STACK
ANNJiX
SYLLABUS
A PROPOSED SYSTEM
OF
LOGIC.
AUGUSTUS DE MORGAN,
F.R.A.S. F.C.P.S.
OF TKINITY COLLEGE, CAMBRIDGE ;
PROFESSOR OF MATHEMATICS IN UNIVERSITY COLLEGE, LONDON.
Ai'-rXnv- 'ifiuffiv 01 ftaQivri; y oa./j,uocra,'
'O 'ygapifiaTvii etTtigof au liXixit /3Xf(T»»v.
LONDON:
WALTON AND MABERLY,
28 UPPER GOWER STREET, AND 2T IVY LANE, PATERNOSTER ROW.
1860.
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PREFACE.
THE matters collected in this Syllabus will be found in those of my
writings* of which the titles follow: — I. On the Structure of the
Syllogism .... (Cambridge Transactions, vol. viii, part 3, 1847).
II. Formal Logic, or the Calculus of Inference, necessary and
probable (London, Taylor and Walton, 1847, 8vo.). III. On the
Symbols of Logic, the Theory of the Syllogism, and in particular of
the Copula, .... (Cambridge Transactions, vol. ix, part 1, 1850).
IV. On the Syllogism, No. iii, and on Logic in general (Cambridge
Transactions, vol. x, part 1, 1858). Of these works the formulae
and notation of the first are entirely superseded ; the notation only of
the second (the Formal Logic] may be advantageously replaced (see
§ 24) by that of the third and fourth and of the present tract. There
is very little in the first three writings on which my opinion has
varied ; but of all three it is to be said that they are entirely based
on what I now call the arithmetical view 6f the proposition and
syllogism (§ 8, 173, 174), extending this term not merely to the
numerically definite syllogism, but to the ordinary form, to my own
extension of it, and to Sir W. Hamilton's departure from it.
* The writings which oppose any of my views at length are the following, so far
as my memory serves. I. A letter to Augustus De Morgan, Esq. . . . on his claim to
an independent rediscovery of a new principle in the theory of syllogism, from Sir
William Hamilton, Bart London and Edinburgh, Longman and Co., Mach-
lachlan and Co., 1847, 8vo. II. Review of my Formal Logic, signed J. S., in the
Biblical Review, 1848. III. Review of my Formal Logic (since acknowledged by
Mr. Mansel, the author of the Prolegomena Lpgica) in the North British Review/or
May 1851, No. 29. IV. Discussions on Philosophy by Sir W. Hamilton; London,
Longman and Co.; Edinburgh, Machlachlan and Co. (1st edition, 1852, 8vo. pp.
621*-652* ; 2nd ed. 1853, 8vo. pp. 676-707) ; in which a letter is reprinted which
first appeared in the Athoneettm of August 24, 1850.
4 PBEFACE.
The relations of my work on Formal Logic to the present
syllabus are as follows. Chapter I, First Notions, may afford previous
knowledge to the student who has hitherto paid no attention to the
subject. Chapter III, On the abstract form of the Proposition may
be consulted at § 93 of this syllabus. Chapters IV, On Propositions,
and V and VI, On the Syllogism, are rendered more easy by the nota-
tion of this syllabus, and are partially superseded. Chapter XIV,
On the verbal Description of the Syllogism, is entirely superseded.
rest of the work may be read as the titles of the chapters
suggest.
A syllabus deals neither in development nor in diversified ex-
ample : and does not make the space occupied by any detail a
measure of its importance as a part of the whole. I have omitted
many subjects which are to be found in all the books, or dwelt lightly
upon them : partly because more detail is contained in my Formal
Logic, partly because any one who masters this tract will be able to
judge for himself what I should have written on the omitted subjects.
I have also endeavoured to remember that as a work of this kind
proceeds, less detail of explanation is necessary.
I should suppose that a student of ordinary logic would find no
great difficulty in understanding my meaning : and that those who
are accustomed to symbolic expression, mathematical or not, would,
even though unused to logical study, find no more difficulty than an
ordinary student finds in Aldrich's Compendium. Either of these
classes, I should think, would not fail to come to the point of under-
standing at which a reflecting mind can allow itself to meditate
acceptance or rejection without latent fear of over-confidence.
Whether a beginner who is conversant neither with ordinary logic
nor with symbolic language will comprehend me is another question :
and one on which those who try will divide into more than two
classes. Such a reader, making concrete examples for himself as he
goes on, and never leaving any article until he has done this, will
either get through the whole tract, or will stop at the precise point at
which he ought to stop, upon the principle of the next paragraph.
Every spoon has some mouths that it can feed ; and some that it
cannot. Every writer has some readers who are made for him, and
he for them ; and some between whom and himself there is a great
gulph. I might prove this, in my own case, by a chain of discordant
PREFACE.
testimonies running through thirty years, if I had leisure and liking
to hunt up extracts from reviews. I will content myself with a
couple which are at hand : observing that I have no acquaintance
with the authors. In 1830, I published my treatise on Arithmetic,
and the following sentences speedily appeared in reviews of it : —
This book appears to us to mystify a
very simple science.
It is as clear as Cobbett in bis lucid
intervals.
In 1847, I published my Formal Logic, and two opponents of- my
views wrote as follows :
Mr. De Morgan I This is an undeniably long extract, and yet we would, did our
is certainly not a j limSts allow, continue it .... "We beg the reader's notice to
lucid writer. the exquisite precision of its language ; to the definiteness of
every line in the picture ; for though it is a description of a
profound mental process, still it is a luminous picture, the light
of which does not interfuse its lineaments .... We are at
very solemn issue with Mr. De Morgan upon this argument . . .
These antagonisms remind us of the stork and the fox, and of the
failure of their attempts to entertain each other at dinner. When an
author's dish and a reader's beak do not match, they must either
divide the blame, or agree to throw it on that exquisite piece of
atheism, the nature of things.
The points on which I differ from writers on logic are so many
and so fundamental, that I am among them as Hobbes among the
geometers, and, mutato nomine, may say with him of Malmesbury : —
In magno quidem periculo versari video escistimationem meam, qui a
logicis fere omnibus dissentio. Eorum enim qui de iisdem rebus
mecum aliquid ediderunt, aut solus insanio ego aut solus non insanio ;
tertium enim non est, nisi (quod dicet forte aliquis) insaniamus
omnes.
Hobbes was in the wrong : the question of parallel or contrast
must be left to time. But though some writers on logic explicitly
renounce me and all my works, yet one point of those same works is
adopted here and one there ; so that in time it may possibly be said
that
Mahometans eat up the hog.
All these differences are not about the truth or falsehood of my
neologisms, but about the legitimacy of their adoption into logic, the
study of the laws of thought. And I cannot imitate Hobbes so far
6 PREFACE.
as to write contra fastum logicorum, seeing that, oblitis obliviscendis,
I have personally nothing but courtesy to acknowledge from all the
writers of known name who have done me the honour of alluding to
my speculations. In return, I endeavour to tilt at their shields and
not at their faces. Should I make any one feel that I have missed
my attempt, I will pray the introduction into these lists of the old
practice of the playground. When I was a boy, any offence against
the rules of the game was held to be nullified if the offender, before
notice taken, could cry Slips! — as I now do over all, to meet con-
tingencies. As to the matter of our differences, I neither give nor
take quarter : it is
I will lay on for Tusculum,
Do thou lay on for Rome !
as will sufficiently appear in my notes.
Now to another topic. I produce a fragment of a well-known
conversation : those who choose may fill up the chasms.
"... I therefore dressed up three paradoxes with some ingenuity .... The
whole learned world, I made no doubt, would rise to oppose my systems : but then I
was prepared to oppose the whole learned world. Like the porcupine, I sat self-
collected, with a quill pointed against every opposer. Well said, my boy, cried I,
.... you published your paradoxes; well, and what did the learned world say to
your paradoxes ? Sir, replied my son, the learned world said nothing to my para-
doxes ; nothing at all, sir. Every man of them was employed in praising his friends
and himself, or condemning his enemies ; and unfortunately, as I had neither, I
suffered the cruellest mortification, neglect."
A friend of the author just quoted remarked that a shuttlecock
cannot be kept up unless it be struck from both ends. I was spared
all the mortification of neglect by the eminence of the player who
took up the other battledore. Of this celebrated opponent I can truly
say that, so far as I myself was concerned, I never looked with any-
thing but satisfaction upon certain points of procedure to which I
shall only make distant allusion. For I saw from the beginning that
he was playing my game, and raising the wind which was to blow
about the seeds of my plant. The mathematicians who have written
on logic in the last two centuries have been wholly unknown to even
the far-searching inquirers of the Aristotelian world : to the late Sir
William Hamilton of Edinburgh I owe it that I can present this tract
to the moderately well informed elementary student of logic, as con-
taining matters of which he is likely enough to have heard something,
and may possibly be curious to hear more.
PREFACE. 7
In controversy — and controversy was to him an element of life
and a spring of action — Sir William Hamilton was too much the
fencer of the moment, too much the firer of to-morrow's article : his
impulses sometimes leap him over the barrier which divides philosophy
from philosophism. Hoot the proofs of this out of his pages, and we
have before us a man both learned and ingenious, profound and acute,
weighty and flexible, displaying a most instructive machinery of
thought as well when judged right as when judged wrong, save only
when cause of regret arises that Oxford did not demand of his youth
two books of Euclid and simple equations. In describing a character
some points of which are at tug of war — especially when liking for
quotation was one of them — Greek may meet Greek. He showed
more fondness than was politic in one so plainly destined to survive
the grave for a too literal version of the motto
But on the other hand, though too much inclined to rule the house,
he was d/xoSss-jroTJi? oW<? Ix^esAAs* be. rov $*iF»vgov etvrov Kattct, x.»t
Trx^ettei. And this as to words as well as things. Magnificent com-
mand of language, old terms rare enough to be new, and new terms
good enough to be old, mask defects and heighten merits. In his
writings against me, it delighted him to enliven the statements of the
accuser by portraits of the mind of his opponent : colouring his notion
of the mathematician from the darkness of his want of notion of the
mathematics, his great and admitted defect as a psychologist. I dealt
with the statements in my last Cambridge paper : I now oppose my
sketch of him to his sketch of me, without the least misgiving as to
which of the two will be pronounced the best likeness.
A. DE MORGAN.
UNIVERSITY COLLEGE, LONDON,
November 12, 1859.
CONTENTS.
*** The references are to the sections.
Objective View. — (1-4) Definitions, &c. (5-32) The arithmetical form
of the cumular proposition. (33-53) The arithmetical form of the cumular
syllogism. (54-56) The sorites. (57-62) Proposition and syllogism of
terminal precision. (63-73) Exemplar proposition and syllogism. (74-86)
Numerically definite proposition and syllogism. (87-104) Doctrine of figure
and wider views of the copula. (105-111) Aristotelian syllogism.
Subjective View.— (112-123) Class and attribute. (124-135) Aggrega-
tion and composition. (136-151) Proposition, judgment, inference, demon-
stration. (152-173) Relation: onymatic relations. (174, 175) The four
readings of a proposition. (176-189) Mathematical proposition and syllo-
gism, (190-205) Metaphysical proposition and syllogism. (206) Arith-
metical reading in intension. (207-222) Extent and intent. (223-226)
Hypothetical syllogism, &c. (227-242) Belief, probability, testimony, argu-
ment. (243) Aggregation and composition of probabilities. (243) Symbols.
Controversial Notes. — (1) Use of logic. (14) Some may be all, denied in
practice by some logicians. (23) Use of analysis of enunciation. (27) Con-
version. (64) Sir W. Hamilton's system ; his account of the author's
system. (69) Limitation of matter of a syllabus. (71) Sir W. Hamilton's
system. (72) Example of error in use of exemplar system. (93) Logician's
mode of supplying the defects of his syllogism. (96) Incompleteness of
common logic : legitimate subtleties. (104) Use of generalisation. (108)
References for old logic. (116) The world has got beyond the logicians.
(124) Logical and physical composition. (131) The logician's areal phraseo-
logy. (138) The logician's form and matter. (139) Contrary and contra-
dictory. (146) The logicians and the mathematicians. (163) Can the
principles of conversion and transition be deduced from those of identity,
difference, and excluded middle? (165) The logician's form and matter.
(170) For logician's aggregation read composition. (172) Metaphysical
notions, why introduced in logic. (175) Dependence of a proposition upon
its universe. (177) Genus and species. (190) Metaphysical terms. (202) _
Predicables. (210) Wrong opposition in universal and particular. (211)
Extent and intent. (212) Scope and force. (214) The modern logician's
extension and comprehension. (216) A word supplied by Sir W. Hamilton.
(220) Class and attribute. (228) Belief. (232) Bias and exhortations to
get rid of it considered.
*** The notes end the articles in which the marks of reference occur.
SYLLABUS,
1. LOGIC * analyses the forms, or laws of action, of thought.
* Logic has a tendency to correct, first, inaccuracy of thought, secondly,
inaccuracy of expression. Many persons who think logically express them-
selves illogically, and in so doing produce the same effect upon their hearers
or readers as if they had thought wrongly. This applies especially to teachers,
many of whom, accurate enough in their own thoughts, nourish sophism in
their pupils by illogical expression.
It is very commonly said that studies which exercise the thinking faculty,
and especially mathematics, are means of cultivating logic, and may stand
in place of systematic study of that science. This is true so far, that every
discipline strengthens the logical power : that is to say, strengthens most of
what it finds, be the same good or bad. It is further true that every disci-
pline corrects some bad habits : but it is equally true that every discipline
tends to confirm some bad habits. Accordingly, though every exercise of
mind does much more good than harm, yet no person can be sure of avoiding
the harm and retaining only the good, except by that careful examination of
his own mental habits which most often takes place in a proper study of logic,
and is seldom made without it. This being done, and the house being built,
the scaffolding may be thrown away if the builder please : though in most
cases it will be advisable to keep it at hand, for use when inspection or
repairs are needed. Some persons make the argument about the utility of
logic turn on .the question whether disputation is or is not best conducted
syllogistically : on which I hold — waiving the utter irrelevancy of the
question — that those who cannot so argue need to learn, while those who can
have no need to practise. It is just the same with spelling.
As to the difficult word form, and the variations of it, I refer to Dr.
Thomson's Outlines of the Necessary Laws of Thought, § 11-15. By a form
of thought I mean a necessary law of action, considered independently
of any particular matter of thought to which it is applied.
2. Logic is formal, not material:- it considers the law of
action, apart from the matter acted on. It is not psychological,
not metaphysical : it considers neither the mind in itself, nor the
nature of things in itself; but the mind in relation to things, and
things in relation to the mind. Nevertheless, it is so far psycho-
B
10 SUBDIVISIONS. — NAMES. [2-6.
logical as it is concerned with the results of the constitution of the
mind : and so far metaphysical as it is concerned with the right
use of notions about the nature and dependence of things which,
be they true or be they false, as representations of real existence,
enter into the common modes of thinking of all men.
3. The study of elementary logic includes the especial con-
sideration of —
1. The term or name, the written or spoken sign of an
object of thought, or of a mode of thinking.
2. The copula or relation, the connexion under which
terms are thought of together.
3. The proposition, terms in relation with one another ;
and the judgment, the decision of the mind upon a
proposition: usually joined in one, under one or
other of the names.
4. The syllogism, deduction of relation by combination of
other relations.
4. The thing which is not of the mind, and can be imagined
to exist without the mind, is the object : the mind itself is called
the subject of that object. Thus even a relation between two
minds may be an object to a third mind. Logic considers only
the connexion of the subjective .and objective: it treats of things
non secundum se, sed secundum esse quod habent in anima.
5. The consideration of names, as names, may be made to fur-
nish the key to the mechanical, or instrumental, treatment of the
ordinary proposition and the ordinary syllogism.
6. For this purpose a name is a mere mark, attached to an
object: a letter painted on a post would do as well for expla-
nation as the name of a notion or concept in the mind attached in
thought to an external object. In this part of the. subject, the
assertion * Every man ig an animal ' is treated as if it were merely
' Every object which has the name m-a-n has also the name
a-n-i-m-a-l.' It answers* to * Every post on which X is painted
has also Y painted on it.'
* Remember that in producing a name, the existence of things to which
it applies is predicated, i. e. asserted : and (§ 16) existence in the universe
of the propositions. There is no conditional proposition intended, as Every
X is Y (if Y exist). Every proposition is a conceivable or imaginable truth,
when its terms are conceivable or imaginable, except only when it announces
a contradiction, as ' Some men are not men,' or when it is its own subject-
matter and denies itself, as ' What I now say is false,' a proposition which is
false if it be true, and true if it be false.
7-14.] PROPOSITION. — QUANTITY. 11
7. The proposition, in this view, is no more than the con-
nexion of name with name, as marks of the same object: the
judgment is no more than assertion or denial of that connexion.
The word is asserts the connexion : the words is not deny it.
8. This kind of proposition belongs to the arithmetical view of
logic : there is in it result of enumeration of similar instances : as
in ' Every X is Y ', ' 50 Xs are not Ys ', i.e. there are 50 (or more)
instances in which the name X occurs unassociated with the
name Y.
9. The first-mentioned name is called the subject : the second
the predicate.
10. The logical quantity (i.e. number of instances) is either
definite, or more or less vague.
11. Definite quantity is either absolutely or relatively definite:
absolutely, as in * 50 Xs (and no more) are Ys ' ; relatively, as in
' 2-sevenths (and no more) of all the Xs are Ys ', and in ' All the
Xs are Ys ', and in * None of the Xs are Ys ', which last is both
absolutely and relatively definite.
12. Quantity may be definite at one end, and vague at the
other : as in ( 50 (or more) Xs are Ys ' ; 'at least 2-sevenths of
all the Xs are Ys ' ; * most of the Xs (i. e. more than half) are
Ys ', which, however, is limited at one end, not dejinite.
13. The only perfectly definite quantities in ordinary use are
all and none. The materials for other absolute or relative nume-
rical definiteness are but seldom found in human knowledge.
The words all and none are signs of total quantity, and make
propositions universal, as ' All Xs are Ys ', ' No Xs are Ys '.
14. The contrary (usually called contradictory} propositions of
the last are ' Some Xs are not Ys ', and * Some Xs are Ys '.
Here f some ' is a quantity entirely vague in one direction : it is
not-none ; one * at least ; or more ; all,f it may be. Some, in
common life, often means both not-none and not-all; in logic, only
not-none. Some is the mark of partial quantity ; and the propo-
sitions which commence with it are particular. The contrary
(usually contradictory) forces of the pairs are seen in ' Either all
Xs are Ys, or some Xs are not Ys ; not both ' ; and in * Either no
Xs are Ys, or some Xs are Ys ; not both.'
* Kemember that some does not guarantee more than one. There is
much distinction between none, (no one) as the logical contrary of some
(one or more), and nothing as the limit of some quantity. In passing from
the proposition ' no man can live without air' to 'there is one man at least
who can live without air' we make a transition which alters the notions
12 QUANTITY. — UNIVERSE. [14-16.
or concepts attached to man : the word man no longer represents entirely
the same idea. But in passing from an indenture of apprenticeship with no
premium at all to one with a premium of one farthing, we make no change
of notion. An Act was once passed exempting such indentures from duty
when the premium was under five pounds sterling : the Court of King's
Bench held that the exemption did not apply when there was no premium at
all, because " no premium at all " is not " a premium under five pounds."
That is, the judges gave to nothing, as the terminus of continuous quantity,
the force of none, as the logical contrary of some; and violated to the
utmost the principle of the Act, which was intended as a relief to those
who could only pay small premiums, and a fortiori, or rather a fortissimo,
to those who could pay none at all. Lawyers ought to be much of logicians,
and something of mathematicians.
f Some may be all. In common language this is or is not the case
according to the speaker's state of knowledge. But in logic there are no
implications which depend upon the matter. When a logician says that
'Every X is Y' he means that ' All the Xs are some Ys' and that 'some
Ys are all the Xs'. Whether he have or have not exhausted all the Ys
he does not here profess to state, even if he know. Again, when he says
' Some Xs are Ys' he does not mean to imply ' Some Xs are not Ys': that
is, his some may be all, for anything he asserts to the contrary. But when
two propositions, each of which contains the vague some, are conjoined, the
mere meaning may render the conjunction an absurdity unless some take the
force of all. Just as in algebra an equation having two unknown quantities
has the values of those quantities vague ; but when two such equations are
conjoined, those values become definite : so in logic, in which the same thing
occurs. Thus the two propositions 'All Xs are some Ys', and 'All Ys are
some Xs', when true together, force the inference that some must in both
cases be all. Forget that some is that which may be all, and these two
propositions appear to contradict one another : very distinguished logicians
have asserted that they do so. Sir W. Hamilton (Discussions, 2nd edition,
p. 688) calls them incompossible propositions, meaning propositions which
cannot be true together. The word is an excellent one, and much wanted :
but not here. These two propositions are not incompossible, unless some
take into its meaning not-all as well as not-none : and some is never allowed
by logicians to mean not-all. It is needless to argue things so plain : it would
have been needless to state them twice, except for the eminence of the
writers who deny them upon occasion.
15. In f All Xs are Ys', Y is partially spoken of; there may
or may not be more Ys besides : the same of ' Some Xs are Ys '.
In ' No Xs are Ys ', and in ' Some Xs are not Ys ', Y is totally
spoken of; each X spoken of is not any one of all the Ys.
16. By the universe (of a proposition) is meant the collection
of all objects which are contemplated as objects about which
assertion or denial may take place. Let every name which belongs
to the whole universe be excluded as needless : this must be parti-
cularly remembered. Let every object which has not the name
16-20.] UNIVERSAL, ETC. — AFFIRMATIVE, ETC. 13
X (of icldch there are always some) be conceived as therefore
marked with the name x, meaning not-X, and called the contrary
of X. Thus every thing is either X or x ; nothing is both : ' All
Xs are Ys ' means ' No Xs are ys ' : ' No Xs are Ys ' means ( All
Xs are ys ' : ' Some Xs are not Ys ' means •' Some Xs are ys ' :
* Some Xs are Ys ' means ' Some Xs are not ys '.
17. The following enlarged definitions include the definitions
above given, and apply to all the uses of terms and relations in
this work. Let a TERM be total or partial according as every
existing instance must or need not be examined to verify the
proposition. Thus in e Everything is either X or Y ', X and Y
are both partial : an object being examined and found to be X,
the proposition is made good so far as that object is concerned ;
that object may also be Y, but if so, it need not be ascertained :
consequently, Y is partially spoken of; and the same may be said
of X. But in ' Some things are neither Xs nor Ys ', X and Y
are both total : we can only verify it by an object which is not
any one of the Xs and not any one of the Ys.
18. Let a PROPOSITION be universal or particular, according as
the whole universe of objects must or need not be examined to
verify it. Thus f Everything is either X or Y ' is plainly uni-
versal : but f Some things are neither Xs nor Ys ' is particular :
the first object examined may settle the truth of the propo-
sition.
19. Let a PROPOSITION be affirmative which is true of X and X,
false of X and not-X or x ; negative, which is true of X and x,
false of X and X. Thus ( Every X is Y ' is affirmative : * Every
X is X ' is true ; ( Every X is x ' is false. But f Some things are
neither Xs nor Ys ' is also affirmative, though in the form* of a
denial : ' Some things are neither Xs nor Xs ' is true, though
superfluous in expression ; ' Some things are neither Xs nor xs ' is
false. Again, * Everything is either X or Y ' is negative, though
in the form of an assertion : ( Everything is either X or X ' is
false ; ' Everything is either X or x ' is true.
* When contrary terms are introduced, it is impossible to define the
opposition of quality by assertion or denial : for every assertion is a denial,
and every denial is an assertion. The denial 'No X is Y' is the assertion
* All Xs are ys.' The necessary distinction between affirmative and negative
is therefore drawn as in the text : these technical terms are retained, though
perhaps they are hardly the right ones for me to use.
20. Affirmative and negative propositions are said to be of
different quality.
14 FORMS OF ENUNCIATION. [21-23.
21. Let X, totally spoken of, be X) or (X : let X, partially
spoken of, be )X or X(. Let a negative proposition be denoted
by one dot; an affirmative proposition by two dots or none, at
pleasure. I follow Sir William Hamilton in calling this notation
spicular (see § 216, note). So far as yet appears, we have pro-
positions with the following symbols, —
Universal Affirmative X))Y All Xs are some Ys.
Particular Negative X(-(Y Some Xs are not (all) Ys.
Universal Negative X)-(Y All Xs are not (all) Ys.
Particular Affirmative X( ) Y Some Xs are some Ys.
Contraries
Contraries
iesli
22. These forms have been denoted by the letters A, O, E, I,
for many centuries ; A and I from the vowels in hffvrmo ; E and
O from the vowels in n&gQ. The word (all), in parentheses, is
not grammatical: the word any should be substituted for all.
The reason why, for the present, I do not use f any ' will appear in
the sequel.
23. Take the four pairs, X, Y; X,y ; x, Y; x, y; and apply
the four forms above to all four. Sixteen results appear; of
which eight are but a repetition of the other eight. Of the eight*
which are distinct, we have four written above : the remaining
four appear among the following, —
From )) we have X))Y, X))y, x))Y, and x))y. Of these
X))y is obviously X)-(Y. And x))Y is x)-(y, a new form : no
not-X is not-Y ; nothing is both not-X and not-Y ; everything is
either X or Y. This being a universal proposition with both
terms partial, and also a negative proposition, let it be marked
X(-)Y. Again, x))y is x)'(Y, or Y)-(x, or no Y is not-X, or
every Y is X, or some Xs are all Ys, or X((Y.
From () we have XQY, XQy, x()Y, x()y. Here X( )y
is X(-( Y ; and x( )Y is Y( )x, some Ys are not Xs, or all Xs are
not some Ys, or X)-)Y. And x()y is a new form, Some not-Xs
are not-Ys ; some things are neither Xs nor Ys. This is a par-
ticular proposition, affirmative, with X and Y both total : let it be
marked X)(Y.
* Any one who wishes to test himself and his friends upon the question
whether analysis of the forms of enunciation would be useful or not, may try
himself and them on the following question: — Is either of the following
propositions true, and if either, which ? 1. All Englishmen who do not take
snuff are to be found among Europeans who do not use tobacco. 2. All
Englishmen who do not use tobacco are to be found among Europeans who
do not take snuff, llequired immediate answer and demonstration.
24-26.]
FORMS OF ENUNCIATION.
15
24. The eight distinct* forms in which X and Y appear
are as follows; the ungrammatical introduction (all) being made
as before, —
Universal propositions Contrary particular propositions
X))Y All Xs are some Ys X(-(Y Some Xs are not (all) Ys
X)-(Y All Xs are not (all) Ys XQY
X(-)Y Everything is either some X X)(Y
or some Y (or both)
X((Y Some Xs are all Ys X)-)Y
For symmetry, X)-(Y might be read ' Everything is either not
(all) X, or not (all) Y ' ; and X( )Y as * Some things are both
some Xs and some Ys'. This will be better seen when we
come to § 206. At present, however, I preserve the ordinary
reading.
* The following is the comparison of the notation in my formal Logic
with that used in my second and third papers in the Cambridge Transactions,
and in this syllabus.
Some Xs are some Ys
Some things are not either
(all) Xs nor (all) Ys
All Xs are not some Ys
For X))Y,
For X((Y,
For X)-(Y,
For X(-)Y,
X)Y and A,
X(Y and A'
X.Y andE,
x.y and E'
For X()Y,
For X)(Y,
For X(-(Y,
For X)-)Y,
XY and I,
xy and I'
X:Y and O,
Y:X and O'
For X»Y, D, and A, + a
For X || Y, D and A/ + A'
For X(c(Y, D' and A' + O,
For X)o(Y, C, and E, + I'
For X |-| Y, C and E, + E'
For X(o)Y, C' and E' + I,
These comparative notations being fixed in the mind, any part of my Formal
Logic may be read in illustration of the present work. And the detailed
character (§ 64, note) of the latest notation is, if I may judge, so much of a
facilitation, that any reader of the Formal Logic will find it easier to trans-
late the notation as he goes on than to confine himself entirely to the
notation of that work : and this especially as to the tests of validity and the
assignment of the symbol of inference.
25. The thirty-two forms which arise from application of
contraries are as now written, all the eight cases above being
used : the four in each line are of identical meaning.
Universals
X))Y , X)-(y , x(-)Y
X)-(Y X))y x((Y
X(-)Y X((y x))Y
X((Y X(-)y x)-(Y
26. The rule of contraversion — changing a name into its
contrary without altering the import of the proposition — is, Change
also the quantity of the term, and the quality of the proposition.
Thus X))Y is X)-(y and x(-)Y. When both names are contra-
Particulars
*((y
X(-(Y
•XQy
x)(Y
. *»y
x(-)y
XQY
X(-(y ,
x)-)Y
, *)(y
*>(y
X)(Y
X)-)y ,
x(-(Y
, *()y
*))y
X)-)Y
, X)(y
x()Y
. xC(y
16
RELATIONS OF PROPOSITIONS.
[26-30.
verted, change both quantities, and preserve the quality : thus
X(-)Yisx>(y.
27. The rule of conversion* — making the names change
places, without altering the import of the proposition — is, Write
or read the proposition backwards. Thus X))Y is Y((X; or
X))Y may be read backwards, Some Ys are all Xs. That is,
make both the terms and their quantities change places.
* Writers on logic have nearly always meant by conversion merely the
change of place in the terms, without change of place in the quantities. Ac-
cordingly, when the quantities are different, (common) logical conversion
is illegitimate. Thus X))Y and Y))X are not the same : but X()Y and
YQX are the same. There is this difficulty in the way of using the word
conversion in the sense proposed in the text : namely, that common logic has
rooted it in common language that 'Every X is Y' is the converse (true
or false as the case may be) of 'Every Y is X.' Leaving the common
idioms for the student to do as he likes with, I shall, if I have occasion to
speak of a proposition in which terms only are converted, and not quantities,
call it a term- converse.
28. Each universal is inconsistent with the universals of dif-
ferent qualities, and indifferent to the universal of different quan-
tities. Thus X))Y is inconsistent with X)g(Y and X(-)Y, and
neither affirms nor denies X((Y. Each universal affirms the
particulars of the same quality, contradicts the particular of
different quantities, and is indifferent to the particular of the same
quantities. Thus X))Y affirms X()Y and X)(Y, contradicts
X(-(Y, and neither affirms nor denies X)-)Y.
Is inconsistent Neither affirms
with nor denies
)•( 0
Is neither affirmed nor denied by
(0 )( ('( )')
)•( 0 )') C(
29. Contrary names, in identical propositions, always appear
with different quantities. We cannot speak of some Xs without
speaking about all xs ; nor of all Xs without speaking about
some xs.
30. A particular proposition is strengthened into a universal
which affirms it (and more, may be) by altering one of the quan-
tities : thus )•) is affirmed in (•) and in )•( Remember § 16.
Affirms
Contradicts
() )(
C(
)') C(
o
)( 0
)•)
Is affirmed by
Contradicts
)'( C)
))
(( ))
X
)) ((
(0
C) )'(
((
31-36.] SYLLOGISM. 17
31. In a universal proposition, if one term be partial, it has
the amount, not the character, of the quantity of the other: if both,
the quantities of the two terms together make up the whole uni-
verse, with the part common to both, if any, repeated twice.
32. In a particular proposition, the quantity of a partial term
is vague, but remains the same through all forms. And when
both terms are total, the partial quantity still remains expressed :
as in X)(Y, or Some things are neither Xs nor Ys ; which some
things are as many as the xs or ys in the equivalents x()y,
X)-)y,andx(-(Y. '
33. If a proposition containing X and Y be joined with a
proposition containing Y and Z, a third proposition containing X
and Z may necessarily follow. In this case the two first pro-
positions (premises) and the proposition which follows from them
(conclusion) form a syllogism.
34. If an X be a Y, if that same Y be a Z, then the X is the
Z. This is the unit-syllogism from collections of which all the
syllogisms of this mode of treating propositions must be formed.
At first sight it seems as if there were another : if an X be a Y,
if that same Y be not any Z, then the X is not any Z. But this
comes under the first, as follows : the X is a Y, that Y is a z,
therefore the X is a z, that is, is not any Z. The introduction of
contraries brings all denials under assertions.
35. Two premises have a valid conclusion when, and only
when, they necessarily contain unit-syllogisms; and the con-
clusion has one item of quantity for every unit-syllogism so
necessarily contained.
36. And all syllogisms may be derived from the following
combinations : —
)) )) or X))Y Y))Z, or All Xs are Ys and all Ys are Zs.
The conclusion is X))Z, All Xs are Zs : there is the unit-syllogism,
This X is a Y, that same Y is a Z, repeated as often as there are
Xs in existence in the universe. Or, X))Y))Z gives X))Z, or
)) )) gives )).
() )) or X()Y Y))Z, or Some Xs are Ys, all Ys are Zs.
The conclusion is X( )Z, Some Xs are Zs : there is the unit-
syllogism so often as there are Xs in the first premise. Or,
X()Y))Z gives X()Z, or () )) gives ().
(( () or X((Y Y()Z, or Some Xs are all Ys, some Ys are Zs.
The conclusion is X( )Z, Some Xs are Zs : this case is, as to form,
c
18 YAHIETIES OF SYLLOGISM. [36-40.
nothing but the last form inverted. Or, X((Y()Z gives X()Z or
(( )) or X((Y Y))Z or Some Xs are all Ys, All Ys are Zs.
The conclusion is X()Z, Some Xs are Zs, as many as there are
Ys in the universe. Or, X((Y))Z gives XQZ, or (( )) gives ( ).
But this case gives no stronger conclusion than () )) or than (( (),
though it has both premises universal.
These are all the ways in which affirmative premises produce
a conclusion in a manner which has no need to take cognisance of
the existence of contrary terms. And since all negations are con-
tained among affirmations about contraries, we may expect that
application of these cases to all combinations of direct and contrary
will produce all possible valid syllogisms.
37. Apply the form )) )) to the eight varieties XYZ, xYZ,
xyZ, xyz, xYz, XYz, Xyz, XyZ, and contravert x, y, z, when-
ever they appear. Thus )) )) applied to x y Z is x))y))Z, or
X((Y combined with Y(-)Z or X((Y(-)Z or (( (•). The con-
clusion is x))Z or X(-)Z. That is, X((Y(-)Z gives X(-)Z; or,
If some Xs be all [make up all the] Ys, and everything be either
Y or Z, then everything is either X or Z. This process applied
to the eight varieties gives the following eight forms of universal
syllogism, that is, universal premises with universal conclusion.
. )) )) (0 » (( (•) (( (( (•) X . )) X X (( X (•)
Here are all the ways in which two universals give different quan-
tities to the middle term.
38. Apply () )) to the eight varieties and we have eight
minor-particular syllogisms, particular conclusion with the minor
(or first) premise particular,
())) )•))) )((•).)((( )•)>( OX (•((( ('((•)
Here are all the ways in which a particular followed by a uni-
versal give different quantities to the middle term.
39. Apply (( (), and we have eight major-particular syllo-
gisms, particular conclusion with the major (or second) premise
particular.
((() XO )))•). )))( )•((•(. (((•( (•))( (•)»
Here are all the ways in which a universal followed by a par-
ticular gives different quantities to the middle term.
40. Apply (( )) and we have eight strengthened particular
syllogisms, universal premises with particular conclusion.
(()) >()) ))O ))(( XX ((>( (•)(( (•)(•)
40-46.] TEST OF VALIDITY, — RULE OF INFERENCE. 19
Here are all the ways in which two universals give the same
quantity to the middle term.
41. There are 64 possible combinations, of which the 32
enumerated give inference. The remaining 32 may be found by
applying the eight varieties to ( ) (( , )) ( ) , ( ) )( and ( ) ) •) : and
in no case does any inference follow. Thus X()Y and Y((Z
are consistent with any of the eight relations between X and Z,
which should be ascertained by trial.
42. The test of validity and the rule of inference are as
follows, —
There is inference 1. When both the premises are universal ;
2. When, one premise only being particular, the middle term has
different quantities in the two premises.
The conclusion is found by erasing the middle term and its
quantities. Thus )•( (•) gives )••) or )) (§ 21). That is ' No X
is Y, and Everything is either Y or Z ' gives ' Every X is Z '.
43. Premises of like quality give an affirmative conclusion : of
different quality, a negative. A universal conclusion can only
follow from universals with the middle term differently quan-
tified in the two. From two particular premises nothing can
follow.
44. A particular premise having the concluding term strength-
ened, the conclusion is also strengthened, and the syllogism is
converted into a universal : having the middle term strengthened,
the conclusion is not strengthened, and the syllogism is converted
into a strengthened particular syllogism. Thus if () )), with
conclusion (), have the premise () strengthened into )), the syllo-
gism becomes )) )) and yields )). But if () be strengthened into
((, the syllogism becomes (( )) and yields only (), as before.
45. A universal conclusion affirms two particulars : if either
of these be substituted in the conclusion" of the universal syllogism,
the syllogism may be called a universal of weakened conclusion
or a weakened universal. Thus X))Y))Z, made to yield only X( )Z
or X)(Z, instead of X))Z, is a universal of weakened conclusion.
No further notice need be taken of this case.
46. Of the 24 syllogisms of particular conclusion, the con-
clusions are equally divided among (), X> )*)' anc^ (*(• ^^e
following table is one of many modes of arrangement of the
whole.
20
ARRANGEMENTS OF SYLLOGISM.
[46-48.
Premises
Affirmative
Negative
Affirmative
Minor
Affirmative
Major
Universal ^Particular
0
((
))
0
0
((
X C)
(•) X
)) X
X ((
))
X
The middle column contains the universals : and each universal
stands horizontally between the two particulars into which it may
be weakened, by weakening one of the concluding terms. And
each strengthened particular stands vertically between the two
particulars from which it may be formed by altering the quantity
of the middle term in the particular premise only.
47. If two propositions give a third, say A and B give C ;
then, a, b, c, meaning the contrary propositions of A, B, C, it
follows that A, B, c, cannot all be true together. Hence if A, c,
be true, B must be false, or b true : that is, if A, B, give C, then
A, c, give b. Or, either premise joined with the contrary of the
conclusion, gives the contrary of the other premise. And thus
each form of syllogism has two opponent forms . But the order of
terms will not be correct, unless the premise which is retained be
converted. If the order of the terms in the syllogism be XY
YZ XZ, we shall have in one opponent XY XZ YZ, which in
our mode of arrangement must be YX XZ YZ, the retained
premise changing the order of its terms.
Thus the opponent forms of )•) )), which gives )•), are as
follows. First, (•(, retained premise converted; ((, contrary of
conclusion; (•(, contrary of other premise; giving (•( (( and con-
clusion (•(. Secondly, ((, contrary of conclusion; ((, retained
premise converted ; (( , contrary of other premise ; giving (( ((
and conclusion ((.
48. The universal and particular syllogisms can be grouped
by threes, each one of any three having the other two for its
48-53.] OPPONENTS. — QUANTITY OF CONCLUSION. 21
opponents. And these groups can be collected in the following
zodiac, as it may be called.
)) J
O w V:^
The universal propositions at the cardinal points are so placed that
any two contiguous give a universal syllogism, whether read
forwards or backwards, as )•( ((, )) )•(. Join each of these
universals with its contiguous external particular, so as to read in
a contrary direction to that in which the two universals were read,
and a triad is formed each member of which has the other two
members for its opponent forms. As in
>((( >(() ())); or as in (•) )) X (( (OX-
49. The strengthened particulars have weakened universals
(§ 45) for their opponent forms. Thus (( )) with the conclusion
() has )) )•( with the conclusion (•( and )•( (( with the conclusion
)•), for its opponents. And (( (( with the conclusion () has )) )•(
with the conclusion ).) and )•( )) with the conclusion )•) for its
opponents.
50. The partial terms of the conclusion take quantity in the
following manner, —
In universal syllogisms. If one term of the conclusion be
partial, its quantity is that of the other term : if both, one has at
least the quantity of the whole middle term, and the other of the
whole contrary of the middle term.
51. In fundamental particular syllogisms. The partial term or
terms of the conclusion take quantity from the particular premise.
52. In strengthened particular syllogisms. The partial term or
terms take quantity from the whole middle term or its whole
contrary, according to which is universal in both of the premises.
53. These rules run through every form of the conclusion in
which there is a particular term. Thus X))Y))Z gives
1. X))Z in which Z has the quantity of X
2. x((z in which x has the quantity of z
3. x(-)Z, xs as many as ys, and Zs as many as Ys
22 SORITES. — COMPLEX PROPOSITION. [53-57.
Again, X(-)Y)(Z gives X(-(Z, X()z, and x)-)z, in which the
quantities of X( and of )z are the number of instances in the
' some things ' of Y)(Z.
Thirdly, X>(Y>(Z gives X)(Z, x(-(Z, x()z, and X», in
which the quantities of x( and )z are the number of instances
in Y.
54. A sorites is a collection of propositions in which the major
term of each is the minor term of the next, as in
X))Y>(Z(-)T))U)(V((W
or All Xs are Ys, and No'Y is Z, and everything is either Z or
T, and every T is TJ, and No U is V, and Some Vs are all Ws.
55. A sorites gives a valid inference, 1. Universal, when all
the premises are universal, and each intermediate term enters once
totally and once partially ; 2. Particular, when one (and one only)
of the two conditions just named is broken once, whether by con-
tiguous universals having an intermediate of one quantity in both,
or by occurrence of one particular without breach of the rule of
quantity.
56. The inference is obtained by erasing all the intermediate
terms and their quantities, and allowing an even number of dots
to indicate affirmation, and an odd number of dots to indicate
negation.
Thus X))Y>(Z()T>(U(-)V))W gives X)-)W
X(-)Y((Z((T(.)U)(V((W gives X(-(W
X((Y(.)Z>(T(-)U>(V gives X((V
57. We have seen that each universal may coexist with either
the universal of altered quantities or with its contrary : which is
a species of terminal ambiguity. Thus X))Y may have either
X((Y or X)-)Y true at the same time. All these coexistences
may be arranged and symbolised as follows ; giving propositions
which, with reference to the ambiguity aforesaid, have terminal
precision.
1. X)°)Y or both X))Y and X»Y All Xs and some things besides are Ys
2. X| | Y or both X))Y and X((Y All Xs are Ys, and all Ys are Xs
3. X(°(Y or both X((Y and X(-(Y Among Xs are all the Ys and some
things besides
4. X)o(Y or both X)-(Y and X)(Y Nothing both X and Y and some
things neither
5. X|-|Y or both X)-(Y and X(-)Y Nothing both X and Y and every
thing one or the other
6. X(o)Y or both X(-)Y and X()Y Every thing either X or Y and some
things both.
58-63.]
COMPLEX SYLLOGISM.
23
58. If any two be joined, each of which is 1, 3, 4, or 6, with
the middle term of different quantities, these premises yield a
conclusion of the same kind, obtained by erasing the symbols of
the middle term and one of the symbols {o}. Thus X)o(Y(o)Z
gives X)o)Z : or if nothing be both X and Y and some things
neither, and if every thing be either Y or Z and some things both,
it follows that all Xs and two lots of other things are Zs.
59. In any one of these syllogisms, it follows that || may be
written for )o) or (°( in one place, or |*j for either )°( or (o) in one
place, without any alteration of the conclusion, except reducing
the two lots to one. But if this be done in both places, the con-
clusion is reduced to || or |-|, and both lots disappear. Let the
reader examine for himself the cases in which one of the premises
is cut down to a simple universal.
60. The rules of contraversion remain unaltered: thus
X(o)Y)o(Z is the same as X(o(y(o(Z &c.
61. The following exercises will exemplify what precedes.
Letters written under one another are names of the same object.
Here is a universe of 12 instances of which 3 are Xs and the
remainder Ps ; 5 are Ys and the remainder Qs ; 7 are Zs and
the remainder Rs.
P P P .P P
Q Q Q Q Q
R R R R R
We can thus verify the eight complex syllogisms
X)')Y)°)Z P(°)Y)°)Z P(=(Q(°)Z P(
XXX
P P
P P
Y Y Y
Y Y
Q Q
Z Z Z
Z Z
Z Z
In every case it will be seen that the two lots in the middle form
the quantity of the particular proposition of the conclusion.
62. The contraries of the complex propositions are as follows :
Contraries.
X(,(Y
X;-;Y
X),)Y
X)°)Y
X| |Y
X(°(Y
X)"(Y
XI-IY
X(°)Y
Both X))Y and X)-)Y
X))Y - X((Y
X((Y - X(-(Y
X)-(Y - X)(Y
X)-(Y - X(-)Y
XQY -XQY
X(-(Y or X((Y or both
X(-(Y - X)-)Y
X)-)Y - X))Y
X()Y - X(-)Y
X()Y - X)(Y
X)(Y - X)-(Y
X;;Y
X),(Y
63. The propositions hitherto enunciated are cumular : each
one is a collection of individual propositions, or of propositions
about individuals; X.)~)Y is ' All Xs are some Ys '. This pro-
position is an aggregate of singular propositions.
24 EXEMPLAR PROPOSITION. [64.
64. There is a choice between this cumular mode of con-
ception and one which may* be called exemplar • in which each
proposition is the premise of a unit-syllogism: as 'this X is one
Y ', f this X is not any Y '. The distinction is seen in ' A II men
are animals ' and ' Every man is an animal ', propositions of the
same import, of which the first sums up, the second tells off
instance by instance. In the second, every is synonymous with
each and with any.
* The late Sir William Hamilton entertained the idea of completing the
system of enunciation by making the words all (or when grammatically
necessary, any) and some do every kind of duty. He thus put forward, as
the system, the following collection : —
Affirmative
1. All X is All Y
2. All X is some Y X))Y
3. Some X is all Y X((Y
4. Some X is some Y XQY
Negative
5. Any X is not any Y X)'(Y
6. Any X is not some Y X)-)Y
7. Some X is not any Y X(-(Y
8. Some X is not some Y
Of the two propositions which are not in the common system (1 and 8)
the first (§ 14, note f) is X| | Y, compounded of X))Y and X((Y : it is
contradicted by X(-(Y and X)')Y, either or both. The second (8) is true
in all cases in which either X or Y has two or more instances in existence :
its contrary is ' X and Y are singular and identical ; there is but one X,
there is but one Y, and X is Y '. A system of propositions which mixes the
simple and the complex, which compounds two of its own set to make a third
in one case and one only, § 57, and which offers an assertion and denial
which cannot, be contradicted in the system, seems to me to carry its own
condemnation written on its own forehead. From this system I was led to
the exemplar system in the text. For Sir W. Hamilton's defence of his own
views, and objections to mine, see his Discussions on Philosophy, &c. Appen-
dix B. In making this reference, however, it is due to myself to warn the
reader who has not access to the paper criticised that Sir W. Hamilton did
not read with sufficient attention, partly no doubt from ill health. The
consequence is that I must not be held answerable for all that is represented
by him as coming from me. For example, speaking of my Table of exemplar
propositions, he says " And mark in what terms it [the table of exemplars]
is ushered in : — as ' a system . . . .' Nay, so lucid does it seem to its inventor,
that, after the notation is detailed, we are told that it ' needs no explanation? "
The paragraph here criticised had two notations, one of which I called the
detailed notation, because there is more detail in it than in the other : the
other is the old notation, augmented. The first had been sufficiently ex-
plained in what preceded ; the second was, as to the augmentations, new to
the reader. Accordingly, the table being finished, I proceeded thus " The
detailed notation needs no explanation. The form given to the old notation
may be explained thus " Sir W. Hamilton represented me as saying
that after the notation [all the notation, I suppose] is detailed, it [table or
notation, I know not which] needs no explanation. I select this small point
64-69.] EXEMPLAK PROPOSITION. 25
as one that can be briefly dealt with : there are many more, which I shall
probably never notice, unless it be one at a time as occasion of illustration
arises. A very decisive case is exposed in the postscript of my third paper in
the Cambridge Transactions.
65. Quantity is now replaced by mode of selection. There is
unlimited selection, expressed by the word any one : vaguely limited
selection, expressed by some one. When we say some one we
mean that we do not know it may be any one.
66. Let (X and X) now mean any one X: let )X and X(
mean some one X.
67. The propositions are as follows : the first of each pair
being a universal, the second its contrary particular.
Exemplar form. Cumular form.
X)(Y Any one X is any one Y X and Y singular and identical
X(-)Y Some one X is not some one EitherX not singular, or Y not singu-
Y lar ; or if both singular, not identical
X))Y Any one X is some one Y All Xs are some Ys
X(- (Y Some one X is not any one Y Some Xs are not (all) Ys
X((Y Some one X is any one Y Some Xs are all Ys
X) •) Y Any one X is not some one Y All Xs are not some Ys
X)-(Y Any one X is not any one Y All Xs are not (all) Ys
X()Y Some one X is some one Y Some Xs are some Ys.
Six of the forms of this exemplar system are identical with six
of the forms of the cumular system. And these six forms are the
forms of the old logic, if we take care always to read X((Y and
X)-)Y backwards, and to count X)-(Y and X()Y as each a
pair of propositions, by distinguishing the reading forwards from
the reading backwards.
68. The two new forms of the exemplar system (the first and
second above) come under the same symbols as the two new
forms of the cumular system, (•) and )(: but the meanings are
widely different. Both systems contain every possible combi-
nation of quantities, as well in universal as in particular pro-
positions.
69. If the above propositions be applied to contraries, we have
a more extensive system of propositions. I shall not enter on this
enlargement, because the peculiar proposition of this system,
X)(Y, is of infrequent* use in thought as connected with the
consideration of X and Y in opposition to their contraries.
* All necessary laws of thought are part of the subject of logic : but a
small syllabus cannot contain everything. The rejection from logic, and the
rejection from a book of logic, are two very different things. It has not been
D
26 EXEMPLAR SYLLOGISM. [69-71.
uncommon to repudiate rare and unusual forms from the science itself, by
calling them subtleties, or the like. This (§ 73) is not reasonable : but as
to the contents of a work, especially of a syllabus, the time must come at
which any one who asks for more Inust be answered by
Cum tibi sufficiant cyathi, cur dolia quaeris ?
As another example : — I have, § 16, required that no term shall be intro-
duced which fills the whole universe. In common logic, with an unlimited
universe, there is really no name as extensive as the universe except object
of thought. But it is otherwise in the limited universes which I suppose.
A short and easy chapter on names as extensive as the universe might be
needed in a full work on logic, but not in a syllabus.
70. To make a syllogism of valid inference, it is enough that
there be at least one unlimited selection of the middle term, and
at least one affirmative proposition. And the inference is obtained
by dropping all the symbols of the middle term. Thus X((Y(-)Z
shows premises which give the conclusion X(')Z: or ' Some one
X is any one Y and Some one Y is not some one Z ' giving ' Some
one X is not some one Z '.
71. There are 36 valid* forms of syllogism, as follows, read-
ing each symbol both backwards and forwards, but not counting
it twice when it reads backwards and forwards the same, as in
XX, (())•
Fifteen in which X is joined with itself or another, —
XX )()) )((( XX )()•) )((*( )(() )((')
Fifteen in which the syllogism is but an exemplar reading of
a cumular syllogism, —
)))) 0)) (()) ))>(. ((>( OX )))') )')))
Six which give the conclusion (•), —
(((') (OO 0)0
* If Sir William Hamilton's system be taken, there are also 36 valid
forms of syllogism, the same as in the text : but the law of inference is
slightly modified, as follows. When both the middle spicula? turn one way,
as in )) and ((, then any spicula of universal quantity which turns the other
way must itself be turned, unless it be protected by a negative point. Thus
)( (), which in the exemplar system gives )), in the cumular system
gives ().
Exemplar system.
Any one X is any one Y
Some one Y is some one Z
Therefore Any one X is some one Z
Cumular system.
All Xs are all Ys.
Some Ys are some Zs.
Therefore Some Xs are some Zs.
This distinction will afford useful study. The minor premise of the exem-
plar instance implies that there is but one X and one Y.
72-75.] NUMERICALLY DEFINITE SYLLOGISM. 27
72. The exemplar proposition is not unknown. It is of very
frequent use in complete demonstration. When Euclid proves
that Every triangle has angles together equal to two right angles,
he selects, or allows his reader to select, a triangle, and shows
that any triangle has -angles equal to two right angles : and the
force of demonstration is for those who can see that the selection*
is not limited by anything in the reasoning. The exemplar form
of enunciation, then, is of at least as frequent use in purely
deductive reasoning as any other ; and is therefore fitly intro-
duced even into a short syllabus. In any case it is a subject of
logical consideration, as being an actual fonn of thought.
* The limitation of the selection by some detail of process is one of the
errors against which the geometer has especially to guard. I remember an
asserted trisection of the angle which I examined again and again and again,
without being able to detect a single offence against Euclid's conditions. At
last, in the details of a very complex construction, I found two requirements
which were only possible togtther on the supposition of a certain triangle
having its vertex upon the base. Now it happened that one of the angles
at the base of this triangle was the very angle to be trisected : so that the
author had indeed trisected an angle, but not any angle ; he had most satis-
factorily, and by no help but Euclid's geometry, divided the angle 0 into
three equal parts, 0, 0, 0. A modification of his process would have been
equally successful with 180°, which Euclid himself had trisected.
73. The following passage, written by Sir Wrlliam Hamilton
himself, should be quoted in every logical treatise: for it ought
to be said, and cannot be said better. " Whatever is operative in
thought, must be taken into account, and consequently be overtly
expressible in logic ; for logic must be, as to be it professes, an
unexclusive reflex of thought, and not merely an arbitrary selec-
tion— a series of elegant extracts- — out of the forms of thinking.
Whether the form that it exhibits as legitimate be stronger or
weaker, be more or less frequently applied; — that, as a material
and contingent consideration, is beyond its purview."
74. The heads* of the numerically definite proposition and
syllogism are as follows, —
Let u be the whole number of individuals in the universe.
Let x, y, z, be the numbers of Xs, Ys, and Zs. Then u— x, u—y,
u — z are the numbers of xs, ys, and zs.
* On this subject I have given only heads of result, the demonstrations
of which will be found in my Formal Logic.
. 75. Let mXY mean that m or more Xs are Ys. Then mXy
means that m or more Xs are ys, or not Ys. And wiYX and
myX have the same meanings as TnXY and wXy.
28 NUMERICAL PROPOSITION AND SYLLOGISM. [76-82.
76. Let a proposition be called spurious when it must of
necessity be true, by the constitution of the universe. Thus, in a
universe of 100 instances, of which 70 are Xs and 50 are Ys, the
proposition 20 XY is spurious : for at least 20 Xs must be Ys,
and 20 XY cannot be denied, and need not be affirmed as that
which might be denied.
77. Let every negative quantity be interpreted as 0 : thus
(6-10) XY means that none or more Xs are Ys, a spurious pro-
position.
78. The quantification of the predicate is useless. To say
that mXs are to be found among nYs, is no more than is said in
raXY. To say that raXs are not any one to be found among any
lot of nYs is a spurious proposition, unless m + n be greater than
both x and y, in which case it is merely equivalent to both of the
following, (m + n — y) Xy, and (m + n — #) Yx, which are equi-
valent to each other.
79. In raXY, the spurious part, if any, is (x + y — w)XY;
the part which is not spurious is (m + u — x — ?/)XY. For each
instance in the last there must be an x which is y. The follow-
ing pairs of propositions are identical.
m XY and (m+u — x — y) xy
mXy and (m+y— T) xY
m xY and (m +x — y) Xy
wixy and (m+x+y — M) XY
80.
Their contraries.
(x+l-m) Xy (y+l_m) xY
(x+l — TW) XY (u+l—y-m)xy
(u+l — ar-m) xy (y + l-wi) XY
Identical propositions.
TO XY (m+u— x— y) xy
m Xy (m+y — x) xY
m xY (m+x — y) Xy
77i xy (m+x+y — u) XY
81. From mXY and wYZ we infer (in + n—y) XZ, or its
equivalent (m + n + u — a: — y — z)xz. The four following forms
include all the cases of syllogism : the first two columns show the
premises, the second two the identical conclusions, —
m XY n YZ (m+n— y) XZ (m+n+u— x—y—z) xz
m Xy 7i YZ (m+n— x) xZ (m+n-z) Xz
m XY n yZ (m+n— z) Xz (m+n— x) xZ
. m Xy n yZ (m+n+y— x— z) xz (m+n+y— w)XZ
82. When either of the concluding terms is changed into its
contrary, the corresponding changes are made in the forms of
inference. Thus to find the inference from mxy and wyz, we
82-86.] NUMERICALLY DEFINITE PROPOSITION. 29
must, in the fourth form, write x for X, z for Z, X for x, Z for z,
u — x for x, and u — z for z.
83. A spurious premise gives a spurious conclusion: and
premises neither of which is spurious may give a spurious con-
clusion. A proposition is only spurious as it is known to be
spurious : hence when u, x, y, z are not known, there are no
spurious propositions.
84. Every proposition has two forms, one of names contrary
to the other, both spurious, or neither. Whenever X()Y is true
in a manner which, by the constitution of the universe, might
have been false, then x()y, or X)(Y is also true in the same
manner. The ordinary syllogism would have two such contra-
nominal forms of one conclusion, and, properly speaking, has two
such forms. When the conclusion is universal, we know it has
them: for X))Z is x((z, X)-(Z is x(')z, &c. These we may see
to be the contranominal conclusions of the numerical syllogism.
For X))Y is #XY, and Y))Z is yYZ, whence O+?/-r/)XZ and
(x+y— y + u — x— ^)xz, or #XZ, which'is X))Z, and (u — z) xz,
which is x((z. Again, let X()Y be wXY, then, Y))Z being
?/YZ, we have (m+y— ?/)XZ, or mXZ, and (m + u— x— z) xz,
its equivalent. If x, z, u, be known, then if m XZ be any thing
except what must be, we have m + u~>x + z, and (m + u—x — z) xz
is x( )z or X)(Z. As it is, x, y} u, being unknown, we have raXZ
certainly true, be it spurious or not, and we can say nothing
of (m+u — x — z) xz.
85 . Syllogisms with numerically definite quantity rarely occur,
if ever, in common thought. But syllogisms of transposed quan-
tity occur, in which the number of instances of one term is the
whole possible number of instances of another term. For example;
— cFor every Z there is an X which is Y; some Zs are not
Ys'. Here we have zXY and wyZ ; whence (z + n — ^)Xz and
(z-\-n — #)xZ. The first is wXz, a case of X('(Z ; some Xs are
not Zs. Thus, ' For every man in the house there is a person
who is aged; some of the men are not aged': it follows, and
easily, that some persons in the house are not men ; but not by
any common form of syllogism.
86. Of terms in common use the only -one which can give the
syllogisms of this chapter is e most '. As in
Most Ys are Xs ; most Ys are Zs ; therefore some Xs are Zs.
Most Ys are Xs ; most Ys are not Zs ; therefore some Xs are
not Zs.
30
FIGURE. CONVERTIBILITY. — TRANSITIVENESS. [86-92.
Most Ys are not Xs; most Ys are not Zs; therefore some
things are neither Xs nor Zs.
87. Each one of our syllogisms may be stated in eight
different ways, each premise and the conclusion admitting two
different orders. Thus X))Y, Y))Z, giving X))Z may be stated
as Y((X Y))Z giving Z((X, or as X))Y, Y))Z, giving Z((X, &c.
All the orders are as follows —
I.
XY YZ XZ
YX ZY ZX
II.
XY ZY XZ
XY ZY ZX
III.
YX YZ XZ
YX YZ ZX
IV.
YX ZY XZ
XY YZ ZX
88. Whenever there is a first and a second, let them be called
minor and major. Write the premises so that the minor premise
shall contain the minor term of the conclusion (though it has
long been most common to write the major premise first), and
we have
I.
XY YZ
XZ
XY
II.
ZY
XZ
III.
YX YZ XZ
IV.
YX ZY XZ
These orders are called the four figures. Thus X))Y Y))Z
giving X))Z is stated in the first figure; X))Y Z((Y giving
X))Z is stated in the second figure ; Y((X Y))Z giving X))Z is
stated in the third figure ; Y((X Z((Y giving X))Z is stated in
the fourth figure.
89. The .first figure may be called the figure of direct transi-
tion : the fourth, which is nothing but the first with a converted
conclusion, the figure of inverted transition ; the second, the figure
of reference to (the middle term) ; the third, the figure of reference
from (the middle term).
90. The first figure is the one which has been used in our
symbols; and it is the most convenient. The distinction of
figure is wholly useless in this tract, so far as we have yet
gone: it becomes necessary when we take a wider view of the
copula.
91. A convertible copula is one in which the copular relation
exists between two names both ways : thus ' is fastened to ' ' is
joined by a road with ' ' is equal to, ' ' is in habit of conversation
with/ &c. are convertible copulse. If ' X is equal to Y ' then ' Y
is equal to X ' &c.
92. A transitive* copula is one in which the copular relation
joins X with Z whenever it joins X with Y and Y with Z. Thus
* is fastened to ' is usually understood as a transitive copula : ' X
92-93.] EXTENSION OF COPULA. 31
is fastened to Y ' and ' Y is fastened to Z ' give ' X is fastened
to Z'.
* All the copulas used in this syllabus are transitive. The intransitive
copula cannot be treated without more extensive 'consideration of the combi-
nation of relations than I have now opportunity to give : a second part of
this syllabus, or an augmented edition, may contain something on this
subject.
93. The junction of names by appiirtenance to one object, the
copula hitherto used, is both convertible and transitive : and from
these qualities, and from these alone, it derives the whole of its
functional power in syllogism. Any copula which is both transi-
tive and convertible will give precisely the syllogisms* of our
system, and no others : provided always that if contrary names
be introduced, no instance of a name can, either directly or by
transition, be joined by the copula with any instance of the con-
trary name. For example, let the copula be some transitive and
convertible mode of joining or fastening together, whether of
objects in space or notions in the rnind, &c. : so that no X is ever
joined with any x, &c. The following are two instances of
syllogism. •
X))Y)'(Z. Every X is joined to a Y; no Y is joined to a Z ;
therefore no X is joined to a Z. For if any X were joined to a Z,
that Z would be joined to an X, and that X to a Y, whence that
Z would be joined to a Y, which no Z is.
X(-)Y)(Z. Everything is joined either to an X or to a Y;
Some things are joined neither to Ys nor to Zs ; therefore Some
Xs are not joined to Zs. For if every X were joined to a Z,
then every thing being (by the first premise) joined either to an
X or to a Y, is joined either to a Z or to a Y, which contradicts
the second premise.
* The logicians are aware that many cases exist in which inference about
two terras by comparison with a third is not reducible to their syllogism.
As ' A equals B ; B equals C ; therefore A equals C.' This is not an instance
of common syllogism : the premises are ' A is an equal of B ; B is an equal
of C.' So far as common syllogism is concerned, that 'an equal of B'
is as good for the argument as 'B' is a material accident of the meaning of
' equal.' The logicians accordingly, to reduce this to a common syllogism,
state the effect of composition of relation in a major premise, and declare
that the case before them is an example of that composition in a minor
premise. As in, A is an equal of an equal (of C) : Every equal of an equal is
an equal ; therefore A is an equal of C. This I treat as a mere evasion.
Among various sufficient answers this one is enough : men do not think as
above. When A = B, B = C, is made to give A = C, the word equals is a
32 EXTENSION OF COPULA. [93-97.
copula in thought, and not a notion attaeJied to a predicate. There are
processes which are not those of common syllogism in the logician's major
premise above : but waiving this, logic is an analysis of the form of thought,
possible and actual, and the logician has no right to declare that other than
the actual is actual.
94. The convertibility of the copula renders the inference
altogether independent of figure.
95. Let the copula be inconvertible, as in 'X precedes Y'
from which we cannot say that ' Y precedes X '. We must now
introduce the converse relation 'Y follows X', and the conversion
of a proposition requires the introduction of the converse copula.
96. This extension, when contraries are also introduced, is
almost unknown in the common run of thought : but it may serve
for exercise, and also to give an idea of one of those innumerable
systems of relation with which thought unassisted by systematic *
analysis would probably never become familiar.
* The uneducated acquire easy and accurate use of the very simplest
cases of transformation of propositions and of syllogism. The educated,
by a higher kind of practice, arrive at equally easy and accurate use of some
more complicated cases : but not of all those which are treated in ordinary
logic. Euclid may have been ignorant of the identity of " Every X is Y" and
" Every not-Y is not-X," for any thing that appears in his writings : he makes
the one follow from the other by new proof each time. The followers of
Aristotle worked Aristotle's syllogism into the habits of the educated world,
giving, not indeed anything that demonstrably could not have been acquired
without system, but much that very probably would not. The modern
logician appeals to the existing state of thought in proof of the completeness
of the ordinary system : he cannot see anything in an extension except what
he calls a subtlety. In the same manner a country whose school of arith-
metical teachers had never got beyond counting with pebbles would be able
to bring powerful arguments against pen, ink, and paper, the Arabic
numerals, and the decimal system. They would point to society at large
getting on well enough with pebbles, and able to do all their work with such
means : for it is an ascertained fact that all which is done by those to whom
pebbles are the highest resource, is done either with pebbles or something
inferior. I have long been of opinion that the reason why common logic is
lightly thought of by the mass of the educated world is that the educated
world has, in a rough way, arrived at some use of those higher developments
of thought which that same common logic has never taken into its compass.
Kant said that the study of a legitimate subtlety (necessary but infrequent
law of though*,) sharpens the intellect, but is of no practical use. Sharpen
the intellect with it until it is familiar, and it will then become of practical
use. A law of thought, a necessary part of the machinery of our minds, of
no practical use ! Whose fault is that ?
97. Let any two names be connected by transitive converse
relations, for an example say gives to and receives from (under-
97-100.] TRANSITIVE AND INCONVEETIBLE COPULA.
33
standing that when X gives to Y and Y gives to Z, X gives to Z)
in the following way, —
No X gives to another X, either directly or transitively, &c.
Every X either gives to a Y or receives from a y, but not both
Every x either gives to a Y or receives from a y, but not both
Every X which gives to a Y, receives from no other Y, &c.
The same of all combinations of names, as Y with X and x, &c.
98. The following are the propositions used, with their
symbols ; and in a corresponding way for any other copula which
may be used, —
X)')Y Every X gives to a Y
X('-(Y Some Xs give to no Ys
X)'-(Y No X gives to a Y
X(')Y Some Xs give to Ys
X('-) Y In every relation, something
either gives to an X or re-
ceives from a Y (or both)
X)'(Y In some relations, nothing
gives to any X nor receives
from any Y
X('(Y Some Xs give to all the Ys
X)'-)Y All Xs do not give to some
Ys
X))'Y Every X receives from a Y
X(-('Y Some Xs receive from no Ys
X)-('Y No X receives from a Y
XQ'Y Some Xs receive from Ys
X(')'Y In every relation, something
either receives from an X
or gives to a Y (or both)
X)('Y In some relations, nothing
receives from any X nor
gives to any Y
X((TT Some Xs receive from all Ys
XyyY All Xs do not receive from
some Ys
99. Propositions are changed into others identical with them by
this addition to the rule in § 26 : — When one term is contraverted,
the relation is also converted: when both, the relation remains.
In the following lists the four in each line are identical, —
X)')Y
X)'-(Y X))'y
X('-)Y X(('y
X('(Y X(-)'y
x(('Y
x))'Y
x)-CY
x('(y
x)')y
XQY
X)'(Y
X()'y
X)('y
x)('Y x)'-)y
x)-)'Y x)'(y
x(-CY x(')y
The relations may be converted throughout.
100. To prove an instance, how do we know that X)')Y is
identical with x(')'Y? If every X give to a Y, the remaining
Ys, if any, do not give to any Xs, by the assigned conditions of
meaning : consequently those remaining Ys receive from xs. As
to ys, none of them can give to Xs, for then they would give to
Ys : therefore all receive from xs. Conversely, if x(-/Y, no X
can receive from y, for then neither could that y receive from
x, nor could that X give to Y : so that there would be a relation
in which neither does any thing give to Y, nor receive from x.
Consequently, every X gives to Y.
34 SYLLOGISM OF INCONVEETIBLE COPULA. [101-105.
101. Let the phases of a figure depend on the quality of the
premises in the following manner: + meaning affirmative, and
— negative, remember the phases in the following order, —
102. For the four figures, let these four phases be the first or
primary phases: thus H is the primary phase of the third
figure. To put the other phases in order, read backwards from
the primary phases, and then forwards.
1 2 34
Figure I. + + — + + —
II. - + ++ + -
III. +- - + + + - -
Thus + — is the third phase of the second figure.
103. In the primary phases, the direct copula may be used
throughout. When one premise departs from the primary phase
in quality, the converse copula must be used in the other ; when
both, in the conclusion. This addition is all that is required in the
treatment of the syllogism of inconvertible copula.
104. Thus,* the premises being X)-(Y Z))Y, we have the
primary phase of the second figure, whence X)-(Z with the direct
copula. That is, if no X give to a Y, and every Z give to a Y,
no X gives to a Z. For if any X gave to a Z, that Z giving to
a Y, that X would give to a Y, which no X does. Now con-
travert the middle term, and we have X))y Z)-(y, the phase of
the second figure in which both premises differ from the primary
phase. Hence Every X gives y, No Z gives y, yields no X is
given by Z. For if any X were given by Z, a y would be given
by that Z, which is given by no Z. But 'no X gives Z' will
not do.
* The reader may exercise himself in the formation of more examples.
The use of such a developement as the one before him is this. Every study
of a generalisation or extension gives additional power over the particular
form by which the generalisation is suggested. Nobody who has ever
returned to quadratic equations after the study of equations of all degrees,
or who has done the like, will deny my assertion that Ol ^i-rti ftxi^nav may
be predicated of any one who studies a branch or a case, without afterwards
making it part of a larger whole. Accordingly, it is always worth while to
generalise, were it only to give power over the particular. This principle, of
daily familiarity to the mathematician, is almost unknown to the logician.
105. The common system of syllogism, which being nearly
complete in the writings of Aristotle may be called Aristotelian,
105-108.] AEISTOTELIAN SYLLOGISM. 35
is as much as may be collected out of the preceding system by
the following modifications, —
1 . The exclusion of all idea of a limited universe, of contrary
names, and of the propositions (•) and )(. 2. The exclusion of
all right to convert a proposition, except when its two terms have
like quantities, as in )•( and (). Thus X))Y must not be read as
* Some Ys are all Xs '. But X))Y may undergo what is called
the conversion per accidens : that is, X))Y affirming X( )Y, which
is Y()X, X))Y may be made to give YQX. 3. The exclusion
of every copula except the transitive and convertible copula.
4. The addition of the consideration of the identical pairs X)'(Y
and Y)-(X, X()Y and Y()X, as perfectly distinct propositions.
5. The introduction of the distinction of figure. 6. The writing
of the major and minor propositions first and second, instead of
second and first: thus X))Y))Z is written 'Y))Z, X))Y, whence
106. There are four forms of proposition : A, or X))Y or
Y))X, not identical; E, or X>(Y or Y)-(X, identical; I, or
X()Y or Y()X, identical; O, or X(-(Y or Y(-(X, not identical.
107. There are four fundamental syllogisms in the first figure,
each of which has an opponent in the second, and an opponent in
the third. There are three fundamental syllogisms in the fourth
figure, each of which has the other two for opponents. Alto-
gether, fifteen fundamental syllogisms. There are three strength-
ened particular syllogisms, two in the third figure, and one in the
fourth : and one weakened universal, in the fourth figure. In all,
nineteen forms.
108. Every syllogism has a word attached to it, the vowels of
which are those of its premises and conclusion. In the first figure
the consonants are all unmeaning; in the other figures some of
the consonants give direction as to the manner of converting into
the first figure. Thus K denotes that the syllogism cannot be
directly converted into the first figure, though its opponent in the
first figure may be used to force its conclusion. S means that
the premise whose vowel precedes is to be simply converted. P,
which occurs in all the strengthened particulars and the weakened
universal, means that the conversion per accidens is to be employed
on the preceding member. M means that the premises must be
transposed in order. Each syllogism converts into that syllogism
of the first figure which has the same initial letter. G is an
addition of my own, presently described ; it must be left out when
36 ARISTOTELIAN SYLLOGISM. [108-111.
the old system is to be just represented. R, N, T, have no signi-
fication. The following are the names put together in memorial*
verse.
* The best attainable exposition of logic in the older form, with modern
criticism, is Mr. Mansel's edition of Aldrich's compendium. Should a reader
of this work desire more copious specimens of old discussion, he may perhaps
succeed in obtaining Crackanthorpe's Logica Libri quinque (4to, 1622 and
1677). Sanderson's Logic is highly scholastic in character. For a compen-
dium of mediaeval logic, ethics, physics, and metaphysics, I have never found
anything combining brevity and completeness at all to compare with the
Precepta Doctrince Logicce, Ethicce, &c. of John Stierius, of which seven or
eight editions were published in the seventeenth century (from 1630 to 1689,
or thereabouts) and several of them in London. There is a large system of
the older logic in the lustitutiones Logicce of Burgersdicius, and a great
quantity of the metaphysical discussion connected with the old logic in
Brerewood de Predicabilibus et Predicamentis. As all these books were printed
in England, there is more chance of getting them than the foreign logical
works, which are very scarce in this country. For more than usual infor-
mation on parts of the history of logical quantity, a subject now exciting much
attention, see Mr. Baynes's New Analytic, in which will be found much
valuable history so completely forgotten that it is as new as if he had
invented it himself.
109. Barbara, Cela^rent, Darii, Feri^oque prioris:
Cesare^r, Camestres, Festino<jr, Baroko secundae :
Tertia Darapt<jri, Disa^mis, Dati^rsi, Felapton,
Bokardo, Ferison habet. Quarta insuper addit
Bramanti^rp, Came<mes, Dimari^s, Fe^sapo, Fre^rsison.
110. I leave the verification of what has been said as an
exercise. As an example of reduction into the first figure take
the syllogism Camestres from the second figure.
A
E
E
111. The letter Gr indicates (§ 103) the member in which,
when a transitive but not simply convertible copula is used, the
copula is to be the converse of the copula employed in the other
members. Thus Cela^rent shows that the minor premise must
have the converse copula. Suppose, for example, that the copula?
are gives to (transitively understood) and receives from. Then
' No Y gives to an X ; every Z receives from a Y ' yield e No Z
gives to an X'. For if any X received from a Z, which (second
II. Camestres reduced into I. Celarent.
Every Z is Y , , No Y is X E
No XisY (s) Every Z is Y A
No
X is Z (s)
No ZisX E
111-116.] NAMES, SUBJECTIVE AND OBJECTIVE. 37
premise) receives from a Y, that X would receive from a Y, which
contradicts the first premise.
112. In the preceding articles I have considered hardly any-
thing but mere assertion or denial of concomitance, of any sort or
kind whatsoever. I now proceed to more specially subjective
views of logic.
o
113. A term or name may be in one word or in many. It
describes, pictures, represents, but does not assert nor deny. Its
object must exist, whether in thought only, or in external nature
as well : and everything which does not contradict the laws of
thought may be the object of a term. But sometimes the thinker's
universe will be the whole universe of thought ; sometimes only
the objective universe of external reality ; sometimes only a part
of one or the other.
114. Terms are used in four different senses. Two objective,
directed towards the external object, or to use old phrases, of first
intention, or representing first notions. Two subjective, directed
towards the internal mind, of second intention, or representing
second notions.
115. In objective use the name represents, 1. The individual
object, unconnected with, and unaggregated with, any other object
of the same name ; 2. The individual quality, forming part of, and
residing in, the individual object. One name may, at different
times, represent both : thus animal, the name of an object, is the
name of a quality of man. In fact, quality is but object con-
sidered as component of another object. The quality white, a
component of the notion of an ivory ball, is itself an object of
thought.
116. The objective* uses of names have been considered as
the bases of propositions and syllogisms, in the preceding part of
this tract.
* The ordinary syllogism of the logicians, literally taken as laid down
by them, is objective, of first intention, arithmetical. I call it the logician's
abacus. When the educated man rejects its use, and laughs at the idea of
introducing such learned logic into his daily life, I hold his refusal to be in
most cases right, and his reason to be entirely wrong. He has, and his peers
always have had, some command of the subjective syllogism, the combination
of relations to which I shall come. He has no more occasion — in most cases
— to have recourse to the logical abacus for his reasoning, than to the
chequer- board, or arithmetical abacus, for adding up his bills. I hold the
combination of relations to be the actual organ of reasoning of the world at
large, and, as such, worthy of having its analysis made a part of advanced
38 CLASS AND ATTRIBUTE. [116-121.
education; the logician's abacus being a fit and desirable occupation for
childhood.
117. In subjective use the name represents, 1. A class, a
collection of individual objects, named after a quality which is in
thought as being in each one : 2. An attribute, the notion of
quality as it exists in the mind to be given to a class. Attribute
is to individual quality what class is to individual object.
Between the notion of a class, and the notion of an object, though
the name be the same in both cases, there is this distinction. The
class-name belongs to a number of objects : the object belongs to
a number of class-names ; for it may be named after any one of
its qualities. We have classes aggregated of many objects : and
objects compounded of many classes. But in this second case the
object is said to have many qualities. The class is a whole of one
kind: the object is a whole of another kind. This distinction
emerges the moment a name begins to be a universal, that is,
belonging to more than one object.
118. Class and attribute are units of thought: a noun of
multitude is not a multitude of nouns. When we are fortunate
enough to get four distinct names, we readily apprehend all these
distinctions. This happens in the case of our own species : the
objects men, all having the quality human, give to the thought
the class mankind, distinguished from other classes by the attri-
bute humanity. Should any reader object to my account of the
four uses of a name, he can, without rejection of anything else in
what follows, substitute his own account of the four words man,
human, mankind, humanity.
119. With grounds of classification, and reasons for nomen-
clature, logic has nothing to do. Any number of individuals,
whether yet unclassed, or included in one class, or partly in one
class and partly in another, may be constituted a new class, in
right of any quality seen in all, by which an attribute is affixed
to the class in the mind.
120. The term X, or Y, or Z may and does denote, at one
time or another, the individual, the quality, the class, or the
attribute. Any one who finds the distinction useful might think
of the individual X, the quality X-ic, the class X-kind, and the
attribute X-ity.
121. Identical terms are those which apply to precisely the
same objects of thought, neither representing more than the other:
121-124.] umvEKSE. — CONTKARIES. 39
so that identical terms are different names of the same class.
Thus, for this earth, man and rational animal are identical terms.
The symbol X||Y will be used (§ 57) to represent that X and Y
are identical. When of two identical terms one, the known, is
used to explain the other, the unknown, the first is called the
definition of the second.
122. The whole extent of matter of thought under consider-
ation I call the universe. In common logic, hitherto, the universe
has always been the whole universe of possible thought.
123. Every term which is used (§ 16) divides the universe
into two classes; one within the term, the other without. These
I call contraries : and I denote the contrary class of X, the class
not-X, by x. When the universe is unlimited contrary names
are of little effective use ; not-man, a class containing every thing
except man, whether seen or thought, is almost useless. It is
otherwise in a limited universe, in which contraries, by separate
definiteness of meaning, cease to be mere negations each of the
other, and even acquire separate* and positive names. Thus, the
universe being property under English law, real and personal are
contrary classes. Logic has nothing to do with the difficulties of
allotment which take place near the boundary: with the decision
upon those personals, for example, which, as the lawyer says,
savour of the realty. The lawyer must determine the classifi-
cation, and logic investigates the laws of thought which then
apply. If the lawyer choose to make an intermediate class,
between real and personal, then real and personal are no longer
logical contraries.
That X and Y are contrary classes is denoted (§ 57) by
* The most amusing instance which ever came within my own knowledge
is as follows. A friend of mine, in the days of the Irish Church Bill, used
to discuss politics with his butcher : one day he alluded to the possible fate
of the Establishment. 'Do you mean do away with the church'? asked the
butcher. 'Yes', said my friend, 'that is what they say'. 'Why, sir, how
can that be'? was the answer ; 'don't you see, sir, that if they destroy the
church, we shall all have to be dissenters'!
124. Terms may be formed* from other terms, —
1. By aggregation, when the complex term stands for every-
thing to which any one or more of the simple terms applies. Thus
animal is the aggregate of (the aggregants] man and brute.
2. By composition, when the complex term stands but for
40 AGGREGATION AND COMPOSITION. [124-129.
everything to which all the simple terms apply. Thus man is
compounded of (the components) animal and rational.
3. By mixture of these two methods of formation.
* The reader must carefully remember that we are now engaged (§ 4)
especially upon the esse quod habent in anima : and, if not accustomed to
middle Latin, he must remember also that esse is made a substantive, mode
of being. Animal cannot be divided into man and brute except in a mind.
Logical composition must be distinguished from physical or metaphysical.
Light always consisted of the prismatic components ; but, before Newton, it
was not a logical quality of light that it is to be conceived as decomposible.
Accordingly, a compound of qualities, though it may constitute a full dis-
tinctive definition of an object of thought, can never be accepted as a full
description : there may be many more. The logician therefore must, in
thinking of a compound, imitate the genial Dean Aldrich, the author of the
Compendium of Logic to which so many have been indebted, in the structure
of his fifth reason for drinking.
125. The aggregate of X, Y, Z will be represented by
(X, Y, Z) : the term compounded of X, Y, Z will be represented
by (X-Y-Z) or (XYZ).
126. The aggregate name belongs to each of the aggregants:
but the compound name does not belong to each of the com-
ponents, necessarily.
127. An aggregate is not impossible if either of its aggregants
be impossible, or if two of them be contradictory : but a compound
is impossible in either of these cases.
128. In these and all other formulae, care must be taken to
remember that the logical phrase implies nothing: the phrases of
ordinary conversation frequently imply, in addition to what they
express. Thus * some living men breathe ' and ' every man is
either animal or mineral ' are colloquially false by what they
imply, but logically true because the logical use implies nothing.
129. A class may be compounded of classes, as well as aggre-
gated : thus the class marine is compounded of the classes soldier
and sailor. An attribute may be aggregated of attributes, as well
as compounded: thus Adam Smith's attribute productive is ag-
gregated of land-tilling, manufacturing, &c. But composition of
classes and aggregation of attributes are infrequent. Any name
may be thought of either as a class, or an attribute (or character,
as it is often called): and it is usual and convenient to think of
class when aggregation is in question, and of attribute when com-
position is in question. So we rather say the marine unites the
cfiaracters of the soldier and the sailor : the productive classes (as
129-134.] EXTENSION AND INTENSION. 41
Adam Smith said) consist of farmers, manufacturers, &c. But
all this for convenience, not of necessity : and the power of un-
learning usual habits must be acquired. All modes of thought
should be considered: the usual, because they are usual; the
unusual, that they may become usual.
130. The words of aggregation are either, or: of composition,
both, and. Thus (X, Y) is either X or Y (or both) ; (XY) is both
X and Y.
131. The more classes aggregated, so long as each class has
something not contained in any of the others, the greater the
extension* of the aggregate term. The more attributes com-
pounded, so long as each attribute has some component not
contained in the others, the greater the intension. Animal has
more extension than man : man has more intension than
animal.
* The logicians have always spoken of 'all men' as constituting the
'extent' of the term man : thus the whole extent of man is part of the extent
of animal. They have chosen that their more and less should be referred to
by phrases derived rather from the notion of area than from that of number.
Hence arise certain forms of speech which, when quantity is applied to the
predicate, are not idiomatic; as 'All man is some animal'.
There are savage tribes which have not sufficient idea of number even
for their own purposes : among them, when a dozen or more of men are to be
indicated, an area sufficient to contain them is marked out on the ground.
The speculative philosophers of the middle ages were in something like the
same position : though the mercantile world was well accustomed to large
numbers, the philosophical world, excepting only some of the mathematicians,
was very awkward at high numeration. The works on theoretical arithmetic
show this well. I have been straining my eye over the twelve books of the
Arithmetical Speculativa of Gaspar Lax of Arragon (Paris, folio, 1515) to
detect, if I could, a number higher than a hundred ; and I have found only
one, the date.
132. The name of greatest extension, and of least intension, of
which we speak, is the universe.
133. The contrary of an aggregate is the compound of the
contraries of the aggregants : either one of the two X, Y, or both
not-X and not-Y ; either (X, Y) or (xy). The contrary of a
compound is the aggregate of the contraries of the components ;
either both X and Y, or one of the two, not-X and not-Y ; either
(XY), or (x, y).
134. The following are exercises on complex terms, —
' Both or neither ' and f one or the other, not both ', are con-
traries. That is (XY, xy) and (Xy, Yx) are contraries. Now
F
42 PROPOSITION AND JUDGMENT. [134-139.
the contrary of the first is (x, y)-(X, Y), which is (xX, xY, yX, yY),
which is (xY, yX) since xX, y Y, are impossible.
X||(A,B)C gives x||(ab,c)
X||(A,B)(C,D) - x|j(ab,cd)
X||AB,C — x||(a,b)c
X||A,B(C,D) gives x||a(b,_cd)
X||(A,BC)(D,EF)- x||(ab,c),(de,f)
X||(A,B,aC) — x||abc
Deduce these, and explain the last.
135. A term given in extension, as (A, B, C), has its contrary
given in intension, (abc); and vice versa. Aggregates or com-
ponents of either only enable us to deduce components or aggre-
gates of the other.
136. A proposition is the presentation, for assertion or denial,
of two names connected by a relation : as ( X in the relation L to
Y.' A judgment is the sentence of the mind upon a proposition:
certainly true, more or less probable, certainly false. Propositions
without accompanying judgment hardly occur: so that propo-
sition comes to mean, by abbreviation, proposition accompanied by
judgment.
137. The distinction between certainty and probability is
usually treated apart from logic, as a branch of mathematics. A
few of the leading results, relative to authority and argument, will
be afterwards given.
138. The purely formal proposition with judgment, wholly
void of matter, is seen in ' There is the probability a that X is
in the relation L to Y'. From the purely formal proposition no
inference can follow. In all elementary logic, the terms are
formal, the relation* material, and the judgment absolute assertion
or denial (or, as a mathematician would say, the probabilities
considered are only 1 and 0).
* The logician calls ' Every man is animal' a material instance of the
formal proposition 'Every X is Y'. He will admit no relation to be formal
except what can be expressed by the word is : he declares all other relations
material. Thus he will not consider ' X equals Y ' under any form except
' X is an equal of Y '. He has a right to confine himself to any part he
pleases : buj he has no right, except the right of fallacy, to call that part the
whole.
139. Contrary propositions are a pair of which one must be
true and one false : as ' he did ', ' he did not '; or as ' Every X is
Y', * Some Xs are not Ys '. Contraries contradict* one another;
but so do other propositions. Thus e All men are strong ' and
' all men are weak ' contradict one another to the utmost : the
second says there is not a particle of truth in the first. But the
contrary merely says there is more or less falsehood : to ' all men
139-146.] INFERENCE AND PROOF. 43
are strong ' the contrary is * There are [man or] men who are not
strong'.
* In the usual nomenclature of logicians, what I call the contrary is
called the contradictory, as if it were the only one. In common language,
when two persons disagree, we say they are on contrary sides of the question :
in the usual technical language of logic, this would mean that if one should
say all men are strong the other says no man is strong. But in common
language, the one who maintains the contrary is he who advocates anything
which the other is opposed to.
140. Every proposition has its contrary : there is no assertion
but has its denial; no denial but has its assertion. Every logical
scheme of propositions must contain a denial for every assertion,
and an assertion for every denial.
141. Inference is the production of one proposition as the
necessary consequence of one or more other propositions. In-
ference from one proposition may be either an equivalent or
identical proposition, or an inclusion. If from a first proposition
we can infer a second, and if from the second proposition we can
also infer the first, the two propositions are logical equivalents.
Thus ' X is the parent of Y ' and ' Y is the child of X ' are logical
equivalents : And also ' Every X is Y ' and ' Every not-Y is
not-X '. But from * Every X is Y ' we can infer f Some Xs are
Ys ', without being able to infer the first from the second : the
second is only included in the first.
142. When inference is made from more than one proposition,
the result is called a conclusion, and its antecedents premises.
143. Inference has nothing to do with the truth or falsehood
of the antecedents, but only with the necessity of the consequence.
When the inference from the antecedents is preceded by showing
of their truth, the whole is called proof or demonstration.
144. Deduction, or a priori proof, is when the compound of the
premises gives the conclusion. One false premise, and deduction
wholly fails.
145. Induction, or a posteriori proof, is when the aggregate of
the premises gives the conclusion. One false premise, and the
induction partially fails.
146. Absolute or mathematical proof is when the conclusion is
so established that any contradiction would be a contradiction of
a necessity* of thought
* Logic considers the laws of action of thoiight : mathematics applies
these laws of thought to necessary matter of thought. That two straight
lines cannot inclose a space is a necessary way of thinking, a proposition to
44 LOGIC AND MATHEMATICS. [146-147.
which we must assent : but it is not a law of action of thought. That if two
straight lines cannot inclose a space, it follows that two lines which do
inclose a space are not both straight, is an example of a rule by which
thought in action must be guided.
Mathematics are concerned with necessary matter of thought. Let the
mind conceive every thing annihilated which it can conceive annihilated, and
there will remain an infinite universe of space lasting through an eternity of
duration : and space and time are the fundamental ideas of mathematics.
Of course then the logicians, the students of the necessary action of thought,
are in close intellectual amity with the mathematicians, the students of the
necessary matter of thought. It may be so : but if so, they dissemble their
love by kicking each other down stairs. In very great part, the followers
of either study despise the other. The logicians are wise above mathematics;
the mathematicians are wise above logic : of course with casual exceptions.
Each party denies to the other the power of being useful in education : at
least each party affirms its own study to be a sufficient substitute for the
other. Posterity will look on these purblind conclusions with the smile of
the educated landholder of our day, when he reads Squire Western's fears
lest the sinking fund should be sent to Hanover to corrupt the English
nation. A generation will arise in which the leaders of education will know
the value of logic, the value of mathematics, the value of logic in mathe-
matics, and the value of mathematics in logic. For the mind, as for the
body, B/av vrogi^ou ircivTeS-i srXjjv \x xaxut.
This antipathy of necessary law and necessary matter is modern. Very
many of the most illustrious names in the history of logic are the names
of known mathematicians, especially those of the founders of systems, and
the communicators from one language or nation to another. As Aristotle,
Plato, Averroes (by report), Boethius, Albertus Magnus (by report), Ramus,
Melancthon, Hobbes, Descartes, Leibnitz, Wolff, Kant, &c. Locke was a
Competent mathematician : Bacon was deficient, for the consequences of
which see a review of the recent edition of his work in the Athenceum for
Sept. 11 and 18, 1858. The two races which have founded the mathematics,
those of the Sanscrit and Greek languages, have been the two which have
independently formed systems of logic.
England is the country in which the antipathy has developed itself in
greatest force. Modern Oxford declared against mathematics almost to this
day, and even now affords but little encouragement : modern Cambridge to
this day declares against logic. These learned institutions are no fools,
whence it may be surmised that possibly they would be wiser if they were
brayed in a mortar; certainly, if both were placed in the same mortar, and
pounded together.
147. Moral proof is when the conclusion is so established that
any contradiction would be of that high degree of improbability
which we never look to see upset in ordinary life. Among the
most remarkable of moral proofs is that common case of induction
in which the aggregants are innumerable, and the conclusion
being proved as to very many, without a single failure, the mind
147-154.] ALTERNATIVES. RELATION. 45
feels confident that all the unexamined aggregants are as true
as those which have been examined. This is probable induction :
often confounded with logical induction.
148. A proof may be mixed : it may be deduction of which
some components are inductively proved : it may be induction, of
which some aggregants are deductively proved.
149. Failure of proof is not proof of the contrary.
150. If any number of premises give a conclusion, denial of
the conclusion is denial of one or more of the premises. If all but
one of the premises be affirmed and the conclusion denied, that
one premise must be denied. These two processes, conclusion
from premises, and denial of one premise by denying the con-
clusion and affirming all the other premises, may be called
opponents.
151. Repugnant alternatives are propositions of which one
must be true, and one only. If there be two sets of repugnant
alternatives, of the same number of propositions in each, and if
each of the first set give its own one of the second set for its
necessary consequence, then each of the second set also gives its
own one of the first set as a necessary consequence. Thus if
A, B, C, be repugnant alternatives, and also P, Q, R, and if P be
the necessary consequence of A, Q of B, R of C, then A is the
necessary consequence of P, B of Q, C of R. If P be true,
neither B nor C can be true ; for then Q or R would be true,
which cannot be with P. But one of the three A, B, C, must
be true: therefore A is true. And similarly for the other
cases.
152. A relation is a mode of thinking two objects of thought
together : a connexion or want of connexion. Denial of relation
is another relation: and the two are contraries. The universe
may have only a selection from all possible relations.
153. The name in relation is the subject: the name to which
it is in relation is the predicate. Thus in 'mind acting upon
matter ' mind is the subject, matter the predicate, acting upon is the
relation. When the relation is convertible, subject and predicate
are distinguished only by order of writing, as in § 9.
154. All judgments (asserted or denied relations) may be
reduced to assertion or denial of concomitance by coupling the
predicate and the relation into one notion. As in f mind is a thing
acting on matter' or * mind is not a thing acting on matter'. In
all works of logic, the consideration of relation in general is
46 RELATION. [154-162:
evaded by this transformation, and the developement of the science
is thereby altogether prevented.
155. If X be in some relation to Y, Y is therefore in some
other relation to X. Each of these relations is the converse of the
other. Converse relations are of identical effect, and neither
exists without the other. In conversion the subject and pre-
dicate are transposed and usually change order of mention : as in
* X is master of Y ; Y is servant of X'.
156. When a relation is its own converse, it is said to be
simply convertible. As in 'X has nothing in common with Y'
and ' Y has nothing in common with X '; or as in ' X is equal to
Y ' and < Y is equal to X '.
157. When the subject of one relation is made the predicate
of another, the first predicate may be made the predicate of a
combined relation: as in the master* of (the- nephew-of-Y), that
is, the-master-of-the-nephew of Y.
* The most familiar relations are those which exist between one human
being and another ; of which the relations of consanguinity and affinity have
almost usurped the name relation to themselves. But hardly a sentence can
be written without expression or implication of other relations.
158. A combined relation may have a separate name, or it
may not. Thus brother of parent has its own name, uncle : but
friend of parent has no name which describes nothing else.
159. A combined relation may be of limited meaning, or it
may not. Thus ' non-ancestor of a descendant of Z ' has a limit-
ation of meaning with reference to Z ; he is certainly non-ancestor
of Z, But ' ancestor of a descendant of Z ' has no such limitation:
any person may be the ancestor of a descendant of any other.
160. When a relation combined with itself reproduces itself,
let it be called transitive: as superior; superior of superior is
superior, the same sort of superiority being meant throughout.
A transitive relation has a transitive converse: thus inferior of
inferior is inferior.
161. Relations are conceivable both in extension and in in-
tension (§ 131), both as aggregates and as compounds. Thus
* child of the same parents with ' is aggregate of ' brother, sister,
self: the relation of whole to part has among its components
1 greater ' and * of same substance with '.
162. If two relations combine* into what is contained in
a third relation, then the converse of either of the two combined
with the contrary of the third, in the same order, is contained in
162-163.] RELATION. IDENTITY. 47
the contrary of the other of the two. Thus the following three
assertions are identically the same, superior and inferior being
taken as contraries, that is, absolute equality not existing. Let
the combination be * master of parent' and the third relation
( superior '.
Every master of a parent is a superior
Every servant of an inferior is a non-parent
Every inferior of a child is a non-master.
From either of these the other two follow. This may be gene-
rally proved : at present it will be sufficient to deduce one of the
assertions before us from another. Assume the second; from it
follows that every parent is not any servant of an inferior, and
therefore, if servant at all, only servant of superior, whence master
of parent must be superior.
* This theorem ought to be called theorem K, being in fact the theorem
on which depends the process (§ 108) indicated by the letter K in the old
memorial verses.
163. The relation in which an object of thought stands
to itself, is called identity ; to every thing else, difference. Every
thing is itself : nothing is anything but itself : and any two things
being thought of, they are either the same or different, and can be
nothing except one or the other. These principles enter into the
distinction between truth and falsehood : but cannot distinguish
one truth from another. They are antecedent* to all nomen-
clature, and to all decomposition.
* Many acute writers affirm that syllogism can be evolved from, and
solely depends upon, three principles: 1. Identity, A is A; 2. Difference,
A is not not-A ; thirdly, excluded middle, Every thing either A or not- A.
Now syllogism certainly demands the perception of convertibility, ' A is B
gives B is A', and of transitiveness, ' A is B and B is C gives A is C'. Are
the two principles deducible from the three ? If so, either by syllogism or
without. If by syllogism, then syllogism, before establishment upon the
three principles, is made to establish itself, which of course is not valid.
Consequently, we must take the writers of whom I speak to hold that
convertibility and transitiveness follow from the principles of identity,
difference, and excluded middle, without petitio principii. When any one of
them attempts to show how, I shall be able to judge of the process : as it is,
I find that others do not go beyond the simple assertion, and that I myself
can detect the petitio principii in every one of my own attempts. Until
better taught, I must believe that the two principles of identity and transi-
tiveness are not capable of reduction to consequences of the three, and must
be assumed on the authority of consciousness.
Should I be wrong here : should any logician succeed, without assuming
syllogism, in deducing the syllogism of the identifying copula 'is' from
48 IDENTITY. ONYMATIC RELATION. [163-167.
what may be called the three principles of identification, I shall then admit
a completely established specific difference between the ordinary syllogism
and others in which the copula, though convertible and transitive, is not the
substantive verb. I should expect, in such an event, to deduce the transi-
tiveness and convertibility of 'equals' from 'A equals A', 'A does not
equal not- A' and 'every thing either equals A or not- A', where A is magni-
tude only.
164. Identity is agreement in every thing and difference in
nothing. Complex objects of thought usually agree in some
things and differ in others. They get the same names in right
of those points in which they agree, and different names in right
of those points in which they differ. And thus, all resemblances
or agreements giving an agreement of names, and all differences
giving a difference of names, all the forms of inference are capable
of being evolved out of those forms in which nothing but con-
comitance or non-concomitance of names is considered (§ 5 to
§73).
165. Relations which have immediate reference to, or are
directly evolved from, the application of names and the mode of
thinking about names in connexion with objects named, or with
other names, may be called onymatic * relations.
* The logician has hitherto denied entrance to every relation which is
not onymatic ; declaring all others to be material, not formal. When the
distinction of matter and form is so clearly defined that it can be seen why
and how no connexions are of the form of thought except those which I
have called onymatic, it will be time enough to attempt a defence of the
introduction of other relations. In the meantime, looking at all that is
commonly said upon the distinction of form and matter, I am strongly
inclined to suspect that there is nothing but a mere confusion of terms ; that
is, that when the logician speaks of the distinction of form and matter, he
means the distinction of onymatic and non-onymatic. Dr. Thomson, in his
Outlines, Sfc. (§ 15, note) observes that the philosophic value of the terms
matter and form is greatly reduced by the confusion which seems invariably
to follow their extensive use. The truth is that the mathematician, as yet,
is the only consistent handler of the distinction, about which nevertheless
he thinks very little. The distinction of form and matter is more in the
theory of the logician than in his practice : more in the practice of the
mathematician than in his theory.
166. The only relation in which a name, as a name, can
stand to an object, is that of applicable or inapplicable.
167. Names may have many grammatical and etymological
relations to one another, but the only relations which are of any
logical import are the relations in which they stand to one another
arising out of the relations in which they stand to objects. Ac-
167-170.] MATHEMATICAL AND METAPHYSICAL DELATION. 49
cordingly we consider two names as having objects to which both
apply, or as both applying to nothing whatsoever.
168. When X, Y, Z, are individual names, and we say <X is
Y, Y is Z, therefore X is Z ', we can but mean that in speaking of
X and Y we are speaking of one object of thought, and the same of
Y and Z, so that in speaking of X and Z we are speaking of one
object. The law of thought which acts in this 'inference is the
transitiveness (§ 160) of the notion of concomitancy : if X go with Y,
and Y go with Z, then X goes with Z.
169. When names denote classes, the primary relation between
them is that of containing and contained, in the sense of aggregate
and aggregant(§ 124): other relations spring out of this, as will
be seen. This relation is mathematical in its character : a class is
made up of classes, just as an area is made up of areas. It is
physically possible to connect the two aggregations : we can
imagine all men on one area, and all brutes on another: the
aggregate of the areas contains the class animal, the aggregate of
man and brute.
170. When names denote attributes, the primary relation is
that of containing and contained, in the sense of compound* and
component (§ 124). This relation is metaphysical in its character :
the mode of junction of components is not mathematical, but is a
subject for metaphysical discussion, though how that discussion
may terminate is of no importance for logical purposes. The
manner in which the sources of the notion rational are combined
with those of the notion animal in the object which is called man
has nothing to do with the laws of thought under which the
compound and the components are and must be treated.
* It is not uncommon among logical writers to declare that an attribute
is the sum of the attributes which it comprehends ; that, for example, man,
completely described by the notions animal and rational conjoined, is the
sum of those notions. This is quite a mistake : let any one try to sum up
animal and rational into man, in the obvious sense and manner in which he
sums up man and brute into animal. The distinction of aggregation and
composition, very little noticed by logicians, if at all, runs through all cases
of thought. In mathematics, it is seen in the distinction of addition and
multiplication ; in chemistry, in the distinction of mechanical mixture and
chemical combination ; in an act of parliament, in the distinction between
'And be it further enacted' and 'Provided always'; and so on.
Hartley has more nearly than any other writer produced the notion of
composition as distinguished from aggregation. His compound idea has a
force and meaning of its own, which prevents our seeing the components in
it, just as, to use his own illustration, the smell of the compound medicine
a
50 METAPHYSICAL NOTIONS AND NAMES. [170-173.
overpowers the smells of the ingredients. But even Hartley represents the
compound of A and B by A + B.
171. When the class X is contained in the class Y, as an
aggregant, the attribute Y is contained in the attribute X, as a
component. Thus the class man is contained in the class animal :
the attribute animal is contained in the attribute man. These
two apparent contradictions are both true in their different senses :
say he is man, you say he is in animal ; say he is man, you say
animal is in him. Class man is in class animal, as aggregant
in aggregate ; attribute animal is in attribute man, as component
in compound.
172. In all things which do not depend on ourselves, we learn
to think of that which always happens as necessarily happening, of
that which always accompanies as being essential,* part of- the
essence, part of the being. This metaphysical notion is always in
thought, in one form or other, whenever undeviating concomitance
of one notion with another is established or supposed.
* Upon this word may be said, once for all, what is to be said concerning
the use of metaphysical terms in logic. We have nothing to do with the
way in which the mind comes to them ; our affair is with the way in which
the mind works from them. Thus it is absolutely essential to the fitness of
three straight lines to be the sides of a triangle that any two should be
together greater than the third ; contradiction is inconceivable. It is
naturally essential to an apple to be round; contradiction is unknown in
nature. It is commercially and conveniently essential to a tea-pot to have
a handle ; any contradiction would be unsaleable and unusable. In all
these cases, and whatever may be the force of the word essential, the mode of
inference is the same : for the logical consequences of Y being an essential of
X are but those of Y being always found whenever X is found. Why then
do we not confine ourselves to this last notion, leaving the character of the
conjunction, be it a necessity of thought, a result of uncontradicted observa-
tion, or a conventional arrangement, &c. entirely out of view? Simply
because, by so doing, we fail to make logic an analysis of the way in which
men actually do think. If men will be metaphysicians — and metaphysicians
they will be — it must be advisable to treat the metaphysical views of the
most common relations, the onymatic, in a system of logic. The metaphysical
notion is a natural growth of thought, and children and uneducated persons
are more strongly addicted to it than educated adults.
173. Out of these onymatic relations arise five different
modes of enunciating the same proposition. One of these, the
arithmetical, already treated, merely states, or sums up, an enu-
meration of concomitances, or non-concomitances : as in * Every
man is an animal ' ; or as in * No man is a vegetable '.
174-176.] VARIETIES OF ENUNCIATION. 51
174. The four subjective modes of speaking which the notions
of relation develope, are, —
1. Mathematical. Here both subject and predicate are notions
of class : the class man contained in the class animal.
2. Physical. The subject a class, the predicate an attribute.
As in 'man is mortal': the class man has the attribute subject to
death.
3. Metaphysical. Both subject and predicate are notions of
attributes. As in * humanity is fallible': fallibility a component of
the notion humanity.
4. Contraphysical. The subject an attribute, the predicate a
class. As* in 'All mortality in the class man', or 'none but men
are mortal ': that is, we must attribute mortality only in the class
man ; or, all of which mortality is the attribute is in the class
man.
* I take a falsehood for once, to remind the reader that with truth or
falsehood of matter we have nothing to do.
175. All these niodes of reading are concomitant : each one of
the five gives all the rest. If all the men in the universe* be so
many animals, then the class man is in the class animal, and has
the attribute animal as one of its class marks ; also, the attribute
animal is an essential of the attribute humanity, and the attribute
humanity is to be sought only in the class animal.
* According to the universe understood, so is the mode of taking the.
meaning of the ouymatic terms. For example, if the universe be the
universe of objective reality, then, all existing men being ascertained to be
animals, it is of the nature of man, as actually created, to be animal : the
attributes of animal are naturally essential to man. If the universe be the
universe of all possible thought, then, if all men conceivable be animals, if,
for whatever reason, it be impossible to think of man without thinking of
animal, then the attribute animal is an essential of the attribute humanity.
And now arises a question of words, with which logic has nothing to do.
Those creatures of thought which occur in the fables, dogs and oxen, &c.
which are rational as well as animal, are they men f Certainly not, according
to the notion which the word represents. Consequently, the phrase rational
animal is a larger term than man, when all the possibilities of thought are
in question. But this is not a question of logic. The logician, as such, does
not know what man is, nor what animal is : ' but he knows how to combine
'every man is animal' with other propositions, so soon as he knows that he
is permitted to use that proposition.
176. We have now to render the proposition and the syllogism
into the four readings, mathematical, metaphysical, or mixed, in-
52 MATHEMATICAL PROPOSITION. [176-183.
venting appropriate terms for all the relations which occur. It
will be sufficient, however, to treat the first and third system, the
wholly mathematical, and the wholly metaphysical.
177. When Every X is Y, X))Y or Y((X, let the class X be
called a species of the class Y, and Y a genus* of X. In the con-
trary case, when some Xs are not Ys, X('(Y or Y)')X, let X be
an exient of Y, and Y a deficient of X.
* In the common use of these words, the species is a part only of
the genus. As here used the species may be the whole genus. This is,
to my mind, the greatest liberty I have taken with the ordinary terms of
logic.
178. When No X is Y, X)-(Y or Y)-(X, let each class be
called an external of the other, or let the two be called coexternals.
In the contrary case, when some Xs are Ys, X()Y or Y()X,
let each be called a partient of the other, or let the two be called
copartients.
179. When every thing is either X or Y, X(-)Y or Y(-)X,
let each class be called a complement of the other. In the contrary
case, when some things are neither Xs nor Ys, X)( Y or Y)(X, let
each class be called* a co-inadequate of the other.
* Punsters are respectfully informed that the reading coin-adequate, and
all jokes legitimately deducible therefrom, are already appropriated, and the
right of translation reserved.
180. The spicular symbols may be made to stand for the
relations themselves. Thus )) means species or genus, according
as it is read forwards or backwards ; (( , genus or species : and so
on.
181. Genus and species are converse relations; as also exient
and deficient : of external, partient, complement, coinadequate, each
is its own converse. Genus and deficient are contrary relations ;
as are species and exient, external and partient, complement and
coinadequate.
182. Genus is both partient and coinadequate ; as also is species.
External is both exient and deficient, and so is complement.
183. These are exercises in the meanings of the terms, and
should be thought of until their truth is familiar; as also the
following, —
The genus has the utmost partience, and may have the utmost
coinadequacy. The species has the utmost coinadequacy, and may
have the utmost partience. The external has the utmost deficiency,
183-186.] MATHEMATICAL SYLLOGISM. 53
and may have the utmost exience. The complement has the utmost
exience, and may have the utmost deficiency.
184. These relations have terminal ambiguity, founded on the
notion of contained having two cases, filling the whole, or filling
only a part. Thus
Genus is either -species or exient
Species is either genus or deficient
External is either complement or coinadequate
Complement is either external or partient.
185. Read the identities in § 25 into this language, as in,
Species is external of contrary, Contrary of species is complement,
Contraries of species and genus are genus and species, &c.
186. The following are the combinations* of mathematical
relation which take place in syllogisms. Each triad in the first
list contains a universal and two particular syllogisms, the three
being opponents (§ 47), connected also by the theorem in § 162.
The second list (§ 187) contains the strengthened syllogisms.
)) )) Species of species is species
(( (• ( Genus of exient is exient
(• ( (( Exient of genus is exient
(( (( Genus of genus is genus
)) )*) Species of deficient is deficient
)•) )) Deficient of species is deficient
)•( (•) External of complement is species
)•( (• ( External of exient is coinadequate
(• ( (•) Exient of complement is partient
(•) )•( Complement of external is genus
(•) )•) Complement of deficient is partient
)•) )•( Deficient of external is coinadequate
)) )•( Species of external is external
(( ( ) Genus of partient is partient
() )•( Partient of external is exient
(( (•) Genus of complement is complement
)) )( Species of coinadequate is coinadequate
)( (•) Coinadequate of complement is deficient
External of genus is external
External of partient is deficient
Partient of species is partient
54 MATHEMATICAL SYLLOGISM. [180-189.
(•) )) Complement of species is complement
(•) )( Complement of coinadequate is exient
)( (( Coinadequate of genus is coinadequate.
* Note that when, and only when, one of the combining words is either
genus or species, the other two words are the same ; and this throughout
the fundamental or unstrengthened syllogisms. What law of thought does
this represent ? And except when one of these words so occurs, the three
words of relation are all different.
187. (( )) Genus of species is partient
)) (( Species of genus is coinadequate
(•) (•) Complement of complement is partient
)•( )•( External of external is coinadequate
(( )•( Genus of external is exient
)) (•) Species of complement is deficient
(•) (( Complement of genus is exient
)•( )) External of species is deficient.
188. When we give what may be called, comparatively,
terminal precision, as in § 57, we may use the following nomen-
clature,—
. )°) A deficient species may be called a subidentical
1 1 A species and genus is an identical
(o( An exient genus may be called a superidentical
)o( A coinadequate external may be called a subcontrary
I* | An external complement is a contrary
(°) A partient complement may be called a supercontrary.
189. The complex syllogisms (§61) may be read as follows, —
)o) )o) A subidentical of a subidentical is a subidentical
(°( (°( A superidentical of a superidentical is a superidentical
)°( (°) A subcontrary of a supercontrary is a subidentical
(o) )o( A supercontrary of a subcontrary is a superidentical
)o) )o( A subidentical of a subcontrary is a subcontrary
(°( (o) A superidentical of a supercontrary is a supercontrary
)o( (o( A subcontrary of a superidentical is a subcontrary
(o) )o) A supercontrary of a subidentical is a supercontrary.
The following modes of connecting the symbols, as applied to
the same two terms, may be useful, —
)°) ))> Species; )•), but not the greatest possible.
(°( ((, Genus; (•(, but not the least possible.
)o( )•( , External ; )( , but not the greatest possible,
(o) (•), Complement; (), but not the least possible.
190-195.] METAPHYSICAL RELATIONS. 55
1 90. I now proceed to the metaphysical relations * between
attribute and attribute.
* The terms of metaphysical relation are picked up without difficulty
in our common language : but those of mathematical relation had in several
instances to be forged. This means that the world at large has more of the
metaphysical than of the mathematical notion in its usual form of thought.
But though the unconnected words essential, dependent, repugnant, alternative,
are constantly on the tongues of educated people, the combinations of these
relations are not made with any security, and when thought of at all, enter
under a cloud of words : while the analysis by which precision of speech and
habit of security might be gained is treated with contempt, as being logic.
A whole drawing-room of educated men may be without a single person
who can expose the falsehood of the assertion that the essential of an
incompatible must be incompatible ; a proposition which I have heard
maintained, though not in those words, by persons of more than respectable
acquirements ; sometimes by actual error, sometimes by confusion between
the essential of an incompatible, and that to which an incompatible is
essential. But even of the persons who are not thus taken in, very few
indeed, when told that the answer to ' the essential of an incompatible is
incompatible' is 'not so much; only independent', will be puzzled by the
juxtaposition of incompatibility and independence as viewed in a relation of
degree. In making these remarks, it will be remembered that I am not
speaking of any words of my own, nor of any meanings of my own. The
words are common, and. I take them in their common meanings ; but it is
not generally seen that these common words, used in their common senses,
are sufficient, in conjunction with their contraries, to express all the
relations which occur in a completely quantified system of onymatic
enunciation.
191. "When X))Y let the attribute Y be called an essential
of the attribute X, and X a dependent of Y. In the contrary case,
X(- ( Y, let Y be called an inessential of X, and X an independent
of Y. Remember that dependent on does not mean dependent
wholly on, or dependent only on.
192. When X)-(Y, let each attribute be called a repugnant of
the other. When X( )Y, let each be an irrepugnant of the other.
193. When X(-)Y let each attribute be called an alternative
of the other. When X)( Y let each be called an inalternative of the
other.
194. When difference of symbols is desired, the square bracket
may be used instead of the parenthesis : thus ] ] may denote
dependent when read forwards, and essential when read back-
wards, &c.
195. Essential and dependent are converse relations; as are
also inessential and independent. Of repugnant, irrepugnant, alter-
56 METAPHYSICAL SYLLOGISM. [ 1 95-200.
native, inalternative, each is its own converse. Essential and
inessential are contrary relations ; as are dependent and inde-
pendent, repugnant and irrepugnant, alternative and inalter-
native. Compare § 181.
196. The essential is both irrepugnant and inalternative: as
also is the dependent. The repugnant is both independent and
inessential: as also is the alternative. Compare § 182.
197. The essential has the utmost irrepugnance, and may
have the utmost inalternativeness. The dependent has the
utmost inalternativeness, and may have the utmost irrepugnance.
The repugnant has the utmost inessential ity, and may have the
utmost independence. The alternative has the utmost inde-
pendence, and may have the utmost inessentiality. Compare
§183.
198. These relations also have terminal ambiguity (Com-
pare § 184).
Essential is either dependent or independent
Dependent is either essential or inessential
Repugnant is either alternative or inalternative
Alternative is either repugnant or irrepugnant.
199. Read the identities in § 25 into this language, as in
Dependent is repugnant of contrary, contrary of dependent is
alternative, contraries of dependent and essential are essential and
dependent, &c.
200. The following are the combinations* in syllogism, ar-
ranged as in § 186.
)) )) Dependent of dependent is dependent
(( (• ( Essential of independent is independent
(• ( (( Independent of essential is independent
(( (( Essential of essential is essential
)) )•) Dependent of inessential is inessential
)•) )) Inessential of dependent is inessential
)•( (•) Repugnant of alternative is dependent
)•( (• ( Repugnant of independent is inalternative
(• ( (•) Independent of alternative is irrepugnant
(•) )•( Alternative of repugnant is essential
(•) )•) Alternative of inessential is irrepugnant
)•) )•( Inessential of repugnant is inalternative
200-203.] METAPHYSICAL SYLLOGISM. 57
)) )•( Dependent of repugnant is repugnant
(( ( ) Essential of irrepugnant is irrepugnant
( ) )•( Irrepugnant of repugnant is independent
(( (•) Essential of alternative is alternative
)) )( Dependent of inalternative is inalternative
)( (•) Inalternative of alternative is inessential
Repugnant of essential is repugnant
Repugnant of irrepugnant is inessential
Irrepugnant of dependent is irrepugnant
Alternative of dependent is alternative
Alternative of inalternative is independent
Inalternative of essential is inalternative.
* Note that when, and only when, one of the combining words is either
essential or dependent, the other two words are the same ; and this throughout
the fundamental or unstrengthened syllogisms. What law of thought does
this represent ? And except when one of these words so occurs, the three
words of relation are all different.
201. (( )) Essential of dependent is irrepugnant
)) (( Dependent of essential is inalternative
(•) (•) Alternative of alternative is irrepugnant
)•( )•( Repugnant of repugnant is inalternative
(( )•( Essential of repugnant is independent
)) (•) Dependent of alternative is inessential
(•) (( Alternative of essential is independent
)•( )) Repugnant of dependent is inessential.
202. I now proceed to form metaphysical* terms expressing
relations of terminal precision (compare § 188). Let an inherent
be an attribute asserted; let an excludent be an attribute denied;
let an accident, which is also non-accident, be an attribute affirmed
of part and denied of the rest. Thus of man, life is an inherent,
vegetation an excludent, wisdom an accident and a non-accident.
* This new formation cannot be overlooked, since it is the extension of
the Aristotelian system of predicables, genus and species (used in the old
sense) and accident, to the system in which contrary terms are permitted.
Otherwise, the relations of terminal ambiguity, compounded, might serve
the purpose.
203. Each of these relations may be either generic or specific.
Either is generic when it applies in as large or a larger degree to
a larger genus : specific, when it does not so apply to any larger
58 PREDICABLES. [203-205.
genus. This being premised, the following relations will be found
correctly stated, —
>. v T ,. i j i . f Specific accident
)o) Inessential dependent is •< *, J .
{^Generic non-accident
Dependent essential is Specific inherent
(o( Independent essential is Generic inherent
)o( Inalternative repugnant is Generic excludent
|-| Repugnant alternative is Specific excludent
, N i, ,. f Generic accident
(o ) Irrepugnant alternative is < _ . .
(.specific non-accident.
204. The following are examples of each of these terms, the
universe being terrestrial animal, —
Specific accident ; generic non-accident. Lawyer is in this rela-
tion to man: accident and non-accident, because an attribute of
some men, and not of others ; specific accident, because not found
in the additional extent of any genus larger than man ; generic
non-accident, for the same reason.
Specific inherent. Rational is in this relation to man: inhe-
rent, because an attribute of all ; specific, because no attribute of
the additional extent in a larger genus.
Generic inherent Biped is in this relation to man; inherent,
because an attribute of all ; generic, because an attribute of the
additional extent of a larger genus.
Generic excludent. Oviparous is in this relation to man; ex-
cludent, because an attribute to be denied of man ; generic, because
to be also denied of the additional extent of some larger genera.
Specific excludent. Dumb (wanting articulate language with
meaning) is in this relation to man; excludent, because to be
denied of man ; specific, because not to be denied of the additional
extent of any larger genus.
Generic accident; specific non-accident. Naked (not artificially
clothed) is in this relation to man; accident and non-accident,
because some are and some are not; generic accident, because an
accident of the additional extent of larger genera; specific non-
accident, because not non-accident of any such additional extent.
205. When either of the relations belongs equally to a term
and its contrary, it may be called universal. Thus an attribute of
both term and contrary is a universal inherent; an accident and
non-accident of both term and contrary is a universal accident and
non-accident; an excludent of both term and contrary is a uni-
versal excludent. But the first and third of these terms are chiefly
205-209.]
EXTENT AND INTENT.
of use in defining the universe : the second is that relation which
we suppose until some contradiction is affirmed.
206. With the arithmetical reading in extension may be joined
that in intension, § 115, 131. In extension, the unit of enume-
ration is one of the objects all of which aggregate into the class :
in intension, the unit of enumeration is one of the qualities all of
which compose the object. The following is the system of arith-
metical reading in intension: naturally connected (§ 129) with
the metaphysical mode of viewing objects of thought. The
inversion of the quantities, presently further described, will be
easily seen ; namely, that )X and X( now indicate that X is taken
completely, *in all its qualities; while X) and (X indicate that X
is taken incompletely, in some (some or all, not known which)
of its qualities. The term any (§ 22) is here introduced when
grammatically desirable.
Arithmetical reading in intension
All qualities of Y are some qualities of X
Some qualities of Y not any qualities of X
All things want either some qualities of X,
or some qualities of Y
Some things want neither any quality of X,
nor any quality of Y
All things have either all the qualities of X
or all the qualities of Y
Some things want either some of the quali-
ties of X, or some of the qualities of Y
Some qualities of Y are all the qualities
of X
Any qualities of Y are not some qualities
of X.
Symbol Metaphysical reading
X))Y , X dependent of Y
X independent of Y
X repugnant of Y
X(-(Y
X)-(Y
XQY X irrepugnant of Y
X(-)Y
X)(Y
X((Y
X)-)Y
X alternative of Y
X inalternative of Y
X essential of Y
X inessential of Y
207. I now proceed to further consideration of the subject of
quantity. No new results can appear, but it will be necessary
both to adapt the old results to the more subjective view of logical
process, and also to consider the distinctions of quantity from
new points of view.
208. The distinction of the two tensions, extension and inten-
sion (§ 131), or, for brevity, extent and intent, may for clearness
(§ 129) be applied only to classes and attributes. The extent
of a class embraces all the classes of which it is aggregated : the
intent of an attribute embraces all the attributes of which it is
compounded.
209. A class may be subdivided down to the distinct and non-
60 QUANTITY OF EXTENT AND INTENT. [209-211.
interfering individual objects of thought of which it is composed :
and here subdivision must stop. But it is not for human reason
to say what are the simple attributes into which an attribute may
be decomposed: the decomposition of the notion rational, for
example, into distinct and non-interfering component notions, is
the subject of an old controversy which will perhaps never be
settled. But this difficulty is of no logical importance.
210. The relation of quantity as exhibited in the arithmetical
view of the proposition (§ 13, 14), giving the distinction of univer-
sal and particular quantity, as it is commonly expressed, or of
total and partial quantity, as I have expressed it, may be in this
part of the subject most conveniently attached to other names.
Let the terms full extent* and vague extent be used to replace
total extension and partial extension : and let full intent and vague
intent replace total intension and partial intension.
* These terms are convenient from their brevity : full extent is shorter
than universal extension. But they are still more useful as avoiding the
ambiguity of the words some, particular, partial, which, as we have seen
(§ 14, note f) misleads even the highest writers. The logical opposition
of quantity is not quantity universal and quantity not universal, but quantity
asserted to be universal and quantity not asserted to be universal. Two words
cannot be found which express the opposition of undertaking to assert and
not undertaking to assert universality. We may therefore be content with
full and vague, which, if they do not express opposition, at least do not, like
universal and particular, express the wrong opposition.
211. Additional extent can only be gained by a new aggregate
containing extent which is not in the collective extent of the
others : additional intent only by a new component which is not
in the joint intent of the others. Thus the extent of the class
animal is not augmented by the aggregation of the class having
volition, if the universe be the visible earth. Again, the intent*
of the notion plane triangle is not augmented by the junction of
the notion capable of inscription in a circle. The distinction
between these real and apparent augmentations is of the matter,
not of the form: and is of no logical import except this, that
when we say that a new aggregant increases extent, and a new
component increases intent, we must be prepared, with the
mathematicians, to reckon 0 among the cases of quantity.
* There is a remarkable difference between extent and intent, which,
though logically nothing at all, is psychologically very striking. Say we
discover extent hitherto unknown, without the necessity of reducing intent
to include it within a class thought of. Columbus did this when he first
was able to add the class American to the classes then known under man.
211-214.] OPPOSITION OF EXTENT AND INTENT. 61
Here is nothing beyond what was possible in previous thought, which could
people the seas to any extent. But when we add intent without diminishing
extent, which knowledge is doing every day, we cannot conceive beforehand
what kind of additions we shall make. A beginner in geometry gradually
adds to the intent of triangle, which at first is only rectilinear three-sided
figure, the components — can be circumscribed by a circle — has bisectors of
sides meeting in a point — has sum of angles equal to two right angles —
and other properties, by the score. The distinction is that class aggregation
joins similars* — but that composition of attributes joins things perfectly
distinct, of which no one can predicate anything merely by what he knows
of another thing. When the old logicians threw the notion of intent out
of logic into metaphysics, they were guided by the material differences of
qualities, and did not apprehend their similarity of properties a* qualities.
212. The distinction of extent and intent has found its way
into common language, in the words scope* and force, which I
shall sometimes use. Thus in * every man is animal' the term
man is used in all its scope, but not in all its force ; a person
incognisant of some of the components of the notion man, that is,
of the whole force of the term, might have the means of knowing
this proposition. But animal is used in all its force, and not in
all its scope. This answers to saying that in 'Every X is Y',
the term X is of full extent and vague intent ; the term Y is of
full intent and vague extent
* The logicians, until our own day, have considered the extent of a term
as the only object of logic, under the name of the logical whole : the intent
was called by them the metaphysical whole, and was excluded from logic.
In our own time the English logical writers, and Sir William Hamilton
among the foremost, have contended for the introduction of the distinction
into logic, under the names of extension and comprehension : Hamilton uses
breadth and depth. Now I say that in the perception of the distinction
between scope and force, as well as in other things, the world, which always
runs after quack preparations, has ventured for itself out of the logical
pharmacopoeia. This certainly in a rude and imperfect way : and without
apprehension of any theorems. I have not found, though I have looked for
it, any such amount of recognition that the greater the scope the less the
force as I could present without suspicion of the aut inveniam out faciam
bias. But I think it likely enough that some of my readers may casually
pick up passages which show a. feeling of this theorem.
213. The quantity considered in the arithmetical view of logic
(§ 5-111) was entirely quantity of extent. I now proceed to the
comparison of extent and intent.
214. In every use of a term, one of the tensions* is full, and
the other vague: the full extent and the full intent cannot be
used at one and the same time ; and the same of the vague extent
and the vague intent. Thus X) and (X must stand for X used
62 OPPOSITION OF EXTENT AND INTENT. [214.
in full extent and vague intent : and )X and X( for vague extent
and full intent.
The proof of this proposition is as follows. When a term is
full in extent, we can abandon or dismiss any aggregant of that
extent we please : the proposition, though reduced or crippled by
the dismissal, is true of what is left : but we may not annex an
aggregant at pleasure. When a term is vague in extent, we
cannot dismiss any aggregant whatever: for we know not by
what aggregant the proposition is made true : but we may annex
any aggregant at pleasure : for we do not thereby throw out what
makes the proposition true, even if we annex no additional truth ;
and we do not, when speaking vaguely, affirm or deny of any one
selected aggregant. And as the extent must be full or vague,
and we must be either competent or incompetent to dismiss an
aggregant taken at pleasure, and must be either competent or
incompetent to annex one, the converses follow (§ 151), namely,
that when we are competent to dismiss, the extent is full, and
when we are incompetent the extent is vague : and also that when
we are competent to annex, the extent is vague, and when not,
the extent is full. Precisely the same proposition may be
established upon the intent of a term, and its components.
Now let a term be of full extent. In diminishing the extent,
which we may do, we can so do it as to augment the intent : and
if we be competent to augment the intent, that intent must be
vague, as just proved. Similarly, if a term be of vague extent, we
are competent to annex an aggregant, that is, to diminish the
intent ; whence the intent must be full. And the same may be
proved in like manner when either kind of intent is first supposed
instead of extent; though by use of § 151 this case may be seen
to be contained in that already treated. And the learner may
gather the whole from instances. Thus A, B))PQ gives A))PQ,
andA))P; but not A,B,C))PQ, nor A,B))PQR. But PQ))A,B
does not give P)))A, B, nor PQ))A, though it does give PQR))A,B
and PQ))A,B,C; and so on. And further, from § 133, this
proposition can be made good of all universals when it is known
of one : and the same of all particulars.
* The logicians who have recently introduced the distinction of extension
and comprehension, have altogether missed this opposition of the quantities,
and have imagined that the quantities remain the same. Thus, according to
Sir W. Hamilton 'All X is some Y' is a proposition of comprehension, but
' Some Y is all X' is a proposition of extension. In this the logicians have
abandoned both Aristotle and the laws of thought from which he drew the
214-216.] DISMISSAL AND TRANSPOSITION OF ELEMENTS. 63
few clear words of his dictum : ' the genus is said to be part of the species ;
but in another point of view («ja«i) the species is part of the genus'. All
animal is in man, notion in notion : all man is in animal, class in class. In
the first, all the notion animal part of the notion man: in the second, all
the class man part of the class animal. Here is the opposition of the
quantities.
215. It appears then that the elements of a tension (aggregants
of an extent, components of an intent) may be dismissed from the
term used fully, but cannot be introduced ; may be introduced
into a term used vaguely, but cannot be dismissed. The dis-
missible is inadmissible : the indismissible is admissible.
216. Elements of either tension may, under the limitations of
a rule to be shown, be transposed from one term of a proposition
to the other, either directly, or by contraversion, without either loss
or gain of import to the proposition. Thus AB)-(Y is the same
proposition as A)-(BY, and X))A,B is the same as Xa))B. The
demonstration of this may best be seen by observing that every
universal is a declaration of incompossibility,* and every particular
a corresponding declaration of compossibility . Thus X)-(Y is an
assertion that X and Y, as names of one object, are incompossible ;
and X( )Y that they are compossible. Again, X))Y declares X
and y to be incompossible ; and so on.
Now it will be seen that AB)-(Y is merely a statement that
the three names A, B, Y, are incompossible ; and so is A)'(BY.
Hence AB)(Y and A>(BY are identical. Similarly AB( )Y and
A()BY are identical, both declaring the compossibility of A,B, Y:
or thus, if two propositions be identical, their contraries must be
identical. Hence we learn that in Y))a, b we have YB))a &c.
Carrying this through all transformations, we arrive at the
following rules: —
1. In universal propositions, vague elements (the elements of
terms of either vague tension) are transposible ; directly in nega-
tives, by contraversion in affirmatives. But full elements are
intransposible.
2. In particular propositions, full elements (the elements of
terms of either full tension) are transposible ; directly in affirm-
atives, by contraversion in negatives. But vague elements are
intransposible.
Thus in X))Y, Y is of vague extent ; if it be (A, B), its
aggregant A is transposible, the proposition being affirmative, by
contraversion: that is X))A, B is identical with Xa))B. The
rules are for comparison and generalisation, not for use. Nothing
64 DISMISSAL AND TRANSPOSITION OF ELEMENTS. [216-219.
can be more evident than that if every X be either A or B, every
•X which is not A is B.
* These good words are Sir William Hamilton's (see § 14, note t), to
whom, in matters of language, I am under what he would have called
obligations general and obligations special. His occasional writing of the
adjective after the substantive is a useful revival of an old practice, tending
much to clearness. As to my obligations special, he, finding the word
parenthesis not enough to erect his reader's hair, described my notation as
"horrent with mysterious spiculae". This was the very word I wanted,
§ 21 : for parenthesis has come to mean, not the punctuating sign, but the
matter which it includes : and parenthetic notation would have been
ambiguous.
217. It has in effect been noticed that for every full term in a
proposition a term of as much or less tension may be substituted ;
and for every vague term a term of as much or more tension.
This is the whole principle of onymatic syllogism, or rather may
be made so: for the varieties of principle upon which all onymatic
inference may be systematically introduced are numerous. Thus
in X()Y)-(Z, giving X(-(Z, all we do is to substitute for Y used
vaguely in extent, the as extensive or more extensive term z. Or
thus; — for Y of full intent, we substitute the as intensive or less
intensive term z. For Y)*(Z or Y))z, shows that z, if anything, is
of greater extent and less intent than Y.
218. There are processes which appear like transpositions, but
are not so in reality. Thus X))PQ certainly gives XP))Q : here
is a universal proposition, in which the element of a full tension is
transposible. But not transposible within the description in § 216,
in which it is affirmed that the 'proposition after transposition is
identical with the proposition before transposition. This is not
the case here; for though X))PQ gives XP))Q, yet XP))Q
does not give X))PQ. Here, since X))PQ gives X))Q and
X))P, the term XP is really X. And further, since X))PQ
gives X))Q from whence XR))Q, be R what it may so long
as XR has existence, the deduction of XP))Q from X))PQ is
a case of something different from mere transposition : for P,
in XP))Q, may be changed into anything else.
219. The dismissal of the elements of terms comes under
what may be called the decomposition of propositions. When the
elements of both terms are of the full tension, the proposition is a
compound of m x n propositions, if m and n be the numbers of
elements in the two terms. Thus A, B))CD gives and is given
by the four propositions A))C, A))D, B))C, B))D.
220-222.] YAGUE QUANTITY IN CONCLUSION. 65
220. Species, external, deficient, coinadequate
Dependent, repugnant, inessential, inalternative
must carry the notion of full extent and vague intent. For
example, the universe being England, farmer is a deficient of
landowner : of all the class * farmer, no part is identical with a
certain part of the class landowner. To know this by extent I
must know the whole class farmer : but to know it by intent, I
need not know all the attributes of the notion farmer. Let there
be but one of these attributes which is not an essential of land-
owner, and the proposition is established.
Genus, complement, exient, partient
Essential, alternative, independent, irrepugnant
must carry the notion of vague extent and full intent. The
symbols will here help the memory of those who have fully
connected them with the words.
* The student must, in any one proposition, be on his guard against
thinking inconsistently of class and of attribute. Either of these modes of
thought may be chosen, but not both together, unless the attribute be
made to distinguish the class, without exceptions. For a remarkable
instance, take the word gentleman : what different things people usually
mean, according as they are speaking by notion of class or of attribute ; the
common attribute excludes a percentage of the class, and admits many who
are not of the class. The reader may be puzzled to make out the text,
unless his character of landowner correspond to his class.
221. The rules of § 50-52 must be translated as follows.
A vague term in the conclusion takes extent or intent (scope or
force) as follows.
1. In universal syllogisms, if one term of conclusion be of
vague scope or force, it has the scope or force of the other ; if
both, one has the scope or force of the whole middle term, the
other of its whole contrary.
2. In fundamental particular syllogisms, the vague term or
terms of the conclusion take scope or force from the vague
premise.
3. In strengthened particular syllogisms, the vague term or
terms of conclusion take scope or force from the whole middle
term or from its whole contrary, according to which is of full
scope or force in both premises.
222. For example, (actual) farming depends on occupation
of land (see the caution in § 191, often wanted in reference to this
i
66 HYPOTHETICAL SYLLOGISM, ETC. [222-225.
very instance) ; and occupation of land is an essential of county
respectability : therefore farming and such respectability are in-
alternative. Here the terms of conclusion are both of vague
intent or force, and the middle term of full intent : the force is
precisely so much as is contained in the notion of occupying land.
Any component either of actual farming or of county respect-
ability which can be possessed by a non-occupier of land is of no
import in the conclusion as from the above premises.
Take the mathematical form of the above: — Farmers are a
species of occupiers of land ; the county respectables are a species
of occupiers ; whence the farmer is a coinadequate of the county
respectable, or both together do not make up the whole universe
(that is, as implied, the population of the county). Here the con-
trary of the middle term, the class of non-occupiers of land, forms
the extent of coinadequacy of the terms of conclusion implied in
the premises.
223. The admission of complex terms, and of copular relations
more general than the word of identification is, enable us to
include in common syllogism all the cases known as hypothetical
syllogisms, conditional syllogisms, disjunctive syllogisms, dilem-
mas, &c. I shall merely take a few cases of these.
If P be true, Q is true; but P is true, therefore Q is
true. This is an hypothetical syllogism, so called. To reduce it
to a common syllogism between * Q', 'true', and a middle term,
we have 'Q' is l a proposition true when P is true'; 'A propo-
sition true when P is true ' is ' true ' (because P is true) ; there-
fore Q is true. Many other ways might be given. In truth,
though the reduction is possible, the law of thought connecting-
hypothesis with necessary consequence is of a character which
may claim to stand before syllogism, and to be employed in it,
rather than the converse. But the discussion of this subject is
not for a syllabus : see § 226. In a similar way may be treated
' If P be true, Q is true ; Q is not true : therefore P is not true '.
224. Say that P is either A, or B, or C ; A is not X, B is not
X, C is not X ; then P is not X. This is the syllogism
tP))A,B,C>(X giving P>(X
a common syllogism with the middle term an aggregate.
225. Either P is true, or Q ; if P be true, X is true ; if Q
be true, Y is true ; therefore either X or Y is true. The truth
is the alternative of the truth of P or Q ; which is the alternative
225-228.] BELIEF. — PROBABILITY. 67
of the truth of X or Y ; therefore the truth is the truth of the
alternative of X or Y.
Various other instances will be found in my formal Logic,
pp. 115-125.
226. In all syllogisms the existence of the middle term is a
datum. If the conclusion be false, the syllogism being logically
valid, and the premises true if the terms exist, then the non-
existence of one of the terms is the error. And if the terms
which remain in the conclusion be existent, the nonexistence of
the middle term may be inferred. When the syllogism is sub-
jective in character, the transition into the objective syllogism
frequently hinges on this point. Suppose success in a certain
undertaking, such success being conceivable, depends both upon
X and upon Z : then X and Z are not subjectively repugnant.
Suppose that in objective reality they are repugnant : their
coexistence being a thing wholly unknown and incredible. It
follows then that success is objectively unattainable; impossible
as things are, people say. The metaphysical premises X((Y))Z,
X, Y, Z, being conceivable, give X( )Z : and if X and Z have
objective existence, and X)'(Z, it follows that Y does not exist ;
for if it did, the premises X((Y))Z would give X()Z. Suppose
a qualification which depends both upon natural talent and early
training; and suppose the talent to be one which cannot be
developed early, as things go ; then, as things go, the qualification
is unattainable.
227. The remaining logical whole of which we have to con-
sider the parts is belief. This feeling is one the magnitude of
which ranges between two extremes ; certainty for, such as we
have as to the proposition * Two and two make four '; and cer-
tainty against, such as we have as to the proposition ' Two and
two make five '. The first has the whole belief, or no unbelief; the
second has no belief, and the whole unbelief. These extremes are
represented by 1 and 0, on the scale of belief: and would be
represented by 0 and 1, if we chose (which is not necessary) to
have a scale of unbelief.
228. That which may be or may not be claims* a portion of
belief and a portion of unbelief: that is, we partly believe in the
" may " and partly in the " may not" Thus if an iirn contain 1 3
white balls and 7 black balls, and nothing else, and I am going to
draw a ball without knowing which, and without more belief in
68 TESTIMONY AND ARGUMENT. [228-231.
one ball than in another ; then my belief in the drawing of white
is to my belief in the drawing of black as 13 to 7, that is, 1 re-
presenting certainty, I have £$ of belief in a white ball, and -^ in
a black ball. This is usually expressed by saying that the odds
in favour of a white ball are 13 to 7, and the chances, or proba-
bilities, of the drawing of white or black ball, are ^ and -/^. I
shall call 13:7 the ratio of belief in a white ball, or of unbelief in
a black ball; and 7:13 the ratio of belief in a black ball, or of
unbelief in a white ball ; and 1 3 and 7 the favourable and unfa-
vourable terms for the white ball.
* Here, as in all other things, there are portions which are too small to
be of perceptible effect. Csesar may not have died in the manner stated : he
mat/, if there were such a person, which may not be true, have been captured
by the Britons, and detained in captivity for the rest of his life. But the
received history absorbs so much of our belief that we have but a mere atom
to divide among all the different ways in which that story may be wrong.
There are two opposite fallacious methods of thinking : first, the confusion
of high moral certainty with absolute knowledge in right of the nearness of
the quantities of belief in the two ; secondly, the confusion of high moral
certainty with matters of practical uncertainty, in right of the want of
absolute knowledge in both.
229. Referring to my Formal Logic, for full explanation on
the subject, I shall here only digest a few rules relative to the
measures of belief and unbelief, in questions especially relating to
logic.
230. Any alteration of our minds with respect to belief or
unbelief of a proposition is derived from two sources, —
1. Testimony, assertion for or against by those of whose know-
ledge we have some opinion. This, when absolutely unimpeachable,
is authority; though this word is used loosely for testimony of
high value. Testimony speaks to the thing asserted, to its truth
or falsehood ; it turns out good if the proposition be true, and bad
if the proposition be false.
2. Argument, reasoning for or against, addressed to the mind
on its own fprce. This, when absolutely unimpeachable, is de-
monstration; though this word is used loosely for reasoning of
great force. Reasoning speaks, not simply to the truth or false-
hood, but to the truth as proved in one particular way. If an
argument be invalid, it does not follow that the proposition is
false, but only that it cannot be established in that one way.
231. When the proposer of an argument believes in its con-
231-233.] TESTIMONY AND ARGUMENT. 69
elusion, he is one of the testimonies in favour of the conclusion,
independently of his argument.
232. Among the testimonies to a conclusion must be counted
the receiver himself, whose initial state of mind enters as the
testimony* of a witness into the mathematical formulas, though a
thing of a very different kind. Suppose that, all circumstances
duly considered so far as he is able, the receiver begins with an
impression that the proposition in hand has 7 to 3 against it, or
(3 : 7) is his ratio of belief at the outset. Upon this belief the
future testimonies and arguments are to act : and the mathe-
matical effect is the same as if the first witness bore testimony
(7 : 3) against the proposition.
* This is the point on which the mathematical study of this theory
throws most light. Simple as the thing may appear, there is not one writer
in a thousand who seems to know that the legitimate result of argument and
testimony depends upon the initial state of the receiver's mind. They
request him to begin without any bias ; to make himself something which
he is not by an act of his own will. Judges request juries to dismiss all
that they know about the case beforehand : and this when the juries know,
and the judges know that they know it, that the mere fact of the prisoner's
appearance at the bar is itself three or four to one in favour of his guilt.
Now the jury do not dismiss this presumption, because they cannot : and
they need not, because the sound remedy against the presumption lurks in
their own minds, and is ready to act. It would not be advisable to discuss
in a short note the method in which common honesty manages to hit the
truth, in spite of prepossession. But I may state my conviction that if the
juryman were consciously to aim at being somebody else, that is, a person
without any preconceived notion, he would give a wrong verdict far more
often than he does. I should recommend him not to think about himself at
all, but to forget himself altogether, or at least not to be active in bringing
himself before himself; and to listen to the evidence. And further, to
remember that the inquiry does not terminate in the jury-box ; that the
trial of the evidence commences when the jury retire ; that the evidence of
eleven other men to the character of the evidence is itself part of the
evidence ; and that the demand for unanimity on the part of the jury is the
expression of the determination of the law that the juryman shall be forced,
if needful, to take other opinions into account. I trust this necessity for
unanimity will never be done away with.
233. In assigning numerical value to degrees of belief, we are
supposing cases which are nearly as unusual in human affairs as
numerically definite propositions (§ 13). But by the study of
accurate data, supposed attainable, we analyse the sources of error
to which our minds are subject in the rough processes which our
state of knowledge obliges us to use.
70 CALCULATION OF TESTIMONY. [234-236.
234. The method of compounding testimonies is by multiply-
ing together all the favourable numbers for a favourable number,
and all the unfavourable numbers for an unfavourable number.
For instance, a person thinks it 10 to 3 against an assertion.
Two witnesses affirm it, for whose accuracy it is in his mind
7 to 4 and 8 to 3 : two witnesses deny it, for whose accuracy it
is in his mind 11 to 5 and 3 to 1. What ought to be his state of
belief after the testimony ?
The several ratios for the assertion are
3: 10, 7:4, 8:3, 5: 11, 1:3
And 3x7x8x5x1 : 10 x 4 x 3 x 11 x 3, or 7 : 33 is the
ratio of belief as it should be after the whole testimony is taken
into account : or 33 to 7 against the assertion.
235. When several arguments are advanced on one side of a
question, of which the several chances of validity are given, the
chance that the side taken is proved, that is, that one or more of
the arguments are valid, is as follows. Take the product of the
unfavourable numbers for the unfavourable number, and subtract
it from the product of the several totals for the favourable number.
Thus if three arguments be advanced on one side, the ratios of
belief in which are (4 : 3), (2 : 1), (3 : 7), the unfavourable num-
ber is 3x1x7, which subtracted from the product of 4 + 3,
2 + 1, 3 + 7, gives the favourable number. Hence (189 : 21)
or (9 : 1) is the chance of the side being established by one or
more of the arguments.
236. Every argument, however weak, lends some force to its
conclusion : for it may be valid, and if invalid does not disprove
the conclusion. But it must be remembered that this conclusion
is modified by the argument on the other side which arises from
the production of weak arguments, or none but weak arguments.
Weak arguments from a strong person themselves furnish an
argument. If an assertion be true, it is next to certain that very
strong arguments exist for it ; if such arguments exist, it is highly
probable that such and such a person could find them : but he
cannot find them ; whence there is strong presumption that the
arguments do not exist, and from thence that the assertion is not
true. This kind of reasoning really prevails, and leads to a
rational conclusion that the production of none but weak argu-
ments is a strong presumption against the truth of their con-
clusion. But when weak arguments are mixed with strong ones,
236-241.] CALCULATIC A 000 094 069
they may rather tend to reinforce the conclusion, though the
general impression is that they only weaken their stronger
companions.
237. If ever an argument be of such nature that according as
it is valid or invalid the conclusion is true or false, that argument
is of the nature of a testimony, and must be combined with the
rest as in § 234.
238. When testimony and arguments on both sides are to be
combined, the result is obtained as follows. Combine all the
testimony into one result, as in § 234, all the arguments for as in
§ 235, and all the arguments against in the same way. Then
form the favourable and unfavourable numbers in the ratio of
belief required, as follows: —
Favourable number. Unfavourable number.
Multiply together Multiply together
The favourable number of the testi- | The unfavourable number of the
mony testimony
The unfavourable number of the ; The unfavourable number of the
argument against argument for
The total of the argument for i The total of the argument against
For instance, testimony giving (7 : 3), argument for, (5:2) and
argument against, (8 : 1), the ratio of belief for the truth of the
assertion should be (7 x 1 x 7 : 3 x 2 x 9) or (49 : 54), that is, it
is 54 : 49 against the assertion being true.
239. When testimony is evenly balanced, (1:1), it may be
altogether omitted. When the arguments for and against are
evenly balanced, the arguments may be omitted. When the
arguments on both sides are very strong, even though not evenly
balanced, the mind may be presumed unable to compare the two
very small quantities which they want of certain validity, and the
arguments may be treated as evenly balanced.
240. When no argument is offered for, let (0 : 1) represent
the ratio of belief which is to be used in the above rule : and the
same when no argument is offered against.
241. When testimony is evenly balanced, and argument
for is (m : n), there being no argument against, we have
(1 x 1 x m + n : 1 x n x 1), or m + n : n for the truth of the
assertion. Thus, on a matter on which our minds have no bias,
an argument which has only an even chance of validity gives
2 to 1 for the truth of the conclusion.
72 TESTIMONY AND ARGUMENT. [242-244.
242. Any one may wisely try a few cases, setting down in
each, to the best of his judgment, or rather feeling, his ratios of
belief as to testimony, argument for, argument against, and final
conclusion. If the last do not agree with the calculation made
from the first three, he does not agree with himself. This he
may very easily fail to do, for, in such matters of appreciation, one
element may have more than justice done to it at the expense of
the rest, on the principle laid down in the Gospel of St. Matthew,
xxv. 29.
243. The distinction of aggregation and composition occurs in
the two leading rules of application of the theory of probabilities.
When events are mutually exclusive, that is, when only one of
them can happen, the chance that one or other shall happen is
found from the separate chances of happening by a rule of
aggregation, namely, by addition. But when events are entirely
independent, so that any two or more of them may happen
together, the chance of all happening is found by applying to
the separate chances a rule of composition, namely, multiplication.
The connexion of the formulae of probability with those of logic in
general has been most strikingly illustrated by Professor Boole, in
his Mathematical Analysis of Logic, Cambridge, 1847, 8vo., and
subsequently in his Investigation of the Laws of Tliought, London,
1854, 8vo. In these works the author has made it manifest that
the symbolic language of algebra, framed wholly on notions of
number and quantity, is adequate, by what is certainly not an
accident, to the representation of all the laws of thought.
244. I end with a word on the new symbols which I have
employed. Most writers on logic strongly object to all symbols
except the venerable Barbara, Celarent, &c. in § 109. I should
advise the reader not to make up his mind on this point until he
has well weighed two facts which nobody disputes, both separately
and in connexion. First, logic is the only science which has made
no progress since the revival of letters : secondly, logic is the only
science which has produced no growth of symbols.
Erratum. Page 55, line 20, for will be puzzled read will not be puzzled.
LONDON :
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