that no other theory accounts for the facts, or that no other
argument can be advanced in support of the theory, then the
matter is different ; but without some reason to believe this, such
inferences are worth nothing. Nevertheless, they are inferences
which we are all very apt to make.2

7. There remains lastly the fallacy of Many Questions. This
consists in putting questions in such a form that any single answer
involves more than one admission. If one admission be true and
another false, and the respondent is pressed for a single answer,
he is exposed to the risk of confutation, whatever answer he makes.
' The execution of Mary Queen of Scots was brutal and sacrilegious
— was it, or was it not ? ' If it was brutal but not sacrilegious, what
is a man to answer ? He will be accused by saying no of denying
the brutality, by saying yes of affirming the sacrilege. Sometimes,
instead of submitting two problems for decision together, the ques-
tion appears to submit only one ; but that is one which would not
arise except on the assumption of a certain answer to another : and
so the respondent again cannot answer it without committing him-
self to more than he intended, or on a matter which has not been
definitely submitted to him. Of this sort is the famous enquiry,
* Have you left off beating your mother ? ', as well as any question
that asks for the reason of what has not been admitted to be true.
It is often recounted how Charles II asked the members of the Royal
Society why a live fish placed in a bowl already full of water did not
cause it to overflow, whereas a dead fish did so ; and how they gave
various ingenious reasons for a difference which did not exist. If

a case of that of Accident (Soph. El. vi. 168b 27). Under it in turn might be
brought Post hoc, propter hoc. If Goodwin Sands were caused by building
Tenterden Steeple, they would have appeared, as they did, so soon as the
steeple was built ; but they might equally have done so, if the building of
the steeple had nothing to do with their appearance.

1 Cf. p. 523, supra.

2 This fallacy is ' logical ', or formal ; it can be expressed in symbols. So
can an argument in a circle sometimes be ; e. g. if it is of the form ' A is B,
B is C ,\ A is C : and B is C because A is C and B is A \

598 AN INTRODUCTION TO LOGIC [chap.

one were to enquire why a protective system encourages the industry
of the country which adopts it, or how dowsers are made aware by
their feelings of the presence of subterranean waters, the fallacy
would be the same. It may be said that a respondent is always able
to give an answer which will save him from any misconstruction ;
to the question ' Have you left off beating your mother ? ' the
answer ' no ' might seem to be an admission of the practice ; but
why should not a man reply ' I never began it ' ? To this it may be
rejoined, first, that in the old disputations, and in some situations,
such as the witness-box, to-day, a man might be more or less precluded
from ' explaining himself ', and required to give a ' plain answer ' to
a question which does not admit of it. With the use of the fallacy
under this sort of duress may be compared the custom of ' tacking '
in the American legislature. The President of the United States
can veto bills, and does veto them freely ; but he can only veto
a bill as a whole. It is therefore not uncommon for the legislature
to tack on to a bill which the President feels bound to let pass a
clause containing a measure to which it is known that he objects ;
so that if he assents, he allows what he disapproves of, and if he
dissents, he disallows what he approves.1 But secondly, even where
no unfair duress is employed, the practice of presupposing a certain
answer to one question in the form of putting another throws the
respondent off his guard, and makes him apt to admit without con-
sidering it what, if it had been explicitly submitted to his considera-
tion, he might have doubted or denied.

The fallacy therefore is not a trivial one ; such questions are a
real source of error, when we put them to ourselves : of unfair con-
futation, when we put them to others. But it is doubtful whether
it is a fallacy extra dictionem. For when a question is so put as
that it must be answered by yes or no, and misconception is unavoid-
able on either answer, the error arises from the way in which the
question is worded ; and the same may be said of the acquiescence
in false assumptions, into which in other cases we are entrapped.

The foregoing remarks have been directed to explain what are the
types of fallacy which have been traditionally distinguished; and are
still many of them very commonly referred to by their traditional
names. The types are not all equally distinct, frequent, or important ;
but the original meaning of each name has been given as far as pos-

1 Bryce's American Commonwealth, Part I, c. xx : vol. i, p. 214

xxvn] APPENDIX ON FALLACIES 599

sible, because nothing but misunderstanding can result when different
writers employ such terminology each in his own meaning, and
there did not for the most part seem sufficient reason to prefer any
later interpretation for a standard. In a few cases later interpreta-
tions which have much to be said for them have been given as well.
No doubt Fallacy is a subject on which successive generations to some
extent need new treatises : not because the principles change, but
because the fields change in which they are most prolific. Many sug-
gestive illustrations of the dominion which fallacy holds in important
subjects of modern thought may be found in the pages of Whately,
Mill, or de Morgan, to which reference has already several times been
made.

INDEX

Abscissio Infiniti, 127.

Absolute terms, 39-40, 156 : A., the,
173.

Abstraction, two senses of, 34 : in
scientific investigation, 477.

Accent, fallacy of, 583.

Accident, as a Head of Predicables,
75-81 : = coincident, 76 : separable
and inseparable a., 108-10 : essential
a., 211. Fallacy of a., 587-8 (cf. 574
n. 1), 595.

Accidental judgements, 210-11.

Aldrich, H., Artis Logicae Rudimenta, ed.
Mansel, cited, on definition of Mo-
dality, 201 n. 1 : on form of Barbara
Celarent, 285 n. 2 : on form of Dictum
de Omni et Nullo, 297 n., 352 n. 4, 580
n. 1 : cf. s. v. ' Mansel '.

Amphiboly, fallacy of, 580.

Ampliative judgements, 208 n. 2, 211.

Analogy, argument from, 532-42 :
original sense of, 532 : origin of
modern sense of, 537 : in Aristotle,
379 n. 1 : no proof by, 538 : relation
to Induction by Simple Enumeration,
540 : involves a general principle,
541-2.

Analysis, in Induction, 459-64 (cf.
492-3), 499.

Analytic judgements, 207 sq. : not =
verbal j., 212 : a. judgements of sense,
213.

Antisthenes, on Judgement, 22 n. 1, 27.

Apelt, Professor O., on Aristotle's
Categories, cited, 52 n. 1.

Apodeictio judgements, 188, 192-6.

A posteriori knowledge, 210 n. 2 :
reasoning, 436-7.

Appellatio, in names, 157 n. 1.

A priori knowledge, 210 n. 2 : reason-
ing, 437 : synthetic judgements a
priori, 210.

Arbor Porphyriana, 130.

Arguing in a circle, 592.

Argumentum ad hominem, 591, 574
n. 1 : ad misericordiam, 590.

Aristotle, his definition of a term,
18 n. 1 : on bpwvvp\a and avviuvv\ia, 31,
47 : on irapwvvua, 133 n. 1 : on ovopia
dopio-Tov, 42, 43 n. 1 : on the Cate-
gories, 48 sq. ; his doctrine contrasted
with Kant's, 61-5; being not a signi-
ficant term, 44 n. 2, 50, 65, 166 : his
conception of Matter, 54-5 ; of genus

as the matter of its species, 88 : on
the Four Elements, 100, 119 : on the
Predicables, c. iv passim : on Defini-
tion, 129 (cf. s.v. 'Definition'); on
constructing Definitions, 129-31 : on
Extension and Intension, 136 : on the
subject in judgement, 167 : on the
contingent, 198 n. 1 : on modal dis-
tinctions, 206 n. 1 : on the quantifica-
tion of the predicate in judgement,
224 n. 1 : on contrariety, 229 n. 1 :
on sub-contrariety, 231 n. 1 : his
definition of Syllogism, 249 ; on the
indirect moods (= the so-called fourth
figure) of s., 258-9,281-2; distinction
of a. Ttkftos and a. 5war6s, 287 ; sup-
posed source of the Dictum de Omni et
Nullo in, 296 n. 1 ; on modal s., 452 n.
2: on Demonstration, 305, 311, 524:
on hypothetical reasoning and the
avWoyiafios i£ vnodiaews, 344 n. : on
Enthymeme, 350 n. 1 : on Sorites,
354 n. 3 : on the inductive syllogism,
378-80 ; on the meaning of (irayuyr)
in Ar., 378 n. 2, 379 n. 1 ; on Induc-
tion in the modern sense (cf. s. v. l Dia-
lectical Reasoning'), 387-91 ; on itiiai
and Koival dpxai, 387-9 (cf. s. ».' Prin-
ciples'); on -napaStiy/xa (= argument
from analogy), 379 n. 1, 540-1 : on
fallacies, c. xxvii passim ; his division
of fallacies, 574-5, 577-9 : his logical
writings, 375-6 : reasons for his ascen-
dancy in the Middle Ages, 374 : his
distinction of formal, material, final
and efficient causes, 488 : his theory
of motion, 514. Cf. also 142, 176 n. 1,
193 n. 2, 194, 211 n. 1, 237 n. 1, 248,
256, 306 n. 1, 313 n. 1, 315 n. 1,
320 n. 1, 326 n. 1, 341 n. 2, 372, 379
n. 1, 381 n. 1, 382 n. 4, 386-8 nn.,
393 n. 1, 399, 408 n. 2, 443 n. 1, 529,
549 n. 1.

Assertoric judgements, 188, 191-2 : a.
propositions, 196.

Association of ideas, 542.

Attributive terms, 28, 36-8.

Augmentative judgements, 208, 211.

Austin, J., Jurisprudence, quoted, 579 n.2.

Averroes, on Fig. 4 of syllogism, 283-4.

Bacon, Francis, Lord Verulam, quoted,
271 n. 1, 370, 373, 375, 392-4 (his ex-
position of Induction), 429, 433 n. 1,

INDEX

601

438 n. 1, 441, 461, 465, 470 n. 1, 527,
528, 565 n. 1, 566 n. 1, 585, 595.

Bagahot, W., Physics and Politics, cited,
374 n. 1.

Bain, Alexander, cited, 137 n. 3, 191,
297 n., 322 n. 2.

Balfour, Rt. Hon. A. J., quoted, 506.

Barbara Celarent, history of, 267 n. 1 :
first form of in Petrus Hispanus, 269 :
in Aldrich, &c, 285 and n. 2.

Begging the question, 591-4.

Boetius, Isagoge, 66 n. 2, 348 n. 1.

Bosanquet, Dr. B., cited, 11 n. 1, 140
n. 1, 150 n. 3, 169 n. 1, 179 n. 1,
199 n. 1, 214 nn. 1, 2, 441 n. 1, 452:
on the nature of Induction, 524-7.

Bradley, Mr. F. H., cited, 10 n. 2, 35, 37
n. 3, 41 n. 1, 68 n. 1, 140 n. 1, 146 n. 2,
161 n. 3, 164 n. 2, 169 n. 1, 177 n. 1,
178 n. 1, 179 n. 1, 185, 194, 213 n. 1,
229 n. 1, 232 n. 1, 240 n. 2, 243 n. 1,
249 n. 3, 251, 252, 395 n. 1, 441 n. 1,
541 n. 2, 554.

Bryce, Viscount, Studies in History and
Jurisprudence, cited, 556 n. 1 : American
Commonwealth, cited, 598 n. 1.

Buridanus, Joannes, on nomina con-
notativa, 157 : on Modality, 206 n. 1.

Categorematic words, 19.

Categorical judgement, 181, 183-4.

Categories, Aristotle's doctrine of, c.
iii : its relation to that of Kant, 61-5

Causation, Law of, c. xix pass. : dis-
tinguished from 'laws of causation',
402 ; these to be discovered, ib. : not
= uniformity of sequence, 404 : how
involving uniformity, 402-21 : as-
sumed in physical science, 401, 544 :
cannot be proved inductively, 421-4 :
grounds of elimination furnished by,
439, 494-7.

Cause, meaning of, 401, 406 : Aristotle's
doctrine of, 488 n. 1 : c. a thing, not
an event, 404, 426 : the relation not
perceived, 428 : non-reciprocating,
c. xxii : involves time, 420-1 : how
related in time to effect, 421 n. 1 :
continuity of with effect, 487 : dis-
continuous causal relations, 488-90 :
plurality of causes, 491-2 : composi-
tion of causes, 491 n. 1, 519 n. 1 :
problem of, how related to doctrine of
the Predicables, 78-81.

Certainty in science, why hard to ob-
tain, 503-5.

Chance, what, 78 (cf. 201-4).

Change, implies something permanent,
13, 487 : continuity of, ib.

Class, nature of a, 83-5, 175 n. 1 : class-
relations the main interest of Symbolic

Logic, 12 n. 2 (cf. 552 n. 8): class-
concept, what, 175 n. 1.

Classification, 115 sq. : relation of to
Logical Division, 115-16, 134: to
Definition, 129 : relation of extension
and intension of terms in, 137 sq. :
c. by type, 103 : compromises in, 1 33-5,
139 n. 1 : varies with different pur-
poses, 459-61.

Coefficient of correlation, 476.

Collective terms, 38 : c. judgements,
177.

Colligation of facts, 469 sq.

Commensurate terms, 72.

Composition of causes, 491 n. 1, 519
n. 1 : fallacy of c, 581.

Concept, its nature, 22-6, 69-71 : alone
definable, 82.

Conceptualism, 25-6, 82.

Concomitant variations, use of in
Induction, 500, 557-62.

Conditional arguments (■= hypothe-
tical and disjunctive), 348 n. 1 : c.
principles in science, 414-20.

Conjunctive ( - hypothetical) proposi-
tions and arguments, 348 n. 1.

Connotation of terms, 147, 150: Mill's
use of the word, 149, 153 : whether
belonging to proper names, 150-3 :
in infimae species of abstracts, 153-4 :
history of word connotare, 156-8.

Consequent, fallacy of the, 596-7, 338-
9, 574 n. 1, 577 n. 3: involved in
taking verification for proof, 523.

Contradiction, Law of, 13, 46, 209,
211.

Contradictory propositions, 229 ;
terms, ib. n. 1.

Contraposition of propositions, 238-40.

Contrary propositions, 229 : terms, ib.
n. 1.

Conversion of propositions, 232-7 : c.
by negation, 238 : inference whether
present in c, 240-4 : conversio syllogismi,
291.

Copula, in a proposition, 161-3 : sig-
nifies existence, 163-6.

Crackenthorpe, K., Logicae Libri Quin-
que, cited, 47, 107 n. 2, 284 n. 2, 296
n. 1, 580 n. 1, 583 n. 1.

Crucial instance, 527, 565 n. 1.

Darwin, C, Origin of Species, quoted,

465-6, 532 n. 1.
Dedekind, R., Was sind und was sollen

die Zahlen, 552 n. 3.
Deduction, meaning of the word, 378

n. 1 : contrasted with Induction,

396-8, 545.
Definition, 72, 86 sq. : of properties

112 n. I : nominal d., 91 sq. : analysis

602

INDEX

of d. into genus and differentia, 82-3 :
no d. of individuals, 81 : a d. not
properly a judgement, 15 n. 1 : diffi-
culties in, 92 sq. : Locke's view of,
92 : rules of, 111-15 : connexion with
Logical Division and Classification,
116: negative differentiae in, 113-14,
122, 131 n. 1 : by dichotomy, 126-31 :
how far arbitrary, 211-13.

Democritus, on space, 172 n. 1.

Demonstration, nature of, 261 n. 1 (cf.
371), 524 : not syllogistic, 308-11, 524 :
contrasted with Dialectic, 398-9.

De Morgan, A., Formal Logic, cited,
44 n. 2, 569 n. 2, 574 n. 1, 577, 585,
589 n. 1, 596 n. 1, 599: Budget of
Paradoxes, cited, 594 n. 1.

Denotation of terms, 146 sq.

Descartes, B., cited, 26, 528.

Designations, 30, 32.

Development, meaning of, 89 (cf. 412).

Dialectical reasoning, 387-9 : opposed
to demonstration, 398-9 : fallacies
incident to, 686-7.

Dicey, Professor A., Law and Opinion in
England, cited, 589 n. 2.

Dichotomy, 121 sq.

Dictum de Omni et Nullo,50 n. 1, 296
sq. : supposed Aristotelian authority
for, 296 n. 1 : interpretation of, 302-10 :
not a premiss of argument, 311-13.

Difference, Mill's ' method of, 433-4 :
not unaffected by 'plurality of causes',
497-9 : how far superior in cogency
to his other ' methods ', 498-9.

Differentia, 74, 82 sq. : constitutive
and divisive d., 180: diagnostic d.,
131 (cf. 112).

Dilemma, 357-65 : destructive d. may
be simple, 360-1.

Disjunctive proposition, 181-8, 346 :
d. reasoning, 344-9 : in induction,
394, 431, 442.

Distribution of terms, 216-21 : in
syllogism, 270-7.

Diversity of effects, 492 sq.

Division, Logical, 115-17, 134: rulesof,
117-21 : basis of, 118, 131 : does not
proceed to individuals, 131-2 : cross-d.,
119 : Physical D. (= Partition), 132 :
Metaphysical D., 132-3 : fallacy of D.,
581.

Downam, G., Comment, in Petr. Rami
Dialectica, cited, 354 n. 4.

Elimination, its place in induction,
430, 524-7 ; grounds of, 439-43 : how
affected by non-reciprocation of cause
and effect, 492-501 : incomplete in
induction by simple enumeration, 531.

Empedocles, 100.

Empirical facts, 883 : e. generalization,
530.

Empiricism, what, 193-4 (cf. 384 n. 1).

Enthymeme, 350-3 : Aristotelian sense
of, 350 n. 1.

Enumerative judgements, 177-8.

Epicheirema, 352.

Episyllogism, 352.

Equipollency of propositions, 237 n. 1,
240.

Equivocation, fallacy of, 579, 583 n. 1,
595.

Essence, 91 sq., Ill : in geometry,
96 sq. : nominal e., 92-4, 303.

Essential judgements, 210-14.

Euler's diagrams, why misleading in
syllogism, 272-3.

Exceptive propositions, 215.

Excluded Middle, Law of, 13, 41 n. 1.

Exclusiva, Bacon's, 392-4.

Exclusive propositions, 215.

Existential import of propositions,
242 n. 1.

Experiment, importance of to induc-
tive enquiry, 476, 493-4 : its relation
to the • method of difference ', 498-9 :
in some enquiries impossible, 555.

Explanation, c. xxiii : its nature, 502 :
contrasted with Dialectic, 398 : not
possible from ' common principles ',
503 : of particular facts and of laws,
the same in kind, 502, 507 : seldom
syllogistic, 507 : as subsumption, 512 :
deductive and inductive reasoning in,
512-21 : examples of, 508-21.

Explicative judgements, 208, 211

Exponible propositions, 215.

Exposition (e«0e<7»s), 320, 327.

Extension of terms, 136 : constituted
by kinds, not individuals, 143-6 : how
distinguished from Denotation, 146 :
some terms without e., 143, 147.

Fallacies, c. xxvii : reasons for dis-
cussing in a treatise on Logic, 567-9 :
difficulty of classifying, 569-73 : defi-
nition of, 566, 577 : in dictione and
extra dictionem, 574-5, 577 n. 3, 578 :
logical and material, 575, 577 n. 3.

False Cause, fallacy of, 594-6.

Figure of Speech, fallacy of, 584.

Figure of syllogism, what, 257-9 :
determination of moods of first f.,
264-9 : do. of the several f., and their
rules, 277-80, 286 : the first f. why
called scientific, 306, 316 (cf. 398-9) :
the third why inductive, 319 : distinc-
tive character of the three f., 330-1 :
Galenian, or fourth, f., 280-5, 325-30 ;
erroneously added, 262 j reduction of
moods of, 290.

INDEX

603

Form and Matter, distinction of, 54-7,
89-91 : in thought, 5 n. 1, 237 : in
judgement, 163, 170 : in argument,
c. xvii, 575.

Forms, Bacon's doctrine of, 392-3.

Fowler, T., his use of symbols in
representing inductive arguments,
441 n. 1.

Freewill, 198.

Fry, Sir Edward, quoted, 364 n. 2.

FundamentumDivisionis, 86, 118,131.

Galen, his (fourth) figure of syllogism,
259, 262, 283.

Genus, 73, 81 sq. : should be an unity,
82-8 : distinguished from class, 83-6 :
the vKtj of its species, 88 (cf. 121 n. 1):
summum, subalternum, and proximum g.,
116.

Geometry, distinction of essence and
property in, 96 sq. : grounds of cer-
tainty in, ib. (cf. 135): use of diagrams
in, 333. Cf. s. v. < Mathematics '.

Goclenius, B,., 354 n. 3.

Godwin, W., Political Justice, cited,
559 n. 1.

Groto, J., Exploratio Philosophica, cited,
68 n. 2.

Grounds of a judgement, what, 205-6.

Hamilton, Sir "William, cited, 137 n. 3,
222, 268 n. , 285 n. 2, 348 n. 1, 354 n. 3.

Heath, Sir Thomas, Elements of Euclid,
cited, 572 n. 1.

Hegel, G. W. F., cited, 11 n. 1, 60,
308 n. 1, 584 n. 1.

Herschell, Sir John, cited, 394-5.

Hippocrates, his attempt to square the
circle, 572.

Historical method, the, 562-4.

History and Science contrasted, 77,
102, 242, 467-8, 508-10.

Hobbes, Thomas, his definition of a
name, 20, 150 n. 1 : thought all infer-
ence syllogism, 252 n. 1 : Nominalism
of, 300.

Hume, David, on causation, 404 : on
rules for detecting causal relations,
895, 440 n. 1, 442 : on the nature of
virtue and vice, 534.

Hypothesis, its place in induction,
464-75 : not to be restricted by Logic,
466 : varies in difficulty and range,
467-8.

Hypothetical judgement, 181-6, 196 :
h. reasoning, 335-9, 348 n. 1 : do., not
reducible to syllogism, 839-44 : h.
necessity, 195.

Idea, meanings of the word, 16 n. 2.
Identity, Law of, 13, 22, 408, 420-1.

Ignoratio Blenchi, 590-1, 570, 576.

Illicit process of major or minor
term, 274-5, 576.

Immediate inference, meaning of
term, 232: processes of, 232-40:
whether real inference, 240-7.

Individual substances, 'Zii, 54-7 : not
in any category, 52 n. 1 : not defin-
able, 81 : whether the ultimate sub-
jects of judgement, 167-8.

Individuality, not displayed in inor-
ganic things, 102.

Individuation. Principle of, 54-7, 81-
2, 90-4, 132, 145.

Induction, c. xviii pass. : meaning of
word, 378 ; confusion in use of, 395 :
Perfect I. ( = I. by Complete Enumera-
tion), 380, 504 ; do. in Mathematics,
543 : confusion in contrast of Perfect
with Imperfect I., 504 : the I. of the
inductive sciences assumes universal
connexions in nature, 401 : deductive
reasoning often involved in, 513-23,
cf. 446 n., 471 : disjunctive reasoning
in, 394, 441-3 (cf. 444-57), 524-7 : how
contrasted with Deduction, 397-8 :
its nature, 399, 401, 430, 435-6:
examples of, 444-57, 462-4, 471-5,
479-82, 489-90 (cf. 77 n. 1) : con-
clusions of I. not intelligible, 436-9
(cf. 205, 505) : other operations than
reasoning in, c. xxi : applicable to
discovery of laws in the same way
as of causes, 483 : I. by Simple
Enumeration, 529-32 ; its relation to
argument from Analogy, 540-1 ; I. in
the third figure of syllogism, 319-25
(cf. 379) : Mathematical I., 543-4 ;
its relation to the I. of the inductive
sciences, 544-9: 'quantitative I.',
557 : theory of I. not a ' Logic of
Truth ', 254-5 : Aristotle's conception
of I., 380-7: Bacon's, 391-4.

Inductive Methods, Mill's so-called,
430-5, 497-9: their basis, 440 n. 1.
Cf. s. v. < Mill, John Stuart '.

Inference, what, 232, 240-1 : all i.
self-evident, 193-4 (cf. 264), 312 n. 1,
549-52 : not all syllogistic, 252, 294-5,
366 : a priori and a posteriori i., 436-9,
556 : i. not from particulars to parti-
culars, 542 : how possible from false
premisses, 331-4 : Mr. F. H. Bradley's
definition of, 554 : immediate i., v.
sub voc.

Infinite terms, 42 n. 2 (cf. 127), 245 :
i. judgements, 42, 173 n. 2, 215.

Instance, prerogative, 433, 499 : soli-
tary, 433 : negative, 434 : crucial,
527, 565 n. 1 : glaring, 566 : original
meaning of instantia, 531 n. 1.

604

INDEX

Intension of terms, 136 : alleged in-
verse relation of i. and extension,
138-43, 146 : in proper names, 151.

Interaction, 421 n. 1.

Intermixture of Effects, 519-21 : ho-
mogeneous and heteropathic, 519 n. 1.

James, William, Principles of Psychology,
cited, 411 n. 1, 574 n. 1.

Jevons.W. S., cited, 18 n. 1, 124, 134,
137 n. 3, 150 n. 3, 222, 228 n. 1, 237
n. 1, 384 n. 1, 395 n. 1, 400 n. 1,
433 n. 2, 441 n. 1, 503-4, 530 n. 3,
557 n. 1, 674 n. 1, 577 n. 3, 596 n. 1.

Joachim, H. H., On the Nature of Truth,
cited, 194.

Joannes Philoponus, his definition of
Logic, 3.

Joannes Sarisburiensis, on the doc-
trine of predicables, 110 n. 1.

Judgement, the true unit of thought,
14, 159 : relation to the proposition,
18 : Antisthenes' objection to, 22 :
general nature of, 159-61, 169 : pro-
perly expressed by the indicative, 160 :
the copula in, 161-6 : logical, gram-
matical and metaphysical subject
of, distinguished, 166-9 : distinction
of j. according to Quality, 171-4 :
do. Quantity, 174-81 : do. Relation,
181-8 : do. Modality, 188-201 : collec-
tive or enumerative and universal j.,
distinguished, 176-9 : modal particu-
lars, 199, 240 : existential j., 167 :
infinite j., 173 n. 2, 215 : indefinite j.,
176: conjunctive j., 181 : simple and
complex j., 181 : pure and modal j.,
192, 200 : ampliative, augmentative,
explicative j., 208, 211 : analytic and
synthetic j., 207-15 : essential and
accidental j., 210-14 : verbal and real
j., 212-14 : exceptive, exclusive, ex-
ponible j., 215 : opposition of j., 228-
31: conversion, permutation ( = ob-
version), contraposition of j., c. x :
tautology not j., 209: a j. does not
assert agreement or disagreement
between its terms, 271 n. 1 : ' sound
j.', 205. Cf. s. v. ' Proposition '.

.Kant, I., his doctrine of Categories,
61-5: on analytic and synthetic
judgements, 207-14 : his canon of
syllogism, 308-10 : ou change, 13,
487 : on Applied Logic, 555 n. 1.

Kepler, J., his hypothesis of elliptic
orbits, 471 : his three laws, 516 (cf
635). V

Klein, Miss Augusta, 39 n. 1, 45 n 2,
73 n. 2, 114 n. 2, 172 n. 1, 303 n. 2.

Knowledge ' of acquaintance ' and

' kn. about ', 68 n. 2 : distinct from
opinion, 160.

Lambert of Auxerre, 284 n. 1.

Lambert, J. H., Neues Organon, on Fig. 2
of syllogism, 316 n. 2.

Lang, Andrew, Custom and Myth,
quoted, 536, 563 n. 1.

Laplace, J. S., Marquis de, quoted.
465.

Lavoisier, A. L., oxygen-theory of,
472-3.

Laws of Nature, 1-2, 886 n. 3, 426-7,
507 : conditional, unconditional, and
derivative, 414-16 : a law not a cause,
414-15 : not in time-relations, 415
n. 1 : their discovery the aim of in-
ductive science, 481-2 : precautions
necessary in formulating, 557 sq. :
L. of thought, 13, 332 : not psycho-
logical, 567.

Leibniz, G-. W. von, cited, 179 n. 1.
355, 506.

' Lewis Carroll ', cited, 312 n 1, 570.

Locke, John, quoted, 3, 35, 249, 257
n. 1, 271 n. 1, 313 n. 2, 574 n. 1 : con-
ceptualism of, 25 : on ' nominal
essences', 92, 303 : on ' mixed modes ',
93 n. 1.

Logic, general nature of, c. i : how far
formal, 4-7, 184, 237, 366-8 : not an
art, 9-10: practical value of, 11:
how concerned with things, 11, 48,
57, 67, 110 : false antithesis between
L. of Consistency and L. of Truth,
254-5, 370-5 : Deductive and Induc-
tive L., wrongly opposed, 371 n. 2,
396-8 : relation of progress in, to
progress of science, 371-2 : Inductive
L., history of, 394-5 : Applied L., 555 :
L. not an empirical science, 566-7 :
Symbolic L., 12 n. 2, 120 n. 1, 228,
241-2, 331 n. 2.

Lotze, H., cited, 21 n. 1, 22 n. 1,
400 n. 1, 413 n. 2, 421 n. 3, 440 n. 1,
466 n. 1, 538 n. 2.

MacColl, H., Symbolic Logic, on the
customary formulation of syllogisms,
331 n. 2.

McTaggart, Dr. J. E., Studies in Hegelian
Cosmology, quoted, 535, 593.

Maier, H,, Die Syllogistik des Aristoteles,
cited, 5S8 n. 1.

Maine, Sir Henry, cited, 508-9, 536-7,
592 n. 2.

Major term, why so called, 259-61,
328-30 (cf. 380) : illicit process of,
274-5 : m. premiss, how far surviving
in complete knowledge, 311, 524 n. 2:
in what sense a memorandum, 310.

INDEX

605

Mansel, H. Ii., Prolegomena Logica,

cited, 184 n. 1 : on dilemma, 360,

863 n. 1 : on' petitio principii', 591

n. 3. Cf. s. v. ' Aldrich '.

Many Questions, fallacy of, 697-8,

674, 587 n. 1.
Marshall, Prof. A., Principles of Econo-
mics, cited, 248 n. 2, 510 n. 2.
Materialism, inadequacy of, 411-12 (cf.

490).
Mathematics, reasoning of, 393, c. xxv :
argues largely from rationes cognoscendi,
305: employs syllogism when ?, 311,
545 : generalization in, 544-9 : its
principles not empirical, 549-52 : in
what sense deductive, 545-6 : mathe-
matical induction, 543-4.
Matter, Aristotle's conception of, 54-5 :
genus the m. of its species, 88 : m. not
the principium individuationis, 90 : ne-
cessary and contingent m., 196, 198.
Measurement, importance of in induc-
tive science, 501, 519-21 : difficulties
in regard to, 557-62.
Mechanism, what, 410.
Mellone, Dr. S. H., quoted, 125 n. 1.
Methodological ass umption.what, 564.
Methodology, 459, 555 : of science,

c. xxvi.
Michael Psellus, 206 n. 1, 267 n. 1 .
Mill, James, Analysis of the Phenomena
of the Human Mind, cited, 31, 157,
164 n. 1.
Mill, John Stuart: on adjectival
terms, 37 : on Definition, 82 : on
Cause, 113, 404-6: on Connotation
and Denotation, 147 sq. (cf. 299 n.) :
on the copula, 164 n. 1 : on Modality,
206 n. 1 : on the alleged 'petitio' in
syllogism, 302 n. 1 (cf. 310 n. 1) : his
misunderstanding of ' nota notae ', 308
n. 2 : on mathematical truth, 384 n. 1 :
on 'Laws of Nature', 386 n. 3, 507 :
his attempt to prove the Uniformity
of Nature, 421-5 : his view of its place
in inductive argument, 443 : on the
meaning of ' phenomenon ', 427 n. 1 :
his ' Inductive Methods ', 430-5, 440
n. 1, 442 ; are in reality one, 435 ;
their basis, 440 n. 1 ; canon for ' Joint
Method ' defective, 435 n. 1 ; his use of
symbols in explaining, 441 n. 1 : on
the formation of hypotheses, 465 : on
'colligation of facts', 469-70: on
Plurality of Causes, 491 n. 1 : error as
to its bearing on his ' Inductive
Methods', 497-9: his 'Deductive
Method of Induction ', 513-24 ; con-
fuses verification and proof, 524 n. 1 :
on Argument from Analogy, 540 : on
the Logic of the Moral Sciences, 370,

555-6 : on mathematical axioms, 549-
52: on necessities of thought, 579: on
Fallacies, 574 n. 1, 595 n. 1, 599 : his
Utilitarianism quoted, 584 : his place in
the history of inductive theory, 395.
Cf. also 12, 370, 400, 458 n. 1.

Minor term, why so called, 259-60 :
illicit process of, 274-5.

Minto.W., Logic, Inductive and Deductive,
cited, 158, 360 n. 1, 372, 586 n. 1.

Mixed modes, 93 n. 1.

Modal adverbs, 206 : m. particulars,
199, 240.

Modality, Kant's categories of, 65 : of
judgements, 188-201, 206-7 : Mill's
view of, 206 n. 1.

Modus ponens, 335-7 : tollens, 335,
337-8 : ponendo tollens, 845 ; how
far valid, 346-7: tollendo ponens,
345.

Mood, of syllogism, 262-3, 277-80:
indirect, in Fig. 1, 257 n. 2, 258-9,
268-9, 279: valid, how determined,
263 sq.: subaltern m., 285: m.-names,
origin of, 267 n. 1.

Morison, Sir Theodore, Industrial
Organization of an Indian Province, cited,
457 n. 1.

Name, Hobbes's definition of, 20, 150
n. 1 ; not = term, 20-1 : proper n., 29,
108 : n. of universals, 32-5 : general
n., n, of what?, 32, 144.

Nature, not a closed mechanical
system, 410-13, 417. Cf. also s.v.
' Law ', ' Uniformity '.

Necessity, not a vox nihili, 406 : its
relation to freedom, 410 : n. in judge-
ment, hypothetical and apodeictic,
192-6 : in causal relations, 406-10.

Negation, nature of, 172-3 : conver-
sion by, 238.

Negative terms, 41-6 : judgements,
170-4.

Nelson, L., Ueber das sogenannte Erkennt-
nisproblem, cited, 215 n. 1.

Nettleship, E. L., Philosophical Remains,
cited, 140 n. 1, 142.

Newton, Sir Isaac, his theory of
gravitation, 513 sq.

Nominalism, 31, 92, 300.

Nota notae, principle of, 307-8.

Observation, difficulties of, 475 : how

related to experiment, 494.
Obversion = Permutation, q.v.
Occam, William of, on nomina absoluta

and connotativa, 156-8 : his ' razor ', 506.
u^ijwfxa and ovvuivv^m., 31, 47.
Opposition of propositions, 228-31.
opos (' term '), meaning of, 18 n. 1.

606

INDEX

Paronymous words;, 133.

Particular propositions, 174, 176, 178-
9 : modal p., 199, 240.

Per accidens predication (opposed to
per se„ 49 ; conversion per a., 234.

Perception, its relation to judgement,
14 : Kant's doctrine of, 61.

Permutation of propositions, 237-40,
245-7.

Per se predication (opposed to per
accidens), 49.

Personal equation, 475.

Petitio Principii, fallacy of, 591-4,
576 : alleged in syllogism, 301 sq.

Petrus Hispanus, 267 n. 1.

Petrus Mantuanus, 284 n. 2.

Phenomenon, meanings of, 427-8,
486.

Plato, on Conceptualism, 25 : on nega-
tion, 173 : on the four elements, 100 :
on space, 172 n. 1 : cited also, 15 n. 3,
16 n. 2, 27 and n. 2, 142, 176 n. 1,
210 n. 1, 333 n. 2, 359, 371, 538 n. 3,
581 n. 1, 582 n. 2, 590 n. 4.

Plurality of Causes, 491 sq., 523, 557 :
J. S. Mill on, 497-9.

Podmore, H., History of Modern Spiritua-
lism, cited, 476 n. 3.

Poincar<§, P., cited, 384 n. 1, 411 n. 2,
527 n. 2, 552 n. 3.

Polysyllogism, 354.

Poor Law Commission, 1834, Report
of. quoted, 454-7.

Porphyry, his list of Predicables,
66 n. 2, 106-10: 'tree of P.', 130,
132 n. 1 : on intension and exten-
sion, 137 n. 2.

Port Royal Logic, quoted, 352 n. 1.

Post hoc, propter hoc, fallacy of,
595-6.

Poste, E., ed. of Sophisfici Elenchi,
quoted, 430 n. 1, 572 n. 2, 588, 592 n. 1,
594 n. 3.

Prantl, Carl, Qeschichte der Logik im
Abendlande, cited, 19 n. 2, 157, 206
n. 1, 267 n. l,283nn. 1,2, 284 nn. 1,2,
374 n. 2.

Predicables, doctrine of, c. iv (cf. 389):
Aristotelian and Porpliyrian lists, 66,
106-10 : its relation to the question of
the meaning of ' cause', 78-81 : Aris-
totle's proof that his list is exhaustive,
126.

Prediction, successful, how far a test
oi a theory's truth, 523 n. 1.

Premiss, what, 254, 256 : major and
minor, 256 : major in Fig. 1, 308-9,
310-11, 524 n. 2.

Prerogative instance, 433, 499.

Prichard, H. A., cited, 29 n. 2, 35 n. 1,
65 n. 7, 399 n. 1.

Principium Individuationis, 54-7, 81-

2, 90-4, 132, 145.
Principles, ' common ' and ' special ',

387-9, 397-8, 457 n. 1, 545, 571-2:

'common' do not explain, 503.
Privative terms, 40-1, 45.
Probability, meaning of, 201-5.
Problematic judgements, 188, 197-

200, 205.
Proper names, 29-30 : are equivocal,

47, 152 n. 1 : whether connotative,
150-3 : have no extension, 155 : have
intension, 151 : as predicates, 153,
167 n. 2.

Property, 75 : its relation to defini-
tion or essence, 91 sq. ; fourfold
division of, 104 : generic p., 104 n. 1,
126 n. 1 : individual p., 107 n. 1 :
diagnostic p., 112 (cf. 131).

Proposition, its relation to judgement,
18: ambiguity in verbal form of, 23-4,

48, 145 : unity of a, 161 : categorical,
101 : p. secundi and tertiiadiacentis, 163
n. 1 : verbal and real p., 212-14 : ex-
ceptive, exclusive, exponible p., 215 :
opposition of p., 228-31 : conversion
of, 232-7 : permutation of, 237-8 :
particular p., two meanings of, 178-
80, 199, 322. Cf. also s.v. 'Judge-
ment '.

Prosy llogism, 352.
■nporepov cpvaa and v. rinTv, 88.
Pseudographema, 572.

Quality of judgements, 171-4.
Quantification of the Predicate,

222-8.
Quantitative methods in Induction,

importance of, 501 (cf. 557-62).
Quantity of judgements, 174-81, 216-

17.
Quaternio terminorum, 270, 576.

Ratio cognoscendi and r. essendi, 205, 305.

Real propositions, 212.

Realism, 31-2 (cf. 54), 70, 107.

Reality, as ultimate subject of every
judgement, 166-9.

Reasoning, probable, 452, 528-9 : of
Mathematics, 543. Cf. s. v. ' Infer-
ence '.

Reduction of syllogisms, c. xiii : in
Fig. 4, 290: indirect, 292-3, 318:
uncalled for, 314 sq., 330.

Reid, Dr. Archibald, cited, 332 n. 4.

Relation, whether instances of a, 27-
8, 35, 145-6 : Kant's view of, 61 :
external r., 194, 538 n. 1 : not in-
dependent of its terms, 533 : distinc-
tion of judgements according to r.f
181-8.

INDEX

607

Belative terms, 39-40, 59.
Ritchie, D. G., Plato, cited, 539 n. 1.
Romanes, J. G., Darwin and after

Darwin, cited, 449-51, 474 n. 2, 489

n. 1, 510 n. 1, 512, 630 n. 2, 535 n. 1,

593 n. 3.
Russell, Hon. Bertrand, on relations,

27 n. 3, 357 n. 2 : on class-thinking,

228 n. 2.

Salisbury, Robert, Marquis of, quoted,
479 n. 2.

Sanderson, T., Compendium A rtis Logicae,
cited, 19 n. 2, 237 n. 1.

Science and History contrasted, 77,
102, 242, 467-8, 508-10.

Second Intentions, 8.

Secundum quid, fallacy of, 589-90,
573, 574 n. 1, 595.

Self-evident, the, 193-5.

Shyreswood, William, 267 n. 1 .

Sigwart, Chr., Logic, cited, 169 n. 3,
213 n. 1.

Singular judgements, 174 : for what
purposes classed with universal j.,
216-17 : s. terms, v. s.v. 'Term'.

Smith, Adam, Wealth of Nations, quoted,
453-4.

Smith, Prof. J. A., cited, 332 n. 2.

Sorites, 354-7 : Goclenian s., 354 n. 3.

Space, non-Euclidean, 548 n. 2.

Species as a head of Predicables, 106 :
s. infima and subalterna, 107 n. 2, 116 :
constituent s., 117.

Spencer, Herbert, 68 n. 3, 89, 388.

Spinoza, Benedict de, 173.

Stapper, R., on the Summulae Logicales
of Petrus Hispanus, 268 n.

Stock, St. George, Deductive Logic, cited,
42 n. 1, 577 n. 3.

Stout, Prof. G. F., on Error, cited,
194 n. 1 : Manual of Psychology, cited,
564 n. 1.

Subaltern genus, 116 : s. species, 107
n. 2, 116: s. mood, 285: s. opposi-
tion of judgements, 229.

Subcvmtrary opposition of judge-
ments, 229.

Subject, grammatical, logical, meta-
physical, distinguished, 166-9.

Subject-concept, what, 23, 80 n. 2.

Substance, 35: first and second s.,
53, 59.

Subsumption, what, 310-11 : cf. 336,
841 n. 2, 347, 397, 512.

avKKoytffudi «£ inroOtatais, 344 n.

Suppositio of names, 19 n. 2, 157 : s.
materialis, 19.

Syllogism, general nature of, c. xi :
Aristotle's definition of, too wide,

249 : problem of, 253 : nomencla-
ture of, 254-63 : figures of, 257-9 :
moods of, 262-3 : their determina-
tion in Fig. 1, 263-7 ; do., indirect,
268-9; generally, 277-85; rules of,
270-7 : rules of the several figures of,
278-80, 284: reduction of imperfect
moods of, c. xiii, 314 sq. : proposed
canons of, c. xiv ; Dictum de omni et
nuUo, 296-301 ; Nota notae, 307-8 ;
Kant's canon of, 308-10 : charged with
petitio principii, 301-10, cf. 381 : Fig. 1 in
what sense prior, 317 ; and scientific,
306, 316, 398-9 : Fig. 2, 315-19: Fig. 3,
319-25, 400 n. 2 : fourth or Galenian
figure of, 259, 280-5 (cf. 269 n. 1),
325-30 : when used in mathematics,
311: not the form of demonstration,
ib., 524 : hypothetical argument not
syllogistic, 339-44 ; Aristotle's a. «£
inroGfCTdiis, 344 n. : s. crypticus, 353 n. 1 :
inductive s., 380 : modal s., 452 n. 2 :
s. unable to generalize, 400.

Symbols, their use and defects in re-
presenting inductive reasoning, 441,
486-7.

Syncategorematic words, 19.

Synthesis, Kant's view on the work of,
61.

Synthetic judgements, 207 sq. : do.,
a priori, 2l0.

Term, in a judgement, what, 17-18 :
how defined, 21 : do., by Aristotle, 18
n. 1 : derivation of, ib. : distinction oft.
and word, 18 : do. of t. and name, 20-1 :
do. of t. verbal and term of thought, 21-2 :
concrete, abstract, and attributive t.,
28, 36-8, 59 ; abstract t. and names
of universals, 34-5, 145-6, 154 n. 2,
299 n. : attributive t., 154-5: singular
and general or common t., 29-31,
59 : general terms names of indivi-
duals, 32, 144 : singular terms as pre-
dicates, 167 n. 2: collective t., 38, 59:
absolute and relative t., 39-40, 59:
absolute and connotative t., 156 :
positive, privative, and negative t. ,
40-6: negative t. in definition, 113-14 :
contradictory t., 41 : infinite t., 42
n. 2, 245 : univocal, equivocal, and
analogous t., 46 : commensurate t., 72 :
intension and extension of, 136, cf.
c. vi pass.; denotation of, 146; con-
notation of, 147 sq. ; the antitheses
compared, 155-6 ; their history, 155-
8: distribution of t., 216-21 : major,
middle, and minor t. in syllogism,
what, 259-62, 328-30, 379-80.

Theophrastus, on the indirect moods
of Fig. 1 of syllogism, 283.

eo8

INDEX

Thinking, as distinguished from judg-
ing, 159 ad fin., 165.

Thompson, Archbishop W., Laws of
Thought, cited, 137 n. 3, 248 n. 1.

Thought, form and matter of, 5 n. 1 :
relation of language to, 15-16.

Topics, what, 389-90 : of cause, 430 :
Aristotle's treatise of the name, 375-6,
389-91 : t. or locus of fallacy, 595.

Trendelenburg, F. A., cited, 60, 352
n. 4.

Truth, whether one can be known
independently of all, 385 n. 1.

Turner, Prof. H. H., cited, 474 n. 1,
517 n. 1.

Unconditional principles in science,
414-16.

Undistributed middle, 270-2: u.
terms, 2-16.

Uniformity of Nature, meaning of,
402 : consistent with variety, 402,
416: do. with spiritual activity in
nature, 420 : importance of, in induc-
tive science, 428-9, 443-4 : cannot be
proved inductively, 421-5 : cf. also
s.v. 'Causation'.

Universals, what, 27, 29 : w. in re, ante
rem, post rem, 32 : names of, 32-5, cf.
145, 154, 299 n. : u. not causally re-
lated, 69 n. 1 : imply uniformity of
nature, 409: u. judgement, 174-81.

Universe, ' limited ', or ' of discourse ',
44 n. 2, 165 n. 1.

Venn, J., Empirical Logic, cited, 120
n. 1, 121 n. 1, 135 n. 1, 441 n. 1.

Verbal propositions, 212.
Verification of a theory, 523 : not «-

proof, ib., 597.
Vernon, Dr. H. M., Variation in Animals

and Plants, quoted, 444-7.

Wallace, A. R., quoted, 509.

Wallis, J., Logic, cited, 239 n. 1,
247 n. 1.

Watts, Isaac, Logic, cited, 114 n. 1,
285 n. 2.

Webb, C. C. J., 110 n. 1.

Welton, Prof. J., Manual of Logic, cited,
395 n. l,459n. 1.

Whately, Archbishop, Logic, quoted,
157, 297 n., 568, 569, 574 n. 1, 575,
577 n. 3, 599.

Whewell, W., his writings on Induc-
tion, 395, 468-73: on Colligation of
Facts, 469.

Wilson, J. Cook, p. vi, 6 n. 1, 23, 55
n. 2, 67 n. 1, 80 n. 2, 185, 241, 332
n. 3, 386 n. 1, 523 n. 1, 547 n. 1.

Wollaston, W., Religion of Nature de-
lineated, cited, 161 n. 1.

Xenocrates, on Aristotle's Categories,
51 n. 1.

Zabarella, Count, de Quarto Figura
Syllogismi Liber, cited, 284 : on Dictum
de Omni, 296 : on reduction of hypo-
thetical arguments to syllogism, 339
n. 1.

Zeno, on motion, 359, 362.

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