A MODERN ELEMENTARY LOGIC

St ebbing Lisssie Susan, 18S5-1943

A modern elementary logic,
5th ed, 3 ^ethuan C 196l 3

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Stabbing, Lizzie Susan,

1885-1943.

A modern elementary

logic /

1961, 1969

A Modern Elementary Logic

This is a straightforward textbook for philo
sophy students in their first year. For this
reason traditional and modern developments
in logic have been combined in a unified
treatment.

Questions and exercises have been added
in an appendix, and solutions to them are
provided.

UNIVERSITY PAPERBACKS
U.P.28

By the same author

A MODERN INTRODUCTION TO LOGIC
PHILOSOPHY AND THE PHYSICISTS

LOGIC IN PRACTICE
THINKING TO SOME PURPOSE (Pelican Books)

A Modern
Elementary Logic

BY

L. SUSAN STEBBING

REVISED BY

C. W. K. MUNDLE

UNIVERSITY PAPERBACKS

METHUEN - LONDON
BARNES & NOBLE NEW YORK

First published August 12, 1943

Reprinted five times
Fifth edition, revised, 1952

Reprinted three times
First published in this series 1961

Printed in Great Britain by

Butler &. Tanner Ltd, Frome and London

Catalogue no. (Methuen) 2/6766/27

University Paperbacks are published

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and BARNES & NOBLE INC

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Reviser s Note to Fifth Edition

In the course of using this book for teaching purposes, I dis
covered a number of logical errors, most of them in the first
five chapters. Although I have now made a systematic search
for such errors I cannot guarantee that none have been over
looked, for several which I had not noticed have since been
pointed out to me by Mr R. N. W. Smith of St Andrews
University. I have limited myself to such alterations to the
text as could be effected by the deletion, amendment or
insertion of not more than a single sentence or group of
logical symbols except in the following cases: the second
paragraph on p. 87 has been rewritten, and the set of
diagrams given on p. 96 has been altered. Regarding this last
change, Miss Stebbing followed J. N. Keynes in acknowledg
ing only seven of the ten possible cases. Keynes gave only
seven diagrams because he was restricting his attention to the
cases where S and P, as well as S and JP, have members, but
Miss Stebbing does not seem to have intended to adopt this
restriction. In the course of looking for logical errors I have
noticed and corrected a few grammatical slips.

C. W. K. MUNDLE

University College
Dundee, ig$2

KfttfSAS CTIY W PUBLIC

<2T 67i^ : * :

Publisher s Note

Readers should bear in mind that this book was written
dialing the Second World War. For this reason a number of
examples have a military flavour.

Preface

The aim of this book is extremely limited. It is definitely a
textbook primarily intended for the use of first-year students
reading Logic for a University examination. The present
conditions of examinations in this country make it necessary
to include some trivial technicalities with which, in time,
teachers of elementary logic will be able to dispense. Already
the situation is much more hopeful than it was a few years
ago. Teachers and examiners have made some progress in the
task of carting away some dead wood. Accordingly, it has
been possible to reduce the discussion of technical trivialities
in this book to a small space, thus leaving the student time to
consider the wider implications of logic as a formal discipline
instead of a depository of antiquities.

Within the limited number of pages available it has not
been possible to treat of methodology, or scientific method,
as fully as is necessary for covering all the topics dealt with in
examinations in elementary logic. Only the last chapter
touches upon these topics and hardly does more than indicate
to the student what the problems are about which he must
seek fuller information elsewhere. This omission is the less to
be regretted in that there are several books on scientific
method well suited to the purpose and obtainable in most
university and public libraries. The case of formal logic is,
however, very different. So far as I know, there is no simple,
introductory textbook on formal logic, written from a modern
point of view, that is both unencumbered with much dead
traditional doctrine and yet meeting the needs of students
preparing for an examination. I have tried to repair this
omission. I have kept especially in mind the student who is
reading by himself, without any guidance from a teacher.
There are, I understand, not a few such students today, many
of them in H.M. Forces. To aid this class of reader I have
included some typical questions, together with a key indicat
ing the way in which the questions should be answered,

vii

VL11 PREFACE

These are given in an Appendix, which can be neglected by
those who are fortunate enough to have someone at hand to
deal with their perplexities. In the answers I have given I
have had in mind the sort of difficulties which years of teach
ing have shown me to be very commonly felt by young
students who are not born logicians and yet can gain some
thing (besides the advantage of passing an examination)
from a course in elementary logic. I am convinced that the
careful consideration of logical principles, even in the simple
form in which they are presented in this book, is well worth
while for any student. This belief, I am aware, may be the
result of my own interest in, and liking for, logical studies; in
other words, I am perhaps allowing myself to be too partial
to my main preoccupation. It remains possible that a state
ment that is not impartial may nevertheless happen on the
truth.

I regret very much that I have given no references to
Formal Logic., by Prof. A. A. Bennett and Prof. C. A. Baylis
(Prentice-Hall, New York). Although I possessed a copy
war conditions deprived me of it, and it is only recently
that I have been able to read it. It can be unreservedly
recommended.

I am much indebted to Mr A. F. Dawn for reading the
Appendix and making many valuable suggestions, and to
Professor D. Tarrant and Miss M. E. F. Thomson for help in
the reading of the proofs. To Professor Tarrant I owe more
than help in proof-reading; her critical mind and sound sense
have saved me from many blunders.

War-time economy forbids a dedicatory page; it is, how
ever, permitted to express the wish to dedicate this textbook
to my students, present and past, in appreciation of the help
they have often given me, unsuspected by themselves.

L. SUSAN STEBBING

Bedford, College
London, May

Contents

REVISER S NOTE TO FIFTH EDITION page v

PUBLISHER S NOTE vi

PREFACE vii

I THE STUDY OF LOGIC i

i. Reflective Thinking. 2. Argument. 3. Validity
and Truth. 4. Form and Logical Forms. 5. Logical
Symbolism and Form

II PROPOSITIONS AND THEIR

RELATIONS 16

i. Propositions and Sentences. 2. Propositions,
Mental Attitudes, and Facts. 3. Assertion, Inference,
and Implication. 4. The Traditional Analysis of
Propositions. 5. Simple, Compound, and General
Propositions. 6. The Seven Relations between Proposi
tions and the Figure of Opposition. 7. Immediate
Inferences

III COMPOUND PROPOSITIONS AND

ARGUMENTS 43

i. Equivalents and Contradictories. 2. Compound
Arguments with One or More Composite Premisses

IV THE TRADITIONAL SYLLOGISM 54

i. Defining Characteristics of a Syllogism. 2. Figures
and Moods of the Syllogism. 3. Reduction and the
Antilogism. 4. Polysyllogisms. 5. Abbreviated Argu
ments and Epicheirema

V INDIVIDUALS, GLASSES, AND

RELATIONS 75

i. Individuals and Characteristics. 2. Classes. 3.
Relations. 4. Class-inclusion and Class-membership;
Single-membered Classes. 5. Subclasses and Empty
Classes. 6. The Universe of Discourse and the Universal
Class. 7. Reconsideration of the Traditional Treatment
ix

X CONTENTS

of Opposition and Immediate Inferences. 8. The
Logical Properties of Relations and the Validity of
Inferences

VI CLASSIFICATION AND

DESCRIPTION page 100

i. Terminological Confusions. 2. Connotation,
Denotation, and Intension. 3. Extension and Con
notation. 4. Classification and Division. 5. The Pre-
dicables. 6. Definition. 7. Descriptions

VII VARIABLES, PROPOSITIONAL
FORMS, AND MATERIAL
IMPLICATION 126

i. Variable Symbols. 2. Prepositional Functions and
General Propositions. 3. Material Implication and
Entailing. 4. Extensional and Intensional Interpreta
tions of Logical Relations

VIII LOGICAL PRINCIPLES AND THE

PROOF OF PROPOSITIONS 146

i. The Traditional Laws of Thought. 2. Necessary
and Factual Propositions. 3. The Necessity of Logical
Principles. 4. Persuasion and Proof. 5. Is Syllogistic
Proof Circular?

IX METHODOLOGY OF SCIENCE 166

i. Inductive Reasoning. 2. Causal Laws. 3. Methods
of Experimental Inquiry. 4. The Nature and Import
ance of Hypothesis. 5. Systematization in Science

APPENDIX l8y

References and Exercises arranged in chapter order
Key to the Exercises

INDEX 211

CHAPTER I

The Study of Logic

9 I. REFLECTIVE THINKING

When we are told something that is startling or unpleasant
we may be moved to ask our informant: How do you know
that? 3 Usually such a question is a demand for reasons: we
want to know the grounds for the statement rather than to
inquire what were the processes of thought through which
our informant was led to make the statement in question; we
are asking for some assurance; we are not willing to accept
the statement without evidence. The sort of answer that
would satisfy such a questioner would take the form: Because
it (i.e. what was originally stated) follows from so-and-so. 5

The reader, it is assumed, will have no difficulty in under
standing the above paragraph; he is already familiar with a
notion of great importance in the study of logic, namely, the
notion of evidence in support of a statement. In this book it
is taken for granted that our interest in logic is, for the most
part, confined to the domain of evidence. Our purpose is to
examine the principles in accordance with which it is reason
able to accept, or to reject, statements made by ourselves or
by other people. During the greater part of our daily life we
unhesitatingly accept what we hear or read and the answers
given to our questions. It seldom occurs to us to question what
generally passes as true; for example, that our cat will pro
duce kittens and not puppies, that if we sow seeds from a
poppy we shall get poppies and not sweet-peas, that a stone
thrown into a pond will sink and ripples will spread outwards
from the spot where the stone struck the water, that - in the
northern hemisphere - we shall never see the sun due north,
that we shall eventually all of us die. Examples could be
added indefinitely. Most of us could give reasons for these
beliefs but usually it does not seem necessary to ask for them.

i

2 A MODERN ELEMENTARY LOGIC

Our common daily activities are for the most part carried on
without reflection: the paper knife will slit the envelope if we
make the usual movements, the upset cup of coffee will stain
the tablecloth, the electric light will come on if we turn the
switch. Unless we could take such things for granted our
more or less orderly lives could not go on as they do.

This unreflective state of mind cannot always be main
tained: our statements are challenged or some unexpected
change occurs in our environment. We may even have
enough leisure and alertness of mind to be curious and thus
to begin to ask questions simply to satisfy our curiosity, as
intelligent children do. To be in a questioning frame of mind
is to be thinking; reflective thinking essentially consists in
attempting to solve a problem and thus in asking questions
and seeking answers to these questions so as to solve the
problem. We distinguish reflective thinking from idle reverie
or day dreaming. In reflective thinking our thoughts are
directed towards an end - the solution of the problem that
set us thinking. Thinking is a mental process in which we pass
from one thought to another. A thought is an element in this
process which requires for its full expression a complete
sentence. When one thought is more or less consciously con
nected with another in order to elicit the conclusion towards
which our thinking is directed we are reasoning.

Reasoning is a familiar activity; we all reason more or less,
badly or well. We connect various items of information and
draw conclusions; we judge that, if certain statements are
known to be true, then certain other statements are also true
and must be accepted. In saying that the latter must be accep
ted we are saying that, provided we are thinking logically, we shall
accept them; that is to say, we should not be rational beings
if we accepted the former statements and rejected the latter.

2. ARGUMENT

Consider the following passage taken from Boswell s Life of
Johnson.

I introduced the subject of toleration. JOHNSON: Every
society has the right to preserve public peace and order,

THE STUDY OF LOGIC 3

and therefore has a good right to prohibit the propagation
of opinions which have a dangerous tendency. To say the
magistrate has this right, is using an inadequate word; it is
the society for which the magistrate is agent. He may be
morally or theologically wrong in restraining the propaga
tion of opinions which he thinks dangerous, but he is
politically right. MAYO : C I am of opinion, Sir, that every
man is entitled to liberty of conscience in religion; and that
the magistrate cannot restrain that right. JOHNSON: Sir,
I agree with you. Every man has a right to liberty of con
science, and with that the magistrate cannot interfere.
People confound liberty of thinking with liberty of talking;
nay, with liberty of preaching. Every man has a physical
right to think as he pleases; for it cannot be discovered how
he thinks. He has not a moral right, for he ought to inform
himself and think justly. But, Sir, no member of a society
has a right to teach any doctrine contrary to what the
society holds to be true. The magistrate, I say, may be
wrong in what he thinks: but while he thinks himself right,
he may and ought to enforce what he thinks. MAYO:
Then, Sir, we are to remain always in error, and truth
never can prevail; and the magistrate was right in persecut
ing the first Christians. JOHNSON: Sir, the only method
by which truth can be established is by martyrdom. The
magistrate has a right to enforce what he thinks; and he
who is conscious of the truth has a right to suffer. I am
afraid there is no other way of ascertaining the truth, but
by persecution on the one hand and enduring it on the
other. *

This conversation is an example of argumentative dis
course. It is argumentative because the thoughts of the
speakers are connected in such a way as to lead to a con
clusion, that is to say, there is direction towards a statement
which logically concludes the argument. Certain statements
were taken for granted from which the conclusion was
obtained; these statements are called premisses. A premiss is a
statement from which another statement, called the conclusion,

* Bosweirst/5 of Johnson (Globe edition, 1922), p. 265.

4 A MODERN ELEMENTARY LOGIC

is drawn. Thus premiss and conclusion are correlatives. Not
every statement is put forward as a premiss, any more than
every man is a husband. But just as men become husbands
by entering into the relation of marriage so a statement
becomes a premiss when it is put into the relation of providing
evidence for a conclusion. Usually more than one premiss is
required to establish a conclusion, and more than one con
clusion can be drawn from the same statement or set of
statements.

Whenever we use such words as therefore 3 , it follows 5 ,
hence , consequently , we profess to have offered premisses
from which our conclusion may be drawn; when we use such
words as because , for , since , for -the reason that , we
profess to offer premisses for a conclusion already stated, that
is to say, we are offering evidence in support of our conclusion.
The premisses are evidence for the conclusion only in virtue
of certain relations in which they stand to the conclusion. The
relation between premiss and conclusion which justifies the
assertion that the conclusion follows from the premiss is a
relation of implication. When this relation holds, the premisses
imply the conclusion, and the conclusion follows from the pre
misses. For example, the conjoint assertion of the two state
ments, Every society has the right to prohibit the propagation of
opinions having a dangerous tendency and These opinions have a
dangerous tendency, implies Society has the right to prohibit the
propagation of these opinions. Provided that the premisses are
true, the conclusion is true. We might refuse to admit that
one of the premisses is true, or we might deny both; in that
case we are not rationally bound to accept the conclusion,
but we must, in our turn, give reasons for denying the premiss
or premisses. To do this is to argue.

If the reader looks back to the conversation reported by
Boswell, he will see that Johnson was engaged in putting
forward premisses to justify his conclusions.* The reader may

* The student should re-read Johnson s argument and attempt to determine
its structure. It should be noticed that Johnson (i) asserts his belief [on the topic
for discussion] and gives a reason for it; (ii) points out (in response to a comment
made by another participant in the discussion) the need to make certain dis
tinctions; (iii) makes further statements on the basis of these distinctions; (iv)
answers an objection to his original contention by accepting the objection as arj
unavoidable consequence.

THE STUDY OF LOGIC 5

very well dissent from Johnson s conclusions; if so, he will in
his turn be engaged in reflective thinking - arguing from
premisses to conclusion or seeking premisses to establish as a
conclusion a statement which, perhaps, had been previously
accepted without question. Johnson s argument was on a con
troversial topic and carried on in a somewhat controversial
manner. This is not the essential characteristic of argument
Even though we often dispute heatedly with one another we
do sometimes enter into an argument solely in order to arrive
at correct conclusions. It is this sense of "argument" which
concerns the logician, and from this point of view an argu
ment is simply a set of statements in which one statement (the
conclusion) is accepted on the evidence of the remaining
statements (the premisses). Frequently the conclusions we
seek to establish do not stand to the premisses in so strict a
logical relation as that of being implied by the premisses; the
premisses may provide evidence in support of the conclusion
without providing logically conclusive evidence , in this case the
relation may be said to be a probability relation. When the con
clusion is implied by the premisses the argument is deductive",
when the premisses do not suffice to imply the conclusion but
nevertheless have some weight as evidence in favour of it the
argument is said to be inductive. In an inductive argument the
premisses may be true and yet the conclusion may be false;
the evidence, however strong, is thus inconclusive. With
arguments of this kind we shall be concerned later. In a
deductive argument the conclusion could not be false and yet
the premiss be true; hence, in this case, the evidence is rightly
called conclusive.

In ordinary discussions we seldom state fully all the
premisses which we should on reflection unhesitatingly admit
to be required for establishing our conclusion; still less do we
recognize exactly how it is that the premisses do suffice (when
they do) to establish the conclusion. In practice our argu
ments are often very much abbreviated; we omit premisses
because they are self-evident or are regarded as accepted by
everyone. This procedure is good enough for most of our
purposes and is further necessary in order to avoid intolerably
long and prolix statements. It is not, however, free from

O A MODERN ELEMENTARY LOGIC

danger, for it may be that the validity of the argument
depends upon an unstated, or implicit, premiss which would
not be accepted had it been made explicit what the required
premiss is. The omission of premisses is, as we shall see later,
a common cause of fallacious arguments.

3. VALIDITY AND TRUTH

We have just used the phrase, c the validity of the argument .
An argument is valid if the truth of the premisses necessitates
the truth of the conclusion; this is equivalent to saying that
the premisses cannot be true and the conclusion false, or, in
other words, that the premisses logically imply the conclusion.
We have just used three alternative expressions to state the
relation holding between premisses and conclusion in a valid
argument. It should be noted that we do not define these
expressions but assume that the reader understands at least
one of them - for example, c the premisses cannot be true and
the conclusion false ; he has then to see that the other two
expressions are alternative ways of saying the same thing. It
is, moreover, taken for granted that we know what is meant
by "true" and "false". The logical relation of implication
holding between a premiss and a conclusion does not deter
mine whether the premiss is true; hence, the validity of an
argument is in no way a guarantee that the conclusion is
true. For example, Boccaccio died before Dante and Dante died
before Voltaire together imply Boccaccio died before Voltaire.
Logical considerations alone suffice to assure us that the
conclusion is true provided that the premisses are, for the
premisses certainly imply the conclusion. In fact, the first
premiss is false, the second is true, and the conclusion is true.
We know this (if we do know it) not from logic but from
historical records. Again, it may be true that Bothwell loved
Mary, Queen of Scots, and also that she loved Bothwell; but,
from Bothwell loved Mary it does not follow that Mary loved
Bothwell; there are, unfortunately, many unrequited lovers.
Both these statements may be true or one may be true with
the other false; hence neither implies the other. But Darnley
married Mary does follow logically from Mary married Darnley,

THE STUDY OF LOGIC 7

and indeed, conversely; if one statement be true, the other is;
if one is false, the other is. It is impossible for A to be married
to B without its being true that B is married to A , this logical
impossibility is involved in the meaning of "married to". But
logic does not determine who marries whom, who loves
whom, nor when men are born or die.

Consider the following examples of arguments:

(1) All Athenians are Greeks and no Greeks are bar
barians; therefore, no Athenians are barbarians.

(2) All Austrians are Germans and all Germans are
Europeans; therefore, all Austrians are Europeans.

(3) No insects have six legs and all spiders are insects;
therefore, no spiders have six legs.

(4) All members of Parliament have great responsibili
ties, and Winston Churchill has great responsibili
ties; therefore, Winston Churchill is a member of
Parliament.

(5) Some poets are not Roman Catholics and all who
acknowledge the authority of the Pope are Roman
Catholics; therefore, none who acknowledge the
authority of the Pope are poets.

We shall examine each of these five examples in order to
answer two questions: (i) Are the premisses true? (ii) Is the
argument valid? [The student should carry out this examina
tion for himself before he reads further.]

We summarize the result of the examination as follows:

(i) Are the premisses true? (Is the conclusion true?) Is the argument valid?

1 . Both premisses true. Conclusion true. Valid.

2. First premiss false. Conclusion true. Valid.

3. Both premisses false. Conclusion true. Valid.

4. Both premisses true. Conclusion true. Invalid.

5. Both premisses true. Conclusion false. Invalid.

In addition to answering the two questions posed to us we
have noted whether the conclusion is true or false. From these
examples we can see that there can be (a) a true conclusion
from a valid argument although the premisses are false; (b)
an invalid argument with both premisses true and the con
clusion true; (c) a false conclusion in an invalid argument

8 A MODERN ELEMENTARY LOGIC

with true premisses. Validity, then, is not dependent upon
truth. On reflection we see that this must be so. Every state
ment has implications, or, as we sometimes say, consequences.
For instance, a scientist may wish to determine whether a
possible hypothesis, which would account for the phenomena
he is investigating, is true or false. An hypothesis is a state
ment of the form If so and so, then suck and such (e.g. If light has
a finite velocity , then the light from different stars reaches us after a
longer or shorter time depending upon the distances of the stars from
the earth) . The consequences are deduced, and, when possible,
tested. If the implied consequence is false, there is no reason
to accept the hypothesis; if the implied consequence is true,
then the hypothesis may be true. When the premisses of a
valid argument are true, then the conclusion must also be
true. When the argument is valid and the premisses are false,
we do not know whether the conclusion is true or not; con
sequently, we should have no reason for accepting the con
clusion as true. When the argument is invalid and the
premisses are true, we again have no reason for accepting the
conclusion; in such a case we might say that the conclusion 5
is not properly a conclusion since it does not follow logically
from the premisses; hence, the argument is inconclusive.*

We had no difficulty in determining whether the state
ments (premisses and conclusions) in our five examples were
true or false, since these statements were about familiar
subject-matter. Anyone who reads this book (it is assumed)
knows that Austrians are not Germans but that both
Austrians and Germans are Europeans; and so on, in each 01
the other examples. The question whether these statements
are true is a question concerning matters of fact, or, as we
shall say, it is a factual question. The question whether the
premisses suffice to prove the conclusion is a question about
the logical form of the argument. As logicians we do not care
whether Austrians are Germans, or whether Athenians are
not barbarians; our concern is wholly with the conclusiveness
of the arguments, for unless our arguments are conclusive we

* As the logician, Augustus de Morgan, has said: It is not therefore the
object of logic to determine whether conclusions be true or false; but whether
what are asserted to be conclusions are conclusions.

THE STUDY OF LOGIC 9

have no logical reasons for accepting the conclusions. If the
conclusion does follow from the premisses, the argument is
valid; if the conclusion does not follow from the premisses the
argument is invalid. The validity of an argument depends
entirely upon the logical form of the argument. What, then,
do we mean by logical form?

4. FORM AND LOGICAL FORMS

We are all familiar with the notion of change of form: butter
left standing in the sun becomes a runny mess; water heated
to boiling-point becomes steam, frozen it becomes ice; an
orderly procession of civilians suddenly charged by mounted
police becomes a disorderly crowd, and so on. What is meant
by and so on , in the last sentence? It is used to invite the
reader to supply other examples, in the confident expectation
that he will be able to do so, for the examples have all been
of the same form: something that in one sense remains the
same and in another sense is different. The crowd and the
orderly processions are composed of the same people but they
have entered into new combinations; the shape made by
them as they march in procession is quite different from the
varying shapes made as they huddle together or push one
another in various directions. Probably we should say that
the crowd was a shapeless mass , for we tend to use the word
shape only when the elements that are shaped stand in
relatively constant relations to one another. But shape is a
matter of degree; when we press a piece of rubber we change
its shape; when we blow into a toy balloon we transform it
from a relatively shapeless bit of stuff into shapes of various
sorts, perhaps ending with a round ball-shape. Shape is the
most common meaning of the word "form", but we fre
quently use it in greatly extended senses. How widely we
understand this notion of form is shown by its numerous
synonyms or partial synonyms, e.g. arrangement, orderliness,
type, norm, standard, design, pattern. The paper pattern for
a dress has the same form, or shape and size, as the material
of the dress when the material has been cut to pattern. This
is what we mean by saying that the paper pattern is a pattern.

10 A MODERN ELEMENTARY LOGIC

The design of the English penny and twopenny stamps
(current issue) is the same but they differ in colour; the
design as well as the colour of the shilling stamp differs from
that of either of the other stamps. A meat-mould, a jelly, and
a blancmange may all have the same shape or form but they
differ in the materials of which they are made. Everyone
understands this distinction between material and form, or, as
we sometimes say, between matter and form. When a child
builds a house out of his toy bricks he is arranging the bricks
(i.e. the matter) in a certain way, viz. in the form of a house;
this is a construct. Not all that is constructed, or has form, is
material. Consider, for instance, musical form. A scale is a
musical form consisting of notes, but these notes cannot be
taken in any order; they must be put together in a certain
definite way. We might use the same notes put in a different
order and thus get a singable melody, quite different from the
scale. We distinguish between hymn form, fugue form, sonata
form; we might say that a symphony is a sonata for the
orchestra.

Why do we call the musical scale a scale? Obviously
because the order of the successive notes is felt to resemble the
order of rungs as one goes up and down a "scale" or "ladder".
A ladder means originally a certain material thing, but we
have come to recognize a ladder-like arrangement in many
other things, e.g. a ladder in a stocking, and even more
abstractly, as when we speak of the educational ladder. Our
manner of speaking shows that we implicitly recognize a
common form in diverse material; we see a relation that is
analogous between the notes of a scale from lower to higher
and the colour-scale from dark to light colours. Analogy is the
recognition of a common form or structure in very unlike
things.

Our thoughts have form. When we are successfully
engaged in reflective thinking our thoughts occur in an
orderly way; what does not fit in is, as far as possible, kept out
of mind. Our languages are adapted, somewhat imperfectly,
to express our thoughts; hence, we have grammatical form-
Words cannot be put into any order to make a sentence. The
student who knows a little Latin but not much finds in

THE STUDY OF LOGIC II

translating unseens that he sometimes knows all the words
but can t get the hang of the sentence , but sometimes he can
get the hang of the sentence but does not know what some of
the words mean. In the first case his knowledge of syntax is
inadequate; in the second case it is his vocabulary that is at
fault. Syntax is the formal structure of a language; vocabulary
is its material.

In learning Latin syntax Balbus murum aedificavit does just
as well as but no better than Caius puellam amamt to illustrate
the use of the accusative case. Analogously, the logician can
use any material to illustrate logical forms. As soon as we are
able to frame sentences correctly we have gained an implicit
knowledge of grammatical form; as soon as we are able to
reason, and to demand reasons, we have an implicit know
ledge of logical form. Our apprehension is at first implicit; if
it were explicit we should not merely apprehend but com
prehend: we should thus understand just why this combina
tion of words, in grammatical form, was right for our purpose,
and just why that special combination of statements was
logically right for sound reasoning. In studying logic we
extract this implicit knowledge from the particular instances
in which it is present, and are thus able to state the logical
principles to which our reasoning must conform if it is
valid. Our interest is wholly in the formal combinations of
statements.

Consider the statement If Jones is a painter, and all painters
are irascible, then Jones is irascible. This is a compound statement
consisting of three statements each of which could be
separately asserted. This compound statement is true in
virtue of its form; if the first two statements are true, the third
must also be true, but, as we have seen, the implication holds
even if the first two statements (joined by and] are false. Thus
the whole compound proposition is true in virtue of its form.
The implication does not depend upon any characteristics
Jones may possess other than his being a painter, we could then
say, If Robinson is a painter, and all painters are irascible, then
Robinson is irascible. It is not difficult to see that we could
likewise replace painters by musicians, by schoolmasters, or by
anything else that would make sense, provided we made the

12 A MODERN ELEMENTARY LOGIC

substitution in both statements; likewise with irascible. Let us
then replace Jones by X, painters by C s, and irascible people by
D s; we thus obtain, If X is a C, and all C s are D s, then X is
a D. We have now no longer a definite statement about
certain persons or classes of things but a logical form or
structure. If we substitute for X, C, D anything yielding a
statement which makes sense, we shall have an instance of a
valid implication instead of an implicationalform. What makes
the implication valid (and thus the statement that it holds
true) is the form of the separate statements and the mode in
which they are combined, i.e. the way in which the three
statements are inter-related.

Logic is a formal science. What exactly is involved in say
ing that logic is formal will be clear only after we have studied
in detail various logical forms. For this purpose we need to
make explicit the forms we implicitly apprehend. Con
sequently, at times we shall need to use special symbols, since
we want to consider forms of reasoning without paying atten
tion to the subject-matter, or material, of specific arguments.

5. LOGICAL SYMBOLISM AND FORM

We are all familiar with such symbols as the national flag, a
flag at half-mast, the wearing of a crown. Language is a
symbolism. We use language not only to express our feelings
but to communicate what we feel and know to others. So
long as men were confined to a spoken language they could
not communicate more than those living were able to
remember. With a written language it is possible to com
municate what we know to those who live centuries after we
and all our contemporaries are dead. We communicate by
using signs to convey our meanings. A word is a special sort
of sign. A sign indicates something other than itself. For
example, a rapid upward movement of the arm until the
tips of the fingers touch the cap is a visible sign conventionally
expressive of courteous recognition of a superior in rank. But
to whom is this sign significant? Only to those who are aware
of this special convention of a salute. Signifying is a relation
requiring three terms: a sign, that of which the sign is

THE STUDY OF LOGIC 13

significant, and an interpreter for whom the former indicates
the latter. The appearance of the sunset-sky is a sign to a
weather-wise countryman of what tomorrow s weather will
be; it is significant for him because he has had experience of
connecting a certain sunset-appearance with a certain kind
of weather next day; to the ignorant townsman there may be
no significance. A symptom, in the medical sense, is a sign
characteristic of a certain sort of disease. These are natural
signs; they are to be contrasted with conventional signs which
owe significance to the actions of men who seek to fulfil their
needs and desires.

The w r ords in our languages are conventional signs.
Aristotle (thinking of the spoken language) called them
sounds significant by convention 3 . They are not merely
sounds but significant sounds; in the written language words are
significant marks , but a word must not be identified with any
particular sound uttered by someone on a particular occasion,
nor with any particular mark written by someone in a
particular place. For example, in this paragraph the mark
sounds occurs more than once, but these separate, numerically
different, but recognizably the same, marks are each an
instance of the one word sounds. In sending a telegram we
count .the number of words in the sense of marks; if the mark
five occurs twice we count it twice in estimating the cost of the
telegram at so much per word; in the sense of the mark s
meaning there is only the one word five. Sometimes a mark
may exemplify more than one word, e.g. patient 5 , bull 3 .
Bull is a mark that may be used to signify a certain kind of
animal or it may be used to signify a certain sort of ludicrous
jest.

A conventional sign is called a symbol- The kind of symbols
we are most familiar with are ordinary words; these are called
verbal symbols. Anyone who knows our language knows what
it is we are referring to when we use words in the language.
For many scientific purposes we find it more convenient to
use non-verbal symbols. There are various kinds of non-verbal
symbols, of which we shall here distinguish only two kinds.
A third will be dealt with later.*

* See Ch. VII.

14 A MODERN ELEMENTARY LOGIC

(i) Shorthand symbols* These are either abbreviations for words
or concise marks substituted for words, directly representing
what they symbolize. For instance this is used as a
road-sign symbolizing that there is a double bend
ahead. This shorthand symbol can be more easily
apprehended by a rapid driver than the words c double bend
ahead . In mathematics shorthand symbols make it possible
to express a complicated idea so briefly that it can be appre
hended at a glance. For example, V is easier to grasp in
a formula than the square root of; similarly, + instead of
c plus 3 , X instead of multiply 5 , and so on. The student will
recognize that shorthand symbols are indispensable in
practice if we wish readily to grasp even comparatively
simple algebraical expressions. For example,

/ , b + Vb*-4ac\ (
= a { x + - ) [
\ 2<2 / \

1 , i, , , - ,

ax 2 + bx + c = a { x + - ) [x +

will be easily read by anyone with the most elementary know
ledge of algebra; if the student attempts to write out this
equation using only English words he will soon find that it
is difficult to keep his head. The choice of appropriate marks,
i.e. shorthand symbols, is often very important. Compare, for
example, the difficulty of working out a long sum in multi
plication using Roman numerals with the ease of working it
out when expressed by means of the Arabic notation.* In
logic we find such shorthand symbols as == to stand for is
equivalent to 5 ; = for equals , ^ for a special sense of
implication, extremely convenient both for brevity and ease of
apprehension. We shall see later that it is a help to use
different shorthand symbols for distinguishing between
different meanings of the word "is".
(ii) Illustrative symbols. Suppose someone contends that all who

* A simple example of a shorthand symbol of great utility is provided by
io ?1 ; this is short and easily apprehended (once the notational rules have been
learnt), but written out in full, in the usual manner, it would require so many
zeros after the unit i that it would be difficult to grasp what number it is. Sir
Arthur Eddington believes that the number of electrons in the universe is
136 X 2 258 , a number which requires i followed by 79 other digits to write out
in full (see The Philosophy of Physical Science, p. 171).

THE STUDY OF LOGIC 15

have been at Public Schools are fair-minded. Someone may
reply, e l don t agree. A, who was at a Public School, is grossly
unfair. Granted the truth of the second statement, the
generalization that all Public School men are fair-minded is
disproved. The symbol C A was used to stand for a definite
person who was not specified. In the trial of blackmailers it
is sometimes necessary to conceal the name of the victim from
the public press; accordingly, he may be referred to as c Mr
A 5 . This device is convenient for it permits an individual to be
uniquely referred to throughout the trial without disclosing
his identity to the general public. The A 5 and the Mr A
used in the above examples are instances of the use of
illustrative symbols. Our purpose in using illustrative symbols
in logic is analogous to the purposes in the examples; we want
to refer to some one definite object but not to an identifiable
object; hence we use capital letters of the alphabet to serve
as an arbitrary, undescriptive name. An illustrative symbol
signifies a definite object, or characteristic, but not a specified
one. The use of x for the unknown in solving algebraic
equations is an example of the use of an illustrative symbol.
The combination of shorthand and illustrative symbols
enables us to exhibit explicitly the forms of our arguments.
To understand why it is that a given argument is valid and
another invalid, we must be able to discern clearly their
respective forms since it is upon their form that their validity
depends.

CHAPTER II

Propositions and Their Relations

I. PROPOSITIONS AND SENTENCES

In discussing examples of argument we have hitherto used
the word statement 5 to refer to what is stated by someone or
other. This word is ambiguous for it may mean either what is
stated or the verbal expression used by a speaker in stating
something. The ambiguous word was deliberately used
because we did not then wish to raise the question of dis
tinguishing these two meanings. The word proposition 3 is
frequently used for the former. A proposition is anything that
can significantly be said to be true or false. A proposition
stated in thought, in speech, or in writing, must be expressed
in words or other symbols arranged in the sort of order which
we recognize as constituting a sentence. A proposition must
not be confused with a sentence; not all sentences express pro
positions. When King Lear exclaims,

Why should a dog, a horse, a rat, have life,
And thou no breath at all?

he is asking a question, not stating anything true or false
although he was certainly presupposing the truth of a
proposition concerning the comparative value of e his fool s
life. Again, when he cries, Tray you, undo this button , he is
making a request, not stating anything. In the context of a
conversation an interrogative sentence may be apprehended
as having the force of a proposition, but, if so, the sentence-
form is simply disregarded. A rhetorical question is intended
to be understood as a statement:

Whafs Hecuba to him or he to Hecuba
That he should weep for her?

In thus passionately asking himself this question Hamlet uses

16

PROPOSITIONS AND THEIR RELATIONS IJ

the question-form to emphasize the inevitable answer - an
answer which his further argument assumes. It is not a
genuine question for the questioning-attitude was not present,
but it is present when, in the same soliloquy, he asks himself,
Am I a coward? 5 This time he is not sure what the answer is.
The same proposition may be stated by using different
sentences, e.g. "I have a dog", "I possess a dog", "Ich habe
einen Hund", "J ai un chien". These four different sentences
all express the same proposition. We shall see later that some
times the same sentence may be used to express different
propositions, for sentences no less than single words may be
ambiguous.

2. PROPOSITIONS, MENTAL ATTITUDES, AND FACTS

The four sentences given above which express the same
proposition have the same meaning; indeed, the proposition
just is what these sentences mean. What the sentence means
can be believed, disbelieved, doubted, or merely entertained
as a supposition. A thinker may have any one of these
attitudes, at different times, to the same proposition. The
preceding sentence expresses a proposition which I, the
author of this book, believe; you, the reader, may be willing
to suppose the proposition to be true in order that you may
further inquire what follows if it is true; you may doubt it and
subsequently resolve your doubt and come to have the
attitude of believing the proposition in question; or you may
disbelieve it.

"Belief 35 as ordinarily used may be ambiguous, for it may
mean the mental act of believing or that which is believed. For
the purpose of this book "belief" will always be used to mean
that which is believed. In this sense a belief means a proposition
that is believed; all beliefs are then propositions, but many
propositions are not believed. Many beliefs are not true but
every belief (being a proposition) either is true or is false and
not both true and false. A proposition, whether believed or
not, is true or false. Whether a proposition is true is deter
mined by what is in fact the case, or, more shortly, by facts.
Facts simply are; they are neither true nor false. If anyone

l8 A MODERN ELEMENTARY LOGIC

were to judge that Sir Walter Scott wrote Marmion, he would
be judging truly; it is in fact the case that Sir Walter Scott
wrote Marmion, and it would still be a fact if no one except
Sir Walter Scott knew that it was so. Obviously no example
can be given of what no one has ever thought of, but there
are many facts that have not been thought of and never will
be thought of.

Philosophers are by no means agreed as to the nature of
truth and falsity or with regard to the relation of facts to
propositions in virtue of which relation we can say that a
given proposition is true or that it is false. The discussion
of this topic belongs to the branch of philosophy called
epistemology or theory of knowledge and lies outside the
scope of this book. We must be content with the dogmatic
assertion that facts determine whether propositions are true
or false.

To disbelieve that Sirius is the nearest star to the earth is to
believe that Sirius is not the nearest star to the earth. Propositions
can always be paired in this way so that one contradicts the
other; that is, one must be true and one must be false. To
disbelieve a proposition is thus logically equivalent to believ
ing its contradictions. We are not at all concerned with the
differences there may be between the mental attitudes of
believing and disbelieving but only with the logical relations
between what is believed and what is disbelieved. Connected
with believing and disbelieving are affirming and denying.
These are mental acts familiar to everyone. If I* am asked:
e ls equality of incomes desirable? and I answer c Yes , then I
am in effect affirming that equality of incomes is desirable; if
I answer c No 3 , then I am in effect denying that equality of
incomes is desirable. Suppose my belief is that the JVb-answer
is correct: ihen I might say Equality of incomes is not
desirable 3 but I might equally well have said c Equality of
incomes is undesirable . In one case I use an affirmative, in
the other a negative sentence to express my belief, but either
sentence expresses equally well that I am denying that equality

* T stands - here and elsewhere in this book - for any thinker, unless it is
explicitly qualified to show that T, in the given context, stands for the author
of this book, viz. Susan Stebbing.

PROPOSITIONS AND THEIR RELATIONS IQ

of incomes is desirable. The distinction between affirming and
denying is fundamental: whether I affirm or deny that such
and such things are related may be of the utmost importance,
and should I pass from denying to affirming I have changed
my mind; the difference, however, between using an affirma
tive or a negative sentence to express either my denial or my
affirmation is not a logical difference; the verbal statements
will be different but both are used to state the same belief or
proposition. Every affirmative sentence can be translated into
an equivalent negative sentence, and conversely, just as I can
translate e j ai un chien by C I have a dog .

3. ASSERTION, INFERENCE, AND IMPLICATION

It is characteristic of the study of logic that at the beginning
we use certain words in the confident expectation that they
will be understood, but, later, we talk about these words,
perhaps raising difficulties that do not ordinarily occur to us
as we go about our daily business, making inferences and
seeing the implications of other people s statements. "State",
"affirm", "deny 53 are instances of this procedure* The reader
has had no difficulty in our using these words. Now, however,
we must inquire what precisely is meant by "stating a proposi
tion" , how does a stated proposition differ from that proposi
tion unstated?

In ordinary conversation when we use a sentence in the
indicative we intend our hearers to understand that we
believe the proposition. If I say The Russians resistance at
Stalingrad is magnificent 5 , I should be understood to be
stating that I believe this proposition, and am not merely
putting it forward for contemplation^ provided that I say the
sentence in the course of a discussion or in silent meditation
about the war situation in September 1942. In teaching logic
we often take examples of propositions merely in order to
investigate the logical relations between propositions of
various forms; it by no means follows from the fact that we
use a given example of a proporition that we wish to assert it.
Our attitude to the example is purely contemplative. We do
wish to make assertions to the effect that a given proposition

2O A MODERN ELEMENTARY LOGIC

(contemplated as an example) does stand in a certain relation
to another proposition (also contemplated). Nearly the whole
of this book consists of assertions which the author believes
and hopes the reader will also believe.

Without assertion there is no argument; this is equivalent
to saying that without assertion there is no inference. Since our
usual attitude is one of making declarations, putting forward
our point of view, informing one another of our beliefs, we do
not ordinarily need to call attention to the distinction between
asserting a proposition and contemplating it. The distinction is
nonetheless of vital importance. Even in ordinary conversa
tion we do not always intend to assert the propositions we
state; sometimes we take to a proposition an attitude of
hypothetically entertaining the proposition in order to see what
follows from it. But we do intend, somewhere or other, to
break the chain of hypothetically entertained propositions
and make an assertion: So this is true*. For example, e lf the
Russians continued resistance implied that the German army
could be defeated by the Russians alone, and the Russians
could continue to resist, then the German army could be
defeated by the Russians alone asserts nothing more than c if a
given implication were true and a given proposition were
true, then a given conclusion would follow . This is not the
sort of statement we should wish to make if we were anxiously
(however amateurishly) considering the possible outcome of
the war. Contrast this with, Since the Russians can continue
to resist, and since their continued resistance implies that the
German army can be defeated by the Russians alone; there-
fore, the German army can be defeated by the Russians
alone. 3 Here two assertions are made: If so-and-so, then such-
and-such is replaced by since so-and-so; therefore such-and-such.
The conclusion has been detached from the if . . . then . . .
statement and has been put forward as true, and thus as
capable of standing by itself. To assert a proposition is to put
forward the claim that the proposition is true; from the point
of view of the speaker the assertion of a proposition is the
putting forward of a belief. That the proposition is asserted
forms no part of the proposition itself. Affirming and denying
are assertive acts. The difference between the assertive and

PROPOSITIONS AND THEIR RELATIONS 21

the contemplative attitudes is fundamental; inference is
assertive. Propositions have implications whether anyone
thinks of them or not; inference involves a thinker.

Inference is a process of thought in which the thinker
passes from a certain proposition (the premiss) to another
proposition (the conclusion) because he apprehends, or
believes himself to apprehend, certain evidential relations
holding between the premiss and the conclusion, in virtue of
which relations he asserts the conclusion. It should be noticed:
(i) that evidential relations are not necessarily conclusive, they
may be probability relations; (ii) a thinker may falsely
believe that he is apprehending an evidential relation, when,
in fact, no such relation is present. He is nonetheless inferring^
but he is not justified in inferring the conclusion unless his
belief that the evidential relations are present is not mistaken.
Unfortunately, we often do make mistakes of this kind. It is
a mistake to define "inference" so narrowly that it covers
only deducing. This mistake is frequently made. It is even
worse to define inference in such a way that "inferring
invalidly" is excluded from the definition. Whether an in
ference is deductive or inductive depends upon the relations
holding between the premiss and the conclusion.

4. THE TRADITIONAL ANALYSIS OF PROPOSITIONS

Aristotle is commonly and justly regarded as the founder of
the science of logic. As Professor A. N. Whitehead says:
Aristotle founded the science by conceiving the idea of the
form of a proposition, and by conceiving deduction as taking
place in virtue of the forms. 5 * Unfortunately his successors,
for nearly two thousand years, studied in detail only a very
few forms of propositions; they tried to express anything that
anyone might want to say in one or other of four prepositional
forms together with a few other forms that were not carefully
studied at all. No clear distinction was made between a
proposition and a sentence so that some important distinc
tions were relatively neglected whilst differences in verbal
statements were treated as differences in prepositional forms.

* Proceedings of the Aristotelian Society, N.S. XVII, p. 72.

22 A MODERN ELEMENTARY LOGIC

In this section we shall be concerned with the traditional
scheme.

Consider the following propositions:

(1) All Cornishwomen are good cooks.

(2) JV0 British Ambassadors are women.

(3) Some poets are pacifists.

(4) Some voters are not householders.

Each of these propositions contains three elements - subject,
copula, predicate - and in addition a sign of quantity. The
subject and predicate are called the terms of the proposition;
the copula (some part of the verb to be) connects the predicate
with the subject; the sign of quantity shows whether reference
is made to all or to some of the members of the class con
stituting the subject-term, (i) and (2) differ in quantity from
(3) and (4)5 the former being called universal, the latter
particular propositions, (i) and (3) are affirmative, (2) and (4)
are negative; this is said to be a difference in quality. This
classification of propositions rests upon the assumption that
any proposition is a statement to the effect that one class is -
either wholly or partially - included in, or excluded from,
another class. Certainly many propositions are quite naturally
expressed in one or other of the four forms exemplified above;
our examples are not at all odd in expression. On the other
hand, many statements do not resemble any of these four
in form and cannot, without distortion of meaning, be put
into one of them. For example, e To know all is to pardon
all/

At present we neglect these difficulties, but they must not
be entirely forgotten. We shall now use the illustrative
symbols, S, P, to stand respectively for the subject and
predicate of the propositions; the four traditional forms can
then be symbolized as follows:

All S is P SaP A Universal affirmative.

JVb S is P SeP E Universal negative.

Some S is P SiP I Particular affirmative.

Some S is not P SoP Particular negative.

The third column gives the letters customarily used to name

PROPOSITIONS AND THEIR RELATIONS 23

these forms; the vowels are derived from the first two vowels
in affirmo (I affirm) and from the vowels ofnego (I deny). They
provide a convenient shorthand symbolism. The second
column shows the quantity and quality of the proposition by-
putting the appropriate vowel between the illustrative sym
bols, S and P. If the terms of the proposition were symbolized
by M and N, then the four propositions would be written as
follows: MaN, MeN, MiN, MoX. The student should
familiarize himself with this shorthand symbolism. It has long
been used for convenience only but it has one special merit
it serves to remind us that we are concerned not with
specified classes, e.g. Cornishwomen and good cooks, but with
any class. The four propositions listed on page 22 are true or
are false, i.e. they really are propositions. The second list is a
list of prepositional forms: All S is P does not assert anything
that is true or that is false; it may be regarded as an empty
schema into which may be fitted a proposition such as no. I
on page 22.

It should be noticed that universal propositions are dis
tinguished from particular propositions in that the former are
unrestricted generalizations and the latter are restricted. In
stating All Archbishops are males., reference is made to every
member of the class archbishops , in stating Some architects are
women reference is not made to every member of the class
architects. This difference is technically named a difference in
distribution. The decision whether a term is distributed or
not is of primary importance in determining the validity of
certain of our inferences. Hence, it is desirable for the student
to familiarize himself with this notion; the following defini
tions should be learnt:

A term is distributed, in any proposition, if reference
is made to every member of the class for which the term
stands.

A term is undistributed, in any proposition, if reference is not
made to every member of the class for which the term stands.
It is easy to see that the subject-terms of universal propositions
are distributed, whilst the subject-terms of particular pro
positions are undistributed. With regard to the predicate-
terms the determination is not so simple. No Eskimos are

24 A MODERN ELEMENTARY LOGIC

sculptors does clearly exclude the whole class of sculptors from
the class of Eskimos no less than it excludes the Eskimos from
the sculptors. Hence, the predicate-term is also distributed.
In the particular proposition. Some socialists are not Marxists., it
is stated that the whole class of Marxists is excluded from some
socialists. Thus the predicate-term is distributed. In the pro
position All Cabinet Ministers are Members of Parliament the
reference is not to the whole class of Members of Parliament ,
consequently the predicate-term is not distributed. Likewise
in the proposition Some policemen are detectives the predicate-
term is not distributed. The following table sums up these
conclusions which we have obtained by considering specific
examples of the four forms:

Proposition

Subject

Predicate

A

All S is P

distributed

undistributed.

E

No S is P

distributed

distributed.

I

Some S is P

undistributed

undistributed.

Some S is not P

undistributed

distributed.

It should be noted that in these forms "some" must be taken
to mean "some at least", which is equivalent to "some and
perhaps all". In ordinary English we most commonly use
"some" to mean "some only"; thus Some A.R.P. workers are
paid would probably be understood to mean that some were
paid and some were not paid. But it might be used to mean
that some at least were paid, leaving it still open to question
whether all were. Now, if we were to interpret "some" in
Some S is P to mean "some only", then this proposition would
be in fact, though not in linguistic form, the conjoint assertion
of both the / and the propositions, for it would assert Some
A.R.P. workers are paid and some A.R.P. workers are not paid. It
is, therefore, desirable to give the minimum interpretation to
"some"; we thus interpret "some" so that it is consistent with
"all" but excludes the meaning of "none". Accordingly pro
positions A and / are consistent, and E and are consistent
as thus interpreted.

If we take S and P to stand for two different unspecified
classes, there are five different relations possible between

PROPOSITIONS AND THEIR RELATIONS 25

them, ranging from complete coincidence to complete
mutual exclusion:

1. The two classes may completely coincide.

2. The first may be wholly included in the second without
coinciding with it.

3. The first may wholly include the second but not
coincide with it.

4. The two classes may partially overlap, i.e. each partially
includes and partially excludes the other.

5. The two classes may wholly exclude each other.

The mathematician Euler (1707-83) represented these
class relations diagrammatically, using circles whose spatial
relations have some analogy with the logical relations of the
two classes. These diagrams, known as Euler s Circles, are:

It is important to notice that there are four prepositional
forms and five diagrams; hence there is not a simple corre
spondence between the prepositional forms and the circles.
This is due to the fact that propositions are used to state what
we know or believe; and what we know is usually not
determinate. If we knew, with regard to some class S and
some other class P, that they were related precisely in the way
in which the two circles in diagram 4 are related, we should
know more than any one of the A, E, /, propositions can
state. Since an undistributed term is indeterminate in its
reference, a proposition containing an undistributed term
cannot be represented by any one of Euler s diagrams. Only

26 A MODERN ELEMENTARY LOGIC

diagram 5 corresponds to a single proposition of the four-fold
scheme, viz. E 9 which is the only proposition in which both
terms are distributed, thus giving information with regard to
the whole extent of each term. To state the information
provided by each of the first four diagrams it is necessary to
affirm conjointly two or more of the propositions. The follow
ing table expresses in terms of Euler s diagrams the informa
tion provided by each of the four propositions:

A allows i, 2; excludes 3, 4, 5.

E allows 5; excludes i, 2, 3, 4.

/ allows i, 2, 3, 4; excludes 5.

allows 3, 4, 5; excludes i, 2.

Unless at least one possibility represented by the five diagrams
is excluded no information has been given; to know that
trepangs are wholly or partially included in, or excluded from,
the class of echinoderms is to know nothing more about tre
pangs than can be known by logic alone. We might just as
well- replace trepangs by T, and echinoderms by E. This is
indeed equivalent to what we have done by using the symbols
S, P, to illustrate any two different classes. If, however, we
are told that trepangs are wholly included in the class of
echinoderms, we know that diagrams 3, 4, and 5 are excluded.
If, now, we further know that trepangs are wholly included in
echinoderms without exhausting that class, we know that their
relation corresponds uniquely to diagram 2. This information
can be given by the conjoint assertion of an A and an
proposition: All trepangs are echinoderms and some echinoderms are
not trepangs.

At this point an intelligent student might well ask such
questions as the following:

1. What about those things which are neither trepangs
(whatever these may be) nor echinoderms? Are they sup
posed to lie outside the circles? If so, where in the diagram
are they represented?

2. If I say, Ghosts are not always draped in sheets , am I
to draw a circle representing ghosts even if there aren t any
ghosts in the world?

PROPOSITIONS AND THEIR RELATIONS 2J

To answer these questions it will be necessary to raise other
questions which go beyond the traditional treatment of
propositions. Accordingly these questions will be answered in
a later chapter.

5. SIMPLE, COMPOUND, AND GENERAL PROPOSITIONS

Among the simplest statements we can make are those which
attribute a characteristic or property to an individual thing,
e.g. That leaf is green, That table is round, Roosevelt is wise. We
shall adopt the convention that such propositions as these are
simple and that they are subject-predicate propositions. The
subject is that to which some characteristic is attributed; the
predicate is that which is attributed to the subject. Simple
propositions are to be contrasted with compound propositions
and with general propositions. Consider the following:

A. (i) The line AE is equal to the line BC.

(2) Aristotle was tutor to Alexander the Great.

B. (3) If the angle BAG is not equal to, or less than, the

angle EDF, then it is greater than the angle EDF.

(4) If Winston Churchill has visited Moscow, then
Stalin will be pleased.

(5) If Tom has matriculated, then he cannot be less
than sixteen.

(6) Either Sirius is not larger than the sun or it is much
farther from the earth than the sun is.

(7) It is not the case both that fuel economy is un
necessary and that also the production of coal is
decreasing.

(8) Paul is in the R.A.F. and Marion has joined the
A.T.S.

According to the convention we have adopted, the proposi
tions of set A, as well as those in the first paragraph above,
are simple. Those of set B are compound. A compound
proposition contains two or more component propositions.
Thus in (4) there are the two components: Winston Churchill
has visited Moscow and Stalin will be pleased. Each of these could
significantly be separately asserted but they are not so

28 A MODERN ELEMENTARY LOGIC

asserted; what is asserted is that the second is consequent
upon the first, hence, the second is called the consequent and
the first is called the antecedent. (3) and (5) are other examples
of this form; they are called hypothetical propositions. What is
common to these three propositions is that each as a whole
asserts that the antecedent implies the consequent, in the
sense that the antecedent cannot be true without the con
sequent s also being true. The antecedent is the implying
proposition, the consequent the implied proposition. The rela
tion between these in virtue of which the implication holds is
different in different cases, e.g. in (3) it is due to certain
definitions in geometry, in (4) to certain political and military
conditions in Europe in 1942, in (5) to certain university
regulations. It should be noted that the truth of the hypo
thetical depends not at all on the truth of the antecedent or
the consequent separately considered but only on the relation
asserted to hold between them. It has sometimes been held
that a hypothetical proposition expresses doubt. This is a
mistake. The intention of anyone who asserts (4), for instance,
is not to express doubt whether Churchill has visited Moscow
but to assert a consequence of the visit were it in fact made.*

(6) is an example of an alternative proposition; it asserts
that at least one of the two component propositions is true, not
excluding the possibility that both are. The component
propositions are called alternants^ there may be any number
of alternants. The interpretation of either ... or ... as non
exclusive has the same logical justification as the interpreta
tion of some, in / and propositions, to mean some at least and
perhaps all; namely, that ambiguous expressions should be
given minimum significance. Common usage of either . . . or
. . . varies. To say Tom is either stupid or idle does not
necessarily exclude the possibility that he is both. On the
other hand, to say either immediate aid must be given to
U.S.S.R. or national unity will be split from top to bottom
would probably be intended to be taken as asserting exclusive
alternatives.

(7) is an example of a disjunctive proposition; it asserts that

* The student who knows some Latin should consider, from this point of
view, the logical basis of the rules for conditional sentences in Latin.

PROPOSITIONS AND THEIR RELATIONS 2Q

not both of two component propositions are true, and is
consistent with neither s being true. The component pro
positions are called disjunct*; there may be any number of
disjunct*.

Compound propositions fall into two distinct kinds: (i)
composite, including hypothetical, alternative, and disjunctive
propositions; (ii) conjunctive propositions. (8) is an example of
a conjunctive proposition. The three forms of composite
propositions are related to one another in such a way that
anything stated in one of these forms can be equivalent^
stated in either of the other two forms. How this can be done
will be explained in 6.

At the beginning of this section we said that certain
propositions, of which examples were given, would be
regarded by us as simple subject-predicate propositions. Set
A provides other examples of simple propositions but they
are not subject-predicate propositions; they are relational
propositions: The line AE is equal to the line BC states that the
relation of equality holds between the two lines named respec
tively AE, BC. There are various kinds of relations which
must later be distinguished. At present it is enough to notice
that a relation requires at least two entities standing in the
relation; the entities between which a relation holds are
called the terms of the relation. In the proposition Andrew is
twin of Mary, the terms are obviously Andrew, Mary.

The notion of a simple proposition is itself not at all simple.
Some logicians consider that, for instance, This is white is an
absolutely simple proposition. We reject this view but must
here be content to say only that we regard a proposition as
simple provided that (i) it does not contain other propositions
as components (ii) and includes in its verbal expression a
word, or set of words, which uniquely indicates an identifi
able object.* The traditional Logicians did not approach the
analysis of propositions from this point of view. They seem to
have assumed that a grammatically simple sentence expressed
always a simple proposition, and that a grammatically com
plex sentence expressed always a compound proposition.

* We shall see later that this Is equivalent to saying that a simple proposition
is one that does not involve any reference to variables in its analysis.

30 A MODERN ELEMENTARY LOGIC

Thus the sentence "All schoolmasters are fallible" and the
sentence "Thomas Arnold is fallible" were regarded as alike
expressing simple propositions; whereas the sentence "If a
man is a schoolmaster, he is fallible" was taken to express a
compound proposition. This is a mistake - "All school
masters are fallible" and "If a man is a schoolmaster, he is
fallible" are verbally different statements of the same pro
position, and it is not simple. The proposition expressed by
"All schoolmasters are fallible" is clearly an A proposition.
Propositions stating that one class is, wholly or partially,
included in, or excluded from, another are general proposi
tions. These, it will be remembered, are the A, E 9 /,
propositions of the traditional schema. It is a complete
muddle to regard such propositions as simple although it is
true that they cannot be analysed into the combination of
two, or more, simple propositions. We must, then, distinguish
these general propositions both from simple propositions and
from the compound propositions with which we have so far
been concerned. We shall see later exactly why it is that
particular propositions (/, 0) are correctly said to be general.

6. THE SEVEN RELATIONS BETWEEN PROPOSITIONS
AND THE FIGURE OF OPPOSITION

We have already seen how the possible truth or falsity of one
or more propositions limits the truth or falsity of others, and
we have had no difficulty in recognizing, in earlier sections,
pairs of contradictory propositions and pairs of equivalent
propositions. Unless we were able to recognize some cases of
contradiction and to discern equivalence in spite of verbal
difference we could hardly begin the study of logic, since
logic arises from reflection upon our attempts to think
problems out. But to be able to recognize logical relations in
some instances is not the same as knowing clearly exactly
what these relations are. In this section we shall be concerned
with seven relations between propositions which are of funda
mental importance. Every discussion concerning valid in
ferences in the book may be regarded as illustrating one or
other of these seven relations; it is thus important that they

PROPOSITIONS AND THEIR RELATIONS 3!

should be thoroughly understood. Consider the following
eight propositions:

(a] Human nature never changes.

(b) If human nature never changes, wars will not cease.
(c} If human nature does change, wars will cease.

(d) Wars \sill not always go on.

(e] Wars will not cease.

(/) Human nature always remains the same.
(g) Human nature can rise to sublime heights.
(h) Human nature does change.

These propositions are either about human nature or about
wars or about the connexion between human nature and war.
But propositions may be about the same subject-matter and
yet not be logically connected, e.g. (a) and (g). These could
both be true or both be false or one true with one false; thus,
the truth or falsity of one is logically independent of the truth or
falsity of the other. Other pairs of independent propositions
are contained in the list, e.g. (g), (K). The student should
select for himself other pairs. Some propositions in the list are
not independent of others in the list; (d) denies what (K)
asserts; these are contradictories of each other. At first sight
it may seem that (b) and (c] are contradictories; a little
reflection, however, will show that this is not the case: there
is no contradiction in saying that wars will go on under
certain conditions (e.g. provided that human nature does not
change) but not under other conditions (e.g. provided that
human nature does change); hence (b) and (c) are also
independent of one another.

Let us now assert (b} together with (a), thus obtaining the
conjunctive proposition: If human nature never changes, wars will
not cease and human nature never changes. What is the relation
between this conjunctive proposition and (e) given above? If
(b) and (a) are both true, then (i) must also be true; but
(e) may be true even though the conjunction of (b} with (d) is
false. Thus the truth of (e) leaves the truth of the conjunction
of (b} with (a) undetermined. Other propositions thus related
will be found in the list; propositions so related that if the
first is true the second is true, but if the second is true the

32 A MODERN ELEMENTARY LOGIC

truth or falsity of the first is not thereby determined, are said
to be in the relation of superimplicant to subimplicant,

(a) and (/) are verbally different but both assert the same
matter of fact; hence, either they are both true or both false.
These propositions are said to be equivalent.

We have now recognized, by means of significant examples,
four of the seven distinct logical relations that may hold
between one proposition, or set of propositions, and another
proposition, or set of propositions. We shall now define these
and the remaining three relations. Using/?, q as illustrative sym
bols for different propositions, the definitions are as follows:

(1) Equivalence or Co-implication: p and q are equivalent, or
co-implicant, when they are so related that if p is true, q is
true, and if q is true, p is true; and \p is false, q is false, and
if q is false, p is false. Thus, p == q, if they are true together
or false together. This is the relation that holds when p
implies q and q implies p. The name co-implicant brings out
this relation.

(2) Superimplication or Super alternation: p is superimplicant to q
provided that if p is true, q is true, but q may be true
although p is false. Thus the truth of q leaves the truth of p
undetermined.

(3) Subimplication or Sub alternation: p is subimplicant to q pro
vided that if q is true, p is true, but p may be true although
q is false. The relation of subimplication is the converse of the
relation of Superimplication; hence, when p is superimplicant
to q, then q is subimplicant to p.

(4) Independence: p is independent of q when neither the truth
nor falsity of p determines the truth or falsity of q\ and
conversely.

(5) Sub contrariety: p is subcontrary to q provided that, if p is
false, q is true, and if q is false, p is true, whilst p and q can
be true together. The excluded case is the conjoint falsity of
p and q.

(6) Contrariety: p is contrary to q provided that, Up is true, q is
false, and if q is true, p is false, whilst p and q can be false
together. The excluded case is the conjoint truth of p and q.

(7) Contradiction: p and q are contradictories of one another
provided that, ifp is true, q is false, and if ^ is false, q is true;

PROPOSITIONS AND THEIR RELATIONS 33

hence, p and q cannot be true together or false together, i.e.
one must be true and one false.

These relations are relations of consistency or inconsistency;
if any of the first five hold between propositions they are
consistent, if either of the last two, they are inconsistent. The
relation of independence combines consistency with com
plete lack of any conditions necessary for inference. This lack
of any possible inferential connexion is clearly shown by
propositions (g) and (rf), for instance, on page 31 ; it is present
equally in the case of (b) and (c) although not so easily
apprehended. Contraries are not less mutually inconsistent,
or incompatible, than contradictories; the former differ from
the latter in that there are non-equivalent alternatives to both
of two contrary propositions.

These seven relations are summed up in the following
table, in which p is true is represented by p s p is false by p, and
likewise with q, and q.

Relation

Given

then q or q Given

then q or q

p equivalent to q

P

9 P

q

p superimplicant to q

P

^ P

undetermined

p subimplicant to q

P

undetermined p

q

p independent of q

P

undetermined I p

undetermined

p subcontrary to q

P

undetermined p

q

p contrary to q

P

q P

undetermined

p contradictory to q

P

3 P

q

In considering these relations between propositions we
have not confined our attention to the traditional schema,
the A 9 E, /, propositions. Since every proposition stands to
every other proposition in one or other of these seven rela
tions, they must be so defined as recognizably to hold
between propositions of any form whatever. The traditional
Logicians, thinking of propositions as differing only in
quantity and quality or both, constructed c the Square of
Opposition 5 . The word "opposition" is here used in a
technical sense which permits compatible propositions to be
opposed. Thus "opposition" must be defined as follows: Two
propositions are opposed if they differ in quantity or in quality
or in both quantity and quality. Those differing in quality

34 A MODERN ELEMENTARY LOGIC

but not in quantity are contraries (if quantity universal), sub-
contraries (if quantity particular}. Those differing in quantity
and quality are contradictories. Those differing in quantity but
not in quality are subaltern. It is easy to construct the Square
of Opposition by taking the diagonals of the Square as joining
respectively the two pairs of contradictories, viz. A and 0,
E and /. The student may be left to work this out for himself.
Here the traditional oppositions will be represented by an
incompletely symmetrical figure, since the perfect symmetry
of a square is not fitted to represent unsymmetrical relations.

Contraries

This Figure of Opposition illustrates the following facts:

(i) No two of the traditional A, E, I, propositions are equivalent and
no two are independent.

(ii) The two universal forms are contraries.

(iii) The two particular forms are subcontraries.

(iv) Universals and particulars differing in quality are contradictories.

(v) The universal form is superimplicant to the particular of the same
quality, the latter being subimplicant to the former.

The traditional Square does not illustrate clearly the important
distinction between superimplication and its converse.

The following table presents in summary form what may
be validly inferred, given the truth or the falsity of these
propositions:

Given

It can be inferred

A true

false

/true

false

E true

A false

/false

true

/true

A undetermined

E false

undetermined

true

A false

E undetermined

/ undetermined

A false

E undetermined

/ undetermined

true

E false

A undetermined

/true

undetermined

/false

A false

E true

true

false

A true

E false

/true

PROPOSITIONS AND THEIR RELATIONS 35

It will be seen that the truth of either of the universal propositions
determines the truth or falsity of the other three; the falsity of either of
the particular propositions determines the truth or falsity of the other
three. But the truth of the particulars leaves two undetermined cases,
and thefalsitv of the universals leaves two undetermined cases.

7. IMMEDIATE INFERENCES

We have already seen that propositions whose verbal state
ment is different may be equivalent. Consider the following
two pairs of propositions: (i) All canned meats are rationed goods :;
jVb canned meats are unrationed goods , (ii) Some Cabinet Ministers
are intelligent , Some Cabinet Ministers are not unintelligent. In each
pair the propositions are equivalent, their subject-terms are
the same but their predicate-terms are contradictories. Terms
are contradictory when they stand respectively for two classes
which are mutually exclusive and together exhaust the wider
class within which both fall. Thus, for example, if the wider
class is goods, then every member of this class falls either under
the subclass rationed goods or under the subclass unrationed goods.
Hence to assert that all canned meats are included in the class
of rationed goods is equivalent to asserting that no canned
meats fall in the class of unrationed goods. It may be objected
that this is not the case with pair (ii) since being intelligent is
not exactly the same as being not unintelligent. This may be
admitted since we ordinarily so use "not unintelligent" as to
suggest a considerable degree of intelligence. This illustrates
the figure of speech meiosis, in which what is said intentionally
gives an impression that something is less than is really the
case; hence the terms may be regarded as contrary rather
than contradictory. To avoid misunderstanding we can
always affix non to the affirmative term, e.g. non-intelligent. It
must always be remembered that in ordinary discourse what
we convey is in part dependent not only upon the context but
also upon intonation, emphasis, and even subtle changes in
facial expression. For the purpose of discussing logical rela
tions we ignore these characteristics of speech.*

It is a distinguishing characteristic of equivalent proposi
tions that one can be substituted for the other, in any

* To ignore them is justifiable in an elementary textbook, but this does not
mean that they do not need investigation.

36 A MODERN ELEMENTARY LOGIC

argument in which either occurs, without affecting the validity
of the argument. Equivalent propositions can be inferred one
from the other.

It has been customary to distinguish inferences as being
either mediate or immediate. Usually a conclusion is inferred
from one premiss together with one or more other premisses;
in such cases the inference is said to be a mediate inference. An
inference is said to be immediate if the conclusion is inferred
from a single proposition. This distinction is not of funda
mental logical importance but it is convenient to retain it.
Certain forms of immediate inference are traditionally
recognized; we shall deal with them briefly.

In inferring one proposition from another care must be
taken to see that the inferred proposition (or conclusion) does
not assert anything not implied in the original proposition
constituting the single premiss; it is, however, legitimate to
assert less. This restriction is a special application of an
important principle of deduction: Do not go beyond the evidence.
Hence, if in the given proposition a term is undistributed,
that term must not be distributed in the inferred proposition.
It has been customary to allow a conclusion having an undis
tributed term to be inferred from a premiss in which that term
is distributed. In such cases the given proposition will be
superimplicant to the conclusion.*

Before we state the immediate inferences customarily
accepted we must consider an assumption upon which their
validity, in some cases, rests. Suppose we wish to consider a
set of students as possessing or not possessing the char
acteristics of being able and hard-working. We should expect
to find the following cases: those who are both able and hard
working; those who are able but not hard-working; those who
are not able but are hard-working; those who are neither
hard-working nor able. We have, then, four mutually
exclusive and collectively exhaustive classes of students.
Using H to stand for hard-working, non-Hfor its contradictory,
A for able, non-A for its contradictory, the four classes can be
symbolized by AH, A non-H, non-AH, non-A non-H. We have
assumed that students are contained in each of these four

* We shall see later that such inferences are not strictly valid.

PROPOSITIONS AND THEIR RELATIONS

37

classes. It might be the case that there were no students who
are both non-H and non-A; the fourth class will then be said
to be empty. If any class contains members, we say that the
class (which will be determined by one of the characteristics)
is existent. Representing any subject-term and any predicate-
term and their contradictories respectively by S 9 non-S, P,
non-P.. the assumption upon wiiich the validity of the tradi
tional immediate inferences is based can be stated as follows:
S 9 77072-5, P, non-P all exist, i.e. no one of the classes is
empty.

Traditional immediate inference depends upon two funda
mental operations, namely, obversion and conversion.
(i) Obversion. To assert S is P is equivalent to denying S is
non-P. Thus it is always possible to obtain a proposition
equivalent to a given proposition by substituting for the
original predicate its contradictory and by changing the
quality of the proposition. Its technical definition is: Obversion
is a process of immediate inference in which from a given proposition
another is inferred having for its predicate the contradictory of the
original predicate.

SCHEMA OF OBVERSION

Original proposition Obverse

A

All S is P

= No S is non-P

E

E

NoSisP

== All S is non-P

A

I

Some S is P

== Some S is not non-P

Some S is not P

= Some S is non-P

I

The symbol == between the original proposition (called the obvertend]
and the obverse shows that they are equivalent: the quality is changed
but the quantity remains unchanged.

Examples of significant obversion:
Obvertend

Obvert

No snobs are welcome guests = All snobs are unwelcome guests.
All Quislings are contemptible == No Quislings are other than con
temptible.

(2) Conversion. By the converse of a proposition is ordinarily

38 A MODERN ELEMENTARY LOGIC

meant another proposition in which the terms are inter
changed. For example. All equilateral triangles are equiangular
and All equiangular triangles are equilateral would be regarded
as converses. But neither can be said to be immediately
inferred from the other since such an inference would violate
the rule that no term may be distributed in the inferred
proposition unless it was distributed in the original proposi
tion. These are both A propositions, in which the subject-
term is distributed but the predicate-term is undistributed.
The technical definition is: Conversion is a process of immediate
inference in which from a given proposition another is inferred having
for its subject the original predicate.

From No snobs are welcome guests we can infer No welcome
guests are snobs. In each of these propositions both terms are
distributed: the propositions are equivalent. From Some Irish
men are air-gunners we can infer Some air-gunners are Irishmen.
These propositions are also equivalent since, in each of the
propositions, both terms are undistributed.

From All landowners are capitalists we cannot infer that All
capitalists are landowners, since the subject of the converse is
distributed but was given undistributed in the original
affirmative proposition of which it is predicate. Hence, such
a converse is illegitimate; we must infer the weaker proposi
tion, Some capitalists are landowners. The proposition thus in
ferred is said to be weaker 3 than the original since it is not
possible to pass back from it to the original; the converse in
the case of an A proposition is subimplicant to the original.
Accordingly, it is said that an A proposition admits only of
conversion by limitation , this is commonly called by the Latin
term conversion per accidens.

From the proposition, Some brachiopods are not bivalves we
cannot infer Some bivalves are not brachiopods, since, in the
inferred proposition the predicate (brachiopods} is distributed,
whereas it was given undistributed as the subject of a par
ticular proposition. It is in fact true that some bivalves are
not brachiopods, and in fact, no brachiopods are bivalves.
But we assert this from information not provided by the
original statement, which was in the form of an 0, not an E,
proposition.

PROPOSITIONS AND THEIR RELATIONS
SCHEMA OF CONVERSION

39

Original proposition

Converse

A AllS is P ->

Some P is S

I

E No SisP ==

No PisS

E

I Some S is P ==

Some P zV S

I

O Some S is not P

None

It should be noted that the converse is the same in quality as the
original. The symbol -> shows that the converse of A is not equivalent
to A but subimplicant to it.

(3) Contraposition. The converse of a proposition can, of
course, be obverted, and an obverse be converted. Hence,
other forms of immediate inference may be obtained by suc
cessively converting and obverting, in either order. There are
two forms which have received special names, namely, con
traposition and inversion.

Contraposition is a process of immediate inference in
which from a given proposition another is inferred having for
its subject the contradictory of the original predicate. From
No mammals are fish we obtain by obversion All mammals are
non-fish, from this, by conversion, Some non-fish are mammals]
and by obverting this we obtain, Some non-fish are not non-
mammals. The two latter satisfy the definition of contraposi
tion, and are obverts of one another.

SCHEMA OF CONTRAPOSITION

Original proposition Contrapositive

Obverted contrapositive

(A) All S is P
(E) No S is P
(I) Some SisP
(0) Some Sis not P

= No non-P is S (E)
-> Some non-P is S (I)
None
33 Some non-P is S (/)

= All non-P is non-S
-> Some non-P is not non-S
None
= Some non-P is not non-S

w

(0)
(0}

It should be noted that / has no contrapositives, since / obverts to 0,
and O has no converse, E has not an equivalent contrapositive, since E
obverts to A, and A has a non-equivalent converse.

(4) Inversion is a process of immediate inference in which
from a given proposition another is inferred having for its
subject the contradictory of the original subject. Thus it is

4-O A MODERN ELEMENTARY LOGIC

required to obtain from a proposition of the form S-P (where
quantity and quality are not specified) a proposition of the
form non-S - non-P, ornon-S-P. By obversion we obtain the
contradictory of the predicate-term. Hence, if we can infer a
proposition having S as predicate, its obvert would have
non-S as predicate; if this proposition admits of being con
verted we should have a proposition of the required form. If
the last proposition is an proposition it cannot be converted.
On trial it will be found that by alternately obverting and
converting (in that order) we can obtain from A a proposition
of the required form; by alternately converting and ob verting
(in that order) we can obtain from E a proposition of the
required form. An inverse cannot be obtained either from the
/ or the proposition, since, in each case, in attempting to
obtain a proposition with non-S as subject we succeed only in
obtaining one with non-S as predicate in an proposition,
which cannot be converted. The process required to obtain
inverses from A and from E are set out below:

A All Sis P. E MS is P.

obv. JVb S is non-P, conv. No P is S.

conv. JVb non-P is S. obv. All P is non-S.

obv. All non-P is non-S. conv. Some non-S is P.

conv. Some non-S is non-P. obv. Some non-S is not non-P.

obv. Some non-S is not P.

The required inverses are the underlined propositions. It will
be seen that the obverted inverse of A is Some non-S is not P.
This inference, therefore, breaks the rule of distribution, since
P was not distributed in All S is P. Yet this inference has been
obtained by using only the processes of obversion and con
version which are taken to be valid. This result ought to
puzzle us. If we take a significant example the result may
well be absurd, e.g. All honest politicians are mortals has, as its
obverted inverse, Some dishonest politicians are not mortals, and
for the other inverse, Some dishonest politicians are immortals. The
result is absurd because, relying on information about the
world not derived from logic, we claim that the original
proposition is true and the inverses are false. But any proposi
tion implied by a true proposition is true; if, therefore, using
only the processes of obversion and conversion we obtain a

PROPOSITIONS AND THEIR RELATIONS 4!

false proposition from a proposition admittedly true, we must
begin to doubt whether these processes are valid. We find it
necessary, then, to examine the assumptions upon \vhich the
validity of conversion and obversion rest. Our reason for
thinking that Some dishonest politicians are immortal is false is
that we do not believe that there are any immortal men;
accordingly, we assented to the statement that all honest
politicians are mortals. If, however, there are immortal men
and honest politicians are wholly included in the con
tradictory class, viz. mortal men, then immortal men must
include dishonest politicians. But it is by no means logically
necessary that every class represented by S, non-S, P, non~P,
should have members; hence, the assumption that none of
these classes is empty must be made explicit. If we assume
Something is not P, then we have an additional premiss in
which P is distributed, but if inversion requires this additional
premiss it can hardly be regarded as a process of immediate
inference in the sense in which "immediate inference" has
been defined. The difficulty we find in the illicit process of the
predicate-term in passing from All S is P to Some non-S is not P
suggests that immediate inferences may not be valid apart
from implicit assumptions, which must be made explicit. The
relevant assumption is that S, non-S, P, non-P are none of them
empty. If this be admitted, then, if All S is P, it follows that

SUMMARY OF IMMEDIATE INFERENCES

Form

A

E

I

Original proposition .

SaP

SeP

SiP

SoP

Converse

PiS

PeS

PiS

Obverse ....

SeP

SaP

SoP

SiP

Obverted converse

PoS

PaS

PoS

Contrapositive .

PeS

PiS

PiS

Obverted contrapositive

PaS

PoS

PoS

Inverse ....

SiP

SiP

Obverted inverse

SoP

SoP

42 A MODERN ELEMENTARY LOGIC

non-P cannot be S 9 so that non-P must be non-S> i.e. some non-S
is non-P. We shall see later that an assumption of existence is
always required to render valid the inference of a particular
proposition from a universal proposition.

The traditional immediate inferences we have been con
sidering may be conveniently summed up in the table given
on page 41. We shall henceforth write non-S as 5, and non-P
as A

CHAPTER III

Compound Propositions and Arguments

I. EQUIVALENTS AND CONTRADICTORIES

In 5 of the last chapter we distinguished two kinds of com
pound propositions, namely, conjunctive and composite proposi
tions. In this chapter we shall be concerned to see what
exactly is asserted by stating any one of these propositions.
We shall begin by considering two propositions, illustratively
symbolized by p and by q respectively, and their con
tradictories, symbolized by p, q. These may be combined
conjunctively as follows: (i) p and q, (2) p and q, (3) p and q,
(4) p and q. The order in which the conjuncts are asserted is
indifferent; for instance, there is no logical difference between
Dickens is a great novelist and Anthony Trollope is a good storyteller
and Anthony Trollope is a good storyteller and Dickens is a great
novelist. Which of the two components in each proposition we
assert first will be determined by the context of the discussion
in which one or other of them happened to be asserted. If one
compound were asserted no one would feel any need to assert
the other.

It may seem easy to state the denial of any proposition; we
all know how to contradict our neighbour. But it is not always
easy to distinguish at once between denial by affirming the
contrary and denial by affirming the contradictory. We sometimes
fly to extremes and thus assert more than we need. In some
cases, in everyday discussion, we even at times mistake two
independent propositions for contradictories.* How should
we contradict Every prospect pleases and only man is vile? This
asserts both conjuncts to be true; to deny it must mean to
assert either that both conjuncts are false or that at least one
is false. The former is the assertion of the contrary of the

* For example, propositions (b) and (c) on page 31. The student should
formulate the contradictories of these propositions.

43

44 A MODERN ELEMENTARY LOGIC

original conjunctive proposition, the latter of the con
tradictory. These are often confused. The contrary is: Neither
does every prospect please nor is man only vile", the contradictory is:
Either not every prospect pleases or not only man is vile. This con
tradictory can be also stated in the form. It is not the case that
every prospect pleases and also that only man is vile. The student
should convince himself that both these contradict the
original proposition. The conjoint assertion of p with q is
equivalent to the denial that// and q can be disjoined; hence
the disjunctive Not both p and q contradicts Both p and q; it is
also clear that if not both of two propositions can be asserted,
then at least one must be denied; hence a conjunctive can be
equally well denied by an alternative proposition.

Ordinary statements in different composite forms can easily
be seen to be equivalent. Consider the following:

(i) Either Martin is stupid or Jones is a bad teacher.

(ii) If Martin is not stupid, Jones is a bad teacher,
(iii) If Jones is not a bad teacher, Martin is stupid,
(iv) Not both Martin is not stupid and Jones is not a bad teacher.

If we write p for Martin is stupid, q for Jones is a bad teacher, and
p, q for their respective contradictories, we can exhibit the
form of these four propositions as follows: (i) Either p or q;
(ii) If p, q; (iii) If q, pi (iv) Not both p and q. These are all
equivalent to one another and are consequently alike con
tradicted by the conjunctive Both p and q.

It will be noticed that we have two hypothetical proposi
tions in the list above and that they are equivalent. The one
is constructed from the other by separately contradicting the
original antecedent and consequent and then reversing them,
so that the contradictory of the original consequent is the
new antecedent and conversely. We saw that the order of the
components of a conjunctive proposition is logically in
different; the same holds of the order of the disjuncts in a
disjunctive, and of the alternants in an alternative proposi
tion. In the case of hypothetical propositions this is not so.
If he is a hard worker, he will be successful is not equivalent to If
he will be successful, he is a hard worker, there are other condi
tions of success - he may be lucky or unusually clever. Using
X to stand for any one statement, and T for any other, we

COMPOUND PROPOSITIONS AND ARGUMENTS 45

must notice that If X, then T is logically independent oflfT,
then X: the former asserts that X is sufficient to the truth of T\
the latter that T is sufficient to the truth of X. These may both
be true, but either may be true \vithout the other being true.
We must also notice that unless ordinarily means if not . . .,
and is not equivalent to only if not . . .; the former states a
condition that is sufficient, the latter a condition that is neces
sary, but a condition may be sufficient without being neces
sary; for example. Unless it is wet, I shall go for a walk asserts
that I shall go for a walk if it is not wet, but this is not
equivalent to saying Only if it is not wet, I shall go for a walk,
for I might go for a walk even if it were wet because I am tired
of staying indoors or I want to please a friend. In an ordinary
conversation the context should suffice to show in which sense
"unless" is being used.

The lack of symmetry in the relation of/? to q in Ifp, then q,
which makes the simple conversion If q, then p invalid, is
again due to our accepting the minimum interpretation of
statements, as in the case of Either p or q. To interpret either
... or ... exclusively is equivalent to asserting Either p or q
and not both p and q, i.e. to the conjunction of an alternative
and a disjunctive proposition. To interpret If p, then q as
asserting that j& is sufficient to the truth of q without at the same
time asserting it to be necessary to the truth of q is to avoid
committing ourselves to the maximum assertion that p is both
sufficient and necessary to the truth of q. If we wish to make this
latter assertion we can do so by the conjunctive If p, then q,
and if q, then p. In science we frequently wish to assert that p
implies q and also q implies p\ i.e. we seek a pair of proposi
tions in which the implying component of one is the implied
component of the other. Frequently, however, this is not
possible: we know that loss of appetite is consequent upon a
certain bodily disease, but it may also be consequent upon a
deep sorrow. Medical scientists seek to find whether there are
common factors, which could be medically treated, in these
two cases, and, if so, what they are; but medical scientists are
not always successful. Hence, we must avoid the mistake of
invalidly inferring If q, then p from Ifp, then q. The conjoint
assertion of these two propositions is of special importance for

46 A MODERN ELEMENTARY LOGIC

the advance of knowledge; they have been called com
plementary propositions. Likewise Either p or q and Not both p
and q are called complementary propositions.

The term complementary [says W. E. Johnson] is especi
ally applicable where propositions are conjoined in either
of these ways, because separately the propositions represent
the fact partially, and taken together they represent the
same fact with relative completeness.*

This point may be further illustrated by the pair of general
propositions represented by SaP, PaS. These are comple
mentary; they are consistent but neither can be validly
inferred from the other. Together they assert that the class S
is wholly included in the class P and the class P is wholly
included in the class S, i.e. the classes S and P are co
extensive; e.g. Every triangle whose base angles are equal is
isosceles and every isosceles triangle has its base angles equal. The
contradictory of the conjunctive proposition SaP and PaS is
Either SoP or PoS. Thus All Germans are Nazis and only Germans
are Nazis is contradicted by Either some Germans are not Nazis
or some Nazis are not Germans. It must be remembered that
either . . . or is interpreted as non-exclusive.

The table below sums up the equivalences between the
composite forms, together with the contradictory in each
case. It should be observed that Ifp, then q and Ifq, thenp are

EQUIVALENCES AND CONTRADICTORIES OF
COMPOSITE PROPORTIONS

Contra-

Equivalent hypoiheticals Disjunctive Alternative

1. Ifp, then q == Ifq, thenp = Not bothp_ and q = Either p or q

2. Iff, then qz=Ifq, thenp == Not bothp and q s= Either p or q

3. Ifp, then q = Ifq, thenp == Not bothp and q == Either p or q

4. If p, then q = Ifq, thenp == Not bothp and q == Either p or q

dictory
p and q
p and q
pandq
p and q

the same in form 3 for it is logically indifferent what letters we
use as illustrative symbols; we used X, T above to illustrate
antecedent and consequent respectively. But, on the assump-

* W. E. Johnson, Logic, Part I, p. 37. Mr Johnson points out that comple
mentary propositions are frequently confused in thought and frequently con
joined in fact 5 . It should, however, be noted that they are sometimes not
conjoined in fact; hence, their tendency to be confused in thought may lead
us astray.

COMPOUND PROPOSITIONS AND ARGUMENTS 47

tion that p stands for some one definite proposition and q for
some other definite proposition, then Ifp, then q is distinguished
from Ifq, thenp as its complementary. Both will, therefore, be
included in the list.

Certain observations on this table are important and
should be carefully noted, (i) Propositions on different lines
are independent; (ii) as any proposition contradicting a
given proposition also contradicts any equivalent proposi
tions,, the proposition on the right of the black line contradicts
all four propositions left of it on the same line; (iii) the pro
positions, on different lines, along the principal diagonal are
stated in terms of p 9 q, and are clearly independent; (iv)
propositions in the same column are the same in form but -
on the assumption we have been making, viz. that p stands
for p is true, p stands for p is false (likewise with q, q} - these
are conveniently distinguished, and have, therefore, been
separately considered.

The significance of the composite forms can be brought out
by formulating specific rules for inferring the various
equivalent propositions when one is given. It will suffice to
give these for the case of the hypothetical Ifp 9 then q. It must
be remembered that If . . . then . . . can be interpreted as
implies, in the sense that, when p implies q, q is true provided p
is true. Given If p, then q:

(1) The denial of the antecedent is implied by the denial
of the consequent; hence, If q, then p.

(2) Either the antecedent must be denied or the con
sequent asserted; hence, Either p or q.

(3) The assertion of the antecedent is not consistent with
the denial of the consequent; hence, Not both p and q.

It is not difficult to formulate corresponding rules for
obtaining equivalents from one of the other two composite
forms. The student should construct significant examples and
transform them in the equivalent propositions; he may then
intuitively apprehend the validity of these inferences. We
shall consider one example.

Example. The British Government in the summer of 1942
desired to impress the people with the need for economizing

48 A MODERN ELEMENTARY LOGIC

in fuel in order that the war industries should not be
hampered for lack of fuel. The Government s exhortations
might be summed up in the short statement. If we waste fuel,
we lose the war. To this proposition the three following are
equivalent: (i) If we do not lose the war., we have not wasted fuel;
(2) Either we do not waste fuel or we lose the war, (3) It is not the
case both that we waste fuel and do not lose the war. In the next
section we shall see that once we have fully grasped these
rules, and have thus understood the precise significance of the
various composite forms, we shall be in a good position for
understanding certain forms of arguments of common occur
rence in everyday reasoning. If we understand these forms
we may be on our guard against mistakes in reasoning which
occur all too frequently from an imperfect apprehension of
what precisely has been asserted in the premisses.

2. COMPOUND ARGUMENTS WITH ONE OR MORE
COMPOSITE PREMISSES

Let us consider the following examples of arguments, taken
from everyday conversation; some are valid, some are invalid.

(1) Two boys are watching the approach of an aeroplane.
One says, That s a bomber; I think it is a Stirling. The other
replies, e lt has four engines and I think it must be a Stirling
or a Liberator, but I don t think it is a Stirling. As the aero
plane approaches nearer, the first boy says, You are right;
it has twin-fins and rudders, so it is a Liberator.

(2) You cannot maintain that after the war there should
continue to be unrestricted competition among the nations
for the world s natural resources and yet, at the same time,
hold that we ought to aim at giving to all nations economic
security. But you do admit the latter; hence, you must reject
unrestricted competition. Moreover, if there is unrestricted
competition, there will be more world wars, and you have
agreed that there must be no more world wars.

(3) If Frock s book deepens our sense of humanitarian
values, it is worth writing even in time of war; but it is
certainly worth writing in time of war, so I conclude that his
book deepens our sense of humanitarian values.

COMPOUND PROPOSITIONS AND ARGUMENTS 49

(4) c lf a man is a coward, he will seek to evade military
duties, but Tobias is not a coward; so he won t attempt to get
out of military duties. 3

(5) Tor a novelist to be sure of getting his books properly
reviewed, he must be either already famous or have written
a really first-rate book; but Jensen is already famous, so his
novel is not first-rate. 3

It is not difficult to determine the structure of these argu
ments.* It will suffice to examine in detail only the first.
It presents a common form of reasoning - something is
recognized as being this or that:, then, some characteristic is
looked for that would suffice to distinguish this from that. The
argument can be formally analysed as follows:

(i) Either the aeroplane is a Stirling or a Liberator;
(ii) If it has twin-fins and rudders., it is not a Stirling, but it has

twin-Jins and rudders; hence, it is not a Stirling.
(iii) Combining (i) and the conclusion of (ii) yields the

conclusion: It is a Liberator.

The logical structure can be exhibited as follows:

Either A or B (i)

IfF, then not-A\ ,.

F :. not-A / W

/. B (iii)

In the following table we set out formally the four modes of
argument corresponding to the four varieties of composite
premisses, adding the Latin name traditionally used in each
case:

COMPOUND MODES

Modus-f form of Composite Premiss

1. Ponendo ponens: Ifp, then q; butp; . . q Hypothetical

2. Tollendo fattens: If p, then q; but q; :. p Hypothetical

3. Ponendo tollens: Not both p and q; but p; .*. q Disjunctive

4. Tollendo ponens: Either p or q; butp; . . q Alternative

* The student should before reading further determine for himself whether
the conclusion, in each case, does in fact follow from the^premisses.

f These barbarous names are derived from the Latin verbs: ponere = to
affirm; toilers = to deny; hence, they can be interpreted as follows: (i) by
affirming, affirms; (2) by denying, denies; (3) by affirming, denies; (4) by
denying, affirms.

5O A MODERN ELEMENTARY LOGIC

The rules for these modes are:

(i) Ponendo ponens: From the affirmation of the antecedent,
the affirmation of the consequent follows. (2) Tollendo tollens:
From the denial of the consequent, the denial of the ante
cedent follows. (3) Ponendo tollens: From the affirmation of one
disjunct, the denial of the other disjunct follows. (4) Tollendo
ponens: From the denial of one alternant, the affirmation of
the other alternant follows.

From these rules it is easy to see that, in the examples given
above, (3) is invalid because the antecedent is affirmed on
the ground of an affirmation of the consequent; (4) is invalid
because the consequent is denied on the ground of a denial
of the antecedent; (5) is invalid because one of the alternants
is affirmed and the other is denied in consequence. These three
fallacies are all due to the failure to appreciate what exactly
the relevant composite premiss asserts. To affirm the ante
cedent because the consequent has been affirmed is to confuse
an hypothetical with its complementary; similarly, in denying
the consequent because the antecedent has been denied. To
deny an alternant because the other alternant has been affirmed
is to confuse an alternative proposition with the comple
mentary disjunctive, or to treat it as though it were the con
junction of the alternative with the complementary disjunc
tive. That this is a confusion should be clear from our previous
discussion of the composite propositions. These invalid modes
of inference can be summarized as follows:

1. Hypothetical: Ifp, then q; but q; , . p (consequent affirmed).

2. Hypothetical: If p, then q; butf; .*. q (antecedent denied).

3. Alternative: Either p or q; but pi . . q (alternant affirmed) .

4. Disjunctive: Not both p and q\ but q\ :. p (disjunct denied) .

Since the same statement can be made in any one of the
four composite forms of propositions, the compound modes
can be reduced to one another.

Equivalent arguments

Ponendo ponens Tollendo ponens

If you paid 2, he over- == Either you did not pay 2

charged you; or he overcharged you;

You paid 2; You paid 2;

/. he overcharged you. /. he overcharged you.

COMPOUND PROPOSITIONS AND ARGUMENTS 5!

In the same manner the ponendo tollens and tollendo tollens
can be obtained, the conclusion in each case being the same.
The Dilemma. As the popular use of the phrase, e l am in a
dilemma 3 shows, the dilemma is a form of argument, the
purpose of which is to prove that from either of two alterna
tives an unwelcome conclusion follows. If skilfully employed,
it can be made effective by an orator and amusing to an
audience; it can also be used seriously. For these reasons, no
doubt, a disproportionate amount of space has been given to
it in books on logic - disproportionate because no new
logical principles are involved. We shall deal with it shortly.
A dilemma is a compound argument consisting of a premiss
in which two hypotheticals are conjunctively affirmed and a
premiss in which the antecedents are alternatively affirmed
or the consequents alternatively denied. If there are three
hypotheticals conjunctively affirmed the argument is called a
trilemma, if four, a quadrilemma, if more than four a polylemma.
These are of rare occurrence; sometimes dilemma is used to
cover all these forms.

Four distinct kinds of dilemma are recognized:

1. Complex Constructive:

Ifp 9 then q, and if r, then t,

But either p or r,

/. either q or t.

2. Simple Constructive:

If/, then y, and if r, then q,

But either p or r,

:.q.

3. Complex Destructive:

If/, then q, and if r, then t 9

But eithei not-y or not-,

/. not-/ or not-r.

4. Simple Destructive:

If/, then y, and if/, then r,

But either not-y or not-r,

.*. not-/.

52 A MODERN ELEMENTARY LOGIC

It is obvious that the rules for the hypothetical and alterna
tive modes of argument directly apply to the dilemmatic
forms, so that they need not be re-stated here.

The dilemma is often regarded as a peculiarly fallacious
mode of argument. This is, however, a mistake; any form
of argument can be, and most are, fallaciously used either
through stupidity or cunning. In so far as there are any
difficulties in using valid dilemmas these arise from the
difficulty of finding premisses both significant and pertinent
which are true and also fulfil the conditions imposed by the
form. The force of the dilemmatic situation presented in the
alternative premiss depends upon the condition that the
alternants must be exhaustive. If there is a third alternative,
we can escape between the horns of the dilemma .* Thus, a
too-anxious parent might argue: Tf my son is idle, he will fail
in his examination; and if he overworks, he will be ill; but
either he will be idle or he will overwork; therefore, my son
will either fail in his examination or be ill. 9 The third alterna
tive is too obvious to require stating; it is, however, just
possible that some people may be as silly as this argument
suggests. An example of a valid dilemma is the following: c lf
you reflected carefully you would have seen your mistake;
and if you were honest you would have admitted it; but
either you do not see your mistake or you do not admit it;
therefore, either you have not reflected carefully or you are
not honest. 3 This is a complex destructive dilemma; the con
clusion can be avoided only by objecting correctly to the
factual truth of the hypothetical premiss. But this way
of rejecting a conclusion is not confined to dilemmatic
arguments.

A dilemma is said to be rebutted if another dilemma be
constructed leading to a conclusion which seems to contradict
the original conclusion. Thus an Athenian mother is reported
to have presented her son with the dilemma:

Tf you say what is just, men will hate you; and if you say

* This phrase emphasizes the fact that the dilemma has been regarded as
essentially a disputatious argument; the speaker seeks to impale his adversary
upon the horns , i.e. the unwelcome alternatives; but we do not always argue
to refute adversaries , we may seek to convince those who oppose our view, even,
sometimes, to convince ourselves.

COMPOUND PROPOSITIONS AND ARGUMENTS 53

what is unjust, the gods wiU hate you; but you must say what
is just or what is unjust; hence either men will hate you or
the gods will hate you.

To this the son replied:

If I say what is just, the gods will love me; and if I say
what is unjust, men will love me; but I must say one or the
other; therefore, either the gods will love me or men will
love me. 5

The rebuttal consists in transposing the two consequents
and contradicting* them. Thus the form of the mother s
dilemma is: If^, then q; and if not-p, then r; but p or not-p;
therefore, q or r.

The son s rebuttal is of the form: If/?, then not-r; and if
not-p, then not-q; but either p or not-p; therefore, either not-r
or not-q.

It is clear that q or r is not contradicted by not-r or not-q;
these propositions are independent. What the son needed to
prove in order to allay his mother s fears was Both men and gods
will love me.

A dilemma is said to be c taken by the horns when the
alternatives are accepted but the consequences drawn from
them are denied. These picturesque modes of argument have
no special logical significance. As tests of our ability to use
logical principles and to discern violations of principles they
have some utility, but not much.

* Miss Stebbing here treats "loving" and "hating" as contradictory terms,
though they would usually be regarded as contrary terms. [C. W. K. M.]

CHAPTER IV

The Traditional Syllogism

I. DEFINING CHARACTERISTICS OF A SYLLOGISM

Formal immediate inference is trivial. When we seem to have
inferred a non-trivial conclusion from a single premiss it is
because we have tacitly made assumptions or have pre
supposed a premiss without noticing that we have done so.
At least two premisses are required for a properly formal
inference which is not trivial. Such inference is mediate in
ference. It is seldom that we state both premisses explicitly,
but it is possible to find examples. Sir Henry Gampbell-
Bannerman was making an informal speech to his neighbours
at Montrose. In the course of it he said: An old friend of
mine, Wilfrid Lawson, was accustomed to say: "The man
who walks on a straight road never loses his way. 55 Well, I
flatter myself that I have walked on a pretty straight road,
probably because it was easier, and accordingly I have not
gone astray. * The conclusion I have not gone astray is implied
by the conjoint assertion of two premisses. The man who walks
on a straight road never loses his way (i.e. does not go astray)
and / have walked on a (pretty] straight road. No one should
have any difficulty in seeing that the conclusion does indeed
follow from the premisses. Arguments of this kind, in which
a conclusion is inferred from two premisses, can often
be stated in a traditional form called the syllogism. For
example:

(i) All human beings are liable to make mistakes.
All philosophers are human beings.

. . All philosophers are liable to make mistakes.

* Quoted by Lord Oxford and Asquith in Fifty Tears of Parliament, Vol. II.
P- 5i-

54

THE TRADITIONAL SYLLOGISM 55

(2) No vain people are trustworthy.
All great leaders are trustworthy.

.". No great leaders are vain.

(3) All policemen are tall.

Some policemen are Cockneys.

.". Some Cockneys are tall.

In each of these examples there are three propositions and
three different terms, each of which occurs twice. The term
which occurs in both premisses but not in the conclusion is
called the middle term; it is connected in one premiss with the
predicate of the conclusion, and in the other with the subject
of the conclusion. The subject and predicate of the conclusion
were called by Aristotle c the extreme terms , because they are
connected by a middle term. The predicate of the conclusion
is called the major term; the subject of the conclusion is called
the minor term. The premiss containing the major term is
called the major premiss , the premiss containing the minor
term is called the minor premiss. The major premiss is tradi
tionally stated first, then the minor, and then the conclusion.
This is the order followed in the three examples above, but
the order of the premisses is logically irrelevant. The line
drawn between the premisses and the conclusion is intended
to mark the difference between them - the premisses are
taken for granted or asserted to be true, the conclusion is
drawn from the premisses.

Aristotle defined che syllogism widely. He said, c a syllogism
is discourse (Aoyog) in which, certain things being stated,
something other than what is stated follows of necessity from
their being so , and he adds, I mean by the last phrase that
they produce the consequence, and by this, that no further
term is required from without to make the consequence
necessary .* But the syllogism has traditionally been more
narrowly interpreted so that an argument, even when valid
and in accordance with this definition, can in various ways
fail to fall into syllogistic form. This narrower specification of

* Analytica Prior a^ 24^ 1 8.

56 A MODERN ELEMENTARY LOGIC

traditional syllogistic arguments can be stated in three defin
ing rules:

1. Every syllogism comprises three propositions.

2. Each proposition in a syllogism must be in one of the A, E, /,
forms.

3. Every syllogism contains three and only three terms.

Comments on these rules: (i) Syllogistic arguments are usually
abbreviated so that one premiss is tacitly supplied by the
context or is, perhaps, presupposed only in the sense that
without it the argument is not valid. A syllogism thus incom
pletely stated is called an enthymeme. Sometimes the conclusion
is omitted, mainly as the rhetorical device of innuendo. The
following are illustrations of enthymemes as they might very
well occur in ordinary conversation although not, as a rule,
so tersely expressed:

(i) Dictators are ruthless for all ambitious men are

ruthless. 3
(ii) No honest men are advertisers because all advertisers

are liars by profession. 5

(iii) Sailors are handy folk, so they are always welcome
guests.

In (i) and (ii) the minor premiss is omitted; in (iii) the major
premiss is omitted.*

(2) The singular proposition, e.g. De Valera is not wholly
Irish, She is reckless, is not excluded by this rule since, for the
purposes of syllogistic inference, singular propositions are
regarded as A or E propositions.

(3) This rule is most commonly violated by equivocation,
i.e. by using the same word or phrase with different meanings
in its two occurrences. When this happens the syllogism has
more than three terms or - as it would be more correct to say
- the argument is not syllogistic although it appears to be so
because one word, or phrase, is being used ambiguously. f

These rules suffice to determine what is to be understood
as a categorical syllogism, but they do not suffice to deter-

* Polysyllogisms are also enthymematic. See below, p. 7 1 .
f On this topic see further, p. 101.

THE TRADITIONAL SYLLOGISM 57

mine the conditions under which an argument conforming to
them is valid. That the arguments given on page 55 are valid
will be easily seen, but such seeing 5 is not proof. We need to
see further how it is that the conclusion of a valid syllogism
is valid and to understand exactly why some of the conclusions
we are tempted to draw in arguing are in fact invalid. For this
purpose we must state certain rules or axioms:

I. Axioms of Distribution.

1. The middle term must be distributed in at least one of
the premisses.

2. A term that is distributed in the conclusion must be
distributed in the corresponding premiss.

II. Axioms of Quality.

3. At least one premiss must be affirmative.

4. With one premiss negative, the conclusion must be
negative.

5. With both premisses affirmative, the conclusion must be
affirmative.

From these axioms we can deduce three corollaries, which
we shall find useful in determining which combinations of
.4, ", /, propositions yield valid syllogisms. Some writers
of elementary textbooks in Logic include these corollaries
among the rules, or axioms, but it is desirable to prove them,
A corollary is a theorem, and a theorem is a general proposi
tion which is proved entirely by reference to the axioms and
definitions. For the three following theorems we shall use the
traditional name corollary.

Corollaries, (i) At least one premiss must be universal. This can be
established by indirect proof; i.e. by supposing that both
premisses could be particular, which is the contradictory of
the theorem asserted.

Proof: There are three cases to be considered, (a) Both premisses
are negative. This violates axiom 3; hence the original supposi
tion is impossible; therefore, its contradictory, the theorem,
is proved.

(b] Both premisses are affirmative. Then, since both are par
ticular (assumed), no term in either premiss is distributed;

58 A MODERN ELEMENTARY LOGIC

hence, the middle term is undistributed; accordingly axiom
i is violated.

(c) One premiss is affirmative, the other negative. Since only one
term is distributed, it must, by axiom I, be the middle term;
but, by axiom 4, the conclusion must be negative [and would
thus have a distributed term, viz. its predicate]; therefore,
axiom 2 is violated.

(ii) Given that one premiss is particular, the conclusion must be
particular.

Proof: There are again three cases: (a) Both premisses are
negative. This is excluded by axiom 3.

(b) Both premisses are affirmative. Since one premiss is par
ticular (given] and both are affirmative,, only one term is
distributed in the two premisses; this, by axiom i, must be
the middle term; therefore, by axiom 2, the minor term can
not be distributed in the conclusion, i.e. the conclusion must
be particular.

(c] One premiss is affirmative, the other negative. Since one
premiss is affirmative and one negative only two terms can
be distributed in the premisses; of these one term, by axiom i,
must be the middle term, and the other, by axioms 4 and 2,
must be the major term; therefore the minor term cannot be
distributed, i.e. the conclusion must be particular.

(iii) Given that the major premiss is particular, the minor premiss
cannot be negative. Since, ex hypothesi, the minor premiss is
negative, then, by axiom 4, the conclusion must be negative,
so that the major term will be distributed in the conclusion.
But the major premiss is particular (given] and affirmative,
by axiom 3; hence, neither term in the major premiss is dis
tributed; therefore, by axiom 2, the minor premiss cannot be
negative if the major premiss is particular.

2. FIGURES AND MOODS OF THE SYLLOGISM

Not all combinations of A, E, I, propositions will yield
valid syllogisms; we must, therefore, determine which com
binations are valid. Let us first, however, consider the four
following arguments:

THE TRADITIONAL SYLLOGISM 59

I. All ruminants are horned. II. No soldiers are pacifists.

All cows are ruminants. All Quakers are pacifists.

.*. All cows are horned. /. No Quakers are soldiers.

III. All film stars are famous. TV. All snobs are obsequious.
Some film stars are frivolous. No obsequious people are

/. Some who are frivolous are financiers.

famous. .*. No financiers are snobs.

The student will have no difficulty in seeing that these
arguments are valid. They differ in form in two ways: (i) in
the position of the middle term; (ii) in the quantity and
quality of the propositions involved.

(i) In I the middle term is subject of the major premiss and
predicate of the minor; in II the middle term is predicate in
both premisses; in III the middle term is subject in both
premisses; in IV the middle term is predicate in the major
premiss and subject in the minor premiss. Using S, M, P, to
stand for minor, middle, and predicate term respectively, we
can symbolize these forms as follows:

I II III IV*

M-P P-M M-P P-M

S-M S-M M-S M-S

:.S-P :.S-P :. S-P .-. S-P

These differences are said to be differences in the figure of the
syllogism. Accordingly, the figure of a syllogism is determined
by the position of the middle term.

(ii) The propositions involved in example I are AAA, in
II EAE, in III All, in IV AEE. This difference is called a
difference in mood. Accordingly, the mood of a syllogism is
determined by the quantity and quality of the propositions
involved. Thus I is in the mood AAA, II in the mood EAE,
and so on.

Consider the argument: All polite people are kind:, Some
customs officers are not polite , therefore Some customs officers are not
kind. Does this conclusion follow from the premisses? A little
reflection should enable us to see that it does not - a man

* The position, of the middle term in the four figures can be easily remembered
by noticing that a line drawn through M in the above schemas gives roughly
a W, viz. \ 1 | /.

6O A MODERN ELEMENTARY LOGIC

may be impolite and yet in other respects kind. If the argu
ment is examined it will be seen that the major term kind is
distributed in the conclusion (being the predicate of a nega
tive proposition) but not in the major premiss; hence axiom 2
is violated. The argument is in figure I and is in the mood
AOO. The invalidity is due to its form; it has nothing to do
with the characteristics of polite people, kind people, and customs
officers. Accordingly, we can assert that the mood .400 is
invalid in figure I, no matter what the propositions involved
may be about. It is invalid because the major term is
illegitimately distributed in the conclusion. This fallacy is
called the fallacy of illicit process of the major term, or, more
shortly, illicit major. Now consider the argument: Some R.A.F.
pilots are artistic ., all R.A.F. pilots are intelligent , therefore All
intelligent people are artistic. This is again invalid; the minor
term is illegitimately distributed; i.e. the syllogism is guilty of
the fallacy of illicit minor. Finally, consider the argument: All
operatic singers are temperamental , all disillusioned poets are tempera
mental , therefore, all disillusioned poets are operatic singers. The
conclusion does not follow; axiom i has been violated, for,
since both premisses are affirmative and the middle term is
predicate in both, the middle term has not been distributed.
This fallacy is known as the fallacy of undistributed middle. It
is of common occurrence in our arguments, but it is not
always easily detected when the argument is less tersely
expressed.

The conventional restriction of the syllogism to the four
traditional categorical forms limits the conclusions to one of
the following, SaP, SeP, SiP, SoP. Negative terms are excluded
so that, for example, we cannot have a conclusion involving
S or P. The major premiss may be any one of the A, E, I, O
forms; so may the minor premiss. There are, then, sixteen
possible combinations. These are written below; the first
letter indicates the major, the second the minor premiss:

AA AE AI AO

EA EE El EO

I A IE II 10

OA OE 01 00

THE TRADITIONAL SYLLOGISM 6l

Some of these combinations can be eliminated at once, by
reference to the axioms. The axioms of quality exclude EE,
EO, OE, 00*; corollary (i) excludes//, 10, 01; corollary (iii)
excludes IE. There are left eight combinations each of which
will yield a valid syllogism in one or more of the figures.
These are AA, AE, AI 9 AO, EA, El, IA 9 OA.

Since the distribution of any term in these propositions
depends upon its position as subject or as predicate, combina
tions not excluded generally by the axioms of distribution
will nevertheless not yield a valid conclusion in every figure.
We have already studied examples of such invalid combina
tions. We have now to deduce from the axioms special rules
for each figure. f

Special Rules of Figure /. Schema M - P

S-M
S-P

(i) The minor premiss must be affirmative. Proof: Suppose the
minor premiss is negative: then the conclusion must be
negative (ax. 4) and the major premiss affirmative (ax. 3).
Then the major term will be distributed in the conclusion but
not in its own premiss, thus violating axiom 2. Therefoie, the
minor premiss cannot be negative, i.e. it must be affirmative.

(ii) The major premiss must be universal. Proof: Since the
minor premiss must be affirmative, the middle term, its
predicate, will be undistributed in the minor premiss; hence,
the middle term must be distributed in the major premiss
(ax. i), of which it is subject; accordingly, the major premiss
must be universal.

From these rules we can directly determine the valid moods
in figure I. Granted the assumption that the classes denoted
by S and P respectively contain members, then, any combina
tion of premisses that justifies a universal conclusion also
justifies a particular conclusion, since, in this case, the

* It should be noted that 00 is also excluded by corollary (i), and OE by
corollary (iii).

f This procedure is elegant and affords a useful exercise. Any student who has
difficulty in following the deduction should turn again to the axioms. It is
important to remember that a term is distributed if it is the subject of a universal
-proposition or the predicate of a negative proposition; it will be undistributed if it
ds the subject of a. particular proposition or the predicate of an affirmative proposition.

62 A MODERN ELEMENTARY LOGIC

particular conclusion would be subirnplicant to the universal
conclusion.

Valid Moods of Figure I. The combinations excluded by the
special rules are: AE, AO excluded by rule (i); I A, OA
excluded by rule (ii) ; accordingly, the valid moods are AAA y
\AAT\, All, EAE, [EAO], EIO. The two moods given in
brackets are the weakened moods, and may be disregarded.
The unweakened moods have been given proper names
which, since the thirteenth century, have been familiar to
students of logic. They are now mainly of antiquarian
interest but are of some use for purposes of reference. Keeping
the same order in which the valid moods have just been listed
and omitting the weakened moods, these names are, Barbara.,
Darii, Celarent, Ferio.*

Special Rules of Figure II. Schema P - M

S-M
S-P

(i) One premiss must be negative.^ This is necessary in order
to secure the distribution of the middle term, which is
predicate in both premisses.

(ii) The major premiss must be universal. This is to prevent
illicit majoi, since the conclusion is always negative as a con
sequence of rule (i).

Valid Moods of Figure II. The combinations excluded by the
special rules are: AA, AI, I A (by rule i), OA (by rule ii);
accordingly, the valid moods are AEE[AEO], EAE[EAO],
EIO, AOO, and their names are Cesar e, Camestres, Festino,
Baroco.

Special Rules of Figure III. Schema M-P

M-S
S-P

* These names were invented for a special mnemonic purpose, viz. to reduce
to a mechanical procedure the reduction of syllogisms in figures II, III, IV to
figure I. It should be noted that the quantity and quality of the propositions
involved in a given syllogism are shown by the vowels contained in the name,
the canonical order of major, minor, conclusion, being preserved, e.g. Celarent.
All other letters can be disregarded. Those interested in the purpose for which
the other letters were used should consult F.L. 258.

t The proofs of these special rules are very easy; in the case of figure I the
proofs have been given in full; for the remaining figures the proofs are merely
indicated.

THE TRADITIONAL SYLLOGISM 63

(i) The minor premiss must be affirmative. This is for the same
reason as in figure I, for the rule is required owing to the
position of the major term, P, which is the same in both
figures, and has no reference to the minor term, S, the posi
tion of which differs in the two figures.

(ii) The conclusion must be particular. This follows from
special rule (i) together with axiom 2.

Valid Moods of Figure III. The combinations excluded by the
special rules are AE, AO (by rule (i)); all other combinations
are permitted but the conclusion must not be universal. For
this reason there are six un weakened moods: AAI, All., IAI,
EAO, EIO, OAO, and their names are Darapti, Datisi, Disamis>
Felapton, Ferison, Bocardo.

Special Rules of Figure IV. Schema P- M

M-S

S-P

(i) The major premiss cannot be particular if either premiss is
negative. Violation of this rule involves illicit major, since the
major term is subject in its premiss.

(ii) The minor premiss cannot be particular if the major premiss
is affirmative. Violation of this rule involves undistributed
middle since the middle term is subject in the minor, and
predicate in the major premiss.

(iii) The conclusion cannot be universal if the minor premiss is
affirmative. Violation of this rule involves illicit minor.

It should be noted that rule (i) combines the two rules of
figure II, rule (iii) the two rules of figure III. Rule (ii) is
analogous to the two rules of figure I but, owing to the
reversed position of the minor and major terms, it is required
that an affirmative major premiss necessitates a universal
minor premiss in order that the middle term should be
distributed.

Valid Moods of Figure IV. The special rules exclude the
combinations AO, OA, AI, and require that A A should have
/as conclusion. Accordingly, the valid moods are: AAI, AEE,
[AEO], EAO, EIO, IAI, and their names are, Bramantip,
Camenes, Fesapo, Fresison, Dimaris.

It will be noticed that in the first three figures there are,

64 A MODERN ELEMENTARY LOGIC

including the weakened moods, six moods in each figure. In
figure III there are no weakened moods, but Darapti and
Felapton have two universal premisses with a particular con
clusion. The middle term is unnecessarily distributed in both
premisses. In figure IV one of the six moods is weakened and
one (Bramantip) contains a premiss (the major) which could
be weakened without affecting the validity of the conclusion;
in this case the mood would be IAI (Dimaris) instead ofAAI.
In Bramantip we have an example of an over-distributed term,
i.e. a term distributed in its premiss but not in the conclusion.
We shall see later that there are difficulties about this mood,
and, indeed, about all weakened moods.* If the same con
clusion can, in any syllogism, be obtained although one of the
premisses is weakened, the syllogism is said to be a strengthened
syllogism^

Figure IV is usually called the Galenian figure, because it is
supposed to have been introduced by Galen; it is seldom
given in books on logic before the eighteenth century. The
following are examples in figure IV:

E No aeroplanes are balloons. A All big men are jovial.

A All balloons are aircraft. E No jovial men are non-smokers.

.*. Some aircraft are not balloons. E .*. No non-smokers are big men.

The student should notice that it would be possible to obtain
the same conclusion, in each of these cases, by a syllogism in
figure I. How this is possible will be explained in the next
section.

3. REDUCTION AND THE ANTILOGISM

By using the syllogistic axioms to deduce special rules for the
figures, thus showing that certain moods must be excluded,
we have not shown demonstratively that the remaining
moods are valid. Aristotle, who may be said to have invented
the theory of the syllogism, did not adopt this method
of justification. He formulated an axiom which directly
guarantees the valid moods of figure I. This axiom is called

* See p. 95, below.

t We shall see later that every syllogism in which there are two universal
premisses with a particular conclusion is a strengthened syllogism, with one
exception, viz. AEO in figure IV.

THE TRADITIONAL SYLLOGISM 65

the Dictum de omni et nullo because it is an axiom concerning
all or none of a class. It has been variously formulated; we shall
formulate it as follows: Whatever is predicated affirmatively or
negatively of every member of a class can in like manner be predicated
of everything contained in that class. Thus, for instance, if All
scholars are inefficient in business affairs and all academic professors
are scholars, then it follows that All academic professors are
inefficient in business affairs. Everyone will admit that, granted
that the premisses (stated in the compound proposition) are
true, then the conclusion is necessarily true. What Aristotle
did was to generalize the grounds of this admission. At the
moment we shall follow Aristotle and admit that the Dictum
is not only true but necessarily true and, further, that it can
be accepted as an axiom. It applies directly only to figure I.
The Dictum permits us equally well to assert that No scholars
are inefficient^ or to assert that Some academic professors are scholars
although, in that case, our conclusion must be an assertion
about some academic professors not about all of them. Hence,
the dictum gives us a schema for figure I:

If Every M is P (or not)
and All (or some) S is M,
then All (or some) S is P (or not) .

From this schema we can directly obtain the two special rules
of the figure and can see clearly why the middle term must
be distributed in the major premiss and why the minor
premiss must be affirmative.

There were reasons, bound up with his metaphysical views,
which made Aristotle content to formulate an axiom for the
first figure alone. Now, if it be granted that the Dictum de
omni is properly axiomatic and, further, that it is the sole
axiom guaranteeing the validity of syllogistic moods, then it
must be admitted that the validity of moods in other figures
than the first can be guaranteed only by showing that these
moods are logically equivalent to first figure moods. This can
be done by showing that a conclusion is obtainable in the first
figure equivalent to or implying the original conclusion and
from premisses equivalent to or implied by the premisses
originally given. The process of thus testing the validity of

66 A MODERN ELEMENTARY LOGIC

moods is known as reduction, of which Aristotle recognized
two methods: (i) direct reduction, performed by converting
propositions or transposing premisses; (2) indirect reduction,
which consists in proof by reductio per impossible. These methods
must now be illustrated,
(i) Direct reduction. Consider the pair of syllogisms below:

(a) (/?)

All Quakers are pacifists \/r No pacifists are soldiers.
No soldiers are pacifists /\ All Quakers are pacifists.

/. No soldiers are Quakers == /. No Quakers are soldiers.

(a) is a syllogism in AEE in figure II (Camestres] ; (/S) is EAE
in figure I (Celarent}\ the two syllogisms are equivalent. In ()
the major premiss is the converse of the minor premiss of (a).
Thus the premisses have been transposed and the original
minor premiss, which has become the new major premiss, has
been converted. Accordingly, since the minor premiss con
tains the subject of the conclusion, the new conclusion must
be converted in order that the original conclusion may be
obtained. It must be remembered that we are assuming that
the validity ofCelarent is established by the Dictum de omni, and
we have thus shown that the mood Camestres in figure II is
valid; we are not contending that the moods of figure I are
superior in self-evidence to the moods of figure II. We are
adopting an attitude of doubting something that seems to be
self-evident, and we resolve the doubt by showing that the
same conclusion can be obtained by means of a mood
guaranteed by the Dictum; in doing so, we have used only
simple conversion - which we have admitted to be valid -
and transposition of the premisses. We shall now give one
more example of direct reduction:

Figure III. AAI AIL Figure I

All pedants are bores. All pedants are bores.

All pedants are scholars. > Some scholars are pedants.
/. Some scholars are bores. /. Some scholars are bores.

We do not need so much information as is provided in figure
III, AAI (Darapti), in order to draw the same conclusion,

THE TRADITIONAL SYLLOGISM 6j

since the middle term is unnecessarily distributed twice;
hence we can convert the minor premiss (A] by limitation (7).
When both the premisses of a valid syllogism admit of
simple conversion it is clear that the order of the terms is
logically indifferent. This is the case when the major premiss
is E and the minor /; hence, the mood 70 is valid in every
figure. This is shown below:

/. Ferio IL Festino IIL Ferison IV. Fresison

MeP == PeM = MeP ~ PeM

SiM = SiM = MiS == MiS

:. SoP == /. SoP ~ /. SoP = /. SoP

These four syllogisms are all equivalent no matter in what
figure they may happen to be. They present indeed four ways
of making the same set of statements. Syllogisms of which the
premisses are A and / (in either order) or A and E (in either
order) are also equivalent, in the sense that the same con
clusion can be obtained from the given premisses, in several
figures providing that transposition of the premisses is
allowed.* These equivalences are exhibited below:

I. Celarent IL Cesare t // Camestres IV. Camsnes

MeX ~ XeM = XeM = MeX

YaM == TaM = TaM = TaM

:. TeX = :. TeX = /. XeT = /. XeT

I. Darii HI. Datisi IIL Disamis IV. Dimaris

MaX ~ MaX = MaX == MaX

TiM = MiT == MiT = riM

:. nx = .-. nx = .-. xir = /. xir

IIL Felapton IV. Fesapo

MeX = XeM

Mar = Mar

:. r x ~

(2) Indirect reduction. The moods Baroco (AOO in figure II)
and Bocardo (OAO in figure III) lie outside this scheme of

* To bring out the equivalences in a brief form the regular order of the
premisses is not always maintained; the minor premiss is that which contains
the subject of the conclusion, the major premiss that which contains the predicate
of the conclusion. Accordingly, the minor and major terms are to be identified
by looking at the conclusion. The order of the premisses is always logically
irrelevant.

t Notice that there is no equivalent argument in figure III hi which the
conclusion must always be particular.

68 A MODERN ELEMENTARY LOGIC

equivalences; they cannot be reduced to the first figure, so
that indirect reduction must be used. It must be remembered
that we suppose ourselves to be proving that the conclusion is
validly inferred and that we have accepted the validity of the
moods of the first figure. It will suffice to exhibit this method
in the case of Bocardo, i.e.

MoP

MaS

:. SoP.

We reason as follows: If SoP is not true, then its contradictory,
SaP, must be true; combining SaP with the minor premiss
MaS, we obtain

SaP

MaS
:. MaP,

which is hi Barbara. But MaP, the new conclusion, contradicts
MoP, which was given true as a premiss of the original syl
logism; hence, its contradictory, MaP, must be false; but
MaP is the conclusion of a valid syllogism in figure I;
hence it is true if its premisses aie true; since it is not
true at least one premiss must be false; this cannot be MaS,
since that was given as true; therefore SaP, its other premiss,
must be false; therefore, SoP is true, and that is the original
conclusion.

The reasoning upon which indirect reduction is based rests
upon the principle that, if the conclusion of a valid syllogism
is false, then at least one of the premisses must be false. This
principle can be stated generally, in the form of an hypo
thetical proposition with a compound antecedent. Let p, q, r,
be illustrative symbols for the major and minor premisses and
the conclusion of a valid syllogism. Then we have: If p and q,
then r. This is equivalent to If not r, then either not-p or not-q\ i.e.
if the conclusion, r, is false, then at least one of the premisses,
p, q, is false. Again, If p and q, then r is equivalent to Not (p
and q) and not-r. This disjunction was called, by Mrs Ladd-
Franklin, an inconsistent triad ., she invented the name antilogism
for the triad of propositions constituted by the two premisses

THE TRADITIONAL SYLLOGISM 69

of a syllogism and the contradictory of its conclusion. The
following is an example of an antilogism:

p Xo pets are ugly.
q All cars are pets.
f* Some cats are ugly.

Any two of these propositions imply the falsity of the third;
hence, we obtained three valid syllogisms.

Celarent Festino Disamis

p No pets are ugly. p No pets are ugly. f Some cats are ugly,

q All cats are pets. f Some cats are ugly. q All cats are pets.

r :. No cats are ugly. q Some cats are not pets, p .*. Some pets are ugly.

These three syllogisms are respectively in figures I, II, and
III. It will be found that, starting with a valid syllogism in
any one of these three figures two other syllogisms will be
obtained, one in each of the other figures, if the contradictory
of the first conclusion is combined first with one premiss and
then with the other premiss; the new conclusion thus obtained
will contradict the omitted premiss. It follows that there must
be an equal number of valid syllogisms in each of the first
three figures, and that they can be arranged in sets of
equivalent triads. ~f

Figure I can be regarded as asserting that a general rule
applies to a particular case; thus, in the example of Celarent
given above, a rule is negatively asserted, viz. No pets are ugly.,
the case of cats is subsumed under it, and the conclusion that
none of them is ugly is deduced. We shall see that, from this
point of view, we can again bring out the interdependence of
the first three figures. For example:

If All great statesmen sometimes lie

and George Washington is a great statesman,

then George Washington sometimes lies.

* f, p, q stand respectively for not-r, not-p } not-q.

f These triads are: Barbara, Baroco, Bocardo; [AAI, AEO, Felaptori}; Celarent,
Festino, Disamis; [EAO, EAO, Darapti\; Darii, Camestres, Ferison; Ferio, Cesare,
Datisi. Triads containing weakened conclusions or strengthened premisses are
included in brackets. Figure IV is self-contained; the equivalent sets are
all in the same figure, and are: \Bramantip, AEO, Fesapo]; Camenes, Fresison,

Dimaris.

70 A MODERN ELEMENTARY LOGIC

Now, if we deny that George Washington sometimes lies but
admit the rule, we must deny that he is a great statesman;
then we get. Denial of Result combined with Rule yields Denial
of Case. This will be a syllogism in figure II. If, however, we
deny that George Washington sometimes lies but contend
that he is a great statesman, we are forced to deny the rule.
Then we get, Denial of Result combined with reassertion of
Case, yields Denial of Rule. This will be a syllogism in figure III.

This interrelation of the three figures suggests that we can
easily formulate dicta for figures II and III analogous to the
Dictum de omni for figure I. Dictum for figure II: If every
member of a class has (or does not have) a certain property,
then any individual (or individuals) which do not have (or
have) that property must be excluded from that class. Dictum
for figure III: If certain individuals have (or do not have) a
certain property, and these individuals are included in a
certain class, then not every member of that class lacks (or
has) that property.

These dicta are self-evident in the same sense as the Dictum
de omni is self-evident; probably they would be most easily
apprehended in the first instance by means of a significant
example, explicitly stated; once the dictum has been clearly
seen to be exemplified in a particular case, it can be
generalized to cover other cases.*

Each of the four figures has certain distinctive character
istics. In the first figure only can all four, ^4, E 9 /, 0, forms
be proved, and only in this figure can the conclusion be A.
It is also the only figure in which the major and minor terms
occupy the same position in their own premiss as in the
conclusion; it is no doubt this characteristic which makes
reasoning in figure I seem to be the most natural. In figure II
the conclusion is always negative, and it is thus specially
adapted to show that an individual (or set of individuals)
must be excluded from a given class. Hence, it is sometimes

* For example, If every member of the class airmen has the property of good-
eyesight, then these volunteers who lack the property of good-eyesight are excluded
from the class airmen. It is quite easy to derive the special rules of the figures
from their respective dicta, as in the case of the Dictum de omni. The fourth figure
can be similarly dealt with, but we shall not include the statement of its dictum
in this book. Anyone who is interested should consult M.I.L., p. 97, or W. E.
Johnson, Logic, Part II, p. 87.

THE TRADITIONAL SYLLOGISM Jl

called the Figure of Exclusion. The third figure, admitting only
of particular conclusions, is specially adapted to show that
not every member of a class has a certain property, or that
two properties are compatible since both are possessed by a
given individual or by a certain set of individuals. When the
middle term is singular, denoting a single individual, this is
the most natural figure to use. For example, Stalin is a dictator,
Stalin has a passionate love of his country implies that to be a
dictator is not incompatible with love of one s country.
Again, Staunton is a great chess-player, Staunton is eccentric might
suggest that there is an essential connexion between being a
great chess-player and being eccentric. Accordingly, figure
III is sometimes called the Inductive Figure. It must, however,
be noticed that the conclusion cannot show us more than that
the two properties are compatible (or, it might be, incom
patible); it w 7 ould then remain to discover some way of show
ing that the compatibility was due to an essential connexion,
and the incompatibility to an essential disconnexion. To
prove such conclusions as these we must go beyond the
syllogism.

4.* POLYSYLLOGISMS

A polysyllogism is a chain of syllogisms in w T hich the con
clusion of one syllogism constitutes a premiss of the next. The
conclusions of all the syllogisms except the last are not stated;
this is the sole peculiarity of this form of argument. The
syllogism whose conclusion is a premiss (unstated) of the next
syllogism is called a prosyllogism; a syllogism one of whose
premisses is the (unstated) conclusion of the preceding syl
logism is called an episyllogism.

The Sorites is a polysyllogism in which only the final con
clusion is stated and the premisses are so arranged that any
two successive premisses contain a common term. For
example:

All dictators are ambitious.

All ambitious men are without compassion.

* This section and the next section may be regarded as concerned with
examination tricks. Those who do not require to pass elementary examinations
in logic, set by old-fashioned examiners, can disregard them.

72 A MODERN ELEMENTARY LOGIC

All men without compassion are relentless.
All relentless men are feared.
All men who are feared are pitiable.
/. All dictators are pitiable.

Two forms of Sorites are traditionally recognized:

(1) The Aristotelian Sorites. The minor premiss is stated first,
and the term common to two successive premisses occurs first
as predicate and then as subject; hence the form is

All A is JB
All B is C
All C is D
All D is E

:. All A is E

The special rules of this form are: (i) Only one premiss,
namely the last, can be negative. (Violation of this rule would
involve two negative premisses in one of the constituent
syllogisms.) (ii) Only one premiss, namely the first, can be
particular. (Violation of this rule would involve undistributed
middle.)

(2) The Godenian Sorites (so-called after Goclenius, who is said
to have introduced this form). The major premiss is stated
first, and the term common to the two successive premisses
occurs first as subject and then as predicate; hence the
form is

All D is E
All C is D
All B is C
All A is B

:. All A is E

The special rules of this form are: (i) Only one premiss,
namely the first, can be negative, (ii) Only one premiss,
namely the last, can be particular. An example of a Goclenian
Sorites is afforded by the following: If those who lack friends are
miserable, and those who are despicable lack friends, and those who
betray their own country are despicable, and those who love power for

THE TRADITIONAL SYLLOGISM 73

itself betray their country, and Quislings love power for itself., then
Quislings are miserable. This Is stated as a set of Implications,
not as asserted premisses,

5. ABBREVIATED ARGUMENTS AND EPICHEIREMA

A syllogism with one proposition omitted is called an
enthymeme, e.g. Whales are not fish because they are mammals.
Here the major premiss, Nofish are mammals, is omitted. This
is called an enthymeme of the first order. If the minor premiss
is omitted, the enthymeme is of the second order; if the con
clusion, the enthymeme is of the third order. These names are
quite unimportant. \Vhat is important is that we should be
able to recognize an enthymeme for what it is, namely, an
argument with an unstated premiss or conclusion. It is
extremely rare for us to state our reasoning in full. We most
often omit the major premiss, for we are apt to state that so-
and-so has a certain characteristic because it is a special case
without bothering to state the rule under which the case falls;
but sometimes we state the rule and the result, taking for
granted that we are dealing with a case which falls under the
rule; less frequently, we state the rule and the case, leaving
the result to be implicitly understood.

An epicheirema is a syllogism in which one or both of the
premisses is stated as the conclusion of an enthymematic
syllogism. For example:

No Marxist scientists are fair to Euclid s achievement,
because they dislike its sociological background;
Professor H. is a Marxist scientist;
. . Professor H. is not fair to Euclid s achievement.

This is a single epicheirema; when both premisses are
stated as the conclusion of an enthymematic syllogism, the
epicheirema is said to be double.

In a reasoned argument we frequently omit not only single
premisses but even a whole syllogism, tacitly presupposed.
Sometimes, indeed, an argument is merely hinted. It is often
not difficult to supply the missing links but the omission of a
connecting premiss may lead to a fallacy which would be

74 A MODERN ELEMENTARY LOGIC

detected if the argument were fully stated. It is for this reason
that the brief examples given in logical textbooks are so
obvious as to seem merely silly - the reader feels he would
never make a mistake like that I Yet elementary mistakes in
reasoning are of common occurrence.

An argument is sometimes put forward as a single premiss,
on the assumption that the missing premiss and conclusion
are too obvious to need explicit stating. For example:

(1) If that boy comes back, I ll eat my head 5 (Oliver
Twist], The hearer supplies the premiss and conclusion
required to complete the tollendo tollens argument.

(2) c lf we are marked to die we are enow to do our country
loss; and if to live, the fewer men, the greater share of honour*
(Henry V}. This dilemma is faulty, since the alternatives
marked to die, marked to live, are not exhaustive; more men
might make the difference between victory and defeat.

CHAPTER V

Individuals, Classes, and Relations

I. INDIVIDUALS AND CHARACTERISTICS

We have seen that the validity of inference depends upon the
relations of implication and not at all upon the truth or falsity
of the premisses. It is sometimes possible to know that an
implication holds between propositions without paying any
attention to the internal structure, or form, of the proposi
tions themselves. For example. If p and q, then r implies Either
(P or ?} or r implies If f, then either p or q^ no matter what kind
of propositions/?, g, rare. Frequently, however, this is not the
case. When we used p., q, r as illustrative symbols for the
premisses and conclusion of a valid syllogism, we were able
to represent the syllogism as an implicational form- If p andq,
then r. But nothing in this form enables us to know that No pets
are ugly, All cats are pets, No cats are ugly are so related that the
first two of these propositions jointly imply the third. We
know this only because we can analyse the propositions into
No Ms are P* s, All S s are M s, No S*s are P ^; these forms show
us that the first two do jointly imply the third.

Traditional logic is wholly concerned with propositions as
analysed complexes the elements of which are not proposi
tions. The terms of the A, E, /, propositions are classes; it
is these that constitute the subject-matter of the propositions.
But not all terms are classes; there are also individuals. Terms
thus fall into two groups: classes and individuals.

No attempt will be made here to define the word "indi
vidual"; it will be taken for granted that we all know how to
use the word; thus, Pius X is an Italian is a proposition about a
specified individual, viz. Pius X, and being an Italian is pre
dicated of this individual. Whenever we make statements
about individuals we say that they have, or do not have,
certain characteristics - this Pope is subtle, that table is

75

? A MODERN ELEMENTARY LOGIC

round, the sunset last night was beautiful, his attitude is
intelligent, this feeling is pleasant, and so on. What we
predicate of individuals is a characteristic, or, as it is sometimes
called, a property. Roundness is an example of a characteristic;
it is logically indifferent whether we say Roundness char
acterizes this table , or This table has the characteristic of
being round , or This table is round 5 . The last is our normal
mode of expression; we think of things as having definite
characteristics without as a rule thinking what it is to be a
characteristic or to characterize. But the three sentences
given above all mean the same.

Characteristics are not always symbolized by single words,
for example, dissolubility in water expresses a characteristic
of sugar; we might also have said c the capacity of dissolving
in water . For certain philosophical problems it is important
to distinguish between different kinds of characteristics and
between degrees of complexity. For our present purpose this
is not necessary. We must notice, however, that characteristics
may characterize other things than individuals, e.g. reducibility
is highly abstract, a certain proposition is true, a certain relation is
difficult to grasp.

An individual has characteristics but does not characterize;
it stands in relations but is not itself a relation. As contrasted
with an individual a characteristic is abstract. Some logicians
use the word concept for what is here called a characteristic. This
has the advantage of not suggesting that a characteristic must
characterize something; there may be characteristics that
characterize nothing, for every characteristic has a con
tradictory characteristic, e.g. perfect - imperfect, justice - in
justice, animality - non-animality. We use concepts with ease
long before we begin to talk about concepts. Unfortunately
when we, as philosophers, begin to talk about concepts we
tend to ask nonsensical questions about them, e.g. What is a
concept? and expect an answer of the same sort as we should
expect to the question, What is a centipede? It is enough
here to say that abstracting is not a highly difficult intellectual
feat; whenever we think we are abstracting, attending to
something and not to something else, recognizing similarities
and differences without necessarily noticing that we are

INDIVIDUALS, CLASSES, AND RELATIONS 77

recognizing similarities and differences. As William James, the
psychologist, has said, C A polyp if it ever thought "Hallo,
thuigemabob again!" would thereby be a conceptual thinker.*
The disadvantage of using the word concept 3 instead of
characteristic is that it tends to suggest that a concept is
dependent upon being thought of. This is a mistake. Complex
characteristics, e.g. man, are conveniently called concepts,
provided that we remember that a concept is entirely
identical with a characteristic or a specifiable complex of
characteristics. When w r e fully understand a concept we are
ably actually to specify these characteristics. What I under
stand by a concept, e.g. justice, home, may be different from
what you understand by it; we can then be said to have
different conceptions of the same concept. Thus Newton certainly
had a different conception enforce from Einstein s, but, so to
speak, they intended to think about the same concept.
Advance in scientific thinking in part consists in clarifying
our conceptions; we aim at abstracting from our personal
habits of thinking, our private attitudes, hopes, and fears,
and apprehending clearly what is constant in significance
throughout repeated instances.

The converse of the relation of characterization is exempli
fication , an object, or entity, characterized by red exemplifies
redness, i.e. is an instance of it. Thus Abraham, Aristotle, John
Bunyan, James Clerk-Maxwell, etc., exemplify man , these
individuals are characterized by the complex characteristic
signified by the word "man".

A characteristic that could be exemplified even if in fact
there are no actual instances of it is said to be existent. This is
the use of the word "existence" in mathematics, as when we
say, c an even prime exists 5 . This sort of existence or being
must be distinguished from the full-bodied (so to speak) kind
of being which individuals have, namely being in time and
space. Bertrand Russell calls the former subsistence, the latter
existence. We shall not make use of the word "subsistence" in
this book; when we say that a characteristic exists we shall
simply mean that it is not inconsistent to assert that it has
instances.

In the case of individuals we must distinguish between

78 A MODERN ELEMENTARY LOGIC

what could consistently exist and what does in fact exist. For
example, there could be a King of the United States but in
fact there is not; there could be a King of Utopia but in fact
there is no such country as Utopia and thus no King of
Utopia. It is easy to indulge in much discussion on this point
and to fall into apparently inextricable difficulties. But we do
understand very well what is meant by saying that God exists
and by saying that God does not exist. The distinction between
what does exist (in the sense in which this, that, or the other,
individual exists) is the distinction between fact and fiction.
Questions of existence are to be settled in two ways. If we
ask, Do just men exist? 3 we may start from the assumption
that certain men called just exist, e.g. Aristeides, but want to
ask whether they are really just. This is a question about the
concept just, i.e. it asks what the characteristic just is. The
answer to this question is given by a definition of the word
"justice", i.e. by clarifying the concept symbolized by
"justice". But, given this clarification, we may still want to
ask whether justice is exemplified in human beings. Such a
question can be answered only by empirical investigation,
just as the question, Do centaurs exist? , must be settled by
looking everywhere to see whether there are any centaurs.
Similarly, the questions, c Does God exist? , Does the Devil
exist? , might be meant in either of these two ways and must
be settled either by clearing up what we mean when we use
the word "God", or the word "Devil", or by appeal to our
experience.*

2. CLASSES

We often want to talk about all the instances of a certain
characteristic, these instances being taken together. When we
refer to all possible exemplifications of a given characteristic
(simple or complex), we are speaking of the class determined
by the characteristic. Those instances of the class which exist
are called the members of the class, or sometimes, the elements
of the class. The class is said to contain its members.

* It must not be assumed that experience is limited to what is given to sense.
Whether this is so, or not } is a metaphysical question lying beyond our scope
as logicians.

INDIVIDUALS, CLASSES, AND RELATIONS 79

We are all familiar with the notion of class and, as we have
seen, Aristotle s logic was primarily concerned with relations
between classes and only incidentally with statements about
individuals. The notions of class, class-membership, class-inclusion
are presupposed by Aristotle s treatment and are not discussed
by traditional Logicians except in the most perfunctory
manner.

A class must be distinguished from its members for, as we
shall see in a moment, a class has characteristics which its
members lack. It must also be distinguished from the word
or symbol used to refer to it. This is not peculiar to classes;
we must always distinguish between a symbol and what it
symbolizes, though in fact w r e do not always keep the distinc
tion clear, especially when talking about classes.

There are two ways of selecting the individuals who con
stitute the membership of a class. One is to enumerate the
individuals one after another, the order of enumerating
being indifferent. For example, we might enumerate the
individuals, Stalin, Mussolini, Hitler, and thus obtain the class
whose members are Stalin, Mussolini, Hitler. The second
way is to select a certain characteristic, e.g. being a dictator in
Europe in 1340, which may belong to many individuals. In
fact, the membership of this class consists of the three in
dividuals named above; there is, however, nothing in the
complex characteristic which determines that it should be
limited to three members.* World-dictator is a characteristic
determining a class which contains no members, though no
doubt Hitler wishes that it contain one member and that he
should be that member.

The enumerative selection of a class is possible only when
the class contains a finite number of members; it is then called
a finite class. An infinite class is clearly not capable of being
enumerated; hence, such a class must be determined by a
characteristic, whilst a finite class is usually but not neces
sarily so determined. For instance, a complete census, free
from errors, of the inhabitants of Great Britain enumerates all

* Indeed this class probably contains more than these members, if General
Franco and Dr Salazar are to be regarded as dictators in their own countries.
The class could be limited to the three members specified if we altered the
characteristic to being a belligerent dictator in Europe in September 1942.

8O A MODERN ELEMENTARY LOGIC

the members of the class inhabitants of Great Britain. We might
enumerate the class containing the following members:
Pompey the Great, FalstafFs red nose, Cleopatra s Needle,
Napoleon s emotion on first seeing St Helena. No one but a
logician, or a fool, would want to select such a class, but we,
for a purpose, have just done so, and the class - which con
tains four members - might be described as c the class I have
just selected , and these members each possess a certain
property possessed by nothing else in the universe, viz. the
characteristic of being either Pompey the Great or Falstaff s red nose
or Cleopatra: s Needle or Napoleon s emotion on first seeing St Helena.
Such artificial classes are seldom useful for scientific purposes,
but this artificial class has the use to which we have just put it.

A given characteristic is said to determine the class each
member of which exemplifies that characteristic. Thus men
determines the class containing the members Adam, Aristotle,
Buddha, . . . Winston Churchill, where the dots indicate
each of the other human beings whom in fact we could not
enumerate though, it is assumed, God could do so; in a
minute another item would have to be added to the enumera
tion, and so on, for every human being that is born. Thus men
includes the dead, the living, and the yet to be born human
being.

A characteristic which determines a class is said to be a
class-property. This phrase is misleading, for a class-property
is a property common and peculiar to all the members of a
class; it is not a property of the class at all. It is a property
of the class men that it has exemplification, but the class men
has not the property of being a rational animal.

We could be, even if we are not, acquainted with the in
dividual Stalin; but we could not be acquainted with the class
determined by being a dictator in Europe in 1940. Accordingly,
the way in which we refer to a class when we use a class-
symbol is quite different from the way in which we refer
to an individual when we use a proper name speaking to
the person named. Class-symbols are descriptive; we can
significantly use class-symbols although no members are pre
sented to us, and even if we do not know whether the class
has members or not. It is for this reason that we can signi-

INDIVIDUALS, GLASSES, AND RELATIONS 8l

ficantly prefix to class-symbols such words as "all", "some",
"any", "a", "the".

When we speak of all the members of a class the word "all"
may be used ambiguously; we may mean "each and every
one member" or "all the members jointly". Usually the con
text suffices to make the meaning clear, but we may some
times be in doubt, e.g. "All the men could not move the cart"
might mean that not one of them alone could move the cart or
it might mean that all together could not. "The police routed
the crowd" means all the members of the police jointly; "The
police carried truncheons" means each member of the police
did. When we use a term to signify each member severally,
then we are said to use it distributively , when we use a term to
signify all together, then w^e use it collectively. The distinction
is a distinction in usage.

In the collective usage of "<z/Z", all the members of a class
constitute its collective membership. For example, if the enemy
army occupies a country, that which occupies is the collective
membership of the class; it is clearly not each individual
soldier who occupies the country nor the class for the class
cannot carry arms nor shoot -it is only individuals that
can act.

Finally, we must keep clear the distinction between classes
and associations, or organizations, such as the Post Office
organization, the T.U.C., the United States, the League of
Nations. The class containing as members the Nations in the
League of Nations must be distinguished from the League of
Nations: being a member of the League of Nations is a class-
property of Great Britain, and of each one of the other
member-Nations, but being-a-League-of-Nations is not a pro
perty of any member. To say that it was would be to talk
nonsense.

3. RELATIONS

All deduction depends upon the logical properties of rela
tions. Relation cannot be defined without using words that
are more or less synonymous. We all recognize that in
dividuals in the universe are not isolated; they stand in

82 A MODERN ELEMENTARY LOGIC

various relations. Physical objects stand in spatial and gravita
tional relations; human beings are related in numerous ways,
e.g. by kinship, by enmity, or by friendship, by precedence,
and so on. In short every individual object, of every possible
sort, is related to some other individuals and also to the
characteristics which they exemplify or which they fail to
exemplify. Characteristics also stand in relations to other
characteristics, e.g. implication, consistency, inconsistency.

Relations relate terms. The most elementary character
istic of a relation is the number of terms it requires in order
to make sense. Father <?/~ requires two terms; loving, governing,
hurting are also two-termed. Such relations are called dyadic.
Relations requiring three terms are triadic, four terms
tetradic, five terms pentadic, and so on. Relations requiring
an indefinite number of terms are polyadic (e.g. among).
Some logicians call any relation requiring more than three
terms polyadic. In ordinary discussion we seldom talk about
relations requiring more than four terms. Giving is triadic:
Tom gave a ball to Mark relates giver, gift, and recipient. Teaching,
between, are other examples of a triadic relation; owing is
tetradic: Jones owes Spencer 10 for this watch. Our discussion
will be confined to dyadic relations.

Every relation has a sense, i.e. a direction in which it goes,
e.g. loving goes from lover to loved, father of from male parent to
child. The term/r0?tt which the relation goes is the referent; the
term to which the relation goes is relatum. In Mary loves
Darnley (as the order of the words in English shows) Mary is
referent, Darnley is relatum. We will substitute the illustrative
symbols x,y for these respectively, and R for the relation; then
we have xRy, which signifies something having a relation to some
thing. It is sometimes convenient to write R(x, y] instead of
xRy, so that the same mode of symbolizing can be used for
triadic relations, and those with more terms than three, e.g.
R(x, y, z), is a relational form into which we could fit the
relational statement Tom gives a penny to Mark, provided that
we have adopted some convention to show the order of the
terms. As we are here concerned only with dyadic relations
we shall use xRy. In what follows R will illustratively sym
bolize some one relation but not a specified relation.

INDIVIDUALS, CLASSES, AND RELATIONS 83

Relations are said to hold or fail of given terms. When R
holds from x tojv, then there is some relation which holds from
y to x, which will be the converse of the original relation. We
might symbolize the converse of R by R. xRj is always
equivalent to yR?x but R and R? are not necessarily the same
relation. For example, x loves y is not equivalent to y loves x
since the loved does not necessarily love in return, and is thus
not also lover of the one who loves. The converse of R is
sometimes written R, as for instance by Bertrand Russell and
A. N. Whitehead in Principle, Mathematica. We shall use R c for
the converse of R since it is more directly suggestive of the
converse of a relation. Which symbol we adopt is logically
indifferent; it is a matter of notation to be decided on grounds
of convenience or taste.

Logical properties of relations are properties which belong to
relations without reference to the terms they may happen to
relate. Many of these properties can be stated only if there are
certain limitations to the possible referents and relata. Hence,
it is convenient to distinguish between the domain, converse
domain, and field of a relation.

If R is any relation, then the domain oFR is the class of terms
that have R to something, i.e. all possible referents of R. The
converse domain is the class of terms to which something has R\
i.e. all possible relata of R. The field of R is the sum of the
domain and the converse domain of R. The domain and the
converse domain may overlap, as, for example, is the case
with the relation ancestor of limited to the field of the direct
descendants of George I. The domain is the class of all those,
in this field, who have descendants; the converse domain is
the class of those who are his descendants. In this field,
Edward VII is referent to George V, George VI, and is
relatum to Queen Victoria, George L

The relations holding between members of a family are
familiar and can be used to illustrate important logical proper
ties of relations. If the reader considers what is the converse
of married to, father of, uncle of, ancestor of, he will easily notice
that sometimes the same relation relates x,y (any two terms)
as relates y, x, and sometimes a different relation. Again, the
father of a father is not a father but a grandfather, but the

84 A MODERN ELEMENTARY LOGIC

ancestor of an ancestor is also an ancestor. These family
relationships suggest to us the importance of distinguishing
relations according to the properties they have. We shall now
consider those properties of relations that are important for
inference.

(1) Symmetry. A relation R is symmetrical when xRy =zyRx.
Thus, if xRy, thenjy^. For example, spouse of, equal to, different
from, brother or sister of.

A relation R is asymmetrical when xRy is incompatible with
yRx. Thus, if xRy, then never yRx. For example, father of,
darker than, greater than, preceding.

A relation R is non-symmetrical when xRy is neither equiva
lent to nor incompatible with y Rx. Thus, if xRy, then perhaps
yRx and perhaps not yRx. For example, implication, friend to,
sister of.

(2) Transitiveness. This distinction is based upon the con
sideration of pairs of terms with reference to some relation R.
A relation R is transitive when, provided it holds from x toy,
and also from y to , it must hold from x to . Thus, if xRy and
yRz, then xRz. For example, ancestor of, exactly contemporary
with } parallel to, implication.

A relation R is intransitive when it is such that if xRy and
yRz, then never xRz. For example, next to, father of, one year
older than.

A relation R is non-transitive when it is such that if xRy and
yRx then perhaps xRz and perhaps not xRz. For example,
sister of, overlapping in time with, cheating, different from.

The properties of symmetry and transitiveness, and their
opposites, are logically independent. Hence, we can classify
relations into the four following groups:

(i) Symmetrical transitive: equal to; matching in colour.

(ii) Symmetrical intransitive: spouse of; twin of.
(iii) Asymmetrical transitive: ancestor of; greater than; above;

before.
(iv) Asymmetrical intransitive: father of; greater by two than.

Relations that are both symmetrical and transitive have
the formal properties of equality. There is a third important
property that belongs to such relations; this property is called

INDIVIDUALS, CLASSES, AND RELATIONS 85

reflexweness. It may be defined as follows: a relation R is
reflexive if it holds between x and itself, i.e. xRx. Identity is
reflexive; as tall as is reflexive, and so on. A relation may be
symmetrical without being reflexive, e.g. spouse of. The only
relation that can be said to be reflexive without limitation is
identity. Reflexiveness, symmetry, transitiveness are formal
properties of identical with., and thus, equal to. Any relations
that have these properties are of the formal nature of identity,
e.g. exactly matching, co-implication, coincidence.

A relation that is both transitive and asymmetrical has also
another property, called aliorelative. A relation R is aliorelative
when it is such that no term x has R to itself, e.g. successor of.
Asymmetrical relations are necessarily aliorelative, but the
converse is not the case, since spouse of, twin of are sym
metrical but also aliorelative. But if a relation is both transi
tive and asymmetrical, it is also aliorelative.
(3) Connexity. Given any relation R and the field of R, it is
not necessarily the case that any two terms in the field are
related by R or R c . For example, given the field human beings
and the relation ancestor of, it does not follow that of every pair
of terms the relation must hold. When, however, this does
hold the terms are said to be connected. Connexity may be
defined as follows: A relation R is connected when, given any
two terms of its field, viz. x,y, then either xRy oryRx (i.e. xRy
or xR c y). If this condition does not hold then R is said to be
unconnected.

A relation that is transitive, asymmetrical and connected
is a serial relation, i.e. it suffices to generate a series, e.g. an
arithmetical progression. Greater than, limited to the field of
natural numbers, is connected, since, of any two numbers one
is greater than the other if actor ofh unconnected. Greater than
suffices to generate the series i, 2, 3, 4 ...

Relations may also be classified according to the number
of terms to which the referent or relatum may stand in the
given relation R. If Jones is a debtor to Robinson, it does not
follow that Robinson alone stands in that relation to Jones,
who may have many debtors; Jones may also himself have
debtors. If Mary has sisters she is not the only daughter of
David but she has only one father. In a monogamous country,

86 A MODERN ELEMENTARY LOGIC

if Mary is wife of James then no other man can be her
husband and no other woman be James s wife. As these
examples suggest, we can distinguish four groups of relations
from this point of view:

(i) Many-many relations: R is many-many when both the
domain and the converse domain can contain more than one
member, and the selection of a term from either does not
determine the selection of a term from the other, e.g. i of
latitude north of, creditor to, sister of.

(ii) Many-one relations: R is many-one when the selection of
a term from the domain determines the selection of the term
from the converse domain, but not reversely, e.g. child of.

(iii) One-many relations: R is one-many when the selection of
a term from the converse domain determines the selection of
the term from the domain, but not reversely, e.g. father of.

(iv) One-one relations: R is one-one if the selection of a given
referent determines the selection of the relatum, and reversely.
There may be many members of the domain and the con
verse domain ofR, but the selection of any one of these terms
as referent uniquely determines the selection of the relatum,
and reversely. For example, eldest son of a father, greater by
one.

It should be noticed that, for instance, parent of is not a
one-many relation since, if x is parent of y, then x may be
either father or mother of y\ hence two terms stand in the
given relation tojy. If, however, the referents be limited to
males, then the relation is one-many, if the relatum be now
limited to eldest son, the relation is one-one. It is important to
observe that mathematical functions result from one-many
relations, e.g. the cosine of x, the logarithm of y. One-one
relations are of great importance in the exact sciences; cor
relations are one-one relations.*

* It may be of interest to notice that relations can be combined. Suppose
there is a relation R such that xRy, and a relation S such that ySz\ then there
is a relation between x and z compounded of the two relations R, S. This relation
is called the relative product of R and S. Bertrand Russell symbolizes the relative
product of R and S by writing R/S. The relative product of sister of and father
of is paternal aunt. The order in which R, S is taken is significant; if their order
be reversed a different relation may be obtained. For example, the relative
product of father o/and sister of is father of. The converse of a relative product
is obtained by reversing thtjorder of the factors and then substituting their
converses: i.e. converse of 3/R is S/R (using R for Re), e.g. the converse of the

INDIVIDUALS, CLASSES, AND RELATIONS 87

4. CLASS-INCLUSION AND CLASS-MEMBERSHIP;
SINGLE-MEMBERED CLASSES

We say, "All Marxists are determinists", and "Professor
Hodd is a Marxist 53 , and are thus led to suppose that are and
is signify the same relation. This is a mistake. In "All Marxists
are deter minists" are signifies the relation of inclusion ^ in
"Professor Hodd is a Marxist" is signifies membership of a class.
These two relations differ in their logical properties: inclusion
is non-symmetrical and transitive, whereas class-membership is
asymmetrical and intransitive. X can be included in T with
out its being the case that Tis also included in X, but it is also
possible that where Xis included in T, Talso is included in X.
Class-membership, on the other hand, is clearly not sym
metrical, and is indeed asymmetrical. Hodd (in the example)
is a member of the class Marxists, but the class Marxists is not
a member of Hodd. All individuals are members of classes, but
no class is a member of an individual. Class-inclusion is
clearly transitive, but class-membership is not. For example,
Fido is a member of the class of my dogs; the class of my dogs is a
member of the class of single-membered classes; but Fido is not a
single-membered class, for Fido, being an individual dog, is
not a class of any kind. When we speak of classes as members
of other classes we are indeed shifting the meaning of "member
of . In this book w r e shall always understand by a class-
membership proposition a singular proposition.

A singular proposition is a proposition about a uniquely
specifiable entity, e.g. David Hume is a philosopher. This is a pen.
A uniquely specifiable entity (e.g. this pen) may be regarded
as the sole member of some class (e.g. the pens now owned by
me}. The traditional logicians treated every singular proposi
tion as being a statement about a class containing only one
member. On this view, David Hume is a philosopher is equivalent
to All David Humes (there being only one) are philosophers. We
mentioned this view earlier (p. 56) without criticizing it. We
must now observe that, in adopting this view, the traditional

relative product of husband of and daughter-in-law is father or mother-oj. The
relative product of R and R is called the square ofR, Thus R/R can be written
J2 2 ; the relative product of father and father is grandfather; the converse of the
square of father is grandchild. The square of ancestor of is ancestor of.

88 A MODERN ELEMENTARY LOGIC

Logicians did not see clearly exactly what they were doing nor
why their analysis of categorical propositions required this
interpretation of singular propositions.

It is obvious on reflection that a class-inclusion statement
is different in kind from a class-membership statement. If we
say H.M.S. Hermes is an aircraft carrier we are stating that a
certain individual is a member of a class, viz. aircraft carriers.
If we say Aircraft carriers are warships we are saying that every
member of the class aircraft carriers is also a member of the
class warships. A ship can, in the proper sense, sail the seas;
a class cannot sail. We must then distinguish between a state
ment about a single-membered class and a statement that the
class has only one member, and similarly we must distinguish
a single-membered class from its sole member. There exists one
and only one number which is a factor of every number in a given finite
collection of positive integers is a statement that a certain class
has only one member; this member is the H.C.F. of the given
collection of numbers. The H.C.F. is the sole member of the
class determined by the above formula when the finite col
lection is given. The class of even primes is a single-membered
class, and its sole member is the number 2. The class of most
virtuous of dogs necessarily contains only one member, for, if
two dogs were equally virtuous neither could be said to be
the most virtuous. The class of my dogs (on the assumption that I
possess only one dog) is single-membered. This class contains
fewer members than the class of my books, but it does not make
sense to say that my dog has fewer members than the class of
my books, or any other class.

We can see from what has just been said that anything which
can be significantly stated about a class cannot be significantly
stated about an individual. Logicians recognize this distinc
tion by saying that an individual and a class are of different
logical types. Accordingly "are" and "is" in the two sentences
given at the beginning of this section differ in meaning.

5. SUBCLASSES AND EMPTY GLASSES

A class a included in another class /? is said to be a subclass
of /?. It is convenient to call the class /? a superclass of a. The

INDIVIDUALS, CLASSES, AND RELATIONS 89

class Frenchmen is a subclass of Europeans; the class Italians is
also a subclass of Europeans. For many purposes it is useful to
be able to distinguish the subclasses of a class. In the next
chapter we shall be concerned with this process of distinguish
ing subclasses. Sometimes we distinguish a subclass and sub
sequently find that it has no members. For example, in the
summer of 1940 certain penalties were laid down by the
British Parliament to be inflicted upon those who spread
alarm and despondency . It seemed good to the British
Government to take this class into account. But it might well
have turned out to be the case that spreading alarm and
despondency was a complex characteristic having no exemplifi
cation, or, to use other words, the class determined by this
characteristic was found to be empty. An empty class is a
class that has no members. In Chapter II w r e noticed that
there are no dishonest immortal politicians. Among a given
class of school children there may be none who are both hard
working and able. We find no difficulty in seeing that complex
characteristics may lack exemplification. In such cases it is
convenient to say that the class determined by the char
acteristic is empty. This is a mode of speaking, or, as we may
say, a convention. It seems strange to extend the meaning of
"class" in such a way that we can speak of empty classes.
But, as the above examples suggest, we shall avoid certain
difficulties if we do so. For instance, if we admit that A, E, /,
propositions are statements about class inclusion and ex
clusion, we shall get into the sort of difficulties we noticed in
the case of inversion unless we admit that a class may have
no members. If we grant that a class may be empty, then we
can bring out the fundamental difference in form between
the universal propositions A, E and the particular proposi
tions /, 0.

Consider the two propositions: All who spread alarm and
despondency will bejined or imprisoned; All women between the ages
of twenty and thirty will be called up for military service. As under
stood by the people of Great Britain in the years 1940 to the
present (September 1942), it would certainly be admitted
that the significance of the first of these propositions does not
depend upon there being any instances of the complex

go A MODERN ELEMENTARY LOGIC

characteristic spreading alarm and despondency. Indeed, the British
Government no doubt hoped that by threatening penalties
the class determined by spreading alarm and despondency would
remain empty. In the case of the second proposition we
unhesitatingly assert There are women between the ages of twenty
and thirty, i.e. we take for granted that the class constituting
the subject-term is not empty. We do so because the proposi
tion is asserted (if, indeed, anyone does assert it) in the con
text of our knowledge about the people in Great Britain. No
one would have any interest in making this assertion if there
were no women between the ages of twenty and thirty. Let
us for a moment neglect what we know; we should have no
difficulty in admitting that in neither case does the significance
of the proposition depend upon there being members of the
class constituting the subject-term of the proposition.

What, then, is the minimum interpretation that must be
given to these propositions in order to render them significant?
The minimum interpretation imports nothing into the pro
position which depends upon knowledge not derived from the
proposition stated. Clearly, then, it is advisable so to interpret
these propositions that their significance should in no way
depend upon there being any members of the class con
stituting the subject-term. This interpretation can be con
veniently formulated in the sentence "If anyone spreads
alarm and despondency, he will be fined or imprisoned", and
analogously for the second proposition. This formulation
brings out that the proposition asserts that a certain class is
empty, viz. the class determined by the conjunction of char
acteristics spreading alarm and despondency without being either
fined or imprisoned. Its significance is to deny that a certain class
has members. Such a proposition is called existentially
negative.

Now consider the propositions Someyoung men are combatants,
Some dishonest politicians are not mortals. Ordinarily we should
unhesitatingly assert that the significance of these proposi
tions depends upon there being members of the classes respec
tively constituting the subject-term. We so use the word
"some" in English that to assert any proposition of these two
forms is to assert that there are members of the given class for

INDIVIDUALS, CLASSES, AND RELATIONS gi

which some is used as a quantifier. Thus, Some trepangs are
echinoderms asserts that there are members of the class tre
pangs, i.e. the proposition is existentially affirmative. The
proposition Some trepangs are not pleasant to eat is likewise
existentially affirmative, whether true or false.

Granted then that the minimum interpretation of universal
propositions does not require that the class constituting the
subject-term should have any members but that particular
propositions do require this, we can formulate the A, E, I 9
propositions as follows:

A Nothing is both S and non-P SP = o

E Nothing is both S and P SP = o

/ Something is both S and P SP 4= o

Something is both S and non-P SP =j= o

The set on the right-hand side presents a convenient mode of
symbolizing propositions from this point of view. SP, SP stand
for the conjunction of two classes in each case: SP stands for
the class constituted by combining S and P, SP for the class
constituted by combining S and non-P\ C = o" signifies that
the class has no members, i.e. is empty; " =f o" signifies that
the class has members, i.e. is not empty.* This symbolism is
convenient, but it must not be supposed that it gives us any
more, or less, information than is given by the corresponding
English sentences on the left-hand side.

It should be observed that if it is true that nothing is both
S and P, then, provided that S has members, P also has
members, or - as it may equivalently be stated - either S
has no members or non-P has members. | For example, if
it be true that nothing is both human and infallible, then either
the class human beings has no members or there are fallible
beings.

The above formulations bring out very clearly that the
universal propositions are fundamentally different in form
from the particulars, whereas the difference between negative
and affirmative propositions is not fundamental.

* This symbol must of course be distinguished from the number o.
f This can be formulated: Either S = o or P =j= o.

Q2 A MODERN ELEMENTARY LOGIC

If we assume that the subject S has members, we can
formulate these propositions as follows:

A SaP S 4= o and SP = o

E SeP S 4= o 0;zd SP = o

/ SiP SP + o

SoP 5P o

Here, again, the difference in form between the universals
and the particulars is made manifest. On the assumption that
in the particular propositions the class constituting the sub
ject-term is not to be interpreted as necessarily having
members, the formulation is:

/ SiP Either S = o or SP 4= o
SoP Either S = o or SP 4= o

6. THE UNIVERSE OF DISCOURSE AND THE
UNIVERSAL CLASS

In the preceding section it was said c we unhesitatingly assert .
For whom does "we" stand? Presumably moderns of Euro
pean culture who are able to read English. The context in
which this book is written and read enables us to take the
reference of "we" as understood. In any discussion that pro
ceeds without serious misunderstanding or ambiguity the
context is understood by all the speakers. If I say, Hamlet
killed Polonius, not Polonius Hamlet , I shall be understood
to refer to the realm of Shakespeare s plays. If I say,. Crom
well was not really like what Scott makes him out to be , I
shall be understood to be contrasting Scott s fictitious present
ment of Cromwell in Woodstock, with Cromwell who actually
lived and was Lord Protector of England in the middle of the
seventeenth century. We contrast the world of fiction with
the Vorld that actually is . But frequently we want to put
some limitation upon the context of our discourse so that
what we are saying shall not be understood to refer to every
thing that has happened or happens everywhere. For
example, Women have the right to vote would usually be
understood to be limited in reference to the country under
discussion or in which the speakers are living; it would also

INDIVIDUALS, CLASSES, AND RELATIONS 93

ordinarily be understood to be limited to a fairly recent
period of time. The context thus understood may be called
the universe of discourse.*

In the language of classes we can say that the universe of
discourse is the class such that all classes discussed are sub
classes of it. Since every member of a subclass is a member of
its superclass, it follows that every member of a class under
discussion is a member of the one universal class. But just as
we can have a different universe of discourse on one occasion
(e.g. fictitious entities) from the universe of discourse on
another occasion (e.g. actual w T orld), so we can have a
different universal class on different occasions. But, granted
the context of the discussion, there is only one universal class.
In a given universal class we can distinguish subclasses which
would have no place in another universal class, f For example,
in the universal class of men throughout the history of the world it
makes sense to distinguish between men acting freely and men
not acting freely., even if we subsequently decide that one of
these classes is empty; hi the universal class of physical entities
such as electrons the distinction between acting freely and not
acting freely may be without sense.

When we are not clear with regard to the limitations placed
upon the universal class (constituted by any discussion) we
are apt to talk nonsense without noticing that we do so.

7. RECONSIDERATION OF THE

TRADITIONAL TREATMENT OF OPPOSITION AND
IMMEDIATE INFERENCES

Once we have admitted that the universal propositions SaP,
SeP are to be interpreted as existentially negative, we can see

* This phrase was introduced by A. de Morgan (Formal Logic, pp. 41, 55)
and G. Boole (Laws of Thought, p. 166). It was thus explained by de Morgan:
4 If we remember that in many, perhaps most propositions, the range of thought
is much less extensive than the whole universe, commonly so-called, we begin
to find that the whole range of a subject of discussion is, for the_ purpose of
^discussion, what I have called a universe, that is, a range of ideas which is either
expressed or understood as containing the whole matter under discussion. ^

f In Pirandello s play Six Characters in Search of an Author, the worlds of fiction
and of reality are deliberately brought together, with dramatic effect, but the
real characters in the play and the six characters are, in. fact (as we say), both
fictitious.

94 A MODERN ELEMENTARY LOGIC

that -we must reconsider the validity of the inferences allowed
by the traditional Logicians. For \ve have also agreed that
particular propositions are existentially affirmative, so that
Some explorers are intelligent implies that there are explorers
and, consequently, also intelligent beings.

Confining our attention to the traditional square of opposi
tion , we find that A and 0, E and /, respectively, are con
tradictories; for, SaP ~ SP = o, and SoP == SP + o. But the
inference from SaP to SiP> and from SeP to SoP is not valid,
since SaP implies only that nothing is SP (i.e. SP = o),
whereas SiP implies something is SP 9 and this means that the
class S is not empty. Again, SaP and SeP are not contraries
since it is not inconsistent to assert SP = o and also SP = o,
on the assumption that nothing is S. The force of asserting both
of them is to deny that there are any members of S. This may
seem absurd, but it is not difficult to give significant examples:
All disinterested leaders are trustworthy. No disinterested leaders are
trustworthy, taken as both true, constitute a denial that there
are any disinterested leaders.* The inference of SiP from SaP 9
and of SoP from SeP does not hold good, since the particulars
imply that the class S is not empty, whereas the universals do
not imply this.

In general, on the assumption we are making, a universal
proposition can be validly inferred from another universal
proposition, and a particular proposition from another par
ticular; but a particular cannot be inferred from a universal.
Hence, the following traditional immediate inferences are
invalid, unless the assertion that S is not empty be added:
(i) conversion of A; (ii) contraposition of E; (iii) inversion.
Likewise, a syllogism with two universal premisses and a par-

* Mrs Ladd-Franklin gives an example in the following quotation: All x isy y
MX isy assert together that x is neither y nor not-jj>, and hence that there is no x.
It is common among logicians to say that two such propositions are incom
patible; but that is not true, they are simply together incompatible with the
existence of x. When the schoolboy has proved that the meeting-point of two
lines is not on the right of a certain transversal and that it is not on the left of
itj we do not tell him that his propositions are incompatible and that one or
other of them must be false, but we allow him to draw the natural conclusion
that there is no meeting-point, or that the lines are parallel* (Mind, 1890,
p. 77 .). This example assumes that on the right and on the left are contradictory
terms; granted this assumption, then the two propositions are of the form JVb S

INDIVIDUALS, CLASSES, AND RELATIONS

95

ticular conclusion is invalid, since the conclusion will imply
that the class S is not empty, whereas this is not guaranteed
by the minor premisses in the cases under discussion. Con
sequently the weakened moods are invalid, together \\ith
Darapti, Felapton, Bramantip> Fesapo, each of wiiich contains a
strengthened premiss. The valid syllogisms, therefore, reduce
to fifteen: four in figure I, four in figure II, four in figure III,
three in figure IV.

These results confirm our contention in Chapter II that
the validity of inversion depends upon the assumption that
the classes S y 5, P, P are not empty, i.e. have existence in the
universe of discourse.

At this point we can return to the two questions raised on
page 26. The assumption that 5, P, S, P, all exist in the
universe of discourse can be represented diagrammatically by

saying that the area outside the circles, in each case, repre
sents everything that is neither S nor P. Let a rectangle
represent the universe of discourse, within which any of the
five diagrams given on page 25 can be drawn. It will suffice
to take one example: we select diagram 4. The compartments
are labelled with the four possible combinations. We could
substitute any diagram instead of 4; hence, in every case
Some non-S is non-P. If this is correct, then every proposition
of the four traditional forms has an inverse, and, indeed, the
same inverse. This is absurd. We must, then, conclude that
there is not always some area outside the circles but included
in the universe of discourse. We need, then, ten notfae diagrams.

96 A MODERN ELEMENTARY LOGIC

These ten may be conveniently given in the form of
rectangles:

(i)

(ii)
(iii)
(iv)
(v)
(vi)

(vii)

[viii)
(ix)
(x)

SP

SP

SP

SP

SP SP

SP

SP

SP

SP

SP

SP

SP

SP

SP

SP

SP \

SP

SP

SP

SP

SP

SP

SP

SP

These diagrams* should be compared with Euler s diagrams
(p. 25). We have now distinguished two ways of interpreting
each of Euler s diagrams, depending on whether or not the
class SP has members. Thus diagrams (i) and (ii) correspond
to Euler s No. i, and so on.

How do we deal with the case when a term, significant in
the universe of discourse, nevertheless signifies nothing in the
actual world? Consider our original example, Ghosts are not
always draped in sheets. This is a particular negative proposi
tion. We shall represent it by

* The student who is interested in this topic may consult T. N. Keynes, F.L.,
Pt. II, Ch. VIII, Pt. Ill, Ch. VEIL See also M.I.L., Ch. V, 4, 5.

INDIVIDUALS, CLASSES, AND RELATIONS 97

The circle that is shaded may be regarded as blacked-out;
t is empty, i.e. no ghosts exist in the actual universe;
ion-ghosts (5), things draped in sheets (D), things not draped in
sheets (D) all exist both in the actual universe and in the
iniverse of discourse; ghosts (G) exist only in the universe of
iiscourse: the class ghosts is empty, but, in the given propo
sition, it is falsely assumed not to be empty. Hence, the
Droposition Ghosts are not always draped in sheets is false; so too
5 the proposition Ghosts are sometimes draped in sheets (i.e. an
I proposition).

8. THE LOGICAL PROPERTIES OF RELATIONS
AND THE VALIDITY OF INFERENCES

[n discussing the traditional immediate inferences (in
Chapter II) we found that in some cases the inferred con
tusion was equivalent to the premiss from which it was
inferred but that in some cases it w r as subimplicant to it. We
:an now see that this difference follows from the logical
properties of the relations involved. The A, E, 7, proposi
tions are statements of class-inclusion or exclusion. Since
inclusion is non-symmetrical we cannot tell from the fact that
X is included in T whether Tis included in Jfor not. Hence
from SaP (interpreted as meaning All S s are P*s, i.e. the class
S is included in the class P) w r e can infer only PiS. Thus the
converse of an A proposition is not equivalent to the original
proposition. But partial inclusion and total exclusion are both
symmetrical; hence SiP and SeP both have simple converses.
The traditional Logicians did not study the properties of
relations, so that their treatment of immediate inferences is
untidy and unpleasing. The conversion of A 9 E, 7, proposi
tions depends entirely upon the symmetry or non-symmetry
of the relation asserted to hold between the class taken for
subject and the class taken for predicate.

The validity of categorical syllogisms depends upon the
transitivity of the relation of class-inclusion. Using a, /?, y as
illustrative symbols for three different classes, the Barbara
syllogism can be represented by If a is included in /J and /? is
included in y, then a is included in y. That the compound

98 A MODERN ELEMENTARY LOGIC

antecedent implies the consequent is manifest from the fact
that included in is transitive.

The case is different in a syllogism where one premiss is
singular, e.g. if All Marxists are determinists and Professor Hodd
is a Marxist, then Professor Hodd is a determinist. As we have seen,
class-membership is an intransitive relation. The validity of
this syllogism depends upon a modified form of the axiom de
omniy which can be stated as follows: Whatever can be affirmed
or denied of every member of a given class can also be affirmed or
denied of any specified member. This principle has been called the
applicative principle;* it may also be called the principle of
substitution.

Consider the following inferences, where a, b, c are illustra
tive symbols for individuals:

(i) a b and b = c> . . a c.
(ii) a is richer than b and b is richer than c 9 .*. a is richer

than c.
(iii) a precedes b and b precedes c 9 .". a precedes c.

No one will doubt that these inferences are valid, whilst the
following are clearly invalid:

(iv) a loves b and b loves , .". a loves c.
(v) a annoys b and b annoys , /. a annoys c.
(vi) a is father of b and b is father of c, /. a is father of c.

The relations in (i), (ii), (iii) are, in each case, transitive;
the relations in (iv) and (v) are non-transitive, in (vi) in
transitive. In (i) the relation is symmetrical, so that the
relation and its converse are the same; in (ii) and (iii) the
relation is asymmetrical. But the validity of the inference
depends upon the property of transitivity, not upon sym
metry. In each case the conclusion establishes a relation
between the first and third of three terms; the second term
stands in the given relation to one of the terms and in the
converse relation to the other term. Since the relation is
transitive the intermediate term can be eliminated.

Whenever premisses are connected by transitive relations,
chains of deduction are possible. Given that the premisses are

* W. E. Johnson, Logic, Pt. II, p. 10.

INDIVIDUALS, CLASSES, AND RELATIONS 99

true, the intermediate term, or terms, can be eliminated and
the conclusion can be asserted. William James has expressed
the principle in virtue of which such elimination is possible as
the axiom of skipped intermediaries ; he says, symbolically
we might write it as a < b < c < d . . . and say, that any
number of intermediaries may be expunged without obliging
us to alter anything in what remains written .* It is in
accordance with this principle that the conclusion of a Sorites
is obtained, and that the middle term in the categorical syl
logism is eliminated. The property of transitivity, as we have
defined it for dyadic relations, is indeed a special case of the
conditions that make elimination in general possible. |

The traditional Logicians by failing to single out the
property of transitivity as essential to such inferences fell into
absurd difficulties in dealing with arguments such as (ii) and
(iii) above. An argument of this kind was called the a fortiori
argument. Absurd attempts were made to restate the argument
in traditional syllogistic form, i.e. in propositions containing
between them three and only three terms, the terms being
connected by the copula is. These attempts were bound to
fail.*

* Principles of Psychology, Vol. II, p. 646.

f Any student who wishes to consider this topic further should consult G.
Boole, Laws of Thought, Ch. VII; cf. also, J. N, Keynes, F.L., pp. 489-94.
J For a discussion of these attempts, see J. N. Keynes, F.L. 9 pp. 384-8.

CHAPTER VI

Classification and Description

I. TERMINOLOGICAL CONFUSIONS

The topics to be discussed in this chapter can be approached
from various points of view; the emphasis placed upon one
topic as contrasted with another varies in accordance with
the point of view adopted. Extension and intension, connota
tion and denotation, classification and division, definition
and description - all these are more or less interconnected
topics, important not only for the formal logician but also
for the purposes of scientific investigation. The traditional
Logicians approached the discussion of these topics from
the metaphysical standpoint of the classical doctrines of
Aristotle s works on logic, modified by the contributions of
the Schoolmen. We shall not attempt to follow this treatment,
and, with one exception,* we shall not keep to the traditional
terminology. The topics to be discussed in this chapter are
involved in all systematic thinking both at the level of com
mon-sense reflection and of scientific thought.

The discussion of interconnected topics is often confused;
it is difficult to distinguish in thought what is not separated
in fact, whilst the adoption of an unsatisfactory terminology
at the outset hinders further advance. Of these difficulties
extension and intension, connotation and denotation present
an example. These two pairs of words have been used some
times as synonyms, sometimes to indicate different meanings.
We shall distinguish between extension and denotation and
between intension and connotation. We shall further have to
make clear to ourselves what it is that has extension, denota
tion, intension, and connotation respectively. It is only too

* See 5, below. The topics included in this chapter are dealt with more fully,
and with more detailed reference to traditional doctrines, in M.I.L., Ch. II, 3,
4; Ch. IX, 2; Ch. XXII. For a good discussion from a strictly Aristotelian
point of view, see H. W. B.Joseph, Introduction to Logic, Ghs. IV, V, VI.

100

CLASSIFICATION AND DESCRIPTION IOI

easy* to confuse, in this discussion, the symbol and what is
symbolized.

In earlier chapters we have frequently used the word
"term"; it is to be hoped that we have done so without
ambiguity. "Term" is, however, ambiguous though not as a
rule inconveniently so, since the context usually suffices to
show whether we mean by "term 35 a word or an element in a
complex, such as the terms of a proposition, of the syllogism,
or of a relation.-]* In this chapter the word "term" will
always be used to mean a word, or set of words, i.e. what
signifies, not what is signified.

The resemblances between individuals and their differ
ences from one another are recognized in ordinary speech by
our use of class-terms. No one has the slightest difficulty in
using many class-terms; numerous instances of them appear
on every page of this book. A class-term signifies a class-
property, e.g. the word "book" signifies the complex char
acteristic which determines the class of individuals each of
which is a book; the word "steel" signifies a certain constant
conjunction of characteristics.

If I say Give me that book , then "that book" is used in
the hope of referring you to a certain individual object which
you will be able to identify because you understand the words
used. If you do not understand "book" reference fails; if you
do understand "book" but no book is findable reference
again fails. We are clearly using the word "reference" here
with a double usage. This double usage is so familiar that it
requires some effort on our part to notice that it is double.
On the one hand words are used to refer to individuals; on
the other hand words are used to refer to characteristics,
simple or complex; these modes of reference are very different.
We can refer to an individual by using words because, and

* As the author knows to her cost; it is not improbable that the reader also
falls into this insidious confusion.

f Curiously enough the traditional Logicians unwittingly illustrated ^the
ambiguity of "term", by giving as one of the rules of the syllogism that the
middle term must not be ambiguous 5 . Violation of this rule was known as the
fallacy of quaternw terminorum (of four terms). But this was already provided
against by the rule that there must be only three terms. Ambiguity is a char
acteristic of language (i.e. the symbols), not of what language refers to (i.e. the
symbolized).

102 A MODERN ELEMENTARY LOGIC

only because, individuals exemplify characteristics which also
characterize, or could characterize, other individuals. An
individual and its characteristics are distinguishable in
thought but not separable in fact. To keep clear the double
reference of words we need as precise a terminology as we can
devise, for we are going to talk about a distinction which
everyone makes with ease but often without paying attention
to the distinction. Our present concern is with words from the
point of view of their logical functions.

2. CONNOTATION, DENOTATION, AND INTENSION

We have seen that a class is determined by a characteristic,
simple or complex; conversely, any characteristic determines
a class. We mention the characteristic, simple or complex, by
using a word or a combination of words. We shall now use
"term" as a synonym for "a word or combination of words
signifying a characteristic or set of characteristics". A term is
thus an element in the triadic relation signifying; thus a term
(as we are here using the word "term") is a term (in the other
sense) going along with the other two terms required for
signifying, viz. what is signified, and the interpreter. To ask
What does such and such a term mean?* is to ask What does
the term signify? These are synonymous interrogative
sentences,

We noticed (in Chapter V [see p. 77], i) that, for
example, the complex characteristic signified by "man" is
exemplified by Abraham, Aristotle, . . ., where dots are used
to indicate each of the other individual objects that could
correctly have the term "man" applied to it. How are these
objects determined? The answer is clear: because each of
these objects has the characteristic, simple or complex, which
"man" signifies. What "man" signifies is technically called
the connotation of "man". Words or terms have connotation.
The connotation of a term is the characteristic, or set of char
acteristics, which anything must have if the term can be
correctly applied to it. What the term applies to is the mem
bers of the class determined by the characteristic, simple or
complex. This constitutes what is called the denotation of the

CLASSIFICATION AND DESCRIPTION 103

term. It should be noticed that the denotation is not the class
but the collective membership of the class. Hence, the denotation
of a term is the collective membership of the class determined
by the characteristic signified by the term. Thus connotation
determines denotation.

"Man" connotes "rational animal 53 * and denotes men, i.e.
the collective membership of the class determined by being a
rational animal. "Triangle" connotes plane figure bounded by three
straight lines and denotes the collective membership of the
class determined by the connotation of "triangle".

A term signifying a characteristic lacking exemplification
has no denotation, since the class determined by the char
acteristic is empty, and thus has no collective membership;
e.g. "centaur", "house made of gold", "house made of
plastics". If, in the future, a house is made entirely from
plastics, then the term "house made of plastics" will have
denotation. There is nothing in the least mysterious about
this once we have granted that a class may be empty.

The reader may not be willing to agree that "man" con
notes "rational animal"; he may object either: (i) men are
not rational anyhow , or (ii) Nationality isn t a good char
acteristic to select for the purpose of distinguishing men from
other animals*. These objections we might be willing to admit,
but must first point out that anyone who does raise them has
clearly understood what is meant by "connotation", which
is the sole point under discussion. The objections, however,
serve to call our attention to two important points: (i) a
characteristic cannot belong to the connotation of a term if
any member of the term s denotation lacks it; (ii) what char
acteristics are signified by a term (and must therefore
characterize anything denoted by the term) is by no means
always easily settled. It is a sheer mistake to suppose that
most words have fixed and quite determinate meanings, so
that anyone who uses the word correctly knows exactly how he
is using it. To this point we shall need to return. f But, as the
second objection emphasizes, one function which we want

* "Man" can also be said to connote man, i.e. the characteristic or concept
signified by the term "man".
| See 6 5 below.

104 A MODERN ELEMENTARY LOGIC

the words we use to perform is to mark off what we are talk
ing about from anything with which it might easily be con
fused. There may arise a moment in a discussion at which
we find ourselves compelled to ask: Well, what exactly do
you mean by this word? One answer to this question would
be to state the connotation of the word.

At this point a third objection might be raised: (iii) Do
not different people mean different things by the same word? 5
The answer is that often they do but sometimes they do not.
It must be remembered that a term signifies something to
someone; it is the signifying element in the relation and
requires an interpreter. When / use the words "tiger",
"montbretia", "home", "intelligent" (to select examples
almost at random), what I happen to think of as the charac
teristics that must be possessed by anything denoted by one
of these words is very likely to differ to some extent from the
characteristics you think of when you use the word. We say, for
instance, "Home" doesn t mean the same for him as it does
for me, or for you. We want to distinguish the meaning of
a word in this sense from the meaning in the sense of
^connotation 5 . Hence, the convenience of using as a technical
term a word not very often used in common speech, and to
which we (in our activity as logicians) have given a precise
meaning. What the word makes me, or makes you, think of
is distinguished from connotation, and is usually called sub
jective intension. We can define "subjective intension" as "the
characteristics which a given user of the term thinks of as
possessed by the members of the class signified by the term".
The phrase just given in inverted commas tells us the conno
tation of "subjective intension" (unless the author of this
book is in error on this point).

"Intension" has been used as a synonym for "connota
tion" but, as the above objections indicate, this is an unhelp
ful usage. The "intension of a term" connotes characteristics
possessed by the denotation of the term, but we must distin
guish these characteristics into three sets: (i) all the charac
teristics possessed by all the members of the class - whose
collective membership constitutes the term s denotation; (2)
the characteristics which anyone may happen to think of

CLASSIFICATION AND DESCRIPTION 105

when using the term, and which, therefore, vary from time
to time and from one person to another; (3) the character
istics which must be possessed by the denotation of the term.
It is convenient to call (i) the objective intension, or the
comprehension, of the term; (2) the subjective intension; (3)
the connotation. Hence, (i) comprises all that could be
meant, (2) all that you or I may happen to mean, when the
term is used. The connotation includes some only of the
characteristics in fact possessed by the denotation; this selec
tion of a minimum of meaning is, we shall find, useful for
certain purposes, as, for instance, in defining.

3. EXTENSION AND CONNOTATION

We saw that the traditional Logicians failed to distinguish
the relation of an individual to the class of which it is a
member from the relation of a subclass to a class which
includes it. Accordingly they said that, for instance, the class
Europeans extends over or includes in its extension 5 the class
Frenchmen and also that the class Frenchmen includes in its
extension all individual Frenchmen. Now that we have seen
that the membership relation is quite different from the class-
inclusion relation we must also see that we cannot use the
same word both for the term signifying the relation of a class
to its subclasses and for the term signifying the relation of a
class to its members. Accordingly, we shall distinguish in
meaning between "extension 53 and "denotation 55 . The exten
sion of a term signifying a class-property of a given class is
all the subclasses collectively. For example, "Man 55 is a term
signifying a certain class; it denotes each individual man;
the extension of "man 55 is the collective membership of all
subclasses of the superclass man, e.g. it comprises white men,
black men, brown men, yellow men, red men. Another way of
saying the same thing is: the extension of a term signifying
a class-property is all the varieties distinguished as sub
classes. The extension, therefore, are classes, not individuals;
the denotation is the membership of the classes, not the classes.
Hence, when a certain man dies, the extension of "man 55 is
in no way affected. The subclasses need not have members

IO6 A MODERN ELEMENTARY LOGIC

although it must be possible that there should be members.
Thus centaurs is an empty class, but there is no logical incon
sistency in supposing that there may be centaurs^ since there
are none, "centaur" lacks denotation, but its extension com
prises wise centaurs and foolish centaurs.

It has been held by many logicians that extension and
intension vary inversely. This doctrine is worth discussing
because the discussion should reveal the confusions which
have been caused by failure to distinguish clearly between
denotation and extension.* Jevons, for instance, says:

When we pass from one term to another by merely adding
some quality or qualities to the connotation, the denota
tion of the new term is less than the old, and when we
pass from one term to another by merely removing some
quality or qualities from the connotation, the denotation
of the new term is greater than that of the old.f

In his Principles of Science, he states the doctrine as follows:
"When the intent or meaning of a term is increased the
extent is decreased; and vice versa, when the extent is in
creased the intent is decreased. 5 J This he calls an all-important
law . He cites as examples: planet, exterior planet. But, he points
out, there must be c a real change in the intensive meaning,
and an adjective may often be joined to a name without
making a change. Elementary metal is identical with metal:, mortal
man with man.^ These quotations suffice to show that there is
considerable confusion in this doctrine. It is not surprising
to find that logicians who have accepted it have worried
themselves over the question whether the intension of man
can be said to increase when a man dies and decrease when
a human baby is born. Obviously not. The question is so
absurd that we may suppose the whole doctrine is nonsense.
If so, it is not downright nonsense, for it suggests something

* We have defined "denotation" and "extension" in such a way that we could
not attempt to use them as synonyms; that they have frequently been so used
is due to the failure to notice the distinction upon which we have insisted.

t Elementary Lessons in Logic, p. 40. Jevons is careful to point out the decrease
is not in exact proportion to the increase. One wonders why,, hi that case ? the
precise phrase inverse variation should have been used.

t op. cit., Ch. XXX, 13.

ibid.

CLASSIFICATION AND DESCRIPTION IOJ

true but in so confused a manner as to lead to nonsensical
questions.

As the connotation of a term is increased, the extension, is
decreased. It is connotation and extension that vary in this
way, not connotation and denotation, nor intension and exten
sion. Since the extension of "ship" is all the subclasses of
skip, it follows that by enriching the connotation, e.g. adding
steam- and thus obtaining steam-ship, the extension is
decreased, for all subclasses of ship not propelled by steam are
now ruled out. Conversely, by changing the connotation of
"plays" so as to comprise cinema plays the extension is in
creased as the connotation has been decreased, for the term
"plays" will have less richness of connotation if it is to com
prise dramas not witnessed by eyewitnesses than the word
"plays" formerly had.*

These examples suggest that the so-called inverse varia
tion of extension and intension relates to terms arranged in
a classificatory series, i.e. that it relates to classes arranged
in a certain order, namely, in which a subclass is grouped
together with other subclasses under a superclass, which is
in turn a subclass of another superclass, and so on. Such an
arrangement of classes constitutes a classification.

4. CLASSIFICATION AND DIVISION

The process of distinguishing the subclasses of a class is
called logical division; the reverse process is classification. The
process of classifying presupposes the grouping of individuals
in classes; it is useful only when the classes to be arranged
in an orderly manner have important characteristics.
Importance is relative to a purpose. All men have needs which
necessitate the making of classifications, e.g. of people into
enemies and friends, of plants into edible and poisonous -
which itself presupposes a distinction between edible and
non-edible - of materials into inflammable and not-inflam
mable, and so on. The earliest classifications are made to

* It should be noticed that I have not written "plays** and "cinema plays"
but "plays" and cinema plays, i.e. the term "plays" is supposed to include in its
connotation cinema plays. If this is not noticed the reader may think that I
increased the connotation of "plays".

108 A MODERN ELEMENTARY LOGIC

satisfy some practical purpose; in using class-terms it is
hardly possible not sometimes to notice that certain classes
are closely associated with certain other classes. The earliest
stage of a science is the classificatory stage: it is not long
since botany passed beyond this stage and sociology has
hardly done so yet,

A class, then, can be assigned a place in different systems
of classification. The arrangement of vehicles, for instance, in
classes and subclasses would be very different if carried out
for the Ministry of Transport from what it would be if done
to satisfy the needs of the Chancellor of the Exchequer.* An
unscientific person is likely to choose obvious characteristics
for the determination of which subclasses are to be associated,
but obvious characteristics are often not important ones
because they are not connected relevantly. Thus a landlady
arranging a student s books is very likely to be guided by such
characteristics as size, colour, style of binding rather than by
the subject-matter or authors of the books. If the books must
be fitted into shelves of different heights, then size is certainly
a characteristic important for that purpose, but it remains
irrelevant for the purposes of the student who uses the books.
Consider the example on page no.

This arrangement of aeroplanes in subclasses, and sub
classes of subclasses, can be looked at either as a classification
or as a division; if the former, then we begin with the smaller
classes and include them in wider classes; if the latter, we
begin with the widest class and subdivide into smaller classes.
Classification and division are fundamentally the same so far
as the logical principles are concerned. These principles can
be most conveniently stated in terms of the process of
division. Subclasses on the same level are called co-ordinate;
on a level above super-ordinate to the subclass below; on the
level below sub-ordinate.

The basis of the division, that is the characteristic by
reference to which co-ordinate subclasses are differentiated
one from another, is usually known by its Latin name-
fundamentum divisionis. The principles in accordance with

* See MJX., pp. 433-4, where the classification of vehicles, from the point
of view of transport, is worked out.

CLASSIFICATION AND DESCRIPTION IOQ

which a sound division should proceed can be summed up
in the following rules:

1 . There must be only onefundamentum dimsionis at each step.

2. The co-ordinate classes must be collectively exhaustive
of the superclass.

3. The successive steps of the division must proceed by
gradual stages.

From Rule i there follows the corollary that co-ordinate
classes must be mutually exclusive. Violation of this rule
results in the fallacy of cross-division^ i.e. there are over
lapping classes. This corollary together with Rule 2 secures
that every member contained in the classes is contained in
one class only and no member in a superordinate class is
omitted in the next level. Hence, the sum of the subclasses
must equal the whole class divided, or classified.

Rule 3 secures that each stage of the division should be
in accordance with the ongmalfundamentum divisionis. If, for
example, we were to divide university students first into science
and arts students, and were then to subdivide science students
into polite and impolite, and art students into dark,, fair,, and
medium-complexioned, the division could serve no useful purpose.

The fallacy of cross-division is of common occurrence. If we
divide the languages of mankind into Aryan, Semitic, Slavonic,
Hamitic, and Ancient Egyptian, we commit this fallacy, since
Ancient Egyptian falls into the Hamitic group, and Slavonic into
the Aryan. This division is also not exhaustive.

Any given class can be subdivided into two mutually ex
clusive and collectively exhaustive subclasses on the basis of
a given characteristic which is possessed by every member of
the one class and is not possessed by any member of the
other class. Thus we can divide civilians into those doing
work of national importance and those not doing work of
national importance. It would be a contradiction to assume
that any member of the one subclass could also be a member
of the other subclass, whilst every civilian must fall into one
or other of the two classes - granted that the criterion - work
of national importance - is sufficiently well-defined. Such
a division is called division by dichotomy (i.e. cutting in

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A MODERN ELEMENTARY LOGIC

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CLASSIFICATION AND DESCRIPTION III

two]. The following is an example of a dichotomous division.
This division formally secures that the subclasses are
mutually exclusive and collectively exhaustive, but this
formal simplicity is attained only by a multiplication of
classes negatively characterized, and it obscures the simpler
relationships that appear only when classes are arranged on
the basis of positive characteristics. In the natural sciences

Animals

1

Vertebrates

i

Invertebrates

L

Mammals Non-mammals

1

Arthropoda

Non-arthropoda

Birds Non-birds
f

Moll

uses Non-molluscs

Reptiles

Non-reptiles

dichotomous division would be of little use. The above sub
division of vertebrates into mammals and non-mammals, and
so on, which results in placing birds on one level and reptiles
on another, obscures the relation that holds between mam
mals, birds, reptiles, amphibians, and fishes as together
exhausting the class vertebrates.

Traditionally, division has been regarded as the division
of a genus into its species*, the genus from which the division
starts is called the summum genus , the species with which it
ends are called infimae species, intermediate species subaltern
genera^ an intermediate genus is called the proximum genus of
its constituent species. These names are not of importance;
what is important is to recognize that the distinction between
genus and species is relative, and depends for its significance
upon the given table of division.*

* In biological classification genera and species are used in a sense fixed by
the hierarchy of classes: subclasses of species are called varieties:, superclasses of
genera arc families; then come the order and class. It should be observed that in
the dichotomous division of animals, given above, the negative class must be
taken as strictly a subclass of the proximum genus at each stage, thus reptiles are
non-birds, non-mammals, and vertebrates.

112 A MODERN ELEMENTARY LOGIC

We have insisted that a division or a classification is
relative to a purpose - classes are subdivided, or are grouped
together in a wider class in order to bring out connexions
between classes that are fruitful for some purpose. In the
sciences the classes we select for orderly arrangement are
natural classes, i.e. classes whose members are characterized
by connected properties.

5. THE PREDICABLES

If we know that an animal is a mammal we know a good
deal about it, e.g. that it has a backbone, is warm-blooded,
and has some kind of hair, and that the female has milk-
producing glands with which to suckle her offspring. Some
mammals, the marsupials, bring forth their young in a very-
undeveloped stage and carry them in pouches; another group
of mammals lay eggs but still suckle their young. This
example serves to suggest to us to classify the characteristics
possessed by the members of a class into three groups: (i)
those which every member possesses and only the members
of the given class possess; (2) those which every member
possesses but which are also possessed by members of other
classes; (3) those which some only of the members possess.
Let us take as an example the class man. Every member of
the class man has the property ofanimality, also the properties
of being mammalian; every member of the class man has also
properties peculiar to man, e.g. a larger brain relatively to
size of body than any other animal, and along with it
rationality. Animality and being mammalian are generic pro
perties of man, rationality is a specific or differentiating
property. "Generic" is here used in the logical, not the
biological sense; if we regard animal as the genus of man
(neglecting the genus mammals] then we can say that the
species (in the logical sense of "species") man is differentiated
from co-ordinate species of animal by the property of being
rational. This is to follow Aristotle s classification. We shall all
agree that along with the property of being rational there go
other properties peculiar to man within the genus animal, e.g.
capable of seeing a joke, or - to take one of Aristotle s favourite

CLASSIFICATION AND DESCRIPTION

examples - capable of learning grammar. We feel that, even if
a parrot and a budgerigar can speak (i.e. utter verbal
sounds), only a man could learn grammar. Such a property,
common to every member of a species (i.e. a subclass
of a genus) and connected with the property that differ
entiates this species from co-ordinate species, is called a
proprium*

There are also properties which every member of a sub
class of man has but which are not possessed by members of
other subclasses, e.g. white-skinned, black-skinned, curly-
haired, straight-haired, dolichocephalic, brachycephalic, and
so on. Such properties are called accidents.

These names -genus, differentiating property or differ
entia, proprium and accidents - are known as the predic-
ables , for Aristotle first distinguished them when he
attempted to answer the question: What different sorts of
predications can be made about a species? His reply was that
we are able to predicate of the species man (for instance) the
genus - animal, the differentia - rational, a proprium - able to
learn grammar, an accidens- white-skinned, f The genus and
differentia taken together constitute the definition, which is
per genus et differentiam.%

The words genus, species, differentiate, property, accidental
characteristic - all come down to us through Aristotle s
treatment of this topic. Professor R. M. Eaton has said,
Aristotle s genius for clear analysis, which enabled him to
give to logic a terminology and form that persisted for two
thousand years, is nowhere better exemplified than in his
theory of the predicables. It is rare to find a modern

* The Latin word proprium (translation of Aristotle s word tStov) is retained
because in this context it is used with a narrower sense than "property" which
is often used as a synonym for "characteristic". The plural of proprium ispropria.

-f It should be noticed that the subject of predication was the species (e.g. man,
or triangle) not the individual (e.g. Socrates or this scalene triangle). Porphyry
(A.D. 233-304) hopelessly muddled Aristotle s doctrine by putting the species in
place of the definition, and taking the subject to be the individual, e.g. Socrates.
He, and later logicians, wasted their time in making further distinctions, utterly
trivial and needlessly elaborate.

J This means by assigning the genus and the distinguishing characteristic .

General Logic, p. 273. Professor Eaton gives by far the best account of
Aristotle s theory of the predicables, from the point of view of the elementary
student of logic, who wishes to know in more detail what Aristotle s theory
actually was.

114 A MODERN ELEMENTARY LOGIC

logician according such praise to Aristotle s work in logic,
but the praise is - in the opinion of the present author - well
deserved. At the same time we must insist, as Professor Eaton
also admits, that Aristotle s theory of the predicables is rooted
in his metaphysic. That metaphysic we reject. It may indeed
be urged that the influence of Aristotle s metaphysic upon
his logic was very unfortunate, and the traditional Logicians
adherence to it and their retention of every mistake Aristotle
made has been disastrous in hindering the development of
logical doctrines. Aristotle s theory is now mainly of his
torical interest to those who are not studying metaphysics.
It would, however, be worth while to follow it in some
detail - did space permit - because it provides a good ex
ample of a rigorous attempt to analyse the sort of state
ments we can make, and to pay serious attention to the
important distinction between essential and non-essential
characteristics.

We may sum up Aristotle s list of predicables by exhibiting
them in the form of a dichotomous division, the basis of
which is the convertibility or inconvertibility of the predicate
with the subject. A predicate is convertible with the subject
if it is common and peculiar to the subject. This statement
does not make sense unless we remember that, in this con
text, subject must be taken to mean species:

Predicate

I

Convertible with subject Not convertible with subject

i I 1 I

Definition Not the definition, An element in Not an element in
i*e. proprium the definition the definition,
[ i.e. an accident

I 1

Genus Not a genus,
i.e. a differentia

The italicized words are the predicables. The definition is
not a fifth predicable distinct from the others but is the

CLASSIFICATION AND DESCRIPTION 115

predication of genus and differentia together. We add an
example taken from geometry:

Circle
(species]

plane figure
(genus)

bounded by a line whose points are
equidistant from a given point
(differentia)

having equal chords equidistant
from the centre
(propnum)

inscribed in a triangle
(accident)

Artistotle held that each species had a fixed and determinate
essence; this was set forth in the definition. The proprium,
although not part of the essence, was nevertheless regarded
as essential to the species; it is derivable from the essence, i.e.
follows from the definition. Thus the distinction between
definition and proprium was taken to be absolute. This view
we must completely reject. The distinction is absolute only
relatively to a given system of concepts. It is most easily seen
in the case of geometry. Euclid regarded geometrical figures
as given in intuition by construction of the figures in space.
This view is now r abandoned, hence we cannot hold that there
is one and only one definition of, e.g. a "circle", which will
set forth its essence. If the definition of "circle" given above
be accepted, then it is a proprium of a circle that with a
given perimeter its area is maximum; if, however, we define
a circle as the plane figure which has a maximum area with
a given circumference, then it follows that all its points are
equidistant from a given point, and thus this is a proprium.
Which we choose as definition is determined by non-logical
considerations; once chosen, then, whatever can be deduced
from the definition is a proprium. It is easy to see that the
propria are the theorems implied by the axioms and defini
tions. They are essential in the clear sense that to accept the
definitions and reject the propria would be self-contradictory.
The distinction between propria and definition, and
between propria and accidents, is much less easy to draw in
the case of natural species, e.g. man, cow, snake. It must

Il6 A MODERN ELEMENTARY LOGIC

suffice to say that a characteristic or property is essential if,
lacking it, the thing in question could no longer be regarded
as belonging to the species. Accidental predicates are predi
cated not of an individual but of an individual as a member
of a species. The characteristics possessed in common by every
member of a species are, in the case of natural classes,
numerous and connected. Hence we seek to discover certain
characteristics which are significant of others, and can thus
be used as the basis of fruitful inferences. To pursue this
topic further takes us beyond anything recognizable as the
theory of the predicables.

6. DEFINITION

We have seen that the traditional rule for definition is that
it should be per genus et differentiam. This is unduly narrow.
What, we must ask, is the purpose of definition? When do
we want a definition, and, if successful, what does a definition
achieve? The student, for instance, who is beginning the
study of logic may want to know what logic is. Is this a
request for a definition? If so, how is it to be met? The
answer to this latter question will depend upon the needs of
the questioner. Is he entirely ignorant of the meaning of the
word "logic", i.e. has he just met it for the first time? Or
does he know that logic is somehow or other concerned with
reasoning and he wants to know further how logic is to be
distinguished from psychology? If the former, then the
answer, Logic is concerned with the principles of reasoning ,
should meet his case, provided he understands how to use
the words in the defining phrase. If the latter is his case, then
the answer must indicate characteristics differentiating a
logical treatment of reasoning from a psychological treatment.
The most satisfactory answer will probably take the form of
a set of statements with illustrative examples. It is seldom
enlightening to be given a definition in a short, crisp state
ment. Occasionally such an answer will suffice. Suppose A
asks B: What does "whatnot" mean? B replies, C A "what
not" is an article of furniture with open shelves, rather wide,
designed for the purpose of putting objects of various kinds

CLASSIFICATION AND DESCRIPTION IIJ

on it. 5 Then A s question is satisfactorily answered provided
that (i) A knows the words B uses in the defining phrase, (ii)
the defining phrase does indeed present the characteristics
that things called whatnots have. Perhaps (iii) should be
added: A wanted an explanation of "whatnot" and not of
"what not", i.e. something added at the end of a list to mean
"and all sorts of things". The context alone can decide
whether A did mean the sort of thing which B understood
him to mean. If he did not, then communication has failed,

Usually our requests for definitions are not so easily dealt
with. We seek definitions as a means of thinking more clearly
about something; we want to think more precisely, to know
exactly what it is that we are saying. For instance, c What is
the policy of appeasement, as understood by Neville
Chamberlain and his supporters between, say, 1936 and
I 939 ? * Clearly something more than a dictionary definition
of "appeasement" is needed to answer this question. But, we
may feel, "appeasement" as used in the question must have
some reference to the dictionary definition of "appeasement".
Or again, c Are you a Communist? , to which the reply may
be that depends upon what you mean by "Communism". 3
The student has probably taken part in conversations such
as the above. At this point he should ask himself what sort of
answer he would find satisfactory. There is not one and only
one way of explaining how words are used; any answer that
enables us to use the word - for a definition of which a
request is made -is so far a satisfactory definition. The
answer generally takes the form of a sentence, i.e. we explain
a word by using other words. Will this leave us in the un
comfortable position of one who endlessly chases his own
tail?

A proper answer to the questions and difficulties sug
gested by the preceding paragraph would require a book,
not a brief section in a chapter.* All that can be done here is

* I should very much like to write such a book, but neither space (owing to
war-time scarcity of paper) nor time is available. The student who is interested
in these topics would find I. A. Richards Interpretation in Teaching both interest
ing and enlightening. A logician may be pardoned for thinking that Professor
Richards is unduly narrow in his apprehension of specifically logical problems
and perhaps unnecessarily dogmatic. But his books are worth careful study.

Il8 A MODERN ELEMENTARY LOGIC

to suggest a very few of the pertinent questions we need to
ask and to indicate, in the case of a few only of these ques
tions, the lines along which answers to them should be
sought.

We use words to talk about things; we use \vords to ask
for definitions and most commonly we use words in giving
the definition. But there must be an attachment of the words
used to life, i.e. to the rest of reality. We cannot here attempt
to give any account of the ways in which a child begins to
learn the language he hears spoken by those who tend him;
we take the miracle for granted. Verbal expressions must, at
certain points, link up with other things than words unless
definition is to remain merely a set of verbal manipulations.
Such linking up can be given by pointing, i.e. by what has
been called ostensive definition. For example, What does
"wink" mean? The most satisfactory answer to this is given
by "Doing this - and, the speaker winks. The questioner will
then surely know what "wink" means; he may very well not
know if he cannot observe someone winking but has to rely
only on his dictionary.* Again, someone asks, What is an
epic poem? , and is answered, The Iliad, The Odyssey, The
JSneid, Paradise Lost, and anything like these. The difficulty
is to know like in what respects 5 . Shall we include The
Dynasts ? The answer does not take us very far, but it is a
beginning. It is an ending also in the case of words such as
"red", "sound of A$ on a violin". We must ultimately
explain the meaning of many words by giving samples, as in
the case of "epic poem" above.]*

The treatment of definition by most logicians has been too
much divorced from the consideration of how we come to

* At this point I consulted the Shorter Oxford English Dictionary, which gives
"A glance, or significant movement of the eye (often accompanied by a nod)
expressing command, assent, invitation or the like", adding that this (meaning)
is obsolete except in proverbs; it gives under the verb "to wink" - "to close one
eye momentarily in a flippant or frivolous manner, especially to convey
intimate information or to express good-humoured interest". The reader who
knows that the Latin "connivere" means "to wink" will find the derivation of
"connive" interesting.

f The sampling method is quite indispensable, but to learn by means of it is
not as easy as it may sound, as anyone who has tried to learn, or to teach, Latin
by the direct method is likely to admit. Here we can only remind the reader that
we can sort and distinguish without knowing how we have sorted and distinguished.

CLASSIFICATION AND DESCRIPTION IIQ

use words, how we learn to understand. Attention has been
concentrated upon what is, from the scientific point of view,
of great importance, namely, what conditions must a satis
factory definition fulfil? In answering this question we need
to remember that satisfactory 5 , like importance , depends
upon the point of view. Let us first consider the traditional
rules, which presuppose that what is needed is an explana
tion in w r ords of how a given word is to be understood. The
word to be defined is traditionally called the definiendum^ the
defining phrase is called the definiens.

A. Rules concerned with the nature of definition.

1. The definiens must be equivalent to the definiendum.
From this rule two corollaries follow: (i-i) * The definiens
must not be wider than the definiendum. (1-2) The definiens
must not be narrower than the definiendum.

B. Rules concerned with the purpose of definition.

2. The definiens should not include any expression that
occurs in the definiendum or that could be defined only in
terms of it.

3. The definiens should not be expressed in obscure or
figurative language.

4. The definiens should not be negative in significance
unless the definiendum is primarily negative in significance.

Granted that the purpose of giving a definition is to make
clear the limits within which a word, or phrase, can be
rightly used, these rules seem obvious enough to require but
little comment. The point to be stressed is that the definition
and the defining phrase must be equivalent, from which
equivalence it follows that the one can be substituted for the
other without alteration in meaning. A definition per genus et
diferentiam fulfils the conditions laid down by these rules,
provided that the expression used for the differentiating
property be not obscure. What is obscure is relative to the
questioner s knowledge; the futility of using in the defining
phrase words more obscure to the questioner than the word
to be defined is too obvious to need further comment, A

* The corollaries are numbered in this way to emphasize their connexion
with rule i ; the decimal point is used to distinguish the two corollaries.

I2O A MODERN ELEMENTARY LOGIC

circular definition also defeats the purpose of defining, e.g.
c "Physical force" means "the power which produces
motion" 5 is circular if "force 53 and "power" are taken to be
synonyms and if, further, the request was for a definition of
"force" rather than of physical force. "Justice is giving to every
man his due" is circular if "what is due to a man" is defined
as "what it is just he should have".

The definition of "orphan" as "one with no father or
mother" is not faulty because it is negative but because it is
unclear whether an orphan thus defined is a Melchizedek.
"One deprived of father and mother" is affirmative in state
ment and negative in significance, which fits the concept
orphan. The student will easily think of words the primary
significance of which is to deny the possession of an attribute,
e.g. "alien", "bachelor".

A question that has been much discussed is whether defini
tion is of words or of things. The question is badly put; words
are used to refer to something; we define the word, but there
is a word to define only because we want to talk about what
the word stands for; we talk with words about something.

A distinction has been made between verbal and real defini
tion. A verbal definition gives in the definiens a word or set
of words that can be used to symbolize exactly what the
definiendum symbolizes. In real definitions the definiens
represents an analysis of the definiendum. A definition is
always an equation: one word, or set of words, is equivalent
to another word or set of words. The definiens may be
analytic, i.e. it may show an analysis of the definiendum.
Analysis in this sense must be contrasted with physical
analysis. For example, in chemical analysis there is both the
unanalysed whole (e.g. water] and the set of constituents into
which it is analysed. In logical analysis there is not first one
thing and then a set of things, but there are two expressions
which mean the same. For example, given the definition:
"danger" means "exposure to harm", there is not a complex
property symbolized by "danger" as well as the set of
properties symbolized by "exposure to harm"; on the
contrary, there is one set of properties which both "danger"
and "exposure to harm" symbolize.

CLASSIFICATION AND DESCRIPTION 121

7. DESCRIPTIONS

Logicians have often defined "definition 5 as "the explicit
statement of the connotation of a word". The objection to
this definition is that it suggests that the connotation of a
word is fixed, and all that needs to be done is explicitly to
state this. In the case of abstracta this is so, e.g. the terms
used in geometry have definitely delimited meanings. Thus
"chiliagon" means "a thousand-sided regular polygon" on
every occasion of its use. The words we have most trouble in
defining are those whose meaning shifts in different contexts;
such words can only be defined relatively to a given sort of
usage, and with reference to explanatory examples of the
word used in some sentence.

We are naturally tempted to ask whether every word can
be defined. If "to define" means "to explain how the word
is used", the answer is that every word can be defined but
few can be defined shortly. If "to define" means "to state
explicitly the connotation", then the answer is that some
words cannot be defined, either because they have no conno
tation or because the connotation cannot be made clear (to
one who does not already know it) by means of other words
alone. To consider the second case first. "Red" connotes red
ness, but "redness" can only be understood by knowing that
redness is the quality of those objects which "red" denotes,
and this can be known only by seeing red objects. Hence, a
man blind from birth cannot know what "red" means.

The other case is that of words which have no connotation.
Whether there are any non-connotative words is a matter of
dispute among logicians. J. S. Mill held that proper names
have no connotation. Let us begin by very briefly considering
how we use a proper name, for example, "Franklin"., as
contrasted with a phrase such as "the man in the moon" or
"the man whom you spoke to just now".

Most of those who hear the name "Franklin" in 1942
would think of the present President of the United States or
of the younger son of the present Duchess of Kent or of
Benjamin Franklin, the American scientist and statesman;
some may think of some other personal acquaintance. The

122 A MODERN ELEMENTARY LOGIC

name "Franklin" does not give us any information with
regard to the objects so named; there is no reason to suppose
that the four objects (assumed to have been referred to by
"Franklin") have in common anything except (i) being
called by the name, (ii) having properties of sufficient interest
to someone to have been given that name. But (ii) is shared
by those called John , Cordelia , Smoodger , and does not,
therefore, suffice to mark out those called "Franklin" from the
rest. Thus "Franklin" lacks connotation in so far as the name
does not signify any characteristics common and peculiar to
the individuals, calied by the name, for "Franklin" may be
the name of a poodle or a motor-car. If this had been all that
Mill meant by saying that proper names are non-connotative
he would have been right. Possibly this is all he meant, but
he certainly spoke as though he denied any kind of meaning
to a proper name.*

A proper name has a meaning because it is given to an
individual to distinguish him (or it) from other individuals.
Its significance is thus known only to those who are personally
introduced to the individual and to those who know a pro
position of the form, "Franklin" is the name of the present
President of the United States , c "Cordelia" is the name of
the girl over there in the red dress - where it is assumed that
c the girl over there in the red dress uniquely identifies an
individual. In practice we give proper names only to indi
viduals in whom we have a special interest which makes us
desire to refer to them frequently * We ask for c my hot-water
bottle 5 , not for Aristarchus unless we have notified that the
hot-water bottle in question has been thus named.

"The man in the moon", "the present President of the
United States", "the author of Adam Bede", "the author of
Ecclesiastes" resemble proper names in one respect, namely,
that each refers to only one individual. They are known as
definite descriptions, for, unlike proper names, these phrases
are descriptive and can be understood by anyone who knows
English. Some logicians have held that definite descriptions
are names but very complicated names. This view is certainly

* See J. S. MUl, A System of Logic, Bk. I, Gh. II, and see further M.I.L.,
Ch.III,3,4.

CLASSIFICATION AND DESCRIPTION 123

mistaken. If "the author of Adam Bede" were simply another
name for the person called "George Eliot", we could refer
to this person as "the person called the author of Adam
Bedi ", just as we can refer to this person as "the person
called George Eliot 5 " Now Mary Ann Evans was called
"George Eliot 3 , and the fact that she so called herself con
stitutes a sufficient title for her to be spoken of as George
Eliot 5 . But however much she called herself or any one else
called her "the author of Adam Bede" she would not have in
fact been the author unless she had actually written Adam
Bede, and that she did write it is what w r e mean by saying
she is c the author of Adam Bede\ Likewise, the President of
the United States is not made President by being called so
but only in virtue of his actually holding the office.

"The man in the moon" raises another difficulty against
the view that definite descriptions are names, for there is no
man in the moon, and it seems absurd to say that a non
existent individual has a name. So, if we use the description.,
"the present King of France 53 or "the crock of gold in the
well 55 -when there is no such crock of gold -we are using
significant phrases, but there is, in each case, nothing
answering to the description. Philosophers have been puzzled
to explain how we can use descriptions that describe nothing,
and if these descriptions were in fact names the puzzle would
be insoluble.

We owe to Bertrand Russell an account of how descrip
tions are used that shows us exactly how it is that we can
significantly make use of descriptions that describe nothing.
The account is given in terms of the theory of classes. A
definite description is analysable into an identification of a
class together with the implication that the class in question
has only one member. Thus "the author of Adam Bede" iden
tifies the class determined by the characteristic having written
* Adam Bedi and implies that the class has only one member.
Since we have reasons for asserting that Adam Bede was
written by one author the description describes him (or in
this case her). Since we have reasons for believing that
Ecclesiastes was written by two authors, the description
describes no one. "The man in the moon" and "the present

124 A MODERN ELEMENTARY LOGIC

King of France" are also descriptions describing nothing.
Since the significance of these descriptions is entirely inde
pendent of there being any exemplification of the charac
teristic determining the class in each case, their significance
is unaffected by the discovery that the class in question is
empty. This theory also shows how two, or more, different
descriptions can be different even though they describe
nothing, or, what is the same thing in different words, even
though the classes corresponding to the descriptions have no
members; it is not upon the denotation that their significance
depends.

On this theory we can analyse what it is exactly that a
proposition such as The author of Adam Bede* is George Eliot
asserts; it is equivalent to the conjoint assertion of three
propositions:

(i) At least one person wrote Adam Bede.
(ii) At most one person wrote Adam Bede.
(iii) There is nobody who both wrote Adam Bede and is not
identical with George Eliot.

If any one of the three constituent propositions is false, then
the original proposition is false.

The proposition The author of the Iliad exists can be
similarly analysed into the conjoint assertion of:

(1) At least one person wrote the Iliad.

(2) At most one person wrote the Iliad.

If either of these constituent propositions is false, then the
original proposition is false; hence, if more than one person
wrote the Iliad, or if no such book was ever written, then The
author of the Iliad? exists is false. Since (i) and (2) are of the
same form as (i) and (ii) above, it is clear that The author of
Adam Bede* is George Eliot asserts that the author of Adam Bede
exists. Thus any statement attributing a property to the author
of Adam Bede* is false unless there really was such a person.
The analysis of The present King of France is bald asserts
conjointly:

(i) At least one person reigns over France now.
(ii) At most one person reigns over France now.

CLASSIFICATION AND DESCRIPTION 125

(iii) There is nobody who both reigns over France now
and is not bald.

Since, of these three constituent propositions (i) is false, it
follows that the original proposition is false.

The definite descriptions we have so far been concerned
with are singular descriptions; they are often expressed by the
form "the so-and-so". We must, however, be on our guard
against supposing that grammatical similarity is a safe guide
to similarity of logical form. "The lion is carnivorous" does
not express a singular proposition; it expresses a proposition
equivalent to All lions are carnivorous, since this latter proposi
tion implies and is implied by The lion is carnivorous. Accord
ingly, the proposition is a universal affirmative proposition.

Definite plural descriptions are used in stating such proposi
tions as The members of the House of Commons are elected. The
members of the Committee have been notified of the complaint. In
these propositions a statement is made about every member
of a certain class specified by the description.

Indefinite descriptions are used in stating such propositions as
A member of the King s household was killed. This is equivalent
to There exists at least one member of the King s household
and he was killed 3 . Such propositions are often expressed by
the verbal form A so-and-so is such-and-such , but we must
again notice that the same verbal form may be used to
express a proposition different in kind, e.g. C A dog likes
bones means Every dog likes bones .

CHAPTER VII

Variables, Propositional Forms, and
Material Implication

I. VARIABLE SYMBOLS

Frequently in the preceding chapters we have used illustra
tive symbols.* The use of such symbols is not logically
essential but it is convenient and is probably psychologically
indispensable to enable us to concentrate our attention upon
the form of propositions. Illustrative symbols are by no
means confined to logic and mathematics. They are em
ployed in ordinary speech when we use pronouns. For
instance, suppose you are listening to the news on the wireless
and are in a room with several people, some of whom are
not anxious to hear what is said. There is a murmured buzz
of low-toned conversation. You say: I can t hear; someone
is saying something; it may be important but can t it wait
till after the news? Here "I 55 stands for the speaker and is
definitely specified for anyone who knows who is speaking;
"someone is saying something" does not specify who says
what; these pronouns are illustrative symbols standing for one
person in the class of persons in the room but an unspecified
one. Suppose now you say, c jack, it is you who are talking ,
then "Jack" names an individual, i.e. the illustrative symbol
"someone" has been replaced by a specified individual s
name "Jack". In contrast with the indefinite pronoun "some
one" we call "Jack" a constant, for it signifies the same indi
vidual throughout every occasion of its use (provided we
make the assumption that there is only one person named
"Jack" in the set to which reference is being made). Personal
pronouns may also be used indefinitely when the person to
whom reference is made is not specified. In this book "I"

* At this point the student may find it convenient to re-read Ch. I, 4, 5.

126

VARIABLES, PROPOSITIONAL FORMS, ETC. 127

and "you" have been thus used to stand for any one person
(the speaker, questioner, etc.) and for any one other person
(the hearer, answerer, etc/,, respectively.* "He" is often so
used to stand for some unspecified murderer (at least by
detectives in fiction), also in legal documents and in various
expositions, and in some places in this book, where "he"
could be interpreted in some contexts as standing for a female.
We are so used to these conventions that we find no difficulty
in understanding what is meant. ("We" in the preceding
sentence is used in an illustrative w r ay even though one person
denoted by "we" is /, Susan Stebbing.) There is no more diffi
culty in understanding variable symbols than in under
standing how pronouns are used. Statements in which pro
nouns are used will be ambiguous unless the context specifies
the range of their application; this is usually the case, but
sometimes difficulties arise through failure in specification.
Consider the following remarks:

(1) Someone is saying something, 5

(2) c He is saying something.

(3) c J a ck is saying something.

(4) Jack is saying that he does not want to hear him. 3

(5) Jack is saying that he does not want to hear Gram
Swing. 3

As we proceed from (i) to (5) specification is becoming more
and more complete, i.e. at each step one more element pre
viously referred to without specification is now specified. In
accordance with the ordinary conventions of the English
language (5) may be taken as completely specified since "he"
unambiguously stands for "Jack".j It is a question of inter
pretation whether we say that (i) states a proposition; if it
be regarded as either true or false, it is a proposition. Some
logicians may consider that (i) is a propositional form and
that to obtain a proposition the indefinite someone , some
thing must be replaced by a definitely specified element. On
this view (2) and (3) must also be regarded as propositional
forms; it is, then, perhaps hard to draw a line between (3)

* See Gh. II, 2, where warning was given that this procedure would be
followed.

f Gf. the use of se and ille in Latin.

128 A MODERN ELEMENTARY LOGIC

and (4), since "him" is not specified, and "he" is specified as
referring to Jack only on the convention that whoever made
statement (4) would have used a proper name instead of
"he" if there had been risk of understanding that Jack was
speaking of, say, Tom as not wanting to hear. Then, on reflec
tion, we may begin to be doubtful about (5). But I^(the
author of this book) regard (5) as completely specified within
the context supplied by the illustration at the beginning of this
section, viz. a set of people some of whom are listening to the
wireless. There are thus good reasons for holding, in the given
context, that (i) to (5) are all propositions since (it is assumed)
each one of them might be stated by a definite person on a
definite occasion, and the statement so made will be true or
it will be false, i.e. it is a proposition. Hesitation on this
point -namely whether any or all of (i) to (4) are proposi
tions or only schemas (so to speak) for propositions - may
help us to see clearly the difference between a proposition
and a propositional form (or schema for a proposition).
Consider the following expressions:

(1) Jack loves Jill. (6) Someone hates Dick.

(2) Jack loves Ben. (7) Someone hates someone.

(3) Tom loves Ben. (8) A hates someone.

(4) Tom hates Ben. (9) A hates B.

(5) Tom hates Dick. (10) x hates y.

Clearly (i) to (5) are examples of propositions; (6) is a form
of expression which might certainly be used to put forward
a proposition by, say, someone who is trying to account for
the happening of frequent disasters to Dick. (7) is an expres
sion that would hardly be used except in some such context
as the above. (8) and (9) are not propositions since it does
not make sense to assert that a letter of the alphabet hates,
and we had not adopted a convention that "A" was shorthand
for Ann or "5" for Ben., or for any other proper name. (10) is
a propositional form; if a constant be substituted for x and
another constant for jy, then the result would be a proposition
- true or false according to the facts of the case stated in the
proposition. In (10) we have an empty propositional form in
which one constant 3 hates, is given along with two variables,

VARIABLES, PROPOSITIONAL FORMS, ETC. I2g

A variable - or more strictly a variable symbol - Is a
symbol which may be replaced by any one of a set of various
constants, each of the constant symbols standing for a differ
ent individual. Thus, if we suppose ourselves to be limited to
five persons whose names appear in propositions (i) to (5),
and if, further, we suppose that these five propositions state
truly the relations holding between them, then, if in (10) \ve
replace x by one of these names andj by another until we
have tried all the possibilities, the result would be in some
instances a true, and in others, a false proposition. The con
stants thus replacing the variables are called values of the
variables.

We may go a step further than in (10); we can let "hate"
vary, and write e xRy t . This is a pure prepositional form; it
is wholly abstracted from particular persons, emotions, and
so on; nothing is specified, but something is represented, namely,
the form common to all propositions that state that two
terms are related. xRy is a dyadic prepositional form. Tom
is taller than Ben, Dante lived before Mazzini, David worshipped
God are instances of this form xRy, and the symbolization
xRy may be regarded as symbolizing all such propositions.

A prepositional form is a schema: symbols are used to
show empty places waiting, as it were, to be filled; when all
the places are filled the result is a proposition. Logically
there are no restrictions with respect to the symbols we can
use provided that the symbols do the work required of them.
But it is convenient to make use of symbols that will be as
easily grasped and remembered as possible. For this reason
logicians use x,y, z (and other letters taken from the end of
the English alphabet when more than three are required) to
show the empty places for values of the variables. R is often
used to stand for an unspecified relation; sometimes 0, or
other Greek capital letters are used, and the relational form
is written <P(#, j>), <&(x, y, z] according to the number of
variables required, i.e. the number of terms needed to make
sense of the relation. may be regarded as an illustrative
symbol.*

* # can itself be taken as a variable, e.g. can be used to stand for any serial
relation; it will then require two variables, so that we should write # (x,y).

I3O A MODERN ELEMENTARY LOGIC

2. PROPOSITIONAL FUNCTIONS AND GENERAL
PROPOSITIONS

Prepositional forms are called by Bertrand Russell preposi
tional functions, for they are In some respects analogous to
mathematical functions. Whether we speak of functions 5 or
forms is not important. One use of propositional forms is to
enable us to give an analysis of propositions involving the
notions of all of a class and some of a doss; in this connexion
it is more convenient to speak of functions than of forms, but it
must be emphasized that a propositional function is a pro-
positional form- a schema which requires specification in
order that a proposition may be obtained.

The propositions, Ann is sad, Ben is sad, Tom is glad, may
all be regarded as having the same form - a characteristic
is predicated of an individual; other examples are. This is
red,* That is square. If we replace the subject-term, in any of
these propositions, by x, then we have a propositional form,
e.g. *x is sad 3 , containing one variable. The values substitut-
able for x are called arguments of the given propositional
function. | The arguments are determinate entities; in the
cases we shall be considering they are individuals, and the
symbols used to name these individuals are called constants.
Sometimes we use a, b, c, or other letters taken from the
beginning of the alphabet, as illustrative symbols for definite
specifiable individuals which are not, in fact, specified. J
Thus 0a, 0(a, b}, each represent an unspecified but constant
value of their respective functions.

* It is possible to argue that the five propositions given above are not subject-
predicate propositions, and that, for example, This is red is a relational proposi
tion, since (it may be contended) red is a term in an irreducible polyadic
relation. I, myself, take this view of red; but the understanding of such a view
presupposes that we understand what is meant by saying that red is a non
relational quality and This is red a simple subject-predicate proposition. As such
we shall regard it here.

^ This is a technical use of the word "argument", which has nothing to do
with "argument" meaning a connected reasoning.

J Such symbols as a, b, c used in this way are analogous to parameters in
mathematics. For example, in ax + by c = o (which symbolizes any linear
correlation), a, b, c stand for variables denoting any numbers, just as x t y do;
but they are to be distinguished from x y y, because a, b, c retain unchanged
values throughout the same set of operations with x, y. Since, however, a, b, c
were not given determinate values, the result is established for any numbers, so
that a, b, c are properly variables (see, on this point, A. N. Whitehead, Intro
duction to Mathematics^ pp. 68-9, 116-17).

VARIABLES, PROPOSITIONAL FORMS, ETC. 13!

There is a further point about notation with regard to
which we may as well be clear, for the sake of accuracy.
Sometimes we want to indicate the number of variables
required by a given function: thus we distinguish 0x* from
^(*> _> )? si&ze the former requires one, the latter two,
variables. If we were to write <Px, we should be indicating
a variable value of <&x, i.e. of the function represented by 0.
We shall not, in this book, need to make use of 0x, but we
must notice the distinction. We might say that <&x represents
something that has the property <&, whereas 0x represents the
property that something has. 0a indicates a constant but un
specified value of the function 0x. We use <P<2, as we used
Ann 5- sad, in the preceding paragraph, merely illustratively;
we were not talking about an actual person named Ann
whom we knew to be sad; we used c Ann as an example.
Thus in C 0#% e < stands for a definite but unspecified
property, V for a definite but unspecified individual that has
the property.

The class of all possible arguments of a given prepositional
function is called the domain of the propositional function. A
possible argument is one which, when used to complete the
propositional form, makes sense. Consider, for instance, x is
French 3 , and a set of possible values for x, viz. Voltaire,
Cervantes, General de Gaulle, Petain, Franklin Roosevelt.
Relying upon our extra-logical knowledge, w r e can say that
the substitution for x of any one of these five names would
yield a significant proposition, but only the first, third, and
fourth would yield a true proposition. Those arguments
which yield a true proposition are said to satisfy the function
- a convenient word taken from the terminology of mathe
matics; the others do not satisfy the function but they make
sense, and must, therefore, be included in the domain. If we
were to substitute for x, in x is French , the word "wittiness",
the result would be a nonsensical set of words. The significant
propositions which are obtainable by substituting values of
the variables are called the range of significance of the pro-
positional function.

Suppose that in a certain class of university students

* "$" may be read "

132 A MODERN ELEMENTARY LOGIC

reading logic in a given year there are twelve members,
denoted respectively by the letters a, b, ... I. On investiga
tion it is found (we shall suppose) that a is a chess-player,
b is a chess-player, and so on to / is a chess-player. This infor
mation could be given by the conjunction of twelve com
ponent propositions: a is a chess-player^ and b is a chess-player
. . . and I is a chess-player. This takes a long time to write or to
say if we mention each of the twelve component conjuncts
separately; the same information could be given by saying
All these logic-students are chess-players. This proposition is
equivalent to the conjunctive proposition with twelve con
juncts, for all these shows not only that each of the students
is a chess-player, but also that we have left none out. Such
a proposition is enumerative, for each of the individuals about
which the statement is made has been separately taken into
account. Clearly this is possible only in the case of a limited
class with all the members of which we can be acquainted.
A class containing an infinite number of members could not
even theoretically be thus enumerated, and a class containing
an indefinitely large number of members cannot in fact be
enumerated. At present we shall neglect these difficulties and
consider only our limited domain.

We must notice that in using the expression "all these
logic-students are chess-players" we have not stated a
properly universal proposition, since "these" is nothing but
shorthand for the names of the twelve students. Let us
then say, "For all values of #, if x is a logic-student, then x
is a chess-player." This expression is unrestrictedly general,
but we claim to assert the proposition thus expressed only
because we know that a, b, . . . / are each arguments
satisfying the prepositional functions, " is a logic-student"
and "x is a chess-player", and we assume we have left no one
out.

Let us now suppose that we know further that among these
students there are some who are musical. We can state this
information in the form Either a is a chess-player and also
musical or b . . . , where the dots show that we need to write
down the remaining ten alternants. We can express this by
"For some value of #, x is a chess-player and is musical."

VARIABLES, PROPOSITIONAL FORMS, ETC. 133

This is equivalent to Some chess-players are musical, where
"some" has its usual meaning "at least one".

It will easily have been recognized that we have been
using expressions which are suitable for expressing the
universal and particular propositions of the traditional
schedule. These are general propositions. It may seem odd, at
the first glance, that a statement made about some members
of a class should be a general proposition; it will not seem odd,
however, as soon as we reflect that, in our example of the
class of logic-students, the statement refers to some members
in the domain, and it refers to them quite generally, Le. it is
not necessary to specify any one member. The assertion is that
somebody in the domain is both a chess-player and musical.
This is a general statement.

So far we have been considering a domain limited to
twelve possible arguments for the prepositional functions,
x is a chess-player 3 , etc. Let us now forget this limitation and
consider any two characteristics, which we shall symbolize
respectively by "$" and "!?", and thus obtain the two
propositional functions, &x, Wx. Let a be some constant
value for @x and for Wx. We can assert, If <&a, then Wa.
If it did not matter whether we chose a> or b, etc., but any
argument in the domain would satisfy both functions, we
can write For all x, if &x, then Wx. It is usual to abbre
viate this to (x} . 0x implies Wx. An example that would
fit into this form is If an animal is ruminant, it is horned,
i.e. (x) . x is a ruminant animal implies x is a horned
animal . This is a proposition and is thus either true or
false.

We have seen that x is used for a variable symbol. There
is an important difference between the way in which x is
used in (x) . <&x implies Wx and in &X. We saw that <&x repre
sents the property that something has; it is analogous to the
traditional notion of an abstract term, e.g. "x is red" is
roughly equivalent to redness, a property that something has.
The form "x is red" is not a proposition; it asserts nothing
until a value is substituted for x, in "x is red". The proposi
tion yielded by the substitution of a value for x will depend
for its truth or falsity upon which value is substituted. If the

134 A MODERN ELEMENTARY LOGIC

page on which this is printed were substituted for x, in "x is
red", the resultant proposition would be false; if the colour of
blood were substituted, the resultant proposition would be
true. Hence, the nature of the term substituted is all-
important for determining the truth or falsity of the resultant
proposition. But in (x) . x is a flash of lightning* implies x
is followed by thunder , the resultant proposition will be true
no matter what value is substituted for x. Hence, in the latter
expression, x is called an apparent variable* because we do
not need to give a specific value to x in order that the
resultant proposition should be true; in "x is red" we do
need to give a specific value, and the x is here called a real
variable.

It is important to notice that (x) . x is a flash of lightning 5
implies *x is followed by thunder , does not apply only to
those terms which are a flash of lightning; what is asserted is
that ifx is a flash of lightning, x will be followed by thunder.
We can express the same point by using the traditional sym
bolism: All S is P. This makes an assertion about what is non-S
as well as about 6*; if this were not so, we could not use the
method of reductio ad absurdam, which consists in using impli
cations where, as it turns out, the antecedent is false. All that
is required is that in (x} . x is an S implies c x is a P\ we should
know what can be significantly substituted for x in the pro-
positional form. What can be significantly substituted depends
upon the meaning of "S" and of "P"; or, if we use the 0, W
symbolism, upon the meaning of "0" and "y".

There is a point about which it is easy to be confused. The
prepositional form, or function, is not a proposition but, as
we have seen, it is an empty schema, which does not assert
anything. But if we can say that the prepositional function
holds for any, or for some, of its possible arguments, then we
obtain a proposition. Thus the difference between a real and
an apparent variable is extremely important; with the former
we assert nothing, with the latter we assert a true or a false
proposition.

We shall conclude this section by writing down the four
traditional propositions in the symbolism associated with this

* The term "apparent variable" is due to Peano.

VARIABLES, PROPOSITIONAL FORMS, ETC. 135

doctrine of prepositional functions. Let S stand for the terms
which satisfy 0x y and P for the terms \vhich satisfy Wx. Then
we obtain

means [x] . &x implies Wx.

means (x} . &x implies not-^Px.

means (3x) . &x and Wx.

means fix) . <E>x and not-Wx.

The new symbol 3 here introduced will be easily read,
since we are already acquainted both with the traditional
symbolism (given on the left-hand side) and with the analysis
of particular propositions as asserting Tor at least one value
of x, <X>x and W. Thus (3*) can be read There is an
x such that . . . or Tor some value of x . . .

These different forms of symbolism are merely notationally
different. But, as anyone acquainted with the history of
musical notation or of mathematical notation knows, a good
notation brings out the essential points hi a way that makes
them easier to grasp. The advantage of the notation with x
is that it shows us clearly that what we assert in these general
propositions is a connexion of properties and that the asser
tion is significant even when we do not know the individuals
characterized by them. Like the notation used in Chapter V
(SP = o, etc.) this notation once more emphasizes that the
difference between affirmative and negative propositions is
unimportant, whereas the difference between particulars and
universals is fundamental. Finally, it reminds us that the A,
, /, propositions are by no means simple propositions.

3. MATERIAL IMPLICATION AND ENTAILING

In our illustration of the class of logic-students, we felt confi
dent in asserting that (x) . x is a logic-student implies x is
a chess-player*, for we were dealing with a very limited domain.
Knowing (as we do long before we began to study logic) that
it is, as we say, c a mere matter of chance* that all those who
studied logic were chess-players, we shall not wish to assert
that it follows from the fact that someone studies logic, that
he is also a chess-player. But, within our domain, we could

136 A MODERN ELEMENTARY LOGIC

assert that If x is a logic-student, then x is a chess-player; this is
equivalent to Either x is not a logic-student or x is a chess-player.
In writing out the A and E forms above, we used "implies".
We saw (in Chapter II) that a proposition of the form If p,
then q can be interpreted as meaning p implies q, in the sense
that p cannot be true and q false. This fits in with our
assertion about the logic-students.

But "cannot 53 might mean "could not" or it might be
interpreted as meaning "cannot, the facts being what they
are". The second gives a much weaker meaning to "p cannot
be true and q false". To this interpretation of If p, then q,
Bertrand Russell has given the name material implication. This
can be defined as follows:

"p materially implies <f* means "either^ is false orq is true".

We shall contrast material implication with a stricter rela
tion illustrated in the following examples: (i}Ifa triangle is
isosceles, then its base angles are equal; (2) If this is red, this is
coloured; (3) If A is father of B, then B is a child of A; (4) If B
and C have the same parents and C is male, then C is brother ofB;
(5) If all detectives are quick-witted and no quick-witted people are
easily hoodwinked, then no detectives are easily hoodwinked. The
relation that holds between the antecedent (i.e. the implying
proposition) and the consequent (i.e. the implied proposition)
in each of the above examples is a relation of necessary impli
cation. It is, it will be observed, the relation that holds
between the premiss (whether simple or compound) and the
conclusion of a valid inference. In all the examples except
(i) the antecedent alone suffices to necessitate the conse
quent; the latter follows logically from the former alone. In
(i) there is presupposed the axioms of Euclidean geometry;
this being understood, we can say of (i), as of the other four
examples, that the antecedent could not be true and the conse
quent false. For this relation Professor G. E. Moore has used
the word entailing, and this word is now used by many
logicians to signify the relation that holds between p, and q
when p could not be true and q be false. But this is what we most
often mean when we say p implies q\ and we so used
"implies" in Chapter I. Hence, in order to distinguish
entailing from the weaker relation, we shall follow Bertrand

VARIABLES, PROPOSITIONAL FORMS, ETC. 137

Russell, and shall call the matter-of-fact relation material
implication.

It should be noticed that If . . . then ... is ambiguous,
since it may be used to signify materially implies or it may be
used to signify entails. Such a sentence as If it is cold to
morrow, I shall stay indoors is quite naturally used to state
that I shall not as a matter of fact go out if it is cold; this
sentence would not normally be understood to mean that its
being cold tomorrow necessitates my staying indoors., however firm
my resolution may be. On the other hand it is not unnatural
to say, c lf Mary and Jane are second cousins, then at least
one parent of each are first cousins 3 , and here the antecedent
does necessitate the consequent, for the former could not be
true and the latter false; i.e. the antecedent entails the con
sequent. It is, accordingly, not surprising that there should
have been a good deal of confusion with regard to the inter
pretation of If. . . then . . ., and a failure to see clearly
that entailing and material implication are different relations.
Material implication is the weakest of all relations in virtue
of which one proposition can in any sense be said to imply
another; it does, indeed, lay down one essential condition of
implication in every sense in which we could say that one
proposition implies another, namely, that if p is true and q
false, then in no sense can p imply q.

At this point it is notationally convenient to introduce some
shorthand symbols. In defining "p materially implies #" we
used the logical notions of either . . , or, and of the negation of
a given proposition; to say "p is false" denies, or negates, p;
hence, we can write the contradictory of p as not-p. Hitherto
we have used the bar-symbol and have written "j?" to mean
"p is false". We shall now use the symbol introduced by
Bertrand Russell in Principia Mathematical thus "not-/>" is
written "++>p". This is merely notationally different from
"", as "IV" is notationally different from "4". The notion
expressed by "either ... or ..." will be written "v", so
that "either p or y" will be written "p v q"* We shall now

* The symbol "v" is derived from the letter D, which is the first letter of uel,
Latin for "or". It is unfortunate that Russell and the symbolic logicians gener
ally call this relation disjunction.

138 A MODERN ELEMENTARY LOGIC

rewrite the definition of material implication in the linguistic
form:

p=> q . = .~pv q df.

The symbol => is shorthand for "materially implies";
",=... df" is shorthand for "is the defined equivalent
of". The student should have no difficulty in reading this
expression. It must be remembered that the expression on the
right-hand side, the definiens, states the meaning given by
definition to the expression on the left-hand side. Whenever
we define an expression we must, if we are to be consistent in
our usage of words, keep to the definition; hence, when we
say c *p materially implies cf\ or write "p => (f\ we mean
exactly what "^ p v q" expresses, viz. that " either p is false or
q is true"; the either ... or is non-exclusive.

Bearing this definition in mind, we shall see that material
implication holds between propositions of which neither
would ordinarily be said to imply the other; ordinarily, we
understand by "implies" a relation that holds between
propositions which are relevantly connected; by relevant con
nexion we probably mean a connexion in the meaning of the
propositions. To this consideration we shall return after we
have examined some examples of material implication. In
stating these examples we take for granted that we know
(independently of anything we have learnt from logic) which
of the propositions is true, which false; we also know that
every proposition either is true or is false.

(a) 2 4- 2 = 4. (e) A triangle has three sides.

(b) Italy is an island. (f) Rome is in England.

(c) A cat has ten legs. (g) 6 + 41 = 57.

(d} Columbia University is in New York, (fi) The Pope is a woman.

The examples have been indexed by small letters of the
alphabet in parentheses, for the purpose of summing up the
results in a small space; hence (a), etc., will be used to name
the propositions.* We can see:

(a) => (e); (b) =>(/); (c) => (g); (d] does not materially
imply (K), since (d) is true and (K) false. But in the other

* In reading the statements that follow the student should mentally sub
stitute for (a), the proposition 2 + 2 = 4, anc * so on for each index letter.

VARIABLES, PROPOSITIONAL FORMS, ETC. 139

three cases cited either the first is false or the second is true,
and, since either ... ar ... is not exclusive, we can admit the
case when both the first is false and the second is true. The
excluded case is when the fast is true and the second is false., for
anything implied by a true proposition is true: this condition,
we saw, is essential to every 7 possible meaning assignable to
the w r ord "implies".

It is easy to see that the eight propositions given provide
other examples, e.g. (a) => (d); (b] => each of the other
propositions and so on.

We can state these considerations in another way. Every
proposition has two possibilities with regard to truth and
falsity, namely, truth, falsity. These are called the truth-values.
There are, with tw r o propositions, four combinations: (i) both
true; (2) both false; (3) and (4) one true, the other false.
Using T for truth, and F for falsity, we will WTite them down
as follows:

p

9

T

T

T

F

F

T

F

F

Using this notation we will write down the compound
propositions, obtained by combining p with g, (i) by ^ , (ii)
by v, (iii) conjunctively, which we shall symbolize by a dot
(.), so that "p . q" means "p and 5".

p * \ p^l

pvg

p.q

T

T |

T

T

T

T

F |

F j T

F

F

T

T

T

F

F

i

i

T

F

F

From this table we can see at a glance that the conjunction
of p with q (i.e. p . q) excludes three of the possibilities; but
p ^> q excludes only one possibility, viz. p true with q false;

140 A MODERN ELEMENTARY LOGIC

p v q also excludes only one possibility, viz. both p and q false.
We are interested in the interpretation ofp ^ q, with regard
to truth or falsity; we see that any proposition, true or false, is
materially implied by any other false proposition, whilst any
true proposition is materially implied by any other proposition,
true or false. This is in accordance with the results we found
when considering the eight significant propositions given in
the list above.

This result has been called paradoxical; indeed, the con
clusions we have just summed up have been called c the
paradoxes of implication . There is, however, no paradox,
for a paradox is a statement apparently absurd or self-con
tradictory but possibly well-grounded. Provided we keep in mind
the definition of "material implication", these results do not even
seem absurd. What is there absurd in saying that, given the
compound proposition either p is false or q is true, then this
compound proposition is itself true if (i) p is false and q true,
(ii) p is true and q true, (iii) p is false and q is false? Clearly
this is not in the least absurd. What is absurd is to define
materially implies as we have done and then to forget the
definition, drop out the qualification indicated by "materi
ally", and thus think of implies as equivalent to entails. These
so-called paradoxical 5 consequences, as Professor G. E.
Moore has pointed out, appear to be paradoxical, solely
because, if we use "implies" in any ordinary sense, they are
quite certainly false 5 .* It is difficult to use a very familiar
word in a wholly unfamiliar and technical sense without at
times falling back into the familiar meaning which has been
excluded by definition. This is the simple mistake committed by
those who allow themselves to be puzzled by apparent para
doxes resulting from the definition of "material implication".

For certain technical procedures in mathematical logic it
is convenient to define "implication 55 in terms of negation and
either . . . or . . .; thus, for these purposes, "implication"
means "material implication 55 . It should be noticed that
whenever it is true that p entails q, then it is true that p => q,
for =5 is a weaker relation than entailing; it holds whenever
entailing holds, but the converse is not true.

* Philosophical Studies, p, 295.

VARIABLES, PROPOSITIONAL FORMS, ETC. 141

It is not essential to define ^ in terms of either ... or; it can
be equally well defined in terms of negation and conjunction;
thus:

This may be read: "p materially implies q" is the defined
equivalent of "It is false that/ is true and q is false."*
The following equivalences are worth noticing:

p 13 q . ==. ~ p v q . ==. -^ (p . r^ qj,

It should be observed that these three equivalences have
already been stated, in Chapter III, i, as normal equiva
lents of composite propositions. These equivalences are in no
way affected by our definition of =>, for the relation of
material implication suffices to yield the equivalent alterna
tive and disjunctive propositions with which we are already
familiar. It is convenient for certain purposes to use the
shorthand symbols that appear above, but it is not essential.

4. EXTENSIONAL AND INTENSIONAL
INTERPRETATIONS OF LOGICAL RELATIONS

Our discussion of material implication should have made
clear that knowledge of the truth or of the falsity of/, q is
alone relevant to determining whether p ^ q: provided p is
false, q can be any proposition; provided q is true, p can be
any proposition. Hence, we are entirely unconcerned with
what p y q may be about; thus, we pay no attention to what
is commonly called the meaning of the proposition. Hence,
we saw, Italy is an island => The Pope is a woman because both
these propositions are false. (The Pope is a man => Italy is an
island] f is a false statement; the first proposition is true, the
second false; hence the first cannot be related by => to the
second. The facts being what they are we discover that The Pope
is a man does not materially imply Italy is an island. If a

* The fact that we can give alternative definitions of p 12 q illustrates the
fact that no one of these definitions is fundamental. We can take our choice
whether we shall regard either ... or or both . . . and as fundamental; then,
combining with negation we get the definitions given above.

t Parentheses are used here to show that the two propositions are combined
into a single statement which is asserted, as a whole, to be false.

142 A MODERN ELEMENTARY LOGIC

geological convulsion broke off Italy from the continent, then
either of these two propositions would imply the other. Thus,
it is what is actually the case that determines whether a
material implication holds. Another way of saying this is to
say that whether a proposition is true or is false depends upon
what the facts are. It is a fact that Italy is a peninsula; hence,
Italy is an island is in discordance with, Italy is a peninsula is in
accordance with, this fact. Looking at a proposition merely
from the point of view of whether it is true or false is said to
be taking the proposition extensionally* We are supposed to
know (how does not matter for our purpose) whether the
truth-value of a given proposition is truth or is falsity. That is
all we need to know.

Suppose, meditating upon the frailty of human nature, we
say To err is human 5 . Let us now make the somewhat rash
assumption that this is equivalent to All men err . What does
this proposition assert?

(i) We attempt to analyse it as follows: Either A is not
human or A errs; and Either B is not human or B errs . . . and Either
X is not human or X errs. The dots show that we have left out
many cases. Now, Either A is not human or A errs is equivalent
(by definition) to A is human => A ens , and so on, in each of
the cases cited. Now A 9 B . . . X belong to the class human
beings , hence, we can drop out our reference to the individuals
A, B, etc., and say x is human => x errs, whatever x may be. This
is an instance of generalized material implication, i.e. a con
junction of singular statements asserting that a material
implication holds. Russell calls this formal implication , in
order to contrast it with the conjunction of singular proposi
tions, true or false, which fulfil the condition required for
material implication. No new concept of implication is
involved in passing from material implications to formal
implications (as thus understood); a formal implication is
simply a collection of material implications, in which the
truth or falsity of the resultant statement depends entirely
upon the truth-values of the singular statements constituting
the components of the compound proposition.

At this point we are forced to ask ourselves whether we
were justified in saying that, since A 9 B, . . . X belong to the

VARIABLES, PROPOSITIONAL FORMS, ETC. 143

class human beings, we can omit further reference to them and
assert that whatever x may be, x is human ID x errs. For this
procedure rests upon the assumption that what is true of a
collection of individuals which are members of a given class
is true of all members of the class, including those not in the
subclass \vhich constituted the original collection. Clearly this
does not hold. For instance, to say * whatever is true of a
subclass of humans is true of all humans 5 is clearly false;
Russians are a subclass of humans, Frenchmen are another
subclass, and there are many things true about Russians that
are false about Frenchmen, and conversely. It is not necessary
to multiply instances.

(2) We thus attempt another analysis. Although it is not
true 3 , we may urge, that all human beings have all the
characteristics true of all Russians, this is irrelevant since the
characteristic we are concerned with is the liability to make
mistakes*, there is an essential connexion between human nature
and liability to make mistakes , it follows from the fact that
human nature is what it is that human beings err.

If we give this answer, then we are taking an intensional
view; we are asserting that there is a connexion between being
human and erring which can be apprehended without examin
ing vast collections of human beings and finding out in each
case that this, that, and the other human being errs. We may be
willing to admit that we should not have noticed this con
nexion unless we had been confronted with actual instances
of it; but that is true also of, say, the connexion between
being an angle in a semi-circle and being a right-angle. But, once
we have noticed it, we are asserting a connexion that is not
merely a statement of the coincidence of true singular
statements.

This second answer suggests that we might reformulate our
original proposition thus: To be human implies to err*. This
reformulation has the advantage of showing that we are
abstracting the characteristics being human, erring, from the
instances which exemplify them; thus, we are considering
these characteristics in a contemplative way, not taking note
of their exemplification in actual cases. Or, as we said just
now, we are regarding the proposition intentionally, as asserting

144 A MODERN ELEMENTARY LOGIC

a connexion of meaning. Clearly , then, "implies" will not be
interpreted as "materially implies". Are we, then, to interpret
"implies" in c To be human* implies c to erf as entails?

This question raises a problem of great importance to
which no decisive answer can be given, and of which no
adequate discussion is possible within the limits of this book.
Enough, perhaps, may be said to make clear the sort of
questions this problem raises.

Let us go back to the examples of entailing, given at the
beginning of 3. We observed in the case of each of the five
examples, that the antecedent could not be true and the con
sequent false; further, that the antecedent alone sufficed to
necessitate the consequent. The word observed 5 , used in the
last sentence, is appropriate. We could not claim to have
done more than to adduce examples which, the reader would
admit were examples of a relation entirely different from
material implication. We can now add that the truth of the
compound propositions adduced as examples is entirely inde
pendent of the make-up of the actual world. That the con
sequent followed, in each case, from the antecedent could be
known without knowing whether the component propositions
were true or false. Consider example (5), for instance: the
entailing relation holds between the compound antecedent
and the consequent; the whole proposition is an example of
a syllogism in CelarenL Thus one example of entailing is the
relation of the premisses to the conclusion in a valid syl
logism. Example (2) - If this is red, this is coloured -is quite
different. This is an example of connected meanings; we so use
"red 55 that to say c this is red and to deny this is coloured is
to say what is self-contradictory.

It can hardly be maintained that this is true of the con
nexion between being human and erring. We conclude that we
cannot hold that To be human entails c to err . Nevertheless,
we need not rest content with the view that All human beings
err can be adequately analysed into a set of material implica
tions stating Either it is false that A is human or it is true that A
errs, and so on, throughout the remainder of the individuals
B . . . X. There is another alternative left us. We shall be
bold enough to maintain that the characteristic of being

VARIABLES, PROPOSITIOXAL FORMS, ETC. 145

human is relevant to the characteristic of erring^ in a way in
which The Pope is a man is not relevant to 2 -r 2 = 4, although
- since they are both true - these two propositions materially
imply one another, and are thus materially equivalent.

What the relation of material implication demands is solely
truth-values; what the relation of entailing requires is a
necessary connexion between that which entails and that
which is entailed. We are now insisting that there is another
connexion that may be found between propositions intension-
ally interpreted, namely, a connexion of relevance: the mean
ing of the premiss must be relevantly connected with the meaning
of the conclusion.

And what, it may be asked, do we mean by being relevantly
connected? Some attempt to answer this question will be
made in Chapter VIII. We shall scarcely be able to claim
that we have done more than to pose the problem; certainly
we shall not solve it. But to see that there is a problem to
folve is to have taken the first step essential to solving it. So
sar as the author of this book is concerned, this first step is
likely to be also the last.

CHAPTER VIII

Logical Principles and the
Proof of Propositions

I. THE TRADITIONAL LAWS OF THOUGHT

In every chapter of this book we have been engaged in
reasoning; we have - to use a popular phrase - put two and
two together and obtained four . We have judged that, if
certain propositions are true., others are also; if certain
propositions are false, others are also; again, if certain
propositions are false, others are true. We have not only
judged that these conclusions are so, but that they must be so.
In Chapter I we pointed out that to judge in this manner is
characteristic of rational beings; it is the mental activity we
call reasoning. When we reason correctly, our reasoning is in
accordance with logical principles.

Three of these principles were formulated clearly by
Aristotle.* They are traditionally known as the three Laws
of Thought 5 . They may be stated as follows:

1. The Law of Identity: Everything is what it is.

2. The Law of Contradiction: A thing cannot both be and
not be so and so.

3. The Law of Excluded Middle: A thing either is or is not
so and so.

This statement of the Laws is appropriate to the considera
tion of the singular proposition This A is B\ Aristotle was
thinking of the most elementary and fundamental character
istics of predication, in its purely formal aspect. They can
be reformulated as they concern propositions, implication,
and truth and falsity:

(i) Every proposition is equivalent to itself (i.e. every

* See Andytica Priora, 470, 9; Metaphysics 10060, 7; De Interpretations, i8b, 1-5.
Cf. M.I.L., Ch. XXIV, 2. For a detailed discussion of the traditional laws,
see J. N. Keynes, Formal Logic, Appendix B, pp. 450-67.

146

LOGICAL PRINCIPLES 147

proposition implies and is implied by itself), Principle of
Identity*

(2) No proposition is both true and false.

(3) Every proposition is either true or false.

This formulation brings out the essential relation of the
three laws; they cannot, however, be reduced to a single
principle, since the deduction of, for instance, (3) from (i) or
from (2) requires the independent notions of falsity,, or of
negation, which cannot be defined without using the principles
themselves. Both (2) and (3) are required in order to define
the relation of contradiction between propositions, since con
tradictory propositions are defined as propositions \vhich can
not both be true but one must be true.

These "three laws of thought" have been subjected to
severe criticism by modern logicians; these criticisms may be
summed up in the somewhat Pickwickian formula: They are
not laws, they are not laws of thought, and they are not the laws
of thought since there are others no less essential/ We shall
examine these criticisms briefly. The first two points may be
taken together. Certainly, the laws of thought are not state
ments of psychological laws, i.e. statements of the ways in
which we do think. Unfortunately, we often contradict our
selves, we often think (or behave as though we believed) that
there is a mean between truth and falsity. The "laws" are
not made true by the way in which men think; they are state
ments of how men ought to think, and will think if, and in so
far as, men are thinking rationally. Accordingly, it is far
better not to use the description laws of thought ; it is better
to call them logical principles . "Laws" suggest at best
uniformities in mind and nature, at worst commands. Un
fortunately, no one has the power to command us to think
logically; even were this not so, we have not always the power
to obey such a command. Our thinking is in part determined
by our emotional attitudes and our deep-seated prejudices.

Certainly the three Laws are not sufficient for regulating
our thinking; it is undoubtedly true that Consecutive thought
and coherent argument are impossible without these laws,

* For reasons given later on this page, it is better to call these Principles and
not Laws.

148 A MODERN ELEMENTARY LOGIC

but the traditional Logicians were mistaken in singling these
out as though they were in any sense more fundamental than
other logical principles. We cannot here attempt to state all
those other principles which are clearly exemplified in
ordinary reasoning; it must suffice to mention only three:

(4) Principle of Syllogism: If p implies q, and q implies r,
then p implies r. This is the principle which underlies the
dicta of the traditional syllogism, but it has a much wider
application.

(5) Principle of Deduction (sometimes called the Principle
of Inference] : Ifp implies q, and p is true, then q is true. This
principle permits the omission of an implying proposition (the
antecedent) provided that the implying proposition is true;
it is in accordance with this principle that conclusions are
drawn from true premisses in valid arguments.

(6) The Applicative Principle (or Principle of Substitution] :
Whatever can be asserted of any instance however chosen can be
asserted about any given instance. W. E. Johnson has said of
this principle that c it may be said to formulate what is
involved in the intelligent use of "every" ? .*

The last three principles are exemplified in all chains of
reasoning, whilst the first three are also exemplified in all
coherent reasoning. These principles do not suffice but they
are all essential to sound reasoning.

Various special criticisms have been made of the three
principles known as the traditional laws of thought , most
of which rest upon extraordinary muddles. Thus it has been
argued that A is not necessarily A, for A is changing all the
time, and anyhow, everyone knows that A is always B\ The
point probably intended in this comment is that things change
and that every thing has various different properties. The
principle is not in the least in conflict with these contentions.
Unless A were identifiable as A, it would make nonsense to
say that A is B. In the form in which this principle concerns
propositions, it is clearly true, since, unless p implies p, p
could be both true and false. This takes us to the principle
of contradiction, so that the principle of identity stands or
falls with it.

* W. E. Johnson, Logic, Pt. II, p. 9.

LOGICAL PRINCIPLES 149

More serious criticisms have been made of the principle of
excluded middle. We shall first, however, consider an objec
tion that is so easily refuted that it should never have been
made by competent logicians.

(i) It is ^ argued that things change insensibly , so that
sometimes it is not possible to assert that the thing has, or has
not, a given characteristic; e.g. this tomato is ripe, this tomato is
not ripe may neither be true, and yet these propositions are
formal contradictories. The point lies in the last statement.
Are the propositions contradictories, or only apparent con
tradictories? That will entirely depend upon what we mean by
"ripe". Is there a criterion of ripeness? If so, then the proposi
tions are contradictory, and there seems no reason to deny
that both cannot be true. If there is no criterion of ripeness,
then "ripe 53 is like "bald", namely, a word used to signify any
one of a range of degrees in which a characteristic may be
present. Some words are properly vague, i.e. are used to
signify a characteristic capable of a continuous series of inter
mediate degrees. It is illogical to demand that a sharp dis
tinction should be drawn between that which possesses and
that which does not possess such a characteristic. We may not
know where c to draw the line , and in some cases no line can
be drawn. But, if it be granted that "bald" can be precisely
defined in terms of number of hairs, then bald and not-bald are
proper contradictories; if it cannot be thus precisely defined
then these are not proper contradictories.*

(ii) The most serious objection to the principle relates to
its use with regard to propositions. It is argued that in addition
to the true and the false there is also the doubtful (or the
undecided).

We may begin by noticing that this looks like a cross
division. The division of propositions into true, false is
dichotomous, i.e. true, false are mutually exclusive and col
lectively exhaustive. It is possible to argue that much discus
sion still centres round the exact meaning of "true" and
"false". This is so, but it is at least clear that in every
ordinary usage the division is dichotomous. We can easily

* I have discussed this point in more detail in TTtinking to Some Purpose^
pp. 138-42 (3rd edition).

I5O A MODERN ELEMENTARY LOGIC

obtain a four-fold division of propositions into: (i) true and
known to be true, (2) false and known to be false, (3) true but
not known to be true or known to be false, (4) false but not
known to be false or known to be true. Now we can certainly
say that (3) and (4) yield the doubtful (or the undecided in the
sense that we are not able to decide whether the proposition
is true or is false). But it is clear that (3) and (4) both fall
under our original dichotomous division. A proposition is
true if in accordance with the facts; false if not in accordance.
We may very well not yet know, or never be able to know,
which of these possibilities is the case, but that we can be thus
ignorant of the facts has not the slightest tendency to show
that any proposition can be neither in accordance with the
facts (i.e. true) nor not in accordance with the facts (i.e.
false).

It must not be supposed that the above remarks are an
attempt to prove the principle of excluded middle; if what has
been said had been offered as a proof it would certainly be
circular. All that has been attempted is to show that the
objection has no point, and is, in fact, guilty of the fallacy of
cross division.

It may, however, be further argued that, even if the
assertion that a proposition is true if in accordance with the
facts, and false if not, be accepted, the principle of excluded
middle still fails, since the facts may be undecided. This con
tention rests upon a sheer mistake. It has been argued most
strongly in connexion with facts about the future. Let us
consider the proposition, Hitler will be a prisoner in London on
March 10, 1943. This proposition is asserted today (Septem
ber 27, 1942) by the author of this book (who would like it
to be true but is afraid it is false) . The comment in parentheses
is the sort of comment we all of us make at times with regard
to propositions about the future. The view we are now con
sidering is that the proposition about Hitler (henceforth to
be symbolized by p] is neither true nor false. There seem to
be two different reasons urged in favour of this view.

(i) p is not known to be true and is not known to be false.
This must be granted, but as we have just seen, this does not
imply that it is neither.

LOGICAL PRINCIPLES 15!

(2) If we argue that p Is either true or false, we are assert
ing that it either is the case that Hitler will be a prisoner in
London on March 10 of next year or it is not the case; and
this presupposes that there are past and future facts which
necessitate that he will be a prisoner in London next March,
if p is true; or it presupposes that there are past and future
acts wiiich necessitate that he will not be a prisoner in
London next March, if p is false. But, it is argued, this
assumes the truth of what is called determinism , namely,
that everything that happens is necessarily determined by
past events. Determinism, it is urged, is open to dispute.

This argument entirely fails to establish the required con
clusion. Whether Hitler s future movements are, or are not,
determined by past and present facts, the statement that he
will be in London on a certain date is a factual statement. I
determinism is correct, then it is factually (or causally)
necessary that he will be in London on the given date; or it
is factually (or causally) impossible that he will be in London
on the given date. Now, whichever of these is the case, either
the facts necessarily determine that p is true or the facts
necessarily determine that p is false; if, however, determinism
is false, then past and present facts in no sense determine
Hitler s future movements, so that he may, or may not,
be in London on the specified date. But whether p is true
or is false is not in any way affected by the answer to the
question: e are there facts now which determine future facts?
To suppose otherwise is to confuse (i) causal necessity with
logical necessity, (ii) truth with our knowledge of the
truth.

Certain logicians have argued that if there is no available
method of determining whether a given proposition is true or
is false, then it is neither. Examples of such undecidable
propositions are: Julius Caesar sneezed as he entered the Senate for
the last time. All numbers of the form 2 2fl+9 + i are factorable.
This contention again confuses truth with knowledge of truth.
Some who have taken this view with regard to undecidable
propositions have, it seems, wished to maintain that unless
the truth of a proposition can be verified or falsified, then it is
neither true nor false. To maintain this is simply to substitute

152 A MODERN ELEMENTARY LOGIC

for the notion of truth the notion of verifiability. Here it must
suffice to assert that this is a question of terminology, and
nothing in the contentions of these logicians suggests that
anything is to be gained by this change in the meanings of
these words.*

2. NECESSARY AND FACTUAL PROPOSITIONS

We saw in the last chapter ( 4), that we can regard proposi
tions from an extensional or from an intensional point of
view. When we adopt the latter point of view we pay atten
tion to the meaning of the proposition, that is, to what the
proposition states; from the former point of view we consider
only its truth or its falsity. The mere fact that two proposi
tions are both true (or both false), which entitles us to assert
that one materially implies the other, does not give to the
combination thus made any unity of meaning. That is why
it surprises us to discover that Italy is an island => The Pope is
a woman, or that 2 + 2 = 4 => A triangle has three sides. We
cannot easily bring the two component propositions together
in thought; the truth of the implying proposition does not in
any way limit the truth or the falsity of the implied proposi
tion; only, if it so happens that the implied proposition is false
and the implying proposition is true, then the former does
not materially imply the latter. Whether ID holds or not we
discover only after we know the truth-values of the com
ponent propositions. As we saw in the last chapter, a certain
geological change in the structure of the continent would
make it true that Italy is an island no longer materially implies
The Pope is a woman, since the latter proposition is false. We
shall, accordingly, say that material implication is a factual
relation} whether it holds or not depends upon the actual
constitution of the world. Entailing, on the contrary, is a
necessary relation.

* This position is that of most Logical Positivists. The questions raised are
more properly philosophical than strictly logical, and cannot be discussed here.
The objections to the principle of excluded middle, discussed above, have been
dealt with in a masterly fashion by Professor C. A. Baylis, in an article entitled,
*Are some Propositions neither true nor false? 3 (Philosophy of Science, Vol. 3,
No. 2, April 1936). This article is so clearly and beautifully written that even
elementary students may be able to profit from reading it.

LOGICAL PRINCIPLES 153

Consider the following propositions:

(1) Every body continues in a state of rest or of moving
uniformly in a straight line, except in so far as it is
subject to external forces.

(2) All planets move in elliptical orbits.

(3) Men must die.

(4) Cows are ruminants.

(5) This red rose is not red.

(6) Water freezes at o c Centigrade.

(7) An angle in a semi-circle is a right angle.

(8) Prices are regulated by the law of supply and demand.

(9) Hitler entered Prague on March 15, 1939.

(10) It rained in Tintagel on September 28, 1942.

(11) An igloo is an Eskimo dome-shaped hut.

It is easy to see that these propositions are of very different
kinds. Should any one of them be disputed, the evidence
required to justify its assertion would be entirely different
from the evidence required in the case of some of the others.
Let us examine them from this point of view. Our first step
should be to sort them out, so as to bring together those
which require the same sort of evidence in order to justify
their assertion. For this purpose we need a principle of
division.*

Ought we not first to inquire in the case of each proposition
whether it is true or false? This is not essential. Consider (10)
for instance: the evidence required to establish its truth (if it
is true) is of the same sort as the evidence required to establish
its falsity (if it is false). I, the author, who am now writing
this sentence, assert that proposition (10) is true. The
evidence I offer is (i) today is September 28, 1942, (ii) I
see rain falling each time I look up from my desk, (iii) I
remember seeing the rain falling this morning. Now both
(i) and (ii) may be questioned, i.e. evidence in support of
these assertions may also be asked for. There is not space
to pursue this illustration in detail here. It must suffice to
say that my evidence for (i) is based upon my acceptance of

* The student is recommended to pause at this point, and to sort out the
propositions for himself.

154 A MODERN ELEMENTARY LOGIC

my calendar as being correctly marked; my evidence for (ii) Is
sense-experience. I quite literally see rain falling. It is not to
be denied that people sometimes think it is raining when it is
not, but the final, and only evidence that can be offered is -
seeing and feeling rain falling, (iii) may seem to be more
dubious, but in fact it is not. My reliance about so recent a
memory is not less great and is not (so far as I can introspec-
tively judge) different in kind from my reliance upon the
direct evidence of my sense-experience. It is characteristic of
the sort of evidence constituted by both (ii) and (iii) that it is
available only for myself. (Here "/" could, under suitable
conditions, stand for some other person who is having the
same sort of experience.) If this be granted, then the truth of
proposition (10) cannot, at a subsequent date, be established
by exactly the same sort of evidence, or rather, there would
be needed in addition evidence of another sort, e.g. an entry in
someone s diary, the report of the meteorological office, and
so on. The entry in the diary could be regarded as reliable
evidence only if the testimony of the writer could be estab
lished as acceptable. And his statement is based (if correct)
upon such evidence as that offered in (ii) and (iii). It is not
unlikely that no entry in anyone s diary, no sufficiently
detailed report from the meteorological office, will be able to
be cited in evidence of proposition (10) by the time this book
is printed; detailed daily reports of the weather in a small
Cornish village are not likely to be made. But, whether this
is so or not, that is the sort of evidence that would be required
to establish the truth of (10) at some date subsequent to the
present.

This is an example of a singular factual proposition; so,
too, is proposition (9). The event stated in (9) is an event of
considerable importance in the history of Europe, and con
sequently, of the world today. It is reasonable to suppose that
there will be an abundance of testimony which can be used
as evidence of its truth. If I (the author*) have made a slip
in the date, exactly the same sort of evidence will establish

* No apology should be needed for the author s intrusion into the text at this
point. The purpose is to call the reader s attention to the need (when occasion
demands) to verify the statements made to him, and to point out to him that
certain propositions need more careful scrutiny than others.

LOGICAL PRINCIPLES 155

its falsity-. In the case of both g, and iOj the sort of evidence
required can be summed up under the three heads: <a] direct
experience, (b) reliance upon testimony which involves fa)
someone else s direct experience, (/J) some method of testing the
reliability of such testimony, (7) general principles of in
ference. Propositions (9) and (io) 3 different though they are,
resemble each other in one important respect, namely, the
evidence for their truth includes, in each case, someone s
direct experience at a specified date. It is probable that for
years to come the indirect evidence of testimony will be
available to establish (9), but not available to establish (10).
This difference has nothing to do with the logical nature of
these propositions; both are singular factual propositions;
their difference has to do with the relative importance of their
truth for the affairs of men. With that difference the logician
is not at all concerned.

Propositions (2), (3), (4), (6) are also factual propositions
but they are not singular propositions; each of them involves
generalization. Without generalization no science is possible.
In the next chapter we shall examine what is involved in
generalization; here it is enough to point out that generaliza
tion involves an inferential leap: it is the passage from direct
observation that certain observed instances of the class Ceach
have the property/, to the conclusion that every member of
C has /. The four propositions now being discussed are the
results of such an inferential process. But they are not all on
the same level. Cows are ruminants, taken thus in isolation from
any context of discussion, may be regarded as a statement
that cows fall within a certain superclass in a biological
classification; or it may be regarded as a generalization from
the observation of particular cows. The latter interpretation
takes the proposition to be at a more primitive level than the
former; by the time we are able to assign a biological class to
its place in a classification a certain amount of systematiza-
tion has been achieved. (2), (3), and (6) may be taken
together, so far as our present purpose is concerned. Of each
it is true that (i) it involves generalization from direct
observation of particular instances, (ii) the evidence for its
truth is in large part derived from its place within the system

156 A MODERN ELEMENTARY LOGIC

of the special science to which it belongs.* (8) is also a
factual generalization but, as every student of economics will
readily admit, it cannot be truly asserted without consider
able qualification. For example, in Great Britain today, the
price of many commodities is regulated by governmental fiat.
Even apart from this complication, questions peculiar to the
so-called social sciences will force themselves upon our
attention once \ve begin seriously to examine what is the
evidence upon which the assertion, Prices are regulated by the
law of supply and demand, rests. f

Proposition (i) would at one time have been regarded
as a generalization from the observed behaviour of bodies
extrapolated to fit ideal (i.e. imagined) conditions in which
no actual body can ever be. The way in which this statement
has been formulated suggests, what is in fact the case, that
proposition (i) is, as used by physicists, no generalization
from experience; it is a mixture of conventions and records
of observation. This proposition is Newton s First Law of
Motion*, the evidence for it is to be found in the whole body
of Newtonian science. Once granted, then proposition (2) can
be deduced from it together with certain premisses about
planets derived by generalization from particular instances.
It must be emphasized that the evidence for Newton s Law
is so fundamentally different in kind from the evidence upon
which a natural law (such as water freezes at o Centigrade) is
based that we feel compelled to put evidence in inverted
commas - a symbolic device commonly adopted to show that
we are using a word in an unusual sense.

Proposition (7) is entirely different from the other proposi
tions we have been considering; nothing that happens in the
world is relevant to its truth or falsity. That an angle in a semi
circle is a right angle follows from the definitions and axioms of
Euclidean geometry; it is a necessary consequence of these.

Proposition (n) may be regarded as the statement of a
definition. We say may be regarded because it depends upon
the context in which it is asserted what exactly the words used

* On this point, see further, Gh. IX, 5.

1 1 much regret that lack of space prevents me from raising, and attempting
to answer these questions. The student should ask himself, in what sense of
"law" is there a law of supply and demand.

LOGICAL PRINCIPLES 157

to express it are intended to convey. Here it is given apart
from a context; it was in fact taken from the Everyman
Dictionary^ at random. "Igloo" means "an Eskimo dome-shaped
hut" has the form of a definition of "Igloo". Even so, it con
tains a factual element, since it is an assertion which involves
the statement that "igloo 55 is the word used by Eskimos to
refer to \vhat in English can be described as "a dome-shaped
hut". The evidence for the truth of this proposition is factual.

Proposition (5) is a self-contradictory proposition, or, as it
is sometimes called, c an inconsistency . It is necessarily false,
and its contradictory, A red rose is red, is necessarily true. To
know that this proposition is true it is necessary and sufficient
to know the meanings of the words used to express it. Such
propositions are usually called tautologies.

If we review our prolonged discussion of the eleven
propositions given at the beginning of this section, we shall see
that we can divide them into tw r o mutually exclusive and
collectively exhaustive classes, the principle of division being
the nature of the evidence required to establish their truth
or falsity; the two classes may be denominated: factual
propositions, non-factual propositions. The latter may be
subdivided into: necessarily true propositions, necessarily
false propositions, or self-contradictories.

Factual propositions are sometimes called contingent proposi
tions, because they can be known to be true (or false) only by
investigating what happens in the world, i.e. their truth (or
falsity) is contingent upon what the world is like, and cannot,
accordingly, be discovered by any careful examination of the
structure of the propositions. The contradictory of a con
tingent proposition is also contingent. We have seen that
contingent (or factual) propositions differ among themselves
with regard to the way in which their truth or falsity can be
established. All alike, however, are ultimately based upon
direct observation of particular instances; that is to say, there
must be an appeal to sense-experience. Facts that can be
known only by sensible observation are called empirical
facts . Such facts constitute the original data of the natural
sciences. Upon them, in the last resort, is built the imposing
structure of the physical sciences.

158 A MODERN ELEMENTARY LOGIC

Necessarily true propositions are usually called necessary
propositions , for necessarily false propositions are self-con
tradictory and thus impossible. Many modern logicians hold
that all necessary propositions are tautologies (i.e. resemble
This red rose is red] . Thus 2 -7- 2 = 4 is regarded as a tautology
on the ground that the truth of the proposition follows from
the definition of the terms involved. On the same grounds
such propositions as An angle in a semi-circle is a right angle are
regarded as tautologies. These logicians usually make dis
tinctions within the class of tautologies. For example, Wealth
is riches, Courage is bravery, are called synonymous propositions.
It is not possible for us to examine these views. It must suffice
to point out that, given that a proposition is such that its
truth is a consequence of the nature of the terms involved in
it, then the proposition is necessary and its contradictory is
self-contradictory. It is impossible for a necessary proposition
to be false. This statement is itself tautologous.

3. THE NECESSITY OF LOGICAL PRINCIPLES

Some contemporary logicians (including those known as
Logical Positivists ) hold that all necessary propositions, in
cluding logical principles, are conventions. Some go further
and maintain that such Laws of Nature as the gravitational
laws are conventions.* To discuss this view properly it would
be necessary to examine the various meanings of the word
"convention", and to show how gradually we pass from the
meaning of "convention" as used in the forms of social inter
course (e.g. c Mrs Johns is not at home 3 ) to its use in connexion
with scientific laws. We not only have not the space to
attempt this here; it must be admitted that a rigorous analysis
of the concept convention has not yet been carried out. We
mention the view simply in order to point out that here is
something for the student to investigate if, and when, he can.
We shall not adopt the conventional view of logical principles
in this book.

It is not easy to make clear exactly in what sense of

* This view is specially associated with the writings of Professor A. S. Edding-
ton on philosophy of science.

LOGICAL PRINCIPLES 159

"necessary", logical principles are necessary.* It is simple
enough to assert that their truth is self-evident, and that a self-
evident truth must be necessarily true. But self-evidence is a
dangerous notion; it seems to combine obviousness and
logical priority-. What is obvious to one person is not to
another; it depends in part upon keenness of mental vision
and in part upon familiarity. Unfortunately, we have learnt
that a proposition which has long been regarded by com
petent thinkers as self-evident turns out to be false. What is
indubitable is not necessarily true; our capacity to doubt
depends upon our previous knowledge and our mental
agility.

Modern logicians have devoted considerable skill and
energy to the construction of deductive systems, in the sense
in which, for example, Euclidean geometry is a deductive
system. Setting out from carefully stated definitions and
postulates, theorems are deduced by a rigorous step-by-step
deduction. Some of these systems have been specially devised
in order to offer proofs of the principles of logic. The most
elaborate construction of the kind is the Principia Mathematica
of Whitehead and Russell. | In this system the principle of
contradiction, for instance, is not included among the
postulates; it is deduced comparatively late in the system.
But this by no means shows that the principle has not in fact
been used throughout the demonstration. What such a system
shows is that logical principles are so closely knit together
that any one principle can be deductively derived from a
finite set of other principles, and can be shown to imply itself.
This procedure may strengthen us in our belief that logical
principles are indispensable for all rational thinking, but it
cannot be regarded as offering an independent proof of the
principles themselves. We must be content to assert here that
logical principles are so fundamental to our thinking that
without presupposing them we could not think at all, and
could not, theiefore, construct systems.

* The difficulty is by no means due solely to the need for brevity, although
this limitation does increase it. The difficulty is, however, in large part due to
unclearness on the part of the author herself.

f See M.I.L., Ch. X. 4. An excellent introduction to the study of Principia
Mathematica is provided by Part III of R. M. Eaton s General Logic.

l6o A MODERN ELEMENTARY LOGIC

4. PERSUASION AND PROOF*

To believe a proposition and to believe it to be true are one and
the same thing; nevertheless, we often believe propositions
which are false. We should like our beliefs to be knowledge;
sometimes we entertain a belief knowing that it is believed and
not known. We can know our conclusions to be true only when
we know both that the premisses are true and that they imply
the conclusion. For this purpose we reason. Unfortunately, in
our haste to resolve our doubts we may be persuaded to
believe by other methods than by reasoning. A sharp distinc
tion is here to be drawn between persuasion and conviction]
they are to be distinguished by the nature of the process
whereby doubt is resolved. The orator frequently uses the
method of persuasion; his aim is to induce belief at all costs
rather than to prove his contentions; his art consists in
persuading his readers (or hearers) to accept conclusions for
which he may have offered no evidence, and which may
even be false. The orator s appeal is not to reason but to
uncontrolled emotion, not to considerations logically rele
vant but to prejudice. We are not infrequently orators to
ourselves.

The method of rational conviction consists in reasoned
proof. A well-constructed argument, designed to convince
the intellect, exhibits the characteristics of clearness, con
nectedness or relevance,, freedom from contradiction or con
sistency, demonstrativeness or cogency. If I seek thus rationally
to convince myself or others that a certain proposition is true,
I must be careful to ascertain whether the premisses are true
and I must aim at constructing a rigorously valid argument.
An argument is valid if the conclusion is drawn in accordance
with the logical rules, e.g. of the syllogism or of the compound
arguments. We may be honestly mistaken in supposing that
our argument is valid; there may be unsuspected ambiguities
in our language; we may use as a premiss a proposition which
we erroneously believe to have been proved. There are many
ways of going wrong. In the ordinary discussions of practical

* Some of the paragraphs in this section have been taken in part from M.I.L.,
Ch. XXTV. A fuller treatment will be found in that chapter.

LOGICAL PRINCIPLES l6l

life, concerning politics, art, education,, religion, careful atten
tion to the form of our arguments is not sufficient to ensure that
our conclusions are true. We make tacit assumptions, which
do not always hold; we have often to rely on but slight
probabilities. Formal logical rules cannot afford us a certain
guarantee that our arguments are conclusive, but a keen
awareness of them, combined with the desire to reason cor
rectly, undoubtedly helps us to detect fallacies and to put
the rules we have learnt into practice.

It is customary in elementary textbooks on logic to include
a chapter (sometimes more than one) on fallacies. We shall
content ourselves with a brief indication of the commonest
kinds of fallacy, and shall make no attempt to classify
them.*

To commit a fallacy is to break one of the rules of logic
which are regulative of sound reasoning. An argument in
which one (or more) of these rules is broken is said to be
fallacious. In learning the rules we must have also learnt the
fallacy that arises from their violation. It will suffice here to
remind the reader of formal fallacies due to violation of the
rules of immediate inference and the syllogism. These may be
briefly listed as follows: (i) the fallacy of wrong distribution,
e.g. by simple conversion of an A proposition, by illicit major
or illicit minor, and the fallacy of undistributed middle term;
(2) the fallacy of affirming the consequent, and the fallacy of
denying the antecedent; (3) the so-called fallacy of four
terms , which consists in using ambiguous language so that
the term indicated by the words used in the premiss is not the
term indicated by the words used in the conclusion, or a
similar mistake with regard to the language used to indicate
the middle term.

(3) differs from (i) and (2) in the important respect that
the fallacy is due to the language used in stating the proposi
tions entering into the argument, so that, unlike the case of

* It would be a serious mistake if the student supposed that the treatment of
fallacies given here is at all adequate. In my opinion fallacies cannot profitably
be dealt with shortly; they need to be illustrated at length. Space does not
permit this, nor should it be necessary. The student ought, after studying the
preceding chapters, to be able to make out his own list. I have given many
examples of fallacious reasoning in my Thinking to Some Purpose, see especially
Chs. XII and XIII.

l62 A MODERN ELEMENTARY LOGIC

(i) and ^2), attention to the formal rules alone will not suffice
to guard us from falling into this fallacy. By the nature of the
case this fallacy cannot be illustrated briefly.*

Fallacies of irrelevant conclusion are extremely common.
A conclusion is irrelevant if it is not the conclusion we set out
to prove and does not imply it. Such a fallacy is called by
logicians ^ignoratio elenchi* (i.e. the mistake of disregarding the
opponent s contention). An example is afforded by the con
tention that post-primary education is useless because some
highly educated men and women are not good citizens. | The
e appeal to authority (called argumentum ad verecundiam) is
sometimes fallacious, namely, when a point in dispute is sup
posed to be settled by showing that some respectable person
has held the disputed view. If, however, the authority in
question is an expert in the subject and the opponent is
ignorant, the appeal to authority is justifiable. Logicians,
however, might notice that progress in logical theory was
delayed for centuries because logicians were too ready to
suppose that what Aristotle had said was both true and the
whole truth of the matter. Another form of this fallacy con
sists in trying to argue that a certain person s contention must
be false because he is a disreputable fellow. A converse error
is to credit someone s opinion on, say, theology or education,
because he (or she) is in the public eye in some other capacity
wholly unrelated to the topic, e.g. a popular novelist or a film
star. The fallacy consists in assuming a relevant connexion
between public fame in one capacity and expertness in quite
another. It does not, of course, follow that the novelist or the
film star is incompetent in these other affairs, but it must not
be taken for granted.

The fallacies of composition and division are converses of
each other: both rest upon the confusion of the collective and
the distributive use of a term or upon the confusion of an
alternative with a conjunctive proposition. Thus the extra
vagant man argues that, since he can afford to buy A, or B,
or C he can afford to buy A and B and C; the niggardly man

* For a fuller treatment see my Thinking to Some Purpose, pp. 127-38, and
also, M.LL.j Ch. II, 2-4.

f I take this example from a discussion at which. I was a participant, and also
the next example, illustrating a circular argument.

LOGICAL PRINCIPLES 163

argues that since he cannot afford to buy A and B and C he
cannot afford A, or J3, or C.

Fallacies of circular argument consist either in flatly
assuming the point at issue or in using as a premiss a proposi
tion which can itself be proved only by using the conclusion
for which it has already been used as a premiss. The arguer
goes round in a circle. For example, it is argued that higher
education is useless because it does nobody any good to study
once he has left school. The premiss simply repeats the con
clusion, but usually in a more subtle and disguised form. If
the diameter of the circle 5 is very large, the fallacy may be
hard to detect, Descartes fell into this fallacy (in a small
circle) when he argued, There cannot be a vacuum, because
if there is nothing between two bodies they must touch .

A fallacy of this sort is known as petitio principii^ i.e. begging
the question. One form of it consists in using question-begging
words, usually in the form of unpleasant epithets. As Mr
A. P. Herbert has said, give your political dog a bad name
and it may do him more harm than many sound arguments 5 .*

5. IS SYLLOGISTIC PROOF CIRCULAR?

Some logicians have contended that all deductive arguments
involve the fallacy of petitio prindpii, because the conclusion
could be deduced from the premisses only if these premisses
somehow contained the conclusion . There may be some
confusion if we use the word "contained" in this context; it
must mean that the premiss implies the conclusion. This is
certainly a condition of all valid deductive argument, but it
does not necessarily involve a circle. It is true that if ^ implies
q, p cannot be true unless q is also true; but there will be a
circular argument only if the truth of q has been used as a
premiss in establishing that p is true. That this is not neces
sarily the case will be recognized when we examine the way
in which we do use deductive arguments, and more especially
the syllogism, in order to obtain a conclusion. If Newton s
physical theories are true, then it follows that, for example,

* What a Word! p. 229. Ch. VIII of Mr Herbert s book contains many
amusing and instructive examples of this fallacy.

164 A MODERN ELEMENTARY LOGIC

a pair of double stars will revolve around their common
centre of gravity in elliptical orbits. Now this statement con
cerning the pair of double stars formed no part of the evidence
upon which Newton s physics is based. But the conclusion
can certainly be validly deduced from premisses afforded by
Newton s physics. We may know that everyone to whom a
V,C. is awarded has performed an act of conspicuous gal
lantry, and subsequently discover that A, whom we had not
supposed to be specially courageous, is a V.C. and we thence
conclude that he has performed an act of conspicuous
gallantry.

It may be objected to this last example that we cannot be
certain that the V.C. is always rightly awarded. Even if this
were true, the objection would be irrelevant. The falsity of a
premiss in no way tends to show that the argument is invalid,
still less that it commits the particular fallacy of petitio
principii. It is important to notice that universal premisses
may be accepted on the basis of evidence which is not con
clusive but has considerable weight; new cases can be sub
sumed under this universal premiss and a conclusion deduced
which certainly did not constitute part of the original
evidence.

J. S. Mill raised this question in its best-known form. He
argued that e in every syllogism, considered as an argument
to prove the conclusion, there is a petitio principii .* The point
of this contention lies in what we mean by proving the con
clusion . Mill looked at it in this way: Every X is 9 This A is an
X 9 therefore This A is a + How do we know that every X is
unless we have already used this A as part of the evidence for
establishing the generalization stated in the major premiss?
As Mill clearly saw, the answer to this question involves an
account of how we come (i) to form, (ii) to justify, empirical

* System of Logic, Bk. II, Ch. Ill, 2. Space is lacking to examine Mill s
doctrine here. It is discussed, but not very clearly, in M.I.L., Ch. XII, 3, and
to that the student may be referred. In that chapter also I have discussed the
question whether we can obtain new knowledge from syllogistic reasoning.
By far the best account of Mill s theory of the syllogism is contained in R,
Jackson s An Examination of the Deductive Logic of John Stuart Mill. The student
cannot do better than read this book, if he is interested in this problem; he must,
however, be warned that it is not an easy book to read and was not written for
the elementary student.

LOGICAL PRINCIPLES 165

generalizations. This question cannot be discussed here, but
it may be pointed out that our inferences, when they are
fruitful, are made within a context of knowledge. To prove
a proposition is to find true premisses by which it is implied.
When our premisses are factual propositions the evidence for
their truth is never conclusive, but this does not imply that
all factual generalizations are of equal value. There are
various sources of knowledge and various criteria for deter
mining what \veight may validly be attached to a conclusion
that has not been demonstrated. Mill wanted to use as
premisses only propositions that are known to be certainly true.
We never can know this when our premisses relate to matters
of fact. It is, however, a mistake to suppose that \ve must
wait until the evidence is - so to speak - all in before we can
assert a proposition and use it as a premiss for deducing
conclusions that we should not have known otherwise. We
cannot by deductive inference guarantee the material truth
of factual propositions, but we can show that conclusions
follow from such premisses and have such probative force as
belongs to the premisses themselves.

CHAPTER IX

Methodology of Science *

I. INDUCTIVE REASONING

If we were confined to deductive reasoning we should be
gravely inconvenienced. To say this is indeed to speak too
mildly. We should not be able to reach any conclusion con
cerning matters of fact that went beyond the present
testimony of our senses, or the records of our memory J .f
Generalization (i.e. going beyond the evidence) is essential
to carrying on the affairs of our daily lives; it lies at the very
foundation of all the empirical sciences. All the sciences
except logic and mathematics are empirical; they are based
upon observation, experiment and generalizations from ex
perience. Generalization from a number of observed instances
of a certain class, which are assumed not to constitute all the
instances of the class, is called Induction by simple enumera
tion . Its logical foim is: All the observed S s are P s; therefore all
S s are P s. This inference is clearly not valid, for, in inferring
from a premiss about some S s a conclusion about all S*s y there
is an illicit distribution of S. Consequently, the premiss may
be true although the conclusion is false. This is an essential
characteristic of inductive reasoning. All valid reasoning is
deductive, but it does not follow from this that inductive
reasoning is unreasonable, unworthy of a clear thinker. What
does follow is that we must find other criteria with which to
check and control our reasoning than the criteria provided
by the rules of deductive reasoning. It is far more difficult
to discover these criteria, to make them explicit, and to
formulate rules than is the case with deduction. To do so

* Within the limits of a short chapter it is impossible even to indicate all the
topics that must be included in any study of scientific method. It is essential for
students who are reading for university examinations to consult other textbooks
on scientific method. See M.LL., Pt. II; Cohen and NageL Introduction to Logic
and Scientific Method, Bk. II.

t Hume, An Enquiry Concerning Human Understanding, Sect. IV, Pt. I.

1 66

METHODOLOGY OF SCIENCE 167

constitutes one of the main problems of what is known as the
methodology of science 5 , i.e. a systematic investigation of the
logical character of the methods employed in the empirical
sciences. It must be admitted that this investigation is still in
a stage that may be described as rudimentary.

It is impossible within the limits of a single chapter to do
more than to indicate some of the chief questions that arise
in connexion with the methodology of science, and in this way
to suggest to the reader how wide is the field for study.*

Everyone makes inferences by simple enumeration. The
statement just made is itself an instance of such a mode of
inference. It is vital to simple enumeration that there should
be no conflicting evidence, that is, no instances of the class in
question which lack the characteristic which has been found
to belong to all the observed instances. A single contradictory
instance at once disproves the conclusion. Many Europeans
who have observed a few instances of the class Japanese^
and have found them all to be dark-eyed, have drawn the
conclusion: All Japanese people are dark-eyed. A single example
of a blue- or a grey-eyed Japanese would disprove this con
clusion. But it might still be reasonable to hold that the
percentage of dark-eyed people among the Japanese is very
high. It would not be very surprising to find that among a
nation, which for centuries did not intermarry with other
nations, there should be a tendency towards one colour of
eyes.

Consider the following statements:

Artists with dark hair and blue eyes almost always paint landscapes,
while short artists with dark hair and dark eyes paint figures.

Blue-eyed painters with relatively broad heads tend to figure painting,
and those with long heads to landscapes.

An exceptionally short head means artistic versatility and the ability
to paint both landscapes and figures.

Women tend more to paint figures than do men.

These statements were made in a short article in the News
Chronicle (Sept. 7, 1938). Perhaps the reader will agree with
the author of this book that the statements are surprising. If

* I have dealt at considerable length with methodological problems in
Part II ofM.LL. The student must consult some textbook about these problems,
for the account in this chapter is nothing but a sketch. He is recommended to
read also J. S. Mill, System of Logic, Introduction, Bk. Ill, Chs. I-X, XIV, XXI.

l68 A MODERN ELEMENTARY LOGIC

so, we should ask ourselves why they are surprising. Variation
in the colour of hair or eyes, in height, and in width of head
do not strike us as likely to be correlated with artistic ability
or with the sort of pictures an artist paints. Especially is this
the case with regard to colouring. If we ask why this should
be so, the answer is not far to seek. We are accustomed to
seeing hens of various colours, and cows, roses, and rabbits;
we think of colouring as an accidens of a species. That there
should be a correlation between colouring and the kind of
picture an artist is likely to paint seems hard to believe.* On
the other hand, we are not surprised to learn that a specific
glandular deficiency is correlated with a specific mental
defect, that a deficiency in vitamin G is correlated with the
disease known as scurvy. We expect the waves to dash against
the rocks after a gale has been blowing. As these illustrations
show we have found in our experience that characteristics
often go together in groups. It is for this reason that we find
class-names indispensable, e.g. artists, cows, politicians, Ameri
cans, measles. Such classes as these differ from the artificial
classes we make at will, such as square scarlet things, black-haired
archdeacons. Cows, for example, possess in common char
acteristics which differentiate them from other classes, such
as horses, buffaloes; whereas black-haired archdeacons probably
have no characteristics in common, except the colour of their
hair, which are not also possessed by other black-haired men
or by other archdeacons. We feel that being black-haired is not
a characteristic in any way relevant to the performance of
archidiaconal functions. This feeling has a respectable basis
in our past experience and in the recorded experience of
generations of men, as handed down to us in their class-
names, and in records of their observations. Such classes as
these may be called natural kinds, to adopt a name from
J. S. Mill.

The nature of induction by simple enumeration can be
stated as follows: Such and such instances of0 have the property !F;
no instances of0 lacking W have been observed] therefore, every & has

* The statements quoted from the News Chronicle are given in a report of the
conclusions reached by Dr Mostyn Lewis after four years of investigation;
his work is described as research in racial psychology 5 . The number of artists
analysed was said to be 1 3 ooo,

METHODOLOGY OF SCIENCE I&)

W. The instances of constitute a class having the properties con
noted by "W*\

Inferences of this sort belong to a very early stage of man s
thinking; without a considerable accumulation of the results
of such inferences science would be impossible. Class-names
enable us to abbreviate and to connect; it is the connexion
of properties that is essential not only to scientific thinking
but also to the ordering of our daily lives. Although some
things c just happen so 3 , we all believe that there are depend
able regularities in the world. Everyone believes that if he is
hungry and eats food, his hunger will be satisfied; that water
will quench his thirst; that fire will warm him; that heat will
melt snow and butter; that day will alternate with night.
Such beliefs as these are held with varying degrees of strength.
They may be mistaken. The thirst of fever is not quenched by
water; a dying man is not \varmed by the fire. Nevertheless,
without believing in some dependable regularities we should
not act as in fact we all do. That our expectations are some
times fulfilled shows that we have learnt that natural happen
ings can be regarded as having some kind of order; that they
are sometimes disappointed reveals our partial ignorance.

We are, then, accustomed to distinguish between occur
rences which we regard as being regularly connected and occur
rences which we consider to be only accidentally, or casually,
conjoined. Occurrences of the first type we shall call uniformi
ties^ of the second type multiformities* Simple enumeration
leads us to discover such minor uniformities as the connexion
between flames and warmth^ drinking water and quenching thirst,
being a negro and having curly black hair. The last example differs
from the first two in that it is a uniformity of co-existing
characteristics, whereas the other two are uniformities of
successive occurrences. The latter may be called causal con
nexions. For the analysis of causal connexions simple enumera
tion does not suffice.

2. CAUSAL LAWS

The earliest stage of a science consists in distinguishing multi
formities from uniformities and in recognizing in some

170 A MODERN ELEMENTARY LOGIC

multiformities characteristics relevantly connected in such a
way that uniformities of higher generality and abstractness
may be discovered. Hence, the first task of the scientist is to
describe and classify. As was suggested in the last section
everyone engages in this type of scientific activity; we pass
insensibly from common-sense knowledge through organized
common sense to knowledge that can be called strictly
scientific. There is no sudden break.* Primitive savages have
to make some effort to control their environment; certainly
knowledge gives power.

The scientist is not interested in singular statements such
as This water has just boiled, I am feeling hot now, This man is
angry, except in so far as the fact each describes can be
regarded as an instance of some type of order. The sciences
are branches of orderly knowledge: the scientist aims at seeing
the connexions between things of certain sorts, natural hap
penings (i.e. events in nature), and organizing them into
systems. The scientist takes note of the particular occurrence,
This water has just boiled, only in order to determine the con
ditions under which it has boiled, the temperature at boiling-
point, the change which occurs as it passes into steam, and so
on. "Water" now signifies a constant conjunction of characteristics,
which we call properties of water. To say this thing has such
and such a property* is a way of saying that this thing under
certain conditions behaves in such and such a way . For example:
Iron has the property of expanding with rise of temperature means
Iron expands when heated; Sugar has the property of solubility means
Sugar dissolves in fluids.

As the above examples suggest and our daily experiences
abundantly show, the way in which something behaves (e.g.
a lump of sugar, a poker) depends both upon the sort of
thing it is and the situation in which it happens to be placed.
This lump of sugar dissolves in water; this poker does not.
The poker thrust into a fire becomes hot; taken out and put

* Consider, for instance, how often the result of a complicated set of psycho
logical experiments (e.g. the formulation of practice curves ) strikes the layman
as just a statement of what everyone knows about the improvement of ability
to perform something as a result of practising. Nevertheless, the scientific
investigation prepares the way for formulating more precise and generalized
statements about human behaviour than is possible at the common-sense level.

METHODOLOGY OF SCIENCE IJI

in the fender It becomes cold again and reverts, approxi
mately, to its former condition. After frequent recurrences of
being heated and left to cool its shape gradually changes:
eventually it may be hardly recognizable as that poker. Each
of these things we recognize as an instance of what we have
called a natural kind^ i.e. a thing having characteristics of a
certain sort which make it the sort of thing it is. Whenever a
certain kind of thing is in a certain definite situation it will
exhibit certain characteristic modes of behaviour; these are
recurrent modes of change. Causal laws are the laws of these
recurrent modes of change.

The recognition that kinds of things behave characteristic
ally leads us to the discovery of causation and conditions.
Similar modes of change recur in situations that differ in
certain respects. Iron becomes red-hot in a furnace, in a fire
in a cottage, in a burning factory, in the muzzle of a cannon
when many cannon-balls have been fired. Thus shortly to
indicate widely different situations in w^hich something very
familiar to us is happening (iron becoming red-hot) will not
serve our present purpose unless we can forget what we are
familiar with. (Think, for example, of Charles Lamb s story
of the Chinaman s discovery of roast pork.) We discover that
there are occurrences to the happening of which much else
that is also happening in the same spatio-temporal situation
is irrelevant. If this were not so there could be no causal laws
and no science. The discovery of a causal law is the discovery
of what is relevant to a given mode of behaviour. It is for this
reason that the discovery of causal laws requires observation
of particular situations. It is only from observation that we
know that sugar dissolves in water and pokers become red-hot
in a fire. Thus causal laws cannot be deduced from a single
situation which is passively observed; they are discovered by
analysis of different situations in which things are brought
into relations with other things; we observe their behaviour
in varying situations. By eliminating factors present in different
situations we can discover which factors are irrelevant to a
given mode of behaviour. In the next section we shall be
concerned with methods by which causal laws can thus be
ascertained,

172 A MODERN ELEMENTARY LOGIC

It is important to distinguish causal laws from the par
ticular causal propositions which state exemplifications of the
laws. A particular causal proposition states a definite causal
occurrence happening once only. For example. This shot
through his heart caused this marts death. In asserting that his
death was caused by the shot we are asserting more than the
historical fact that two particular occurrences were con
joined. When anything is happening there are always multi
tudes of other things happening simultaneously and in close
succession. To say that the man s death was caused by the
shot must mean that whenever a bullet passes through a
man s heart there follows the cessation of the beating of his
heart, i.e. he dies. The form of such a causal law is: Whenever
an occurrence having the characteristic happens at a time t to a
thing of the kind k^ then an occurrence having the characteristic W
happens at a time t 2 to a thing of the kind k%. It may be the case
that (i) and ^are the same sort of characteristic., k and k 2
are the same thing; (iii) t : and t 2 are the same time. It is the
causal law that is fundamental, not the particular causal
proposition stating an instance of causation.

When we ask for the cause of an occurrence, e.g. the break
ing of this window, we expect an answer that would hold good
in other cases. On reflection, at least, we should agree that
whatever caused this window to break would also cause other
windows to break. But we are not always thinking at the same
level when we ask questions about the breaking of the win
dow. What broke the window? 3 is a question which would
probably be satisfied by the answer, c An air-raid , or by C A
bomb . The first answer is extremely abstract, but it does
indicate one important element in any satisfactory answer to
the question, for it cites an occurrence without which (it is
presumed) that particular window would not have been
broken when and as it was. The second answer cites an
essential factor in the particular situation. But it would
unhesitatingly be admitted that the mere presence of a bomb
in the neighbourhood would not have sufficed to do the
damage. An unexploded bomb might be harmless. A third
answer might be, c The explosion of a bomb . However, there
are (we assume) other windows in the same neighbourhood

METHODOLOGY OF SCIENCE 173

which are not broken. A fourth answer, The impact from
the blast of an exploded bomb , approaches the scientific
level of thinking. In ordinary life, What broke the window?
is probably asked at the level of thought of the first or
second answer; the last two state the circumstances more
carefully.

This example may suffice to show that c the cause of an
occurrence, A\ is an ambiguous expression. The reader
should ask himself what sort of answer would satisfy a medical
officer of health who inquires, What is the cause of this
outbreak of typhoid in my district? 5 He does not want an
answer in terms of bacilli; he knows that wherever people
have contracted typhoid a bacillus is present; his interest is in
the source that carried the bacilli; is it the water, or the milk,
or the meat, or what? But this knowledge had to be gained
by long and patient research. This involved at the beginning
a careful examination of complicated situations in which
people were ill with typhoid; their circumstances had to be
carefully noted and one type of situation compared with
another. The form of the question that controls this activity
of thinking is: What factor is present in these situations
which is such that whenever it is present typhoid occurs? 3 The
word "factor" here must not be assumed to stand for some
thing simple.

We may, then, say X causes T 5 means Given that X
happens, then ^happens . We shall see later that this is not
accurate, but it is sufficiently accurate to guide investigation
in its earliest stage. * Cause 5 and effect* are names used for
the referent and the relatum, respectively, of the causal rela
tion. This relation is asymmetrical; in certain usages of the
word "causes" it is also a many-one relation.

3. METHODS OF EXPERIMENTAL INQUIRY

J. S. Mill attempted to formulate with some precision various
methodical procedures for the purpose of ascertaining the
causes of specified phenomena (i.e. occurrences). He did not
achieve all that he believed himself to have achieved but his
methods , with certain qualifications, show the ways in

174 A MODERN ELEMENTARY LOGIC

which we must prepare the material in order to obtain an
answer to the question, What is the cause of 27 (where
T is an illustrative symbol). They have the merit of making
clear the fundamental part played by elimination in causal
inquiry. Our statements of Mill s methods must be very
brief.*

The methods rest upon two principles fundamental to the
concept of cause: (i) Nothing is the cause of an effect which
is absent when the effect occurs; (2) Nothing is the cause of
an effect which is present when the effect fails to occur. These
are acceptable to common sense; indeed, Mill s methods do
little more than organize the procedures of plain men when
they seek to find the answers to such questions as: What
makes the drawer stick? , c Why won t the car start? , c Why
is honey so scarce in this district this year?

In stating the methods we shall assume throughout that we
are searching for the cause of an occurrence, T (called by
Mill a phenomenon ). In the next section we shall notice
how large are the assumptions tacitly made as we proceed
on our investigations. Plain men always make large, tacit
assumptions.

We have to prepare our material in order to investigate
the cause of T; the two principles of causation given above
show that we shall do well to: (i) compare different situations
in which T is present; (ii) compare situations in which T
occurs with other situations in which T does not occur in
spite of similarity in various respects.

(i) The Method of Agreement. Rule: If two or more instances
of the occurrence of T have only one factor in common,
then this factor, in which alone all the instances agree, is the
cause of T.

For example, all the patients suffering from typhoid (in a
given district) may be found all to have used the same water

* I shall not state the methods in Mill s own words, mainly in order to be
briefer, but also to avoid certain mistakes in his formulation which he probably
did not notice. The student must read Mill s own account (see System of Logic,
Bk. Ill, Ch. VIII). It would also be advisable to read M.I.L., Ch. XVII,
especially 2, where a detailed example of an experimental investigation is given.
To read this section may suffice the lazy or over-worked student; others are
advised to work out for themselves, in detail, some example of an experimental
inquiry. The snippety examples often given in textbooks are of little value.

METHODOLOGY OF SCIENCE 175

supply; hence, the water is causally connected with the
patients having typhoid.

It will be noticed that the example does not fit the rule.
Picture to yourself what happens when several people - in
the same district - fall ill of typhoid (or any other disease).
They all, ex hypotkesi, live in the same neighbourhood; but
certainly some will be men, some women, some fat, some
thin, some fair-haired, some dark, some may be agricultural
labourers, some plumbers, some university students, and so
on. This and so on 5 is justified, for we can all quite easily
fill up the details. We know that some of the patients will
agree in being males, others (or some of the former) agree 5
in being labourers, others agree 5 in being fair-skinned, etc.
It is not possible to find instances in which all the circum
stances but one differ. We cannot begin to use the rule until
we have made an immense number of judgements of irrele
vance. When we have done so then we may find that only
one factor is always present in the set of instances in which
Tis present; in that case we are justified in asserting that this
factor is the cause of T. But in most cases we cannot be sure
that our judgements of irrelevance are correct; hence, we
should at the level of practical common sense, begin to look
for cases in which J*was absent even though these resembled
the former to a considerable extent. Hence, we use the next
method.*

(2) The Joint Method of Agreement and Difference. Rule: If a set
of instances of the occurrence of T have only one factor, A,
in common while several instances in which T does not occur
have distributed among them the other factors that were
present with T except A 9 then A is probably causally con
nected with T.

This method suggests that we must find a set of instances
in which T is present conjoined with a number of factors,
but in any two instances only one factor A is present in both.
These are called the positive instances. We then find a set of
instances resembling the first as much as possible but all

* Mill gives next the Method of Difference. For reasons which, I hope, will
be made clear in the text, I have put the Joint Method second. In formulating
the Rule for the Joint Method I have departed widely from Mill s formulation.
(For the reasons for this, see M.I.L., pp. 336-7.)

176 A MODERN ELEMENTARY LOGIC

agreeing in the absence of T. These are the negative instances.
Comparison between the two sets of instances shows that
when A is present T occurs, when A is absent T does not
occur. Hence, in accordance with the two fundamental prin
ciples we can conclude that A causes T or is at least con
nected with its causation.

For example, in the typhoid investigation, it may be sus
pected that the water is the source of the typhoid infection.
If all those who have typhoid have used the same water
supply, it will be a help to consider people living in the
district who have not got typhoid, and have water from
another source, and to inquire whether some of these have
meat from the same butcher as some of the typhoid patients,
and if some of the former have milk from the same dairy as
the latter. If so, then we can judge that the meat and the milk
are irrelevant factors.

This method is well adapted to such inquiries as the follow
ing: Is the direct method of teaching Latin satisfactory? Are
hasty marriages likely to end in divorce? Are limes as good
as oranges as a protection against scurvy?
(3) The Method of Difference. Rule: If an instance in which T
occurs and an instance in which T does not occur have in
common all factors except A, and A occurs only in the
instance when T occurs, then A is the effect, or the cause,
or an indispensable part of the cause of T.

This method is clearly more cogent in its conclusion than
either of the other two. The method of agreement might lead
us to conclude that two concomitant occurrences, e.g. the
sounding of a hooter in a factory and the ringing of a bell in
a school, were cause or effect of each other. People have often
assumed that a patent medicine is a cure for a disease because
of the evidence offered in unsolicited testimonials printed
in advertisements; they forget that those who were not cured
did not write to the proprietors. If we can find a negative
instance resembling the positive instance in all relevant
factors but one, then that factor is undoubtedly causally
connected with T.

It is clear that it is difficult to secure these conditions, for
it must be possible to introduce, or to withdraw, A without

METHODOLOGY OF SCIENCE 177

there being any other change than the non-recurrence of T.
If, however, we can be reasonably sure that the two instances
differ in only one re j evant respect, then the method is applic
able under certain experimental conditions* For example,
dropping a piece of blue litmus paper into an acid; it turns
red; we conclude that the acid is the cause of the change in
colour. We put sugar into a cup of tea, and it tastes different;
the sugar is the cause of the different taste.

These examples are artificially adapted to illustrate the
method; we know what examples to choose. But if we see,
from the example, how the method is used, we may be able
to put it into practice when we really are investigating an
occurrence to discover its cause and not merely talking about
someone else s investigation. Only we must be very careful
to see that our judgements of irrelevance are justified. These
remarks can be applied to all the methods, but are most
obviously illustrated by the Method of Difference.

It should be noticed that we never use the Joint Method
if we can secure the more stringent conditions required by
the Method of Difference.

(4) The Method of Concomitant Variations. Rule: If, in a com
plex situation containing both A and T, the factor Y varies
in some manner whenever A varies, then A is causally con
nected with Y.

We reason in accordance with this method when we con
clude that applying heat to a tube filled with mercury is the
cause of the rise of mercury in the tube. This method is
important in connexion with the investigation of quantita
tive variation; it requires data derived from measurement.
If we seek to examine the effects of a rise in the price of
tobacco upon the consumption of tobacco, we should be
applying a principle of concomitant variation. But the
variation would probably not be precise; there might be
many disturbing factors (e.g. as the price rises, it so hap
pens, that more people are at leisure, or there are air-raids
and people smoke more during the night) which prevent us
from being certain how much an increase in price would
decrease consumption on occasions when those factors are
absent.

178 A MODERN ELEMENTARY LOGIC

(5) The Method of Residues. Rule: If in the case of a complex
occurrence, certain factors, W 9 V, T, are known, from
previous investigations, to be the effects of C, E, H, then the
residual effect % is caused by the only other factor, A, which
is jointly present with W, F, T.

There is no good ground for considering this as an inde
pendent method. In so far as it is applicable it uses the
method of difference theoretically to establish a conclusion
that is dependent upon previous investigations.* The argu
ment is in fact deductive.

In thus summarily stating Mill s methods we have inci
dentally suggested that they suffer from grave defects if they
are to be regarded as complete methodical procedures for
establishing causal connexions. The following points should
be noted: (i) Each method presupposes that judgements
of irrelevance have been correctly made. (2) This means
that the investigator is already in a position to formulate
an hypothesis of the form: In this situation the possible
cause of T must be found among the factors A, B, C, D.
But this step is one of the most difficult, and nothing in
Mill s account of the methods shows that he recognized
either its difficulty or its importance. (3) Each method,
when it can be properly used, gives some grounds for the
conclusion that is drawn, but these grounds are far from
being conclusive.

The value of Mill s methods lies mainly in the fact that
they lay down what may be described as minimal conditions
for the investigation of causes of occurrences. By using them
with due care we eliminate factors that might seem to be
possible causes because these factors have been present when
the investigated effect was at first observed. They show that
A cannot be the cause of T unless (i) A is regularly followed
by T, (ii) A is never present when T is absent, (iii) A and T
vary together.

* The favourite example of this method Is afforded by the discovery of the
planet Neptune, as the result of calculating the orbit of Uranus from the known
effects of the known planets, and finding a discrepancy between the calculated
orbit and the observed orbit. It was suggested that another planet must be the
cause of the residual effect. This reasoning is clearly deductive (see M.I.L.,
pp. 346-7).

METHODOLOGY OF SCIENCE 179

Mill himself recognized a practical difficulty in applying
the Method of Agreement, namely, that on one occasion A
may indeed cause T but on another occasion T may be
caused by B. There can be no doubt that as cause is used
in ordinary discussion there can be such a plurality of causes.
It is well known that men die from many different causes.
That is to say the causal relation is assumed to be many-one.
For practical purposes it is certainly convenient to know
that there are many ways of encompassing the death of one s
enemy or of giving enjoyment to one s friends. But is it in
fact true that different causes result in exactly the same effect?
The procedure of a coroner s court is based upon the denial
that the causal relation is many-one; it is assumed that if the
characteristics of the effect Y (viz. this person s death) be
carefully analysed, then it will be seen that variation in the
complex situation described by "2"" is in one-one correlation
with variation in the complex situation described by "A".
This assumption is plausible. At the same time it must be
admitted that it accords ill with our common-sense use of the
concept cause. If we admit that the plurality of causes is pos
sible, then we cannot agree with Mill that it affects only his
Method of Agreement. It is true that a rigorous use of the
Method of Difference can assure us that in the given case no
other cause was possible; but this does not suffice to show
that in another situation the effect T may not be the result
of quite other factors.

As the admission that there may be a plurality of causes
shows, Mill s methods are insufficiently analytic. He did not
sufficiently recognize the truth of Francis Bacon s remark:
the force of the negative instance is greater . If we have
reason to suppose that A is the cause of T, it is vitally
important that we should seek for instances of the occurrence
of T conjoined with A in which the factors other than A are
varied as much as possible.* Repetition of instances of the
conjunction of A with T have little value unless these
instances vary widely among themselves.

* J. M. Keynes has laid stress upon the importance of such variation: not
resemblance but unlikeness is what we must look for. See A Treatise on Prob
ability, Chs. XIX-XX, and cf. M./.L., Ch. XTV, 3.

l8o A MODERN ELEMENTARY LOGIC

4. THE NATURE AND IMPORTANCE OF
HYPOTHESIS

If we are interested in the process whereby scientific dis
coveries are made, we can hardly over-emphasize the part
played by the formulation and development of hypotheses.
An hypothesis is a proposition suggested by the evidence
available to establish the conclusion but insufficient to demon
strate the conclusion. Hypotheses are formed when we seek
to ask why something has happened. Why, for instance, are
booms followed by slumps? (If booms are not followed by
slumps, cadit quaestio.} Why does water not run uphill and
yet it rises in a pump? Why does water not rise in a pump
to a height greater than thirty-three feet at sea-level? Why
are some people so much afflicted by nightmares?

The question asked by "Why 53 may require an answer in
terms of human, or divine, purpose, or it may require an
answer in terms of what previously happened on account of
which this (which initiated the question) happened. The
first is a demand for a teleological explanation, the second
demands how things are connected independently of any
one s purposes and desires. This is often called scientific
explanation; it would be a mistake, however, to suppose that
scientific explanations cannot involve reference to purposes;
they must involve such reference when actions, as distinct
from natural occurrences, are involved.

It must be noticed that an intelligent question beginning
with Why or with How cannot be asked except on the basis
of some knowledge about the situation which prompted the
question. It cannot be answered without considerably more
knowledge than the questioner possessed. The question and
answer may be formulated by the same person; in that case,
he is first seeking for knowledge, later he is in possession of
the knowledge sought - assuming that he has answered the
question correctly. Even a cursory acquaintance with the
history of a scientific discovery suffices to show how indis
pensable is a background of relevant knowledge.* In this
short sketch we take the possession of relevant knowledge for

* See M.I.L., Ch. XIII, 3-4; XVI, 1,2.

METHODOLOGY OF SCIENCE l8l

granted, but it must not be forgotten that we have done so.

The method of using an hypothesis to answer a question
is commonly regarded as consisting of four steps: (i) Aware
ness of a complex familiar situation in which something is
felt to call for explanation. (2) Formulation of an hypothesis;
i.e. the statement of a proposition which connects the unex
plained occurrence with data derived from previous obser
vations, the proposition being such that, if it is true, then the
given occurrence, together with other occurrences not yet
observed, could be deduced. (3) Deduction from the hypo
thesis of its consequences; these consequences must include
both the given occurrence and other supposed occurrences
which will happen provided the proposition is true. (4)
Testing the hypothesis by appeal to observable occurrences.
This last stage is usually called the Verification 3 of the hypo
thesis. The name is not very fortunate, since what is verified
is that the consequences take place., rather than that the original
proposition - the hypothesis - is true. Various hypotheses
may be consistent with the happening of the occurrence
which is being investigated.

To state a simple example, we will suppose that someone
asks: Why is there no meat in the meat-safe, for I put the
week s ration there this morning? First hypothesis (A x ) : Per
haps someone got in and has stolen it. If so, then you must
have seen someone pass the window for [the meat-safe is in
the back garden and no one could get over the back-garden
wall; the only approach is by a path down the side of the
house; anyone who goes there passes the window of the front
sitting-room]. But you did not see anyone pass the window;
we shall conclude that no one did, as you always notice a
shadow - at this time of day - falling across the window.
Perhaps the maid has put the meat in the scullery (A 2 ). If so,
then it will be there still; but it is not there. Perhaps a dog
jumped the wall and stole the joint (A 3 ). If so, there will be
scrabbles on the wood of the safe; there are marks of
scrabbling; therefore, a dog got in and stole the joint.*

* This records an instance which actually happened. The remarks in brackets
give information to the reader that was taken for granted by the occupiers of
the house.

l82 A MODERN ELEMENTARY LOGIC

The form of this reasoning is as follows: If h then p(A)
(where "A" is shorthand for the alleged occurrence deduced
from h^ and "p(A)" is shorthand for the proposition that the
occurrence took place. Analogous shorthand symbols are
used throughout). But not-^(^). If A 2 , then p(B); but
not-p(B). If A 3 , then p(C); but p(C]. The rules of formal
deduction show that if A x implies p(A), then not-p(A] implies
h^ Hence, the truth of not p(A] (i.e. the falsity of p(A})
justifies us in asserting that h^ is false. The formal procedure
is the same with regard to A 2 . But in the case of h z the
position is different; here we have: If A 3 , then^(C); but/>(C);
therefore A 3 . This commits the fallacy of the consequent. We
can, therefore, accept h z only on condition that A ls A 2 , A 3
together exhaust the possible hypotheses; we should then
have the following valid argument: (where "p(0}" is short
hand for the proposition The meat has disappeared}.

(i) Ifp(0) 9 then either A x or A 2 or A 3 ;
h l or k 2 or h 3 = A x and A 2 and 7z 3 is false.

(ii) If A!, then/>(^4); but j)(4) is false; /. A x is false.
(iii) If A 25 then^(-B); but _p(JB) is false; . . A 2 is false.
(iv) Ifp(0) 9 then either A x or A 2 or A 3 ; but not h^ or A 2 ;

It may be said that in the investigation of matters of fact
it is never possible to assert a proposition of the form (i)
above; we cannot be sure that we have exhausted all possible
hypotheses. Thus the assertion that our hypothesis is Verified
by the consequences does not amount to the assertion that
the hypothesis is certainly true\ rather we should say that the
deduced consequences are verified and the hypothesis is
confirmed.

When the deduced consequences are not verified (i.e. the
proposition stating that such and such an occurrence has happened
is false), it is by no means always the case that the original
hypothesis is totally discredited; it may be that it can be
emended in such a way that the original deduced conse
quence is no longer implied.

A successful prediction is often regarded as of very great
importance in establishing an hypothesis. It is, however, easy

METHODOLOGY OF SCIENCE 183

to overestimate its significance, as we shall see if we remember
that more than one hypothesis may be consistent with the
facts. Those who rely on the predictions of newspaper
astrologists forget this; they seem to think that the only
hypothesis consistent with the successful prediction is that
the astrologer obtained his information from the stars.

5. SYSTEMATIZATION IN SCIENCE

Although a science begins with such piecemeal discoveries
as, e.g. Water rises in a pump, It becomes more difficult to breathe
as one ascends higher and higher up a mountain, it does not
advance very far until sets of discoveries (established by
means of the method discussed in the last section) can be
connected together. The discovery that air has weight con
nected the rise of mercury in a barometer, the rise of water
in a pump, the difference in the boiling-point of water at
sea-level and on the top of Snowdon, etc. To be brief,
Newton s great physical synthesis connected together the
fall of unsupported bodies^ the phenomena of high and low
tides and of neap and spring tides, the movements of the
moon, the revolution of the planets round the sun and . . .,
for the list could be much extended. Discoveries made in one
small department of one branch of science are relevantly
connected with those made in another department of the
same branch; discoveries in one branch of science (say,
chemistry) are connected with discoveries in another branch
of science (say, physiology) ; the outcome may be a body of
specialized knowledge reaching the dignity of a new branch
of science (say, biochemistry). The metaphor of "branches"
- if not pressed - is significant, for it suggests that the various
sciences tend to develop and grow together, so that discoveries
in one reinforce discoveries in another. This is all too brief,
and, if it be forgotten that we are here engaged merely in
making comments on a vast subject, what has just been said
may be downright misleading. The point to be insisted upon
is that, with many qualifications, we can assert that natural
occurrences are interconnected in such a way that, for
example, a thorough understanding of how it is that sap

184 A MODERN ELEMENTARY LOGIC

rises in trees would involve taking note of the law of gravita
tion and the behaviour of living matter.

We might put the point in this way: On what grounds are
we justified in believing that water runs downhill? That we do
believe it is not questioned. The child s answer is: Because
water always does run downhill 5 ; a more advanced answer is,
Because water seeks its own level ; another answer is,
Because water is a very good example of a fluid. Each of
these answers does something to connect the behaviour of
water with something else; even the child s answer asserts
that this water running down this hill is not to be regarded
as an isolated phenomenon. Perhaps the answer we should
give today is: That water runs downhill follows from the prin
ciples of mechanics. Accordingly, either there is something
wrong with the principles of mechanics or water runs down
hill. To dispute the principles of mechanics is to upset a
whole domain of ordered knowledge. This may have to be
done; to some extent it has been done as the result of the
work of Einstein, but this work would not have been accept
able unless it had fulfilled two conditions: (i) the new hypo
thesis is in accordance with all the observed occurrences
including those hitherto accounted for satisfactorily by the
Newtonian scheme and those discrepant with it; (2) the new
hypothesis does itself offer fruitful deductions guiding sub
sequent experimental inquiry. It is well known that Ein
stein s theory satisfies these conditions.

The method of science is sometimes called hypothetico-
deductive. There is some merit in this appellation. Einstein has
said: Theory is compelled to pass more and more from the
inductive to the deductive method, even though the most
important demand to be made of every scientific theory will
always remain that it must fit the facts. The more advanced
a theory is the more its exposition assumes deductive form; in
consequence, an advanced science is an immense system of
interconnected facts; new discoveries are fitted in to the
system, even if at times the system must be modified to
accommodate them. Our trust in any one generalization
(which may have begun with the precarious and childish
method of simple enumeration) is in no small part dependent

METHODOLOGY OF SCIENCE 185

upon our trust in the system as a whole. We have trust in
the fidelity- of the system to the observable occurrences
because we find that it works it guides us to further experi
mental observations; it connects what had hitherto been
isolated and thus unexplained; finally, it shows us what are
the right questions to ask if we seek to understand the world
in which we live. To understand a statement is to know what
implied it and what it implies. Little though the activities
of men can be thought to be in harmony with Aristotle s
belief in homo sapiens^ we understand how he could have
entertained it when we reflect upon the fact that man alone
(so far as we know) asks questions - occasionally - solely in
order that he may know the intellectual satisfaction of having
his questions answered.

A P P E N Y D I X

Containing References for
Further Reading, Exercises,, and a Key

REFERENCES FOR FURTHER READING,
AND EXERCISES

Abbreviations used in citing the titles of books to which frequent
reference is made will be given in square brackets after the first citation
of the book in question. The references have been kept to a minimum;
the student who consults any of these books will find further guidance
in them for his future reading. Those references marked with an asterisk
may be regarded as alternative readings on the same topic; those
marked with a dagger are intended for more advanced students.

CHAPTER I

REFERENCES

Stebbing, L. Susan. A Modern Introduction to Logic (Methuen: 2nd or
3 rd edition only). Chs. I, II, XXIV, i. [Jlf.7.LJ

* Cohen, M. R. and Nagel, Ernest. An Introduction to Logic and Scientific
Method (Geo. Roudedge & Sons Ltd, 1934). Ch. I. [C. and N.]

* Eaton, R. M. General Logic (New York: Charles Scribner s Sons,
1931). Pt. I, i, 2, 8. [Eaton.}

Keynes, John Neville. Studies and Exercises in Formal Logic (Macmillan:
4th ed., 1906). Introduction. [F.LJ]

Chapman, F. M. and Henle, P. The Fundamentals of Logic (Charles
Scribner s Sons, 1933). Pt. I 3 Ch. I.

t Joseph, H. W. B. An Introduction to Logic (Oxford University Press:
2nd ed. 3 1916). Ch. I.

EXERCISES

1. In the case of each of the following statements, find two statements
from which the given statement would follow: (a) Some taxes are un
economic, (b) Mr Crab is a bore, (c) Cora ripens in the sun. (d) Some
monkeys can be taught tricks.

2. Find an example of argumentative discussion (taken from any book
or newspaper) ; set out the conclusion the writer seeks to establish, and
specify the premisses given in support of it.

3. Distinguish between validity and truth.

187

l88 APPENDIX

CHAPTER II

REFERENCES

* M.I.L. Chs. IV, V.
*C. andji. Chs. II, III.
Eaton. Pt. I, Ch. V, i.
f FJL. Pt. II, Chs. Ill, IV.

EXERCISES

4. What is the purpose of restating categorical propositions in regular
A 9 E, I 3 forms? Try to restate each of the following statements in one
(or more) of these forms; indicate whether anything has been lost in the
restatement:

(1) Only metals are good conductors of heat.

(2) He that fights and runs away may live to fight another day.

(3) Sometimes all our efforts fail.

(4) Who drives fat oxen should himself be fat.

(5) No admittance except on business.

(6) Man alone repines.

(7) A man may smile and smile and be a villain.

(8) To be great is to be misunderstood.

(9) Nothing ever becomes real till it is experienced.
(10) He who praises everybody praises nobody.

(i i) Where you see a Whig you see a rascal.

(12) Popular preachers are not always sound reasoners.

(13) All that glisters is not gold.

(14) To the pure all things are pure.

(15) Humour is not given to all great teachers.

5. Construct a set of propositions to illustrate the square of opposi
tion. Which terms in these propositions are distributed, and which are
undistributed?

6. Determine the logical relation holding between each pair of the
following propositions:*

(1) All cruel actions are unjustifiable.

(2) All unjustifiable actions are cruel.

(3) Some justifiable actions are not cruel.

(4) No justifiable actions are cruel.

(5) Some justifiable actions are cruel.

(6) Some cruel actions are not unjustifiable.

(7) Some actions that are not cruel are not unjustifiable.

7. Give the obverse and the contrapositive (where possible) of the
following: (i) All are not saints that go to Church, (ii) Only small
children love tin soldiers, (iii) No shrimps are obtainable today.

8. Restate the following propositions in such a way that without being
weakened they may all have the same subject-term and the same

* In answering questions of this kind the student will probably find it helpful
to formulate the propositions in various ways (e.g. obverse, etc.) so that
equivalent and non-equivalent propositions can be easily recognized by means
of immediate inference.

APPENDIX 189

predicate-term: (i} All F is not-C; (ii) Some not-F is C; ^iii , JVb not-F is C;
(iv) Some F is C.

9. Granted that Some sailors are patriotic is true ? show which of the
following statements may be inferred to be true, which false, and w lich
doubtful:

(1) Some who are not sailors are unpatriotic.

(2) No patriotic people are sailors.

(3) Some patriotic people are not other than sailors.

(4) No unpatriotic people are sailors.

(5) Some sailors are not unpatriotic.

10. Give the contradictory and a contrary of: No man can be a politician
except he be first a historian or a traveller.*

1 1 . Show that Some aeroplanes are bi-planes is the subimplicant of the
contradictory of the subimplicant of the contrary of the contradictory of
the subcontrary of itself.

12. Consider whether there are any ambiguities in the following state
ments: (i) All are not just that seem so. (ii) Some of the soldiers were not
afraid, (iii) All the fish weighed 4 Ib. Assign the contradictory of each
of the interpretations you give.

CHAPTER III

REFERENCES

M.I.L. Chs. V, VII.

C. andN. Ch. II, 3. Ch. Ill, 3, 5.

f F.L. Pt. II, Chs. IX, X.

f Johnson, W. E. Logic. Pt. I, Ch. II.

Joseph, H. W. B. An Introduction to Logic. Ch. IX.

EXERCISES

13. Give the contradictory of: *Man is born free; and everywhere he
is in chains.*

14. In the case of each of the following propositions give three other
composite propositions equivalent to the original:

(i) If wages are increased, prices will rise.

(ii) Either the child was badly taught or he is exceptionally stupid,
(iii) You cannot both eat your cake and have it.
(iv) If a man will begin with certainties, he shall end in doubts.

(v) Either we are not responsible for our actions or our actions are

within our own power,
(vi) If C is D, then Q.is not R.

15. Suppose you wish to choose a tutor who will teach you enough
logic to pass your examination. You have the following evidence with
regard to four tutors, -4, B, C, D:

(a) Either a student is not taught by A or he fails to pass.

(b) Unless a student is not taught by B, he fails to pass.

(c) Only if a student is not taught by C does he not pass.

( d) Only if a student is not taught by D does he pass.

How can you decide which tutor to select?

APPENDIX

1 6. Construct an argument in the Modus tollendo tollens:, obtain the
same conclusion from equivalent premisses but stated in (i) modus tollendo
ponens. (ii) modus ponendo tollens^ (iii) modus ponendo ponens.

17. Exhibit the logical structure of the following arguments, adding
any premisses that may be required; determine in each case whether the
argument is valid:

(i) If Abraham Lincoln were alive today, a just and reasonable
peace would be made. But, since he is dead, a just and reasonable
peace will not be made.

(ii) "If the law supposes that", said Mr Bumble, "the law is an
ass - a idiot".

(iii) Either the Pythagorean theorem in geometry is true or it is not
worth the labour of studying it; but it is true; therefore, it is not
worth while to study it.*

(iv) Trices only fall if there is over-production. But if there is not
over-production, factories close; if factories close, the number of
the unemployed increases. If more people are unemployed, there
is dissatisfaction and social unrest. Consequently, if prices fall,
there is dissatisfaction and social unrest.

(v) This author is certainly muddle-headed; for, if I follow his
argument, he is certainly muddled, and if I do not follow it,
he is obscure hi his statement of the argument.

(vi) If your uncle is rich, you will not be afraid of asking him for
a loan. But you are not afraid. Consequently, I conclude that
your uncle is rich.

(vii) It is an undertaking of some degree of delicacy to examine into
the cause of public disorders. If a man happens not to succeed
in such an inquiry, he will be thought weak and visionary; if he
touches the true grievance, there is a danger that he may come
near to persons of weight and consequence, who will rather be
exasperated at the discovery of their errors, than thankful for the
occasion of correcting them. If he should be obliged to blame
the favourites of the people, he will be considered as the tool of
power; if he censures those in power, he will be looked on as an
instrument of faction. But in all exertions of duty something has
to be hazarded (Burke).

1 8. Select from the following, those statements which are equivalent:

(1) Where you see a Whig you see a rascal.

(2) If you see a Whig you don t see a rascal.

(3) If you see a Whig you see a rascal.

(4) Either you see a rascal or you don t see a Whig.

(5) Only if you see a rascal do you see a Whig.

(6) Only if you do not see a rascal do you not see a Whig.

(7) Unless you see a rascal you do not see a Whig.

19. Give the contradictory and a contrary of each of the following:

(1) If poetry comes not as naturally as leaves to a tree, it had better
not come at all.

(2) I am certain that you are wrong.

(3) All endogens are all parallel-leaved plants.

APPENDIX igi

CHAPTER IV

REFERENCES

ALI.L. Chs, VI, VII.

C. andN. Ch. IV.

f F.L. Pt. Ill, Chs. I-VIII.

f H. W. B. Joseph. An Introduction to Logic. Chs. XII-XVL

EXERCISES

20. State the rules that are necessary and sufficient to ensure the
validity of a categorical syllogism. Prom directly from these rules:*

(1) That the mood EIO is valid, and the mood IEO invalid, in every
figure.

(2) That O cannot be a premiss in figure I, a major premiss in figure
II, a minor premiss in figure III, nor a premiss in figure IV.

(3) That, if the major term is predicate in its own premiss, the minor
premiss cannot be negative.

(4) That an A proposition can be proved only in figure i.

(5) That if the middle term is distributed in both premisses, the con
clusion must be particular.

21. Show, by means of the general rules of the syllogism, in how many
ways it is possible to prove a proposition of the form SeP.

22. (i) All intelligent people are competent,
(ii) No unintelligent people are reliable.

(iii) Not all competent people are unreliable,
(iv) Some unreliable people are not competent.
Determine whether (iii) and (iv) are implied by (i) and (ii) jointly.

23. Determine the mood and figure of a valid syllogism which con
forms to the conditions: (i) the major premiss is affirmative; (ii) the major
term is distributed both in the conclusion and in its own premiss; (iii)
the minor term is undistributed in both premiss and conclusion.

24. Construct a significant syllogism in Bocardo\ restate the argument
so as to obtain an equivalent conclusion from equivalent premisses in
the mood Darii.

25. Given the special rules of figure I, show by reductio per impossibile
that, in figure II the conclusion must be negative, and in figure III the
conclusion must be particular.

26. Construct a valid Sorites consisting of five propositions and having
Some young men are not shy in advising their elders for its conclusion. Name
the form of the Sorites you give.

27. If C is a sign of the presence of A, and B is likewise a sign of Z>,
and if B and C never co-exist, can it be validly inferred that A and D
may sometimes not be found together?

* It should be carefully noted that the proof asked for is to be a deduction
from the general rules of the syllogism, not from the special rules for^each figure;
thus (i) cannot be proved by examining each of the four figures in turn; it is
necessary to show that the validity of EIO and the invalidity of IEO follows
directly from the general rules irrespective of the position of the terms, i.e. with
out reference to the special rules.

APPENDIX

28. Examine the validity of the following arguments, supplying any
premiss that is implicit:

(1) His generosity might have been inferred from his humanity, for
all generous people are humane.

(2) Of course the U.S.A. is an Anglo-Saxon nation, in spite of its
mixture of races; for all Anglo-Saxon nations are devoted to freedom,
and devotion to freedom is nowhere more evident than in America.*

(3) *I cannot help you to do this because I am not able to do it
myself.

(4) Only sensitive people resent criticism and, since only sensitive
people are musical, it follows that all musical people resent criticism.

(5) Two bodies must touch each other when there is nothing between
them; consequently a vacuum is impossible. 5

(6) You cannot consistently maintain that no one who does not work
ought to have money that he has not earned, for you hold that a man
should be permitted to leave his sons and daughters his whole fortune,
and in many cases this suffices to maintain them in idleness for the rest
of their lives.

(7) He cannot maintain that all wars are unjustifiable, since he denies
that persecution is justifiable, and it is sometimes not possible to prevent
persecution except by making war upon the persecutors.

(8) Only pacifists are Quakers, but not all pacifists are Quakers; only
Socialists - and not all of them - are Marxists; among both pacifists and
Socialists you will find those who support the raising of the school-
leaving age. Hence we can conclude that no Quakers are Marxists, but
not all non-Marxists are Quakers; further, some of those who are not
Quakers and also some who are not Marxists support the raising of the
school-leaving age.

(9) If you deny that industry and intelligence are incompatible, and
I deny that they are inseparable, we can nevertheless agree that some
industrious people are intelligent/

(10) The country needs clever politicians; a clever politician is one
who knows how to control his party-machine; anyone who knows how to
control his party-machine is apt to engage in shady practices. Hence,
we conclude that the country needs those who are apt to engage in
shady practices.

(i i) Whatever is desired by all is desirable; all men desire their own
happiness; therefore every man desires the happiness of all men, so
universal happiness is desirable.

(12) Some fashionable views are not true, for no fashionable views
are subtle and some true views are subtle.

(13) To be wealthy is not to be healthy; not to be healthy is to be
miserable; therefore, to be wealthy is to be miserable.

(14) It is impossible to prove that industry can flourish without com
petition unless you can also prove that the lack of any competition does
not lead to decreased effort on the part of the workers; for it is certainly
the case that when the efforts of the workers decrease, industry does not
flourish.

(15) Most of those present at the meeting were in favour of opening
a second front now, and most of those present were Conservatives;
hence, some Conservatives are in favour of opening a second front* now.

APPENDIX 193

CHAPTER V

REFERENCES

M.I.L. Ch. I, i. Ch. IV, 5, 6. Cb. VII, 5. Ch. IX, i. Ch. X,
i, 2, 3.

*C. and*. Ch. VI, i, a, 3.

* Eaton. Pt, I, Ch. VIII.

Chapman, F. M. and Henle, P. The Fundamentals of Logic. Chs. Ill,
VII,

Johnson, W. E. Logic. Pt. I, Chs. VIII, X, XIII.

Langer, S. K. An Introduction to Symbolic Logic (Geo. Allen & Unwin,
1937). Chs. I, II.

| Russell, Bertrand. An Introduction to Mathematical Philosophy (Geo. Allen
& Unwin, 1920). Ch. V.

EXERCISES

29. Construct a significant example of each of the relations listed
below, and assign the logical properties of the relation in each case:
greater than, twin of, ancestor of, married to, factor of, exactly matches
in coloiiTj aunt of, in debt to, imply 3 lover of.

30. Give examples of: (i) many-one relation- (ii) one-one relation;
(iii) relative product. Construct three propositions each of which con
tains the converse of one of your examples.

31. What is a class? How can there be (i) empty classes, (ii) single-
membered classes?

32. Formulate the following propositions existentially:

(1) Some Italians are not Fascists.

(2) None but the brave deserve the fair.

(3) No butterflies live long.

(4) Only legal experts can draft an act of parliament.

33. S A11 deductive inference depends on the logical properties of
relations. Discuss.

34. Discuss the validity of the inference of Some not-S is not-P from
the premiss All S is P. Illustrate your answer by using the proposition
All far-sighted statesmen have failed to find a means of abolishing war.

35. Given that universal propositions are existentially negative and
particular propositions are existentially affirmative, determine the
validity of the following inferences: (i) SaP /. PoS; (ii) MaP and SaM,

/. SiP; (iii) PeS :. SiP.

CHAPTER VI

REFERENCES

M.I.L. Chs. Ill, XXII.

C.andN. Ch. XII.

Eaton, Pt. II, Chs. VI, VII.

Joseph, H. W. B. An Introduction to Logic. Chs. IV, V, VI.

j Russell, Bertrand. An Introduction to Mathematical Philosophy. Ch. XVI.

Mill, John Stuart. A System of Logic. Chs. II, VIII.

194 APPENDIX

EXERCISES

36. Distinguish between extension and denotation, giving examples.

37. With regard to each of the following terms cite not less than six and
not more than ten subclasses: plane figure, symbol, vehicle, university student,
metal.

38. \Vhat do you understand by "connotation"? How would you
answer the question asked by a schoolboy: What is "rationalize"?

39. Assign the various predicables for (i) aviator, (ii) sonnet, (iii)
schooner, (iv) paviour, (v) communique.

40. Which of the following definitions seem to you to be faulty? For
what reason? Suggest an emended definition in any two of the examples:

(i) A square is a rectangle; (2) spinster means one who spins cotton;
(3) negligence is want of proper care; (4) twinkle means scintillate; (5)
a soldier is a man of military skill serving in the army.

41. Illustrate, by reference to the term ship, what is meant by the
inverse variation of extension and connotation.

42. Arrange the following in an orderly manner: lyric, novel, literary
work of art, sonnet, epic poem, comedy, narrative prose work, historical
work, scientific treatise, ode, Origin of Species, Ly ell s Principles of Geology,
fiction, triolet, Moll Flanders, drama, Alice in Wonderland.

43. How do you account for the omission of ordinary proper names
from a dictionary? Discuss the logical characteristics of such names.

CHAPTER VII

REFERENCES

MJ.L. Chs. II, VIII, IX, i, 2. Ch. X, 5.

C. andN. Gh. VI, 4.

f Johnson, W. E. Logic. Pt, II, Ch. III.

Langer, S. K. An Introduction to Symbolic Logic. Ch. II, 3-6.

| Russell, Bertrand. An Introduction to Mathematical Philosophy. Ch. XV.

EXERCISES

44. Explain the use of illustrative symbols, giving examples. Distin
guish illustrative symbols from variables.

45. Explain and illustrate: propositional form, variable proposition,
values of a function, range of significance of a propositional form.

46. Define " ID ", and give examples.

47. What is an extensions! interpretation of logical relations?

CHAPTER VIII

REFERENCES

M.I.L. Ch. XXIV. Ch. IX, 4. Chs. X, XI, XII.

C.and N. Chs. VII, IX.

Eaton. Pt. II, Ch. V, 5, 6.

Johnson, W. E. Logic. Pt. I, Chs. Ill, IV.

APPENDIX 195

EXERCISES

48. What is meant by "the laws of thought"? Comment upon the
statement, Logic is the science which investigates the general principles of
valid thought*, with special reference to the words italicized.

49. Indicate the kind of evidence required to establish each of the
following statements:

(1) There is a cathedral in Salisbury.

(2) A square has four right angles.

(3) Iron expands when heated.

(4) Jack is taller than Tim implies Tim is shorter than Jack.

(5) Red roses are red.

(6) There are mountains on the other side of the moon.

(7) Light waves are electromagnetic.

(8) There are three feet in a yard.

(9) A married man has a wife.

(10) No two people have the same finger-prints.

50. What is a circular proof?

51. Distinguish between persuasion and proof.

52. Give examples of (i) contingent, (ii) tautological, (iii) self-contra
dictory statements.

53. How would you define "logic"?

CHAPTER IX

REFERENCES

M.LL. Pt. II.
C.andN. Ghs. X-XIV.

Joseph, H. W. B. Introduction to Logic. Chs. XVIII-XXIV.
MiU, J. S. A System of Logic. Introd. Bk. II, Ch. I, Bk. Ill, Chs.
I-XIV, XXI.

KEY TO THE EXERCISES

Full answers are given only to those questions which admit of a definitive
solution.

1. (a) All taxes which are costly to collect are uneconomic; Some taxes
are costly to collect, (b) All people whose conversation is mainly about
their own exploits are bores; Mr Grab s conversation is mainly about his
own exploits, (c) All cereals ripen in the sun; Corn is a cereal, (d) Any
animal that is attentive and imitative can be taught tricks; Some
monkeys are attentive and imitative.

Note. - These are examples of premisses fulfilling the condition speci
fied in the question. It should be noticed that, in every case, the terms
in the conclusion each appear in one premiss.

2. See Ch. I, 2.

3. See Ch. I, 3.

4. See Ch. II, 3. The purpose of restating any proposition is to
exhibit clearly the way in which its constituent elements are put together;
if we can find certain formulations that can be taken as standard forms,
we can more easily see how different statements are related logically to
one another. So-called reduction to logical form is a matter of con
venience, but convenience is important; we need aid in deciding what
inferences are permissible. Thus, for example, 8# 2 = 3* 8 is usually
rewritten 8# 2 $x + 8 = o in order to bring out its resemblance to
ax* -f bx -h c = o, which is the standard form.

(1) All good conductors of heat are metals. (This statement can .also
be restated No non-metals are good conductors of heat.}

(2) All who fight and run away are among those who may live to
fight another day. (This restatement has less force, since the significance
of the verb may is weakened when it is used in an adjectival sentence.)

(3) Some failures are failures of all our efforts.

(4) All who drive fat oxen are properly themselves fat. (In replacing
should be by are properly the significance is weakened.)

(5) All who are allowed to be admitted are those on business.

(6) No non-hum an creature is one who repines. (Alternatively, All
who repine are human and none who are non-human are those who repine.)

(7) Some who smile and smile are villains. (This restatement loses
the implication that smiling and villainy seem to be incompatible but are
not so in fact.)

(8) All who are great are misunderstood. (This restatement fails to
bring out the implication that being misunderstood is a consequence of
being great. In the traditional reformulation of propositions the A, E, /,
forms are interpreted as existentially affirmative, i.e. it is assumed that
classes determined by the subject- and predicate-terms have members.
The statement All S*s are P s may be asserted as the result of an examina
tion of the members of the class ; this leaves open the possibility that
every member of S happens to be also a member of P even though there
is no essential connexion between S and P. See page 25 above.)

196

KEY TO THE EXERCISES

(9) Nothing not-experienced is real. (Alternatively, Ail that is real is
experienced.)

(10) All who praise everybody are praisers of nobody. (See comment
on (8;.)

(n) All Whigs are rascals. (This is much less emphatic than the
original. See further, exercise 18.)

(12) Some popular preachers are not sound reasoners.

(13) Some glistering things are not gold. (Note that All ... not
. . /, in the example, is used so as to distribute gold things but to leave
glistering things undistributed.)

(14) All who are pure are those who find all things pure. (Alterna
tively, All things are pure to those who are pure.)

(15) Some great teachers are not endowed with humour.

5. (i) All sea-gulls are greedy; (ii) No sea-gulls are greedy; (iii) Some
sea-gulls are greedy; (iv) Some sea-gulls are not greedy.

(i) and (iv) are contradictories; (ii) and (iii) are contradictories;
(i) and (ii) are contraries; (iii) and (iv) are subcontraries; (i) is super-
implicant to (iii), (ii) is superimplicant to (iv} 5 whilst (iii) is subimplicant
to (i) and (iv) subimplicant to (ii); (iii) and (iv) are subcontraries.
Hence the four given propositions illustrate the square (or figure) of
opposition.

6. (Note. - The answer given here provides an example of the pro
cedure recommended in the note added to the question. It should,
however, be observed that the question is fully answered once the name
of the logical relation in each case has been assigned.)

Let C, U, C, O represent cruel actions, unjustifiable actions, and their
contradictories, in accordance with the usual convention. We shall first
write down each proposition with, on the same line, some immediate
inferences from it; we then set out the full answer to the question as
stated:

(1) CaU == CeO (obv.) = OeC (com. ofobv.}.

(2) UaC = UeC (obv.) == Cell (com. ofobv.) == CaO (obv. of com.

ofobv.).

(3) OoC s= OiC (obv.) E= CiO (com. ofobv.} = CoU (obv. of com.

of obv.} .

(4) UeC === OaC (obv.) -> CiO (com. ofobv.).

(5) OiC ss CiO (com.} = CoU (obv. of com.).

(6) CoU = CiO (obv.) 35 OiC (com. ofobv.) == OoC (obv. of com.

ofobv.).

(7) CoU 53 CiO (obv.) = OiC (com. ofobv.) == OoC (obv. of com.

of obv.}.

i and 2 independent (complementary); 3 subimplicant to i; i and 4
equivalent; i and 5 contradictories; i and 6 contradictories; i super
implicant to 7 (inverse); 2 superimplicant to 3 (inverse); 2 and 4
independent; 2 and 5 independent; 2 and 6 independent (contra-
complementary); 2 superimplicant to 7 (inverse); 3 subimplicant to 4;
3 and 5 subcontraries; 3 and 6 subcontraries; 3 and 7 equivalent; 4 and

5 contradictories; 4 and 6 contradictories; 4 superimplicant to 7; 5 and

6 equivalent; 5 and 7 subcontraries; 6 and 7 independent.

7. (i) = Some who go to church are not saints. Obverse: Some who go
to church are other than saints; Contrapositive; Some who are other than

ig8 KEY TO THE EXERCISES

saints go to church, (ii) =s All who love tin soldiers are small children.
Obverse: None who love tin soldiers are other than small children;
Contrapositwe: None other than small children love tin soldiers, (iii)
Obverse: All shrimps are unobtainable today; Contrapositive: Some things
unobtainable today are shrimps.

8. (i) FaC S3 FeC == CeF.
(ii) FiC S3 OF = CoF.
(iii) FeC 53 CeF = CaF.
(iv) FiC 35 OF.

The required forms are CeF, CoF, CaF, CiF.

g. By reformulating these five propositions as immediate inferences,
we can exhibit their relation to one another:

(1) SiP (using S for sailors, S for its contradictory, P for patriotic people,
P for its contradictory).

( 2 ) p e s == SeP.

(3) PoS - PiS 1 (obv.) SE &P (.)

(4) PeS 53 &P ss iP.

( 5 ) .&J5 == ^p.

Thus (2) to (5) form the square of opposition (omitting the o proposition),
whilst (i) is an inverse of (4); hence, given SiP is true, then (i) and (4)
are doubtful; (2) is false; (3) and (5) are true.

10. Contradictory. Some man can be a politician without being either
a historian or a traveller.

Contrary. All men can be politicians without being either historians
or travellers.

1 1 . Let A stand for aeroplanes, B for bi-planes, then the given proposition
is AW. The following diagram shows what is required:

Aab > AeB

AiB > AoB

The four propositions are assumed to stand at the corners of the figure
of opposition. The arrows show the passage from AiB to its subcontrary
AoB, to AaB, contradictory of AoB, and so on, in accordance with the
numbered steps.

12. (i) This statement might mean that no one who seems just is
just (an E proposition), or it might mean some are not (an
proposition).

(ii) This statement may mean that some of the soldiers were and
some were not afraid, i.e. "some" may be used for "some only";
it might also be used to assert that at least some and perhaps all
were afraid.

(iii) This statement may mean either that the fish together weighed
4 Ib. or that each fish weighed 4 Ib. The contradictories (given
in the order of interpretation) are:
(t) Some who seem just are just. All who seem just are just.

KEY TO THE EXERCISES 199

(w) Either no soldiers were afraid or all soldiers were afraid. No

soldiers were afraid.
(Hi) The total weight of the fish was less than, or more than, 4 Ib.

Some of the fish weighed less, or more, than 4 Ib.

13. Either man is not born free or he is not everywhere in chains.

14. (i) If prices do not rise, wages are not increased.

Either prices will rise or wages will not increase.

It is not the case both that prices will not rise and wages will

increase,
(ii) If the child is not badly taught, then he is exceptionally stupid.

If the child is not exceptionally stupid, then he is badly taught.

It is not the case both that the child w T as not badly taught and

also that he is not exceptionally stupid,
(iii) Either you do not eat your cake or you do not have it.

If you eat your cake, you do not have it.

If you have your cake, you do not eat it.
(iv) Either a man will not begin with certainties or he will end in

doubts.

If a man shall not end in doubts, he will not begin with

certainties.

It is not the case both that a man will begin with certainties

and also not end in doubts,
(v) If we are responsible for our actions, then our actions are within

our own power.

If our actions are not within our own power, then we are not

responsible for our actions.

It is not the case both that we are responsible for our actions

and that our actions are not within our own power,
(vi) Either C is not D or Q,is not R.

IfQisR, then C is not ZX

It is not the case both that C is D and Q,is R.

15. The four given statements can be reformulated as hypothetical
propositions as follows:

(a) If a student is taught by A, he fails.

(b) If a. student is taught by B, he fails.

(c) Fa student is taught by C, he passes.

(d) If a student is taught by D, he fails.

By successively affirming the antecedents of (a), (b) and (d) 9 we affirm
in turn their consequents; hence, the tutors A 9 B, D are each excluded;
the antecedent of (c) being affirmed, we can affirm its consequent, viz.
he passes. Thus we decide that C is the tutor who will ensure that the
student passes the examination.

Note. -The student will find it useful to study the following^ equiva
lences, assuring himself (by means of the intuitive apprehension of a
significant example) that these equivalences hold:

Ifp 9 fanqsslfq, thenf ss Either p or q = Only iff, q = Only if q, p
== Unless p, q = Unless q,p.

1 6. Modus iollendo tollens: If civilians are cowardly, then factories stop
work in an air-raid: but factories do not stop in an air-raid; .*. civilians
are not cowardly.

2OO KEY TO THE EXERCISES

Equivalences:

(i) Either civilians are not cowardly or factories stop work in an
air-raid.

But., Factories do not stop work in an air-raid;
.". Civilians are not cowardly.

(ii) It is not the case both that civilians are cowardly and that
factories do not stop in an air-raid.
But, Factories do not stop work in an air-raid.
. . Civilians are not cowardly.

(iii) If factories do not stop work in an air-raid, then civilians are
not cowardly.

But, Factories do not stop work in an air-raid.
.*. Civilians are not cowardly.

Note. - In the example given above the antecedent and the consequent
of the original argument are both affirmative statements; this is by no
means necessary.

17. (i) If Abraham Lincoln were alive today, then a just and reason
able peace would be made. Abraham Lincoln is not alive

today;

/. A just and reasonable peace will not be made.

Invalid: fallacy of denying the antecedent.
(ii) If the law supposes that, the law is an ass, a idiot.

(But the law supposes that);

.*. The law is an ass, a idiot.

Valid (provided the premiss in parentheses is granted) .
(iii) Either the Pythagorean theorem ... of studying it.

But the Pythagorean theorem ... is true;

/. It is not worth while to study it.

Invalid: fallacy of affirming an alternant.

(iv) (a) If prices fall, then there is over-production; and if there is
not over-production, then factories close;
(But either there is over-production or there is not over
production) ;

.*. Either prices fall or the factories close.
Invalid: The omitted premiss is almost certainly the premiss
given in parentheses. But this premiss affirms the conse
quent of the first proposition and the antecedent of the
second, whereas, what is required for establishing the con
clusion is the alternative affirmation of both antecedents.

(b) If factories close, the number of unemployed increases;
If the number of unemployed increases, there is dissatis
faction and social unrest;

( .*. If factories close, there is dissatisfaction and social
unrest) .
Valid.

Although these two arguments are valid, the conclusion given in the
original argument, viz. If prices fall, there is dissatisfaction and unrest-
does not follow. The conclusions of (a) and (b) taken together warrant
only the conclusion: Either prices fall or there is dissatisfaction and social
unrest.

KEY TO THE EXERCISES 2OI

(v) If I follow his argument he is muddled; if I do not follow his
argument, he is obscure in his statement,
(But either I follow his argument or I do not} ;
.*. Either he is muddled or obscure in his statement.
Valid. Note, however, that the speaker has made the doubtful
assumption that his inability to follow the argument could not
be due to any other cause than the author s obscurity in
statement.

(vi) If your uncle is rich, you will not be afraid ... a loan.
But you are not afraid;
.". Your uncle is rich.

Invalid: fallacy of affirming the consequent. (Probably the speaker
has in mind the premiss, Only if your uncle is rich . . ., and
this is equivalent to If you are not afraid, then your uncle is . . .)
The argument would then be a valid modus ponendo ponens.

(vii) (a) If a man happens not to succeed . . ., he will be thought
weak and visionary; and if he succeeds (touches the true
grievance) , he may come near . . . correcting them.
(But he will succeed or not succeed) ;
.". Either he will be thought weak ... or come near . . .
correcting them.

(b) If he should be obliged . . . people, he will be considered
. . . power, and if he censures those in power, he will be
looked on ... faction.

(But either he will be obliged to blame the favourites or
will censure those in power) ;

.". Either he will be considered the tool of power or he will
be looked on as an instrument of faction.

(c) If anyone is thought weak ... or comes near ... be ex
asperated, or is considered the tool of power or as . . .
faction, then he is engaged in an undertaking of some
degree of delicacy.

(But anyone who examines the cause of public disorders is
thought weak ... or comes near ... or is considered the
tool of power or as ... faction) ;

.*. Anyone who examines the causes of public disorders is
engaged in an undertaking of some degree of delicacy.

(d) If anyone is engaged in an undertaking of some degree of
delicacy, he has to hazard something;

(If anyone is exerting himself in duty, he has to engage in
an undertaking of some degree of delicacy) ;
/. If anyone is exerting himself hi duty, he has to hazard
something.

These four arguments are valid, provided that the implicit
premisses enclosed in parentheses are granted.

18. Statements (i), (3), (4), (5), (7) are all equivalent; each is
equivalent to the categorical statement, All Whigs are rascals. Statement
(2) is equivalent to the categorical statement, No Whigs are rascals. (6)
is independent and equivalent to All rascals are Whigs.

202 KEY TO THE EXERCISES

19. (i) Contradictory: It is the case both that poetry does not come as

naturally as leaves to a tree and that it had better come than
not come. Contrary: If poetry comes as naturally as leaves to
a tree, it had better come.

(2) Contradictory: I am not certain that you are wrong.
Contrary: I am certain you are right.

(3) Contradictory: Either some endogens are not parallel-leaved or
some parallel-leaved plants are not endogens.

Contrary: No endogens are parallel-leaved plants.

20. For the rules, see p. 57.

(i) To prove that EIO is valid in every figure:

Since the major premiss is universal its subject is distributed, and since
it is also negative, its predicate is distributed; . . both major and middle
terms are distributed in this premiss whether it is of the form P-M or
M-P. Since the conclusion is particular the minor term is not distributed;
accordingly the minor premiss SiM, or MiS, can be combined alterna
tively with PeM, or MeP. The mood EIO is thus valid in every figure.

To prove that IEO is invalid in every figure:

Since the major premiss is particular affirmative, the major term will
be undistributed whether it is subject or predicate; but, since the minor
premiss is negative, the conclusion must be negative; hence, P, the major
term, will be distributed in the conclusion. Thus the mood IEO involves
illicit major, no matter what the position of the major term may be hi
its own premiss; accordingly, IEO is invalid in every figure.

(2) * (a) O cannot be major premiss in figure I, for, if it were the
minor premiss must be affirmative; in that case, M will be undistributed
"* ^ e Pi 001 premiss, so M must be distributed in the major premiss of
which it is subject. But is particular; hence its subject is undistributed;
.". cannot be the major premiss in figure I.

(b}^0 cannot be minor premiss in figure I, for, if it were, the major
premiss must be affirmative and the conclusion must be negative. But
P is predicate in the major premiss, and would be undistributed if this
premiss were affirmative; thus, there would be a fallacy of illicit major;
.". O cannot be the minor premiss in figure L

(c) cannot be major premiss in figure II, since one premiss must be
negative (in order to distribute M, which is predicate in both premisses),
and, in consequence, the conclusion will be negative with a distributed
predicate, viz. P. But P is subject in the major premiss which must, then,
be universal to secure the distribution of P; /. cannot be the major
premiss in figure II.

(d) cannot be a minor premiss in figure III, for the same reason as
in figure I (see b above).

- * i?t stu l ent should notice that there are a variety of slightly different ways
in which such proofs as these can be given. The exact wording is not important-
consequently, in the following answers variations are deliberately introduced
? ?!r t0 S that the relevant P * 311 * can be differently stated. Henceforth
5 M l .y 111 ** used to stand respectively for minor, middle, and major terms.
Proofs will be less and less fully stated, since, once a student has grasped the
procedure, he should have no difficulty in fitting in the indications provided in

KEY TO THE EXERCISES 203

(e) cannot be a major premiss in figure IV, for the same reason as
in figure II (see c above).

(/; cannot be a minor premiss in figure IV, for the same reason as
in figure I, except that, in this case^ the illicitly undistributed term would
be M, which would be subject of a particular minor premiss, and predi
cate of an affirmative major premiss, and thus would not be distributed
in either premiss.

(3) This theorem can be proved from the considerations already
adduced in the answer to (2) . (Note that, if P is predicate (i.e. the major
premiss is M-P), it can be distributed only if the major premiss be nega
tive; but, if either premiss is negative, P will be distributed in the con
clusion.)

(4) To prove an A proposition both premisses must be affirmative, and
the minor must be universal to distribute S; hence the minor premiss
must be SaM. In this premiss M is undistributed; it must, therefore, be
distributed in the major, which is affirmative; hence, the major premiss
must be universal affirmative with M as subject. The syllogism is, there
fore, MaP, SaM, . . SaP, and no other combination of premisses will
yield SaP.

(5) There are three cases: (a) Both affirmative; Since M is to be distri
buted in both, it must be subject of both, and the premisses must be
universal; S will be predicate of an affirmative premiss, and will thus
be undistributed; hence, the conclusion must be SiP.

(b) One affirmative and one negative premiss: Together these can distribute
three terms; of these terms two must be M, and the remaining term P
(since the conclusion must be negative). Thus S cannot be distributed,
i.e. the conclusion must be SoP.

(c) Both premisses negative: excluded by general rules of quality.

21. To prove SeP.

Both premisses must be universal, with one affirmative 3 and one
negative; i.e. the premisses must be A and E in either order.

(i) Let the major be E, i.e. either MeP or PeM. The minor must then

be affirmative, with S distributed; . . it must be SaM.
(ii) Let the minor be E, i.e. either SeM or MeS. The major must then
be affirmative, with P distributed; .". it must be PaM.

Accordingly, SeP can be proved in four different moods, viz.:

(i) MeP (2) PeM (3) PaM (4) PuM

SaM SaM SeM MeS

:. SeP :. SeP :. SeP :. SeP

(Note. -In (i) and (2) the major, and in (3) and (4) the minor,
premisses are simple converses of each other.)

22. Let / stand for intelligent, / for unintelligent, people; and let R
stand for reliable, R for unreliable, people. Then the four given proposi
tions can be represented as follows:

(i) laC, (ii) leR, (iii) CoR, (iv) RoC.

Now (ii) leR = Rel (com.) = Ral (obv.). Combine Ral with (i) IaC 3
and thus obtain the Barbara syllogism: laC, Ral, /. RaC.

Now (iii) CoR = CiR (obv.), which is the converse per acddens of RaC;
hence (i) and (ii) jointly imply (iii).

2O4 KEY TO THE EXERCISES

Now (iv) RoC 555 RiC (obv.) 9 and RiC is the inverse of RaC\ hence (i)
and (ii) jointly imply (iv), provided that R and C exist.

23. By (i) the major premiss is affirmative, and by (ii) the major term
is distributed in this premiss, of which it must, therefore, be the subject
and the premiss must be universal; hence the required premiss is PaM.
By (ii) the major term is given as distributed in the conclusion, which
must, therefore, be negative, and, since by (iii), the minor term is
undistributed in the conclusion, the conclusion must be SoP. Since M is
undistributed in PaM, it must be distributed in the minor premiss, which
must be negative, with S undistributed (by iii); hence, the minor premiss
is SoM. The required syllogism is thus PaM, SoM, . . SoP (i.e. AOO in
figure II).

24. Bocardo: Some archers are not graceful., All archers are athletes, /. Some
athletes are not graceful. In order to obtain an equivalent conclusion from
equivalent premisses in the mood Darii, we require the A proposition as
major premiss with the subject and predicate reversed. This cannot,
however, be done, since A converts to /, which is non-equivalent, and
would not, with another particular premiss, yield any conclusion at all.
There is the further difficulty that an proposition has no convert.
Hence, to obtain equivalent premisses we must use obversion as well as
conversion. The required steps are as follows: (i) obvert the original
major; (2) convert this obvert; (3) transpose the premisses; (4) draw a
conclusion from the premisses thus obtained. This syllogism will be in
Darii; (5) convert the new conclusion; (6) obvert the convert; this yields
the original conclusion.

(ij Some archers are not graceful = Some archers are ungraceful.

(2) Some ungraceful people are archers.

(3) (Major) All archers are athletes.

(Minor) Some ungraceful people are archers;

(4) .*. Some ungraceful people are athletes,

(5) = Some athletes are ungraceful,

(6) 5= Some athletes are not graceful.

25. See pp. 62-4, above. Since we are given that the major premiss
is universal and the minor is affirmative, we find that the moods in
figure I must fit into the scheme:

If every (or some) Xis T (or not),

And every (or some) is X;

Then, every (or some) is T (or not).

In reductio per impossibile we deny the conclusion; we thus obtain the
schema, Every (or some) is not T (or is) . Combining this successively
with the schemas for the two premisses, we obtain:

(i) If every (or some) is not T (or is) minor premiss,

and every X is T (or not) major premiss,

then, every (or some) is not X conclusion.

(ii) If every (or some) < is not T (or is) major premiss,

and every (or some) is X minor premiss,

then, some X is not T (or is) conclusion.

(i) yields the moods of figure II, in each of which the conclusion must
be negative; (ii) yields the moods of figure III, in each of which the
conclusion must be particular.

KEY TO THE EXERCISES 205

26. No self-confident people are shy in advising their elders.
All good administrators are self-confident.

All Civil Service officials are good administrators.
Some young men are Civil Service officials,
.*. Some young men are not shy in advising their elders.

This is a Goclenian Sorites.

27. The information provided can be stated in the premisses:

CaA
BaD
BeC.

To establish the desired conclusion, we must be able to deduce from these
premisses at least one of the propositions AoD or Do A. But neither D nor A
is distributed in the original premisses, whilst D is distributed in AoD,
and A in Do A; therefore, neither of these conclusions can be obtained.
Accordingly, the answer to the question is in the negative.

28. (Note. - In answering this question only brief indications of the
premisses will be given.)

(1) All generous people are humane. (Invalid, ; undistributed middle.}
He is humane;

.*. He is generous.

(2) All A.-S. nations are ... to freedom. (Invalid, . undistributed
middle.)

The U.S.A. is ... to freedom.
.*. The U.S.A. is an A.-S. nation.

(3) This argument is invalid because it assumes that what I cannot do
alone, I cannot do with others. The fallacy is analogous to the fallacy of
composition.

(4) All who resent criticism are sensitive. (Invalid, ". undistributed
middle.)

All musical people are sensitive,
.*. All musical people resent criticism.

(5) Invalid, for the conclusion, Two bodies with nothing between must touch,
assumes the point to be proved, viz. There cannot be nothing between bodies,
i.e. a vacuum is impossible. Thus the argument commits the fallacy of
begging the question.

(6) Tou admit: A fortune can be left to a man s children which is
sufficient to keep them in idleness, i.e. it is permissible for heirs to a
fortune to have unearned money without working.

Tou maintain: No one ought to have unearned money without working.
These two statements are contradictory.
The argument is valid.

(7) Persecution is not justifiable.

. . Whatever is needed to prevent persecution is justifiable.
The conclusion does not follow; accordingly, the remainder of the argu
ment is irrelevant.

(8) Using P, Q 3 5, M, R, for pacifists, Quakers, Socialists, Marxists, and
those who support the raising of the school-leaving age, respectively, the informa
tion given can be summed up in the premisses:

QaP, Pod, MaS, SoM, PiR, SiR.

The conclusion is said to be: QeM and fidoQ;, QiR and MiR.

2O6 KEY TO THE EXERCISES

On examination it will be found that the conclusion does not follow,
although none_of the four constituent propositions is inconsistent with
the premisses QiR == RoQ } and MiR == RoM, but any attempt to con
nect Q, and AI, or Q, anc ^ R? or M anc * ^? or their contradictories, by
combining the premisses in any order would involve illicit distribution.

(9) Let S represent those who are industrious and P represent those who
are intelligent. Then, you deny SeP, I deny SaP and PaS.

Now, denying SeP = affirming SiP;
and denying SaP and PaS = affirming either SoP or PoS.

Question is whether these two denials can be said to agree that some
industrious people are intelligent*, i.e. whether SiP is true. Either SoP or
PoS neither implies nor is implied by SiP, but these are consistent. Hence,
if agree that SiP is true 5 means do not assert SeP , then you and I
agree; if, however, agree, etc. means assert that SeP is false , we do
not agree.

(10) This argument is valid only on the assumption that to need X,
when X is inseparable from T, implies needing T also. This assumption
is manifestly untrue.

(i i) All men desire their own happiness does not imply that each man desires
the happiness of all. Hence, even if it be granted that whatever is desired by all
is desirable, it does not follow that the happiness of all (i.e. universal happi
ness) is desirable. The conclusion is consistent with the premisses (provided
it is assumed that it is possible both to desire one s own happiness and the
happiness of all other people) ; but to assert that the premisses imply the
conclusion is to commit the fallacy of composition.

(12) No fashionable views are subtle.
Some true views are subtle;

/. Some fashionable views are not true.
This argument is invalid: it commits the fallacy of the illicit major.

(13) Using initial letters for class-names, these propositions may be
symbolized as follows: WoH and HaM, .*. WaM. _ _

Now WoH == WiH-, then we have the syllogism, HaM, WiH, .*. WaM,
which involves the fallacy of illicit minor. But to be wealthy is not to be
healthy* is ambiguous; it may be used to assert WeH which obverts to
Watt, and WaH and HaM implies WaM.

(14) This argument may be briefly formulated as follows:
If decreased effort, then industry does not flourish.
If no competition, then decreased effort;

.*. If no competition, then industry does .not flourish.
It is valid. It should be noted that the validity depends upon the
assumption that "competition" has exactly the same force in both
statements. It may well be relevant to stress the difference between
competition between different firms and competition between different workers in
the same firm (as in piece-work).

(15) This argument is of the form: Most M is C, Most M is S; :. Some
S is C. This is valid, since "most" means "more than half", so that,
taking the two premisses together, the middle term, M, is referred to
in its whole extent, i.e. is distributed.

29. His income is greater than yours: asymmetrical, transitive.
Castor is twin of Pollux: symmetrical, intransitive.
Henry VII is ancestor of Elizabeth: asymmetrical, transitive.

KEY TO THE EXERCISES 2O7

Othello is married to Desdemona: symmetrical^ non-transitive .

7 is a factor 0/42: asymmetrical^ non-transitiie.

This ribbon exactly matches in colour that dress: symmetrical, transitive.

Jane is aunt of Thomas: asymmetrical, intransiiiie.

Tom is in debt to Dick: asymmetrical^ non-transitize.

The falsity of the conclusion implies the falsity of at least one

premiss in a valid syllogism: non-symmetrical , transiiizv.
John is a lover of Mary: rum-symmetrical, non-transitive.

30. (i) servant of; child of; (ii) eldest son of a father; double of;
(iii) cousin of; step-father of.

(i) Edward is Jacob s master; (ii) 10 is the half of 20; (iii) Marina is
the cousin of George,

31. See pp. 78-81; 87-9.

32. (i) Non-Fascist Italians =^= 0.

(2) Non-brave people deserving the fair = 0.

(3) Long-lived butterflies = 0.

(4) Non-legal experts able to draft an act of parliament = 0.

33. See pp. 81-4; 93-9.

34. See pp. 88-93.

35. (i) SaP, on the given assumption, states that SP O, whilst PoS

states SP 4= O. But SaP does not imply the existence of P or of
S; hence, the inference is invalid.

(ii) MaP states MP = and SaM states SSl = 0, whereas the
conclusion SiP states SP 4= 0. But the premisses do not suffice
to establish the existence of S (i.e. the minor term); hence the
inference is valid.

(iii) PeS states PS = 0, whereas SSP states SP =J= O; but if nothing is
both P and S, either P = or S =J= 0; consequently P&S implies
S =i= unless nothing is P. But, if P = 0, then P 4= 0. It
follows that fflP, and thus the inference is valid.

36. See pp. 102-7.

37. See pp. 107-12.

38. See p. 102. In answering a schoolboy s question: What is
"rationalize"?*, it would be necessary to ascertain the context, since the
verbal form rationalize has three wholly distinct meanings in common use,
and a fourth meaning from which, in devious ways, the other three
meanings have been derived. Context alone can settle which meaning
is relevant. (See any Dictionary for these meanings, viz. original, used in
mathematics, in economics, in psychoanalysis. To explain properly the
meaning of a word it is essential to give examples illustrating its use,
for we do not understand a word until we know how to use it in different
sentences.)

39. It should be remembered that various definitions of a word can
be given, and that there are various propria and accickns. The following are
illustrative examples:

Genus Differentia Propria Accidens

(i) (Amator),

man. Able to pilot an Having know- Member of

aeroplane. ledge of R.A.F.

altimeters.

208

Genus
(ii) (Sonnet),

KEY TO THE EXERCISES

Differentia

Propria

Accidens

poem.

Having 14 deca

Having a rhyme

Having the

syllabic lines,

scheme.

rhyme

expressing

scheme

one idea.

abba cdcdcd.

(iii) (Schooner),

sailing-ship.

Fore-and-aft

Having masts.

Having a

rigged.

Scots

skipper.

(iv) (Pamour),

workman.

Employed to lay

Having arms.

Being English.

pavements.

(v) (Communique),

announce

Official.

Concerning

Depressing in

ment.

matters of

content.

national
importance.

40. (i) Too wide; it requires the differentia - having four equal sides.
(2) Too narrow, since the spinning need not be confined to cotton.
There is an additional (and now distinct) meaning, viz. "unmarried
woman". (3) Satisfactory. (4) Errs by defining the unknown by what
is likely to be more unknown. Def.: "Shine with quivering or inter
mittent light". (5) Too narrow, since military skill may be lacking. Def.:
"A person serving in an army".

41. Ship is a class-name used for a variety of sea-going vessels; hence
there are numerous subclasses constituting the extension of ship, the con
notation of which is "large sea-going vessel". If we arrange the sub
classes in an orderly classification, then any subclass has smaller exten
sion than its superclass, but has increased connotation, since its connotation
will contain the property (or properties) differentiating one subclass
from a co-ordinate class and from the superclasses. For example, sailing
ship excludes steam-ship, etc., and adds differentiating property sailing.
Again, the subclass brigantine excludes schooners and brigs, etc., and adds
to sailing ship, the differentia having two masts a brig s foremast, square-
rigged, a schooner s main-mast, fore-and-aft rigged (see pp. 107-12, above).

42. We clearly need a class not included in the list, under which
literary work of art and scientific treatise may find an appropriate place.
The following is a possible arrangement:

non-literary prose work

literary work of art

historical

scientific

i

i .

narrative
1

fiction

i

poem drama

rV r-

epic lyric comedy

?r

i?

of

Moll
Flanders

I

Alice in
Wonderland

Origin of
Species

i

Principles
Geology

1

I

ode sonnet triolet

KEY TO THE EXERCISES 2CK)

This is a logically unsatisfactory arrangement, but it is difficult to see
what good purpose could be served by classifying the various classes given
in a single classificatory table. To mark the criticism, queries have been
put to indicate omission of essential superclasses. To include in the
table individuals, e.g. Origin of Species , is to make a muddle of any classifl-
catory scheme (see p. 105 above;.

It must be noticed that familiarity with the nature of the subclass is
essential for the purpose of classifying.

43. See pp. 103-5, above. Main points to be noted: (T; the sense in
which ordinary proper names lack connotation, whereas dictionary
meaning is usually the connotation; (ii) the significant use of ordinary
proper names depends upon the speaker s knowledge that many descrip
tions do in fact describe the individual so named (cf. M.LL. 9 Gh. Ill,
2).

44. See above, pp. 14-15 and pp. 126-9.

45. See above, pp. 130-5.

46. See above, pp. 135-41.

47. See above, pp. 141-4.

48. See pp. 146-52.

49. (i) Empirical proposition: Granted that there is agreement with
regard to the mining o f "cathedral", the evidence required is observa
tional. Testimony may be used to establish it, but those who testify to
its truth must have relied upon observation at some stage.

(2) This statement is true by definition; hence, the evidence required
is given provided "square" has been defined.

(3) Causal law: Observation and assumptions with regard to natural
happenings provide the evidence.

(4) The second of these two propositions follows from the first, since
the meaning of "taller than" necessitates the second.

(5) Tautology.

(6) Observation would suffice to establish this proposition, in the same
way as in example (i). It might also be established by indirect observa
tional methods, depending upon measurement of shadows. It is not in
fact possible for anyone dwelling on the earth to test its truth or falsity,
since there is no practicable way of observing the other side of the moon.
This fact does not in the least affect the logical status of the evidence
required.

(7) Observation and experiment, together with mathematical deduc
tion.

(8) This is a tautology, true by definition.

(9) Similar to (8).

(10) This can be established only by induction by simple enumeration
(see Gh. IX, i). It is not logically impossible that two people should
have the same finger-prints, but the amount of evidence suffices to make
the acceptance of the proposition reasonable.

50. See p. 163.

51. See p. 160, and cf. M.I.L., Ch. XXIV, i.

52. (i) It will rain tomorrow. Examples are provided by nos. (i), (3),

(6), (10), and (7), in question 49.

(ii) A right-angled triangle is right-angled. Examples given in
nos. (2), (4), (5), (8), (9), in question 49.

2IO KEY TO THE EXERCISES

(iii) Red roses are not red. The widower s wife has called. Five

times six is forty.

53. Note. -Your definition must cover all the topics which -you con
sider should be dealt with by logicians, and exclude any topic lying
outside their scope.

Index

The paragraph-headings giren in the Table of Contents
should also be consulted. \ o references are given to the
names of authors or titles of books cited in the Appendix

Abstracting, not difficult, 76; and
recognizing similarities, 76

Abstraction, 80

Accident, 113

Affirmation, and affirmative and
negative sentences, 18; and denial,
1 8; and Tes and j\b, 18

Affirmative sentence, and affirmation,
1 8; and denying, 18

"All", ambiguous use of, 81

Alternant, def., 28

Antecedent, def., 28

Antilogism, def., 64

Argument, a fortiori, 99; conclusive-
ness of, 8; deductive and inductive,
5; examples of, 2, 3, 7; validity of,
7 seq., 75, 160 _ .

Argument of prepositional function,
130

Argumentative discourse, def., 3

Arguments, compound modes of, 49;
well-constructed, 160

Aristotle, 21, 55, 64, 65, 113, 146, 162

Assertion, 19 seq.

Axiom de omni, 65

Axioms, of distribution, 57; of
quality, 57

Bannerman, Sir H. Campbell, 54

Baylis, C. A, viii, 152 n.

Belief, and assertion, 19 seq.; and

proposition, 17, 20; and truth, 160

seq.

Bennett, A. A., vin
Boole, G., 93 n., 99 n.

Causation, and conditions, 171

Cause, of an occurrence, 1 72 seq.

Causes, plurality of, 179

Chains of deduction, 0$

Characteristic, and determination of
class, 80; and pr opri ion, 113 n.

Characteristics, and class, 78, 102;
and exemplification, 77; and exist
ence, 77; and words, 76

Class, and characteristics, 102; ele

ments, 78; empty, 37, 88 seq. t
103; enumerativc selection of, 79;
existent, 37; -inclusion, 79, 87, 97;
-membership, 79, 87; sub-c., 88,
1 08; super-c., 88; universal class,
92; and words, 78

Class-property, 80

Classes, and associations, 81; and
existence, 78

Classification, 107 seq.; and sub
classes, 1 08

Collective membership of a class, 81;
and denotation, 103

Composite propositions, 29; alter
native, 28; disjunctive, 28

Conclusion, and premiss, 3 seq.;
evidence for, 4; irrelevant, 162

Conjunctive proposition, 29

Connotation, and definition, 121; and
proper names, 121 seq.; of words,
102 seq.

Consequent, 28

Consistency, 33

Construct, and form, 10

Context, of propositions, 35

Contradiction, 32; def., 32; law of,
146

Contradictory terms, 35

Contraposition, def., 39; schema of,

39

Contrariety, def., 32

Conversion, def., 38; and logical pro
perties of relation, 97; per accident,
38; schema of, 39

Conviction, rational, 160; and truth,
160

Copula, 22

Deductive form, science and, 184; and

theory, 184
Definiendum, 119
Definiens, 119
Definition, and connotation, 120;

ostensive, 116; per genus et differ"

entiam, 115; real, 120; rules of, 119;

verbal, 120
de Morgan, A., 93 n.

211

212

INDEX

Denial, and affirmation, 19; by
affirming contradictory or by af
firming contrary, 43

Denotation, 102; and class, 103; and
collective membership of class, 103

Descriptions, definite plural, 125; in
definite, 125; singular, 125

Differentia, 112

Dilemma, 51 seq.

Disjunction, 137 n.

Distribution, of terms, 23

Division, Logical, 107; dichotomous,
109; rules of, 109

Eaton, R. M., 113, 113 n., 159 n.

Eddington, A. S., 158 n.

Einstein, A., 184

"Either . . . or . . .", interpretation
of, 49; 45 seq.

Entailing, 136, 140, 152

Enihymeme^ 56

Enumerative selection, 79

Epicheirema, 73

Equality, and identity, 84; of rela
tions, 84

Equivalents, 30, 31, 34 seq. , in moods
of syllogism, 61

Euclid, 115

Euler s circles, 25

Evidence, i; premiss as e for con
clusion, 4; e for different kinds of
propositions, 153 seq.

Exemplification, 77

Exist, 77

Existence, see Class

Existentially affirmative, 91

Existentially negative, 91

Extension, def., 105; inverse variation
with intension, 106

Fact, and proposition, 18; and true,
false, 1 8

Fallacy of, circular argument, 163;
composition, 162; division, 162;
four terms, 161; illicit distribution,
161; illicit major, 161; illicit minor,
161; irrelevant conclusion, 162;
ignoratio elencki. 162; petitio principii,
163

Form, and construct, 10; gram
matical, 10; implicational, n;
logical, 9 seq., 122; and material,
10; musical, 10; and pattern, 9; and
shape, 9; and validity, 15

Fundamentum dwisionis, 108

Galenian figure, 64
Generalization, 155, 166
Genus, 112 seq.

Hypothesis, def., 8; nature of, 180;
and prediction, 182; verified, 182

Hypothetical proposition, 44; com
plementary, 46

"I", explained, 18; use of, 126 seq.

Identity, 85

Implication, def., 4; extensional and

intensional interpretation of, 142;

formal, 142; material, 135 seq., 138,

140; necessary, 136
Imply, and entails, 136
Inconsistent triad, 68
Individual, 76; and class, 88; naming

an, 126; specifiable, 130
Induction, by simple enumeration.

1 66 seq.

Inductive methods, 1 74-6
Inference, and assertion, 19; def., 21;

and evidential relations, 21; im
mediate, 36; mediate, 36
Intension, 104. See also Extension,

Connotation
Inversion, def., 39; validity of, 94

Jackson, R., 164 n.
James, William, 77, 99
Johnson, Samuel, 2 seq.
Johnson, W. E., 46
Joseph, H. W. B., 100 n.

Ladd-Franklin^ C., 68, 94 n,

Laws, causal, 169, 172; of Nature, 158

Laws of thought, 146

Logic, a formal science, 12

Logical form, and grammatical simi
larity, 122. See also Form

Logical Positivists, 152 n., 158

Logical principles, necessity of, 158
seq. See also Principles

Material implication, 135 seq. , para
doxical consequences of, 140; and
truth, 141 ; and truth-values, 145

Mill, J. S., 121, 122 n., 164

Moore, G. E., 136, 140

Multiformities, 169

Necessity, of logical principles, 158
seq.; and meaning, 158; and self-
evidence, 159

Negation, 140

Negative sentence, see Affirmative
sentence

Obversion, def. of, 37; schema of, 37
Opposition, of propositions, 33; figure
of, 34; square of, 33

INDEX 213

Parameters, 130 n.

Peano, 134 n.

Pirandello, 93 n.

Ponendo ponens, 49 se q.

Ponendo tollens, 49 seq.

Porphyry, 113 n.

Predicables, 112

Predicate, of proposition, 22, 27

Principle, applicative, 985 148; of
contradiction, 146; of deduction,
148; of excluded middle, 146,, 149
seq. , of identity, 146; of skipped
intermediaries, 99; of substitution,
98, 148; of syllogism, 148

Proof, and conviction, 160; reasoned,
1 60; syllogistic, 163

Proper names, 121 seq.

Proposition, alternative, 28; and
assertion, 19 seq.; and belief, 17;
compound, 27, 29; contemplated,
20; contingent, 157; def., 17;
diagrammatic representation of, 25;
disjunctive, 28; factual, 152, 154,
157; general, 27, 30, 133; hypo
thetical, 28; hypothetically enter
tained, 20; independent, 32; neces
sary, 152; non-factual, 157; par
ticular, 22 seq.; relational, 29;
self-contradictory, 157; simple, 27,
29; singular, 154; stated, 19; and
statement, 16; subject-predicate,
27; undecidable, 151; universal, 22

Prepositional form, and function, 130;
and proposition, 128, 134; and
schema, 129; and specification, 127
seq.

Prepositional forms, and verbal state
ments, 21

Prepositional function, 130; range of
significance of, 132; and traditional
schema of propositions, 135

Propositions, traditional analysis of,
21

Propriiffn, 113

Quality, of propositions, 22
Quantity., of propositions, 22

Range of significance, 132

Reasoning, def., 2; deductive, 166;
inductive, 166 seq. See Argument

Reasons, demand for, I

Reduction, direct, 66; indirect, 67

Referent, 82

Relation, aliorelative, 85; asym
metrical, 84; connexity of, 85;
converse of, 83; domain of, 83;
dyadic, 82; field of, 83; intransitive,
84; logical properties of, 83, 97;

many-many, 82 a 86; many-one, 86;
non-s\Tnrnetrica!j 84; non-transi
tive, 84; one-many, 86; one-one,
86; polyadic, 82; sense of, 82; sym
metrical, 84

Relation^ 82

Relevant connexion, and meaning,
*38, 145

Richards, I. A,, 117 n.

Russell, Bertrand, 83, 123, 130, 136,
137, 142, 159

Satisfying a function, 131

Sentence, and proposition, 16 seq.

Sign, conventional, 13; natural, 13; of
quantity, 22; and symbol, 13

Signifying, relation of, 102

4k Some", interpretation of, 24

Sorites, 71-2, 99

Species, 113

Specification, of individuals, 126 seq.

Statement, ambiguity of, 16; com
pound, n; def., 3; implications of,
6; and premiss, 3

Stating, a proposition, 19

Subcontraries, 34

Subcontrariety, 32

Subimplicant, 32

Subject, of proposition, 22, 27

Superimplicant, 32

Syllogism, Aristoue s definition of, 55;
corollaries of rules of, 57; defining
characteristics, 54 seq.; distinctive
characteristics of figures of, 70;
figures, 58; major, minor, middle
term of, 55, 59 n.; moods of, 58;
polysyllogism, 71; principle of the,
148; special rules of figures of, 61
seq.; strengthened, 64; terms of, 55;
valid moods of, 62 seq.; weakened
moods, 62

Symbolism, traditional, 22; use of, 23,
129

Symbols, and class, 79; constant, 126,
130; and conventional sign, 12;
illustrative, 14, 126, 129; natural
sign, 13; shorthand, 14; variable,
126 , 129; &x } 131; 0x 9 131; 3*,

135; => 138; ~, 137; v > J 37;

"...= ... df", 138
Syntax, and formal structure, 1 1

Tarrant, Z>., viii

Tautologies, 157 seq.

"Term", ambiguity of, 101

Terms, class-t, 101; distribution of,
23, 6 1 n.; of a proposition, 22; rela
tion, 29; signified, signifying of a
syllogism, 55, 59 n.

INDEX
Thinking, logical, 2; solving problem, Universe of discourse, 92, 95

Thomson, AI. E, F., viii
Tollendo ponens, 49
Tollendo tollens^ 49
Truth, and validity, 6
Truth-table, 139
Truth-values, 139

Uniformities , 169

Validity, and form, 15; and truth,

6 seq., 75
Variable, apparent, 134; real, 134;

values of a variable, 129

Watehead, A. JV., 21, 83, 130 n.
Word, understanding a, 118 seq.;
using and talking about, 19

8302 3C2

1 06 376