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UNIVERSITY OF TORONTO
DEPARTMENT OF PSYCHOLOGY
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Ethics, A Manual of. By J. S. MACKENZIE, Litt.D., M. A., late
Professor of Logic and Philosophy in the University College of
South Wales and Monnioutlisiiiie, formerly Fellow of Trinity
College, Cambridge, Examinci in the Universities of Cambridge
and Aberdeen. Fiffh Editiou, Evhvged.
" A tborougli and indeiiendent discussion of moral science and philosophy
written with gi'eat care, and witli a freshness and originality that take the work
quite out of tlie. category of tlie ordinary text-book." — Journal oi Edvcatiov.
Logic, A Manual of. By Jamks VVelton, D.Lit.. M.A., late Pro-
fessor of Education in the University of Leeds. 2 vols.
Vol. I. contains the whole of Deductive Logic, except Fallacies,
which are treated, with Inductive Fallacies, in Vol. II.
"A clear and compendious summary of the views of various thinkers ou important
and doubtful points." — Journal of Education.
Logic, Intermediate. By James Welton, D.Lit., M.A., and A. J.
MONAHAN, M.A. With Questions and Exercises.
This book is of the standard of University Intermediate Exami-
nations.
"This admirable manual will be welcomed by all students of mental science.
The names of the joint authors are sufficient guarantee of the excellence and
reliability of the m.itter." — Schoobi, aster.
Logic, Groundwork of. By James Welton, D.Lit., M.A.
Suitable for London Matriculation and similar Examinations.
" Prof. Welton has ventured to re-ari'ange the traditional oi'der of his subject .so
as to put 'induction' in its natural place before 'proof,' and shines in his
illustrations, which are good, modern, and often topical." — Mind.
Logic, Exercises in. By F. C. Bartlett, M.A., Fellow of
St. John's College, Cambridge.
Logic, Questions on, with Illustrative Examples. By HENRY
HOLMAN, M.A., late H.M.I., and M. C. W. IRVINE, M.A.
Second bJdition.
Psychology, A Manual of. By (i. F. Stout, LL.D., M.A., Fellow
of the British Aeadeni}', Professor of Logic and Metaphysics
in the University of St. Andrews. Third Edition, Revised and
Enlarged.
"The Manual is now the best of the very few good books on Psychology that
have been written in modern times."— iV/i'/irf.
Psychology, The Groundwork of. By Prof. G. F. Stout.
" All .students of philosophy, both beginners and those who would describe them-
selves as 'advanced,' will do well to 'read, mark, learn, and inwardly digest' this
hoi.ik."—0.ciiird Mayazine.
innfvcrsft^ tutorial ©rcss 1^.
A
MANUAL OF LOOIC
VOL. L
A
MANUAL OF LOGIC
7.Y TWO VOLUMES
^S BY
J. WELTON, D.LiT., MA.
SOMETIME PROFESSOR OF EDUCATION IN THE UNIVERSITY OF LEEDS
AUTHOR OF "groundwork OF LOGIC," "THE LOGICAL BASES OF EDUCATION'
" THE PSYCHOLOGY OF EDUCATION," THR ARTICLE ON EDUCATION IN
THE ENCYCLOPAEDIA BRITANNICA, " PRINCIPLES AND
METHODS OF TEACHING," ETC.
VOLUME I.
Second Edition {Sixth Impression)
London : W. B. CLIVE
(Untt>erfiitg tuiotUf ^vcsb £^.
High St., New Oxford St., W.O.
1922
600048
PREFACE TO THE FIRST EDITION.
This treatise aims at giving a fairly complete view of
logical doctrine, as generally accepted at the present
time. Of necessity, in a subject where so many
conflicting opinions are advanced, some parts of the
treatment are somewhat controversial ; but it is hoped
that every view which is criticized is fairly stated.
The discussions of disputed points have been mainly
confined to the consideration of the theories of logical
doctrine advanced in well-known works. It was felt
that such a consideration would help in making clear
those general principles which can be philosophically
justified, and which form the ultimate foundation of
the science.
A new method of diagrammatically representing
categorical propositions is suggested in § 106, and it
is hoped that some freshness of treatment will be
found in other parts of the book.
vi PREFACE.
The work is primarily intended for the use of
students preparing for the Examinations of the
University of London ; but it is hoped that it will be
of service to those who are reading for any examina-
tion of which Logic forms a part.
The paragraphs are of three classes. First, those
entirely unmarked, which contain the outlines of the
subject ; Second, those marked with an asterisk, whose
contents are somewhat less elementary ; Third, those
printed in smaller type, which contain fuller discus-
sions of particular points and the treatment of more
difficult topics. It is recommended that these last be
omitted on a first reading by all to whom the subject
is a new one. On a second reading, it is hoped they
may be found both of use and of interest.
The marginal analysis has been made sufficiently
full to serve for a final rapid revision immediately
before an examination. Such an analysis is, probably
most useful when it accompanies the text, as any
point which it leaves obscure, or which has been for-
gotten, can then be immediately, and without trouble,
referred to.
I wish to acknowledge the help I have derived from
the courses of lectures by Dr. Venn, and Mr. W. E.
Johnson, which I have had the privilege of attending
at Cambridge. I have also received most valuable
PREFACE. VU
assistance from the works of Dr. Venn, Dr. Keynes,
Lotze, Ueberweg, Mr. Bosanquet, Mill, Whewell,
Mr. Bradley, Miss Jones, Dr. Eay, Professor Bowen,
Jevons, and Professor Fowler. In addition to these,
the works of other logicians — especially those of Mr.
Stock, Professor Bain, Mr. Abbott, the late Archbishop
Thomson, Mansel, Mr. A. Sidgwick, Hamilton, and
Whately, as well as the Port Boyal Logic, have been
frequently consulted.
My best thanks are due to my friend, Mr. H,
Holman, B.A., late Scholar of Gonville and Caius
College, Cambridge, for his kindness in reading the
proof sheets and suggesting improvements. Mr.
Holman has also prepared a book of exercises and
illustrative examples, with a Key containing references
to this work, which I strongly urge all students of
this Manual to use as a companion to it ; for the
working of exercises — especially on the more formal
parts of the subject — is essential to a thorough mastery
of Logic.
J. W.
LOO. I.
PREFACE TO THE SECOND EDITION.
Though the general character, and much of the
matter, of the volume remains unchanged, yet this
Edition differs from the First in two important points.
In the first place, the book has been considerably
shortened by the omission of much of the controver-
sial and historical matter. It is hoped that this cur-
tailment— which appears to be generally desired by
those who have used the book — will render the work
more manageable by beginners in the study of Logic,
and will make the task of obtaining a connected and
consistent view of logical doctrine an easier one. At
the same time it is believed that nothing of real
importance to the students for whom the book is
primarily intended has been omitted.
In the second place, an endeavour has been maae
to give greater prominence to the distinctions of
thought which underlie those distinctions of language
PREFACE. IX
to which the traditional logic so largely confines itself,
This endeavour has led to considerable alterations in
the Book on Propositions, especially in those sections
which deal with Hypotheticals. It is hoped that
these alterations will make clearer the true character
of the mental act of judgment, and will thus lead the
reader to a more thorough grasp of the fundamentals
of the subject. In making these alterations I have
received material assistance from my friend Professor
Mackenzie, of the University College of South Wales,
who kindly read them through in manuscript, and
made some valuable suggestions.
I would also acknowledge my indebtedness to my
critics, especially to Dr. Keynes, who in the third
edition of his valuable work on Formal Logic has
drawn attention to several points on which I had failed
to make my position clear, and also to several errors.
In all these cases, I have endeavoured to profit by the
criticism, and, in particular, I have entirely altered
my treatment of Eductions of Disjunctive Propositions
(§ 106 in this Edition, § 120 in the First Edition). The
omissions in this Edition will account for any failure
to notice criticisms directed towards those parts.
J. W.
CONTENTS OF VOL. I.
JNTBOD UCTION.
CHAPTER I.
THOUGHT AND LANGDAQE.
1. Relation of Language to Logic
2. Chief functions of Language
(i. ) It gives the power of analysing complex wholes
(ii.) It enables us to form general concepts
(iii.) It abbreviates the processes of thought
(iv.) It serves as a direct means of communicating
thought
(v.) It is a means of recording thought
8. The ambiguities of language cause confusion in thought..
FAOB
1
3
3
3
5
5
5
CHAPTER XL
DKFINITION AND SCOPE OF LOGia
4. Origin of Logic
5. Definition of Logic ... ...
6. Is Logic a Science or an Art I
7. Uses of Logic
8. Divisions of Logic ...
(L) Views as to Conception of
(a) the Realists ...
(6) the Nominalists
(c) the Conceptualieta
10
10
12
13
13
16
16
16
17
xu
CONTENTS.
9.
10.
(ii.) Views as to Judgment
(a) the Nominalist View...
(6) the Conceptualist View
(c) the Material or Objective View
(iii.) Remarks on Inference
View of Logic adopted in this Work
Pure or Formal and Material or Applied Logic
CHAPTER IIL
RELATION OF LOQIO TO OTHER SOIKNOES.
11. General relation of Logic to other Sciences
12. Logic and Metaphysics
13. Logic and Psychology
14. Logic and Rhetoric ...
16. Logic and Grammar ...
PAGE
17
17
17
18
19
19
20
24
25
26
27
28
CHAPTER IV.
THE LAWS OF THOUGHT.
16. General character of the Laws
17. The Principle of Identity
18. The Principle of Contradiction
19. The Principle of Excluded Middle...
20. The Principle of Sufficient Reason...
21. Hamilton's Postulate
22. Mathematical Axioms
80
31
33
34
37
38
89
BOOK I.
TERMS.
CHAPTER I.
GENERAL REMARKS ON TBRMS^
23. Definitions of Term and Name
24. Names or Terms may be Single-worded or Many-worded
25. Categorematic and Syncategorematic Words
40
41
42
CONTENTS. XUl
CHAPTER II.
DIVISIONS OF TERMS.
PAOB
26. Table of Divisions of Terms 44
27. Individual and General Terms 45
(i.) Individual Terms ... .. ... ... ... 45
(a) Proper Names ... ... 45
(6) Significant Individual Terms ... ... 46
(ii.) General Terms ... ... ... ... ... 48
Collective Terms 49
Collective and Distributive Use of Terms ... 50
Substantial Terms ... ... ... ... 51
28. Connotative and Non-connotative Terms ... ... ... 51
(i.) What names are connotative ... ... ... 51
(ii.) Different limits assigned to Connotation ... ... 54
(iii. ) Difficulties of assigning definite Connotation .. 56
(iv.) Denotation of Terms ... 57
(v.) Relation between Connotation and Denotation ... 60
(vi.) Synonyms of Connotation and Denotation... ... 64
29. Positive and Negative Terms ... ... ... ... 64
Incr>mpatibility of Terras ... .. ... ... ... 64
(i.) Contradiction ... ... ... ... ... ... 65
(a) Material 65
(6) Formal 67
(ii.) Contrariety ... ... ... ... ... ... 70
Privative Terms ... 70
(iii.) Repugnance ... ... ... ... ... ... 71
SO. Concrete and Abstract Terms ... ... ... ... 72
(i.) Relation between Concrete and Abstract Terms ... 72
(ii.) Abstract Terms are either Singular or General .. 74
(iii.) Connotative and Non-connotative Abstract Terms 74
31. Absolute and Relative Terms ... 75
CHAPTER III.
THE PREDI0ABLE3.
32. Definition of Predicable 78
33. Aristotle's Four- fold Scheme of Predicabled 78
34. Porphyry's Five-fold Scheme of Predicables ... ... 80
aav
CONTENTS,
36. Genus and Species ...
36. Diflferentia
37. Proprium
38. Accidens ... ...
39. The Tree of Porphyry
40. General remarks on the Predicables
PAOS
81
83
84
85
86
88
CHAPTER IV.
THB CATEGORIES OB PREDIOAMKNTS.
41. The Categories are a Classification of Relations ...
42. Aristotle's Scheme of Categories
43. Objections to Aristotle's Scheme of Categories ...
(i. ) By the authors of the Port Roynl Logic ...
(ii.) By Kant
(iii.) By Lotze
(iv.) By J. S. Mill
44. Answers to these objections
45. Hamilton's Arrangement of Aristotle's Scheme . . .
46. Other Schemes of Categories similar to Aristotle'.^
47. J. S. Mill's Scheme of Categories
48. Kant's Scheme of Categories
89
90
93
93
94
94
94
95
99
100
101
103
CHAPTER V.
DEFINITION OF TERM.S.
49. Functions and use of Definition
50. De&nition per Genus et diferentiam
51. Limits of Definition
52. Rules of Definition ...
53. Kinds of Definition ...
(i.) Nominal and Real
(ii.) Substantial and Genetic
(iii.) Analytically-formed and Synthetically-formed
(iv.) Essential Definition and Distinctive Explanation
107
108
no
114
118
118
120
121
121
CONTENTS.
XV
CHAPTER VI.
DIVISION AND OLASSIFIOATION.
54. Logical Division
(i. ) General Ciiaracter of Logical Division
(ii.) Logical Division is indirect and partially material
(iii. ) Operations somewhat resembling Logical Division
(a) Physical Partition
(6) Metaphysical Analysis
(c) Distinction of meanings of equivocal terms
Rules of Logical Division ...
Division by Dichotomy
57. Purely Formal Division
58. Material Division or Classification...
'Artificial' and 'Natural' Classification ...
Classifications for Special Purposes
Classifications for General Purposes
Classification is not by Types
63. Classification by Series
64. Scientific Nomenclature
65. Scientific Terminology
55.
56
59
60
61
62
PAOK
123
123
125
126
126
126
126
127
130
133
134
136
137
139
144
145
146
150
BOOK II.
ruOPOSITIONS.
CHAPTER L
DEFINITION AND KINDS OF PROPOSITIONS.
66. Definition of Proposition
67. Kinds of Propositions
Categorical Propositions.
68. Analysis of the Categorical Proposition ...
69. Analytic and Synthetic Propositions
70. Quality of Propositions
154
155
156
160
161
XVI
CONTENTS.
71. Quantity of Propositions
(i. ) Universal
(a) Singular
(b) General
(ii.) Particular
Indesignate Propositions
72. The Four-fold Scheme of Propositions
Distribution of Terms ...
73. Other Signs of Quantity
(i. ) Numerically Definite Propositions ..
(ii.) Any
(iii.) A few ...
(iv. ) Plurative Propositions, Most and Feio ; Hardly
any ; Scarce
74. Propositions with Complex Terms
(i.) Explicative modifying clauses
(ii.) Determinative or Limiting modifying clauses
75. Compound Categorical Propositions
(i.) Compound in Form
(a) Copulative
(6) Eemotive
(c) Discretive
(ii. ) Exponible, i.e., Compound in Meaning
(a) Exclusive
(h) Exceptive
(c) Inceptive and Desitive
Hypothetical Propositions.
76. Nature of Hypothetical Propositions
77. Relation of Hypothetical to Categorical Propositions
78. Quality and Quantity of Hypothetical Propositions
(i.) Quality
(ii.) Quantity
PACE
163
163
163
164
167
169
171
172
173
173
174
174
174
176
177
177
178
178
178
178
178
179
179
179
180
181
184
186
186
186
Disjunctive Propositions.
79. Nature of Disjunctive Propositions ... 187
80. Relation of Disjunctive to Hypothetical and Categorical
Propositions ... ... ... ... ... ... 190
CONTENTS.
XVll
81. Quality and Quantity of Disjunctive Propositions
(i.) Quality
(ii.) Quantity ...
82. Modality of Propositions
PAGE
192
192
192
192
CHAPTER II.
IMPORT OF CATEGORIOAL PROPOSITIONS.
83. Predication ...
84. The Predicative View
85. The Class-inclusion View ...
86. QuantiBcation of the Predicate
87. The Comprehensive View ...
88. The Attributive or Connotative View
89. Implication of Existence
196
197
198
200
208
209
211
CHAPTER III.
DIAGRAMMATIC REPRESENTATION OF PROPOSITIONS.
90. Nature and Use of Diagrams ... ... ... ... 215
91. Euler's Circles 216
92. Lambert's Scheme 219
93. Dr. Venn's Diagrams ... ... ... ... ... 220
94. Scheme Proposed ... 222
BOOK III.
IMMEDIATE INFJSESNCES.
CHAPTER I.
GENERAL REMARKS ON IMMEDIAra INFBBENOEH.
95. Nature of Immediate Inferences ...
96. Kinds of Immediate Inferences
226
227
XVIU
CONTENTS.
CHAPTER IL
OPPOSITION OF PB0P0SITI0N8.
97. Opposition of Categorical Propositions
(i.) Subalternation
(ii.) Contradiction...
(iii.) Contrariety .,,
(iv.) Sub-contrariety
98. The Square of Opposition
99. Summary of Inferences from Opposition
100. Opposition of Hypothetical Propositions
101. Opposition of Disjnnctive Propositions
PAGE
228
229
232
234
23G
239
240
244
246
CHAPTER IIL
EDUCTIONS.
102. Chief Eductions of Categorical Propositions ... ... 248
(i.) Obversion 251
Material Obversion ... ... 254
(ii.) Conversion ... ... ... ... ... ... 255
(a) Of A propositions ... ... ... ... 256
(6) Of E propositions ... 258
(c) Of I propositions ... ... ... ... 259
(d) Of 0 propositions ... ... ... ... 260
(e) Obverted Conversion... ... ... ... 261
(iii.) Contraposition and Obverted Contraposition ... 262
(iv.) Inversion and Obverted Inversion ... ... ... 265
103. Summary of Chief Eductions 267
104. Less Important Eductions... 268
(i.) Inference by Added Determinants ... ... ... 268
(ii. ) Inference by Complex Conception ... ... ... 270
105. Eductions of Hypothetical Propositions ... ... ... 271
(i. ) From A propositions ... ... ... ... ... 271
(ii.) From E propositions 272
^iii.) From I propositions' 273
(iv.) From 0 propositions ... ... ... ... ... 273
106. Eductions of Disjunctive Propositions 274
(i.) From Universal Disjunctives 274
(ii.) From Particular Disjunctives ... ... ... 274
CONTENTS. XIX
BOOK IV.
SYLLO G ISMS.
CHAPTER I.
OENKBAL NATUBK OF SYLLOGISMS.
PAOX
107. Definition ot Syllogism .. ... .„ 275
108. Kinds of Syllogisms 280
CHAPTER n.
AXIOMS AND CANONS OF PCBE STLL00ISM3.
109. Basis of Pure Syllogistic Reasoning ... ... ... 282
110. Axioms of Categorical Syllogisms ... ... ... 283
(i.) Axioms applicable to all forms of Categorical
Syllogism 283
(a) Whately's Axioms 283
(6) Hamilton's ' Supreme Canon ' ... ... 284
(c) Thomson's ' General Canon ' 284
(ii.) Axioms applicable to only one form of Categorical
Syllogism 285
(a) The Dictum de omni et nulla 285
(b) The Notanotce 286
111. General Rules or Canons of Categorical Syllogisms ... 287
(i. ) Derivation of the Rules from the Z>ic<«»i .. . ... 287
(ii.) Examination of the Rules ... ... .„ ... 288
(iii.) Simplification of the Rules ... ... 298
(iv.) Corollaries from the Rules ... ... ... ... 302
112. Application of the Rules to Pure Hypothetical and Pure
Disjunctive Syllogisms ... ... ... ... 304
(i.) Pure Hypothetical Syllogisms .. 304
(ii.) Pure Disjunctive Syllogisms ... ... ... 305
xz
CONTENTS.
CHAPTER IIL
FIGURE AND MOOD.
113.
Distinctions of Figure
PAGK
... 306
114.
Axioms and Special Rules of the Four Flo
(i. ) The First Figure
ures ... ... 307
307
(ii.) The Second Figure
308
(iii.) The Third Figure
(iv.) The Fourth Figure
309
:}10
115.
(v.) Classification of Special Rules
Characteristics of each I'igure
311
312
(i.) The First Figure
(ii.) The Second Figure
312
313
116.
(iii.) The Third Figure
(iv.) The Fourth Figura
(v.) Summary
Determination of Valid Moods ...
314
314
314
315
(i.) Direct Determination, by reference
315
(a) to Fundamental Principles of
(6) to General Rules of Syllogism
Thought ... 316
s 319
(ii.) The Mnemonic Lines...
322
117.
118.
119.
Fundamental and Strengthened Syllogism
Subaltern Moods or Weakened Syllogisms
Valid Moods of the First Figure
s 322
323
324
(i.) Barbara
324
(iL) Celarent
328
(iii.) Darii
328
(iv.) Ferio ...
330
120.
Valid Moods of the Second Figure
330
(i.) Cesare
(ii.) Camestres
330
331
(iii.) Festino
332
(iv.) Baroco
332
121.
Valid Moods of the Third Figure
(i.) Darapti
(ii.) Disamis
3.33
333
334
(iii.) Datisi
335
(iv.) Felapton
335
CONTENTS.
XXI
PAQE
(v.) Bocardo
. 336
(vi.) Ferison
. 336
122.
Valid Mnods of the Fourth Figure
. 337
(i.) Bramantip
. 337
(ii.) Camencs
. 338
(iii.) Dimaris
. 338
(iv.) Fesapo
. 339
(v.) Fresison
. 339
123.
Syllogisms and Implications of Existence
. 340
124.
Representation of Syllogisms by Diagrams
. 341
(i.) Euler's Diagrams
. 341
(ii.) Lambert's Diagrams
. 346
(iii.) Dr. Venn's Diagrams
. 346
125.
Figure and Mood in Pure Hypothetical and Pure Dis
junctive Syllogisms
. 348
(i.) Pure Hypothetical
. 348
(ii.) Pure Disjunctive
. 350
126.
127.
128.
129.
CHAPTER IV.
REDUCTION OF SYLLOGISMS.
Function of Reduction
Explanation of the Mnemonic Lines
Kinds of Reduction
(i.) Direct or Ostensive ...
(ii.) Indirect
Reduction and Implications of Existence
130. Reduction of Pure Hypothetical Syllogisms
352
353
355
355
358
359
360
CHAPTER V.
MIXED SYLLOGISMS.
131. Mixed Hypothetical Syllogisms
(i.) Basis of mixed syllogistic reasoning from hypothe
tical major premise
(ii.) Determination of Valid Moods ...
(iii.) Examples
(iv.) Reduction to Categorical Form
362
363
363
368
370
XXll CONTENT&
fAOE
132. Mixed Disjunctive Syllogisms 371
(i. ) Basis of mixed syllogistic reasoning from a disjunc-
tive major premise ... ... ... ... 371
(ii.) Forms of Mixed Disjunctive Syllogisms 371
(iii.) Reduction of Mixed Disjunctive Syllogisms ... 373
(iv.) Examples 374
(v.) Disjunctive Syllogisms in the wider sense 375
133. Dilemmas ... ... ... ... ... ... ... 376
(i.) Forms of Dilemma 376
(a) Determination of Forms ... ... ... 376
(b) Mutual Convertibility of Fornix 379
(c) Other Views 381
(ii.) Reduction of Dilemmas ... ... ... ... 383
(iii.) Rebutting a Dilemma ... ... ... ... 384
CHAPTER VI.
ABRIDGED AND CONJOINED SYLLOGISMS.
134. Enthymemes 387
135. Progressive and Regressive Chains of Reasoning ... 390
136. Sorites 393
(i.) Kinds of Sorites 393
(ii.) Special Rules of Sorites ... ... ... ... 396
(a) The Aristotelian Sorites 396
(b) The Goclenian Sorites 397
(iii.) Figure of Sorites 398
(iv.) History of Sorites ... ... ... ... ... 399
137. Epicheiremas , ... 400
CHAPTER VII.
FUNCTIONS OP THE SYLLOGISM.
138. Universal Element in Deductive Reasoning 402
139. Validity of Syllogistic Reasoning 405
140. Limitations of Syllogistic Reasoning ... 409
INTRODUCTION.
CHAPTER T.
THOUGHT AND LANGUAGE
1. Relation of Language to Logic. Intr.
Cli I
Logic treats of the processes of thought by means of which — '
knowledge in its most general aspect is attained. Now when of'know-^^**
we examine into what we mean by knowledge we find we can ledge in
analyse it into the matter known and the activity of mind
by which it is known. We do not mean that these can be ^o can
separated, so that either can exist apart from the other, km)wledge
but that we can distinguish these two aspects in thought. }^^)°^- ^
The matter known we speak of in general as the objec- known
tive or external world, which we can perceive, but whose stitutive*"''
existence is independent of this or that act of perception. 2'^''^^^^^°^
Moreover, when a portion of it is perceived by any indi-
vidual it must be perceived in one particular way, and is so
perceived under like conditions by every normally-consti-
tuted mind. This then we regard as reality^ and we may say Reality is
that this constraining power is the characteristic of the real. constT^ins^
But to say this is to say that reality can only be conceived ^^ activity
by us as in essential relation to consciousness. When we
say, for instance, that we regard our houses as continuing to
exist unmodified by our absence, we mean that we believe
that if we could then perceive them, they would appear to
us in their ordinary aspect ; for in this way only can we
make our experiences consistent. But we mean no more
LOG. I. 1
INTRODUCTION.
Intr.
Ch.1.
and cannot
be conceived
out of rela-
tion to
thought.
Logic deals
with
thought in
general,
and is,
therefore,
closely con-
nected with
language ;
than this : existence out of all possible relation to conscious-
ness is meaningless to us. Hence we see that reality is not
independent of thought, but is, on the contrary, constituted
for each one of us by that synthetic activity of thought
which alone makes sensuous experience intelligible. It is
thought which by synthesizing impressions received through
different senses gives us ideas of material 'things'; it is
thought which grasps the relations of substance, identity,
time, space, mutual interaction, etc., in which these things
exist, and so constitutes for each one of us an idea of a single
and self -consistent world. The world for each one of us
exists only as thought.
But the world as thought exists in the same way for all.
In so far as the concept of the world held by different
persons varies, the variation is due to differences in the
thoroughness with which it is thought. It is in general
agreement, indeed, that we find the test of accuracy in the
interpretation by individuals of the impressions they receive.
If the distinction between red and blue cannot be perceived
by a particular individual, we say he is ' colour-blind,' i.e., we
assert that though the two impressions to him are but one,
yet they are two in reality, because that is the common
testimony of mankind. It is not with the knowledge or the
thought of individuals that Logic is concerned but with
knowledge and accurate thought in general. But to deal
with this would be impossible were there no commonly
received and understood means of expressing and communi-
cating thought. Such a means is language.
Language we may define as a system of bodily actions, with
a sensible effect at every moment to guide it, which is used for the
purpose of carrying on and of expressing thought. The simplest
form of language is purely imitative as exemplified in those
natural and expressive gestures to which man is sometimes
compelled to resort when desiring to communicate with
persons between whom and himself there is no common
speech. But interesting as this is from a psychological point
of view it does not here concern us. The only language with
which Logic is concerned is the ordinary conventional language
THOUGHT AND LANGUAGE.
Intr.
Ch. I.
but only aa
far as dis-
tinctions of
language
correspond
with dis-
tinctions of
thought.
of spoken or written words ; and it is concerned with that
only in so far as the distinctions of language correspond with
distinctions of thought. But here we must acknowledge a
temptation from which Logic has by no means always kept
free — the temptation to substitute the symbol for the
substance, to deal with language and the distinctions of
language rather than with the underlying reality of thought.
*2. Chief Functions of Language.
Language we have seen to be both an instrument for think- Lang-K-cge
J . • i- ii , 1 has Jive main
mg and a means or expressing and communicating thought functions.
{see § 1). We may analyse its functions under the former of
these heads into three, and under the latter into two, main
classes. We will treat each of these functions separately.
(i.) Language gives the power of analysing complex
wholes.
Unless we could mark any particular element of a complex (l.) it makes
impression by some sign — as a name — we should find it analysis of
impossible to fix attention upon that element to the practical complex
'■ , ^ ^ wholes.
exclusion of others from consideration. We receive an
impression as a whole ; for instance, we see a man writing.
But, with the aid of language, we can separate this into two
ideas — the man and the act of writing — and we connect these
two under the relation of agent and action. In other cases
the relation may be that of subject and attribute, as when
we separately dwell upon an action and its moral character.
(ii.) Language makes possible the formation of Con-
cepts.
Bound up with this power of analysis is the ability to (il.) it gives
form concepts, i.e., intelligible syntheses of the attributes formrnT*^"'
and relations which constitute the essential nature of a class Concepts.
of things. Such concepts involve no sensible mental images ;
it is by being named that their elements gain that definite-
ness and independence which is necessary to enable us so to
group them. Nor is it only in the formation of concepts
1-2
4 INTRODUCTION.
Intr. that names are necessary. The concept, when formed, would
^^' ^' soon become vague, and tend to disintegration, if it were not
held together, and made definite, by a name. The name fits
it to be an object of thought, and preserves it for future use
without the necessity of repeating the whole process of its
formation.
(iii.) Language shortens the process of thinking.
(iH.) It Even without language all thinking is more or less sym-
processes oi '^olic ; for the ideas present to full consciousness derive their
thought. full import from their relation to other ideas to which no
direct attention is paid. With thought aided by language,
especially conventional language, this process is carried much
further. We scarcely ever make the meanings of the words
we use explicit to our minds, though they are implicitly
present, as is shown by the mental shock consequent on hear-
ing, or reading, a pi'oposition connecting incongruous ideas,
as ' The victors .sued for peace.' In the case of many con-
cepts, indeed, our knowledge of what they involve is always
more or less hazy. This may either be because of their great
complexity, as, for instance ' The British Constitution'; or,
more frequently, because we learn to use the name through
its application to individual objects whose special qualities
we have no need to investigate ; this is the case with the
names of very common objects, as ' dog' or 'horse.' In all
such cases the name necessarily stands for a number of attri-
butes of which our idea is more or less shadowy. It is, in
fact, only in the case of scientific terms well known to us,
as ' square,' ' triangle,' that our concepts are, as a rule, per-
fectly distinct, and, in such cases, the attributes implied are
generally more or less consciously apprehended whenever the
word is used. But in ordinary speech it is not so, and this
symbolic use of words for the ideas, or groups of ideas, they
imply is an abbreviation of thought very similar to the
shortening of mathematical operations by the aid of symbols.
This abbreviation makes it possible to carry on trains of
thought infinitely more complex than would otherwise be
possible.
XnOUGHT AND LANGUAGE.
(iv.) Language is a direct means of communicating Intr
thought. chj
This is the most obvious function of languaare. We always ('^P ^* '^
think of it as chiefly useful for this purpose, though, of means of
course, it must be primarily an instrument for conducting caUi^"'''
thought, as, otherwise, we should have nothing to communi- thought.
cate. It is this function of language which makes all social
intercourse possible, and enables each person to profit by the
knowledge acquired by others with whom he is brought in
contact. Thus, mental development is facilitated and made
infinitely more speedy, and of greater breadth and richness,
than would be possible if each mind were condemned to
exist in isolation.
(v.) Language is a means of recording thought.
This is the great use of writing and printing, and is, evi-
dently, an extension of the function last described. By this
means we can benefit by the experience and share in the
knowledge and thoughts, not only of those few persons
whom we may chance to meet, but of men of all times and
all places who have given us, in their writings, a record of
their intellectual work. It is often an advantage, too, to
record our own thoughts and discoveries for future reference ;
and, thus, written language is an aid to the development of
our owu thought as well as a means for communicating it to
others.
*3. Ambiguities of Language.
It has often been pointed out that the exact signification
of a word depends on the context, and that this flexibility of
meaning is necessary if words are to express ideas equally
fluctuating. But it is evident that this indeterminateness
may be a frequent cause of confusion. For, not only may a
word be understood in a slightly different sense from that in
which it is employed, but the identity of the word may cause
the person using it to think his idea remains the same when
it may have undergone even considerable modification. It is
not meant that words have no fixed meaning — they have a
(v.) It en-
ables us to
record
thought.
Indetermin-
ateness in
the meaning
of words
leads to ecu
fusion of
thought and
misunder-
standing.
6
INTRODUCTION.
Intk.
Ch. I.
The same
word used
to express
entirely
different
ideas is not
likely to
lead to
error.
Synonyms
may cause
confusion.
Living
languages
both grow
aua Lie cay.
definite kernel of meaning, (which in names is called their
connotation, see § 28) which is fixed for the time, though
even this may change gradually — but that the full force of
the word is only grasped when the context is known.
But, in addition to this necessary modification of the fixed
kernel of meaning which the shades of thought render neces-
sary, there are, in all languages, other causes of ambiguity.
The use of the same verbal symbol to express entirely
different ideas, (often due to derivation from quite different
roots), though it may give rise to puns, probably never
causes any real misconception. No one is likely to confound
rein with reign, because they have the same sound ; nor the
noun tear with the verb spelled in the same way but meaning
to rend. Not even when spelling and pronunciation both
agree is confusion likely to be caused. No hesitation could
possibly be felt on reading the word vice in a passage as to
whether a moral fault, or an instrument for holding things
firmly, was referred to. These are only apparent ambiguities,
but they show plainly how much the meaning may, in
extreme cases, depend on the context.
The presence of synonyms in a language often leads to
delicate shades of meaning being overlooked or confused, so
that the idea conveyed is not exactly the same as that
intended. This richness of vocabulary frequently leads to
confusion of thought in another way. People often think
they understand a thing merely because they can give it two
names, each of which they use to define the other. Thus
'Truthfulness is veracity'; but 'What is veracity?' —
' Truthfulness.' So thought often remains nebulous when
it is believed to be definite and distinct.
Every living language is subject to processes of growth and
of decay. New words are invented to express new ideas,
and, whilst new, they are generally the most definite in mean-
ing of all the words in the vocabulary. At the same time
old words drop out of use or undergo a gradual modification
of meaning. In this last process we have a fertile source of
ambiguity and confusion. Such a gradual change of mean-
ing may be brought about by the word being either gene-
THOUGHT AND LANGUAGE.
ralized or specialized, or by an alteration in the way in which
the idea it represents is generally regarded. The word virtue
is an instance of the last. It bore a very different meaning
in the mouth of a pagan philosopher from that which it bears
when uttered by a Christian moralist. Not only did it for-
merly include elements, as pride, which would now be regarded
as utterly repugnant to it, but it excluded others, as humility,
which form an important part of the modern notion.
Generalization occurs when the same word is extended to
cover different ideas not before included under it. This fre-
quently happens when something new has to be named ; the
tendency is to give it the name of that familiar object which
it most nearly resembles. So it may happen that, after
several such extensions, the same verbal symbol represents
ideas or things which have little or nothing in common. We
may take the word court as an instance. It originally denoted
what we now call a court-yard. From that, the name was
transferred to the palace to which the yard was attached,
then to the inhabitants of that palace who surrounded the
sovereign. From this use of the noun was formed the verb
to court, meaning to practise the arts in vogue at court ; soon
this was generalized to cover all cases of seeking favour.
Finally, it has been specialized, in one of its meanings, into
denoting seeking in marriage.
The transfer of names by analogy to objects which bear a
real or fancied resemblance to those to which they first
belonged is another example of the generalizing process.
Thus, sounds are called sweet, and griefs bitter, on the analogy
of tastes. In fact, all our higher mental pleasures and pains
are described by words taken by analogy from the physical
world ; we speak of a sharp pain, a light heart, a heavy
trouble.
The word oil is a good instance of generalization. It
originally meant, as the Latin name, oleum, shows, olive- oil
only ; but its application has been gradually extended, till it
is now used to denote many substances, animal, vegetable,
and mineral, which resemble the original ' oil ' in some
qualities. In such a case as this, too, the name decreases in
Intb.
Ch. I.
The mean-
ing of a word
may change
because the
complex
idea has
changed.
Oeneraliza-
tion extends
the applica-
tion of
words and
80 lessens
their fixed
meaning,
and thus
allows the
same word
to have
different
senses.
Names are
transferred
by analogy.
8
INTRODUCTION.
Intk.
Ch. 1.
Some cases
of general-
ization lead
to con-
fusion.
Words with
no definitely
fixed mean-
ing are un-
suited for
use as scien-
tific terms.
Specializa-
tion restricts
the applica-
tion of a
word, but
increases its
meaning.
fixed meaning, and depends more and more on the context,
though its various senses do not differ so much as in such
a word as ' court.'
None of the above examples can be regarded as likely
causes of confusion in thought, or of misunderstanding ; but
all cases of generalization are not so harmless. The word
law may be cited as an instance where confusion has arisen
and has led to much error and controversy. Its original
meaning was the command of a superior, and this is still its
signification in Theology and Politics. As such a command
led to uniformity of conduct in some particular on the part
of the subjects, the word ' law ' was generalized so as to
cover all cases of uniformity in the occurrence of phenomena.
Thus arose the term ' Law of Nature.' But from this use of
the word the idea grew up that a Law of Nature meant
something more than a mere uniformity ; and, thus, con-
fusion of thought was caused by this ambiguity of language
and led to much fruitless controversy.
In many cases words which have been generalized have a
meaning so indeterminate and fluctuating that they may call
up very different ideas in different minds, or in the same
mind at different times. Such terms are particularly un-
suited to scientific discussion, and, when they are used in it,
they generally lead to misunderstanding and dispute.
Political Economy is the most striking example of this ; the
use of such ambiguous words as ' capital,' ' wealth,' ' rent,'
* labour,' etc., has led, not only to endless arguments, but to
contradictions and errors through writers using the words in
varying senses and assuming what is true in one meaning to
be true in all.
The opposite process of Specialization is due to an
occasional meaning being gradually imposed on the general
meaning, and, perhaps, gradually substituted for it. If a
word is often applied in a special manner, that which was
merely an occasional part of its meaning may become an
essential part. Thus fowl meant originally any bird, but is
now restricted to one particular domesticated species.
Vitriol originally denoted any crystalline body with a certain
THOUGHT AND LANGUAGE.
degree of transparency, but is now restricted to one or
two such substances, and its fixed meaning embraces many
qualities besides the two originally implied by the name.
Similarly, most of the ecclesiastical terms used in the Chris-
tian Church have attained their present signification by a
process of specialization ; for instance, a bishop was originally
any overseer ; a priest was an elder ; a deacon an admini-
strator.
These processes are continually going on side by side, and
very often the same word is subjected to both in turn.
Pagan is a good instance of this. Originally it denoted a
villager. Then, when Christianity spread through the
Roman Empire, and the old heathen faith lingered in the
country districts long after it had practically disappeared
from the towns and cities, the name became associated with
heathenism, and so was specialized. Gradually this became
the most important part of the meaning of the word, and
then generalization became easy, and the word was used to
denote any heathen, the original signification being entirely
forgotten.
Thus, we see that though language is indispensable to
thought, yet it may sometimes lead to confusion and mistake.
The remedy is to continually check the symbolic use of
language by a reference to, and examination of, the ideas
underlying it.
Intr.
Ch. I.
The same
word may
be, in turn,
generalized
and special-
ized.
Language
should be
continually
checked by
reference to
the ideas it
expresses.
CHAPTER n.
Inte.
Ch. II.
A Science of
Logic is pos-
sible be-
cause men
can examine
their
thoughts.
Logic ia the
science of
the princi-
ples which
regulate
valid
thought.
DEFINITION AND SCOPK OF LOGItt
*4. Origin of Logic.
Although men possess the power of thought, they do not
always employ that power so as to fulfil the great object of
thought — the ascertainment of truth. The action of men's
minds is not infallible ; and, thus, false judgments are formed,
or false inferences drawn even from true judgments. In
other words, men reason sometimes well, sometimes ill.
But, they are not only able to think and reason about
external objects, they can reflect on those thoughts and rea-
sonings. Now, as false reasoning generally leads to conclu-
sions which are seen to be erroneous because they are rejected
by others, by comparing the mental processes which led to
the untrue results with those which, at other times, led to
true results, the reasons why the former processes were
invalid, and the latter valid, become manifest ; and thus
general principles are discovered to which thought conforms
whenever it is valid. The collection of these principles into
a systematic whole forms the Science of Logic.
5. Definition of Logic.
Logic is the science of the principles wMch regulate
valid thought.
A Science is, in all cases, a systematic body of knowledge
relating to some particular subject-matter. Knowledge of
isolated facts is not science — it can only become so when
such isolated facts are brought under general laws forming
part of a consistent whole. The subject-matter of each science
DEFINITION AND SCOPE OF LOGIC. 11
is some definite part of the material of human knowledge ; Intb,
of Algebra it is the relations and properties of numbers, of Ch^i.
Botany, vegetable life, and of Logic, thought.
A Principle (or Law) is the statement of a general truth ;
that is, a truth which holds good universally in that science,
as contrasted with a particular truth, which holds good in
some cases only. Thus, it is a principle in Physics that all
material bodies attract each other in direct proportion to
their mass and in inverse proportion to the square of their
distance, and this principle we call the Law of Gravitation ;
but, that metals sink in water is not such a general truth,
for it does not hold true in all cases.
Thought is used to denote both the process and the product
of thinking. Logic is concerned primarily with the validity
of the process, that is, of reasoning. But it also takes
account of that of the products ; of the concept, whether it
is true or in agreement with reality ; and of the conclusion
of an argument, which is expressed in a judgment, whether
it is consistent with itself, and whether it expresses the
relations existing between the things concerning which it is
made.
Thought is valid, in the narrower sense, when it does not
involve self-contradiction in any one of its processes. In the
wider and truer sense, it is valid when it agrees with the
objective world — when things are thought of as holding that
relation to each other which they really do hold. And such
thought is knowledge, and, therefore, our definition is
equivalent to saying that 'Logic is the science of the
method of knowledge.' But validity of thought in this
wider sense also resolves itself into consistency ; for, as we
saw in § 1, reality is for us just that mode of thinking our
experience which is forced upon us by the attempt to make
that experience consistent. This wider validity is, therefore,
a wider consistency ; a consistency not limited to any one
process of thought, but embracing the whole of experience
and the whole of mental life. Yalid thought in the wider
sense is that which constitutes the world for us as a consistent
and systematic universe.
12
INTRODUCTION.
Tntr.
Ch. II.
Logic is a
Science and
not an Art,
though it
has a prac-
tical as well
as a theoret-
ical side.
According to the view of validity taken by different
writers on Logic the scope of the science has been enlarged
or contracted [see §§ 8, 9, and 10]. As thought may be
invalid, it follows that the principles of Logic do not regulate
all thought, as the law of gravitation applies to all material
bodies. Invalid thought only becomes indirectly part of the
subject-matter of Logic, when the cause of the invalidity —
or Fallacy — which is always, of necessity, a violation of one
of the principles of valid thought, is investigated.
6. Is Logic a Science or an Art ?
Much dispute has arisen on the question whether Logic is a
Science, an Art, or both. The writers of the Port Royal Logic
called it ' The Art of Thinking,' and were followed in this by Aldrich
and others. Mansel and Thomson, on the other hand, denied that
it is an art at all, whil-it Whately combined the two views, and
defined it as ' The Art and Science of Reasoning.' Mill agreed that
it is both an Art and a Science, and it was called both by ancient
Greek writers.
There is no doubt that it is a Science, as it is an organized system
of knowledge (see § 5). Whether we call it an Art depends on our
use of that term. That Logic has a practical as well as a theoretical
side can hardly be denied ; for, by the very fact of laying down the
principles of valid thought, it furnishes rules for avoiding and de-
tecting false reasoning ; and it provides, moreover, principles for
investigating the relations between things. Thus, the scholastic
writers distinguished between the Logica doctns, or purely theo-
retical part, and the Logica vtens, or practical application of the
former. But here we have a Practical Science, not an Art in the
strict sense of the term. Rightly understood, an Art is a body of
precepts for performing some work, and is, thus, not limited to one
object-matter ; for instance, the art of music involves a knowledge
of musical instruments, of the human voice and its management,
etc., in addition to the knowledge of musical theory, which last
alone can be called Science. Logic is not an art in this sense ; if it
were it would, of necessity, be the widest of the arts — Ars Artium,
as it has, indeed, been called — and would embrace special rules for
reasoning in every branch of knowledge. This it does not do ; its
principles, and the rules drawn from them, are quite general. It does
not even profess to teach men to reason accurately ; it only givea
DEFINITION AND SCOPE OF LOGIC.
13
the principles and rules to which accurate reasoning conforms. A
man may be able to arrive at true conclusions and yet have never
learned Logic ; and, on the other hand, one may know all logical
principles and yet reason falsely.
It is true that, if thought be guided by logical rules and principles,
the result will be valid reasoning, but this is not the object for which
the principles and rules are stated ; they are rather general results
arrived at from an examination of valid processes of thought.
7. Uses of Logic.
One important use of Logic follows from what has been said
above. Though it does not make false reasoning impossible, it does
furnish rules and principles by which error can be detected and,
therefore, avoided. By bringing an argument to the test of Logic
we can ascertain whether or not reason has, in that instance, been
employed rightly ; and, if not, what was the cause and origin of the
error. This is by no means unimportant ; a man may feel sure that
a conclusion is false, and yet be unable to say ivhy it is so. This is
frequently the case with men of 'sound common-sense,' who often
see at once that an argument is invalid, but are unable to point out
where the fallacy lies ; in which case there is, of course, no guarantee
against committing the same error in reasoning again, when, per-
haps, the conclusion may not be so obviously at fault. But, if the
source of the fallacy can be traced, there is hope that similar mis-
takes will be avoided in the future.
But the chief use of Logic is found in the fact that it is pre-
eminently a mental discipline ; and to train the mind should be the
one great object of all study. One's object should never be so much
to acquire knowledge of various facts and sciences, as to develop
and perfect the reasoning powers. As Hamilton said, ' In the world
there is nothing great but man, and in man there is nothing great
but mind ;' and, truly, a well-balanced and evenly-developed mind
is the noblest possession man can enjoy. And a man who, by a
study of Logic, has trained his mind to reason justly, will reap the
advantage in every department of study or practice to which he
may devote himself.
8. The Divisions of Logic.
The aim of thought is to arrive at knowledge, and know-
ledge, as we have seen {see § 1) is the mental assertion of
what we are constrained to regard as true. In other words.
Intr.
Ch. II.
Logic is
a guard
against
error in
reasoning,
and the best
mental dis-
cipline.
Judgment is
the essential
form of
thought.
14
INTRODUCTION.
Tntb.
Ch. II.
We may
analyse
judgments
iatoConcepts
and combine
them into
Inferences.
the essential form of thought is judgment, and every judg-
ment is a partial interpretation of reality. The simplest form
of judgment is an interpretation of an isolated fact of percep-
tion, as when we say • It rains,' or, ' This is a rose.' But as
thought advances, judgments become more and more complex,
and involve a wider and wider range of reality. Indeed, the
ideally perfect judgment would embrace and interpret all
reality. That judgment, of course, we cannot make ; it would
involve complete and perfect knowledge. But every actual
judgment makes an assertion of some kind about reality.
It is plain that in judging we both analyse and construct.
We analyse, in that we decompose complex reality into
> elements standing to each other in certain relations, such as
substance, identity, space, time, causality, etc. ; and we
synthesize in that we construct each of these elements out of
a complex of such relations. These elements of reality are
thought as concepts, and our concepts are true in so far as
they express the nature of reality. As every actual judg-
ment asserts some relation between elements of reality thus
mentally held apart, it is obvious that we may regard judg-
ment as the assertion of a relation between two corcepts.
But it must be kept in mind that this is only half the truth,
the other half being that the whole judgment itself is one
act of thought, refers to one single aspect — though it may
be a complex one — of reality, and may itself be included
in one concept richer than either of those, regarded separately,
which it connects.
But judgment is in no case a simple aflSrmation of percep-
tion ; it always involves more or less interpretation. Now,
interpretation is inference. This is implicit in most cases of
direct perception ; for instance, I inhale a certain odour, and
say, ' There are some violets near.' But when the grounds of
the interpretation are clearly set forth, we have that explicit
inference with which alone logic is concerned. Now such
explicit inference obviously involves a synthesis of judgments,
for each of the grounds of the inference is expressed in a
judgment. These propositions are called the premises, and
the further judgment of which they are the ground, and which
DEFINITION AND SCOPE OP LOGIC.
15
follows necessarily from them, is called the conclusion.
When we have only one premise, the only inference possible
is interpretative, and renders explicit what was implicit in
the original proposition. Such inferences are called Imme-
diate, But when we have two premises the inference is
called Mediate, because the conclusion is only possible
through the unioa of two judgments dealing with one
common element of reality. All cases of Mediate Inference
which involve more than two premises can always be
analysed into series of steps of inference, in each of which
only two premises are concerned. Mediate Inference is
commonly said to be of two kinds, according as we approach
the aspect of reality to be interpreted from the side of some
established general principle, or from that of individual
facts. In the former case the inference is called Deductive,
and in the latter case Inductive. The relation of these to each
other will be more fully considered later on [see § 146 (iii.)].
Lastly, inferences are not isolated, but proceed in trains
towards some definite end. The orderly arrangement in dis-
course of such trains of inferences, is named Method. Here
also the one fundamental aim is the unification of experience,
and the more perfect comprehension of the universe.
It thus appears that the four commonly accepted divisions of
logical doctrine — Conception, Judgment, Inference, Method —
are rather different aspects under which we may regard the one
fundamental act, always in its essence the same, of interpre-
tation of experience by thought. When we regard them
from the view of language the distinction between them is
more marked, and appears as one of varying complexity.
The verbal expression of a judgment is a proposition^ and in
such expression the two concepts into which the judgment
can be analysed are necessarily made distinct and repre-
sented by separate terms. Indeed so distinctly are the terms
held separate that there is always a danger of being misled
into forgetting the fundamental unity of the judgment itself.
Again, when an inference is expressed in language the three
judgments into which it can be analysed stand out distinct
and separate, and there is a similar danger of forgetting that
Intb.
Ch. II.
Inferences
may be
(1) Immedi-
ate.
(2) Mediate.
(a) Deduc-
tive.
(6) Induc-
tive.
Method ia
the orderly
arrange-
ment of
inferences.
The four
divisions of
logical doc-
trine,
(1) Concep-
tion,
(2) Judg-
ment,
(3)Inference,
(4) Method,
are only as-
pects of ona
hmdamen-
tal process.
16
INTRODUCTION.
Intk.
Ch. II.
an inference also is a single act of thonght referring to a
single aspect of reality. Similar remarks apply to trains of
inferences. •,
Different
views as to
the nature
of a con-
cept.
(a) Realism
— that the
universal
had a real
objective
existence.
(6^ Nominal-
ism— that
the name is
the only
general, and
is repre-
sented by
images.
(i.) Views as to Conception.
Very different views have been held by philosophers as to the
nature of General Notions or Concepts. These opinions may be
divided, broadly, into three classes : Realism, Nominalism and
Conceptualism.
(a) The Realists held that some real substance existed in nature
corresponding to every general notion, which combined its con-
stituent properties ; that, for instance, corresponding to the concept
* horse,' there existed something which was no horse in particular,
but in which every individual horse participated — a universal horse
which consisted as it were of horse-essence, and which was the
only real horse existence. This school of thought, to which many
of the Schoolmen belonged, is now quite obsolete.
(6) The Nominalists go to the other extreme and hold that
General Notions are mere matters of words ; that a class is consti-
tuted by its name alone, and that the name is the only general
element. They hold that, every time a General Name [see § 27
(ii.)] is used, an image is present to the mind. They say that the
class is thought of either under the image of one individual member
of it, with a kind of mental reservation that the particular attri-
butes of this representative are to be disregarded, or else by a rapid
succession of images of various members of the class. On this view,
it is difScult to see how we could have any concept of abstract
qualities, such as truth and justice, for of such we can certainly
form no images. Hobbes, Berkeley, and Prof. Bain may be cited
as advocates of this view.
J. S. Mill held a modified view which made some approach to
Conceptualism. He says, " We have a concrete representation,
" certain of the component elements of which are distinguished by a
" mark [i.e. , the class name], designating them for special attention ;
" and this attention, in cases of exceptional intensity, excludes all
" consciousness of the others " (Exam, of Hamilton, p. 323. But
he adds, " there is always present a concrete idea or image, of which
" the attributes comprehended in the concept are only, and cannot
" be conceived as anything but, a part " (ibid., p. 337). He, there-
fore, held that a concept is a mere generic image.
DEFINITION AND SCOPE OF LOGIC.
17
(c) The Conceptualists, to which school most modern writers,
including Kant, Mansel, Dr. Ward^ and Mr. Stout, belong, hold
that a concept involves no image. It is an intelligible, not a sensible,
synthesis of attributes [cf. § 2 (ii.)]7 and thought is constantly carried
on by means of concepts without the accompaniment of any images,
whether particular or generic, except those auditory or motor
images of the names which symbolize the concepts of which the
train of thought is composed.
(ii.) Views as to Judgment.
Equally widely divergent views are held as to the nature of a
judgment, and these views lead to differences of opinion as to the
validity of thought with which Logic has to deal (see § 5), and hence
as to the scope of the science. There are three chief schools : —
(a) llie Nominalists hold that propositions are merely statements
about names, and that the whole scope of Logic is bounded by names
and their relations. Whately is a representative of this school. He
limits the science to a regard for mere verbal consistency ; but, on
the other hand, Mill and Prof. Bain, though Nominalists in their
views of the concept, yet take the Objective view of the scope of
Logic.
(6) The Conceptualists regard Logic as concerned not with lan-
guage but with the thought it represents. The chief representatives
of this school are Kant, Mansel and Thomson. They define Logic
as ' the science of the pure (or formal) laws of thought,' or as ' the
science of thought as thought,' meaning by this, of thought entirely
separated from, and independent of, the things thought about. The
most extreme and consistent writers of this school (as Mansel) hold
that all which can be expressed by a judgment is that one concept
is contained in, or forms part of, another, so that no judgment can
ever do more than unfold and make explicit the content of a con-
cept ; it can never be a statement involving additional information.
Logic, from this point of view, is a mere ' Logic of Consistency '
(as Hamilton called it) ; it can have no concern with the real rela-
tions between things. In other words, it takes the narrower view
of validity of thought (see § 5).
It will be seen that the words ' Nominalist ' and ' Conceptualist
are ambiguous. They may refer to the views held of a concept, or
to the views held as to the nature of a judgment. It is quite pos-
sible for a Nominalist or a Conceptualist in the former sense to bold
LOG. I. 2
Intb.
Ch. n,
(r:) Conceptu
alism — that
a concept is
an intelligi-
ble synthe-
sis of attri-
butes.
Dififerent
views as to
the nature
of a judg-
ment.
(a) The
Nominalist
— that judg-
ments are
about names
only.
(b) The Con-
ceptualist—
judgments
are about
concepts
only.
18
INTRODUCTION.
Intr.
CluII.
(c) The Ob-
jective— that
judgments
are about
things only.
the Objective view of Logic, as, in fact, most modern writers do, in
a more or less modified form.
(c) Those holding the Objective or Matericd view of Logic give it
a much greater scope, for they take the wider view as to the validity
of thought (see § 5). Some, indeed, would make it coincident with
the whole realm ot reality. They hold that propositions do not
express relations between mental concepts, but between the things
those concepts represent. If, for instance, we say ' Grass is green,'
we do not mean to say that our concept of grass contains, or agrees
with, our concept of green, but that the thing grass possesses the
attribute of greenness. Mill takes this view. He defines Logic,
in his Examination of Hamilton (p. 388), as " the Art of Thinking,
" which means of correct thinking, and the Science of the Conditions
"of correct thinking " (p. 391); and by 'correct thinking ' he ex-
plains himself to mean thought which agrees with the reality of
things (pp. 397-8). In his Logic he adopts the definition, "Logic
" is the science of the operations of the understanding which are
" subservient to the estimation of evidence " {Logic, Introd., § 7),
which clearly makes the science conversant with reality ; and a
material treatment is adopted throughout the entire work.
Whewell takes a similar view. He says, " The Logic of Induction
'* is the Criterion of Truth inferred from Facts, as the Logic of
' ' Deduction is the Criterion of Truth deduced from necessary prin-
•' ciples " (Novum Organon Benovatum, p. 98).
Mr. H. Spencer takes an extreme view, and regards all reference
to thought as of quite minor importance. He defines Logic as the
Science which " formulates the most general laws of correlations
" amongst existences considered as objective" {Principles o/ Psycho-
logy, vol. ii., § 302). He further says, " The propositions of Logic
"primarily express necessary dependencies of things and not neces-
" eary dependencies of thought ; and, in so far as they express
'* necessary dependencies of thought, they do this secondarily —
" they do it in so far as the dependencies of thought have been
"moulded into correspondence with the dependencies of things"
{ibid.).
G. H. Lewes draws a distinction between Subjective and Objec-
tive Logic. The former " is occupied solely with the codification
'* of the processes of Proof," whilst the latter is synonymous with
Metaphysics, and is concerned with " the codification of the niost
"abstract laws of Cause" {Problems of Life and Mind, vol. i.,
p. 75).
DEFINITION AND SCOPE OF LOGIC.
19
(iii.) Remarks on Inference.
Of course, writers who hold that logic is only concerned with
the formal self-consistency of isolated processes of thought, deny
that it can really treat of inductive inferences at all, for all such in-
ferences are essentially material. Such logicians, therefore, con6ne
the science to deductive inference and to these so-called * Perfect
Inductions' which consist of a mere summing-up of individual
judgments of perception.
As Induction furnishes many of the general propositions which
are the bases of deductive reasonings, it would seem natural to treat
it before Deduction. But this branch of the subject has only been
treated fully in comparatively recent times. For a long time De-
duction was looked upon as synonymous with Logic. Thus, it is
customary to treat Deduction first, and the usual plan will be
followed here ; because the limits of Deduction are more clearly
defined, because it is simpler, and because a knowledge of its prin-
ciples is necessary to the understanding of Induction.
It may be pointed out that the whole doctrine of Concepts
(Terms), Judgments (Propositions), Immediate Inferences and De-
ductive Mediate Inferences, is frequently spoken of, somewhat
loosely, as ' Deductive Logic,' whilst Inductive Inference is called
' Inductive Logic' It seems better and more accurate to restrict
the terras ' Deductive ' and ' Inductive ' to Inference, to which
alone they rightly belong.
9. View of Logic liere adopted.
It follows from what has been said in previous sections
(c/". §§ 5 and 8) that we regard Logic as dealing not with
processes of merely abstract and symbolic thought, nor with
mere processes of an external reality out of all necessary
relation to thought, but with reality as known, i.e., as inter-
preted by thought. The growth of inference depends on the
difference between knowledge and ignorance ; were all reality
known, there would be no room left for fresh inference ;
were all reality unknown, inference could not begin, for it
would have no starting point. The possibility of inference
is found in the fact that the world is a rational and syste-
matic unity and can, therefore, be understood — partially at
any rate — by a mind which is itself a rational unity. With
2—2
Intr.
Ch. II.
Deduction
will be dis-
cussed be-
fore Induc-
tion.
The terms
' Deductive
and ' Indue
tive' apply
to Inference
only.
Logic is con-
cerned with
reality as
presented in
thought.
20
INTRODUCTION.
Intr.
Ch. II.
We shall
Bpeak of
Terms ami
Proposl-
tions or of
Concepts
and Judg-
ments ac-
cording to
the shade of
meaning
to be ex-
pressed.
Fm-mal Lngic
is concerned
with the
self -consist-
ency of
thonght ;
Material
Logic with
its objective
truth.
the raw material of knowledge — sensations and sense
impressions— Logic does not deal ; with the unconscious
inference involved in perception it only deals indirectly and
in so far as the process is, at bottom, one with the explicit
inference with which it does deal. It is with the conscious
judgments and inferences by which rational mind interprets
sensuous experience, that Logic is concerned. We cannot say,
then, that Logic is either purely subjective or purely objec-
tive ; indeed subjective and objective are only aspects dis-
tinguil^hable in thought of that reality which exists for us only
as embracing them both. We do not, then, restrict Logic to
that merely formal and barren validity of thought which
consists in the absence of self-contradiction in each of its
processes, regarded in isolation from the rest ; we hold that
it must take that wider test of validity which is found in
a complete and consistent system, and which tries every
process of thought by reference to that system.
As to the nomenclature which will be adopted it may be
observed that there seems no good reason for adhering
strictly to the language either of the Nominalists or of the
Conceptualists. We shall very frequently speak of Terms
and Propositions, as the reference will, in many cases, be
more especially to the verbal expression of thought. But we
shall feel at perfect liberty to use the terms Concept and
Judgment, when they seem to be the more appropriate ; that
is, when the reference is chiefly to the mental idea or process.
10. Pure, or Formal, and Material, or Applied, Logic.
Though Logic takes note of the wider sense in which
thonght can be said to be valid, it does not lose sight of the
narrower sense {nee § 5). It must furnish principles to test
the consistency of thought, or it will be useless as an instru-
ment to determine the worth of thought when exercised on
reality. These principles compose that Pure or Formal
Logic, or Logic of Condstency, which the Conceptualists
regard as forming the whole of the science [_see § 8 (ii.) (6)1.
When we apply Logic to the investigation of Objective
reality, we are in the domain of Material or Applied Logic.
DEFINITION AND SCOPE OF LOGia
21
All Induction is, of necessity, material, for its end is to
determine the actual truth or falsity of propositions about
things. A great deal of the subject-matter of Book I is
also material; for instance, the doctrines of Definition,
Connotation of Terms, Predicables, Categories, and, to a
great extent. Division and Classification. The validity of a
Concept — of which the Term is the verbal symbol — is, of
course, a material question. As the doctrine of Propositions
(Bk. II) deals to a great extent with the form in which the
thought is expressed, it is largely formal ; but in so far
as reference is involved to the reality of which the judgment
is made, as in considering the Import of Propositions
(Bk. II, Ch. II), it is material. Most of Deductive Rea-
soning (Bks. Ill and IV) can be treated formally ; for
its purpose is to determine the relative truth of proposi-
tions ; that is, what propositions can be inferred from
others ; and this relative truth depends solely on the
form of the argument, and is entirely independent of the
matter.
By the matter of thought is meant the thing or things
thought about ; by its form^ the way in which the mind
thinks about them. The matter may vary whilst the form
remains the same. In a similar way, many pieces of music
may be written in the same measure and have the same
rhythm, though the series of notes or chords may vary
infinitely ; the rhythm or measure is the form, the notes or
chords are the matter. On the other hand, the same thought
may be expressed in different ways ; that is, the form may
vary whilst the matter remains unchanged. So, to revert to
our illustration, the same series of chords or notes may be
adapted to several different rhythms.
As the validity of formal reasoning depends on the form
alone, we may express our terms in symbols ; and there is a
great advantage in doing this, as the attention is thus fixed
solely on the form, and we are not led to think a piece of
reasoning is necessarily correct because the conclusion which
has been drawn is, as a matter of fact, true. If we say, for
instance,
Intb.
Oh. II.
Deductive
Reasoning
and much of
the doctrine
of Proposi-
tions is
Formal ; the
rest of the
Science is
Material.
Matter of
thought is
the thing
thought
about ;
Form of
thought is
the way the
matter is
considered.
Formal
Reasoning
may be
expressed
symboli-
cally.
22 INTRODUCTION.
Intb. All metals are fusible
^^^- Some substances are not metals
,', Some substances are not fusible
we assert, as a conclusion, a proposition •which is un-
doubtedly true ; but the argument is invalid, for that proposi-
tion does not follow from the premises. Oar purpose here is
to decide, not as to the objective truth of the conclusion but,
as to the validity of the reasoning. Had we written the pre-
mises symbolically, thus
All M's are P
Some 5's are not M
we should have been more likely to examine the argument,
as we should have no prejudice in favour of the conclusion
Some S's are not P, as we have in favour of Some substances
are not fusible. And such an examination would show that
conclusion to be unjustifiable by the premises. For they do
not assert that nothing is P except the A/'s, and, therefore,
we do not know that the some S's which are not A/'s are, in
consequence, not P's. In such a symbolic manner all propo-
sitions may be written in formal reasoning, where S, M, and
P, stand for any matter whatever ; and the validity of pro-
positions and arguments so expressed can, evidently, depend
on their consistency alone. Were we to assert that No S is S,
the proposition would be formally invalid, for it is self-con-
tradictory ; or, if we have accepted as true the proposition
No S is P, we should fall into formal contradiction were we
to assert that Some S is P, whilst S and P remain unchanged
in their reference. When we pass from propositions to argu-
ments the same thing holds. If we assert
Every M is P
Every S is M
we cannot avoid the conclusion that Every S is P without
self-contradiction. Of course, we may give such meanings
to S, M and P that the conclusion is materially false, but
this can only be the case when material error is to be found
in one or both the premises. If we say
DEFINITION AND SCOPE OP LOGIC.
23
All volcanoes are mountains
All geysers are volcanoes
and draw the conclusion All geysers are mountains, our argu-
ment is formally valid ; for, if expressed in symbols, it is
identical with the one just considered ; but the conclusion is
materially false because the premises are false. But, to
establish the truth of the premises is the province of Induc-
tion ; Deduction simply furnishes principles and rules for
drawing consistent conclusions from premises which are
given us.
The consideration of such examples as these shows the
necessity for embracing both Formal and Material Logic in
our science, if we would use it as an instrument for attaining
a knowledge of truth. It may be thought, and has, indeed,
been said, that Deduction is useless, and that Logic should
be Inductive only. This is not the case, for when once a
general proposition has been arrived at by Induction, Deduc-
tion enables us to apply it to numerous cases which were not
before known to be instances of it ; and, thus, the use of
Deduction dispenses with the need of innumerable new
Inductions. For example, when the general proposition
which sums up the Law of Gravitation — that all material
bodies attract each other in direct proportion to their mass
and in inverse proportion to the square of their distance —
was arrived at by Induction, it was immediately applied to
explain not only the fall of unsupported bodies to the earth,
but the motions of the planets and their satellites, the
occurrence of tides, and many other phenomena which were
not previously suspected to be instances of the same law of
nature. Induction and Deduction are not so much two
mutually helpful processes as two aspects of one process.
Intr.
Ch. II.
Logic must
be both
Formal and
Material ;
and must
embrace
both Deduc-
tion and In-
duction.
CHAPTER IIL
Tntr.
Ch. III.
The princi-
ples of Logic
must regu-
late thought
in all
branches of
knowledge.
RELATION OF LOGIC TO OTHER SCIENCES.
11. General Kelation of Logic to other Sciences.
Logic has often been called the 'Science of Sciences,' be-
cause it treats of those regulative principles of thought to
which, however various may be their methods, all branches of
knowledge must conform. Logic does not profess to furnish
rules or means of investigating any particular branch of
science ; its province is purely general, and is confined to
that common basis of all science — the laws which must be
universally observed by all valid thought. Logic has, thus,
not simply an absolute value, as scientifically an end in
itself ; but, through the influence which as the science of
thinking it exerts upon the process of thinking, a relative
value also. For the very enunciation and examination of the
regulative principles of thought further their practical
application, since they are certain to be more fully and
exactly employed by those who are scientifically conscious of
them than by those who reason by the simple light of nature.
As De Morgan says : " I maintain that logic tends to make
" the power of reason over the unusual and unfamiliar more
" nearly equal to the power over the usual and familiar than
"it would otherwise be" {^Budget of Paradoxes, p. 330).
Logic also aids scientific investigation by pointing out the
most appropriate procedure for arriving at conclusions from
the premises with which observation has furnished us. This
practical value of Logic has earned for it the name Ai's
Artium — the Art of Arts — as well as that of 'Science of
Sciences ' {cf. § 6).
RELATION OP LOGIC TO OTHEK SCIENCES.
25
Though thus related to all sciences, yet Logic has closest
relations with those sciences which treat of Being, of Mind,
and of Language, for it investigates thoughts about things
expressed in speech. We will, then, consider more at length
its connexion with Metaphysics, with Psychology, with
Rhetoric, and with Grammar.
12. Log^ic and Metaphysics.
Metaphysics enquires into the nature of Reality as such.
It thus goes beyond the various sciences, each of which deals
with some branch of the phenomena or appearances in which
Reality appeals to our senses. Each science makes certain
assumptions ; e.g. that matter has some kind of existence
that 'things' are constant in their nature, and exist in space
and time, that the changes observed in the world are not
random accidents but are regular and connected together in
causal relations. It then goes on to ask what causal uni-
formities are to be found in its special province, and what
uniformities of nature can be discovered in the ' things' with
which it deals, its aim being to establish a connected body of
doctrine concerning one portion of the contents of human ex-
perience. Every science is thus only a partial description of the
phenomenal world, and rests on assumptions which it does not
verify. The investigation of the validity of these assumptions
is the province of Metaphysics. Metaphysics, therefore, deals
with the presuppositions which underlie all experience, and these
presuppositions it tries to arrange into a system by showing that
they are necessary deductions from one ultimate first principle.
Hence, Metaphysics does not aim at knowing all things, but at
explaining all knowledge by making explicit the very forms
of all existence.
Logic holds a sort of intermediate relation between Meta-
physics and the special sciences. For Logic aims at knowing
the process by whch knowledge is attained. It assumes that
there is an absolute standard of truth, and that our thought
can grasp, at least in part, the true nature of Reality, or, in
other words, that our mental construction of the world is
Intb,
Ch. III.
Logic has
the closest
relations
with Meta-
physics,
Psychology,
Rhetoric,
and
Grammar.
Metaphysics
inveatiijates
the nature
of Reality.
Logic leaves
its ultimate
assumptions
to Meta-
physics.
26
INTRODUCTION.
Intk.
CIlIII.
Logic ac-
cepts cer-
tain meta-
pliysical
postulates.
Logic is
regulative
and ideal ;
Psychology
is empirical
and actual.
neither a mere mechanical copy of the phenomena around us,
nor an arbitrary synthesis of ideas evolved from our inner
consciousness. In brief, it assumes that experience can be
analysed and known. It is thus wider than any of the special
sciences, as it systematizes the formal conditions of all
knowing. But it must hand over all these ultimate assump-
tions to Metaphysics. Indeed that very relation of thought
to phenomena which forms experience, and which Logic
accepts as given, is itself the very central problem of Meta-
physics. Further, Logic accepts without question the
assumptions of the special sciences as to the existence and
relations of ' things. ' Its province is to state these principles
exactly and definitely ; it must leave to Metaphysics the
question as to their validity.
13. Logic and Psychology.
Psychology is the science which investigates the actual
phenomena of the mind and their development. It is wider
than Logic in that it takes account not only of thought, but
of all mental processes ; though, in that all its investigations
must be conducted in accordance with logical principles it is
narrower than Logic, which, as has been said {see § 11), analyses
the methods of all sciences. Psychology is essentially em-
pirical: it enquires into the genesis and character of all
mental activities — whether of thought, of feeling, or of will-
ing—and their relation to each other. It investigates the
mental processes subsidiary to thought and the nature of
thinking, turning its attention in all to what actually occurs
in mind. It seeks to arrange its results as uniformities, and to
deduce therefrom knowledge of the way in which, in reality,
concepts are formed, judgments made, and inferences carried
out, on what conditions mental states depend and what is the
nature of those states.
Logic, on the other hand, is normative ; it furnishes cri-
teria by which false reasoning may be discriminated from
true. It does not enquire how men do think, but lays down
laws in accordance with which they should think ; it is ideal
whilst Psychology is actual. Logic does not take account of
RELATION OP LOGIC TO OTHER SCIENCES. 27
all the ways in which men reach conclusions ; it does not Intr.
enquire how ideas are recalled by the laws of association, or
how belief arises from such association ; of the actual process
of reasoning it takes no account. It is concerned with reason-
ings only in respect to their validity ; with the dependence
of one judgment on another only so far as it is a dependence
of proof. Given certain laws, it determines the form correct
thinking ought to exhibit, but does not enquire whether
men's actual thoughts do conform to that standard.
Though the provinces of Psychology and Logic are thus
distinct, yet the latter can only be satisfactorily studied in
connexion with the former. To thoroughly understand
Logic it is necessary to know what is the nature of the think-
ing mind, what are its limitations, what is the character of the
process of thought, and how it unites with the other mental
elements to form those concepts and judgments which are
the materials with which Logic deals.
14. Logic and Rhetoric.
Logic is connected with Rhetoric in that both have a common Logic deals
object — to lead to the formation of certain conclusions. But they RhetOT-Ic" '
proceed about this in very different ways : Logic appeals to the with per-
reasoning faculty alone, whilst Rhetoric rather aims at stirring up
the emotions ; the former attempts to convince, the latter, by an
appeal to the passions, to persuade. Whilst it is true that such an
appeal will be more powerful if at the same time it is based on valid
arguments, yet by an adroit flattery of men's prejudices it is often
found possible to Instil an opinion which not only is supported by
no sound reasoning, but is actually repugnant to it. Rhetoric is
connected with Psychology in so far as the latter deals with the
emotional side of mind, whilst Logic touches it where it treats of
the intellectual or thinking side. The province of the science of
Rhetoric is to investigate the principles on which discourse should
be founded in order that it may be persuasive — its main ends are,
not the ascertainment of truth but, the enforcement of conclusions
without regard to their validity, and the incitement to action ;
Logic, on the other hand, as dealing with the relation of truths,
investigates the principles on which discourse should be founded in
order that it may be convincing, and its main end is the ascertain-
ment of truth.
suasion.
28
INTRODUCTION.
Intb.
Ch. III.
Logic is
related to
universal —
not to
particular —
Grammar.
Universal
Grammar
deals with
the general
laws of all
language.
Logic is only
indirectly
concerned
with
language.
Logical
analysis of
language
ends with
the Term,
and takes no
note of Parts
of Speech.
*15. Logic and Grammar.
Logic and Grammar are connected through the medium of
language, which is the general instrument of thought. In
considering their relation we must, of course, have regard
only to Universal Grammar ; for the history and idiomatic
peculiarities of particular languages are obviously not directly
connected with a general science like Logic.
Universal or General Grammar is the science of the uni-
versal laws which all languages must observe. It is distinct
from special grammar, which is the application of those laws
to a particular language, under the influence of the habits and
idiosyncrasies of a particular people. Or it may be defined
as " the science of the relations which the constituent parts of
speech bear to each other in significant combination" (Stod-
dart). Whilst particular grammar partakes largely of the
nature of an art, universal grammar is a science, and is
evidently connected primarily and necessarily with language,
which is, indeed, its subject matter ; its province is to trace
the connexion between certain given signs and the thoughts
they are supposed to represent.
With Logic this is reversed ; it is concerned primarily with
thoughts and concepts, and their relations to each other and
to reality ; it is concerned only secondarily with language, as
the means by which these thoughts, concepts, and relations
are invariably expressed. Logic thus considers language
simply as the instrument of thought, and only analyses it to
that point which is necessary to express the simplest element
of thought — the concept. All non-significant words, i.e.,
words which cannot by themselves express a concept, are,
thus, beyond the range of Logic, whilst all words which can
express a concept are regarded by it as of the same class —
they can form Terms, and it is immaterial whether they are,
in grammatical language, substantives, pronouns, adjectives
or verbs. Logically, the form of attribution belongs to all
characteristics of a subject-matter which are not self-depen-
dent, whether they are expressed by nouns, as Caesar was a
Roman, by adjectives, as CiEsar was amliiious, or by a verb,
RELATION OP LOGIC TO OTHER SCIENCES.
29
I
as Caesar conquered. It may be pointed out that all verbs
are logically reducible to the verb ' to be ' and a participle,
and when this analysis is made the attributive force of the
verb appears plainly, as CaBsar was conquering. Personal
pronouns are for Logic the same as substantives, demonstra-
tive pronouns the same as adjectives, whilst, as adverbs bear
exactly the same relation to verbs which adjectives do to
nouns, they are not a distinct form of the content of thought.
Again, Logic requires that relations should be expressed
between things and between concepts, but it is immaterial to
it whether those relations are expressed by inflexions of
words or by a preposition. Thus, Logic takes no note of
that division of words into parts of speech which is so
marked a feature in the grammatical analysis of language.
Again, though the proposition is the unit of thought-
expression both in Logic and in Grammar, yet its treatment
is different in the two sciences. Whilst Grammar acknow-
ledges no sentence in which subject and predicate are not
distinctly expressed, but, given that, deals with sentences of
the most varied construction, Logic accepts judgments made
in any form, even so rudimentary a one as the Exclamatory,
but demands the power of re-stating the meaning of all in
one fixed and simple logical form {see § 68). The logical
analysis of the sentence, too, often differs from the gram-
matical. The grammatical subject is always the noun or
pronoun in the nominative case, and is thus definitely fixed in
each sentence. But there is no such fixity in the case of the
logical subject, which is that known part of the experience
spoken of from which the judgment starts, whilst all the rest
of the sentence is the predicate. Frequently, then, only the
context can determine how much of any given sentence is
interpretative, i.e. belongs to the predicate, though the more
accurately our sentences express our thoughts, the more
closely do the logical and grammatical analyses agree
Intr.
Ch. III.
Logic and
Gram\ nar
treat and
analyse
seutences
differently.
CHAPTER IV.
Intr.
Ch. IV.
The Laws of
Thought are
the funda-
mental,
necessary,
formal, and
d priori
forms which
regulate all
valid think-
ing.
THE LAWS OP THOUGHT.
16. General Character of the Laws.
The Laws of Thought, Regulative Principles of Thought, or
Postulates of Knowledge are those fundamental, necessary?
formal, and a priori mental laws in agreement with which
all valid thought must be carried on. They are d priori, that
is, they result directly from the processes of reason exercised
upon the facts of the real world. They ase formal ; for, as
the necessary laws of all thinking, they cannot, at the same
time, ascertain the definite properties of any particular class
of things, for it is optional whether we think of that class
of things or not. They are necessary, for no one ever does, or
can, conceive them reversed or really violate them, because
no one ever accepts a contradiction which presents itself to
his mind as such. It is true that fallacious reasoning is com-
mon enough, but this springs from a misapprehension of the
meanings of terms, or from a confused use of terms, for
which the ambiguities of language give abundant scope {see
§ 3). Especially in long and involved reasonings, the force
of terms is often unconsciously modified, and even entirely
changed, with the result of invalidating the chain of argu-
ment ; but, at no stage of the process does the reasouer
consciously accept a contradiction. As always really obeyed
by all minds, they are laws in the scientific sense of uni-
formities ; when applied practically to govern and test
arguments, they are laws in that other sense of the word
in which we speak of laws of the land {see § 3). They are
Postulates of Knowledge because they are involved in all
attempts at interpreting experience, i.e. they are assumptions
without which thought cannot even begin the work of
THE LAWS OF THOUGHT.
81
I
reducing to order the chaos of sense impressions. Into the
justification of these postulates Logic does not enter ; that is
the task of Metaphysics (see Bk. I, Ch. Ill, § 12). Logic
assumes them because it finds them assumed in every piece of
correct thought, and it aims at expressing them as perfectly as
possible.
Much dispute has arisen amongst logicians as to the number and
expression of the necessary laws of thought, and as to the place
they should occupy in an exposition of Logic. Mill says they
should, at the earliest, be placed at the beginning of the treatment
of Judgment {Exam, of Hamilton, p. 416), and Lotze gives them
the same position. XJeberweg puts them still later, at the beginning
of Inference, and calls them "Principles of Inference." As, how-
ever, they are the necessary forms of all thought, and are, conse-
quently, required for the full comprehension of Concepts as well as
of Judgments and Reasonings, and as they also form the basis for
all logical division of Terms, we prefer to treat them as introductory
to our consideration of the science. Moreover, the Concept is
nothing apart from the Judgment, and therefore the treatment of
the Proposition really begins with the consideration of its elements
— Terms.
With regard to their number, formal logicians generally recognize
only three such laws of thought — the Principles of Identity, of
Contradiction, and of Excluded Middle. But on the view of Logic
we are advancing the Principle of Sufficient Reason must be added
to these.
We will now consider, in some detail, each Principle, and
the various forms in which it has been expressed.
17. Tlie Principle of Identity.
The simplest statement of this law is the formula A is A
or, as Leibniz put it, * Everything is what it is.' It has also
been expressed, 'Whatever is, is,' (Jevons) ; 'Every object
of thought is conceived as itself' (Mansel). It demands
that, during any argument, we use each term in one un-
varied meaning. On this principle rests the justification of
the Judgment.
No difficulty can be experienced in understanding, and
assenting to, such propositions as A is A, B is B, But, in
Intb.
Ch. IV.
They are
treated here,
as they are
necessary
forms of all
thought.
Principle of
Identity—
A is A.
32
INTRODUCTION,
Intk.
Oh. IV.
All Identity
exists
amidst di-
versity.
such statements there is conveyed no real information. To
say a thing is itself tells no more about it than does the bare
mention of its name. Identity must be interpreted in such
a way as to cover such propositions as A is B, which we are
continually making, and which experience tells us are justi-
fied by facts. We say ' Gold is yellow,' ' Lions are fierce,'
and such statements are caj)able of conveying real informa-
tion. No doubt, if fully analysed, such propositions may be
brought to the form A is A. ' Gold is yellow,' does not
mean that all yellow things are gold — that is, that gold and
yellow are convertible terms ; nor yet that gold is any yellow,
but only gold-yellow. But this analysis is not actually made
in thought, nor is it necessary. Identity is really expressed
in the proposition A is B, viz., the identity of the things to
which both names, A and B, can be applied. But this iden-
tity is expressed amidst a diversity of meaning ; the two
names have not the same signification, and, hence, the propo-
sition, in which they are conjoined, is capable of giving real
information. In truth, it is only amidst some diversity that
we know identity at all. I am the identical person I was ten
years ago, and yet I have changed ; individual men all differ
from each other in many points, yet all share in the common
nature of humanity. When, then, we say A is A we mean
that a thing remains itself even amidst change, and that
a common nature is manifested in different individual
instances.
This view of the Principle, from the subjective side, is brought
out in the statement adopted by Archbishop Thomson [Laws of
Thought, p. 212) : " Conceptions which agree can be united in
" thought, or affirmed of the same subject at the same time." Mr.
Bradley regards the principle to be affirmed as that "Truth is at all
" times true," or " Once true always true, once false always false,"
and he adopts the statement, " What is true in one context is true
"in another." Or, "If any truth is stated so that a change in
"events wiU make it false, then it is not a genuine truth at all"
{Princ. of Logic, p. 133). Ueberweg gives an Axiom of Con-
sistency, which he regards as akin to that of Identity. He ex-
presses it, "/I which is B is B ; i.e., every attribute which belongs
THE LAWS OP THOUOHT.
33
" to the subject-notion may serve as a predicate to the same," for
"the attribute conceived in the content of the notion inheres in
*' the object conceived through the notion, and this relation of in-
"herence is represented by the predicate" (Logic, Eng. trans.,
pp. 231-2). Mill's expression of the Principle {Exam, of Hamilton,
p. 409) : " Whatever is true in one form of words, is true in every
" other form of words which conveys the same meaning," though
an indispensable postulate is really a law of expression rather than
of thought.
18. The Principle of Contradiction.
This Principle, which would be better named The Prin-
ciple of non- Contradiction, is most simply expressed by
the formula A cannot both be B and not he B.
The law has been enunciated in various other ways, a considera-
tion of some of which may help in making its scope and meaning
clear. Thus : A cannot be both B and non-B ; A is not non-A ;
Nothing can both be and not be (Jevons) ; The same attribute can-
not be at the same time affirmed and denied of the same subject
(Aristotle) ; The same subject cannot have two contradictory attri-
butes ; No object can be thought under contradictory attributes
(Mansel) ; Judgments opposed contradictorily to each other (as A
is B, A is not B) cannot both be true (Ueberweg) ; The attribute
cannot be contradictory to the subject, or, A predicate does not be-
long to a thing which contradicts it (Kant) ; What is contradictory
is unthinkable (Hamilton) ; Denial and affirmation of the self -same
judgment is wholly inadmissible (Bradley). Mill's statement, "The
" affirmation of any assertion and the denial of its contradictory are
" logical equivalents which it is allowable and indispensable to make
"use of as mutually convertible" {Exam, of Hamilton, p. 414) is,
again, rather a postulate referring to expression than a principle of
thought.
On this axiom, together with that of Identity, is based all
Immediate Inference from AflBrmative Propositions, It
denies that the same thing can, at the same time, both
possess a certain attribute and not possess it ; and, as thought
must agree with reality, that we can conceive a thing as at
once both possessing and not possessing the same attribute.
The same statement cannot be, nor can we conceive it as
LOO. I. 3
Intb.
Ch. IV.
Principle of
Contradic-
tion—
A cannot both
be B and not
beB.
Contradic-
tory propo-
sitions refer
to tiie same
subject at
tlie same
time.
34
INTRODUCTION.
Intr.
Ch. IV.
Principles
of Ideutity
Hiid Contra-
diction are
proved in-
i iroctly.
Principle ol
Excluded
Middle—
4 either is, or
is not, B.
being, at the same time both true and untrue ; nor can the
same thing at once be strong and yet not be strong. Dif-
ferent parts of the same object may, of course, possess
incompatible attributes ; one end of a bar of iron may be
hot, and the other, in common parlance, cold, but the same
end cannot at once both be hot and not be hot to the same
person ; and our propositions must refer to the same end, as
otherwise, not being made of identically the same subject,
they would not be contradictory of each other. Similarly,
the same end of the bar may at one time be hot, and, at
another time not be hot ; but there would be no contradic-
tion in asserting this, for judgments referring to the same
subject at different times are not the same judgment. A
judgment does not change with time, but once true is always
true. Contradictory judgments, therefore, must refer to
identically the same subject at identically the same time ;
they must assert incompatible attributes as standing in the
same relation (including that of time) to the same subject.
Of course, there must be perfect sameness of sense both in
the single terms of the contradictory propositions and in their
affirmation and negation ; the propositions must be contra-
dictories not merely apparently and in words, but in reality
and meaning.
It has been disputed whether this axiom and that of Identity are
really underivable. Ueberweg thinks they can be deduced from
" the idea of truth, i.e., the consistency of the content of perception
"and thinking with existence" {Logic, p. 238). Anyhow, as
Aristotle said, the validity of the axioms can only be proved in-
directly, viz., by showing that no one can help recognizing them in
actual thinking and acting, and that, were they destroyed, all
distinctions of thought and existence would perish with them.
19. The Principle of Excluded Middle.
The Principle of Excluded Middle between two contradic-
tory judgments is most clearly expressed by saying A either
is, or is noty B.
Other expressions of it are : A is either B or non-B / Two contra-
dictories cannot both be false at the same time ; Everything must
THE LAWS OF THOUGHT.
35
either be or not be (Jevons); Either a given judgment must be true
or its contradictory, there is no middle course (Thomson) ; Of con-
tradictories one must be true and the other false ; Of two contra-
dictories one must exist in every object ; Judgments opposed as
contradictories (such as h is B, A is not B) can neither both be fals3
nor can admit the truth of a third or middle judgment, but the one
or other must be true, and the truth of the one follows from the
falsehood of the other (Ueberweg) ; The double answer, Yes and
No, cannot be given to one and the same question understood in the
same sense (Ueberweg) ; Of contradictory attributions we can only
affirm the one of a thing, and if one be explicitly affirmed the other
is denied (Hamilton). Mill again asserts a corresponding postulate
of expression : "It is allowable to substitute for the denial of either
"of two contradictory propositions the assertion of the other"
{Exam, of Hamilton, p. 416).
Intr.
Ch. IV.
This principle of Thought has been questioned, and even Contradic-
denied, by writers who have confounded contradiction with no^MJ^j'""'
other forms of incompatibility, especially contrariety (see alternative,
§ 22). But, whilst contrary terms mark the utmost possible trariesdo.
divergence, contradiction is simple negation. There are, of
course, many intermediate stages of grey between the con-
trary attributes, blacli and white ; and many varying degrees
of warmth between the contraries, hot and cold. Thei-e are,
then, many alternatives besides the propositions, This paper
is white — this paper is black, This water is hot — this water
is cold. But there is no third alternative whatever be-
tween the contradictory assertions, This paper is white
— this paper is not white, This water is hot — this water
is not hot. It has been urged, as proof that contradiction is
not thus exhaustive, that there is a mean between plus and
minus, viz., zero; but here again, we have contraries, not
contradictories. A mathematical quantity must either be
positive, or not be positive ; and, if the latter, it may be either
zero or negative. Similarly, one thing need not be either
greater or less than another given thing, because ' greater' and
* less ' are not contradictories, and there is a mean, * equal
to,' between them ; but a thing must either be greater or not
be greater than another given thing, and, if it be not greater,
3—2
36 INTRODUCTION.
Intr. it may be either equal to it or less than it. Mill thought he
Ch^'. jjg^jj discovered a mean between the true and the false, which
are both contradictory and contrary terms, viz., the unmean-
ing : " Between the true and the false there is a third pos-
" sibility, the Unmeaning" ( Logic, Bk. II., ch. vii., § 5). But
to this it has been answered that an unmeaning possibility is
no possibility at all ; "a proposition which has no meaning
"is no proposition ; aad ... if it does mean anything it is
"either true or false" (Bradley, Prin. of Logic, p. 145). In
short, great care is necessary to avoid confusing judgments
whose predicates are contrary terms with those whose predi-
cates are contradictories ; it is so easy to make the negation,
which should only deny a strict agreement in all points, imply
a thorough-going and complete divergence. If a man is
declared not guilty of a certain crime people are inclined,
thereupon, to attribute to him perfect innocence ; whereas
there may have been any degree of approximation to full
guilt which yet fell short of it. The denial of guilt as the
accusation 2}uts it leaves open the possibility of some less
degree of guilt ; in many cases, further enquiry is invited
rather than barred.
Non-B is not Lotze objects to expressing the principle of Excluded Middle by
ceptT '^°^' ^^^ formula A is either B or non-B instead of by the formula A
either is, or is not, B, because, be says, non-B is really unmeaning as
it embraces everything in the universe except B / for instance,
' notgreen ' would not only embrace all other colours, but all other
qualities and things whatsoever — as hot, cold, long, etc. — which are
not included under the term ' green.' With this understanding we
may correctly say ' Honesty is not-green ' ; but the proposition is
practically meaningless.
In practice This is certainly true, and when we do use such a negative term
negative ^^ non-B, not-green, which is but seldom, we really, in intention,
usually confine it to the genus of which o is a species ; in the case of not-
liiiiitod in gj,ggQ |;q ti^g genus of colour. Of course, with this limitation it is
not possible to affirm either green or not-green about every subject,
but only about those which possess the attribute of colour.
The Axiom of Excluded Middle is necessary, in addition
to those of Identity and Contradiction, to form a basis for
THE LAWS OP THOUGHT,
37
some forms of Immediate and Mediate Inference. It limits
the thinkable in relation to affirmation, and declares the
necessity of affirming one or other of two opposed contradic-
tory judgments, but it does not decide which of them is true.
Of course, the same limitation to a definite point of time
holds here as in the Principle of Contradiction (see § 18).
By the Principle of Contradiction we are forbidden to think
that two contradictory attributes can be together present in
the same subject ; by that of Excluded Middle we are for-
bidden to think they can both be, at once, absent ; but no
help is given us to decide which must be present and which
absent. i
From the point of view of language the three principles
above discussed may be summed up by saying that whenever
we use a name we must be understood to use it in its full
meaning both (1) positively and (2) negatively, and (3) it
must either be given or denied to everything whatever.
That is, the use of a name asserts all the attributes it implies
and denies all others which are incompatible with those ; and
everything must either possess all those attributes or be
without some, or all, of them.
20. The Principle of Sufficient Reason.
The Principle of Sufficient Reason was first distinctly
formulated by Leibniz in the words, "In virtue of this
" principle we know that no fact can be found real, no pro-
" position true, without a sufficient reason, why it is in this
" way rather than in another ;" and again, " Whatever exists,
"or is true, must have a sufficient reason why the thing or
" proposition should be as it is and not otherwise " {cf.
Monadologle, §§ 31-39). Other statements of the principle
are : Every judgment must have a sufficient ground for its
assertion (Mansel) ; Every proposition must have a reason
(Kant) ; A judgment can be derived from another judgment
(materially different from it), and finds in it its sufficient
reason, only when the (logical) connexion of thoughts
corresponds to a (real) causal connexion (Ueberweg).
As we necessarily regard reality as a systematic unity we
38
TNTKODUCTION.
Tnti?.
Ch. IV
altribute the external invariable connexion between different
phenomena to an inner conform ability to law. This may be
symbolically expressed A + B=C, where we mean that any
subject A, together with the condition by which it is
influenced B, is identical in content with the consequent C,
which is the subject itself as thus altered. For example, if
v4=giinpowder and J5=the high temperature of a spark,
then A -f-Z?=C which is the explosion of that powder. This
relation between the reason {A-]-B) and the consequent (C)
we necessarily conceive as universal ; we could not conceive
A -\-B as a reason for C at all, if it did not always produce C.
The principle, in brief, expresses the necessary j ostulate
of knowledge, that explanation is attainable, and hat the
explanation of any element of reality must be soug'iit in its
relation to other elements, and ultimately to the whole
system of reality. The law of causation is the aspect of the
principle of Sufficient Reason which is most frequently
appealed to (see Bk. v., cji. i.).
The acceptance of the principle necessitates that if we
grant the reason we must accept the consequence which
follows from it, and it is, thus, one of the foundations of
syllogistic, and, indeed, of all other, reasoning. It follows
from this, moreover, that logical necessity is not absolute but
hypothetical ; a consequence appears >/ — and only if — the
appropriate conditions are secured.
tlamilton's
Postulate—
JFc may state
explicitly in
language all
that is impli-
titly con-
tained in the
thought.
21. Hamilton's Postulate
Hamilton, in his Lectures (vol. iii., p. 114) thus states what he
regards as a necessary postulate of Logic. " Before dealing with a
"judgment or reasoning expressed in language, the import of its
" terms should be fully understood ; in other words, Logic postulates
'•to be allowed to state explicitly in language all that is implicitly
" contained in the thought." Some of the consequences which Hamil-
ton deduced from this postulate will be noticed in the chapter on
the Import of Propositions, and reasons given for dissenting from
them (see § 86). But the Postulate itself may be accepted, and
\aken to assert that it is permissible to vary the mode of etating a
judgment so long as the meaning is left unchanged, for the mean-
i
THE LAW3 OP THOUGHT. 39
ing, and not the form of words in which it is expressed, is the im- Intr.
portant point. Read in this sense it becomes practically the same Cli^IV.
as Mill's statement of Identity (see § 17). iSuch. variation in the
wording is frequently necessary in order to reduce the sentences of
ordinary discourse to the strictly logical form, as propositions con-
sisting of Subject, Copula, and Predicate [see §§ 8 (ii. ), 68].
22. Mathematical Axioms.
Valid arguments need not be based entirely on the Principles of All valid
Identity, Contradiction and Excluded Middle, though they must arenot" ^
alwaj's be in conformity with them. Equally cogent are those treated in
founded on mathematical axioms, such as the argumentum d, Logic.
fortiori ; If A is greater than B, and B is greater than C, then A
is greater than C ; or the axiom, Things which are equal to the same
thing are equal to one another. But arguments based on these,
though perfectly valid, are not expressed in that form of reasoning
which is treated of in formal Logic.
Other logical principles, such as the Dictwn de omni et nnllo, the
Canons of the Syllogism, the Postulates of Induction, will be dis-
cussed in connexion with those parts of Inference to which they
respectively apply.
BOOK I.
TERMS.
CHAPTER I.
GENERAL REMARKS ON TERMa
ch. I. * 23. Definitions of Term and Name.
A logical
proposition
consists of
Subject,
Predicate
and Copula.
A Term is the
Subject or
Predicate of
a logical
proposition.
The simplest element of thought is the judgment, and the
verbal expression of a judgment is a proposition (c/l § 8).
When a proposition is expressed in its perfect logical form
— S is P or S is not P [see §§8 (ii.) and 68] — it is seen to con-
sist of three parts : —
(a) Something of -which the assertion is made, called
the Subject, and denoted in the symbolic form of
the proposition by S.
{&) Something affirmed or denied of the subject, called
the Predicate, and symbolized in the formal
statement by P.
(c) The verb is, either alone or accompanied by not,
by means of which the assertion is made, called
the Copula.
The Subject and Predicate are called the Terms (from
Lat. terminus, or boundary) of the Proposition, and they are
the verbal representatives of the things, and of our concepts
of them, between which the judgment affirms a relation
{see § 9). Both, therefore, must be names of substances or
GENERAL REMARKS ON TERMS.
41
of attributes. This leads to a wider use of the word Term as
synonymous with Name, whether forming part of a proposi-
tion or not ; but Logic considers names only when they are
regarded as actual or possible terms in the stricter sense.
* The usually accepted definition of a name is that of
Hobbes, and is as follows : " A Name is a word taken at
" pleasure to serve for a mark which may raise in our minds
" a thought like to some thought we had before, and which,
'• being disposed in speech and pronounced to others, may be
" to them a sign of what thought the speaker had or had not
" before in his mind " (^Computation or Logic, eh. ii.). " Had
not " is included in order to embrace purely negative terms
which simply imply the absence of an idea {see §§ 19 and 29).
To this definition it has been justly objected that, on no known
theory of the origin and growth of language, can it be said
that names are words " chosen at pleasure " to denote things ;
there has been no voluntary and arbitrary affixing of certain
words as signs to certain things, but a natural and gradual
growth of language ; those words had better, therefore, be
omitted from the definition. Some phrase should, also, be
added to it to bring within its scope such ' many- worded
names ' as ' First Lord of the Treasury,' which, though con-
sisting of five words, is yet only one name as it denotes onJy
one object of thought. We may then, finally, say that,
A Name is a word, or combination of words, serving as a mark
to recall to our own minds, and to raise up in the minds of others,
the idea of some object of our thought.
24. Single-worded and Many- worded Terms.
The simplest names consist of a single word, as * horse,'
* London.' Such names are given to all the more important
objects with which we are acquainted and which we require
to name most frequently. But the multitude of things in
the world is so enormous that, not only can we not give each
a separate name of its own, but we cannot even form them all
into definite classes, each with its own name. Many of them
we must name by a kind of description ; thus, many,
perhaps the majority, of names or terms consist of a
Book I.
Ch. I.
A Name is
the verbal
mark of the
idea of a
thing.
A Name may
consist of
any uiinil.cr
of words.
42
TERMS.
Book I.
Ch. I.
combination of several words, and are, consequently, called
Many-worded Names, Such names always contain one or
more words which, if used in a different context, would be
themselves names, but with these are usually other words
which cannot be used as names. For instance, in the
proposition ' The First Lord of the Treasury is the present
leader of the House of Commons ' both the terms are many-
worded names, and both contain words — lord, treasury,
leader, house, commons — which are capable, by themselves,
of forming either the Subject or the Predicate of a proposi-
tion ; and others— first, present — which can be used as pre-
dicates though not as subjects. At the same time, there are
other words — the, of — which cannot, by themselves, form
terms at all.
A Categore-
matic Word
can form a
Term.
A Syncate-
gorematic
Word can-
not form a
Term.
This is the
only divi-
sion of
words recog-
nized by
Logic.
25. Categoreraatic and Syncategorematic Words.
We have, thus, in Logic, two, and only two, classes of
words : —
(a) A Categorematic word is one which can, by
itself, be used as a term.
(ft) A Syncategorematic word is one which cannot,
by itself, form a term; but can only enter,
with one or more categorematic words, into
the composition of a many-worded term.
(The terms ' Categorematic ' and 'Syncategorematic' are
derived from the Greek (cnrijyopfw, / predicate, and avv^
with.)
* This division is exhaustive ; every word must fall into
one or other of these two classes ; and no word, used in the
same sense, can fall into both. It is the only division of
words, as words, recognized by Logic, for that Science pays
no regard to the grammatical division into parts of speech
(."jce § 15). All words which can form a Terra belong to one
logical class though they may be distinguished by grammar
as Substantives, Pronouns, Adjectives or Participles ; all
those which cannot form Terms belong to the other class
though Grammar calls them Adveibs, Prepositions, Con-
GENERAL REMARKS ON TERMS. 43
junctions and Interjections. This division apparently con- Book I.
tains no place for verbs ; the reason of this is that formal ^'^' ^'
Logic recognises only the verb is (or are) which forms the
Copula of all propositions expressed in true logical form [see
§§23 (c) and 68] ; all other verbs are, therefore, represented
in formal logic by is or are and a participle. It is plain
that the Nominative and Possessive Cases of nouns and pro-
nouns are Categorematic, and the Objective Case is Syncate-
gorematic ; thus the logical division of words cuts across
the grammatical. Adjectives and Participles, like the pos-
sessive cases of substantives and pronouns, can always be
used as predicates, but not as Subjects except by an ellipsis,
as when we say ' The virtuous are happy ' where the full
subject is ' Yirtuous People.' Adjectives and their equiva-
lents form true terms (predicates) and are, therefore, Cate«
gorematic words. From the very nature of the case,
Adverbs, Prepositions, Conjunctions and Interjections, as
such, cannot be used as Terms.
Some logicians have called many-worded names Mixed Terms,
because they contain syncategorematic as well as categorematic
words ; but no object is gained by this, for Logic regards a many-
worded name as a whole. Symbolically it is expressed by a single
letter, as S or P, exactly as a single-worded term would be.
Of course, as the division of words into categorematic and
syncategorematic is really into terms and non-terms, it is as absurd
to speak of syncategorematic terms as it is tautologous to speak of
categorematic terms ; for, by definition, every term must be cate-
gorematic. Yet, at least one writer on Logic has been found who
divides Terms into Categorematic and Syncategorematic (see Jevons,
Ekm. Less, in Log., p. 26; Studies in Deductive Logic, p. 9),
forgetting that to speak of a ' Syncategorematic Terra ' is to
violate, in language, the Law of Contradiction (see § 18).
CHAPTER IL
DIVISIONS OP TERMS.
Book I.
ch. II. 26. Table of Divisions of Terms.
Terms may-
be divided
on five
bases.
Table of
divisions of
Terms.
Terms may be divided in various ways according to the
point of view from which we regard them. The following
Table sets forth these different divisions, and the principle
upon which each is founded. Of course, each group is
exhaustive and independent ; every Term must fall under
one or other of the members of each. It may be remarked
that (i.), (ii.) and (iii.) are the only divisions which are logic-
ally important, for they alone are founded on logical con-
siderations.
(i.) Individual and General — as names of individuals or
of members of classes.
(ii.) Connolative and Non-coiuiotatii:e — as names capable
or incapable of definition.
(iii.) Positive and Negative — as names implying the pre-
sence or the absence of some quality.
(iv.) Concrete and Abstract — as names of objects or of
attributes and relations.
(v.) Absolute and Relative — as names implying or not
implying a mutual determination of meaning.
A further division into Univocal and Equivocal terms is
sometimes made. But this is entirely a matter of language.
Whenever the same word serves as name for two or more
distinct classes of things — as, e.g., sleeper, which may mean
either an individual asleep or the support of rails on a rail-
road— we have logically a plurality of terms, for the word in
DIVISIONS OF TERMS.
45
each of its meanings is a distinct and separate term, and
represents a distinct and separate concept.
Book I.
Ch. II
27. Individual and General Terms.
(i.) An Individual Term is one which can be affirmed
in the same sense of only one single thing. Thus,
' London ' cjn be used in the same sense of only one place,
though more than cne place may have this same name ;
' honesty ' denotes only one quality, though it may be pos-
sessed by many individuals ; ' this book ' is limited to one
single volume, and can only be understood by a person who
knows what particular book the speaker is indicating. So,
' The present Queen of England,' ' The richest man in the
world,' 'The longest river in Europe,' are all Singular or
Individual names. But an examination of these and similar
examples will show that, though they are all names of
individuals, yet they differ from each other in that, whilst
some of them tell us of some quality possessed by the thing
they denote, others do not. Of the latter kind are * Honesty '
and 'London.' Such terms as honesty will be discussed in
§ 30, under the head of Abstract Terms.
(a) 'London' belongs to that subdivision of Singular
Terms called Proper Names, which may be thus defined : —
A Proper Name is an arbitrary verbal sign whose sole
province is to indicate an individual object.
It may be thought that such names tell us a great deal
about individuals ; that ' London,' for instance, tells us that
the object spoken of is a large city, situated on the Thames,
the Capital of the British Empire, and many other particulars
with which we may happen to be acquainted about it ; but
this is to confuse our knowledge of the thing obtained from
all kinds of sources with the meaning implied by the
name. The word 'London' informs us of none of these
things ; it may siiggest them by the law of Association of
Ideas, in the same way as hearing a song which we have
An Indi'
vidual or
Singular
Term can be
applied in
the Slime
sense to only
one object.
A Proper
Name is an
arbitrary
verbal sign
which
merely indi-
cates the
object of
which it ia
the name.
46
TERMS.
Book I.
Ch. II.
More than
one object
may bear
the same
Proper
Name, but
not in the
same sense.
A S>gnlfica7ii
Individual
Name is a
General
name limit-
ed, by some
■word, in its
application
to one indi-
vidual.
heard before may suggest the room in which we first heard
it or the person who then saug it ; but ' London ' no more
means these suggested particulars than the melody of the
song means a place or person. Care must then be taken in
considering Proper Names to distinguish between implication
and suggestion. Suggestion is purely a psychological fact.
Logically, the point is that a proper name is not given on
account of a certain meaning, i.e., on account of the posses-
sion of certain attributes, but as a mark of recognition. No
doubt, Proper Names originally had implication, and are con-
tinually tending to assume such meaning : but to the extent
to which they succeed they cease to be purely Proper.
The fact that more than one object may receive the same
Proper Name is no objection to the a.?sertiou that all such
names are Singular or Individual. Thousands of men may
be named Brown, and the same name may be borne by many
dogs, horses, and other things ; for instance, a town could
be named Brown as appropriately as Washington, Gladstone,
or Peel, all of which names have been thus emplojed. But
the name is given to no two of these objects in the same
sense. As it is simply a mark of identification, it does
not matter logically to how many people or things it is
applied.
(6) Significant Individual Terms.
Proper Names are the simplest kind of Singular or Indi-
vidual Terms, But the individual things we may wish to
refer to are too numerous for us to give each of them a
Proper Name of its own, and, sometimes, when a Proper
Name has been given it is unknown to us. We are, there-
fore, often driven to use a General Name [see § 27 (ii.)] with
a limiting word to make definite its applicability to only one
object. The simplest means of doing this is to use a demon-
strative word — as This pen is bad ; Let us go for a walk by
the river. Here we are, in both cases, referring, in a per-
fectly determinate sense, to only one object, and the name
is, therefore, Singular. No doubt, in the latter case, the
river has a Proper Name of its own ; but, in speaking of
DIVISIONS OF TERMS.
47
very familiar objects, we often use such a limited General Book I,
Name in preference to the Proper Name. Again, we may Gli^l.
use a manj'- worded name because it is our only means of
definitely indicating the object we wish to refer to, as its
Proper Name may be unknown to us ; thus, if we speak of
' The inventor of the Mariner's Compass,' ' The Avriter of the
Letters of Junius,' or ' The man in the Iron Mask,' we are
using the only means in our power of exactly designating the
person we mean ; for, in none of these cases, is the Proper
Name of the individual referred to known. In other cases,
such a many- worded name may be used because there is no
Proper Name applicable ; as when we say ' The leader of the
House of Commons,' or ' The present leader of the House
of Commons.* Regarded from a point of view limited in
respect of time this name can only refer to one deGuite
person, and is, therefore. Individual. Had we said simply
'Leader of the House of Commons' the name would not
have been Singular, but General ; for it could, then, be
applied in the same sense to many individuals ; the pi-efix-
ing ' The ' or ' The present ' limits its application so long as
we restrict ourselves to one point of time. In other cases
this limitation in time is unnecessary, as when we speak of
' The first King of Prussia ' or ' The highest mountain in
Asia,' where the reference is plainly to one individual object
of such a kind that no lapse of time can make it applicable
to any other. But were we to say ' The highest known
mountain in Asia ' we should again bring in the limitation
of time to the present ; as it may well be that the highest
known mountain is not really the highest in the continent ;
but it is to this last alone that the former name is cor-
rectly aj)plicable.
It is evident that Singular Names of this second kind
have meaning ; they are Significant — for they not only point
out one member of a class, but, at the same time, inform us ^^^^^ ia
that it does belong to that class, and has at least one attribute "^^'
which marks it out from every other member of the class.
In fact, these names are richer in implication than any other
class of names [see § 28 (i.)].
Such names
are the rich-
est of all
48
TERMS.
Book I.
Ch. II.
A General,
Common, or
Class Term
can be ap-
plied in the
same sense
to an indefi-
nite number
of things.
Such a name
denotes
things
indirectly
through
their posses-
sion of
certain
attributes.
The applica-
tion of a
General
Name may
be merely
potential.
Every Term
must be In-
dividual or
General.
(ii.) A General, Common, or Class Term is one which
can be applied in the same sense to each of an indefinite
number of things ; as book, man, dog. Subjectively con-
sidered, a General Term is the name of a General Notion or
Concept. Whilst a Proper Name indicates an individual
directly, a General Name does so indirectly, for such a name
is given because the individuals to which it is applied, and
from an examination of which the concept is formed, possess
some attribute or attributes in common [see § 2 (ii.)]. The
name, then, implies the possession of certain common
qualities by every individual object which bears it, and,
thus, has a meaning in itself ; it not only indicates certain
objects but it informs us that those objects possess certain
qualities. This likeness constitutes the similar objects a
class, and, hence, a General Term is often called a Class
Term.
It is not necessary for a true General Term that it should
be really applicable to a plurality of objects, or indeed to
any real physical object at all ; it is sufficient for it to be
potentially thus appUcable ; that is, for it to represent a
possibly real, or even an absolutely imaginary, class of things,
because of their possession of some common quality or
qualities. For instance, ' Conqueror of England,' ' Emperor
of Switzerland,' and ' Centaur ' are true General Terms ;
though the first is really applied to only one historical
individual — William I., and the second is not applicable to
any individual at all in the present or in the past, though
both may, conceivably, have an actual application in the
future ; whilst the third is the name of a purely imaginary
being. It is this potentiality of application to a class which
distinguishes General Terms from the second class of
Singular Terms ; for the latter, though they are significant—
that is, have implication- are not applicable, even potentially,
to more than one individual. There is thus an antithesis
between Individual and General Terms, and every Term
must be one or the other.
In Class Terms the unity which in Individual Names is
one of application is restricted to content or meaning ; in
DIVISIONS OF TERM3.
49
applicatioa it is overshadowed by the idea of plurality. la
Collective Terms both these ideas are equally prominent.
These Collective Names are sometimes treated as a separate
division of Terms, co-ordinate with Singular and General,
but this is not desirable ; for, as has been said, every Term
must, of necessit}', be either Singular or General. Collective
Terms are found in each class, and there is, therefore, no
opposition between them on the one hand and either In-
dividual or General Terms on the other. A short examina-
tion of such terms will make this clear,
A Collective Term is one given to a group of similar
units. It thus implies a plurality in unity ; as an army, a
flock, a library. It is not every group of individual objects
which can receive a Collective Name ; the constituents of the
group must bear a general resemblance to each other ; thus,
an alphabet is composed of letters, a navy of ships, a library
of books, a museum of objects of interest. We could find no
use for a name denoting a group composed partly of ships,
partly of books and partly of men, or any other fortuitous
concourse of heterogeneous objects ; and, though we could,
no doubt, manufacture such a name, yet it would not be a
true Collective Term ; for it would not imply that the con-
stituents of the group were all of the same kind.
As a rule, Collective Terms are not Proper Names, but a
few instances may be found, chiefly amongst geographical
names, in which they are. Thus we speak of the Alps, the
Pyrenees, the Himalayas, the Hebrides, the Marquesas, the
Antilles, the Orkneys, all of which are true Proper Names,
for they give us no information whatever about the groups
of natural objects to which they are applied, and are yet
Collective, for they denote a group of similar units.
When the application of an ordinary Collective Term is
limited — in the way illustrated in (i.) {b) of this Section — to
one particular instance of the groups it denotes, it becomes
a Significant Individual Terra. Thus we can speak of ' The
German Navy,' 'The Greek Alphabet,' 'The Bodleian Lib-
rary,' ' The British Museum,' ' The French army which
fought at Waterloo.'
Loa. L ^
Booif I.
Ch. 11.
CollectiTe
Ttrms do
not form a
clussoo-ordl-
nate with
Singular and
General.
A Collecliie
Term is ona
given to i
group of
similar
units.
But ftw
Collective
Terms are
Proper
Names, and
those are
chiefly geo-
graphical
names.
A Collectiva
Term may
bo a Siiruifi-
cant indi-
vidual
Term.
50
TEBMS.
Book I.
Ch. II.
Collective
Terms are
usually
General.
The group
denoted by
a Collective
Term may
be a unit in
a group
whose name
is also a
Collective
Term.
A General
Term, not
by itself Col-
lective, may
be used in a
collective
sense.
There is an
antithesis
between the
collective
and the dis-
tributive
use of
Terms.
In the Col-
lective use
the asser-
tion applies
to the group
as a whole ;
In the Dis-
tributive
use to the
individuals
which com-
pose the
group.
Without such limiting words a Collective Term is General
with regard to the class of which it denotes a member, as well
as Collective in respect of the units of which the group is
composed. Thus ' navy ' is Collective as regards the ships
which form it, but General, as denoting a member of the class
'navies'; 'alphabet' Collective as indicating a group of
letters, General as the name of a member of the class
' alphabets.' We have as true concepts, in fact, of navy and
alphabet as we have of ship and letter, and the former terms
imply attributes equally with the latter. The group denoted
by a Collective Term may even itself be a unit in a larger
group which bears a collective name of wider generality ; so
we may have a series of terms, each, except the first,
collective as regards the preceding one, and each, except the
last in generality, forming a constituent of the group denoted
by the following one : e.g. soldier, company, regiment,
brigade, army. Thus the term Collective is relative in its
meaning. At the same time, a General Term, which taken
by itself is not Collective, may, if in the plural number, be
used in a Collective sense by the prefixing of such a word
as ' All ' in the sense of ' All together,' us ' All these books
weigh several tons.' The true antithesis is, therefore, not
between Collective and General Terms, but between the
Collective and Distributive Use of Terms. When we
use a term Collectively our assertion will only apply to the
group as a whole ; when we use it distributively we assert
something about each member of the group individually.
Thus, if we say ' Half the fleet was lost in a storm,' ' The
regiment was decimated by fever,' ' All the novels of
Thackeray would fill a small bookcase,' ' The books filled six
large boxes,' we are evidently using the terms which form
the subjects of our propositions, whether they are ' Col-
lective' or 'General' in a collective sense; and, equally
clearly, if we say 'The fleet separated,' 'The army was
scattered,' ' All the men were fatigued,' ' All the novels of
Thackeray can be read in a day,' we are using the terms
distributively. The full sense of the separate words is again
seen to depend on the context (c/. § 3).
DIVISIONS OP TERMS.
51
* Substantial Terms. The question has been raised as to
whether names of substances, or ' Substantial Terms' as they
are sometimes called, such as gold, oil, water, are Singular or
General. On the one hand, it is urged that such words denote
the entire collection of one species of material ; on the other,
that when we use them, we do not refer to the whole but to
some definite, or indefinite, portion of the whole. But the
question is not very pertinent, as in such terras the element
of content or meaning is predominant, whilst that of appli-
cation to this or that object is subordinate. In so far as we
do consider this latter aspect, we may say, with Dr. Venn
{Empirical Logic, pp. 170-1), that such terms are a peculiar
kind of Collective Terms, with the special characteristics of
theoretically infinite divisibility, and, at the same time,
perfect homogeneity. It is this which makes them different
from ordinary Collective Terms. We can divide and sub-
divide a number of pieces of gold into any number of parta,
and again reunite them ; and any one part is a fair specimen
of the others. In this they differ as much from ordinary
General Terms as they do from true Singular Terms ; we
can divide an animal, but we cannot reunite it.
28. Connotative and Non-connotative Terms.
(i.) What names are Connotative. In the last section
Terms were considered according to their applicability to one
or more objects, but that division could not be intelligibly
discussed without reference to a fundamental distinction
which is, to some extent, bound up with it. It was shown
that, whilst an Individual Name may be a mere indicative
sign, implying no attribute, all names which are applicable to
a plurality of objects are essentially significant, and imply
some atti-ibute or attributes possessed in common by those
objects ; for, only on this principle, could the same word be,
in any intelligible sense, the name of each member of a class
of things. This distinction between significant and merely
indicative names is expressed by the terms Connotative and
Non-connotative, which we may, therefore, thus define,
Book I.
Ch. II.
Sulstantial
Terms — or
names of
substances
— areapecu
liar kind of
Collective
Terms.
Some Singu-
lar Terms
imply
attributes,
and all
General
Names do sc
nearly in the words of Mill {Logic, Bk. I., ch.
u.
§5):-
4— a
52
TERMS.
Book L
Ch. II.
A Connota-
live Term de-
notes a sub-
ject and im-
plies an at-
tribute or
attributes.
A Xon-Con-
noiative
Term merely
denotes a
subject.
All General
Names are
counotative.
All Adjec-
tives are
General
and, there-
fore, Conno-
ta tive
Teims.
Collective
Terms are
connota tive,
except they
are Proper
Names.
Significant
Singular
Terms are
counotative.
A Connotative Term is one which denotes a subject
and implies an attribute or attributes.
A Non-connotative Term is one which merely denotes
a subject.
When we speak of a subject in this connexion we mean any-
thing which can possess an attribute ; whilst under attribute
we include all that belongs to the subject, not only the out-
ward marks by which it is known — as its shape, size, colour,
weight, etc. — but all its properties and relations whatsoever.
From what has been already said [§ 27 (ii.)], it is evident
that all General Terms are connotative, for they all denote —
or are applicable to — certain objects, and imply that those
objects agree in possessing some attribute or attributes in
common ; in fact, it is the possession of these attributes
which entitles any particular object to bear the name. Thus,
if we use the name ' horse,' we not only refer to an indefinite
number of animals which are so styled ; but we imply that
they all agree in possessing certain well defined charac-
teristics ; any new animal brought under our notice which
possessed those attributes we should, without hesitation, call
a horse. Under the head of General — and, therefore, Con-
notative— Terms must be included all Adjectives, for they
express qualities regarded solely as exhibited by things, and,
if we wish to use them as subjects of propositions we must
name the things they qualify (of. § 25). So with all those
Collective Names which are not Proper Names ; they are all
Connotative, for they are General when viewed as members
of a class [see § 27 (ii.)] ; for instance, ' army ' implies the
aitiibutes of being composed of soldiers, armed, trained, and
maintained for warlike purposes, as well as denotes each col-
lection of men which possesses these attributes. When any
General Name, whether Collective or not, is restricted in its
application by some limiting word or phrase, of course its
implication is not lost. Indeed, that implication is increased,
and thus we have the class of Significant Individual
Names [see § 27 (i.) (6)], which, though they denote only
one object, yet imply the possession of many attributes
DIVISIONS OP TERMS.
53
by that one object. Thus, if we speak of 'a mountain ' we
imply the attributes * height ' and * composition of rock ' ; if
we add 'in Asia ' we increase the number of characteristics,
though we limit the number of things to which the name
applies ; by adding ' high ' we carry both these processes a
step further; and if, finally, we make the term Singular
and speak of ' The highest mountain in Asia ' we, manifestly,
retain all the attributes previously implied, and add to them
uniqueness. All these attributes are implied by the name,
and anybody using the name must be supposed to intend to
convey them to his hearers.
But, were we to use, instead of this significant name, the
Proper Name 'Everest/ which, in our present state of know-
ledge, we believe to be the name of the same object, no such
information would be given. To anybody who knew the
geographical fact that Everest is the highest mountain in
Asia, the name 'Everest' would, doubtless, suggest all that
the words ' The highest mountain in Asia ' imply. But a
word is not Connotative because it may suggest facts or attributes
otherwise known, but because it implies them, so that the name
by itself is, when understood, sufficient to impart the know-
ledge that they are possessed by every object it denotes. This
distinction between suggestion and implication is the dis-
tinction between connotative and non-connotative. No terms
are without some meaning to those who use them, but only in
connotative terras is the meaning a group of implied qu^alitioF!.
No doubt Proper Names were originally significant, and
implied attributes. Thus, Avon in old English meant water;
Jacob meant a supplanter ; Smith or Butcher, one who fol-
lowed a certain trade. But even as so given their main
function was distinction, and the name was retained even
though the attribute it at first implied was removed ; to
deduce connotation from this original descriptive character
is to confuse connotation with etymology. With surnames
there is a very strong suggestion, amounting almost to im-
plication, of family relationshii>. This was even stronger
with old Roman names. For instance in the name Caiua
Julius Caasar, whilst the praanomen, Caius, was non-signi-
BooK L
Ch. II.
Proper
Names .-vre
non-conno-
tative, for
they can
only giiggeft
not imply
attribut;~«.
54
TEKM3.
Book I.
cn. II.
Proper
Names used
typically
Viecome
General.
The Conno-
tation of a
name em-
braces those
attributes,
and those
only, on ac-
count of
which the
name is
given, and
wanting any
of which it
would be
denied.
ficant, the nomen, Julius, indicated the gens, and the cog-
nomen, CjBsar, the family in that gens, of which the individual
was a member. But as a surname can be changed at will, it
seems clear that now, at any rate, its true function is merely
to distinguish the individual, and that it has no necessary
implication of meaning.
* The absence of real implication in Proper Names,
especially to denote pt;rsons, is, probably, to bo partially
explained by the fact that an individual possesses such an
innumerable number of different attributes that no one (or
more) is specially identified with him ; directly some attri-
bute does show itself as a more marked characteristic, the
use of a descriptive ' nick-name ' is likely to become common,
as every schoolboy can testify. In a similar way we may
account for the use of the names of some prominent his-
torical personages to imply the possession of the quality
which the type possessed in an exceptional degree ; thus we
speak of ' a Cicero,' ' a Napoleon,' * a Caligula,' etc., but the
names have then ceased to be true Proper Names and have
become General, and applicable to all objects showing the
indicated qualities in a marked degree.
If, then, we distinguish between implication and sugges-
tion, we must come to the conclusion that the definition given
in the last section of a Proper Name as " an arbitrary verbal
sign " is strictly accurate, or, in other words, that Proper
Names are non-connotative.
The only class of names which remain to be examined in
this connexion are Abstract Names, and it will be more con-
venient to postpone our consideration of this point till we
have discussed the nature of those terms [see § 30 (iii.)].
(ii.) Limits of Connotation. All the attributes directly
implied by a name form its Connotation, and it is clear from
what has been said above that this does not include all those
which are common to all the members of a class denoted
by a General Name, nor, consequently, all those which are
possessed by the individual object to which a Significant
Singular Term is applied, but onli/ those on account of the pos-
DIVISIONS OF TERMS.
55
session of which the name is given, and wanting any of which it
would he denied. The disputes which have arisen about the con-
notation of terms generally, and especially of Proper Names,
have owed their origin to an ambiguous use of this word.
Some writers hold that the Connotation of a name includes
all the attributes common to the members of the Class of
which it is the name. Thus Mr. E. C. Benecke says : "Just
"as the word 'man' denotes every creature, or class of
" creatures, having the attributes of humanity, whether we
" know him or not, so does the word properly connote the
" whole of the properties common to the class, whether we
"know them or not" (il/mrf, vol. vi., p. 532). But this
usage would have many logical inconveniences ; it would
divorce connotation and definition, and make connotation a
matter, not of knowledge, but entirely of objective existence,
and it is with such existence only as known that logic is
concerned. If, to avoid this objection, it is said that the
connotation should embrace all the known attributes common
to a class, then it must be pointed out that some of these
cannot be regarded as essential ; for instance, though every
kangaroo is an Australian animal, yet were such an animal
found elsewhere it would not be excluded from the class of
kangaroos. Similarly an animal which chewed the cud would
be regarded as ruminant even though it did not agree with
all known ruminants in possessing cloven feet. The name
cannot, therefore, be said to strictly imply the possession of
those attributes. Again, some attributes are derivative from
others. Thus, that an equilateral triangle is equiangular,
that a right-angled triangle is inscribable in a semi-circle, etc.,
are attributes derivable from those primary ones which the
name directly implies (c/. § 37). Such attributes are, then,
indirectly implied, and are a necessary consequence of those
directly implied. But it is most convenient not to regard
them as forming part of the Connotation ; as that is to
confuse primary with secondary implication. It would be
convenient to use the term Content to express all the attributes
which are either directly or indirectly implied by a name.
It must be granted that the limits of connotation are con-
BOOK I.
Ch. II,
If Connota-
tion em-
braced all
common at-
tributes it
would
become
entirely a
matter of
objective
existence,
not of know.
ledga
56
TERMS.
Book I.
Ch. IL
Connotation
Is a matter
of know-
ledge.
ventional ; but this 5s made necessary by the imperfection of
knowledge. It is essential that our terms should have a
fixed and definite value at any given time, and that this value
should express the kuowledge which has been attained.
It is thus Been that the question of connotation is, in
essence, a question of knowledge. It is neither entirely
objective, nor entirely subjective, but has both an objective
and a subjective reference. To make it purely subjective
would be to say that every individual should consider the
connotation of a name to consist of these qualities which he
himself may know to be common to the class. But, as
Dr. Bosanquet well remarks, " Surely the question for logic
" is never what a name means for you or me, but always,
" what it ought to mean " {Knowledge and Reality, p. 60).
And what it ought to mean is determined by the fullest
knowledge attained, and is expressed in definitions accepted
by all competent authorities. Without such common agree-
ment as to the correct implication of words, language would
soon cease to be available as a medium of the communication
of anything like exact thought.
Connotation
is conven-
tional but
practically
definite
enough in
most cases.
(iii.) Difficulties of assigning Connotation. When we
say the connotation embraces all those attributes, and those
only, which are directly implied by the name, and that this
is determined by the fullest knowledge attained, and is ex-
pressed in definitions accepted by competent authorities, we
undoubtedly show that connotation is not only conventional,
but may be in some cases, especially in comparatively new
branches of knowledge, somewhat vague. No doubt, in the
case of most terms, it is found sufiiciently definite ; it is, for
instance, clear that the connotation of the term ' square' is
that it is a plane, rectilineal, right-angled, quadrilateral
figure with equal Bides ; none of the other numerous
qualities of squares are included in the connotation ; we need
not think of them when we use the term, they do not form
part of our concept of a square. Still, in some cases, it is,
undoubtedly, far from easy to decide how much a particular
term does or does not connote. When this is the case with
words in common use it may lead to much confusion, but
DIVISIONS OP TERMS.
67
with many names it is an advantage for the connotation not
to be too rigidly limited, so that they may be applied to
newly-discovered objects which most closely resemble those
which already bear the name. And, at all times, the conno-
tation of terms must be subject to revision, should occasion
arise through the discovery of such objects.
In § 3 instances were given of the way in which connota-
tion becomes vague and difficult to assign through the trans-
ference of a name by analogy, metaphor, or partial resem-
blance, to things other than those to which it was originally
applied. Through these processes the same name may come
to denote objects entirely different, but must then be re-
garded as separate and distinct terms accidentally wi'itten and
spoken alike, e.g., ' post,' meaning a piece of wood inserted
in the ground, and * post ' meaning the conveyance of lettera.
This process may be represented symbolically thus: A class of
objects X possesses the common attributes ahc ; the class Y pos-
sesses the common attributes ade ; the class Z possesses d/g.
Now, the name of X is transferred to Y because of the common
attribute a, it is then passed on from Y to Z because of the com-
mon attribute d, and so the same word denotes X and Z which
have not a point in common. As their connotation is then entirely
different, they must be regarded as entirely different terms — the
name of different classes. But shall Y be regarded as forming part
of the class X or as part of Z P It could only do so by restricting
the connotation in the former case to a and in the latter case to d ;
it would, tVierefore, probably be regarded as constituting yet a third
distinct class, and its name would be a third and independent term
for all logical purposes— the history of its origin and development,
though interesting from the point of view of Philology, is of no
value from that of Logic. All cases of ambiguity in language are
instances of indeterminate connotation of names, and a vast number
of fallacious reasonings are due to this indefiniteness.
(iv.) Denotation of Terms. A comparison of the last
section with the present will show that significant names
may be viewed in two lights — their implied meaning or
connotation and their range of application to a number of
objects — this latter aspect is called their Denotation, which
may, therefore, be defined as the number of individual things
Book I
Ch. II.
Connotation
is made in-
definite by
transference
of the name
to objects
not at first
denoted by
it.
This process
can be repre-
sented sym-
bolically.
Denotation
of a Term —
the objects
to which it
i-s appbcable
iu the same
sense.
58
TERMS.
Book I.
Ch. II.
Logically,
Denotation
is fixed by
Connota-
tion, but
practically,
they deter-
mine each
other.
All terms
have Deno-
tation.
to which the term is applicable in the same sense. From what
Las been already said it is clear that the denotation is
logically fixed by the connotation ; objects receive a certain
name, and so form part of the denotation of that name,
because they agree in its connotation. Nevertheless, practi-
cally each helps to determine the other. The connotation
expresses the concept which is formed after an examination
of part, at least, of the denotation ; and, at all times, not only
is the connotation likely to be modified by an increase in the
denotation, but also conversely, making the connotation
more definite or more elastic may decrease or enlarge the
denotation [see (v.)]. In truth, neither is absolutely fixed,
though, for the purposes of Formal Logic, it is necessary to
regard the connotation as strictly invariable, at any rate
throughout the same argument, or the Law of Identity
would be violated (see § 17).
According to the definition of Denotation given above, it
follows that all terms have denotation whether they have
connotation or not, though in the case of Proper Names and
of some Abstract Names the denotation is reduced to the
least possible limit — the unit.
Dr. Venn
holds that
purely men-
tal notions
have no de-
not;itiou.
Dr. Venn, however, dissents from this view. He says : " The
" conception of Denotation becomes appropriate only when we are
" concerned with objects whose existence is limited in some material
" way" (Emp. Log.,]p. 178). If we speak of the denotation of a purely
mental concept, such as a perfect mathematical figure, e.g. a circle,
the only meaning we can give to the term is to say that it embraces
every circle which ever has been, or could be, conceived ; for every
one of them would possess the full connotation ; but none with
material existence, for no perfect circle has ever been drawn. Thus
the denotation of such a term would be absolutely infinite, entirely
notional, and, in large part, merely potential. ^ To include in the
denotation with these perfect imaginary circles, the actual ' circles '
traced on paper would be to include objects which only approximate
more or less roughly to the connotation. These latter, being material
objects, give, of course, a denotation to the word ' circle ' when it
is understood to refer to them, and not to be strictly limited to the
mathematical concept. This denotation embraces all such figures
which are now iu exibtence ; those which have been drawn formerly
DIVISIONS OF TERMS.
59
but have since been destroyed, or those which may be drawn in the
future, cannot be said to form part of the denotation of the word
now, though they did, or will do so, at a different point of time.
Thus the denotation of physically real things is limited in time, and
Bo varies with time ; for instance, the denotation of ' man ' includes
all human beings now living, and none else. If we speak of an
£xtinct animal " like the Dodo or Moa, then I do not think we can
"avoid a leference to the element of time, and must say that it has
" now no denotation " (Venn, Empirical Lorj'ic, p. 179). Though
there is force in this, it seems better, on the whole, to ho!d to the
ordinary view — that is, to make 'Denotation' wide enough to cover
all things to which the name can be correctly applied. If we do not,
it becomes necessary to employ some other word — such as Applica-
tion or Denomination — in this wider sense.
But we may speak of creatures purely fabulous, as dryads,
centaurs, or griffins ; can the terms we then use be said to
have denotation ? We agree with Dr. Venn (o^. clt.^ p. 180)
that they can, and that their denotation must be sought in
the appropriate sphere of existence — that of mythology,
fable, or heraldry, as the case may be. This is, of course,
using the word 'existence' in a somewhat wider sense than
is common in ordinary speech, but it does not seem to do
violence to it, and the extension is necessary to enable it to
include entities having an existence only in thought or
fancy, such as the characters of romance.
It may be said, then, that the Denotation of a Term is the
aggregate of all which, when presented to us, we should
mark by the name ; and that this aggregate must be sought
in the appropriate realm, whether of fact or of fiction (c/.
Venn, Emih Log.^ p. 176).
* In connexion with the subject of Denotation it will be
well to mention what is called the Universe of Discourse, ihsit
is " not the whole range of objects to which a general term
"can be correctly applied — which is the denotation — but
" merely the restricted range to which the speaker at the
"time being intends his remarks to apply" (Venn, Ein]}.
Log.y p. 180). Of course, were terms always precisely used,
their denotation would coincide with this universe, but in
speech they are always modified and limited by the context
Book I.
Ch. II.
The Denota-
tion of fabu-
lous objects
is to be
sought in
mythology,
etc.
Universe of
Discourse —
the limited
sphere witb
in which a
term is
intended to
apply at any
particular
time.
60
TERMS.
Book I.
Ch. II.
Ab a geneml
rule an In-
crease in
Connotation
reduces tho
number of
subclasses
of tbiiit^a de-
noted, aud
vice vertid ;
an increase
in iiuriiherof
sul'-clHSses
deiio yd de-
cre;ises tlio
Conuota-
tiuu, and
vict Vfrs'U
expressed or understood (c/. § 3), and thus, both speaker
and hearer constantly restrict the application of a term to
some portion only of its denotation. If, for instance, we say
'Everybody says so' we certainly do not intend the term
'everybody' to be taken in its full ezten\; '^o as to embrace
all the inhabitants of the world ; wo probably refer, aud are
understood to refer, to a very few persons. Similarly the
term ' Europeans ' is restricted, in most cases of its use, to
human beings. The limits of this universe, which are purely
arbitrary, are left to be tacitly understood, and cannot, of
course, be expressed symbolically ; the restriction is material,
not formal. Nevertheless, it is important to boar in mind
that terms are continually joined together into propositions
in a sense narrower than the words themselves warrant, aud
that such propositions are only intended to apply witbin this
limited sphere.
The phrase is not, however, always used in a limited
sense. It always denotes the whole idea under considera-
tion, and this may coincide with the denotation of the term.
Thus Boole says : " The universe of discourse is sometimes
"limited to a small portion of the actual universe of things,
"and is sometimes co-extensive with that \ini\erse" (Laws
of Thought, p. 166). And Mr. Keynes adds: "It must be
" clearly understood that the universe of discourse is by uc
" means necessarily identical with the region of what we
" ordinarily call 'fact' ; it may be the universe of dreams,
" or of imagination, or of some particular realm of imagiua-
" tiou, e.g. modern fiction, or fairy-land, or the world of the
"Homeric poems" (Formal Logic, 3rd Edition, p. 183).
(v.) Relation between Connotation and Denotation.
As Conuotation implies attributes, and Denotation refers to
the individual objects which possess those attributes, and
which usually form various sub-classes, it is evident that,
as a general rule, an increase in either one will cause a
decrease in the other. As we augment the number of
attributes implied by a name we diminish the number of
things to which that name is applicable, for we exclude
some of the sub-classes ; there are, for instance, fewer white
DIVISIONS OF TERMS.
61
horses than horses. Conversely, if we wish to include under
a name a group of things not before included under it, and
so to enlarge the borders of the class which the term denotes,
we can, usually, only do so by removing from the implication
of the name those attributes which before marked the
difference between the two classes, or, in other words, by
decreasing its connotation. For instance, if we unite the
classes, white men ' and ' not-white men ' we must omit from
the connotation of the common term all specification of
colour ; similarly, if we wish to include both sailing ships
and steam ships under one common name, we must omit the
points of difference, 'sailing' and 'steam,' and retain only
the term ' ship,' Avhich will be applicable to all the members
of both classes but which implies less than the separate
name of either. In short, generally speaking, the less
a name implies, the more groups of things it is applicable
to, and the more it implies the narrower is its range of
application. If, as a very simple instance, a word is taken
which connotes only one attribute, such as ' white,' it is at
once evident that no increase of meaning is possible which
will not decrease its denotation, for now it embraces all
white things whatever, but the addition of any attribute
must limit it to some only of those objects, as, for instance,
if we speak of white animals, white cloth, white paper, etc.
It was shown in contrasting Significant Individual Names
with Proper Names how the continued addition of attributes
increases the connotation and decreases the denotation of
a term, till at length the latter is reduced to unity, and the
former has become the fullest which that term is capable of
bearing, so that connotative singular terms are the most
Eignificant of all names [see § 27 (i.) (&)].
Book I.
Ch. II.
This general relation between the connotation and denotation of
terms can be represented symbolically. If the connotation of the
term h be wxyz, of the term 5 be pwxij, of the term C hepwxz, and of
the term D be ptoyz, then the union of any two of the classes denoted
by those terms decreases the connotation by one element, the addi-
tion of a third class reduces it by one more, whilst the union of all
four classes into one causes the connotation to become w alone, for
This can ba
represyjQted
symholio.
aUy.
62
TERMa
Book I.
Ch. II.
that is the orly element common to the connotation of all. Thus the
connotation of the class A + B is wxy, oi B + D is pu-i/, ot A+B + C
is wx, and B + C + D is jjw, and so on wilh each different combinaticn
of the classes denoted by /, B, C, D. Conversely, if we have the
connotation w it will embrace all the groups of things denoted by
A + B + C + D, if we increase this conmtation by adding x to it,
the denotation is reduced to A + B + C, a further enlargement of
the connotation to ua-y again diminishes the denotation to A + B,
and the final add! lion of a to this connotation makes the name
applicable to the class A alone. But we do not know how many
individuals are included under A, B, C, or D, nor can we tell how
many attributes may be included under the symbols p, w, x, y, z,
each of which may represent a whole group.
Connolation
and Denota-
tion do not
vary in in-
verse ratio.
Every acidt-
tion to the
Connotation
of a Term
does not de-
crease the
Denotation.
It cannot, therefore, be said that Connotation and Deno-
tation vary ia inverse ratio to each other ; such a mathe-
matical conception is quite inappropriate. "We can speak
intelligibly of halving or of doubling the denotation of a
term, but it is meaningless to talk about doubling or halving
its connotation ; and even could we do so there would be
no ratio maintained in the variation of the two aspects of
the term. The application of a term is limited by the
addition of some attributes much more than by that of
others; thus, to add 'white' to man would not limit the
denotation nearly so much as to add ' red-haired ' for there
are many more white men than there are red-haired men.
Similarly, in the example we before considered [§ 27 (i.) (&)]
— mountain — mountain in Asia — high mountain in Asia —
the highest mountain in As-ia — it is evident that some of
the additions to the connotation of ' mountain ' decreased
its denotation much more than others. Moreover, it is not
true that an addition to the connotation of a term will
always cause a decrease in its denotation ; for as a name
does not usually connote every attribute common to a class,
the addition to the connotation of any number of these
common attributes not included in it will not affect the
denotation ; there are, for instance, as many mortal men aa
there are men ; so, though ' mortal ' is not part of the con-
notation of man, yet to speak of ' mortal men ' does not
DIVISIONS OP TERMS
63
Attributes of things
are in Nature very often found in groups, so that where one
is found others are found too ; and it is evident that, when
this is the case, the addition of any of these attributes to
the connotation of a term will not limit its denotation so
long as the one member of the group with which they are
all connected already forms part of that connotation. Thus,
when a triangle is equilateral it is also equiangular, and to
speak of equiangular equilateral triangles does not, therefore,
limit the denotation given by equilateral triangle. So tc
add to ' right-angled triangle ' the attribute ' having the
square on the hypothenuse equal to the sum of the squares
on the sides ' brings in no fresh limitation, for that attribute
is one of a group necessarily found wherever the property
' right-angled ' is joined to triangle. There may, thus, be
many additions to the connotation of a word which will
have no effect on its denotation.
It is, perhaps, scarcely necessary to point out that the
idea of an opposite variation of Connotation and Denotation
is only applicable to classes which can be arranged in a
series of varying generality, so that each smaller class forms
a part of the next larger ; such as, figure, plane-figure, plane
rectilineal figui'e, plane triangle, plane isosceles triangle,
plane right-angled isosceles triangle ; vehicle, carriage,
railway carriage, saloon railway carriage, first class saloon
railway carriage, first class dining saloon railway carriage.
It would be absurd to say that an increase or decrease in
the number of members of any one class affects the connota-
tion of the class name ; that, for instance, the birth of every
baby must decrease the number of attributes implied by
the term 'human being,' and that the death of each man,
woman, and child, must increase that number. It is only
when we add an attribute not common to the whole class
that we exclude some members of the class from participa-
tion in the class name and so decrease the denotation ; or
when we introduce into a class some things not possessing
all the attributes connoted by the class name, that we have
to omit part of its meaning, that it may cover the whole of
Book I.
Ch. II.
The oppo-
.-ite varia-
tion of Con-
notation
and Deno-
tation only
exists in a
series of
classes.
64
TERMS.
Book I.
Oh. II.
this more extended class ; and thus we decrease the connota-
tion. The increasing the connotation and thereby limiting
the application of the term is a process of Specialisation, the
opposite process of decreasing the connotation so as to
embrace a larger number of objects is one of Generalization
(c/. §3).
Of the Syno-
nyms of
Connotation
and Denota-
tion, Inten-
sion and Ex-
tension are
the only two
■which have
been gener-
ally used.
Connotation
and Denota-
tion are the
mostexpres-
* (vi-) Synonyms of Connotation and Denotation.
Both the Connotation and the Denotation of Terms have
been spoken of in Logic under a great number of names.
Thus, instead of Connotation, we find the terms Intension,
Intent, Comprehension, Depth, Implication, and Force ;
whilst the Denotation has been correspondingly styled
Extension, Extent, Sphere, Breadth, Application and Scope.
None of them have come into general use except Intension
and Extension, "which are often used to express from the side
of the concept what Connotation and Denotation express
from the side of the term. Denotation (Lat. de, down ;
nofare, to mark) and Connotation (Lat. con, with ; notare, to
mark) have the advantage of expressing by their etymo-
logical meaning exactly what we want to express when we
use them ; we * mark down ' the objects which we name and
we 'mark with' them their attributes. Thus these terms
are most expressive and appropriate when we deal with the
forms of language in which thought is expressed. But it
would be undoubtedly convenient could such a term as
Content or Intension be used to express both the direct and
indirect implication of a term (c/. ii.).
Inenmpatible
Terms are
those which
Imply attri-
butes which
cannot co-
exist in the
Bame sub-
29. Positive and Negative Terms.'
The formal distinction of Terms into Positive and
Negative is a particular case of the IncompatibilitTj of Terms.
All Terms whatever which imply attributes which cannot
co-exist in the same subject are incompatible. This incom-
1 In the whole of this section the treatment follows, generally, that of the
same Bubject by Dr. Venn {Emp. Log., pp. 181-5).
DIVISIONS OF TERMS.
65
patibility may be expressed either by Contradictory, by
Contrary, or by Eepugnant Terms. The division into
Positive and Negative is the formally logical means of
marking the iirst of these ; that distinction will, however
be more clearly understood if all three kinds of in-
compatibility are considered.
Book I
Ch. II.
(i.) Contradiction— For two terms to be contradictories
it is necessary that they be mutually exclusive and at
the same time collectively exhaustive in denotation;
that is, they must be incapable of being predicated at the
same time about the same subject, and between them they
must embrace everything in the Universe of Discourse.
Now, it is evident that this contradiction may be marked in
two ways — knowledge of the matter may tell us that two
terms are contradictories ; or the very form in which the
terms are expressed may imply this ; the former is Material
Contradiction, the latter is Formal or Logical.
* (a) Material Contradiction. In some important instances
where two groups of things which fulfil both the conditions
of contradiction are equally important, and equally possess
a great richness of meaning, they have each a distinct name.
These names do not, in any way, imply the contradiction,
which can only be known by examination of the facts, for
the names are not constructed for the purpose of indicating
the contradictory relation which exists between them.
Each is connotative, that is, each is the name of a true class
of things which, like all other true classes, is marked by the
presence of attributes common to every member of it. To
understand the contradiction, therefore, we must know the
connotation of each term. It is not necessary that these
connotations should be entirely different from each other —
they may and generally do possess common elements —
but the attributes, or groups of attributes, which are not
common must be such that every individual in the universe
of discourse possesses one or the other, but no single
individual possesses both. Such instances are, of course,
rare, and nature is so diversified and so limitless that when
LOG. I. 5
Contradic-
tory Terms
are mutu-
ally exclu-
sive and col-
lective'y ex-
haustive in
denotation.
Material
Contradic-
tion can
only be
known by
examina-
tion of the
facts.
Material
Contradic-
tories, as a
rule, parti-
ally coincide
in connota-
tion.
66
TERMS.
Book I.
Ch. II.
They are
generally
limited in
Rjijilication
Each term
is pusitive,
i.«.,coiiiioies
the presence
of attri-
butes.
Words with
negative
prefixes are
not, in most
c:ises, true
coutradic-
tories of the
corre.spiiiid-
ing simple
terms.
they do occur they never embrace the whole of existence —
their application is always tacitly limited to the Universe of
Discourse which was described in the last section. This
Universe may be very wide, as when we speak of male
and female, but usuallj' it is somewhat narrow, and the more
it is limited the more numerous are the pairs of material
contradictories which can be found within it. Thus, when
we speak of British and Foreign we are using terms which
are contradictories in the realm of material things, but which
would be utterly inapplicable to abstract ideas — no meaning
can be attached to ' British Justice' or 'Foreign Honesty'
except when by the words ' Justice ' and ' Honesty ' we
mean, not the abstract quality which is the same everywhere
and always but, certain acts of Justice or of Honesty.
Hence, we see that the attribute 'material' is part of the
connotation of both British and Foreign ; as ' living organism '
is part of that of both male and female ; and this illustrates
what was said above, that there may be common elements in
the connotation of material contradictories. If we restrict
our universe still more we have the contradictory terms
British and Alien, which are applicable only to human
beings ; and a further limitation of the universe to English
human beings furnishes the pair Peer and Commoner. With
each limitation we see that the common part of the conno-
tations of the contradictory terms increases, as might be
anticipated from the general relation between denotation
and connotation discussed in the last section ; for it is the
denotation of the universe we are decreasing directly, and
only indirectly through that the denotation of the contra-
dictories. However, for Logic, such material contradictories
are all of the same class — they are all positive, for they all
have a connotation which implies the possession of certain
attributes; and, as has been said, they are comparatively few
in number.
As the need of many more contradictories than these
words su])ply was felt, common language began to form
them by the addition of a negative prefix or affix. We have
many instances in English in words beginning with the
DIVISIONS OF TERMS.
67
prefixes in- tm-, non-, mis-, etc., as insincere, unkind, nonsense,
misfortune, or ending in -less, as senseless.
But these words have, in most cases, ceased to be true
ccntradictories of the corresponding sim^ le terms, and have
only remained so when, in instances such as equal — unequal,
no intermediate idea is possible. When such an inter-
mediate idea can be formed the connotation of the negative
words has tended to become more remote from that of the
positive words and has itself taken on positive elements.
Thus, 'happy' and 'unhappy' are not contradictories,
because they leave an intermediate state of indifference
between them ; we are often neither happy nor unhappy,
for the latter word does not simply imply the absence of
happiness but, in addition, the presence of positive misery.
Common language, in fact, very seldom expresses sharp
distinctions, and the meaning of a term adopted into
common speech tends to approximate to a kind of average of
the things to which it is applicable. Thus, as 'unhappy'
would apply to anything from the simple negation of
positive happiness to the most intense misery, the word
gradually took into its connotation some elements of dis-
comfort, and now signifies a state intermediate between
indifference and deep misery. So with unkind, unholy,
senseless, misfortune — they are no longer the simple
negations of kind, holy, sensible, fortune ; for neither pair
between them exhausts the universe. The same remarks
are applicable to nearly all this class of words, which are
therefore, although incompatible, not contradictory terms.
It must also be noticed that negative prefixes and affixes
sometimes do not imply the negation of any attribute at all
— thus ' shameless ' is not the negation of ' shameful ' bat
almost a synonym with it, and ' invaluable ' means valuable
in the highest degree.
(6) Formal or Logical Contradiction, The necessity of
excluding any intermediate ground is the justification of
logical contradiction, which consists in prefixing not- or ncm-
to the term — thus not-happy simply excludes ' happy,' ' not-
white' shuts out white, *not-man' removes 'man' and
Book L
Ch. II.
Thtsir CODDO-
tation has
positive
elements.
Formal
logical cna-
tradiction
consists in
prefixing
not- or non-
to the Term
66
TERMS.
Book I.
Ch. II.
They are
generally
limited in
Rpplication
Each term
is positive,
i.e., connotes
the presence
of attri-
butes.
Words with
nCffative
prefixes are
not, in most
c:i8es, true
coutradic-
tories of the
correspond-
ing simple
terms.
they do occur they never embrace the whole of existence —
their application is always tacitly limited to the Universe of
Discourse which was described in the last section. This
Universe may be very wide, as when we speak of male
and female, but usually it is somewhat narrow, and the more
it is limited the more numerous are the pairs of material
contradictories which can be found within it. Thus, when
we speak of British and Foreign we are using terms which
are contradictories in the realm of material things, but which
would be utterly inapplicable to abstract ideas — no meaning
can be attached to ' British Justice' or 'Foreign Honesty'
except when by the words ' Justice ' and ' Honesty ' wa
mean, not the abstract quality which is the same everywhere
and always but, certain acts of Justice or of Honesty.
Hence, we see that the attribute 'material' is part of the
connotation of both British and Foreign ; as ' living organism'
is part of that of both male and female ; and this illustrates
what was said above, that there may be common elements in
the connotation of material contradictories. If we restrict
our universe still more we have the contradictory terms
British and Alien, which are applicable only to human
beings ; and a farther limitation of the universe to English
human beings furnishes the pair Peer and Commoner. With
each limitation we see that the common part of the conno-
tations of the contradictory terms increases, as might be
anticipated from the general relation between denotation
and connotation discussed in the last section ; for it is the
denotation of the universe we are decreasing directly, and
only indirectly through that the denotation of the contra-
dictories. However, for Logic, such material contradictories
are all of the same class — they are all positive, for they all
have a connotation which implies the possession of certain
attributes ; and, as has been said, they are comparatively few
in number.
As the need of many more contradictories than these
words su])ply was felt, common language began to form
them by the addition of a negative prefix or affix. We have
many instances in English in words beginning with the
DIVISIONS OP TERMS.
67
prefixes in- nn-, non-, mis-, etc., as insincere, untind, nonsense,
misfortune, or ending in -less, as senseless.
But these words have, in most cases, ceased to be true
contradictories of the corresponding sim; le terms, and have
only remained so when, in instances such as equal — unequal,
no intermediate idea is possible. When such an inter-
mediate idea can be formed the connotation of the negative
words has tended to become more remote from that of the
positive words and has itself taken on positive elements.
Thus, 'happy' and 'unhappy' are not contradictories,
because they leave an intermediate state of indifference
between them ; we are often neither happy nor unhappy,
for the latter word does not simply imply the absence of
happiness but, in addition, the presence of positive misery.
Common language, in fact, very seldom expresses sharp
distinctions, and the meaning of a term adopted into
common speech tends to approximate to a kind of average of
the things to which it is applicable. Thus, as 'unhappy'
would apply to anything from the simple negation of
positive happiness to the most intense misery, the word
gradually took into its connotation some elements of dis-
comfort, and now signifies a state intermediate between
indifference and deep misery. So with unkind, unholy,
senseless, misfortune — they are no longer the simple
negations of kind, holy, sensible, fortune ; for neither pair
between them exhausts the universe. The same remarks
are applicable to nearly all this class of words, which are
therefore, although incompatible, not contradictory terms.
It must also be noticed that negative prefixes and affixes
sometimes do not imply the negation of any attribute at all
— thus ' shameless ' is not the negation of ' shameful ' but
almost a synonym with it, and ' invaluable ' means valuable
in the highest degree.
(b) Formal or Logical Contradiction. The necessity of
excluding any intermediate ground is the justification of
logical contradiction, which consists in prefixing not- or non-
to the term — thus not-happy simply excludes ' happy,' ' not-
white' shuts out white, 'not-man' removes 'man' and
Book L
Ch. II.
Their conno-
tation has
positive
elements.
Formal
logical con-
tradiction
consists in
prefixing
not- or nnn-
to the Term
68
TERMS.
Book I.
Ch. II.
The conno-
tation of
sucli terms
is i)urely
negative.
A Positive
Term im-
plies the
presence of
attributes.
A Negative
'J'erm sin'\'p]y
implies
their ab-
sence.
A logical
negative
term is
limited, in
practice, in
its applica-
tion.
applies to all else in our universe of discourse. The conno-
tation of such terms is, therefore, simply negative ; they
imply nothing but the absence of all, or some, of the attributes
connoted by the term to which they are prefixed. Thus,
formal differs from material contradiction in this, that
whereas in the latter the meaning of each term has to be
apprehended separately, in the former the meaning of one is
enough — it furnishes us, also, with the meaning of the other.
As this negative particle can be applied to any name what-
ever, it is clear that this distinction is exhaustive of all
terms ; those without the negative particle are Positive,
those with it are Negative. Thus formally, unhappy is
positive as well as happy, not-unhappy is negative, as is
not-happy. Of course, in the case of such terms as equal
and unequal, where there is no third alternative, unequal is
negative for it is simply the same as not-equal ; it negates
equal but it implies nothing else, as it has been shown such
words as unhappy, unkind, etc., do.
We may say, then, that a Positive Term implies the presence
of an attribute or group of attributes, and a Negative Term
simply implies the absence of the attributes connoted hy the
corresponding positive term, hut implies the presence of no
attributes whatever.
As this negation is purely formal, and the class denoted by the
negative term is wholly arbitrary with the presence of no common
attributes implied by its name, it is evident we may make this class
as wide as we please. Many Logicians, in fact, and Mill amongst
the number, make it include all existence except those things which
are denoted by the positive name. Thus Mill says, in reply to Bain,
who would restrict the application of e.g. not-white to coloured
things, " In this case, as in all others, the test of what a name de-
•' notes is what it can be predicated of : and we can certainly
"predicate of a sound, or a smell, that it is not white" [ijogic,
Bk. I., ch. ii., § 6, note). It is true we can form the combination,
' This sound is not-white,' but is it not absolutely unmeaning ?
If we represent any term by A, then its negative is represented by
nan- A, and to ask us to form an idea of non-A which shall embrace
everything in existence except those things which are A, is to
DIVISIONS OV TERMS.
69
demand an impossibility. We may form combinations, but, like
the one just quoted, they will be, for the most part, absolutely
meaningless ; and we decline to regard as a true expression of a
logical judgment a meaningless jumble of terms simply because they
are connected by the copula. This is, no doubt, the formal mean-
ing of such a purely negative term, but we have already given
reasons (see § 19) for holding that its application is not, in practice,
thus infinite, but is restricted to the universe of discourse just as
much as material contradictories are ; i.e. that formal and material
contradictories have practically the same denotation, and differ only
in connotation. In the universe of human beings, not-British =
alien; not-alien = British ; in that of material things generally,
not-British = foreign ; not-foreign = British, with regard to their
denotation. And so with every term ; not-white applies to all
colours except white, but it has no meaning when applied to things
which possess no extension in space. There is no doubt that this
limitation is always intended in common speech, and it can always
be gathered from the context when it would be otherwise ambiguous
— not-light may, for instance, belong to the universe of weight, or
to that of brightness. Of course, if we are given simply the symbols
A and non-A, we cannot tell what the limits of our universe are
meant to be, but we know that if we had to translate those symbols
into ordinary language the limitation would become apparent.
Though non-A simply negates the possibility of A, yet to be real
this negation must rest on the fact that some quality is present in
non-A which is incompatible with that connoted by A. Now,
it is obvious that heavy is certainly not-yellow if that term is to be
extended from the universe of colour to embrace all existing things
and attributes except yellow ; yet the attributes ' yellow ' and
'heavy' are by no means incompatible — they may, and do, exist
in the same subject, e.g. in gold.
We hold, then, that the so-called Infinite or Indefinite
Terms which simply mark an object by exclusion from a
class, and are supposed to embrace all existing or conceivable
things except those contained in that class, as not-white to
include sounds, tastes, hymn-tunes, half- holidays, etc., etc.,
are not merely logical figments, but are absolutely useless and
positively misleading — they take the garb of general terms
but there is no concept corresponding to them.
Book I.
Ch. II.
Its denota-
tion is the
same as that
of a material
contradic-
tory of tlie
same terra.
Infinite or
Indefinite
Terms sim-
ply mark au
object by
exclusion
from a class.
These are
logical fig-
ments.
70
TERMS.
Book I.
Ch. II.
* This idea of logical contradiction in terms has no real
application to Proper Names, as they imply no attributes to
be negated. Such a term, therefore, would obviously be a
mere sham ; no idea can possibly correspond to it.
Contrary or
Opposite
Tei-ina ex-
press the
greatest
possible
divergence
in the same
universe.
This dis-
tinction is
material.
A Privative
Tei-m im-
plies the
abeence of
an attribute
in a subject
capable of
possessing
it.
(ii.) Contrariety. Whilst logical contradictions simply
negate each other, common speech can do more than this ; it
can express degrees of divergence, as we saw in the case of
the terms ' happy ' and ' unhappy ' as contrasted with happy
and not-happy. When two terms express the greatest
degree of difference possible in the same universe they
are said to be Contrary or Opposite Terms : thus black-
white ; wise — foolish; strong — weak; happy — miserable are
pairs of contraries. This distinction is ent'rely material
and cannot be represented symbolically ; Formal Logic can
only take notice of and express formal contradiction. The
idea of contrariety rests on the assumption that we do not
simply divide our universe into two classes as in Formal Con-
tradiction, but into a series of groups which have no sharply
defined boundaries, as pleasant, indifferent, unpleasant, pain-
ful, where the extreme terms are contraries.
Under this head may be included Privative Terms, which
are often defined as those which imply the absence of an
attribute which the subject either has had or might be
expected to have, as deaf, a word which is equivalent to not-
hearing and which would not, in ordinary every-day life, be
applied to an object unless, as a rule, things of that class
possessed the capacity for hearing — not to rocks or trees or
other things which never possess that attribute {see Mill,
Logic, Bk. I., ch, ii., § 6). Other instances of such terms are
blind, dumb, lame, etc. Thus understood. Privative Terms
are of absolutely no importance, and it has, therefore, been
proposed to slightly contract the connotation of Privative
so as to make it signify " the absence of an attribute in a sub-
'■^ject capable of possessing it " (Stock, DecL Log., pp. 35-7), but
with no presumption as to the probability of its presence.
This contraction is so slight and the corresponding extension
of its denotation is so convenient that we shall adopt it. In
DIVISIONS OF TERMS.
71
this sense, then, Privative Terms include most of those words
formed with negative prefixes or affixes, as unkind, unhappy,
as well as terms similar to those already mentioned, for we
should not apply 'unkind,' except to a morally responsible
being and one who is, therefore, capable of being kind, nor
' unhappy ' to a being incapable of enjoying happiness. It
should be noticed that the connotation of a Privative Term is
partly negative, in that the word implies the absence of a
certain attribute, and partly positive, in that it always implies
the presence of some attributes which are compatible with
that denied as well as very often of some others which are
incompatible with it. Thus ' unhappy ' implies absence of
happiness, capacity for feeling happiness, and presence of
some degree of misery.
POOK I.
Ch. II.
* (iii.) Repugnance. Terms are repugnant to each
other when, without being directly contrary, or con-
tradictory in the logical sense of exhausting the uni-
verse between them, they are yet incompatible in that
they are mutually exclusive. Very often, when we examine
the denotation (in the restricted sense advocated above) of a
negative term, as not- white, we find it embraces several well-
defined groups, as green, red, blue, etc., no two of which can
be predicated of the same thing at the same time ; these
terms are repugnant to each other. So we may speak of
articles of furniture as being either of wood or not-wood,
and when we examine the latter group we find it contains
things made of gold, of silver, of brass, of steel, of iron, of
stone, and of many other materials ; all these, again, are
repugnant terms, for no two of them are either contradic-
tories or contraries and yet no two can be predicated of the
same thing. This is as entirely a material distinction as is
that of Contrariety, and we see, then, that of all the forms of
incompatibility of terms employed in common speech. Formal
Logic only recognizes that of Formal Contradiction, i.e., the
division of terms into Positive and Negative — into A and
non-A.
Repugnant
Terms arc
incompat-
ible witli
each other,
though
neither Con-
traries nor
Contradic-
tories.
The distinc-
tion is
materiaL
72
TERMS.
Book I.
Ch. If.
A Concrete
Term is tho
name of an
object.
An Altstract
Term is the
name of an
attribute
considered
by itself.
30. Concrete and Abstract Terms.
(i.) Relation between Concrete and Abstract Terms.
The division of Terms into Concrete and Abstract is founded
upon psychological and grammatical rather than upon logical
reasons. It is, however, usual to consider it as part of the
logical doctrine of Terms. The following definitions express
the difference : —
A Concrete Term is the name of an object.
An Abstract Term is tbe name of an attribute con-
sidered by itself.
In the above definition the word ' object' ia used widely
to denote everything, whether material or not, which can be
regarded as having a more or less separate existence as a
whole whose parts or elements are in essential relation to
each other and to the whole which comprises them. It thus
includes such names as Logic and Ethics ; and as point,
line, etc., in their strict mathematical sense.
* Abstract Terms are formed by the process of abstracting
the attention from all the qualities of a thing, except some
particular one, or group, to which the name is then given.
Thus, by attending to one quality only of a tree, we form
the idea of greenness ; by considering only tho moral quality
of a number of good actions we gain the concept of virtue.
But it is evident that all General Terms represent con-
cepts which are formed by Abstraction ; we must not,
therefore, regard this process as a sufiicient ground for call-
ing a term Abstract. If we did, we must include in that
class all Terms whatever except Proper Names. A Term
can only be called Abstract when it denotes a quality which,
though it can only exist in some object, is yet thought of
apart from all objects whatever. Thus, we can think of
'strength' by itself, although we know there can be no
strength except as an attribute of strong things ; or, of
virtue, though it cannot exist apart from good actions. ,
The terms ' Concrete ' and ' Abstract ' have been used in various
senses by logicians. This has caused some objection to their use in
Logic altogether, and Miss Jones has suggested Substantial and
DIVISIONS OF TERMS.
73
Attribute terms as more appropriate names ; the reference to indi-
vidual objects being prominent in the former, and to attributes in
the latter (Elem. of Logic, pp. 12-15 ; 37-39). The distinction
drawn exactly coincides with that taken here, and it may be ab
once granted that thanew nomenclature, if generally adopted, would
be a distinct improvement.
If it be borne in mind that an Abstract Name is not simply
the name of a quality, but of a quality considered by itself, and
apart from the objects which possess it, it will be immediately
seen that Adjectives are not Abstract Terms ; for they name
qualities only indirectly, and considered in connexion with
the things to which they belong. If we say ' Gold is yellow,'
we do not mean that gold is a colour, but that it is a thing
which possesses a certain colour. It is the colour of gold
which we call yellowness, not gold itself. ' Yellowness ' is,
then, the name of the colour or quality ; ' yellow ' is the
name of all objects which possess that quality. Whether
a name is Abstract or Concrete will often depend on the
sense in which it is used ; for the same word may be Con-
crete in one sense and Abstract in another.
The importance of the distinction between Concrete and
Abstract, however, from a logical point of view, lies in those
pairs of terms wherein one is the Abstract and the other
Concrete, as strong, strength ; man, humanity ; square,
squareness ; etc., and in these cases there can never be any
doubt as to which is Concrete and which is Abstract. In all
such pairs the Concrete term is General, and the older
Logicians confined the distinction to such pairs of terms ;
but, in order to make the division exhaustive of all Terms,
we may regard all Individual names, including Proper Names,
as Concrete, though they have no Abstract terms corre-
sponding to them. Every Concrete General Term has not,
in fact, an Abstract corresponding to it, but one is always
theoretically possible, and new ones are continually being
coined as occasion arises. On the other hand an Abstract
Term can only be expressed in Concrete terms, and that but
imperfectly, by an awkward periphrasis. Thus, instead of
strength we can speak of * strong things considered only in
Book I.
Ch. II.
All Adjec-
tives are
Concrete
Terms.
The distinc-
tion be-
tween Con-
crete .and
Abstract is
not a fixed
one.
The distinc-
tion is only
logically im
portant
where the
terms form
pairs.
74
TERMS.
Book I.
Ch. II.
Abstract
Terms ab-
breviato
thought.
Abstract
Names of
single attri-
butes are
Singular.
Abstract
Names of
groups of at-
tributes are
General.
their aspect of being strong.' Abstract Terms are thus a
great help towards abbreviating and systematizing thought ;
the proposition ' Union is strength ' is a much neater and
more universal expression than 'Things which are joined
together, considered only with respect to their being joined,
are strong,' but only by such an awkward and involved sen-
tence can we express approximately the meaning conveyed
by the three words of the original proposition. If we simply
say ' Things which are joined together are strong ' we do not
indicate that we are concerned in the subject only with the
attribute ' being joined together,' and this is the very point
which the proposition ' Union is strength ' emphasizes.
Similarly ' Justice is a virtue ' would receive a fair expression
only in some such sentence as * Just acts, in so far as they
are just, are virtuous.'
(ii.) Singular and General Abstract Terms. Every
Abstract Term which is the name of a single attribute is
Singular, for though the attribute may be possessed by many
objects yet it is conceived by us as one and indivisible. Thus
though there are many square things, yet the attribute
' squareness,' which is common to them all, is evidently only
one, we cannot imagine different species of ' squareness.' So
with ' equality ' and all other terms which are the names of
simple attributes. But when we have a group of attributes,
such as colour, which embraces red, green, white, etc. ; or
humanity, which includes animality and rationality ; then its
name will stand for all these and we must regard it as
General with respect to these included notions.
(iii.) Connotative and Non - connotative Abstract
Terms. It has been much disputed amongst logicians
whether Abstract names can ever be connotative. Mill held
that they could. He says : " Even abstract names, though the
" names only of attributes, may in some instances be justly
" considered as connotative ; for attributes themselves may
" have attributes ascribed to them ; and a word which denotes
•'attributes may connote an attribute of those attributes"
{Logic, Bk. I., ch. i., § 5). Jevons, however, objects to
DIVISIONS OF TERMS.
75
Names are
ni>n-nonuo-
tative.
this and holds that no Abstract Name can be connotative. Book I.
Singular Abstract Names are generally regarded as non-conno- ^^- ^ ■
tative : they denote the attribute which the Concrete Names All singular
connote and there seems nothing left for them to connote
But when an Abstract Term is the name of a group of
attributes there seems no good reason for denying that it is
connotative. Thus, virtue is a name common to justice,
benevolence, veracity, and other qualities of conduct which
men agree in regarding as praiseworthy ; it, therefore,
denotes those good qualities, and connotes the attribute
'goodness' which they possess in common. So, 'colour'
denotes redness, blackness, blueness, etc., and connotes the
power of affecting the eye in a certain way : ' figure '
denotes roundness, squareness, triangularity, etc., and
connotes shape and extension in space. We thus reach the
conclusion that all General Names, Abstract as well as
Concrete, are connotative, and this view seems the only one
which is compatible with the nature of a General Name.
For, the same name can only be applied to a number of
objects of thought, whether things or attributes, because
they agree in the possession of some common quality ; the
name must connote this common property and denote the
objects of thought which possess it ; hence, every General
Name must, of necessity, have both connotation and denota-
tion, that is, it must be connotative. Those, therefore, who
deny that any Abstract Names can be connotative, must
also, in consistency, deny that any can be General.
31. Absolute and Relative Terms.
This division of terms is based on the fact that the
relations of things differ from their other attributes, in
that they involve direct reference to more than one object.
If, for instance, we speak of a man as strong we can confine
our attention to that one individual, but if we speak of
him as a friend we must at once extend our view to include
some other person who stands to him in the relation of
friendship. We may say, then, that
76
TERMS,
Book 1.
Ch. II.
An Aiisolute
Term im-
plies no re-
ference to
anytliing
else.
A Relative
Term im-
plies a refer-
ence to an
object re-
lated to that
which it de-
notes. These
two are Cor-
relatives.
To be corre-
latives, the
Terms must
imply the re-
lation exist-
ing between
the things
denoted.
An Absolute Term is a name which in its meaning
implies no reference to anything else.
A Relative Term is a name which, over and above
the object which it denotes, implies in its signifi-
cation another object which also receives a name
from the same fact or series of facts which is the
ground of the first name.
Each one of such a pair of terras is called the correlative
of the other. In some cases each correlative has the same
name, as friend, companion, nartner, like, equal, near ; in
other cases the names are different, as parent, son ; king,
subjects ; governor, governed ; cause, effect ; greater, less ;
north of, south of. But they are always found in pairs, and
they always owe their names to the same fact or series of
facts. Such pairs are equally common among Adjectives
and Substantives, for attributes may be thus related as well
as things. Neither member of the pair can be thought of
alone ; for the existence of each depends on, and implies,
that of the other ; there can be ro meaning in parent,
except in reference to son or daughter, nor in son or
daughter except as implying parent ; the Absolute Name
for the same individuals would be human being.
It must be remembered that it is the fact that the terms
imply the relation in which the objects stand to each
other which makes them Relative, not the mere existence of
the relation ; thus a king governs men, but king and man
are not correlatives, for the terms do not imply this relation ;
king and subject are correlatives because they do imply it.
In one sense it may bo said that all terms are relative, for
all things in Nature are interconnected — there is nothing
which exists utterly by itself and out of all relation to every-
thing else. This is true, but it is not the sense in which
'Relative' is used in Logic; if it were, we should have
everything entering as a member into innumerable pairs of
correlatives. In Logic terms are only relative when the
existence of the correlative is implied by the meaning of the
term itself.
DIVISIONS OF TERMS. 77
All Relative Terms must connote the fact or series of Book I.
facts which is the basis of the relation and which is, there- Ch^l-
fore, called the fundamentum relationis. Thus, each member The Funda-
of a pair of correlatives connotes the same fact viewed from lationis ia
a different standpoint ; paternity and sonship are not two ^\iic^°*gthe
different facts but the same fact viewed from two different basis of the
sides, and connoted both by parent and by son. So rule and pressed by'
subjection imply the same condition of things regarded from Correlative
the point of view of the ruler and of the subject respectively.
The abstract terms, then, which are the names of the fact "^"l^ S""*
c TG lg J. or hi s
or facts on which the relation depends, cannot be regarded can be Rela-
as correlatives ; for they do not denote two facts but one, *^^®'
and a thing cannot be correlative with itself. Hence,
parent and son are correlatives, but paternity and son-
ship are not. This will be seen more clearly in the case
of those pairs of correlative terms in which each member
has the same name ; friend is correlative with friend, but
the abstract term denoting the connecting bond is friend-
ship, no matter from which side we regard it ; so with
partnership, which is the fmidamentum relationis of the
correlatives partner, partner. Similarly, equality and like-
ness are not Relative Terms though equal and like are ;
for it requires two things to be either equal or like, but the
fact of equality (or likeness) is one. Thus, only concrete An object
terms can be Relative, and of them only those which denote ^ a°Reki.tivo
things whose existence absolutely alone is inconceivable ; it '•'^V'? '=^'^-
. ..,,.. , , ' not be con-
IS mipossible to nnagme a parent as the only bemg who had ceived as
ever existed in the world, for then only an absolute name, absolutely
as human being, would be applicable. All terms which do «lone.
not thus necessitate reference to some things other than that
of which they are the names are Absolute. The distinction
is not, however, of much importance in Logic.
CHAPTER III.
Book T.
Ch. III.
The Predic-
ables are a
classifica-
tiou of the
relations of
the predi-
cate to the
subject of a
logical pro-
position.
A Term does
not always
belong to
the same
Predicable.
Aristotle
drew out a
four-fold
scheme,
founded on
Laws of Con-
tradiction
and Exclud-
ed Middle :
(a) Drfiiii-
tion.
(b) Proprium.
(c) Genus.
(d) Accidenx.
THE PREDICABLES.
32. Definition of Predicable.
The Predicables are a classification of the relations
of the predicate to the subject of a logical proposition.
They do not express what a term is by itself, but only what
relation it bears to the subject of the proposition of which
it forias the predicate. We cannot absolutely refer any
General Term to one definite Predicable ; for the same
term must be assigned now to one and now to another of the
predicables according to its relation to the subject of which
it happens, in any particular proposition, to be predicated.
The consideration of the Predicables might, therefore, have
been postponed till we had examined the doctrine of Propo-
sitions, and many logicians do so postpone it. The point is
of no practical importance, for the whole doctrine of Terms
is of logical interest only because terms are constituents of
propositions ; and, as this subject is concerned with those
constituents and not with propositions as wholes, it seems
better to treat it here.
33. Aristotle's Four-fold Scheme of Predicables.
The first classification of the Predicables was a four-fola division
made by Aristotle which was thus evolved. In every proposition
the predicate must either agree or disagree in denotation with the
subject. If it agrees it either has the same connotation or a diflferent
connotation ; in the former case the Predicate is a Definition, in the
latter case a Proprium. Thus in 'Man is a rational animal,' we
have a definition ; for ' rational animal ' agrees with ' man ' both in
denotation and connotation ; whilst in ' Man is an animal which
THE PEEDICABLES.
79
cooks its food,' the predicate is a Proprium, for though 'animal
which coolcs its food' agrees exactly in denntation with man — i.e.
refers to the same individuals and to no others — yet * to cook food'
is no part of the connotation, or meaning, of the word 'man.' If,
on the other hand, the Predicate has not the same denotation as the
Subject, then its connotation is either partly the same or entirely
different. It evidently cannot be entirely the same, for then there
could be no difference in denotation ; for, as a name denotes all
things which possess all the attributes which it connotes and no
others, it follows that if two names have identically the same conno-
tation, they must be applicable to precisely the same things, i.e.
have exactly the same denotation. If, then, whilst differing in
denotation from the Subject, the Predicate partially agrees with it
in connotation, it is a Oenus ; if it entirely disagrees, it is an Acci-
dens. For example, in the proposition, ' Man is an animal,' the
predicate is a Genus; for, whilst the denotation of 'animal' is
greater than that of man, its connotation is less ; but in the propo-
sition, 'Some men are woolly-haired,' we have an Accidens ; for
the predicate, 'woolly-haired things,' differs from the subject 'men*
both in connotation and in denotation. These four heads of Pre-
dicables may be thus defined :
A Definition is the wjgregate of all the attributes which fully
explain the nature of the subject.
A Projjrium is a mark which invariably belongs to the subject,
and to nothing else, but is not the attribute which would be mentioned
to explain the nature of the subject.
A Genus is a mark or attribute which invariably belongs to the
subject, but not to that alone.
An Accldeiis is an attribute which may or may not belong to the
subject.
The whole scheme may be thus summarized :
1. Agreeing in
notation with
Book I.
Ch. III.
ri.
Predicables.
Agreeing in
Denotation
with Subject.
con-
sub-
^
II
Differing in
Denotation -j
from Subject. 1
]ect.
2. Differing in con-
notation from sub-
ject.
3. Partially agreeing
in connotation with
subject.
4. Wholly differing in
connotation from
subject.
Definition.
Proprium^
Genus.
Accidens.
80
TERMS.
Book I.
Ch. III.
This scheme, being founded on the Laws of Contradiction and
Excluded Middle, is evidently exhaustive of all General Terms
This scheme when used as predicates. It has, however, been practically super-
^'^^ ^Ta ^" seded by a five-fold division, which was first advanced by Porphyry
Porphyry's, in his ' Introduction to the Categories of Aristotle,' written in the
third century, and which has ever since occupied a prominent place
in the traditional logical doctrine.
Porphyry's
Five-fold
scheme :
(a) Genus,
(h) Species.
(c) Differen-
tia.
(ci) Proprinm.
(e) Accidens.
34. Porphyry's Five-fold Scheme of Predicables.
The traditional classification of predicables is that of
Porphyry, and is closely connected with the subjects of
logical definition and division. It is as follows : —
Predicables are-
n.
2.
3.
4.
5.
Genus
Species
Differentia
Pi-opriuin
Accidens
tlSoQ
Sta<l>opa
iSiov
ffvfifStfirjKoQ^
of the Subject.
We will now briefly define these five Heads of Predicables
or ' Five Words,' as they are frequently called, and will then
consider them more in detail.
A Genus is a wider class which is made up of narrower
classes.
A Species is a narrower class included in a Genus.
A Differentia is the attribute, or attributes, by which
one species is distinguished from all others contained
under the same Genus.
A Proprium is an attribute which does not form part
of the connotation of a term, but which follows from it,
either as effect from cause or as a conclusion from
premises.
An Accidens is an attribute which neither forms part
of the connotation of a term nor is necessarily connected
with any attribute included in that connotation.
TliE PREDICABLES.
81
S5. Genus and Species.
These are not absolute terms, but purely correlative. A
Genus has no meaning apart from the two or more species
into which it is divided ; nor has a Species apart from the
containing genus. The same term may be, at the same time,
a Species of the next more general class, and a Genus to the
less general classes it contains ; no term by itself can be
styled a Genus or a Species. Thus, in the example quoted
in § 28 (v.) to illustrate the relation between Connotation
and Denotation of Terms, each intermediate term is a
species to the preceding term, and a genus to the succeeding.
If a term is so general that it is not a species of any more
general term it is called a Highest Genus or Summum Genus;
and if it cannot be further divided into species, but only
into individuals, it is a Lowest Species or Infima Species.
The Aristotelian logicians held that there were ten sunima genera,
which they called Categories or Predicaments {see Bk. I., oh. iv.),
and each such Summum Genus, with a series of terms below it
following in the order of less and less generality down to an Infima
Species, was called a Predicament al Line (Linea Predicamentalis).
As every genus must have at least two species included under it, it
follows that each Predicament must have a plurality of predicamental
lines ; as many, in fact, as there are infimce species. In every such
predicamental line, the general inverse relation between the conno-
tation and denotation of terms is exemplified : the Summum Genus
is the widest in denotation, but the most meagre in connotation of
the whole series ; on the other hand, the Infima Species has a smaller
denotation and a richer connotation than any other term in the
series ; whilst, of the intermediate terms, each is greater in conno-
tation, and less in denotation, than the one preceding it. In every
such line, the nearest genus to every term, of which that term is
itself a species, is called the Proxinmm Genus ; and every term in
the line, except the Summum Genus and the Infima Species, is
termed a Subaltern Genus or Species (Lat. sub, under, alter, the
other of two).
The tenswmwia genera of the Aristotelian logicians are universally
disputed by modern writers, and it is doubtful whether there can
be any true swrnnum genus except Being in general, or Reality ;
but, for practical convenience, the term is employed to denote the
LOG. I. 6
Book I.
Ch. III.
Genua and
Siiecies are
correlative
Terms.
A Genus is a
wider class,
containing
two or more
narrower
classes called
Species.
A Summum
Genus is not
a species of
any wider
term.
An Infima
Species can
only be di-
vided into
individuals.
A Predica-
mental Line
is a series of
subaltern
Genera and
Species
reaching
from a Sum-
VI am Genus
to an Infima
Species, and
passing
through
Subaltern
Genera and
Species.
62
tERMS.
Book I.
Ch. III.
Cognate
Sjiecies are
sub-classes
uuder the
same genus.
Cognat e
Genera are
classes of
varying
generality
under which
a species is
contained.
widest class of things comprehended in any science ; as, for instance,
material substance in Chemistry, Thus it is possible that the same
things may form the summum gemis in one Science but not in
another ; for example, Man is the summum genus of Sociology,
but is only a species under Animal in Zoology ; and Animal, again,
only a species of Living Organism in Biology.
Two or more classes which rank as species under the same Genus
are called Cognate Species. A Cognate Oemis is any one of the
higher classes under which the same species falls. Thus in the
following example (from Fowler's Deductive Logic, p. 61) : —
Figure.
Curvilinear.
Rectilinear.
Triangle.
QuadrilateraL
Polygon.
GeiiuB and
Species are
the only
classes re-
cognized in
Logic ; but
Botany and
Zoology give
them fixed
places in a
hierarchy of
classes.
Equilateral. Isosceles. Scalene.
Equilateral, Isosceles, and Scalene, triangles are cognate species
of the subaltern genus Triangle ; and Triangle, Quadrilateral
Figure, and Polygon, are cognate species of the subaltern genus
Rectilinear Figure. So, Triangle, Rectilinear Figure, and Figure,
are all cognate genera of Equilateral Triangle.
When a General Term is predicated of another General
Term, it is a Genus and the subject a Species ; but when a
General Term is predicated of a Singular Term, it is a
Species, for it is under Infimce Species that individuals are
directly included.
The logical use of the terms Genus and Species must be distin^
guished from the use of the same terms in Zoology and Botany,
where a species, till recently, meant a class of animals, or plants,
supposed to be descended from common ancestors, and to be the
narrowest class possessing a fixed form, whilst a genus meant the
next highest class. If, however, the Theory of Evolution is correct,
many genera and species are really descended from a common stock,
THE PREDICABLES.
83
and the distinction of genus and species becomes partly arbitrary,
and dependent upon such points of resemblance as naturalists
believe important. In these sciences, too, in order to express the
relation of container and contained, other terms are employed
besides the old logical genus and species. Thus, according to its
position in a system of classification, a group is spoken of as
Kingdom, Sub-kingdom, Division, Sub-division, Class, Sub-class,
Cohort, Sub-cohort, Order, Sub-order, Tribe, Sub-tribe, Genus,
Sub-genus, Section, Sub-Section, Species, Sub-species, Variety,
Sub- variety, Variation, Sub-variation. In the language of formal
logic all the intermediate classes are subaltern genera to the sum-
mum genus Animal or Plant.
It must be noted that the ancient logicians considered as
genera and species those classes only which were parted from
each other by an unknown multitude of differences, and not
merely by a few known and determinate ones. Where the
differences were few they were considered as belonging to
the Accidentia of the things, but where they were practically
infinite in number, the distinction was held to be one of
kind, and spoken of as an Essential difference. Mill adopts
the same view, and speaks of species as * Real Kinds.' It is
on this distinction that the next three Predicables are
founded.
Book I.
Ch. ui.
Only classes
separated
from others
by innumer-
able quali-
ties were
formerly
called
Genera and
Species.
36. Diflferentia.
It has been pointed out above that a species is wider in
connotation than the genus under which, in denotation, it is
contained. The excess of the connotation of a species over that of
it s proximate genus is called the Differentia or Difference of that
Species. Thus,in connotation, the sum of genus and differentia
gives species. It is plain, however, that there can be no such
thing as an absolute genus or differentia, for the same attribute
may be differentia in one case and part of the connotation of
the genus in another. Thus, if we have three classes of
things with the respective connotations ab, ac, be, whilst a is
the genus of the first two, and b and c differentiae ; b is the
genus of the first and third, and a and c differentiae ; and c
13 the genus of the second and third, and a and b differentise.
G— 2
Differentia —
the attri-
bute, or
group of
attributes
wliicb dis-
tiiiguisli one
species from
all others
contained
under the
same gui iiia
84
TERMS.
Book I.
Ch. III.
As the connotation of a General Name only embraces
those attributes which it implies, and not all those which are
possessed in common by the things denoted by the name
[see § 28 (ii.)], it is evident that the determination of what
attributes form the differentia of any term depends upon the
definition which unfolds the connotation (see § 49).
Dififerenlise are spoken of as Specific and Generic. A Specific
Differentia is that which distinguishes cognate species from each
other, whilst a Generic Differentia is common to the whole class to
which those cognate species belong, and is, to them, part of the con-
notation of the genus ; it is only a differentia with regard to the yet
liigher genus of which this genus is a species, and with regard to
that, it is, of course, a Specific Differentia. So, every specific differ-
entia of a higher class is a generic differentia with r-ispect to the
classes below it. Of course, summa genera have no differentiae.
Symbolically, if the summum genus x includes the cognate species
xy, xz ; y and z are specific differentiae. But, if we find xy to be a
genus to the subaltern species axy, hxy, then y is a generic differ-
entia with respect to those classes, their specific differentiae being
a and 6.
Fropnuiii—
an attribute
which docs
not form
part of the
connotation
of a term,
but which
uecessaiily
follows from
it.
Whether an
attribute is
Differentia
or Proprium
depends on
the defini-
tion of the
term.
37. Proprium.
Those attributes which are common to every individual which
hears the class name, and which are not included in its connota-
tion, though necessarily connected with it, are called its Propi-ia
or Properties. Propria need not, however, be peculiar to the
members of this class, for they may flow from a par'j of the
connotation which is also part of the connotation of some
other class name.
* The distinction between Differentia and Proprium is
rather founded on the conventions of language than on the
nature of things ; for there is often no valid reason why some,
rather than others, of the common attributes of a class should
be implied by the class name. Thus, with our definition of a
triangle, the attribute ' three-sided ' is the differentia which
distinguishes that sjtecies of plane rectilinear figures from
others, and ' three-angled ' is a proprium ; but if we defined
a triangle — as the etymology of the name, indeed, suggests —
THE PREDICABLES.
85
as a ' three-angled figure ' then the attribute ' three-angled '
would become the differentia, and ' three-sided' the proprium.
This is not so, however, in every case ; and, always, propria
are attributes which flow from the whole, or part, of the con-
notation either as effect from cause or as a conclusion from
premises. Thus, that roan is a tool-using animal flows from
his rationality, as effect from cause — the attribute ' tool-
using ' is therefore a proprium ; whilst, that ' the square on
the hypothenuse of a right-angled triangle is equal in area to
the sum of the squares on the sides containing the right
angle' is a proprium ; for it is an attribute common to all
right-angled triangles, and which can be shown, by reasoning,
to be a necessary consequence of the connotation.
Book I.
Ch. III.
As Propria are common to every individual bearing a class name
we may have Generic Propria, which are common to every species
in a genus, and which flow from the connotation of the name of the
Genus, and Specijic Propria which are attributes flowing from the
differentia of the name of a species, and common to every individual
included in that species. As in the case of Differentiae, the same
attribute is a Specific Proprium of a higher class, but a Generic
Proprium of a lower.
The connexion of a proprium with the connotation is a
necessary one ; that is, its not following would be inconsistent
with some law which we regard as part of the constitution
either of the universe, of our minds, or of both._
38. Accidens.
In this class are included all those attributes which are
neither oonnoted by a term nor are connected with its conno
tation ; that is, which are included under neither of the heads
Genus, Differentia, or Proprium. We have no real defini-
tion of what an Accidens, or Accident, is ; we can only say
what it is not. An Accidens may be described as an attribute
which can be removed from the class, or individual, without
necessitating any other alteration j whilst to remove a
Accidens -
an attribute
which
neither
forms part
of the con-
notation of
term, nor if
necessarily
connected
with any at-
tribute in-
cluded in
that conno-
tation.
86
TERMS.
Book I.
Ch. Ill
Accidentia
aro either
Separable or
Inseparable.
An Insepar-
able AcciJms
is common
to every
member of a
A SejiwrabU
Accidens is
not common
to every
member of a
ulafw.
proprium or differentia would be to destroy the individual,
or class, or at least to fundamentally change its character.
There may be accidentia of a Class or of an Individual, and
in both cases they may be Sepai-able or Inseparable. An
Inseparable Accidens of a class is one which belongs to every
member of the class. It is, of course, difficult to distinguish
such accidentia from propria, and a more extended investiga-
tion into the nature of things is always likely to remove an
attribute from the former class to the latter. But, where
there is no apparent reason why the attribute should always
be found in the individuals of a class, it is called an Accidens.
Thus, that all European ruminant animals are cloven-footed
appears to be an invariable rule, but, as there is no apparent
connexion between chewing the cud and having a cloven
hoof, we regard having a cloven hoof as an Inseparable
Accidens of the class European ruminant. White was long
regarded as an Inseparable Accidens of swans, but the dis-
covery of black swans in Australia has shown that it is only
a Separable A ccidens, that is, 07ie not common to every member
of a class. When we come to individuals the words Separable
and Inseparable have, necessarily, a somewhat different
meaning. An Inseparable Accidens of an individual is one
which belongs to him at all times and can never be changed,
as the date and place of a man's birth, whilst a Separable
Accidens is one which is sometimes present and sometimes
absent or which can be changed, as a man's trade, his acts or
postures. These individual accidentia are of no logical
importance.
* 39. The Tree of Porphyry.
The "iTee of
Porphyry
exemplifies
tlie chief
rredicabl(;8.
An example of a portion of this scheme of Predicables la
furnished by a table known as the Tree of Porphyry because
it was first set forth by Porphyry. It is also called the
Ramean Tree from the prominence given to it by Ramus (a
sixteenth century writer on Logic). It is as follows : —
THE PREDICABLES.
Substance
87
Corporeal
Animate
Sensible
Rational
Socrates
Incorporeal
Body
Inanimate
Living Being
Animal
Insensible
Irrational
Man
Plato
and others.
Here we have the Summum Genus, Substance — and the
Infima Species, Man, which cannot be divided into any
narrower species but only into individuals. The inter-
mediate terms down the centre of the ' tree ' — Body, Living
Being, Animal, are Subaltern Genera and Species ; each is a
genus as regards those below it in the list, and a species with
respect to those above it. The attributes Corporeal, Animate,
Sensible {i.e., able to feel). Rational, are differentiae which
divide each genus into species. Of course, the corresponding
negative attributes are also differentiae, but the species they
would give rise to are omitted for the sake of simplicity ;
their existence must not, however, be forgotten, for every
genus must be divisible into at least two species.
Book I.
ch. iir.
88
TERMS.
Book I.
Ch. III.
The Predic-
ables do not
consider
Singular
Terms as
Predicates.
A Proposi
tion is
Analytic
when the
predicate is
a genus or
differentia
of the sub-
ject ; Syn-
ilietic, if a
proprium or
accidens is
predicated
* 40. General Remarks on the Predicables.
With regard to this five-fold scheme of Predicables it may
be remarked that no provision is made in it for Singular
Terms as Predicates. In fact, by the older logicians singular
terms were never regarded as predicates, and such proposi-
tions as * Lord Salisbury is the present Prime Minister of
England ' were looked upon as outside the scope of Logic.
A Predicable was only another name for a Universal — the
same term regarded in denotation was a Predicable, as being
applicable to several different things ; considered in conno-
tation it was a Universal, as the attributes implied were to
be found in several other and different notions.
When a Genus or Differentia is predicated, the j^rojjosition is
said to be Analytic or Verbal as the Predicate only states
explicitly part of what is implicitly contained in the sub-
ject; but when a Proj)ritim or Accidens is predicated the
proposition is syntlMic or real, as the predicate then asserts
an additional fact, which no analysis of the subject would
reveal. Other names for the same distinction are — Essen-
tial and Accidental, Explicative and Ampliative. Strictly
speaking, a Species is only predicated of an individual ; when
the individual is denoted by a Proper name the proposition
is, of course, synthetic, as the Proper Name implies nothing;
but when the subject is a Significant Singular Name such a
proposition is often analytic, as the Significant Singular
Name frequently contains the species in its connotation.
Thus 'Socrates is a man' is a synthetic proposition, but
' This great Greek philosopher is a man ' is an analytic pro-
position ; for, 'philosopher' implies 'man,' but 'Socrates'
does not.
There is much that is valuable in this scheme, for all classi-
fication depends on the formation of genera and species, and
one of the chief aims of all science is to classify accurately,
and to decide what attributes are essential to the inclusioo
of any individual iu a given class.
CHAPTER IV.
THE CATEGORIES OR PREDICAMENTS.
41. The Categories are a Classification of Relations.
The word Category is derived from the Greek Karijyopelv,
which, in Lo^ic, meant ' to predicate,' and Predicament is the
exact Latin equivalent for that term. The Categories were
intended by Aristotle, who first drew out a list of them, as a
classification of all the possible predicates of any individual
subject. Thus, though he called them yevi] twv wtiov, or kinds
of being, they were really not a classification of things, but
of the relations between things. There were thus, however,
the germs of two views of the nature of Categories in
Aristotle — a classification of existences and a classification of
relations. Of these the former was seized upon and developed
by his immediate followers and by the scholastic logicians,
and hence Categories were traditionally regarded as a classi-
fication of all possible things with no reference to their use
as predicates of a proposition. Thus considered, they were
based on the erroneous notion that the great aim of thought
is to reach ultimate and independent orders of being, under
which all things may be classed. On this view the establish-
ment of a valid scheme of Categories would be the end of
knowledge.
But there is the other and the truer view of Categories —
that which regards them as those relations conceived by the
mind and applied to the interpretation of all experience,
without which all knowledge would be impossible. Thus
looked at they are the beginning instead of the end of know-
ledge.
Book 1.
Ch. IV.
Categories
were re-
garded by
Aristotle as
a classifica-
tion of pos-
sible predi-
cates.
Categories
are wrongly
regarded as
a classifica-
tion of all
nameable
things ;
they are
forms of re-
lation essen
tial to
knowledge.
92
TERMS.
Book I.
Ch. IV.
Under each
Category in-
formation of
some kind
may always
be given
respecting
any indi-
vidual per-
BOO.
ordinary propositions. The other Categories can be predi-
cates only.
* The whole scheme may be thus illustrated : " What is
this individual, Sokrates ? He is an animal. What is his
Species ? Man. What is the Differentia, limiting the
Genus and constituting the Species ? nationality, two-
footedness. What is his height and bulk ? He is six feet
high, and is of twelve stone weight. What manner of man
is he ? 13.Q is, flat-nosed, virtuous, /^ai/eni, brave. In what
relation does he stand to others ? He is a father, a pro-
prietor, a citizen, a general. What is he doing ? He is
digging his garden, lAoughing his field. What is being done
to him ? He is being rubbed with oil, he is having his hair
cut. Where is he ? In the city, at home, in bed. When do
you speak of him ? As he is, at this moment, as he was,
yesterday, last year. In what posture is he ? He is lying
down, sitting, standing up, kneeling, balancing on one leg.
What is he wearing ? He has a tunic, armour, shoes, gloves.
" Confining ourselves (as . . . Aristotle does in the Cate-
gories) to those perceptible or physical subjects which
everyone admits, and keeping clear of metaphysical enti-
ties, we shall see that respecting any one of these subjects
the nine questions here put may all be put and answered ;
that the two last are most likely to be put in regard to some
living being ; and that the last can seldom be put in regard to
any other subject except a person (including man, woman,
or child). Every individual person falls necessarily under
each of the ten Categories ; belongs to the Genus animal,
Species man ; he is of a certain height and bulk ; has cer-
tain qualities ; stands in certain relations to other persons
or things ; is doing something and suffering something ; is
in a certain place ; must be described with reference to
a certain moment of time ; is in a certain attitude or pos-
ture ; is clothed or equipped in a certain manner. Infor-
mation of some kind may always be given respecting him
under each of these heads. . . . Until such information is
gieen, the concrete individual is not known under condi-
tions thoroughly determined. Moreover, each head is
THE CATEGORIES OE PREblCAMENtS. 93
" separate and independent, not resolvable into any of the Book I.
" rest, with a reservation ... of Relation in its most com- Ch^v.
"prehensive meaning. . . , The ninth and tenth are of
" narrower comprehension, and include a smaller number of
" distinguishable varieties, than the preceding ; but they are
"not the less separate heads of information" (Grote's
Aristotle, pp. 77-8)
■* A few further observations may be offered to render this
scheme perfectly clear. Under the fourth category the older
logicians only included substances between which a relation
exists ; if this restriction is neglected this category will not
differ from some of the others.
The seventh and ninth categories should be carefully dis-
tinguished.
Under the eighth category come only answers to the
question ' when ?' Answers to the questions * how long ?'
come under the second category.
The tenth category must be distinguished from the
' Habit ' which is included under the third ; we have the
same ambiguity in the word ' habit ' in ordinary language ;
as ' Habits are hard to break ' ; ' a riding-habit.'
The whole may be thus related to the parts of speech, on
which many suppose the scheme was founded : a Predicate
may oe
{a) A Substantive when it is the name of the kind of thing.
(b) An Adjective of quantity, quality, or comparison.
(c) An Adverb of time or place ; no others imply exist-
ence as these do, and so no others can be used as
predicates.
(d) A Verb either active, passive, or neuter, or express-
ing the result of an action.
43. Objections to Aristotle's Scheme of Categories.
Many attacks have been made on this scheme by moderr.
writers. For example : — ^i'^,^'".'^V''\
^ of the ' Port
* (i.) The authors of the ' Port Royal Logic ' speak of the RoyaiLogic'
divisions as of little use, and even injurious, because " they are this scheme*
" altogether arbitrary', and are founded only in the imagina- *'* -""^rbitrary
leading.
94
TERMS.
Book I.
Ch. IV.
Kant ob-
jected that
the scheme
was not con-
fined to
forms of the
pure under-
standing.
Lotze ob-
jected that
the divi-
sions were
un philoso-
phical and
empirical.
" tion of a man who had no authority to prescribe a law to
" others, who have as much right as he to arrange, after
" another manner, the objects of their thoughts, each accord-
"ing to his own method of philosophising"; and because
" the study of the categories . . . accustoms men to satisfy
" themselves with words, and to imagine that they know all
" things when they know only arbitrary names, which form
" in the mind no clear and distinct idea of the things '■
(Eng. Trans., pp. 40-1).
* (ii.) KantS criticism was founded on a misconception
of Aristotle's design in drawing out the Categories. Look-
ing upon it as the same as his own — that is, to enumerate
the pure or ci priori forms of the understanding — he objects
to the scheme as founded on no principle ; as containing forms
of sensibility {Quaiido, Ubi, Situs) as well as of the under-
standing, and thus confounding empirical notions with pure
notions ; as classing together deduced concepts {Actio, Passio)
with original ; and as omitting altogether some original
elements.
* (iii.) Lotze in his Metaphysic (Eng. Trans., vol. i., pp.
24-5) says : " In the sense which Aristotle himself attached
" to his Categories, as a collection of the most universal pre-
" dicates, under which every term that we can employ of
*' intelligible import may be subsumed, they have never ad-
" mitted of serious philosophical application. At most they
" have served to recall the points of view from which ques-
" tions may be put in regard to the objects of enquiry that
" present themselves. The answers to those questions always
"lay elsewhere — not in conceptions at all, but in funda-
" mental judgments directing the application of the coucep-
'■ tion in this way or that. . . . Aristotle may have had the
" most admirable principles of division ; but they do not
" prove that he has noticed all the members which properly
" fall under them."
(iv.) J. S. Mill — assuming the Categories to have been in-
tended as " an enumeration of all things capable of being
" named ; an enumeration by the summa genera, i.e., the
THE CATEGORIES OR PREDICAMENTS.
95
" most extensive classes into which things could be disti-i-
" buted ; which, therefore, were so many highest Predicates,
" one or other of which was supposed capable of being
"affirmed with truth of every nameable thing whatsoever"
{Logic, Bk. I., ch. iii., § 1)— objects to the list as unphilo-
sophical and redundant and defective. He says : " It is a
" mere catalogue of the distinctions rudely marked out by
" the language of familiar life, with little or no attempt to
" penetrate, by philosophic analysis, to the rationale even of
" those common distinctions. Such an analysis, however
" superficially conducted, would have shown the enumeration
"to be both redundant and defective. Some objects are
" omitted and others repeated several times under different
" heads. It is like a division of animals into men, quadru-
"peds, horses, asses, and ponies " {ibid.). He goes on to say
that Action, Passion and Local Situation {Situs) ought to be
included under Relation, together with position in time
{Quando) and in space (Ubi), and he regards the distinction
between Ubi and Situs as merely verbal. On the other
hand all states of mind are omitted entirely as " they cannot
" be reckoned either among substances or attributes " {ibid.).
To Prof. Bain's objection to this criticism that the Cate-
gories were not intended as an enumeration of things
" capable of being made predicates, or of having anything
" predicated of them " but " as a generalization of predicates,"
Mill replies, " In Aristotle's conception . , . the Categories
" may not have been a classification of Things ; but they
" were soon converted into one by his scholastic followers"
{ibid., note).
44. Answers to Objections.
To some of these objections more or less valid answers
have been made.
* (i.) In reply to the ' Port Eoyal ' criticism Prof. Baynes
says the criticism of the writers " proceeds wholly on mis-
" apprehension ; for the categories, instead of being arbitrary
" names, or imaginary attributes, are all affections of real
"being." He points out "that they are of metaphysical
Book I.
Cli. IV.
Mill ob-
jected to the
scheme as
unphiloso-
phical, aud
both re-
dundant
and defec-
tive.
Prof. Baynes
denies that
the Catego-
ries are ar-
bitrary, and
regards
them as of
Jtetaphy-
sical import.
%
TERMS.
Book I.
Ch. IV.
Mansel says
they were
not in-
tended as a
cla>sifica-
tiou of d
priori forms
of thought.
It is urged
that they
are useful as
an aid to the
exiiiiiina-
t i o u of
nature.
Mansel re-
g.uds them
as founded
on Gramma-
tical con-
siderations.
" rather than logical concernment," and acknowledges that
" these heads, though full in their enumeration, are not, as
" given by Aristotle, co-ordinate among themselves, and, as
*' a consequence, their arrangement is unsymmetrical " {Port
Royal Lofj., Eng. Trans., p. 384).
* (ii.) Mansel points out that Kant is "mistaken in sup-
" posing that Aristotle had any intention of classifying the
" pure forms of the understanding, independent of experi-
" ence. On the contrary, the Categories belong to the matter
" of thought, are generalized from experience, and leave alto-
" gether untouched the psychological question of the existence
" of elements a priori. Any objection, therefore, based on
" the inclusion of empirical or the exclusion of original
" elements, is untenable, and rests on a misapi^rehension of
"the philosopher's design " (Mansel's Ed. of Aldrich, Artis
Logicce Rudimenta, 3rd ed., p. 176).
* (iii.) It is argued that it is no more an objection to say
the number of the Categories is arbitrary than it would be to
bring against a methodical arrangement of books in a library
the charge that for such an arrangement another might be
substituted. No scheme of Categories can, by itself, enable
men to understand the nature of things, and Aristotle's, like
any other orderly classification, is not only not useless, but is
often of much use as an aid to a due examination of nature.
Any orderly arrangement of the innumerable subjects of
thought is better than none, though it must be allowed that
Aristotle's scheme needs re- arrangement, and the last nine
Categories should be classed under the one head ' Attributes '
{see Walker's Commentary on j\lurray^s Logic, pp. 31-2).
(iv.) Mansel replies to Mill's criticism that Aristotle did
not design " a classification of all things capable of being
" named ; at least not in that point of view in which things
" are regarded according to their real characteristics as pre-
" sented to consciousness. The Categories are rather an
" enumeration of the different modes of naming things,
" classified primarily according to the grammatical distiuc-
" tions of speech, and gained, not from the observation of
" objects, but from the analysis of assertions. . . . The
THE CATEGORIES OR PREDICAMENTS. 97
" proposition, as the only assertion capable of truth and Book I.
" falsehood, appears to be regarded as the unit of speech, of Ch^v.
" which the simple term is but a fractional element. It is
" therefore probable that the Aristotelian distinction of
" Categories arose from the resolution of the proposition
"and a classification of the grammatical distinctions indi-
" cated by its parts. . . . The omission, therefore, in the
" Aristotelian list, of separate heads of classification for
" mental states, cannot be charged as a defect in this point
" of view, so long as mind and its various states (whatever
" may be their difference in other respects) are represented
" by the same verbal forms as substances and attributes.
" And accordingly we find various mental states . . . classi-
" fied together with corresponding affections of body, under
" the head of qualities. . . . We might fairly describe the
" Aristotelian Categories as an enumeration of the different
" grammatical forms of the possible predicates of a proposi-
" tion, viewed in relation to the first substance as a subject.
" . , . The Categories are enumerated, not as an exhaustive
" catalogue of existing things, but as a list of the different
" modes of predicating by the copula. They thus originally
" belong to Grammar, rather than to Logic or Metaphysics,
" though the treatment of later philosophers, perhaps in some
" degree sanctioned by Aristotle himself, has brought them
" into closer connection with the latter sciences, and over-
" looked their proper relation to the former " (Hansel's
Aldrich Art. Log. Rud., 3rd ed., pp. 175-8).
It may be doubted, however, whether the origin of the
Categories was an examination of the parts of speech, for
that division of words was by no means sufficiently developed
in Aristotle's time to favour this idea. Aristotle has dis-
tinguished not so much parts of speech as parts of the
sentence (subject, predicate and different forms of the
predicate).
Prof. Bain says in reply to Mill : The Categories "seem Bain says
" to have been rather intended as a generalization of predi- analysis o"
" cates, an analysis of the final import of predication, iuclud- *ortf°/'\!™'
" ing Verbal as well as Real predication. Viewed in this dicatiou.
LOG. I. 7
98
TERMS.
Book I.
Ch. IV.
Grote de-
nied that
Situs and
Ubi are
identical ;
and would
class mental
states under
Qualitas and
T'atsio.
light, they are not open to the objections offered by Mr.
Mill. The proper question to ask is not^ — In what Cate-
gory are we to place sensations, or any other feelings or
states of mind, but — Under what Categories can we predi-
cate regarding states of mind ? . . . Aristotle seems to
have framed the Categories on the plan — Here is an indi-
vidual : what is the final analysis of all that we can predi-
cate about him?" {Ded. Log., p. 265). However he grants
that they are not adapted to any logical purpose ; they can-
not be made the basis of logical departments " {ibid., p. 266).
Grote, in commenting on Mill's criticism, says : " Among
the many deficiencies of the Aristotelian Categories, as a
complete catalogue, there is none more glaring than the
imperfect conception of irpoc n (the Relative), which Mr.
Mill here points out. But the Category KsiaQai (badly
translated by commentators Situs, from which Aristotle
expressly distinguishes it, . . .) appears to be hardly open
to Mr. Mill's remark, that it is only verbally distinguished
from TTOv, Ubi, KeTaQaiis iniendieA to mean posture, attitude,
etc. It is a reply to the question. In what posture is
Sokrates ? Answer — He is lying down, standing upright,
kneeling, etc. This is quite different from the question,
Where is Sokrates ? In the market-place, in the palaestra,
etc, KeToOm (as Aristotle himself admits . . .) is not easily
distinguished from irpoc n. . . . But Keiadai is clearly
distinguishable from irov, Ubi.
" Again, to Mr. Mill's question : ' In what Category are
' we to place sensations or other states of mind — hope, fear,
' sound, smell, pain, pleasure, thought, judgment,' etc. ?
Aristotle would have replied (I apprehend) that they come
under the Category either of Quale or of Pati — voiSrrjTEg
or TrdOri. They are attributes or modifications of Man,
Kallias, Sokrates, etc. If the condition of which we speak
be temporary or transitory, it is a -jraOoQ, and we speak of
Kallias as vdaxo'v n ; if it be a durable disposition or
capacity, likely to pass into repeated manifestations, it is
iroioTTjg, and we describe Kallias as ttowq tiq. . . . This
equally applies to mental and bodily conditions, . . . The
THE CATEGORIES OR PREDICAMENTS.
99
line is dubious and difficult between TrdOoQ and Troion/f, but
one or other of the two will comprehend all the mental
states indicated by Mr, Mill. Aristotle would not have
admitted that ' feelings are to be counted among realities '
except as they are now or may be the feelings of Kallias,
Sokrates, or some other Hie Aliquis — one or many. Ho
would consider feelings as attributes belonging to these
irpwrai Ovaiai [First Essences or Individuals] ; and so in
fact Mr. Mill himself considers them after having specified
the Mind (distinguished from Body or external object) as
the Substance to which they belong. . . . We cannot say,
I think, that Aristotle, in the Categories assigns no room
for the mental states or elements. He has a place for
them, though he treats them altogether objectively. He
takes account of Az'mseZ/" only as an object — as one among
the -n-pwrai ovffiai, or individuals, along with Sokrates and
Kallias " (Grote's Aristotle, pp. 90-1, note).
Book I.
Ch. rv.
* 45. Hamilton's Arrangement of Aristotle's Categories.
Hamilton largely meets the objections against Aristotle's ^^'"^^*^'lj
scheme, as wanting in arrangement, by casting it into a form Categories
of successive grades of subordination (Hamilton's Ed. of sutortoa"
Reid, p. 687, note). The arrangement is : —
'Per se, i.e., (1) Substance.
tion.
Being ,
(ejts). ^
fAbsolutejeither/^^^"^'"'.^-^-' \l\ Q^^^}}^y-
' \^Form, z.e., (3) Quality.
V
Per accidens -{
i.e., mode of
Substance
or Attribute.
Relative,
i.e., (4) Relation.
(5) Action.
(7) Where?
(8) When ?
(9) Posture.
JIO) Habit.
(6) Passion.
Of course, this arrangement does away with their claim to
be summa genera, for as Nos. 5-10 are species of 4, and as 2,
3, 4, are species of Attribute, whilst Substance and Attribute
are themselves species of Being, we are reduced to one
Summum Genus to which all the others are subaltern.
7-2
100
TERMS.
Book I.
Ch. IV.
Other
schemes of
Categories
have becu
given by
various
philo.sniihera.
46. Other Schemes of Categories.
Other schemes of Categories have been put forward by various
Philosophers : —
(i. ) The Stoics reduced the ten Aristotelian Categories to four,
which they called ' The Most Universal Kinds,' and be-
lieved to be forms of objective reality : —
1. TO vTzoKiijiBvov ... ... The Substrate.
2. TToiuv ... ... ... (Essential) Property.
3. Trojf txov ... ... (Unessential) Quality.
4. TTjoof -I TTtif txov ... ... Relation.
They subordinated all these Categories to the most universal of all
notions — that of ov or being ; and regarded the order as necessary,
each being subordinate to those before it.
(ii.) Others arranged the Categories in seven classes : —
1. Mens — mind, or the substance which thinks.
2. Materia — body, or substance extended.
3. Mensura — greatness or smalluess of each part of matter.
4. Positura — their situation in relation to each other.
5. Figura — their figure.
6. Motus — their motion.
7. Quies — their rest or lesser motion.
(iii.) Descartes and Spinoza give Substantia, Attrihutum,
Modus ; and Locke suggests Substance, Mode, and Re-
lation. Both these schemes are related to the Stoic
doctiine of Categories.
(iv.) Archbishop Thomson gives {Laws of Thought, p. 276) :—
/Substance.
Conceivable
things are
Quantity.
VAttribute. -/Quality,
^Relation.
rof Time.
, Space.
, Causation.
, Composition.
, Agreement and Re-
pugnance.
, Polar Opposition.
, Finite to Infinite.
THE CATEGORIES OR PREDTCAMENT3,
101
47. Mill's Scheme of Categories.
Mill, after criticizing Aristotle's list, gives the following
preliminary list of Categories or enumeration of the classes
of nameable things : —
I, Feelings, or states of Consciousness or mind —
sensations, emotions, ideas, voli-
tions.
'Bodies - occupying space — the unknown
external cause to which we
ascribe our sensations.
Minds - the unknown internal subject of
all feelings.
(Qualities.
Quantities,
Relations.
IV, Certain Relations of our Feelings — co-existences,
sequences, similarities and dis-
similarities.
II. Substances-
Book T.
Ch. IV.
Mill starts
with the
preliminary
list:
1. Feeling
2. Sub-
stance.
3. Attri-
butes.
Certain
ReUi-
tions of
Feel-
ings.
4.
He then argues that " For logical purposes the sensation is He th en
■•' ..... _ argues that
Attributes
are reducible
to Feelings,
the only essential part of what is meant by the word
[quality] ; the only part which we ever can be concerned
in proving. When that is proved, the quality is proved ;
if an object excites a sensation, it has, of course, the power
of exciting it" {Logic, Bk. I., ch. iii., § 9). Hence " all the
attributes of bodies which are classed under Quality or
Quantity, are grounded on the sensations which we receive
from those bodies, and may be defined, the powers which
the bodies have of exciting those sensations. And the
same general explanation has been found to apply to most of
the attributes usually classed under the head of Relation "
hid.^ § 13). The exceptions are the relations " of succession
and simultaneity, of likeness and unlikeness. These, not
being grounded on any fact or phenomenon distinct from
the related objects themselves, do not admit of the same
kind of analysis. But these relations, though not, like
other relations, grounded on states of consciousness, are
themselves states of consciousness : ^-esemblance is nothing
102
TERMS,
Book I.
Ch. IV.
and finally
gives :
1. Feelings.
2. Minds.
3. Bodies,
including
Attri-
butes.
4. Succes-
sions, Co-
exist-
ences.
Likeness
and Un-
likeness.
" but our feeling of resemblance ; succession is nothing but
"our feeling of succession" (ibul.). "The attributes of
" minds, as well as those of bodies, are grounded on states of
"feeling or consciousness. . . . Every attribute of a mind
" consists either in being itself affected in a certain way, or
" affecting other minds in a certain way. Considered in
" itself, we can predicate nothing of it but the series of its
" own feelings. ... In addition, however, to those attributes
" of a mind which are grounded on its own states of feeling,
*' attributes may also be ascribed to it, in the same manner
" as to a body, grounded on the feelings which it excites in
"other minds" (ibid., § 14). "All attributes, therefore, are
" to us nothing but either our sensations and other states of
" feeling, or something inextricably involved therein ; and to
" this even the peculiar and simple relations just adverted to
" are not exceptions. Those peculiar relations, however,
" are so important, and, even if they might in strictness be
" classed among states of consciousness, are so fundamentally
"distinct from any other of those states, that it would be a
" vain subtlety to bring them under that common description,
" and it is necessary that they should be classed apart " {ibid.^
§ 15). As the final result, therefore, of his analysis he gives
the following four-fold scheme (ibid.) : —
Feelings, or States of Consciousness.
The Minds which experience those feelings.
" III. The Bodies or external objects which excite certain
"of those feelings, together with the powers or
" properties whereby they excite them ; these
*' latter (at least) being included rather in com-
" pliance with common opinion, and because their
" existence is taken for granted in the common
" language from which I cannot prudently devi-
" ate, than because the recognition of such powers
" or properties as real existences appears to be
" warranted by a sound philosophy.
" IV". The Successio7is and Co-existences, the Likenesses and
" Unlikenesses, between feelings or states of con-
" sciousness."
"I.
"II.
THE CATEGORIES OR PREDICAMENTS.
103
This whole analysis is grounded on Mill's metaphysical
position that external objects are nothing but ' Permanent
Possibilities of Sensations ' and Mind merely a ' Perma-
nent Possibility of Feeling' (see Exam, of Hamilton,
ch. xi., xii.), and cannot be accepted by any who reject that
ultimate position. To one who believes that things really
exist in themselves the resolution of Attributes and Rela-
tions into Feelings becomes impossible. Even if Mill's view
be accepted this scheme is not satisfactory. The whole
argument aims at reducing attributes to Feelings, yet they
are fiaally included under Bodies under a plea which, cer-
tainly, has no philosophical weight. Nor is his reason for
making Successions, etc., a separate class valid after he has
once reduced them to Feelings. Finally, if Bodies and
Minds be nothing more than Mill thinks them, it is difficult
to see why they are not also included under Feelings. Thus,
were he consistent. Mill would have included all existence
under the one Category ' Feeling ' ; for that is, necessarily,
his only Summum Genua.
Book I.
Ch. IV.
Tbe natural
result of
jM ill's argu-
ment is to re-
duce every-
thing to the
one Cate-
gory ' Feel-
ing.'
48. Kant's Scheme of Categories.
Kant, in his scheme of Categories, designed, not to classify things
but, to enumerate the true root-notions of pure understanding, or
d priori forms of thought, which are immanent in the intellect and
essential to the interpretation of every impression received from
without. He endeavoured to attain a complete system of them by
an analysis of the faculty of thought as expressed in the logical
judgment, believing that the primitive notions of the understanding
could be completely ascertained by a thorough examination of all
the kinds of judgment. Though, then, we treat of this scheme here
to render our review of the Categories complete, the discussion
necessarily requires an acquaintance with the Book on Propositions,
and its study should be postponed till that book has been read.
By an analysis of tbe forms of Judgment, Kant arrived at a
division into four species, each with three sub-classes, and from
each form of Judgment he deduced a Category. The whole scheme
is as follows : —
Kant's Cate-
gories are
intended to
be an
enumera-
tion of the d
priori fortns
of thought.
He derived
them from
the forms of
logical judg-
ment.
104
TERMS.
Book I.
Forms of J\
udgment.
Categories.
Ch. IV.
I. Quantity.
h
Quantity.
(i.) Singular
. This Sis P.
(i.) Unity.
(ii.) Particular
. Some S is P.
(ii.) Plurality.
(iii.) Universal
. All S is P.
(iii.) Totality.
II. Quality.
II.
Quality.
(i.) Affirmative ...
. S is P.
(i.) Reality.
(ii.) Negative
. S is not P.
(ii.) Negation.
(iii.) Infinite
. S is non-P.
(iii.) Limitation.
iir. Relation.
m.
. Relation,
(L) Categorical ...
. Sis P.
( i . ) Substantiality —
Substance and
Attribute.
(ii.) Hypothetical ...
. If A is B, S is
P.
(ii.) Causality —
Cause and Ef-
fect.
(iii.) Disjunctive ..
. S is either P or Q.
(iii.) Reciprocity.
i\. Modality.
IV
. Modality.
(i.) Problematic ...
. S may he P,
(i.) Possibility and
Impossibility.
(ii.) Assertory
SiaP.
(ii.) Existence and
Non-existence.
(iii.) Apodeicticor
(iii.) Necessity and
Necessary ...
S must be P.
Contingency.
These Categories, or pure notions, are the d. priori possession of
the intellect, i.e., they are relations under one or other of which all
objects of sense are presented, and are, therefore, necessarily and
universally valid. Moreover, as all our perceptions are sensuous,
these Categories are only valid when applied to sensuous impressions,
which are thus raised into experience. As the Categories are valid
for the whole sphere of perception, they enable us to unite it into
a connected whole, and thus to obtain a coherent experience of the
external world ; which would be impossible if sensations were not
acted upon and synthesized by the understanding. As Mansel
says : " Every complete act of consciousness is a compound of in-
" tuition and thought ; and the portion which is due to the act of
"thought as such . . . will be the form of the representative con-
^' sciousness. Now, by the act of thought, the confused materials
THE CATEGORIES OR PREDICAMENTS.
105
•' presented to the intuitive faculties are contemplated in three
' ' points of view : as a single object, as distinguished from other
" objects, and as forming, in conjunction with those others, a com-
" plete class or universe of all that is conceivable. We have thus
" the three forma (or, as they are called by Kant, categories) of
^' unify, plural{(2/ Sind totality; conditions essential to the possi-
' ' bility of thought in general, and which may, therefore, be re-
' ' garded as d priori elements of reflective consciousness, derived
" from the constitution of the understanding itself, and manifested
" in relation to all its products. They are thus distinguished from
' ' the matter, or empirical contents, by which one object of thought
" is distinguished from another. The Matter of thought is derived
*' from the intuitive faculties, and consists in the several presented
'^phenomena which form the special characteristics of each object "
(Metaphysics, pp. 192-3),
It has been objected that these Categories were derived from the
forms of logical judgment, which should be their applied form,
according to Kant's doctrine, and that this derivation is often
forced and arbitrary. Some of the distinctions, too, in the original
table of judgments are of doubtful value, if not altogether false —
such are the distinctions between the Negative and the Infinite
Judgments (see § 70), and that between the Assertory and the
Apodeictic, which will be further considered in the section on the
Modality of Propositions (see § 82). To again quote Mansel :
" Besides these three [Unity, Plurality, Totality] which are classi-
"fied as categories of quantity, Kant enumerates nine others — viz.,
" three of quality, — reality, negation, and limitation ; three of rela-
" tion, — inherence and subsistence, causality and independence,
" and community or reciijrocal action ; and three of modality, —
"possibility or impossibility, existence or non-existence, and neces-
" sity or contingence. But the Kantian categories are not deduced
" from an analysis of the act of thought, but generalized from the
" forms of the proposition, which latter are assumed without ex-
" amination, as they are given in the ordinary logic. A psycho-
" logical deduction, or a preliminary criticism of the logical forms
" themselves, might have considerably reduced the number. Thus
"the categories of quality are fundamentally identical with those
" of quantity ; — reality, or rather affirmation and negation, being
" implied in identity and diversity, and limitation in their mutual
" exclusion. The remaining categories are, to say the least, founded
" on a very questionable theory in logic ; and the two most import-
Book 1.
Ch. IV.
Some of the
distinctions
in the forms
of judgment
on which
the scheme
is founded
are, at least,
of doubtful
value ;
heuce, the
number of
Categories
could be re-
duced.
106
TERMS.
Book I.
Ch. IV.
Lotze ob-
jects to the
scheme that
it is empi-
ri>^al.
" ant — those of substance and cause — present features which die-
" tinguish them from mere forms of thought " {ibid., p. 193).
Lotze also objects to Kant's scheme, on the ground that it is just
as empirical and as wanting in "a principle to warrant their com-
"pleteness " as Aristotle's. He says " It may be conceded to him
" [Kant] that it is only in the form of the judgment that the acts
" of thought are performed by means of which we aflSrm anything
" of the real. If it is admitted further as a consequence of this
" that there will be as many different primary propositions of this
" kind as there are essentially different logical forms of judgment,
" still the admission that these different forms of judgment have
" been exhaustively discovered cannot be insisted on as a matter,
" properly speaking, of methodological necessity. The admission
"will be made as soon as we feel ourselves satisfied and have nothing
'' to add to the classification ; and if this agreement were universal,
" the matter would be practically settled, for every inventory must
" be taken as complete, if those who are interested in its complete-
" ness can iind nothing new to add to it. But that kind of
" theoretical security for an unconditional completeness, which
" Kant was in quest of, is something intrinsically impossible, "
[Metaph., Eng. trans., vol. L, p. 25.)
CHAPTER V.
DEFINITIONS OP TERMS.
49. Functions of Definition.
Definition is the explicit statement of the Connotation
of a term, i.e., of all the attributes, and of those only, which
are recognized by common agreement of competent thinkers
as implied by the name (see § 28 (ii)). Every definition is,
therefore, an analytic proposition (see § 40), or, rather, a
series of analytic propositions, as a new proposition is required
to state each separate attribute, and in every one of these pro-
positions the predicate simply states in so many words what was
already implicitly contained in the subject. A complete defini-
tion will exhaust the total number of analytic propositions
that can be made with the defined term as subject, for it will
state its whole connotation. Moreover, each one of these
propositions must be universal, i.e., the predication must be
made of every one of the things denoted by the subject term,
and the proposition must be of the form Every S is P; for the
definition must necessarily be applicable to each object which
bears the class name.
Were the necessary logical assumption — that all words have
exactly the same distinct meaning for all who use them —
universally true, Definition would be unnecessary. But our
ideas are often clear without being distinct or adequate ; that
is, we can apply a name accurately enough to the things de-
noted by it without having distinctly present to our minds
all the attributes on account of which it is bestowed upon
them. The use of definition is to give distinctness to these
clear ideas and to make them adequate — to enable us not
only to use the name accurately as regards its denotation,
Book I.
Ch. V.
Definition is
the explicit
statement of
the connota-
tion of a
term
A definition
sums up all
the analytic
predications
which can
be made of
the term de-
fined.
Definitions
make our
ideas dis-
tinct and
adequate.
108
TKRM9.
Book I.
Ch. V.
The defini-
tion com-
pletes the
process of
conception.
hut to employ it with an intelligent apprehension o£ its
exact implication. It is evident, then, that to form a
good definition is a work of no small difficulty, and one
calling for no small sagacity. It involves careful observa-
tion, comparison and analysis of the things observed, abstrac-
tion of the mind from their differences, and generalization,
besides the power of distinguishing primary from derivative
qualities. In short, the definition is the perfecting and com-
pletion of the process of conception [see §§ 2 (ii.) ; 8 (i.)]-
Moreover, the preliminary process of seeArm^ for a definition is
often more important than the finding it. " What we gain by
" discussing a definition is often but slightly represented in
" the superior fitness of the formula that we ultimately
" adopt ; it consists chiefly in the greater clearness and f ul-
" ness in which the characteristics of the matter to which
" the formula refers have been brought before the mind in
" the process of seeking for it. "While we are apparently
" aiming at definitions of terms, our attention should be
" really fixed on distinctions and relations of fact. These
*' latter are what we are concerned to know, contemplate,
" and as far as possible arrange and systematize. . . . And
" this reflective contemplation is naturally stimulated by the
" effort to define ; but when the process has been fully per-
" formed, when the distinctions and relations of fact have
" been clearly apprehended, the final question as to the mode
*' in which they should be represented in a definition is reaUy
*' — what the whole discussion appears to superficial readers —
" a question about words alone " (Prof. H. Sidgwick, Prin-
ciples of Political Economy, pp. 49-60).
Definition is thus essentially practical, and is, therefore, a
part of Applied Logic ; we only need define a term when we
require to use the definition as an aid to the expression of
some truth.
50- Definition per Genus et Differentiam.
In unfolding the complete connotation of a name it is
often practically impossible to express it in terms which de-
p.ote simple attributes only ; and, in nearly every case, to do
DEFINITIONS OF TEEMS.
109
so would make the definition needlessly long and involved. It
is, in all cases, allowable to employ terms expressive of groups
of attributes. Hence we have the time-honoured rule that
definition should be per genus et differentiam. In mentioning
the genus we use a term which implies all the attributes
common to the species whose name is the term to be defined
and to all other co-ordinate species of that genus ; and, by
adding the differentia, we complete the statement of the
connotation by giving those attributes which differentiate
that species from all such co-ordinate species. In other
words, when we have to define a term, we first decide what
class of things it belongs to, and then we mark the attribute, or
group of attributes, which distinguishes it from other members
of that class. The name of the class is the Genus, the dis-
tinguishing attribute, or group of attributes, forms the Dif-
ferentia. The genus selected must be a proximate genus (see
§ 35), as, otherwise, our definition will omit part of the conno-
tation of the term we are defining. If, for instance, we defined
' man ' as ' rational being ' we should omit the attributes
connoted by the word ' corporeal,* and our definition would
allow the name to be applied to other possible beings. Or,
symbolically, if we define a class term whose connotation is
ahcd by referring it to the genus a (instead of to the proxi-
mate genus, abc), and adding the differentia d, we plainly
omit the attributes he from our definition.
Book I.
Ch. V.
It is not
necessary to
state the
Connotation
in simple at-
tributes. It
is enough to
give genua
and differ-
entia.
The genus
must be
proximate.
Which of the attributes form the genus, and which the differentia,
must depend upon the classes with which we compare the term. The
definition of the same name may legitimately vary in mode of ex-
pression, though, when each term employed is fully analysed, all the
modes of expression will be seen to be really identical. For the
object defined is the same and its essential attributes are not
affected by the mode of abbreviating the definition. Thus ' Man '
may be defined as ' rational animal,' where the comparison has
been with other animals ; or, as ' embodied spirit ' where the com-
parison has been with other rational creatures which are not cor-
poreal. But, if the word ' animal,' ' rational,' ' embodied,' * spirit,'
are each expressed in terms denoting simple and distinct attributes,
the definitions will be seen to be identical. This may be shown in
The defini-
tion of a
term may be
variously
expressed,
but these
expressions
must be, at
bottom,
IdeuticaL
110
TERMS.
Book I.
Ch. V.
symbols. Let the essential attributes of P be abed, and X, Y, Z, be
genera under either of which it may be classed. Let the connotation
of X be abc ; of Y be bed ; of Z be acd. Then P can be defined as
Xd, or as Ya, or as Zb. But, if we analyse all these definitions into
their simplest elements, we get, in all cases, that the definition of P
is abed. What, then, in a definition, is regarded as the genus and
what as the differentia depends upon that process of comparison
which, as was pointed out in the last section, is a necessary pre-
liminary to definition.
A definition per genus et differentiam assumes that the meaning
of the name of the genus is known ; but such an assumption is
necessary to the science of Logic, which must regard the require-
ment of a definition of any particular term as an exception to the
general rule that men are acquainted with the meaning of every
term they use.
Definitionis It must be remembered that when defimtion per genus et
not pergemis ^[ff^rentiam is spoken of, it is not meant to imply that the
etdijfferentias m i .... i; j.j. •
hntpfr genus differentia is a single attribute ; it may be a group or attri-
etdifferen- , , , oc n ao\ tt-^^u „„„„;„„ ^„^ Uo^,^ K„+ ^^«
tiam.
Logic re-
gards the de-
mand for a
definition as
exceptional.
butes
{see §5
5 41,
have but
, 43). Each species can have but one
differentia— i.e., one set of attributes to distinguish it from
the co-ordinate species — when referred to any one particular
genus. Hence it is inaccurate to speak, as Mill suggests,
(Logic, Bk. I., ch, viii., § 3) of definition ^jcr genus et
differentias.
51. Limits of Definition.
Only Proper As Definition is the unfolding of the meaning implied by
names oT'^ a name it follows that every significant name can be defined,
simple attri- ^nd that the only terms incapable of definition are Proper
uud^finabie. Names which have no signification [see § 27 (i.) (a)] and
singular Abstract Terms which are the names of simple
attributes which as forming the ultimate limit of our analysis
cannot be expressed in terms more elementary than them-
selves [see § 30 (iii-)]- ^o words can enable one who has
never experienced pain or whiteness to conceive what either is.
In such cases the utmost we can do is to clearly mark out the
notion from others by a process of abstraction and isolation,
and to indicate it by some accidens or accompanying mark.
DEFINITIONS OF TERMS.
Ill
The Scholastic logicians, insisting upon the necessity for all
valid definition to be per genus et differentiam, and holding
that summa genera and wfimce species were absolutely fixed,
denied that the names of either individuals or summa genera
were definable ; because the former had no differentia but
only accidentia ; and because there was no higher genus under
which the latter could be subsumed. Each of the ten Cate-
gories of Aristotle they, therefore, regarded as incapable of
definition, as well as all sub-classes included in infimoB species.
Thus, ' negro ' would be undefinable, as it is a sub-class of
the AnfimcB species, man. But the modern view does not
impose this restriction ; it regards 'A negro is a black man*
as being, practically, a definition per genus et differentiam, the
rigidity of the notion of infimcs species being relaxed. All
Significant Individual Names are also held capable of defini-
tion, as their peculiar attributes can be specified in addition
to those connoted by the name of the class of which they are
members [see §§ 27 (i.) (6), 28 (i.)].
Some terms are manifestly much more easily defined than
others. Those in which the connotation is the more import-
ant element — such as technical terms, whose sole value lies
in an exact meaning — are much more easily defined than
those in which the denotative element predominates. Ex-
amples of the latter are the names of most common objects
— e.g., chair, horse, dog — where we learn to apply the names
without any distinct idea of the attributes those names con-
note [cf. § 2 (iii.)] . That this is the case anyone may easily
discover who contrasts the ease with which the connotation
of such a term as 'rectangle,' for example, can be stated,
with the difiiculty of writing down the essential attributes
of 'dog.'
* But, though all significant names can be defined with
more or less difiiculty, a great number of such definitions
can only be regarded as provisional. Fresh advances in
knowledge may alter our estimate of the relative importance
of attributes, may change propria to differentiae, or differentias
to propria, and so may revolutionize the connotation of the
term, and thus necessitate a revised definition. In fact, with
Book I.
Ch. V.
Terms
whose Con-
notation is
their more
important
element are
more easily-
defined than
those whose
denotation
is predomi-
nant.
Xfost defini-
tions, espe-
cially those
of scientific
terms, are
provisional,
and subject
to modifica-
tion with
advance of
knowledge.
112
TERMS.
Book I.
Ch. V.
Deflnitious
are also mo-
dified by a
change in
the point of
view from
which a,
term is re-
garded.
The bound-
ary line
marked by a
definition is
necessiirily
vague, but
tliisdoesnot
destroy its
value.
scientific terms the growth of knowledge must cause constant
modifications of the definitions ; were they fixed in any science,
that science would cease to advance. Discovery and definition
must go hand in hand, and finality in the latter is not to be
looked for ; it could only be possible with complete and per-
fect knowledge. So emphatically is this the case that it has
been well said : " The business of Definition is part of the
" business of discover}'. When it has been clearly seen what
" ought to be our Definition, it must be pretty well known
" what truth we have to state. The Definition, as well as the
" discovery, supposes a decided step in our knowledge to have
*' been made. ... If the Explication of our Conceptions ever
" assume the form of a Definition, this will come to pass, not
" as an arbitrary process, or as a matter of coarse, but as the
•' mark of one of those happy efforts of sagacity to which
"all the successive advances of our knowledge are owing"
(Whewell, Novum Organon Renovatum, pp. 39-40).
Not) only the growth of knowledge, but a change in the point of
view from which a term is regarded may cause a change in the
accepted connotation. Examples of this are most common in
mathematics. Thus, an ellipse was originally defined as a conic
section with the differentia that the cut goes quite across the cone,
not at right angles to the axis. But in modern works it is defined
as the line traced out by a point moving so that its distance from a
fixed line bears always a certain ratio to its distance from a certain
fixed point. Then the fact that such a curve is a conic section is
deduced by a long and intricate argument ; is, in fact, degraded
from forming part of the connotation to the position of a proprium.*
Again, changes in the denotation of a term caused by its applica-
tion to new classes of objects because of a real or fancied resemblance
to the things of which it is originally the name (see § 3) results in a
certain vagueness of connotation, which, of course, reacts on the
denotation, and gives rise to an indefinite zone of the contents of
which it is difficult to say whether they have a right to the name or
not. Most common words will, if carefully examined, be seen to be
thus more or less vague as to the boundary line of both their conno-
tation and their denotation ; and especially is this the case with
• (y. Dr. Venn, Empirical Logic, pp. 2S4-6,
DEFINITIONS OP TERMS. 113
terms used in the sciences which deal with social phenomena, such Book I
as Political Economy. This vagueness of boundary does not, how- Ch^.
ever, destroy the value of the definition. On this point the late
Prof. Cairnes well remarked : " In controversies about definitions
" nothing is more common than to meet objections founded on the
" assumption that the attribute on which a definition turns ought to
" be one which does not admit of degrees. This being assumed, the
" objector goes on to show that the facts or objects placed within the
" boundary line of some definition to which exception is taken,
"cannot in their extreme instances be clearly discriminated from
" those which lie without. Some equivocal example is then taken,
" and the framer of the definition is challenged to say in which
"category it is to be placed. Now, it seems to me that an objec-
" tion of this kind ignores the inevitable conditions under which a
" scientific nomenclature is constructed alike in Political Economy
" and in all the positive sciences. In such sciences nomenclature,
"and therefore definition, is based upon classification, and to admit
" of degrees is the character of all natural facts. As has been said,
" there are no hard lines in nature. Between the animal and vege*
" table kingdoms, for example, where is the line to be drawn ? . . .
" I reply that I do not believe there is any absolute or certain dis-
" tinction whatever. External objects and events shade oflf into
"each other by imperceptible differences; and consequently defi-
' ' nitions whose aim it is to classify such objects and events must of
" necessity be founded on circumstances partaking of this character,
"The objection proceeds on the assumption that groups exist in
" nature as clearly discriminated from each other as are the mental
"ideas formulated by our definitions ; so that where a definition is
" sound the boundary of the definition will have its counterpart in
"external facts. But this is an illusion. No such clearly cut
" divisions exist in the actual universe. ... It is, therefore, no
" valid objection to a classification, nor, consequently, to the defini-
" tion founded upon it, that instances may be found which fall or seem
" to fall on our lines of demarcation. This is inevitable in the
" nature of things. But, this notwithstanding, the classification,
" and therefore the definition, is a good one, if in these instances
" which do not fall on the line, the distinctions marked by the defi-
' ' nition are such as it is important to mark, sucli that the recognition
"of them will help the inquirer forward towards the desiderated
"goal" (Logical Method of Political Economy, pp. 139-141).
LOG. I. 8
114
TERMS.
Book I.
Ch. V.
A Definition
must be
adequate,
precise, and
clear, and
neitbertaut-
ologous nor
negative.
* Further difficulty is sometimes due to the fact that a
name is frequently used in scientific language with a mean-
ing different from that which it bears in ordinary speech.
It is, really, a different term ; but the identity of the verbal
symbol causes the scientific meaning and the ordinary signi-
fication to be more or less confused.
52. Rules of Definition.
The following rules must be observed in framing a good
definition : —
I. It should contain neither more nor less than the con-
notation of the term defined.
II. It should he clearer than the term defined, and should
not, therefore, he expressed in unfamiliar, figura-
tive, or amhiguous, language.
III. It should not consist of a term synonymous with that
defined.
IV. It should never be negative when it can be affirma-
tive.
Or, to sum the rules into one,
A Definition should be (i.) adequate, precise, and (ii.)
clear, and should not be (iii.) tautologous or
(iv.) negative.
It should be noticed that in Eule IV the term ' Definition '
applies to the whole proposition which states the meaning of
the term, but in the first three rules it denotes the predicate
only of that explanatory proposition. The term is, there-
fore, slightly ambiguous.
We will now discuss each rule in detail.
Toaddapro-
priimi or in-
separable
accirteiis to
a definition
siiggests the
existence of
objects
which pos-
sess all the
attributes
but those.
Rule I. If the definition embraces more than the connota-
tion of the term defined it must include either some of its
propria or some of its accidentia. Of course, in the cases
where either propria or inseparable accidentia are added to
the connotation, the denotation of the definition remains the
same as that of the name defined ; but the very fact of
adding these extra attributes would suggest that they were
necessary to the true definition ; and that, therefore, other
DEFINITIONS OF TERMS.
115
objects exist which possess all the attributes mentioned
except these very ones ; which is, in fact, not the case. If,
for example, an equilateral triangle were defined as ' a triangle
which has three equal sides and three equal angles,' this,
though perfectly true of all equilateral triangles and of no
other figures whatever, would yet be a faulty definition ; for
it suggests that there are triangles which may have three
equal sides and yet not have their angles equal. But if a
separable accidens is added to the connotation of a name as
part of its definition, a graver fault is committed. In this
case the definition will not refer to the whole denotation of
the name defined, for some only of the things which cor-
rectly bear the name possess the attribute in question. The
definition is said in this case to be too narrow. If, for
instance, a triangle were to be defined as ' a plane rectilinear
figure having three equal sides ' the definition would be too
narrow ; for it would apply only to a section of the figures
which are correctly called triangles, the attribute 'equal-sided
being the differentia which marks off the species ' equilateral
triangles ' from the other co-ordinate species which are in-
cluded in the genus ' triangle,' and, therefore, only a separ-
able accidens of that genus. Or, if a labourer were to be
defined as ' one who performs manual work for wages ' the
definition would again be too narrow, as, by the addition of
the separable accidens ' for wages,' it excludes all slaves from
the class labourers of which they indubitably form a part.
In all these cases the definition is redundant, and, therefore,
not sufiiciently precise.
If, on the other hand, the definition contains less than the
connotation of the name it is too wide, for evidently it will
be applicable to a greater number of things than are in-
cluded in the denotation of the term defined (c/. § 50). If,
for instance, an equilateral triangle were defined as 'a plane
rectilinear three-sided figure ' the definition would include
all triangles. In other words it would refer to the genus
instead of to the species only, and would be inadequate.
This, it may be noticed, is the most common fault of so-
called ' definitions.'
8-2
Book L
Ch. V.
To add a
separable
accidens to
a defiuitiou
makes it too
narrow, i.e.,
limits it to
a part only
of the deno-
tation of the
term.
If the defini-
tion con-
tains less
than the
connotation
it is too wide,
i.e., it ap-
plies to ob-
jects not in-
cluded in the
denotation
of the term.
lie
TERMS.
Book T.
Ch. V.
A Definition
should be
expressed in
plain, mi-
ambiguous
and non-
figurative
iMiguage.
The use of a
sjTionym
doos not give
the meaning
of a term.
In all cases, then, the denotation of the definition must be
exactly the same as that of the term defined, and this can
only be secured with certainty by its unfolding all the con-
notation of the term and embracing nothing else.
Rule II. The violation of the rule demanding clearness
in a definition is known as defining ignotum per ignotius or
per ccque ignotum — explaining the unknown by the more, or
equally, unknown. Dr. Johnson's definition of a net as
'a reticulated fabric, decussated at regular intervals ' is an
amusing instance of this. To say that 'Eccentricity is
peculiar idiosyncrasy ' or that ' Fluency is an exuberance of
verbosity ' is, in each case, to give a definition which is cer-
tainly not clearer than the term defined. The so-called
definitions which are expressed in figurative language are a
variety of this fault. To say ' The lion is the king of
beasts,' * Bread is the staff of life ' or ' Necessity is the
mother of invention ' gives no explanation of the meaning
of the terms ' defined.' This rule, however, is not violated
if a name is defined, for the purposes of a special science, in
terms which to one not a student of that science would be
less clear than the name itself ; as, for instance, if for the
purposes of Conic Sections a circle were defined as ' a section
of a cone parallel to the base.'
Rule III. The violation of the rule against tautology in
a definition is called circulus in definiendo, or 'a circle in de-
fining.' It is evidently no addition to our knowledge to
'explain' a term by itself or by a synonym {cf. § 3). To
say that ' Truth is veracity in speech and act' is simply to
affirm that ' Truth is truth,' and this, though perfectly
obvious, is also perfectly useless. The great number of
synonyms in English, due to the presence in the vocabulary
of words derived from both Teutonic and Latin sources,
offers many opportunities for committing this fault ; and,
it may be added, these oiiportunities are by no means spar-
ingly used. But it is by no means confined to English.
Ueberweg quotes the following example from the German
writer, Maass : " ' A feeling is pleasant when it is desired
DEFINITIONS OF TERMS.
117
" because of itself.' ' We desire only what we in some way
" represent to be good.' * The sensibility takes that to be
" good which warrants or promises pleasure, and affects us
" pleasantly ; — the desires rest on pleasant feelings.' The
" pleasant feeling is here explained by the desire, and the
" desire again by the pleasant feeling " (Logic, Eng. trans.,
p. 175). It is, in fact, in cases of long and involved defini-
tions, such as the above — where the three sentences are
taken from different parts of the book — that a ' circle ' is most
frequently found. It is, however, by no means uncommon
to meet with such * definitions ' as ' Life is the sum of vital
functions,' ' Force is a motive power,' ' Man is a human
being.' The definition once given by a Church dignitary
that ' An archdeacon is one who exercises archidiaconal
functions ' is a very neat example of circulus in clefiniendo.
There is no violation of this rule when the name of the
genus is repeated in defining a term which denotes a sub-
ordinate species which has no distinct name, but is specified
by the addition of some limiting attribute to the name of
the genus. There is no tautology, for example, in defining
an equilateral triangle as ' a triangle which has three equal
sides ' ; for the species ' equilateral triangle ' has no separate
name, and is distinguished from the species of the genus
triangle which are co-ordinate with it merely by the limiting
adjective 'equilateral.' Euclid, before giving this definition,
has, of course, defined the name of the genus, ' triangle,*
This word when it occurs in the definition of equilateral
triangle is simply the name of the genus, not that of the
thing defined at all ; and the definition is strictly one per
genus et differentiam.
From this third rule it follows that a term which is the
name of a simple and elementary attribute cannot be de-
fined, as it can only be explained by a synonym or by
itself ; for instance, ' White is that attribute of sensible
objects which occasions us to experience the sensation of
whiteness.' We can only, in truth, describe such terms by
analysing the conditions under which the sensations they
denote are produced (c/. § 51).
Book L
Ch. V.
Th e rule
ugainst tau-
tology is not
violated
when tbe
name of the
species con-
tains that of
the genus.
From this
rule it fol-
lows that
names of
simple attri-
butes cannot
be defined.
118
TERMS.
Book I.
Ch. V.
The name of
a negative
notion
should be
defined
negatively.
But a nega-
tive ' defini-
tion ' of the
name of a
positive
notion is too
indefinite
to give its
meaning.
Nominal and
Jieala.re\i8e(i
by modern
writers in a
sense differ-
ent from
that of the
older logi-
cians.
Rule IV. Negative definitions are always less satisfactory
than those expressed in positive terms, unless they are defi-
nitions of the names of negative notions, in which case they
are to be preferred. It is, for example, simplest to define
an alien as ' one who is not a citizen of the British Empire ';
for the name ' alien ' represents a notion whose sole differ-
entia is just this negative attribute. The definition of
Accidens is also, necessarily, of a negative character (see
§§ 34, 38). But, to ' define ' virtue as ' the opposite of vice,'
or liquid as ' that which is neither solid nor gaseous,' is not
to say in either case what attributes constitute the class
notion, and are, therefore, connoted by the name ; but to
state attributes which are not so connoted. This, besides
giving no positive information, is certain to lead to inde-
finiteness. For the number of attributes which a thing may,
conceivablj', possess is infinite, and to merely exclude a few
of these is by no means to give a clear indication as to how
many, and which, of the innumerable remaining ones must
be possessed by any individual thing to qualify it to receive
the name in question. Many of the objections to thorough-
going Negative (or infinite) Terms apply, in fact, to negative
definitions [see §§ 19, 29 (i.) (&)]. There is a breach of this
rule in Euclid's definition of parallel straight lines as ' those
which lie in the same plane, and which, being produced ever
80 far both ways, never meet.* Another example of the
same fault is when Euclid defines a point as ' that which has
no parts and which has no magnitude.' This rule is really
involved in Rule I, as the connotation of a positive term
cannot be expressed negatively.
53. Kinds of Definition.
From various points of view we get different divisions of
Definitions into classes.
(i.) Nominal and Real. The traditional division of
definitions was into Nominal and Real; and these terms
have been retained by many modern writers on Logic who
have, however, used them in a sense very different from that
in which they were used by the Scholastic writers on the
DEFINITIONS OF TERMS.
Ill)
science. With them a Nominal Definition was one which
unfolded the meaning of a word, and a Real Definition one
which explained the nature of a thing.
Hamilton defined the terms thus : " By Verbal Definition
" is meant the more accurate determination of the significa-
" tion of a word ; by Real, the more accurate determination
" of the contents of a notion. The one clears up the rela-
" tion of words to notions ; the other of notions to things "
(Ed. o/Reid, p. 691).
Ueberweg (Logic, Eng. trans., p. 164) gives the following
meaning to " Nominal and Real Definitions. The former de-
" fines what is to be understood by an expression. The
" Real Definition has to do with the internal possibility of
" the object denoted by the notion, and thus with the real
" validity of the notion ; for it either contains the proof of
" its real validity in the statement of the way in which the
" object originated, or was based upon such a proof." He
thus justifies this use of the term. " The terms Nominal
" and Real Definition are not thoroughly expressive ; for
" every definition defines not the name, nor the thing, but
" the notion, and with it the name and the thing as far as this
" is possible. But so long as the real validity of the defined
" notion is not warranted, it is always possible that a notion
" may have been defined which is only apparently valid, and
" is in truth only a mere name or a feigned notion cor-
" responding to nothing real. On the other hand, the defini-
" tion of an objectively-valid notion serves at the same time
" to give a knowledge of the thing denoted by the notion.
"Considered in this sense these terms justify themselves"
{ibid., p. 167).
Mill held that " all definitions are of names, and of names
" only ; but, in some definitions, it is clearly apparent, that
" nothing is intended except to explain the meaning of the
" word ; while in others, besides explaining the meaning of
" the word, it is intended to be implied that there exists a
*' thing, corresponding to the word. Whether this be or be
'* not implied in any given case, cannot be collected from the
" mere form of the expression" {Logic, Bk. I., ch. viii., § .5).
Book I.
Ch. V.
The Scholiis.
tic writers
said:
A Nominal
Definition
explained
a word.
A Real Defi-
nition ex-
plained a
tiring.
Modern
writers 1k)!J
that a R''al
Definition
implies the
existence of
an object
bearing tht-
name, whilst
a Nominal
Definition
does not do
so.
120 TEKMS,
Book I. Definitions, however, are not arbitrary, but must be
^^ ^' grounded on a knowledge of the corresponding things {see
ibid., § 7). This view of Mill does not seem fundamentally
different from that of Ueberweg, when allowance is made
for the Nominalism of the former, and it is, practically, that
set forth originally by Aristotle that Nominal Definitions
are those in which there is no evidence of the existence of
the objects to which the name is applied. Most English
logicians of the present day agree with Mill that definitions
are of names only, but, of course, the name is merely the
verbal symbol of the notion. On this view Hamilton's two
kinds of Definitions merge into one, and we are left with
the distinction drawn by Mill. But, as he himself says,
both kinds are expressed by the same formula. How then
shall we distinguish them ? The very use of a term gives
a presumption — though a presumption only — that the thing
of which that term is the name exists in the external world.
But the province of Logical definition is not to verify, or to
disprove, this presumption ; but to analyse the concept or
* notion ' which exists in the mind, and which is expressed
by the name. Hence Ueberweg says " every definition de-
" fines . . . the notion." This analysis can be carried ou""
safely only when it is accompanied by a continual referenca
to the things denoted by that name ; and from an examina-
tion and comparison of which the notion was formed (c/l
But thedis- gg 49 ^^^j f^^). It would seem better, then, to finally discard
only leads to this distinction from Logic; for it simply tends to confuse
con us . ^^^ whole object of logical definition by importing into it
considerations with which the process of framing a definition
is not rightly concerned.
ASubitrntial (n.) Substantial and Genetic Or Constructive. In the former, the
envimerates essential attributes of the class are enumerated as they exist in the
attributes ; complete concept ; in the latter, a process is indicated by which they
Definition niay be secured. The Genetic Definition is not a statement of the
indicates a ^y^y j^ which the concept correspondins' to the name has been formed
which they in the mind, but of the way in which, by indirect means, we may
may bo se- form a concept, or a mental picture, of the notion, when it is incon-
venient to Bay directly what it is. The method is chiefly applied in
DEFINITIONS OF TERMS,
121
Mathematics, but there it is frequently the simplest and clearest.
For instance, the easiest way to define a ring is to say ' Let a circle
revolve round a fixed axis in its own plane but outside it.' Such a
definition as this necessarily postulates the possible existence of the
thing whose name is defined ; unless, indeed, the process be one
which it is impossible to carry out. Outside the realm of Mathe-
matics all definitions are Substantial.
(iii.) Analytically-formed and Synthetically-formed. The former
is the giving clearness and exactness to the commonly received
meaning of a word, which is the ordinary work of definition ; the
latter is the giving a new and arbitrary meaning to an old term,
or the equally arbitrary fixing of the connotation of a newly invented
term, to serve the purposes of some special discussion. Such defi-
nitions can only be regarded as legitimate when a new technical
term is absolutely necessary in a science ; and then it is far better to
invent a new term than to give a new and arbitrary meaning to an
old one ; for, in the latter case, both the writer and his readers are
apt to revert, more or less unconsciously, to the ordinary signification
of the term. As examples of Synthetic Definitions may be instanced
many of those adopted by botanists and other naturalists, which are
not statements of the ordinary connotation of the terras, but are
based on a selection of attributes supposed to be either more
fixed, or connected with a greater number of attributes, than are
those included in the ordinary connotation.
These Synthetically-formed Definitions were called ' Nominal ' by
the writers of the Port Royal Logic — another meaning of that
much -abused word.
This distinction, which was introduced by Kant, is psychological
rather than logical ; for it is based on the origin of the definition in
its inventor's mind and not on the form in which it is expressed.
(iv.) Essential Definition and Distinctive Explanation.
The preceding divisions have been between true definitions,
but we have here a distinction between definitions and pro-
positions which are definitions in appearance only.
The Essential Definition gives the connotation, either com-
pletely, when it is a Perfect or Complete Definition, which is
really the only true logical definition ; or incompletely, when
it is Imperfect or Incomplete; but this latter is not a true defini-
tion at all, though it is often spoken of as one. In this case
Book I.
Ch. V.
An Analyti-
cally-formed
Definition
expresses
the ordinary
meaning of
a term ;
A Syntheti-
cally-formed
Definition
gives a new
meaning.
No Proposi-
tion which
does not
give the
whole con-
notation,
and nothing
else, is a
true logical
definition.
122 TEEMS,
Book I. either only a portion of the differentia is given, or the full
Ch^. differentia is added to a genua higher than the proximate
genus, as if one were to define an equilateral triangle as ' a
plane rectilinear figure with equal sides.'
In the Distinctive Explanation likewise we have no real
definition, but a proposition in which propria are given
instead of the connotation. This must be distinguished from
the case discussed under Rule I {see § 52), where propria
were added to the connotation ; there we had a definition,
though a faulty one as it suggested false inferences ; here we
have no definition at all, for the attributes given are not
connoted by the name. The proposition will not even
enable us to identify the objects which bear the name it
explains ; for though they all possess those attributes yet
they need not be the only objects which do so {see § 37).
Under this last head we may place all still less accurate
means of identifying things by an enumeration of some of
their attributes. Such are Descriptions, where inseparable
accidentia are often used, with or without some of the
propria, to enable us to recognize the objects denoted by the
name. Such a proposition is no more a definition than
would be the act of pointing out a member of the class in
question and saying ' I mean something like that,' which is
really a kind of the so-called Definition by Type. Still,
though in no sense definition, description is by no means
useless ; it serves the very useful function of enabling us to
easily identify anything which bears the name. We must
not, however, speak of describing a word ; we define the
name, and describe the thing which bears the name. The
main object of the former is to make distinct our concepts
of things, and so to lead to a greater clearness and definiteness
of thought and language ; that of the latter is to furnish a
rough and ready means of making others recognize the
objects of which we are speaking. It may be pointed out
that the so-called Definitions given in the ordinary Dictionary
belong almost invariably to one or other of the spurious
kinds discussed in this sub-section.
CHAPTER VI.
DIVISION AND CLASSIFICATION.^
54. Logical Division.
(i.) General Character of Logical Division — Logical
Division is the analysis of the denotation of a term. By
this is not meant an enumeration of the individuals which
form the class of which the term is the name, but a statement
of the sub-classes into which that class can be divided. In
other words, it is the splitting up of a genus into its con-
stituent species [cf. § 35). The genus which is to be divided
is called the totum divisum (divided whole), or dividend ;
the species into which it is analysed are styled the memhra
dividentia (dividing members). In dividing a genus we
think of an attribute which is possessed by some of its
members and not by others, and this suggests the funda-
mentum divisionis, or basis of the division. The same genus
may obviously be divided on several different bases into
different sub-classes, according to the attributes on which
the division is founded. Thus, triangles may be divided into
equilateral, isosceles, and scalene, where the fundamentum is
the relation of the sides to each other in length ; or into
right-angled, obtuse-angled, and acute-angled, where it is the
size of the angles. So, the various divisions of terms (see
§ 26) are analyses of the same genus on different bases.
When the same genus is thus divided in different ways the
process is called Co-division; and the classes obtained by
euch a co-division more or less overlap each other, for every
Book I.
Ch. VI.
Logical Divi-
sion is the
analysis of
the denota-
tion of a
term.
The same
genus may
be variously
divided ;
this is Co-
division.
' The treatment of this subject is largely based on that adopted in Pr.
Venn's Empirical Logic
124
TERMS.
Book I.
Ch. VI.
Suh-divUion
is a continu-
ing the pro-
cess of cUvi-
8ion for
more than
one stage.
Here each
step is on a
new basis.
Division
must pro-
ceed one
step at a
time.
member of the genus must fall into one class in each
division, and the classes obtained on one basis are sure not
to correspond exactly, if at all, with those resulting from
another.
When the classes resulting from an act of division are
themselves again divided into their sub-classes we perform
an act of Sub-division. These sub-classes may be again sub-
divided, and so the process may go on till we reach infimm
species — classes, that is, which are only capable of being split
up into individuals. Of course, in every step of a sub-
division, we must have a new fundamentum divisionis, for the
first step exhausts the original basis. Thus, having divided
triangles into equilateral, isosceles, and scalene, it is evident
that we cannot subdivide any of these classes on the basis of
the relative lengths of their sides. But if we take a new
fundamentum we may continue the analysis ; for instance,
we may subdivide both isosceles and scalene triangles on the
basis of the size of their angles into right-angled -isosceles,
obtuse-angled-isosceles, acute-angled-isosceles ; right-angled-
scalene, obtuse-angled-scalene and acute-angled-scalene. Or,
if our original basis was the size of the angles, then we may
subdivide acute-angled triangles into acute-angled-equilateral,
acute-angled-isosceles, and acute-angled-scalene ; whilst right-
angled and obtuse-angled triangles may be divided into right-
angled-isosceles, right-angled-scalene ; obtuse-angled isosceles
and obtuse-angled-scalene.
Every division must be progressive ; it must proceed one
step at a time, and must omit no intermediate species ; the
division must be an enumeration of the species of the
proximate genus. Hence the old logical rule Divisio non
facial saltum (Division must not make a leap). If this rule
is broken we must not be surprised to find that some of the
members of the totum divisum find no place at all in any of
the membra dividentia; for in omitting an intermediate class
the distinctive marks of that class will probably be at the
same time overlooked, and thus, individuals having those
marks, but not possessing the distinctive marks of the lower
species contained in the division, may be omitted.
DIVISION AND CLASSIFICATION.
125
(ii.) Logical Division is indirect and partially material.
But few words are needed to show the utility of Division.
Every subject is much more easily treated and thoroughly
comprehended when its various parts are arranged in an
orderly way. Division, in fact, adds clearness to our notions,
as definition makes them distinct (c/". § 49). As compared
with Definition, however, Logical Division must be regarded
as a secondary and indirect process, for it is a necessary
assumption of Formal Logic that the connotation of a class
term determines its denotation, and not vice versa [cf. § 28
(iv.)] . We do not select a number of objects indiscriminately,
and then seek for some attributes common to them all, which
may form the connotation of the class name we affix to
them ; but we first get a more or less definite idea of the
connotation of the class name, and then include or exclude
individuals from that class in accordance with their posses-
sion of, or want of, those attributes which form that
connotation. This is practically much less the case in some
instances than in others {cf. § 51), but, formalh/, we must
assume the class to be always determined by the connotation
and not by the denotation. Besides, Division must pre-
suppose more or less complete definitions of the names of
the species into which a given genus is to be divided ; for it
is only by appeal to such definitions that we can determine a
fundamentum divisionis ; whilst every definition of a species-
term per genus et differentiam suggests such a fundamentum.
Hence, we also see that no Division can be purely formal,
i.e., involve no appeal to knowledge outside the matter
given. If we are simply given a genus we cannot even begin
to divide it ; for, of necessity, the attributes which separate
one species from another can form no part of the connotation
of the genus. Every such attribute must be a separable
accidens of the genus, and can only be known by an appeal
to sources of information other than the connotation of the
name of that genus. Moreover, for the Division to be of
any practical use, this appeal must be to the objects them-
selves which are included in the genus ; for only thus can we
be sure that we are dealing with really existing classes of
Book I.
Ch. VI.
Division de-
pends on
Definition.
Hence no
division is
purely
formal.
126
TERMS,
P.OOK I.
Ch. VI.
Only Gene-
ral Terms
can appear
in a Logical
Division.
Physical
Partition
divides an
individual
into parts.
Metaphysical
Analysis
enumerates
the attri-
butes of a
thing.
Verbal Dis-
tinction
separates
the mean-
ings of an
equivocal
vpord.
In Logical
Division
only can the
divided
whole be
predicated
of each
dividing
member.
things. Thus, every Division contains, at least to some
extent, a material element (c/. § 10).
(iii.) Operations somewhat resemHing Logical Division.
As a logical division is only the analysing of a genus into its
species it follows that only general terms [see § 27 (ii.)] can
appear in it. A singular term cannot be divided, for it is a
name applicable in the same sense to an individual only ;
and the logical meaning of an ' individual ' is that which is
incapable of logical division. And the division must stop at
infimcB species {see § 35) ; for to go further would be to
enumerate individuals, which, it has been already pointed
out, is not Logical Division. Hence, Logical Division must
be carefully distinguished from
(a) Physical Partition, which is the splitting up of an
individual into its constituent parts ; as, for
instance, a ship into hull, masts, sails, rigging^
etc.
(&) Metaphysical Analysis, or the enumeration of the
attributes of a class or of an individual ; as when
we name whiteness, ductibility, malleability, etc.,
as the attributes of silver.
(c) Distinction of the various meanings of an equivocal
term (cf § 26) : as when we distinguish between
* vice ' meaning a moral fault, and ' vice,' a
mechanical tool.
In a Logical Division the genus can be predicated of each
of the species, and of each individual member of those
species. This follows necessarily from the fact that the
definition of the species involves the genus (c/. § 50). For
instance, if we divide animals into men and brutes, we can
predicate of each man and of each brute that he, or it, is an
animal. In none of these other processes, however, can the
whole be predicated of the parts. We cannot say ' A mast is
a ship,' or 'Whiteness is silver.' In the case of Distinction,
of course the same verbal symbol can be predicated of each of
the meanings : ' This tool is a vice,' or ' This fault is a vice.'
DIVISION AND CLASSIFICATION.
127
But the same definition— that is, the same connotation —
cannot be so predicated, which shows that it is not the saTiie
logical term which is predicated- in each case. This proves
anew the accuracy of what was said before ((/ § 26) — that
an equivocal term is really and logically two or more terms.
55. Rules of Logical Division.
The rules to which a Logical Division must conform to
ensijre validity may be gathered from the preceding section.
They are : —
Book I.
Ch. VI.
L Each act of division must have only one basis,
II. The stib-classes must be together co-extensive with
the whole.
III. If the division be a continued one (i.e., embrace more
than one step), each step must be, as far as possible,
a proximate one.
Or, more briefly : The division must (i.) avoid cross division,
be (ii.) exhaustive, and (iii.) step by step. It will be noticed
that for each separate act of division the first two rules
are sufficient.
Other rules are frequently given, but, on examination, they will
be seen to be redundant. Thus : —
(a) Only class terms can enter into a division. This is involved
in the very definition of Logical Division.
(6) The sub-classes must be mutually exclusive. Rule I
secures this, for if there is only one basis of division it
is impossible for any individual to fall into more than
one sub-class.
(c) The whole must be predicable of each of the sub-classes.
This is provided for by Rule IT, for, if there were a sub-
class of which the name of the whole could not be
predicated, then, evidently, the denotation of the sub-
classes would be together greater than that of the whole
instead of coextensive with it. If the whole and the
sum of the sub-classes form two sides of an equation,
clearly the same name is predicable of both.
Yery few words will be needed to illustrate these rules.
A Logical
Division
must avoid
cross
division,
be exhaus-
tive, and
step by step.
128
TERMS.
Book I.
Ch. VI.
If the Divi-
sion is on
more than
one basis
it will
probably
contain part
of the deno-
tation more
than once,
or be too
narrow.
or coiuiuit
both faults.
If wo omit
species the
Division is
too narrote.
Rule I. — Evidently a division made on more than one basis
would be worthless. It would be nearly certain to include
some individuals in more than one sub-class, so that the total
denotation of the species would be apparently greater than
the denotation of the genus. If, for example, triangles were
divided into isosceles, scalene, and acute-angled, every possible
triangle would fall into one or other of these classes (for
every equilateral triangle is acute-angled), but some would
fall into more than one ; viz., those which are acute-angled
isosceles, and those which are acute-angled scalene. But' we
have no guarantee that the opposite fault will not be com-
mitted and the division be made too narrow by the exclusion
of some individuals from every sub-class. If we divided
triangles, for instance, into equilateral, obtuse-angled, and
right-angled, we should not, indeed, include any individual
twice, but we should exclude all acute-angled-scalene, and
acute-angled-isosceles triangles. Our division is too narrow.
Very probably both faults will be committed ; some in-
dividuals will be included more than once and others omitted
altogether. Thus, if we divide triangles into equilateral,
isosceles, and right-angled we include right-angled-isosceles
triangles twice, and exclude obtuse-angled-scalene, and acute-
angled-scalene triangles altogether. It may, indeed, occa-
sionally happen that a division may be made on two principles
and yet be practically accurate. But it is only in the excep-
tional instances when one attribute solely involves, and is
Bolely involved by, another that this can occur. For example,
a division of triangles into equiangular, isosceles, and scalene,
would be both exclusive and exhaustive ; but that is simply
because all equilateral triangles, and they only, are equi-
angular, and so the division coincides with one made on the
single basis of the relative lengths of the sides. Only when
we have but one basis of division can we be sure that our
sub-classes are necessarily exclusive of each other — that no
individual can be placed in more than one.
Kule II. — We have seen that a violation of Rule I is apt
to lead to a too narrow division, that is, to the exclusion of
part of the denotation of the whole from each of the sub-
DIVISION AND CLASSIFICATION.
129
Book I.
Ch. VI.
classes. Of coarse, the same fault may be committed, even
if Rule I be rigidly adhered to ; for it is possible to omit
one or more of the sub-classes in any division. We should
get too narrow a division, for example, by dividing triangles
into equilateral and scalene, and omitting isosceles. In such
a simple case as this the fault is not likely to be committed,
but when we are dealing with matter as complex as nature
continually presents to us it requires great care to ensure
that we have made a complete enumeration of all the species
contained under a genus. Other examples of too narrow
divisions are of men into good and bad, of books into instruc-
tive and amusing, of objects into useful and ornamental.
The opposite fault is to make the division too wide ; that is
to include among the species some objects not denoted by the
genus. This, again, is not likely to occur in simple cases ;
few, for example, would think of dividing coins into gold,
silver, bronze and banknotes. But an indistinct apprehen-
sion of the connotation (i.e., of the definition) of any of the
terms we employ in our division may lead to this fault when
we are dealing with complex matter.
It is plain that if this rule is broken we have not really
divided the genus at all — but either only a part of it (when
the division is too narrow) or the genus and something else
as well (when the division is too wide). In a true division
the sum of the denotations of the species must exactly
coincide with the denotation of the whole ; and only when
this is the case has the genus given been really and accurately
divided.
Rules I and II may be thus expressed symbolically —
If the genus G is divided into the species ^j ^2 ^3»
then No g^ must be ^2 or ^3^
No ^2 .. 1. ^1 or^gV Rule I.
No ga „ „ ^, or ^gJ
and ^1 + go + ^3 must = G - Rule IT.
Rule III. — This has been already discussed when it was if a con-
remarked divisio non facial saltum [see § 54 (i)] , and it was gion is not
pointed out that a violation of it usually leads to a division ?tep by step
being too narrow. too narrow.
LOG. I. 9
If w c 1 n -
elude classes
which do not
fall under
the genus
we are divid-
ing, the divi-
sion is toa
wide.
In either
case the
genus has
not been
divided.
130
TERMS
Book I.
Ch. VI.
Dichotomy is
division at
each step
into corre-
sponding
Positive and
Negative
Terms.
Dichotomy
is cumbrous,
and, BO far as
it is formal,
is purely hy-
pothetical.
56. Division by Dichotomy.
To ensure that none of the rules of Division are violated,
many Formal Logicians have insisted that all valid Division
most be by Dichotomy {Grk, Sixa, in two, Ttfivw, I cut), or
division at every step into a positive term and its correspond-
ing negative. This process is founded on the Principles of
Contradiction and Excluded Middle.
A strictly dichotomous or bifid classification can always be
thus formed, but it lies open to the objections
(a) That, at each step, one of the sub-classes — and
that frequently the largest ; viz., that denoted by
the negative term — is entirely undefined in its
extent [cf. §§ 19, 2'J (i.) (&)] ; and, no matter how
far the process of subdivision is carried, the last
term must always be formally left thus indefinite.
(&) That, in so far as it is formal, it is entirely hypo-
thetical ; the division does not guarantee the
existence of any of the sub-classes.
(c) That it is excessively cumbrous. It seems absurd
to divide a genus into two classes when it evi-
dently falls naturally into some other, and equally
definite, number of species, and to do so obscures
the fact that these species are co-ordinate.
This process may be represented symbolically. Given
a genus X, it may be divided into two species A and non-A,
according as the objects possess or do not possess a certain
attribute. [We use A, B, etc., to denote non-A, non-B, etc.]
Either the positive or the negative member at either step
may be subdivided. Thus : —
(1.) (ii.)
X X
1
1
1
1
A
A
A
A
I
1
1
1
I
1
B
B
P
P
1
1
1
1
1
1
c
C
Q
Q
etc.
etc.
DIVISION AND CLASSIFICATION. 131
Or to take material examples Book I.
(iii.)
Men
Europeans non-Europeans (J/a?i2/)
Frenchmen non-Frenchmen {if any)
soldiers non-soldiers {if any)
Men ('"•)
Europeans non-Europeans {if any)
Asiatics non- Asiatics {if any)
Africans non-Africans {if any)
I
I I
Americans non -Americans {if any)
Polynesians non-Polynesians {if any)
It is evident that this, like all other Division, in so far as
it is not hypothetical, possesses a material as well as a formal
element. It is by appeal to the matter that we know that
some men are Europeans ; and that some X are A. Even
then, the existence of A and of non-Europeans is hypothetical
unless we make a further appeal to outside matter. Similarly,
* Frenchmen ' is not part of the connotation of Europeans —
we must again appeal to fact. Every step is, therefore,
either partly material, or wholly hypothetical, and it is
evident that a purely hypothetical division is of but little
practical use. In all strictly dichotomous division we must,
at least, finish with a hypothetical term of whose existence, or
non-existence, the division leaves us absolutely ignorant.
9—2
Ch. VI.
132
TERMS.
Book I.
Ch. VI.
Every division may be reduced to dichotomy, but, as was
said above (c), it is absurd to do this when we know de-
finitely the number of sub-classes o\xv fund amentum divisionis
will give rise to. Thus we may make the division
Triangles
equilateral non -equilateral
isosceles
non-isosceles
The Tiee of
Porphyry is
an instance
of Division
by Dichoto-
my.
Dichotomy
is chiefly
valuable as
a test of the
complete-
ness of a
division.
scalene non-scalene {if any)
We know that the last class does not exist, for equilateral,
isosceles, and scalene, form a complete enumeration of the
species of the genus triangle on this basis of division, and
these species are co-ordinate. This shows, however, that
every division on one basis into more than two sub-classes
may be expanded into several successive divisions on slightly
different bases.
The Tree of Porphyry {see § 39) is, omitting the last step,
a good instance of dichotomous division as treated by the
older logicians. We there see that only one — the positive —
term in each dichotomy is sub-divided ; the division proceeds
along the predicamental line (see § 35) towards a certain de
finite end — the species ' man.' We also see its material as
well as its formal character. The attribute ' animate,' for
instance, is not included in the connotation of ' body,' and
so with all the qualities the possession, or want, of which
form the various bases of division ; they are only known by
an appeal to experience.
The real value of a division by dichotomy is to test the
validity of our analysis — particularly to discover if it is
exhaustive — and to find the position of any assigned class.
Thus, in the Analytical Key prefixed to Bentham's British
Flora, which is intended to enable anyone who has a speci-
men of a certain plant before him to discover its species
and its technical name, the arrangement is nearly entirely
DIVISION AND CLASSIFICATION.
133
dichotomous, and, for such a purpose, this form is the most
useful. But to adopt dichotomy as a final arrangement
would be absurd. A botanist, for example, starts at once
with three classes of the siirnmum genus ' plant,' viz., exogens,
endogens, and acrogens, and each of these is subdivided into
varying numbers of orders, and these again into still further
varying numbers of genera, and so on, with little or no
regard to dichotomy, the object being to make the classifica-
tion agree with the distinctions existing in the plants them-
selves. It may be added that every Definition per genus et
differentiam suggests a division by dichotomy, and, conversely,
every such Division supplies us with a Definition of that
kind.
Book 1
Ch. VI.
57. Purely Formal Division.
It has been shown in the preceding sections [§§ 54 (ii.), 56]
that every logical division is partly formal and partly material,
and these two elements continually hamper each other. The
absolutely formal process of perfect and complete dichotomy is
prevented by the desire not to form classes which have no real
denotation, but are purely imaginary. When we discard these
material considerations and develop dichotomous division to the
fullest extent we enter into the domain of Symbolic Logic.
Here we employ only formally negative terms, not, as in the Tree
of Porphyry, (see § 39) terms such as 'rational and irrational,'
which however contradictory they may be in meaning are not so in
form [see § 29 (i.), (ii.)]. These formal contradictories are fully de-
veloped, the number of sub-divisions being determined solely by the
number of terms we are dealing with. Thus, if we are concerned
with three terms S, M, P, we have eight sub-divisions, viz. S M P,
SMP, SMP, SMP, SMP, SMP, SMP, SMP {v/here S,M,P, de-
note non-S, non-M, non-P, respectively). But these are not regarded
as classes, for some of the combinations may be non-existent, but as
mere class-compartments — a framework, as it were, into which existent
cla-sses may be fitted. Every universal proposition is then regarded
as asserting the emptiness of one or more of these compartments ; that
is, as denying the existence of one or more classes. Thus the proposi-
tion No M is P declares that M and P are never found together, i.e. , that
the class-compartment A//' is empty, and this includes the sub-compart-
A full de.
velopment
of formal
division
leads to
Symbolic
Logic.
134
TERMS.
Book 1.
Ch. VI.
ments S M P and S M P. Similarly, All S is M denies the existence
of the class S M, that is, of the sub-classes S M P and S M P. The
combination of these two propositions, therefore, will leave us with
only the compartments S M P, S M P, S M P,S M P occupied ; that is,
those are the classes which, under the terms of our propositions, may
possibly exist. It is evident that such a process as this is purely
formal, and is based entirely upon a development of bifid classifica-
tion. It may be thus represented — X standing for the universe
[see § 28 (iv.)] to be divided, which may include all existing things
if necessary : —
X
I
s
I
s
SM
SM
SM
SM
A develop-
ment of the
material
side of divl-
fiiun leads to
Classification,
SMP SMP SMP SMP SMP SMP SMP SMP
This may, of course, be continued indefinitely, each additional
division doubling the number of compartments. As all the com-
partments are divisions of X we might write X before each of them ;
for the sake of clearness this has been omitted.
Were these compartments regarded as classes, or sub-species of
the genus X, the process would be invalid ; for a thing is not simply
the sum of its attributes, but those attributes must stand in a certain
definite relation to each other. This necessity makes some classes
impossible. For example, a triangle has three sides and three
angles, but every combination of these — e.g. right-angled equilateral
— is not possible. But regarded, as they are in Symbolic Logic, as
mere class-compartments, this objection does not hold, and the
process is formally valid. For an account of the developments of
Logic to which it leads the student may be referred to Dr. Venn's
excellent work on Symbolic Logic.
58. Material Division or Classification.
A similar development of Logical Division on the material
side leads to the Theory of Classification. The object of
classifying is to so arrange in order the facts with which we
are dealing that we can the most easily acquire the greatest
DIVISION AND CLASSIFICATION.
135
possible command over them, and can economize statement
— and so lighten the task imposed on memory — by being
enabled to convey a large amount of information in a few
words. To do this with success evidently requires a con-
siderable knowledge of the phenomena we are engaged in
classifying — it is nearly entirely a material process. To
some extent our classification is already done for us by the
mere use of language. Every giving of a General Name is
a classification so far as it marks the formation of a group,
constituted by the agreement of its members in the possession
of certain attributes. Logic cannot tell us which attributes
should form the basis of our primary, or secondary, divisions
of a genus ; all it can do is to warn us not to select those
which are common to many different kinds of things, and
which do not exercise any recognizable influence on the rest
of their nature. It is only by special knowledge of, and
reference to, the objects to be classified that we can select
those attributes which carry with them the greatest number
of other attributes as the basis of our first division, and can
further decide which properties shall determine the subse-
quent divisions.
It does not follow that because a certain attribute is im-
portant in the division of one species of a genus it will be
important in all the co-ordinate species. We may divide
the genus X on the basis of the attribute a into the species
M, M, 0 ; and then sub-divide M on the basis of b into the
sub-classes P, Q, R. It may be, however, that b is of no im-
portance in the species N and 0 ; for N it may happen that
c is all important, and that on this basis N should be divided
into S, T, U ; whilst for 0 the important difference may be
found in modifications of d which may lead to the sub-species
V, W, Y. The determining of these facts, so as to make our
classification practically useful for the purposes of scientific
study and investigation, is a purely material process. If we
thus adopt different bases of division for each new species
it is evident we can have no formal guarantee that our classi-
fication is exhaustive. In fact, as Lotze remarks, " Classifica-
" tion does not create the complete material, but assumes its
Book I.
Ch. VI.
Every classi-
fication
necessitates
special
knowledge
of tiie things
to be classi-
fied.
There can be
no logical
guarantee
that a classi-
fication is
exhaustive.
136
TERMS.
Book I.
Ch. VI.
A 'Natural'
Classifica-
tion was
held to be
one which
followed
strongly-
marked di-
visions of
nature, the
attribute
on which it
is founded
involving
many other
attributes,
whilst
An ' Artifi-
cial ' Classi-
fication was
regarded as
arbitrary.
But the di-
visions of
nature are
not thus
definitely
markeu,and
every classi-
fication is
both
Natural and
ArtificiaL
"completeness to be guaranteed elsewhere" (Logic, Eng,
trans., vol. i., p. 1 67). Every such classification must, there-
fore, like scientific definition (c/. § 51) be regarded as subject
to revision with the advance of knowledge.
59. ' Artificial ' and ' Natural ' Classification.
If we take a genus and proceed to divide it on the basis of
some single attribute, it may be that this division will not
enable us to assert anything further about the species than
that they possess, or do not possess, that particular attribute.
But if we select some other attribute, we may find that the
presence, or absence, of this property involves a great deal
besides, as other attributes are invariably found where it is
present, and are wanting where it is absent. It has been
customary to call a classification of the former kind Artificial
and of the latter Natural, The former was held to be simply
an arbitrary grouping of natural objects for a certain definite
purpose, but the latter was thought to involve a recognition
of divisions really existing in nature. The leading modern
upholder of this view was Mill, who based it on his doctrine
of ' Natural Kinds.' These he regarded as classes of things
existing in nature which were sharply and definitely sepa-
rated from each other by an unknown and indefinite number
of differences. Practically these ' natural kinds ' were be-
lieved to be due, in the animal and vegetable kingdoms, to
separate acts of creation — all the members of the same
* natural kind ' might be regarded as descended from the
same parents. The spread of the doctrine of Evolution has
largely modified this view, for it is seen that many — if not
all — these ' natural kinds ' are descended from one stock.
Moreover, natural objects do not fall into such well-defined
and separate groups (see § 51). There is thus seen to be no
essential difference between an ' Artificial ' and a ' Natural '
Classification : in neither case are we dealing with ready-
made groups presented to us from without ; but in both we
are grouping objects — or rather our ideas of them, for we
very seldom deal with the objects themselves — in the way
most convenient for our purpose. The words * Artificial '
DIVISION AND CLASSIFICATION.
137
and ' Natural ' are thus misleading ; for every classification is
'artificial' in the points just mentioned, and, on the other
hand, every classification should aim at being ' natural,' in
the sense of being based upon attributes of the objects
themselves. It would be better, therefore, to discard these
terms and to speak of Classification for a special purpose,
and Classification for general purposes, instead of 'artificial'
and ' natural ' respectively. This would remove the stigma
which the use of the word 'artificial' tends to throw upon
classifications which are of the greatest value for certain
special purposes.
As the same objects possess many attributes, and as any of
these may be made the basis of a classification, it is evident
that the same things may be classified in many different
ways. But it should be noticed that it is only in arranging
the intermediate classes that differences of grouping can
come in — the summum genus — e.g., plants in Botany —
and the infimce species — e.g., the tulip, hyacinth, etc. — are
given us by the language of ordinary life, and must be the
same in every scheme of classification. But this potential
variety of grouping is of the greatest utility ; the botanical
arrangement of plants, for example — the ' Natural ' classifica-
tion of Mill — would be of little use for medicinal purposes,
where another and special grouping is required on the basis
of quite different attributes, and neither would serve the
purpose of the farmer or gardener. For each distinct pur-
pose we require a new classification, and the most appro-
priate one is the most ' natural ' for that purpose. We may
give, then, as a fundamental rule in classifying (in addition
to those given in § 55) —
Rule IV. The Classification should be app'opriaie to the
purpose in hand.
60. Classifications for Special Purposes.
The simplest kinds of classifications are those intended
for special purposes ; for in them attention need be fixed
on only one attribute or group of attributes : we are not
concerned with the amount of general resemblance or differ-
BooK I.
Ch. VI.
It is better,
therefore, to
speak of
Classifica-
I ion for
general pur-
poses or for
a special
purpose.
Only the in-
termediate
groups ad-
mit of varied
classifica-
tion.
Every classi-
fication
must be ap-
propriate to
the purpose
in band.
A classifica-
tion for a
sjiecial pur-
pose is based
on only one
attribute,
or group of
attributes.
138
TERMS.
Book I.
Ch. VI.
Such a
classifica-
tion is use-
ful as a key
to a general
classifica-
tion.
An alpha-
betical ar-
rangement
is the
simplest
example.
Buch an ar-
rangement
is useful
only as a
key.
ence. Such a special classification is frequently employed to
serve as a key or index to the classification of the same
objects for general purposes, which is based on our know-
ledge of their whole nature. A very familiar instance is
the alphabetical arrangement which is found to be most
serviceable in library catalogues and in indexes of books,
and whose sole purpose is to indicate where a certain book,
or statement, is to be found. Several attempts have been
made to draw up catalogues of the books in large libraries,
classified according to the subjects treated of, but this in-
variably leads to doubt as to where to look for the title of
any particular volume, owing to the impossibility of mark-
ing off the subject matter of many books by rigid lines. If
there is a class of Historical books and another of works on
Philosophy, under which head should be entered those which
treat of the History of Philosophy, or of the Philosophy of
History ? And would the ' psychological ' or the religious
novel find its most appropriate place in the department of
fiction or in those of philosophy and theology respectively ?
Every such attempt has proved a more or less complete
failure, and it is now pretty generally agreed that the alpha-
betical arrangement is practically the only satisfactory one
for a catalogue. The same holds in the case of an index
to a book ; we can most readily find where to look for the
discussion of any particular question treated of in its pages
by means of an alphabetical list of the subjects discussed.
Useful as such an arrangement is, however, as a key, there
its utility ends. It does not enable us to make a single
general statement about any one of the groups formed,
whose members may, indeed, have practically nothing in
common except that their names happen to commence with
the same letter of the alphabet. It is, therefore, valueless
for the purposes of scientific investigation and research.
Its value, too, is absolutely confined to the language in which
it happens to be written ; if translated, it ceases to be a
classification at all, for the names of the same objects in
two different languages by no means necessarily have the
same initial letter.
DIVISION AND CLASSIFICATION.
139
A less simple arrangement but one whose chief use is still
to serve as an index to a more general classification is such
an Analytical Key as is prefixed to Bentham's British Flora
{cf. § 66). Its object is to enable us to find out the name of
a plant of which we have a specimen before us. As a
bifurcate arrangement is found to offer the most ready
means for attaining this object, that arrangement is almost
exclusively adopted. Thus, flowering plants are first divided
into those whose flowers are compound and those which are
not compound. Then the former are sub-divided into those
with one seed and those with more than one ; and the latter
into those in which the perianth is single and those in which
it is double. Flowers with a double perianth are then sub-
divided into those in which the corolla consists of one piece
and those in which it does not ; into those whose ovary is
free and those in which it is not free ; and so on. In every
case the endeavour is to set forth two easily discriminated
alternatives. Such an arrangement as this is less conven-
tional than the artificial one discussed above, for it is founded
on properties of the objects themselves — though not neces-
sarily on those which directly involve others. Hence, it will
bear translation into another language without losing any of
its value as a classification.
Other classifications for special purposes are not thus in-
tended to serve as keys to a general classification. But
they all agree with the one last discussed in that no attempt
is made to show by the grouping the full resemblances of the
things classified, but only their agreement as to possessing
more or less of the attribute, or group of attributes, im-
portant for the purpose in hand.
Book I.
Oh. VI.
Analytical
Keys, as in
Botany, are
chiefly bi-
furcate.
They are less
artificial
than an al-
phabetical
arrange-
ment as
they are
grounded on
real proper-
ties.
All special
classifica-
tionsare not
intended as
keys to a
general
classifica-
tion.
61, Classifications for General Purposes.
In all the cases we have so far considered, the arrange- ?^ ^^^^ °^
IGCts SLTG
ment is intended to be on paper only, and the intermediate grmipeditia
classes have been of no importance in themselves — they CTcetogere-
simply serve to point out where we may find that of which '^i rescm-
we are in search. The class of books whose authors' names
begin with M, for instance, has no interest for us as a class —
140
tERMS.
Book I.
Ch. VI.
The same
end is aimed
at in a the-
oretical
classifica-
tion for
general
purposes.
Such a
classifica-
tion seeks
fa) to class
objects oa
the basis
of greatest
general
resem-
blances ;
(6) to simi-
larly ar-
r ange
classes.
the books have not of necessity any other point in common
— the arrangement is only useful as enabling us to find some
particular work which we wish to refer to, say Macaulay's
Essays. But the books themselves on the library shelves,
or the plants in a Botanical Garden, will not be arranged in
the alphabetical order, but in groups which have as much in
common as possible. On the same shelves will be found,
for instance, all the books relating to Logic ; and in the
same part of a Botanical Garden we should expect to find
all plants of the lily kind. This bodily proximity of objects
having much in common is an important aid to study ; it
saves time, and by presenting similar objects side by side
renders comparison more easy and complete. Here we see
every group is of importance in itself and is not merely a
kind of index-finger, or meaningless label.
What is attempted partially in such cases as the above in
grouping the objects themselves, it is sought to attain com-
pletely in a theoretical Classification for General Purposes.
Every special classification is, at least, to some extent sub-
jective or personal ; but here it is attempted to form a
classification entirely objective — one which will appeal to all
minds alike. The object to be attained is twofold : —
(o) So to group individuals into classes that each class
name may have the greatest possible connotation
— that is, that the members of each class may
resemble each other in as many points as possible.
(&) To arrange these classes into larger groups on the
same principle.
Were such an ideal classification thoroughly attained the
resultwould be that the whole world of thinkable thingswould
be systematically organized. There would be one supreme
genus with innumerable branching lines of species and sub-
species, and each species Avould be so placed that it would
show by its position the amount of resemblance it bears to
all other species whatsoever. Nothing approaching to this
has, however, ever been attempted ; in all probability it
never will be. The utmost that has hitherto been done is
DIVISION AND CLASSIFICATION.
141
Book I.
Ch. VI.
Only in
Botany and
Zoology has
this been
done with
any com-
pleteness.
to aim at such a classification in separate departments of
knowledge only ; and it is only in Botany and Zoology that
this more modest task has been accomplished with any com-
pleteness. In attempting such a task we may either begin
with the summum genus, e.g., plants, and carry the division
downwards by successive steps— and this is the usual way of
conducting the process — or we may commence with the
injimce s])ecies, roses, lilies, etc., and go on to form gradually
wider and wider classes by a process of aggregation. The
resulting groups will be the same in both cases.
But when the task is attempted many difficulties present
themselves, as will appear from a brief examination of the
special rules (in addition to the general rules given in
§§ 55 and 59) usually given to guide this kind of classifica-
tion. They are : —
BiUle A. The higher the group the more important should
he the attributes by which it is constituted.
Rule B. The classification should be graduated, so that the
groups with most affinity ivith each other may be
nearest together, and so that the distance of one
group from another may be an indication of the
amount of their dissimilarity.
Another Rule — that all groups should be so constituted
as to difi'er from each other by a multitude of attributes —
owed its origin to the doctrine of Natural Kinds. The
more nature is understood the more it is seen that the Law
of Contiguity is everywhere to be traced — one species fades
into another by almost imperceptible degrees {cf § 51), and
it is impossible to constitute our groups in accordance with
this rule, which must, therefore, be discarded.
Rule A. Of the two rules given above it ia evident that Everything
the first is the more fundamental ; for everything depends on deteimfn"'^
our ability to determine which attributes are the most ' im- atfribur'^
portant.' In a special classification, of course, that attribute are most im-
must be considered important which has most connexion with ^°' **" '
the purpose in hand. But here our purpose is general, and the
important attributes have been considered as " those which
The wider
group
should be
determined
by the mora
important
attributes,
and the
classifica-
tion should
be gradu-
ated.
142
TERMS.
Book I.
Ch. VI.
" contribute most, either by themselves or by their effects,
" to render the things like one another, and unlike other
" things ; which give to the class composed of them the most
" marked individuality ; which fill, as it were, the largest
" space in their existence, and would most impress the
" attention of a spectator who knew all their properties but
" was not specially interested in any." (Mill, Logic, Bk. iv.,
ch vii., § 2.) But how shall we determine these important
attributes ? The attributes of an object have no in-
dividual existence apart from our own mental analysis of
them. They do not stand side by side like books on a book-
shelf, but are merely our way of describing things. It,
therefore, largely depends on us how far we analyse these
inextricably connected phenomena and mentally hold them
apart from each other as separate properties. Moreover, we
can, probably, never know all the properties of anyone thing,
i.e., understand its full and complete nature ; consequently
every General Classification must be regarded as always
subject to revision with the advance of knowledge.
'Affinity'
between
classes in
Botany and
Zoology is
used, under
the evolu-
tion hypo-
thesis, in
its primary
meaning.
Qualities
which
furnish
evidence of
descent are
important,
but often
neither
striking nor
of great pre-
sent utility.
Kule B. When we consider Rule B we are at once forced
to consider what is meant by the word affinity. Mill, and
the other upholders of Natural Kinds used ' affinity ' in
a merely metaphorical sense to imply resemblance, but not
anything corresponding to that family relationship which is
the primary meaning of the word. Now, however, under
the influence of the doctrine of Evolution, affinity is regarded
in Botany and in Zoology as meaning just this very relation-
ship ; the word is no longer used metaphorically, but in its
primary meaning. The same doctrine has also led to a
modification of what is meant by ' important ' characteristics,
by causing the evidence of descent to be regarded as an
element of importance. The properties which bear witness
to this are by no means necessarily, or even usually, those
which are of most moment for the welfare of the individual
at present. The latter bear witness rather to the more
recent conditions in which the species has existed ; for these
conditions continually modify those properties which are
DIVISION AND CLASSIFICATION.
143
necessary to vitality and health. But qualities of little con-
sequence to the immediate well-being of the species may
remain through many generations unchanged — though,
doubtless, in the course of very loug periods of time they
tend to disappear. No matter how trivial a property may
be, yet if it appears in generation after generation, it is
good evidence of descent and is, so far, an important attribute ;
but the most obvious and striking attribute, even though it
involves many others, will be no safe guide in classification
if it is subject to modification with changing circumstances.
The habit of climbing in plants, for example, will not
determine species ; for striking as this characteristic is, it is
due to external surroundings, and it is found that even ferns
can climb as successfully as the ivy, if climbing is necessary
to their existence.
It is clear, then, that it is no easy matter to form a
General Classification, but it does not follow that it is im-
possible. On the contrary, in Botany and in Zoology, and,
to a less extent, in Chemistry and in Mineralogy, this has
been done with very considerable completeness, and the first
object mentioned above has been fairly attained. The body
of such a work as Bentham's British Flora, or any other
systematic treatise on Botany or Zoology will furnish an
example. But only in these Sciences has such an elaborate
attempt at classification been made. In them it is felt to
be an aid to discovery and investigation, and it is this end
which has been sought ; but where the classification is seen
to lead to nothing beyond itself, the work has not been con-
sidered worth the labour it entails.
When we are provided in any science with such a General
Classification, we need some help in assigning to any in-
dividual object its place in that scheme. " This operation
" of discovering to which class of a system a certain specimen
"or case belongs, is generally called Diagnosis" (Jevons,
Princ. of Science, p. 708). Any conspicuous and easily dis-
criminated property which is peculiar to the class may be
selected as a guide to the class to which an object belongs,
and a scheme of classification based on these characteristic
Book I.
Ch. VI.
A General
Classifica-
tion has
only been
attempted
in sciences
where it is
seen to be an
aid to dis-
covery.
A General
Classifica-
tion requires
an Analy-
tical Key.
144
TERMS.
Book I.
Ch. VI.
properties forms an analytical Key such as was described
in the last section. This key itself is called by Whewell the
Diagnosis (Novum Organum Eenovatum, p. 23). Such a
diagnostic system should be, as far as possible, bifurcate
(cf. § 60), and each characteristic on which it is based should
be possessed by every member of the class of which it serves
as the sign, and by no other object whatever.
As the type
of a class is
an indi-
vidual it
cannot
furnish a
basis for
forming the
class.
62. Classification is not by Types.
The fact, which has been already more than once referred to {see
§§ 51, 59 and 61), that species, especially in the vegetable and animal
kingdoms, are not separated from each other by rigid and definite
lines, together with the further fact that even between members of
the same species differences exist, led Whewell and other writers to
advance the theory that classification is by Types and not by
characters. Whewell defined "the Type of any natural group " as
" an example which possesses in a marked degree all the leading
"characters of the class" {Nov. Org. Ben., p. 21). He then went
on to say, " A Natural Group is determined, not by a
" boundary without, but by a central point within ; — not by what it
" strictly excludes, but by what it eminently includes ; —by a Type,
"not by a Definition " {ibid., p. 22). But this is not classification
at all. As Jevons remarks : " The type itself is an individual, not
" a class, and no other object can be exactly like the type. But
" as soon as we abstract the individual peculiarities of the type and
" thus specify a finite number of qualities in which other objects
" may resemble the type, we immediately constitute a class. If
"some objects resemble the type in some points, and others in other
" points, then each definite collection of points of resemblance con-
" stitutes intensively a separate class. The very notion of classifica-
" tion by types is in fact erroneous in a logical point of view "
{Prin. of Science, p. 724). It is, in truth, regarding the connotation
as secondary to the denotation, which is an inversion of the true
logical method [see § 54 (ii.)]. We can no more classify by types
than we can define by types [see § 53 (iv.)]. We can use the mental
image of a type as an illustration of a class, and in this way the
conception of a typical example is useful. But it is typical because
the idea of the class is already formed ; not because it is the one
determining factor in that formation.
DIVISION AND CLASSIFICATION.
145
63. Classification by Series.
So far we have considered mainly the grouping of in-
dividuals ; we will now examine the grouping of the classes
thus formed. --^ Our object here is similar to what it was
in the former process ; we desire so to group our classes that
their position may show the amount of similarity, or dis-
similarity, existing between them. Now, any attribute and
its relations to other attributes may vary quantitatively in
different individuals. Hence, it may happen that one species
may pass over into another species, and this into a third,
and that we thus get a serial arrangement. To take an
example from mathematics. On the one side the ellipse
passes into the circle, when its diameters become equal ; on
the other it passes into a straight line, when the conjugate
diameter becomes nil. In such a case we should regard the
typical ellipse as intermediate between these extremes. If
we represent the transverse diameter by x and the conjugate
by y, then in the former case x—y=o,a.nd in the latter
x~y = x. If now we add these -two equations together we
get 2x—2y=x. That is, we may regard that ellipse as
typical and most perfect whose transverse diameter is exactly
twice the length of the conjugate diameter ; for in this case
the attributes and their relations are most characteristic of
the figure.
But when we come to consider natural objects we find
our arrangement is not usually serial ; it rather resembles
a series of concentric circles on a globe, as many groups are
at the same distance from the typical one. Nor is this type
always a kind of average. It may be that the species is
most perfect, not when it is halfway between the two
extremes, but when it is just on the point of passing over
into another species. Thus, the species M may be most
perfect and typical just at the point where it tends to
become N, and similarly N may reach its highest develop-
ment just when ready to pass into 0 and so on. That is,
our basis of grouping is some attribute which gives per-
fection to each species in exact and direct proportion to the
fulness with which it exists. In Zoology, for example,
LOG. I. 10
Book I.
Ch. VI.
We may
group
classes on
the same
principle as
individual
objects.
Some classes
form a
series,
but with
natural ob-
jects the ar-
rangement
of classes is
more com-
plex.
The basis of
grouping is
the attri-
bute on
which the
perfection of
the species
depends,
146
TERMS.
Book I.
Ch. VI.
and fhere
are many
series of
species
branching
from one
point.
the attribute which forms the basis of the grouping of our
species is the possession of aninaal life ; and it is evident
that each member of any species is a more or less perfect
representation of that species in exact proportion to the
degree in which it shares in this vitality. In such a case a
species becomes more perfect as it becomes less its average
self, and the whole series tends towards one highest species,
in which the attribute is found in fullest perfection. In
the animal kingdom, this crowning species is, of course,
man. We attain then, finally, a classification in which the
different species are not simply placed side by side, but in
which they follow one another in a definite order till they
culminate in one point. Nor is there one series only, but
rather a web of series all branching from one point ; the
horse is a different species from the dog, yet they probably
occupy about the same relative position with regard to man.
Were the whole of nature classified we should have an
arrangement of the whole world of thinkable things, in
which species followed species in a definite and determinate
direction, and in which all would be connected into one
harmonious whole.
A Nomencla-
ture is the
system of
Class Names
used in any
science.
Classifica-
tion and No-
menclature
are essential
to each
other.
64, Scientific Nomenclature.
A Nomenclature is a system of names for the groups
of which a classification consists. No classification could
long remain fixed without a corresponding nomenclature,
and every good nomenclature involves a good system of
classification. The two are indissolubly connected. As
Whewell remarks : " System and Nomenclature are each
" essential to the other. Without Nomenclature, the system
" is not permanently incorporated into the general body of
"knowledge, and made an instrument of future progress.
" Without System, the names cannot express general truths,
" and contain no reason why they should be employed in
" preference to any other names " (Novum Organon Reno-
vatum, p. 288). It follows that only those sciences which
have a fairly complete and generally received classification
DIVISION AND CLASSIFICATION.
147
possess a true general nomenclature — the sciences, that is,
of Botany, Zoology, and Chemistry. As the classification
must precede the nomenclature it follows that the latter is
a consequence rather than a cause of extended knowledge.
To give a thing a name which marks its position in a system
implies that its attributes are known, and that a system
exists suflBciently elaborate and regular to receive it in the
place which belongs to it, and in no other. Any system of
names of the classes in a systematic classification is a nomen-
clature ; there may, therefore, be nomenclatures depending
on special ('artificial') or on general ('natural') classifica-
tions, but the latter are, by far, the more important.
" Every nomenclature dependent on artificial classifications
" is necessarily subject to fluctuations ; and hardly anything
" can counterbalance the evil of disturbing well-established
" names, which have once acquired a general circulation.
" In nature, one and the same object makes a part of an
" infinite number of different systems — an individual in an
"infinite number of groups, some of greater, some of less
" importance, according to the different points of view in
" which they may be considered. Hence, as many different
" systems of nomenclature may be imagined as there can
" be discovered different heads of classification, while yet it
" is highly desirable that each object should be universally
" spoken of under one name, if possible. Consequently, in
" all subjects where comprehensive heads of classification do
" not prominently offer themselves, all nomenclature must
" be a balance of difficulties, and a good, short, unmeaning
" name, which has once obtained a footing in usage, is pre-
" f erable to almost any other " (Herschel, Discourse on Natural
Philosophy, § 132). When, however, the science does admit
of comprehensive heads of classification, then the names
should not be unmeaning, but should recall both the resem-
blances and the differences between classes. Such a nomen-
clature prevents our being overpowered and lost in a wilder-
ness of particulars. The number of species of plants, for
example, is so enormous that if each had a name which
expressed no relation with any other, memory would find it
10—2
Book I.
Ch. VI.
Only Bot-
any, Zoology
and Chemis-
try, have
true nomen-
clatures.
A nomencla-
ture may
depend on
either a
special or a
general
classifica-
tion, but the
latter is the
more im-
portant.
A nomencla-
ture should,
if possible,
suggest reliv-
tions be-
tween
classes.
148
TERMS.
Book 1.
Ch. VI.
In Botany
and Zoology
relations are
expressed
by combin-
ing names
of higher
and lower
generality.
In Chemlstiy
relations are
expressed by
modifications
of form in the
names.
impossible to retain anything more than a very small fraction
of the whole number. The nomenclature should, therefore,
be so constructed as to suggest these relations. There are
two main ways of doing this : —
(1) The names of the lower groups are formed by com-
bining names of higher and lower generality,
(2) The names indicate relations of things by modifica-
tions of their form.
The former method is that which, since the time of
Linn^us, has been adopted in Botany and Zoology. In
Botany, for instance, the higher groups have distinct names,
Dicotyledon, Rosa, Geranium, etc. The species is marked
by adding a distinctive attribute to the name of the genus,
as viola odorata, orchis maculata, etc. These distinctive
attributes are not the logical differentia of the species, so
the specific name is not a definition. They are, on the con-
trary, formed from all kinds of more or less important con-
siderations. Sometimes the name is given in honour of an
individual, as Rosa Wilsoni ; sometimes from a country in
which the plant was first observed, as Anemone Japonica ;
sometimes from some peculiarity of the plant, as Geranium
sanguineum. Some are purely fanciful ; for instance " Lin-
" n«us . . . gives the name Bauhinia to a plant which has
*' leaves in pairs, because the Bauhins were a pair of brothers.
*^ Banisteria is the name of a climbing plant in honour of
" Banister, who travelled among mountains. But such names
"once established by adequate authority lose all their in-
" convenience and easily become permanent " (Whewell,
Novum Organon Renovatum, p. 308). Of course names
which, in themselves, describe some peculiarity in the plant
are at first of most value, but any easily remembered name
serves the purpose. The names of varieties, sub-varieties,
etc., are formed on the same principle as those of Species.
The second method of constituting a nomenclature is
employed in Chemistry. This system of names is founded
on the oxygen theory. It " was constructed upon . , , the
" principle of indicating a modification of relations of
DIVISION AND CLASSIFICATION. 149
" elements, by a change in the termination of the word. Book I.
'' Thus the new chemical school spoke of sulphwrjc and ch^l.
" sulphwj-OMS acids ; of sulpha/es and sulphides of • bases ; and
" of sulphwre^s of metals ; and in like manner, of phosphoric
"and phos^AoroMS acids, of phosphates, phosphites, phos-
'•'■phurets. In this manner a nomenclature was produced, in
" which the very name of a substance indicated at once its
" constitution and place in the system.
" The introduction of this chemical language can never cease to
" be considered one of the most important steps ever made in the
" improvement of technical terms ; and as a signal instance of the
"advantages which may result from artifices apparently trivial, if
" employed in a manner conformable to the laws of phenomena, and
" systematically pursued. It was, however, proved that this lan-
"guage, with all its merits, had some defects. The relations of
"elements in composition were discovered to be more numerous
" than the modes of expression which the terminations supplied.
" Besides the sulphurous and sulphuric acids, it appeared there were
"others ; these were called the hyposulphurous and hyposulphuric :
" but those names, though convenient, no longer implied, by their
" form, any definite relation. The compounds of Nitrogen and
" Oxygen are, in order, the Protoxide, the Deutoxide or Binoxide ;
'* Hyponitrous Acid, Nitrous Acid, and Nitric Acid. The nomen-
" clature here ceases to be systematic. We have three oxides of
" Iron, of which we may call the first the Protoxide, bub we cannot
" call the others the Deutoxide and Tritoxide, for by doing so we
* ' should convey a perfectly erroneous notion of the proportions of
" the elements. They are called the Protoxide, the Black Oxide,
' ' and the Peroxide. We are here thrown back upon terms quite
" unconnected with the system.
•'Other defects in the nomenclature arose from errors in the
"theory ; as for example the names of the muriatic, oxymuriatic,
"and hyperoxymuriatic acids; which, after the establishment of
" the new theory of chlorine, were changed to hydrochloric acid,
^^ chlorine, and chloric acid.
* " Thus the chemical system of nomenclature, founded
" upon the oxygen theory, while it shows how much may be
" effected by a good and consistent scheme of terms, framed
" according to the real relations of objects, proves also that
150
TERMS.
Book I.
Ch. VI.
A Termin-
ology is a col-
lection of
terms neces-
sary in de-
scribing
individual
things.
Terminology
is essential
to classifica-
tion.
Botany is
the only
science
which pos-
sesses a com-
plete Ter-
minology.
" such a scheme can hardly be permanent in its original form,
" but will almost inevitably become imperfect and anoma-
" Ions, in consequence of the accumulation of new facts, and
"the introduction of new generalizations. Still, we may
" venture to say that such a scheme does not, on this account,
"become worthless ; for it not only answers its purpose in
" the stage of scientific progress to which it belongs : — so far
" as it is not erroneous, or merely conventional, but really
" systematic and significant of truth, its terms can be trans-
" lated at once into the language of any higher generalization
" which is afterwards arrived at. If terms express relations
"really ascertained to be true, they can never lose their
"value by any change of the received theory" (Whewell,
Novum Organon Renovaium, pp. 275-277).
65. Scientific Terminology.
But we require not only a system of names to designate
classes but a collection of terms which will enable us to
describe individual objects. This is a Terminology, and
it will embrace names of the properties — shape, colour, etc.
— and of the parts of the objects recognised in the science.
As both classification and nomenclature depend upon the
knowledge of the qualities of objects, which knowledge is
the result of comparison and the noting of points of agree-
ment and difference, it follows that, unless we can express
the qualities by suitable names, our nomenclature cannot be
fixed and stable. Terminology is, in brief, the language in
which we describe objects, and, without description, there
can be no classification. All the names which form a
terminology are general names ; though, by their combina-
tion, we can describe individuals.
Botany is the only science which, as yet, possesses a
complete terminology ; this, as well as its nomenclature, it
owes to Linnaeus. " The formation of an exact and exten-
" sive descriptive language for botany has been executed with
" a degree of skill and felicity, which, before it was attained,
"could hardly have been dreamt of as attainable. Every
" part of a plant has been named ; and the form of every
DIVISION AND CLASSIFICATION. 151
" part, even the most minute, has had a large assemblage of Book I.
"descriptive terms appropriated to it, by means of which Ch^i.
" the botanist can convey and receive knowledge of form and
" structure, as exactly as if each minute part were presented
" to him vastly magnii5ed. . . .
" It is not necessary here to give any detailed account of
"the terms of botany. The fundamental ones have been
"gradually introduced, as the parts of plants were more
"carefully and minutely examined. Thus the flower was
"successively distinguished into the calyx, the corolla, the
*' stamens and the pistils ; the sections of the corolla were
" termed petals by Columna ; those of the calyx were called
" sepals by Neckar. Sometimes terms of greater generality
" were devised ; as perianth to include the calyx and corolla,
" whether one or both of these were present ; pericarp for
" the part enclosing the grain, of whatever kind it might be,
"fruit, nut, pod, etc. And it may easily be imagined that
"descriptive terms may, by definition and combination,
" become very numerous and distinct. Thus leaves may be
*' called pinnatifid, pinnatipartite, pinnatisect, pinnatilobate,
*^ palmatijid, palmaiipartite, etc., and each of these words
" designates different combinations of the modes and extent
" of the divisions of the leaf with the divisions of its out-
" line. In some cases arbitrary numerical relations are
" introduced into the definition : thus a leaf is called bilobate
" when it is divided into two parts by a notch ; but if the
" notch go to the middle of its length, it is bijid ; if it go
" near the base of the leaf, it is bipartite ; if to the base, it is
" bisect. Thus, too, a pod of a cruciferous plant is a silica if
" it be four times as long as it is broad, but if it be shorter
" than this it is a silicula. Such terms being established,
" the form of the very complex leaf or frond of a fern is
" exactly conveyed, for example, by the following phrase :
" ' fronds rigid pinnate, pinnse recurved subunilateral pinna-
" tifid, the segments linear undivided or bifid spinuioso-
" serrate.'
" Other characters, as well as form, are conveyed with the
" like precision : Colour by means of a classified scale of
152
TERMS.
Book I.
Ch. VI.
Technical
terms em-
ployed in a
Termino-
logy are un-
ambiguous ;
terms of
common
speech must
be made so
by conven-
tion.
■' colours " but " the naturalist employs arbitrary names. . .
" and not mere numerical exponents, to indicate a certain
" number of selected colours '' (Whewell, Novum Organon
Renovatum.^ pp. 315-317).
* In the above examples we have illustrations of both the
kinds of terms of which a terminology consists — names of
parts of the plants, as pistil, stamen, calyx, frond, and names
of properties, as bipartite, silica, pinnate. Most of these
technical terms being peculiar to the science have a perfectly
clear and definite meaning, but when terms in use in common
life — as the names of colours — are required to form part
of a terminology, their meaning must be precisely, though
arbitrarily, fixed by convention. For there must be no doubt
as to the exact meaning of the terms used in a scientific
description, as, otherwise, our scientific language will be
incapable of expressing all the shades of difference which we
recognize in the objects we are examining and comparing.
To again quote Dr. Whewell : " The meaning of [descrip-
" tive] technical terms can be fixed in the first instance only
" by convention, and can be made intelligible only by present-
" ing to the senses that which the terms are to signify. The
" knowledge of a colour by its name can only be taught
•' through the eye. No description can convey to a hearer
"what we mean by apple-green or French-grey. It might,
*' perhaps, be supposed that, in the first example, the term
" apple, referring to so familiar an object, sufficiently suggests
" the colour intended. But it may easily be seen that this is
" not true ; for apples are of many different hues of green,
" and it is only by a conventional selection that we can
" appropriate the term to one special shade. When this
" appropriation is once made, the term refers to the sensation
" and not to the parts of the term ; for these enter into the
" compound merely as a help to the memory, whether the
" suggestion be a natural connexion as in ' apple-green,' or
*' a casual one as in ' French-grey.' In order to derive due
" advantage from technical terms of this kind, they must be
"associated immediately with the perception to which they
" belong ; and not connected with it through the vague
DIVISION AND CLASSIFICATION. 153
" usages of common language. The memory must retain Book I.
" the sensation ; and the technical word must be understood Ch^i.
" as directly as the most familiar word, and more distinctly.
" When we find such terms as tin white or pinchbeck-brown,
" the metallic colour so denoted ought to start up in our
"memory without delay or search.
* " This, which is most important to recollect with respect
" to the simpler properties of bodies, as colour and form,
" is no less true with respect to more compound notions. In
*' all cases the term is fixed to a peculiar meaning by conven-
" tion ; and the student, in order to use the word, must be
" completely familiar with the convention, so that he has no
" need to frame conjectures from the word itself. Such
" conjecture would always be insecure, and often erroneous.
" Thus the term papilionaceous, applied to a flower, is
" employed to indicate, not only a resemblance to a butter-
" fly, but a resemblance arising from five petal.s of a certain
" peculiar shape and arrangement ; and even if the resemb-
*' lance to a butterfly were much stronger than it is in such
*' cases, yet if it were produced in a different way, as, for
"example, by one petal, or two only, instead of a ' standard,'
" two ' wings,' and a ' keel' consisting of two parts more or
"less united into one, we should no longer be justified in
"speaking of it as a 'papilionaceous' flower" (^History of
Scientific Ideas, vol. ii., pp. 111-113 ; Novum Organnn Reno-
vatuni, pp. 314-315).
BOOK 11.
PROPOSITIONS.
CHAPTER I.
Book II.
Ch. I.
A Proposition
is the verbal
exi^ression
of a truth or
falsity.
Deductive
Logic does
not examine
into the
truth of Pro-
positions,
but Induc-
tive Logic
does.
DEFINITION AND KINDS OF PROPOSITIONS.
66. Definition of Proposition.
A Proposition may be briefly, but suflBciently, defined
as tlie verbal expression of a truth or falsity. From
this it follows that not every grammatical sentence is a
logical proposition. The latter implies belief in the state-
ment made, and claims assent ; whilst the former may be
the expression of a command or a wish, or some other of
the many forms taken by human speech, without necessarily
making a distinct statement challenging assent or dissent.
Not only does every proposition express a truth or falsity,
but this is the only way in which truth or falsehood can be
expressed ; a logical proposition is the one form of words
of which it can be said ' This is true ' or ' This is false.'
Nevertheless, it is no part of the business of Deductive
Logic to examine into the truth or falsehood ot any in-
dividual proposition ; it accepts those offered to it as true,
and determines what inferences can be drawn from them.
Inductive Logic, on the contrary, has for its sphere the in-
vestigation of this very point.
DEFINITION AND KINDS OF PROPOSITIONS. 155
This definition of the proposition makes clear that it Book II.
is the translation into language of the judgment, which, Gh^.
as the essential form of thought, is the ultimate subject- Judgment is
matter of logic (see § 8). All knowledge is expressed in matter of °
affirmations made by thought about reality, and such affir- Logic,
mations expressed in words are propositions. In investi-
gating propositions, therefore, we must constantly go behind
the form of words to the judgments which are more or less
perfectly expressed by them.
67. Kinds of Propositions.
Propositions are traditionally divided into different classes Kinds of
on the bases of Relation of Subject and Predicate, of tioM."^*'
Quality, and of Quantity.
I. Relation.
(i.) Categorical - S is P : Sis not P.
(a) Analytic.
(b) Synthetic.
(ii.) Hypothetical - If A, then G.
(iii.) Disjunctive - Either X or T.
n. Quality.
(i.) Affirmative - S is P.
(ii.) Negative - S is not P.
III. Quantity.
(i.) Universal —
(o) Singular - This 8 is P.
(6) General - Every S is P.
(ii.) Particular - Sotne S's are P.
The nomenclature of the classification under Relation is in a
confused state. Some writers make a twofold division, subsuming
Hypotheticals and Disjunctives under a wider class which they call
Conditional, though by others this use of the terms Hypothetical
and Conditional is reversed. The threefold division is, however
needed to mark important differences between the forma of
propositions.
156
PROPOSITIONS.
Book II.
Gh. I.
Quality and
Quantity
apply pri-
marily to
Categorical
Proposi-
tions.
ACategorical
Proposition
simply as-
serts or de-
nies a fact.
Categorical
Propositions
should he
considered
before Hy-
pothetical or
Disjunctive
Proposi-
tions.
The Demon-
strative and
the Imper-
sonal Judg-
ments are
simpler
forms than
the com-
plete logical
proposition
S is P , but
they can he
expressed
in this com-
plete form,
which is the
only one
adopted in
formal
Logic.
Ab the divisions under Quality and Quantity apply
primarily to Categorical Propositions, it is proposed to treat
them under that head, and to consider how far they are
applicable to Hypothetical and Disjunctive Propositions
when we treat of those forms.
CATEGORICAL PROPOSITIONS.
68. Analysis of the Categorical Proposition.
A Categorical Proposition is one which simply asserts
or denies some fact
not rash.'
as ' Grold is yellow'; ' True bravery is
The consideration of this class of propositions naturally precedes
that of either Hypothetical or Disjunctive propositions. For we
can only require to make the assertion that S is P dependent on a
condition when we have already had experience of the presence of
P in some instances of S, and desire to find the reason for that
connexion. Nor can we say that 5 is P or Q unless we know that
P and Q may be subsumed under a wider genus M ; thus, the Dis-
junctive proposition is a more specialized form of the proposition
S is M.
It was seen in § 8 that the most elementary form of a com-
plete judgment is a simple interpretation of an actual ex-
perience. From this, thought increases in complexity and
generality, until the point is reached when we have a proposi-
tion of the form S is P, in which subject and predicate are
distinct terms {cf. § 23). Now, all categorical propositions are
capable of being expressed in this form, and, for the sake of
simplicity, formal Logic so expresses them. This possibility
is all that is meant when it is said that the complete logical
form is 'involved ' in every judgment. We do not say tha'i;
an exclamation, such as 'Fire!' — which expresses a true
judgment — is a worn-down form of some such statement as
* That property is on fire,' but that it may be expanded into
such a form without change of meaning. The Impersonal
Judgment may be similarly expanded ; e.g., * It rains ' may
be written ' Rain is falling.' Such a reduction makes mani-
fest the artificial character of the formal proposition with
DEFINITION AND KINDS OP PROPOSITIONS.
157
its emphasis of the distinction between subject and predicate.
For in the Impersonal Judgment — which may be regarded
as the most elementary attempt to explain reality — the real
subject is not made definite at all, but is simply the vague
mass of present impressions. The whole force of the
judgment rests in the predicate, which, indeed, as being
the interpretative element may be regarded as the most
essential element in every judgment. In making a reduction
of all categorical judgments to one fixed form of proposi-
tion, formal logic makes a simplification which is not alto-
gether justified by either thought or expression.
* We must now examine this form more closely. As
was pointed out in § 23 it consists of three parts — subject,
copula, and predicate — two terms and the expression of a
relation of agreement, or disagreement, between them. When
we say S in P, ' Gold is yellow,' we mean that we are referring
to identical things under different names implying different
attributes (see § 17). On the other band when we affirm
S is not P, ' Corn is not poisonous,' we mean that the terms
used are applicable to entirely different things, and that
both can never be correctly applied to any single object ; for
all the attributes connoted by the one term are never found
conjoined with all those connoted by the other.
* The Copula. The relation between the terms is expressed
by the Copula ; which is the verb is or are, by itself in affirma-
tive propositions, and conjoined with the particle not in nega-
tive ones. Nothing but this bare relation of agreement or
disagreement is expressed by the copula which, in itself, in-
volves no assertion of existence. It is true that the verb
'to be ' sometimes has this meaning of ' exists,' as when we
say ' Evil is.' But in all such propositions is is not the copula
but the copula and predicate combined, and may be expanded
into is existent — ' Evil is existent — where the is has its merely
relational value (cf. § 89).
* The copula is always in the Present Tense, Every act
of judgment is a present one and expresses a present belief.
Moreover, a proposition which is once true must be always
true ; no change of time can affect it, for it refers to the
Book II.
Ch. 1.
Analysis of
the forms
Sis P,
S is not P.
The Copula
merely ex-
presses
agreement
or disaj^ee-
ment be-
tween the
terms,
and is
always in
the Present
Tense,
158
PROPOSITIONS.
Book II.
Ch. I.
which is the
only one
free from
ambiguity,
and which
marks our
belief that
at Dribiites
co-fxist in
the same
subject.
moment in which it was first made. If, then, we express a
judgment about a past or future event in a formal proposi-
tion, we, as it were, put ourselves at that point of time.
Onl}' when propositions are expressed in the Present Tense
can they enter into formal arguments, for only then are they
unambiguous. If we say M loas P and 5 was M we can draw
no conclusion at all, for the time when S was M may have
been quite different from that in which M was P, and so we
are not justified in inferring that at any one moment S ivas
P. Of course, when we use such arguments, as we constantly
do in every-day life, it is with the tacit understanding that
all the propositions refer to exactly the same point of time.
But the verb was does not express, or even imply, this ; and
the same difficulty meets us in the use of the Future will be.
This difficulty can only be avoided, and this tacit assumption
— that all the propositions in an argument refer to exactly
the same time — expressed, by writing each in the Present
Tense, for that tense is the only one which expresses one
simple, exact, and unmistakable point of time. This restric-
tion to the Present Tense also marks our belief that attributes
co-exist in the same subject. We cannot apprehend them
all at once, for attention can, at most, be fixed on two or
three of them at any one moment ; yet we can vary the
order in which we thus experience them ; and we believe
that the successive knowledge of them is necessitated by the
nature of our minds, not by the nature of the things them-
selves. We recognize that a piece of gold is at once yellow
heavy, and malleable, though we probably perceive those
attributes successively in the order named. By saying ' Gold
IS yellow,' 'js heavy,' 'is malleable,' we emphasize this fact of
the co-inherence of those attributes in the substance cold.
And the same form of speech is adopted when the attributes
immediately apprehended are believed to necessarily involve
the presence of another in the future ; as when we say of
the wound of a still living man that it is mortal, or of a
poison still in the vial that it is deadly.
* Subject and Predicate. As the copula expresses a
relation between the two terms, every afiirmative proposition
DRFINITION AND KINDS OP PROPOSITIONS.
159
sets forth a process of synthesis. This differentiates the
Proposition from the Term, in which the product of the
synthesis is alone regarded. But the mere fact that we are
in a judgment engaged in giving meaning to some aspect
of reality, necessitates our regarding one Term as the more
or less permanent and determined centre to which the other
is to be attached, and which it will qualify. In other words
it is a process of affirming attributes of a thing. This fixed
and determined centre, or thing, logically comes first in
thought, and its name forms the Subject of the proposition ;
whilst the name of the attribute, or interpreting notion,
which we affirm or deny of it, is the Predicate. It by no
means follows that because the Subject is logically first in
thought it is always expressed first in language. This is
very frequently not the case, and which term is subject and
which is predicate must be decided by the meaning of the
sentence rather than by the position of the words in it. If
one term clearly tells us something about the other — as
every Adjective does — it is the predicate. On the other
hand, if one Singular Term occurs, or a General Term is
explicitly used in its whole denotation in an affirmative pro-
position, that term is necessarily the subject (cf. § 72). But
no general rules can be given for distinguishing in aU cases
between Subject and Predicate ; the meaning of the sentence
must decide.
But although a proposition may thus be analysed into two
terms and a copula, it must be borne in mind that language
here emphasizes one aspect of judgment to the exclu-
sion of another and equally important one {cf. § 8). The
judgment is always one act of mind, the interpretation of one
aspect of reality. It is not a comparison of separate things,
nor a connexion of two independent concepts, as is suggested
by the verbal form in which it is expressed. The copula
represents no separate element of thought, and in Aristotle,
as now in common language, copula and predicate wer©
expressed by the same word. The Subject marks the point
of reality which is being interpreted, and the Predicate
expresses the interpretation ; but neither reality nor inter-
BOOK II.
Ch. I.
The Subject
is the fixed
centre or
thing.
Tlie Predi-
cate is the
attribute
aCBrmed of
the subject
Every judg
ment is a
unity.
160
PROPOSITIONS,
Book II.
Ch. I.
Proposi-
tions whose
terms are
syuonyms
express no
true predi-
cation.
The Imper-
sonal Pro-
position is
most
natural
where there
is no defi-
nite subject.
pretation can exist as elements of knowledge apart from each
other.
This Subject and Predicate form of Proposition is so
general that it is used even in cases where there is no real
distinction between the terms, as when they are synonyms
such as ' Mercury is quicksilver,' ' Queen Victoria is Empress
of India,' ' Bismark is Duke of Lauenburg.' In such cases
there is no predication of attributes whatever, but simply a
statement that the same object bears the two names. If we
express this, and say ' The same metal is called both
mercury and quicksilver,' we do get a really significant pre-
dication, but we have changed the whole form of the
proposition in order to do so.
Finally, it may be noted that when we are expressing a
judgment about some group of events which we cannot
easily connect with a definite and fixed centre, we most
frequently and naturally use the Impersonal form of pro-
position, as • It snows.' This, however, as was shown in
the earlier part of this section, can be expanded into the
Subject and Predicate form, and should be thus expanded
before beincf used in formal reasoning.
The distinc-
tion be-
tween Ana-
lytic and
S5Tithetic
Proposi-
tions is one
of r/rigin.
The distinc-
tion Is not a
subjective
one.
* 69. Analytic and Synthetic Propositions.
The distinction between these was pointed out in § 40.
It applies, of course, only to aflBrmative propositions, and is
really a matter of the origin of each judgment. If the
judgment can be obtained by an analysis of a concept
already formed, or — which comes to the same thing — of the
definition of a class name (c/. § 49), then it is Analytic ;
if the predicate asserts an attribute which does not form
part of the connotation of the subject, the judgment is
Synthetic.
It is often objected that this division of judgments is
purely subjective ; that every judgment is at first synthetic,
and by familiarity becomes analytic ; that, e.g., ' Lions are
carnivorous' is a synthetic judgment for any person whose
knowledge of a lion did not hitherto embrace that attribute,
but is thereafter for him, as for all who know the nature of
DEFINITION AND KINDS OF PROPOSITIONS.
IGl
a lion, an analytic judgment. But this is to confound the
personal history of an individual's mind with the general
method of knowledge with which alone Logic is concerned.
The basis of the distinction is the fundamental postulate of
knowledge that reality has a constant nature, that there is a
unity in the world which finds expression in a uniform con-
stitution of things. This is at the basis of the very idea of
connotation and definition, that is of the very possibility of
classification. When a proposition simply states explicitly
what is regarded as the constant and essential nature of the
subject, it is Analytic ; when it makes some additional affir-
mation, it is Synthetic.
A judgment which was analytic in the fullest sense would
make an explicit statement of the full connotation of the
subject. But this demand is not made : any proposition
which states any part of the connotation is held to be
analytic. But as the connotation of a term may change with
increase of knowledge, it is evident that a judgment may
pass from one of these classes to the other ; the distinction,
therefore, is not sufficiently fixed to be of great logical
importance.
Moreover, in another, but a very real sense of the words,
every judgment is both analytic and synthetic. It is analytic,
for it sets over against each other and distinguishes elements
in the one and indivisible mental act of judgment ; and it is
synthetic in that it brings together this present element of
reality and the universal idea which gives it meaning.
70. Quality of Propositions.
In expressing the relation of predicate to subject only
two courses are open to us. We must either affirm that the
subject possesses the attributes connoted by the predicate,
or we must deny this (c/l § 19). On this basis, therefore,
propositions are classed as (i.) Affirmative and (ii.) Negative.
In a Negative Proposition we do not deny that the Subject
has any of the attributes connoted by the predicate, we only
deny that it has them all. Some it must have, for the genus
of which the denied predicate is a species is always under-
LOO. I. 11
Book 11.
Ch. I.
Every jiidg.
ment is, in
a sense, both
analytic and
synthetic.
Affirmation
and Nega-
tion are the
only pos-
sible kinds
of Quality.
162
PROPOSITIONS.
Book TT.
Ch. 1.
In Negative
Proposi-
tions the P
which is de-
nied always
belongs to a
genus which
is affirmable
of the S.
Negation is
rtuc to S pos-
sessing an
attribute in-
compatible
with the
proposed P,
and this is
implicit in
the Negative
J UUglliCUt.
An Infinite
Judgment is
an attempt
to express
negation as
affirmation
--S i» non-P,
but this as-
sumes a
previous
deniaL
Moreover, as
non-P is not
a true con-
cept, there
is no real
affirmation.
stood tc> be affirmable of the subject. If this were not so the
proposition would be meaningless, as if we should say 'Virtue
is not blue,' where there is no real predication, for the
notion of colour is absolutely foreign to an unextended and
abstract concept such as ' virtue.' But if we say ' Those
berries are not poisonous ' a real meaning is conveyed, for it
is understood that they are 'edible' \cf. remarks on the
Universe of Discourse in §28 (iv.)].
Pure negation lias no existence in fact, and cannot be really
thought. The denial of any number of attributes of a Bubject S
can only be grounded in, and justified by, the fact tiiat S possesses
some attribute which is incompatible with the proposed P, so that
if P were added, S would at once lose its character and cease to be
S at all. It is this incompatible attribute which is the real basis
of the negation, though we may not even know what it is, and may
only feel that if S were to receive P it would at once cease to bo
itself. This, though it is not made explicit, must be regarded as
implicitly contained in the negative judgment.
* Some logiciJins have endeavoured to reduce negative
propositions to an affirmative form by regarding the negation
as part of the predicate, and writing S is non-P instead of
S is not P. Such judgments were called Infinite by Kant,
who, however, retained the true negative judgments as well,
though it is evident that to have two forms to express the
same fact is not only superfluous but misleading, as it sug-
gests a distinction which does not exist. But this simplifi-
cation is only apparent. For, " in order to know that S
" accepts non-P, must we not already have somehow learnt
" that S excludes P ? And, if so, we reduce negation to
" affirmation by first of all denying, and then asserting that
" we have denied, — a process which no doubt is quite legiti-
" mate, but is scarcely reduction or simplification " (Bradley,
rrinci'ples of Logic, p. 111). Besides, as has been already
said {see § 19), if non-P be taken — as strictly and formally it
should — to include everything which is not P, then non-P is
not conceivable at all, for we cannot possibly form a concept
which wJU embrace all the heterogeneous elements in the
DEFINITION AND KINDS OF PROPOSITIONS.
163
universe which are excluded from P. In S is non-P we have,
then, the form of an affirmation without the reality. Upon
analysis it will be found that all we can possibly mean by
asserting the absence of P is to deny its presence, and it is
better to do this explicitly. In short, Affirmation and
Negation are fundamentally different, and it can only lead to
confusion to treat the distinction as if it were only verbal,
as is done by expressing the negative proposition 5 is not P as
a sham affirmative 5 is non-P.
Book II.
Ch. I.
Tlie distinc-
tion be-
tween Affir-
mation and
Negation is
not verbal
merely, but
fviuda-
mental.
71. Quantity of Propositions.
The Quantity of a Proposition depends upon whether
the predicate is explicitly affirmed, or denied, of the
whole of the subject or not. This gives a two-fold
division into :
(i.) Universal Propositions, in which the subject is dis-
tributed— i.e., explicitly stated to be used in its
whole denotation.
(ii.) Particular Propositions, in which the subject is
undistributed — i.e., the extent of the denotation
referred to is left absolutely indefinite.
The marking of this distinction evidently necessitates a
fourth element in the Proposition ; viz., the mark of the
quantity of the subject.
(i.) Universal Propositions.
Under this head we have two sub-classes of propositions.
In the one the definite whole which forms the subject is
indivisible, i.e., is an individual ; in the other it is simply
undivided, i.e., it is a class, of every member of which the
predication is made.
(a) Singular Propositions. In a Singular Proposition the
subject is a single individual directly indicated by a Proper
Name or by a General Name with a distinctive limiting
mark attached to it restricting it definitely to the one in-
dividual indicated, such as, • This man,' ' That man,' * The
man of whom I sjwke to you yesterday.' The symbolic ex-
11-2
A Term is
distributed
when expli-
cit reference
is made to
its whole
denotation.
It is undis-
tributed
when the
extent of
denotation
referred to
is left in-
definite.
Universal
Proposi-
tions distri-
bute their
subjects ;
Particular
Proposi-
tions du not.
Universal
Proposi-
tions aie of
two kinds :
('() Singulnr
— where the
SisadeOnite
individual ;
164
PROPOSITIONS.
Book II.
Ch. I.
(6) General —
where the $
consists of
every mem-
ber of a
class.
These two
kinds of pro-
positions
;ire funda-
mentally
one in mak-
ing explicit
reference to
the whole
extent of
the S.
The signs of
quantity of
an Affirma-
tive General
Proposition
are Each,
Every and
All— the
latter
always used
distribu-
tively.
Proposi-
tions with
Collective
Subjects are
Singular.
pression of such a proposition in Logic is — Tin's S is P;
This S is not P.
(i) General Propositions. In a General Proposition the
subject is the whole class which bears the General Name,
of every individual member of which the predication is made.
Each individual is here indicated indirectly through the
General Name, not as a definitely specified individual, but
merely as a member of a class to which it belongs in virtue
of possessing the attributes connoted by the Class Name.
The general symbolic expression of such propositions is —
Everi/ S is P; No S is P.
The name ' Universal ' is often restricted to this latter
class ; but, from the point of view of denotation, there is no
fundamental difference between them and Singular Pro-
positions, for it is absolutely indifferent whether the subject
be small or great in extent so long as, whatever that extent
may be, the whole of it is explicitly referred to. It is this
definiteness of application which distinguishes both kinds of
propositions from the Particular, and, therefore, it is usual
to class them under one common name.
The common signs of quantity for an Affirmative General
Proposition are Each, Every, All. The latter word is am-
biguous, as it may be understood either in a distributive or in
a collective sense [c/. § 27 (ii-)]- 1° ^ General Proposition,
however, it must always be interpreted distributively (^omnes,
not cuncti), for the predication is made of each individual
member of the class, not of the class as a whole ; thus ' All
lions are fierce ' means * Every lion is fierce.' To avoid
ambiguity we shall generally use ' Every ' or ' Each ' in pre-
ference to ' All '; but whenever ' All ' is used it must be borne
in mind that, unless the context shows the contrary, it is
equivalent to ' Every.' The distinction may also be marked
by writing the General Proposition All S's are P, and the
Collective All S is P.
A Collective subject, indeed, gives us a Singular Pro-
position, for the predication is there made, not of indi-
viduals but, of one group — as when we say ' The Romans
conquered Gaul,' where it is evident we are referring, not
DEFINITION AND KINDS OP PROPOSITIONS.
165
to individual Romans but, to the Roman army as a body.
So, if it is said ' All the books in this library weigh several
tons,' the reference is plainly to the whole body of books, and
is equivalent to ' This collection of books weighs several tons,'
where we get at once the typical Singular form. This S is P.
If we regard only the verbal form of the Universal Propo-
sition it would appear on the face of it to be merely a
summing up of a number of singular judgments — 'This,
that, and the other S is P' — one of which has been made of
each member of the class S. Thus, in form, the proposition
Every S is P claims to be the result of a complete enumera-
tion of instances. But in meaning it is generally something
very different. If we compare two such propositions as
* Every book on these shelves treats of Logic ' and ' Every
right-angled triangle is inscribable in a semi-circle ', we see
that they are really very different in essence and in im-
portance. The former refers to only one collection of objects
here and now ; it is universal only in the sense that if the
collection of books remains unchanged the same proposition
will hold true of it throughout the lapse of time. But in
the ordinary sense of the word we cannot say that such
a judgment is necessarily true independently of limitations
of time and space. It resembles, indeed, and that very
closely, singular judgments of fact, such as are most exactly
represented by propositions such as * London is the largest
city in Europe,' where the subject is the Proper Name.
Such judgments are in a sense, universals, for the predicate
is affirmed of all the subject, but they are concrete uni-
versals. If, on the other hand, we examine the latter judg-
ment— 'Every right-angled triangle is inscribable in a
semi-circle' — we see at once that its ground is not an
enumeration of instances, but that the proposition is true
because the nature of right-angled triangles is such that the
afiSrmed predicate must hold true of them, and that this can
be shown by rigorous demonstration. The basis of such a
proposition is found, therefore, in connexion of content, not
in constant experience in perception. And this will be seen
to be the case in all judgments which we feel to be really
Book TT.
Ch. 1.
A Universal
Proposition
is not a
summing up
of singulars
but is the
denotative
expression
of a Oenerii
Judgment —
S is P—
which as-
serts con-
nexion of
content.
IfiS
PROPOSITIONS.
Book II.
Ch. I.
Tlie Generic
Judgment
has both an
abstract and
a concrete
aspect.
The Nega-
tive of 'J his
S i« P is 2'kis
S is not P,
but All S's
are not P and
Every Sis not
P are not
universal,
Imt mean
Snme S's are
not P.
universal, uot only in the sense of always holdiug true of an
examined concrete totalitj', but in that of applicability to
whatever in the universe falls at any time under the subject-
concept. But as the true ground of the universal proposition
is thus seen to consist in the nature of the reality dealt with,
it is better to mark this by expressing the judgment in a
form of proposition which does not suggest a false origin.
This we find in the Generic Judgment — S is P, e.g., Right-
angled triangles as such are inscribable in a semi-circle ; Man
is mortal, — where the connotation of the terms is obviously
the more prominent element. Of such judgments the
ordinary categorical proposition may be taken as a statement
in denotation.
There is thus in the Generic Judgment both an abstract
and a concrete I'eference. It is abstract in that it states an
essential relation of content without direct reference to the
particular instances in which that connexion exists in reality ;
it is concrete in that such relation is always regarded as being
actually so existent.
When we wish to express a Negative Singular Proposition
we need only add the sign of negation to the copula of the
affirmative — This S is not P. But when we require to
express a Negative General Proposition we cannot do it by
saying All S's are not P, or Every S is not P. Each of these
expressions simply means that it is not allowable to affirm
P of every S. but does not mean that P cannot be asserted of
<my S at all. The majority of S's may be P and yet it remain
true that Every S is not P, which, indeed, holds if only one S
is not P. Thus, it appears that All S's are not P, and Every S
is not P, do not possess the definite character which dis-
tinguishes Universal Propositions, but are quite indefinite
in quantity ; that is, they are really Particular, and should
assume the special symbolic form of Negative Particular
Propositions— /So/rte S's are not P. To mark the absolute
and entire separation between P and S required in a General
Negative Proposition we must clearly express the fact that
not a single individual which possesses the attributes con-
noted by 5 also possesses all those connoted by P, and the
DEFINITION AN» KINDS OF PROPOSITIONS.
167
only unambiguous way of doing this is by saying No S is P.
This, then, is the most exact symbolic form of a General
Negative Proposition. The force of this distinction will
appear more clearly if a few examples are considered. Thus,
it is true to say ' Every Englishman is not a lawyer,' for this
only implies that there are some Englishmen who do not
follow the profession of law ; but it would be false to say
' No Englishmen are lawyers.' Similarly, ' All Englishmen
are not brave ' is true ; bui • No Englishmen are brave ' is
false. Of course, if A II is taken collectively, then All Sis not P
is a true negative of All S is P; but, as was remarked
above, ambiguity had better be avoided by using another
form of words. For instance, it would be better to say
' This collection of books is not five tons in weight ' than
to say ' All these books are not five tons in weight ;' for the
strict formal interpretation of this last expression is that
some individual books in this collection do not weigh five
tons ; though this, of course, is not what is meant.
Of course the ultimate justification of such a proposition
as No S is P is found in the fact that the content of S includes
one or more elements which are incompatible with P (^cf. § 70).
The basis of the negative as of the affirmative universal is,
therefore, not an exhaustive examination of instances, as the
verbal form suggests, but a knowledge of content which
finds appropriate statement in the Negative Generic Judg- .
ment 5 is not P, of which the form No S is P is merely the
denotative expression.
(ii.) Particular Propositions.
The distinguishing characteristic of a Particular Pro-
position is the perfect indefiniteness of the application of
the subject. Its general symbolic form is — Some S's are P ;
Some S's are not P. (Some is always used in the sense of
aliqui never in that of quidam.) Now, in using this word
Some in Logic its absolute indefiniteness must always be
borne in mind. Usually, no doubt, in common talk, when
we say ' Some ' we mean to refer to less than all. But, if
the idea underlying the word be analysed, it will be found
that this is not really involved in it. It would be perfectly
Book II.
Ch. I.
The true ex
pression of
a Negative
Universal is
No S U P,
and its justi-
fication is
the Nega-
tive Generic
Judgment
S is not p.
In a Particu-
lar Proposi-
tion the ex-
tent of the 5
is perfectly'
indefinite.
In Logic
some ia
alvsrays the
mark of ab-
solute inde-
tinitcuess.
168
PUOPOSITION3.
Book II.
Ch. I.
Science aims
at Universal
Proposi •
tions,
but common
life often re-
quires inde-
finite ones.
Some does
not exclude
All from its
scope,
and is a con-
fession of
limited
knowledge.
Every inde-
finite sub-
ject, even if
singular in
reference,
gives a
Particular
Proposition.
accurate to say ' I saw some of your friends at the theatre
yesterday ' when the speaker has no knowledge as to whether
the individuals he saw included all the friends of the person
he is addressing or not. In fact, this idea will probably not
be present to his mind at all ; he simply states imperfect
knowledge in an appropriately indefinite form. So a
scientist, when he has observed the concurrence of certain
phenomena in several instances, but knows of no necessary
law connecting them, would only be justified in positively
affirming Some S's are P, even though he had never met with
an S which was not P, and might even think it highly
probable that Every S is P. Of course. Science cannot rest
satisfied with anything short of a definite Universal, but
the indefinite Particular is quite allowable as a stepping-
stone to the more perfect stage. This may be reached
either by finding a necessary connexion which enables us to
affirm that S is P, or by discovering that when S is limited in
a certain way it is P. Thus, it may be that MS is P. A new
class name, /?, would, probably, then be found for MS, so that,
ultimately, we should get the generic judgment R is P. But,
in coalmen life, our knowledge is often avowedly indefinite,
and should, therefore, be expressed in a truly indefinite
form. As this indefiniteness must be absolute we cannot
agree with those few logicians who would depart from the
traditional use of 'Some' so as to exclude *A11' from its
possible range, thus reducing slightly (but, of course, not
abolishing) its indefinite character. Nor would we define
it as ' not none ' for this leads us to a circle, as our only
definition of 'none' must be *not some.' It is better to
say at once that 'Some' is a confession of limited know-
ledge, and means 'I know I can make this predication of at
least one S, but of what part of the denotation of S it
holds good I do not know.' Every indefinite subject gives
us, therefore, a Particular Proposition. We may even know
that the predication can only be made of one individual,
still if that individual is merely referred to as a member of
a class and not definitely marked out from the other members
of that class — as An S is P — the proposition is indefinite,
and, therefore, Particular. In such a case we do not know
DEFINITION AND KINDS OF PROPOSITIONS.
169
which S ia P and the proposition is equivalent to Some (one)
S is P. Particular Propositions, indeed, would be more
appropriately named ' Indefinite ' Propositions, but it is not
advisable to alter the old and long established nomenclature
of the Science.
Particular Propositions in their form express an indefinite-
ness in denotation only; in other words they suggest that
their origin is an enumeration of instances, either avowedly
incomplete, or, at least, not known to be complete. But, as
this suggestion of enumeration was found to be misleading
in the case of the universal proposition, so it is here in most
cases. Sometimes, no doubt, the judgment is the outcome of
a more or less extended experience, but, even then, the
fundamental point of uncertainty is not whether the
enumeration is complete — that is an aspect of the question
quite out of the range of scientific, i.e. exact, thought — but
whether the determination of the content of S is complete.
And the form of proposition which most clearly expresses
this doubt is what is most appropriately called the Modal
Particular, whose affirmative form is S may he P, and its
negative form, S need not be P. The meaning is that the
content of S has not been sufficiently determined to make it
clear whether or not it contains the full and sufficient ground
for P. Of these forms the traditional particular propositions
are the denotative expressions,
Indesignate Propositions. What has already been said
will enable us to deal with those propositions to whose
subjects no sign of quantity is attached, as ' Birds are
feathered.' These are called by Hamilton Indesignate or
Preindesignate Propositions. Some writers have held
that they are quite inadmissible in formal Logic, and it is
true that the traditional Logic, with its undue deference to
distinctions of mere language, does not acknowledge them.
But we have seen that they are the fundamental form of the
judgment, without reference to which the traditional forms
cannot be justified. We have also seen that they can all be
translated into the traditional forms of denotative expression.
When the subject term of an indesignate proposition ex-
presses a general concept, and the predicate makes an
Book II.
Ch. I.
Particular
Proposi-
tions are tlie
denotative
expression
of the Modal
Particulars
S may be P,
S need not
be P.
170
PROPOSITIONS.
Book II.
Ch. I.
In Affirm.
Indesignate
Proposi-
tions if the
P is part
of the con-
notation or
a proprium
of the S —
Universal.
If the P is a
sep.accideiis
of the S —
Particular.
It the P is an
insep. accid.
of the S—
strictly Par-
ticular, but
practically
regarded as
Uuiversal.
Indesignate
Proposi-
tions when
Particular
refer to most
of the S.
assertion which is grounded in that content we have, of
course, the true Generic Judgment, as when we say ' Sin is
worthy of punishment,' where the predicate is a necessary
consequence of a true conception of the subject. Whenever
we know the predicate in an affirmative Indesignate Propo-
sition to be a part of the connotation, or a proprium, of the
subject, we know the proposition is Universal (cf. §§ 35-37).
If the predicate is a separable accidens, we know the Pro-
position is Particular {cf. § 38). If the predicate is an
inseparable accidens we are, strictly speaking, only justified
in affirming it as a particular, as we know no reason for the
connexion of P and S and cannot, therefore, be sure that it is
really invariable. In other words, the only judgment as to
connexion of content we should be justified in making would,
in this case, be the Modal Particular S may he P. Of course,
if uncontradicted experience is the ground on which we make
the judgment, the wider and more varied that experience is,
the greater is the probability that no instances to the con-
trary exist at the present time, whatever may have been the
case in the past or may be the case in the future. But
certainty can never be attained, and though for common
practical purposes the proposition expressing such experience
would be usually regarded as general, yet in logic we have
no right to raise it to the dignity of a true universal, whose
very essence is that it must hold true always and everywhere.
Often, however, the context so limits the subject that we
know it is true universally. Thus, if we have the proposition
* Crows are black,' and interpret it as ' All crows are black '
we have a proposition which is probably, but not certainly,
true. But if the context shows that only English crows are
meant — or even hioum crows — we know the proposition is a
really general. On examination, indeed, it will be found
that Indesignate Propositions, when they are not true
universals because their predicates are separable accidentia of
their subjects, are yet only employed when the predication
can be made of the majority — generally the great majority
— of the members of the class denoted by the subject, as
'Frenchmen are vivacious,' 'Italians are musical.' Such
propositions were called hy the old logicians ' Moral
DEFINITION AND KINDS OF PROPOSITIONS.
171
TTniversals,' -whilst the really general propositions were
termed 'Metaphysical Univervsals.' But, of course, a 'Moral
Universal ' is only a Particular, and if stated universally is
false, as it would be to say ' Every Italian is musical.'
If an Indesignate Proposition is Negative, the predicate
must evidently be either a separable accidens of the subject
or an attribute never found in that subject. In the former
case, the propositions belong to the ' Moral Universals ' just
discussed, and are Particular ; as ' Englishmen are not
cowardly.' In the latter case they are, of course, universal,
as 'Englishmen are not negroes.'
72. Tlie Four-fold Scheme of Propositions.
If we combine the divisions under Quality with those
under Quantity we get a four-fold Scheme of Categorical
Propositions ; viz., Universal Affirmative, Particular Affirma-
tive, Universal Negative, Particular Negative, These it is
customary to indicate by the letters A, I, E, 0, respectively,
those letters being the first two vowels of the Latin verb
affirmo (I affirm), which represent the Universal Affirmative
(A) and the Particular Affirmative (I) ; and the vowels of
the Latin verb nego (I deny) which stand for the Universal
Negative (E) and the Particular Negative (0). By writing
these letters between S and P we obtain a brief symbolic
mode of expressing propositions. Thus : —
A
- Every S in P
SaP.
I
• Some S's are P
S i P.
£
. NoSisP
S e P.
0
• Some S's are not P -
S 0 P.
Book IT.
Ch. I.
Negative
Indesignate
Proposi-
tions are
Particular if
P is a Sep.
accid. of S :
if not, Uni-
versal.
Combin:i-
tion of
Quality and
Quantity
gives :
A. Universal
Affirmative.
I. Particular
Affirmative.
S. Universal
Negative.
0. Particular
Negative.
These may
be briefly
written :
A -- Sa P.
I ■ -S i P.
E ■ -S e P.
O ■ -S o p.
These four forms appear to be naturally dictated by the
common needs of human speech, in which we require either
to affirm or to deny, and to do both either definitely or in-
definitely. They do not quantify the predicate, for that is
regarded as an attribute and read in connotation ((/. § 84).
As, however, every term has denotation [see § 28 (iv.)] , it
is possible, if we wish, to consider the denotation of the
predicate, and to ask whether, if we do so, we are to consider
it as distributed or undistributed in each of the above four
forms.
Aa the Pre-
dicate is
read in con-
notation it
is not
quantified,
but as it lias
denotation
we may con-
sider its di.s-
tribution.
172
PROPOSITIONS.
Book II.
Ch. I.
AffirmatiTe
Propositions
do not dis-
tribute their
predicates,
but Nega-
tive Propo
sitlons do.
Distribution of Terms. Now, in every affirmative pro-
position, whether universal or particular, we assert that a
certain subject possesses an attribute P, but we make no
assertion as to the full extent of the denotation of P. We
do not consider whether or not other objects exist of which
P can also be predicated. In some cases there are such
objects — as when we say ' All lions are fierce,' for there are
certainly other fierce animals ; in other cases there are not —
as when we say ' All diamonds are pure crystallized carbon.'
But in no case is any explicit reference made to the full de-
notation of P ; the extent of its application in each proposition
is determined indirectly by that of the subject of which it is
afiirmed. In every affirmative proposition, therefore, whilst
the predicate is asserted in its full connotation, it is left in-
definite as to its denotation, and is, therefore, undistributed.
In a negative proposition, on the other hand, as was said
above (see § 70), every part of the connotation of the predi-
cate is not denied of the subject, but only the connotation
as a whole. But when we look at the predicate in denota-
tion we find that, in every case, it is distributed. For it is
only when explicit reference is made to every object which
can be included in the denotation of the predicate that a
proposition has any negative force at all. If the subject is
not definitely separated from the whole extent of P, it may
at least partially agree with it, and then there is no negation.
And this is independent of the extent of the subject. If
we deny P of only one individual — as when we say ' This S
is not P ' or ' An S is not P '—yet we must deny every P of
the S in question, or we have evidently denied nothing at all.
For we must needs deny not only that ' This S ' is this or that
particular P but that it is any P whatsoever. A negative pro-
position must assert that no individual included under the
subject can possess the attributes connoted by the predicate
in their entirety. But this is equivalent to excluding from
S every individual object called P, for only those objects
which possess all those connoted attributes can possibly bear
that name.
If we now sum up our results as to the distribution of
DEFINITION AND KINDS OF PROPOSITIONS.
173
each of the terms in each kind of proposition when read in
denotation, we have : —
their Subjects;
predicates ;
(1) Universals (A and E) distribute
Particulars (I and 0) do not.
(2) Negatives (E and 0) distribute their
Affirmatives (A and I) do not.
Thus, E distributes both subject and predicate.
A distributes its subject only.
0 distributes its predicate only,
1 distributes neither term,
* 73. Other Signs of Quantity.
This ; Each., Every, All, No ; Some ; are the only signs of
quantity which are recognized by Logic. If any others
occur they must be reduced to these before the propositions
can be used in strictly logical reasoning. Other marks of
quantity are, however, in common use in ordinary speech,
some of which it will be well to briefly examine.
(i.) Numerically definite statements of quantity. We
occasionally have such propositions as ' Three-fourths of the
S's are P,' ' Sixty per cent, of the S's are P,' ' Sixty per cent.
of the bullets hit the target.' If these are to be taken in
their strict and literal meaning, they must imply that every
S has been examined, and hence they involve a negative pro-
position in addition to the affirmative statement which is
explicitly made. In this sense they come under the head of
Exponible Propositions which are treated in section 75 (ii.).
Thus, 'Three-fourths of the S's are P' would imply that 'One-
fourth of the S's are not P.' If, however, such predications
are of any general importance a new name will soon be found
for the S's which are P, as distinguished from those which
are not P, and the propositions will, thus, become universal.
But this is seldom the case. Such statements generally refer
to some individual occurrence, and are of no general interest ;
for they in no way tend to the advancement of knowledge.
Or the numerical statement is only meant as a rough ap-
proximation ; and then, of course, it can be expressed with
little loss of meaning by the indefinite logical ' some.'
Book IT.
Ch. I.
Universal
Propositions
distribute S,
Negative
Propo-
sitions p.
Other signs
of quantity
must be re-
duced to one
of :— This ;
Each, Ever I/,
All, No;
Some.
Strictly de-
finite nume
rical state-
ments are
Exponible
Proposi-
tions.
If of any
general im-
portance
they tend
to become
Universal,
but this is
seldom the
case.
They are
often mere
approxinia
tions and
then mean
• some.'
174
PROPOSITIONS.
Book TI.
Ch. I.
Any in a
categorical
propositiou
means
' every.'
A few means
'some.'
When A few
is Collective
the proposi-
tion is
Singular.
Plurative
Propositions
are of the
form Most
S's are P.
If Most and
Few are in-
terpreted
strictly the
propositions
are E x -
ponible,
but this Is
not their
logical
sense.
(ii.) Any. Any S is P, 'Any house is a shelter in a
storm.' In such cases ' any ' is evidently exactly equivalent
to ' every,' and such categorical propositions are universal.
For we cannot assert that any S which may be taken at
random will be found to be P unless we know that every S
is P. Each is a denotative expression of the Generic Judgment.
(iii.) A few. This must be regarded as equivalent to
' some.' For when it is asserted that A few S's are P it need
not be meant to limit in any way the number of S's which
may be P, but simply to imply that only a small number of
instances of 5 have been examined, though every one of
those instances may, possibly, have been P. Sometimes it
is Collective and then means ' a small number,' as when it is
said 'A few Greeks defended the Pass of Thermopyl^,'
which they evidently did as a body. Such a proposition
would be better expressed, formally, in the form 'A small
body of Greeks defended the Pass of Thermopyte,' as this
shows its really Singular character.
(iv.) Plurative Propositions. In Ifost and Few we have
signs of quantity which it is possible to interpret either
strictly or vaguely, as we found to be the case even with
numerical statements. If taken in the strictest sense of the
words, they imply that every instance — or, at least, an ex-
tremely large number of instances — of 5 has been examined,
and that in the one case a number less than half (^Few S's),
in the other case a number greater than half but less than
all {Most S's) have been found to be P; but that the other
instances of S have been found not to be P. Thus, Few S's
are P would imply that Most S's are not P, though, at the
same time, A small number of S's are P ; and Most S's are P
would mean that though The majority of S's are P yet still
A small number of S's are not P. Such propositions would,
therefore, belong to the class Exponibles [see § 7.5 (ii.)].
But it does not appear that so strict a meaning is usually
intended ; and, therefore, Logic, restricting itself to that
minimum amount of assertion which a proposition necessarily
implies, can only regard Most as meaning ' more than half,
DEFINITION AND KINDS OF PROPOSITIONS.
175
but as not excluding 'all.' To express this, propositions
of the form Most S's are P are called Plurative. Thus, 3Iost
5's is perfectly indefinite beyond ' half,' and that limit but
roughly estimated. For instance, after being at a political
meeting an observer might say ' Most of the people present
wore a blue rosette,' and only mean to imply that the wear-
ing of the rosette appeared to him to be general, not
necessarily that he saw a few persons present who had no
rosette. Most is, then, equivalent to ' Some more than half '
and may in Logic be generally replaced by ' Some,' so that
Plurative Propositions may be regarded as Particular. It
has, however, been pointed out that, though from two really
particular propositions — Some M's are P, Some M's are S —
nothing can be inferred, yet from two pluratives — Most M's
are P, Most M's are S — the conclusion Some S's are P can be
drawn.
Few S's is indefinite up to the limit of * half ' (again in-
terpreted loosely), as Most Sis is above that limit. But when
we say Few S's are P we usually mean to imply simply that
Most S's are not P, and not that any S's necessarily exist
which are P. Few does not, therefore, necessarily exclude
' none'; for, in that case, Ifost would exclude ' all,' which
we have seen it does not logically do. In fact, to assert that
' Few novelists have ever been superior to Thackeray in
humour' by no means implies that any have ; such a sentence
simply expresses, in a most forcible way, the opinion that
'Most novelists are inferior to Thackeray in humour.'
Hence, Few S's are P must be regarded as really a Negative
Plurative Proposition, and as meaning 3fost S's are not P ;
whilst Few S's are not P is really the AflBrmative Plurative
Most S's are P. Logically, then. Few S's are P must be
treated as an 0 proposition, and expressed Some S's are not P;
whilst Feio S's are not P finds its logical expression as an
I proposition — Some S's are P. From Few M's are not P,
Few M's are not S we can infer Some S's are P in the way
noted above ; for the two given propositions are only nega-
tive in appearance, and their true force is expressed by
Most Ms are P, Most M's are 8.
Book II.
Ch. 1.
Most is per-
fectly inde-
finite above
' half," and
does not ex-
clude 'all,'
and is logic-
ally ex-
pressed by
' Some.'
Few is inde-
finite below
' half ' and
does not ex-
clud e
< none.'
But Few is
only another
way of ex-
pressing
Most . . . not,
and Few . .
not of ex-
pressing
Most.
Thus, Few
S's are P is
really nega-
tive,—O ;
and, Feio S's
are not P
really aflBr-
raative, — I.
176
PROPOSITIONS.
Book II.
Ch. I.
Hardly any
and Scnrce
mean Few.
Though all
terms can be
expressed
by S and p.
yet terms
are fre-
quently
complex.
A. term may
contain a
subordinate
clause,
introduced
by a Rela-
tivePronoun
and equiva-
lent to an
Adjective.
There are
two kinds
of these sub-
ordinate
clauses :
Hardly any S's are P, and The S's which are P are scarce,
are both exactly equivalent to Few S's are P ; that is, both
are logically expressed by Some {=zMost) S's are not P.
74. Propositions with Complex Terms.
As has been already stated (see § 68) all Categorical Pro-
positions can be expressed in one of the forms S is P or
S is not P with a definite, or indefinite sign, of quantity affixed
to the subject (§ 71). But it is by no means generally the
case that the propositions in use in common speech are as
simple in their structure as are those which, for the sake of
clearness, we have employed as examples. It is frequently
necessary to qualify or limit either the subject or the pre-
dicate. Hence, either term, or both terms, may be of any
degree of complexity, and care must be taken to determine
what the true predication really ia. If there is but one
verb there is no difficulty in this ; a many-worded term is
as easily recognized as a single- worded one. Thus 'The
highest mountain in Europe is Mont Blanc ' is obviously of
the form S is P, though subject and predicate have been
written in inverred order. But frequently the qualifications
required are expressed by subordinate clauses embedded in
the proposition, which then, of course, contains more than
one verb. In such cases the predications contained in the
subordinate clauses must be carefully distinguished from
that of the proposition as a whole. The question is, however,
of grammatical, rather than of logical, bearing, as it concerns
merely the various ways in which a given meaning can be
expressed in words. Still, a short discussion of it will pro-
bably conduce to a clearer understanding of some of the
more complex forms which a proposition may take.
A subordinate qualifying, or limiting, clause is introduced
by a Relative Pronoun, expressed or understood, and is
equivalent in meaning to an adjective or adjective phrase,
by which, indeed, it may be replaced. Thus ' In Memoriam
is a poem which contains many beautiful thoughts ' may be
equally well expressed '//j Memoriam is a poem containing
many beautiful thoughts.' Of these subordinate clauses
there are two kinds : —
DEFINITION AND KINDS OF PROPOSITIONS.
177
(i.) Explicative. In this case the qualificatioa contained
in the subordinate clause applies to every individual
denoted by the class name to which it is attached. The
sentence quoted above is an example of this, where the
qualification belongs to the predicate. In ' The natives
present, who all wore garlands of flowers, greeted us kindly '
we have a similar qualification of the subject, the true pre-
dication being evidently contained in 'greeted.' Whenever
a subordinate clause is Explicative, the name which is
qualified can always be substituted for the Relative Pronoun,
and the proposition thus formed remains true when removed
from its context. Thus, in the example given above, 'The
natives present aU wore garlands of flowers,' is a statement
whose truth is guaranteed by the given sentence as a whole.
Book II.
Ch. I.
Explicative,
wliich
qualify the
whole deno-
tation of the
term ;
(ii.) Determinative or Limiting. Here the subordinate
clause restricts the name it qualifies to a certain part of
its denotation. Thus, in ' All men who are over six feet in
height are eligible for enlistment in the Life Guards'
the qualifying clause evidently curtails considerably the
denotation of the subject. In such cases the name quali-
fied cannot be substituted for the Relative Pronoun ; to say
' All men are over six feet in height ' would be obviously
false. These limiting clauses always really affect the sub-
ject, even when it is not immediately apparent that they
do so ; for the subject is the more definitely determined
term in every proposition (see § 68). The occurrence of
a limiting subordinate clause is, therefore, a guide in
deciding what is the logical subject of an involved state-
ment. Thus, ' I have read all the books in this library
which treat of Politics' is logically expressed by 'All the
books in this library which treat of Politics are books which
I have read.' The subordinate clause which here appears
in the predicate is, of course, explicative. The general form
of propositions containing such limiting clauses is, there-
fore, Every S which is M is P, and their meaning is If any
S is M that 8 is P, which shows there is no hard and fast
distinction between Categorical and Hypothetical Judgments.
LOO. I. 12
and Deter-
mhiativf,
which liuiit
that denota-
tion.
These last
always
affect the
subject.
Its
rROPositioNs.
Book II.
ch. 1.
A Compound
Ca tegorical
Proposiiion
— two or
more propo-
sitions
joined in a
single state-
ment.
*75. Compound Categorical Propositions.
We have now to consider cases ia which two or more
propositions are really involved in a single statement, and
not merely apparent!}' so as in the instances considered in
the last section. This compound structure may be apparent
either from the form or only upon analysis of the meaning.
We thus have two kinds of Compound Propositions :^^
A Copulative
Proposition
— simple
union of
Affirmative
Proposi-
tions.
A Remotive
Proposition
— union of
Negative
Proposi-
tions.
A Discreiive
Propnsition
—union of
two affirma-
tive proposi-
tions by an
adversative
conjunc-
tion.
(i.) Compound in Form. These will require but few
words. They are of three classes : —
(a) Copulative Propositions, where we have a simple com-
bination of two or more Affirmative Propositions. There are
two or more subjects, or two or more predicates, or a
plurality of both ; as S and M are P; S is P and R ; S and M
are P and R and Q, etc. These are evidently merely briefer
ways of expressing each predication separately ; S is P, M is
P ; S is P, S is R, etc. Thus ' Gold and silver are precious
metals ' is plainly equal to ' Gold is a precious metal, and
Silver is a precious metal.' There will, of course, be as
many simple propositions as the product of the number of
separate subjects into the number of separate predicates; for
each predicate is united to each subject to form a distinct
proposition.
(6) Remotive Propositions, where we have a similar union
of two or more Negative Propositions, as No S nor M is P ;
No S is either P or R ; No S nor M is either P or R or Q; etc.,
which are equivalent to No S is P, No M is P ; No S is P, No
S is R, etc. Everything said above of Copulative Propositions
applies to Remotives.
(c) Discretive Propositions, where two affirmative pro-
positions are connected by an adversative conjunction — but,
nevertheless, although, etc. Here some opposition is implied
between the propositions joined, which are not expected to
be true together. Thus 'He is poor but honest' would
imply that most poor people are not honest. The pro-
position may evidently be contradicted by denying either
the poverty or the honesty, a sure proof of its compound
DEFINITION AND KINDS OP PROPOSITIONS.
179
chatacter ; for every simple proposition admits of but one Book II.
contradictory. ^^- '•
(ii.) Exponible Propositions, i.e., those whose composition
is not obvious from their form, and which, therefore, require
explanation to show what this hidden composition really is.
For example, all Numerically Definite Propositions [§ 73 (i.)]
and Plurative Propositions [§ 73 (iv.)] if strictly interpreted
would be Exponibles, for each implies both an affirmative
and a negative proposition ; but, as was said above, such
strict interpretation is logically incorrect in the latter case,
and by no means universal in the former. Any proposition,
however simple it may at first sight appear, which can be
contradicted in more than one way, is really compound, and
falls under this head.
Exponible Propositions may be classed as : —
(a) Exclusive Propositions. These contain a word such
as alone, which limits the subject, as ' Graduates alone are
eligible.' This is equivalent to the two propositions * Gra-
duates are eligible ' and ' No non-graduate is eligible,' and it
can be contradicted either by denying the eligibility of
graduates or by affirming that of others. It must be noted
that such an exclusive form distributes the predicate but not
the subject. It can, therefore, be expressed by the A pro-
position ' All eligible persons are graduates,' but this is
really an Immediate Inference from the original proposition
[see § 102 (ii.)]. If we would refrain from inverting subject
and predicate we must use both the propositions Some S is P,
No non-S is P to express the Exclusive 5 alone is P.
(6) Exceptive Propositions. These exclude a portion of
the denotation of the subject-term from the predication by
some such word as except, unless, as Every S except MS is P,
or Every S is P unless it is M. If the exception is purely
indefinite the proposition is particular. Thus ' Every man
except one assented ' does not permit us to assert of any
individual whether he assented or not, for the one dissentient
is unknown. We can, therefore, only say 'Some ( = all but
one) assented' and 'Some ( = an unknown one) did not assent.'
12—2
AnExponihh
Proposition,
is a com-
pound pro-
position
wiiose com-
position is
not obvious.
All proposi-
tions which
can be con-
tradicted in
more than
one way are
compound.
An Exclusive
Proposition
— SalO'.eii P
An Exceptive
Proposition
— Everi/Sis P
unkis itisM.
180
PROPOSITIONS.
Book TI.
Ch. 1.
Bxclusivcs
and Excep-
tives are iii-
tercliange-
able forms.
Inceptive
and Desitive
Propositions
state the be-
ginning or
«nd of some-
thing.
If, however, Ibe exception is cleai-ly specified we can make ;i
definite assertion about each individual, negative or positive
according as he falls within or without the excepted part.
Thus 'Every Frenchman is bound to perform military
service unless he is physically incapacitated ' enables us to
say definitely ' No physically incapacitated Frenchman is so
bound ' and ' All other Frenchmen are so bound.' And the
contradictory of either of these propositions denies the
original statement.
It must be noticed that Exclusives and Exceptives are only
two somewhat different forms for expressing the same mean-
ing. Either can, therefore, be changed into the other, the
excepted part of the one becoming the exclusive subject of
the other, or vice versd, and the quality being changed. Thus
* The miser does no good except by dying ' may be expressed
exclusively as ' Of all the acts of the miser his death alone
does good'; so the Exclusive 'Virtue alone can render a man
truly happy ' is the same as the Exceptive ' Nothing can
render a man truly happy except virtue.'
(c) Inceptive and Desitive Propositions. In these something
is said to begin or to end. They are I'esolvable into two
l)ropositions, the first declaring the state of things before
the change and the second the state after the change. Thus,
' After the Black Death there was a great dearth of labourers
in England ' implies (1) There was no such dearth just
before the time mentioned, (2) There was such a dearth after
it. Similarly, ' Ploughing by oxen has been discontinued in
England for many years ' implies (1) Ploughs were formerly
drawn by oxen in England, (2) The practice has been dis-
continued for many years. As these propositions make two
assertions relating to two different times they may be con-
tradicted by a denial referring to either time. Thus, the
last example may be contradicted by ' Ploughs were never
drawn by oxen in England' or bj ' The practice has not been
discontinued for many years.'
DEFINITION AND KINDS OF PROPOSITIONS.
181
HYPOTHETICAL PROPOSITIONS.
76. Nature of Hypothetical Propositions.
A Hypothetical Proposition is one in which the pre-
dication made in one proposition is asserted as a
consequence from that expressed by another. The pro-
position containing the condition is called the Antecedent
or Protasis, and is introduced b}'^ some such word as If;
that containing the result is termed the Consequent or
Apodosis. For example, in the sentence ' If all prophets
spoke the truth, some would be believed,' the Antecedent
is ' If all prophets spoke the truth,' and the Consequent is
* some (prophets) would be believed.'
The most general symbolic expression of the hypothetical
proposition is If A then G, where A and C represent not
terms but propositions. Other forms frequently given may
be included under this general expression. Of these one of
the most common is If A is B, C is D, but the one which most
truly represents the nature of the judgment expressed by
the proposition is If S is M it is P, where both the protasis
and the apodosis have the same subject. This form indi-
cates that the essence of the judgment is the explicit assertion
that the ground of the attribution of P to S is found in the
fact that S is M.
When the proposition contains four terms, and, therefoi'e,
falls at once under the form If A is B, C is D, analysis of the
meaning frequently shows that this is a mere accident of ex-
pression, and that the real subject of thought is the same in
both antecedent and consequent, so that the judgment may
be equally well expressed by a proposition of the form
If S is M it is P. Such reduction from one form to another is
always allowable when it does not affect the meaning of the
proposition, i.e. the real judgment. For example, the judg-
ment involved in the proposition * If the government of a
country is good, the people are happy ' finds perfect expres-
sion in * If the people of a country are well governed, they
are happy.' So, ' If the barometer falls, we shall have rain '
is reducible to ' If the state of the atmosphere causes a fall
Book II.
Ch. I.
A Ih^iwthtli
cal Proposi-
tion asserts
that the
Consequent
13 grounded
in the Ante-
cedent.
The most
general
symbolic
form is
ff A then 0,
but the most
expressive
is //" S ia M
it is P.
Proposi-
tions con-
taining four
terms can
generally be
reduced to
this form.
182
PROPOSITIONS.
Book II.
Ch. I.
and only by
such reduc-
tion can the
essential
unity of the
judgment
be made
explicit.
of the barometer, that atmospheric state will bring rain.'
' If we ascend a mountain, the barometer falls ' is equivalent
to ' If a barometer is taken up a mountain, it falls.' 'If a
child is spoilt, its parents suffer ' may be resolved into • If a
child is spoilt, it brings suffering on its parents.' ' If you
take a large dose of arsenic you will be killed ' is expressed
by ' If arsenic in undue quantity is taken into an animal
organism, it causes death in that oi'ganism.' *If patience is
a virtue, some virtue may be painful ' is the same as ' If
virtue includes patience, then virtue may be painful.'
In other cases the reduction is not so easy, and in order
that it may be effected links have to be supplied which are
not explicitly stated in the original proposition. In other
words, a hypothetical expressed with four terms conceals the
essential unity of the judgment it expresses, as there is in
the symbolic statement no obvious point of union between
the antecedent and the consequent. But the union is always
there in thought when the proposition is expressed — as all
real judgments always are — in significant words and not in
mere empty symbols, and is generally found without diffi-
culty. For example the point of unity involved in the
judgment * If some agreement is not speedily arrived at be-
tween employers and workmen, the trade of the country will
be ruined * is the recognition of the injurious effect of strikes
on trade, and the whole judgment may be expressed 'If
trade continue to be injured by strikes, it will soon be
ruined.' Sometimes the union is found in the recognition
that the subject of the apodosis is a species under the wider
subject of the protasis, as in * If demagogues are mischievous,
this stump orator is mischievous,' 'If violent emotion is
followed by a reaction, your fit of anger will lead to a re-
action ' ; * If all savages are cruel, the Patagonians are cruel.'
In other cases, both are recognized as species under the same
genus, as in ' If virtue is voluntary, vice is voluntary.'^ But,
in every case, where the judgment is really hypothetical — i.e.
asserts the consequences of a supposition — such unity ia
present. No doubt, the hypothetical form of proposition ia
occasionally used when no such judgment is really involved,
DEFINITION AND KINDS OF PROPOSITIONS.
183
as when Mr. Grimwig in Oliver Twist said, " If ever that boy
returns to this house, sir, I'll eat my head " ; which was only
a forcible mode of asserting disbelief in the realization of
the supposition stated in the protasis ; and was, therefore,
in its essence, categorical. Such propositions are obviously
of but small value in a theory of knowledge.
In discussing the universal categorical proposition we
found that its justification must be sought in a relation of
content which is most appropriately expressed in the Generic
Judgment, S is P, and that this form of judgment has both
an abstract and a concrete reference [see § 71 (i.) (&)]. If
such a judgment is true, it is because there is something
in the nature of S of which P is the necessarj' consequence.
If we make this explicit we have the hypothetical judgment
If Sis M it is P, where the sufficient ground for P is found in M.
Such a relation is nearly as explicitly stated in a Generic
Judgment of the form S which is M is P, a fact which shows
that the categorical and hypothetical forms are not separate
and distinct species of judgment, but merge into each other,
and are distinguished chiefly by the highly abstract character
of the latter [cf. § 74 (ii.)]- For in the hypothetical judg-
ment we have got away from the concrete ; our proposition
is an abstract universal, and deals with only one element in
a complex whole. The judgment, if true, is necessarily and
universally true, and yet may be incapable of concrete reali-
zation. This, indeed, is so with geometrical judgments, such
as ' If a triangle is right-angled, it is inscribable in a semi-
circle/ for no concrete diagram is ever a perfect right-angled
triangle or a perfect semi-circle. Still more clearly, perhaps,
is this seen in such a judgment as ' If a body is given a
certain movement, and if no counteracting conditions arc
operative, it will continue for ever to move in the samo
direction and with the same velocity.' This is impossible of
realization in sensuous experience, and yet is a fundamental
law of physics ; that is, a necessary element in our mental
construction of the material world.
It is evident from what has been said that the hypothetical
judgment is essentially abstract, and, as such, states con-
Boon II.
Ch. I.
The Hypo-
thetical
Judgment
makes ex-
pUcit the
ground of
connexion
of content
implied in
the Generic
Judgment,
and is f.n
abstract
universal.
A Hypo-
thetical
Judgment
may be true
though its
realization is
impossible.
184
PROPOSITIONS.
Book IT.
Ch. I.
Many Hypo-
thetical
Judgments
justify Enu-
merative
Conditional
Judgments,
whicli as
asserting
connexion
of pheno-
mena con-
tain a cate-
gorical ele-
ment.
The transi-
tion from
the cate-
gorical to
the hypo-
thetical is
gradual.
nexion of content. But as the generic judgment finds an
enumerative or denotative expression in the universal cate-
gorical proposition, so many hypothetical judgments can be
represented by what we may, perhaps, call the concrete con-
ditional proposition, whose general symbolic expression is
If any S is M thai S is P, or Whenever an S is M that S is P.
The latter statement is to be preferred, as the use of If in
the former suggests an abstract connexion of content, rather
than that simultaneity of occurrence in experience, which is
what the denotative form explicitly asserts. The denotative
form, then, has a distinct reference to occurrence in time and
space ; it expresses connexion of phenomena, and is, there-
fore, only appropriate when such occurrence is possible. In
other words, it contains a distinctly categorical element, and
is practically equivalent to the proposition Every S which is
M is P. In form, like the proposition Every S is P, it suggests
that its basis is enumeration of instances, but its real justi-
fication is connexion of content expressed by the pure
abstract hypothetical, and found to be realized in pheno-
mena.
77. Relation of Hypothetical to Categorical Proposi-
tions.
We have seen that as judgment becomes less occupied
with concrete and complex phenomena regarded as wholes,
and concerns itself more and more with abstract relations of
content, it gradually passes from the categorical to the hypo-
thetical form (see § 76). But the fact that the generic
judgment mediates this transition, whilst the denotative con-
ditional form mediates a transition in the opposite direction,
shows that no strict line of demarcation can be drawn be-
tween them as modes of thought. With their form as
propositions, the case is, of course, different ; here language
makes fixed and definite a distinction which is far from being
so fixed in thought. Sometimes, it is an accident whether a
judgment is expressed in the hypothetical or the categorical
form ; for instance ' Right angled triangles have the square
on the hypoteneuse equal to the sum of the squares on the
DEFINITION AND KINDS OF PROPOSITIONS.
185
sides ' really gives the ground for attributing the predicate
to the subject and would appropriately take the hypothetical
form ' If a triangle is right-angled, the square on the hypote-
neuse is equal to the sum of the squares on the sides.' But,
in all cases, it should be considered whether the categorical
or the hypothetical form is the more appropriate, and this
depends upon the degree of abstraction involved in the
judgment.
The question whether the categorical and hypothetical
forms can be reduced to each other without change of mean-
ing has been much disputed. From what has been said
above it is evident that though the one essential nature of
judgment pervades both, yet that each emphasizes just that
aspect which is only implicit — and often but vaguely so — in
the other. Thus, the categorical emphasizes reference to
concrete reality existing in space and time ; whilst the
hypothetical brings into prominence the element of relation
of content which is the implicit justification of the categorical.
The two forms cannot, therefore, be regarded as interchange-
able. The element of supposal which is prominent in the
hypothetical disappears if the judgment is written in the
categorical form ; and on the other hand it is introduced
ah extra when a categorical proposition is translated into the
hypothetical form. In all cases where the categorical is
abstract, such translation is, no doubt possible ; but in many
cases, especially when the proposition is a definition, it is
inappropriate. We can say ' If gold is, it is yellow,' or ' If
a lion is, it is carnivorous,' but the form suggests the possi-
bility of the non-existence of the subject, and is, consequently,
not an adequate expression of the judgment really made.
At the same time it must be granted that it is not always
easy to say which kind of proposition will most appropriately
express a given judgment, for most human knowledge is
neither entirely in the realm of concrete facts in all their
particularity, nor in that of pure abstract relation, which
exists only in idea. There are both categorical and hypo-
thetical elements in most of the judgments men actually
make, as is, indeed, shown by the frequency with which the
Book II.
Ch. I.
The two
forms can-
not be re-
duced to
each otiier,
as each
makes pro-
minent an
element
only latent
in the other
186
PROPOSITIONS.
Book II. generic form of judgment — in which we have, side by side,
^^- '• both an abstract universal and a concrete character — is
adopted.
Hypotheti-
cal Proposi-
tions may
be negative.
The most
general form
is — //" A. then
not d.
True hypo-
theticals
are univer-
sal,
but the ex-
plicit Modal
Particular —
//' Sis HI it
may be P —
has a hypo-
thetical
form.
All Modal
Particulars
have deno-
tative
forms.
78. Quality and Quantity of Hypothetical Propositions.
(i). Quality. Hypothetical Propositions admit of dis-
tinctions of quality. Of course, a negative antecedent does
not make a hypothetical proposition negative ; for the con-
sequent is still asserted to follow as a result of the antecedent.
Thus // S is not M it is P is affirmative — 'If a swan is not
white, it is black.' It is when the connexion of the apodosis
with the protasis is denied that the proposition is negative.
The most general symbolic form is 7/" A then not 0, and the
most expressive If S is M it is not P, e.g. ' If a man is honest
he will not deceive his fellows.'
(ii). Quantity. The essence of true hypothetical judg-
ments is their abstract, and necessarily universal character.
But cases may arise in which though a connexion is estab-
lished between P and A/, yet M may not be the full ground of
P, or, though it is the complete ground, may not be universally
operative, or may be liable to be counteracted by other
conditions. In such cases the appropriate proposition takes
the general form If S is M it may be P, and negatively
//' S is M it need not be P, which are more explicit expressions of
tlie Modal Particulars than were considered in an earlier
section [see § 71 (ii.)] . The corresponding denotative forms
— which in these cases can always be found — are Sometimes
when an S is M, it is [or is not'\ P. As examples we may take
' Sometimes when men are much worried, they commit
suicide ;' ' If a man is punished for a crime, he, perhaps, will
not transgress again' ; 'Sometimes when a target is aimed
at, it is not hit ' ; ' Although a man tries his hardest, he may
not succeed.' The last three examples are particular negative,
the first particular affirmative. The * Sometimes ' in the deno-
tative examples, it must be remembered, is as purely indefinite
as is 'Some 'in a particular categorical [see § 71 (ii.)] ; it must
not, therefore, be regarded as excluding ' always.'
DEFINITION AND KINDS OF PROPOSITIONS.
187
Book IT.
Ch. I.
The great characteristic of all particular propositions is
their imperfect and incomplete character. They, on their
very face, therefore, challenge completion. They are but
stepping stones on the way to that more exact and complete tionsrepre-
knowledge which finds expression in the true and universal fe" *j\^g!'^'
hypothetical.
Particular
Proposi-
lueuts.
DISJUNCTIVE PROPOSITIONS.
79. Nature of Disjunctive Propositions.
A Disjunctive Proposition is one which makes an
alternative predication. The most general symbolic form
is Either X or Y, where X and Y represent propositions.
But as the most expressive form of the hypothetical is that
which makes explicit the unity of the judgment, so here the
symbolic form S is either P or Q representing the prescription
to the same subject of an alternative between a definite
number of predicates, most truly expresses the nature of
the judgment. In the simplest cases these alternative
predicates are known to be contained under a wider pre-
dicate M which can be asserted of S {cf. § 68). For example
* He is either a doctor, a lawyer, a clergyman, or a teacher '
may be expressed in the simple categorical proposition ' He
is a member of a learned profession.' So, we may say ' Any
swan is white or black ' where the wider predicate is the
possession of colour. But, though such subsumption is
always theoretically possible, in most cases the alternative
predicates have never been brought under such a wider
predicate ; for occasion has not arisen to make such a wide
and indefinite assertion about any subject. For instance, we
may say ' The election will turn either on the Eight-Hours
Question or on the Question of Home Rule,' but we have no
word which would exactly cover these two cases, and yet be
sufficiently significant to express our meaning if we affirmed
it as a predicate of the given subject.
The effect of sin alternative predicate is to increase the indefinite-
ness of its extent. As all disjunctive propositions are affirmative
A Vhjunciivt
Proposition
makes an
alternative
predication
— S is either P
or Q.
These alter-
natives may
always bo
theoretic-
ally brought
under a
■wider predi-
cate,but this
is seldom
done in
practice.
An alterna-
tive in-
creases the
indefinite-
ness of the
scope of the
predicate.
188
PROPOSITIONS.
Book IT.
Ch. 1.
Proposi-
tions with
alternative
subjects are
not classed
as Disjunc-
tives.
Logicians
differ as to
whether or
not the dis-
junctive
form neces-
sitates the
mutual ex-
clusiveness
of the alter-
native pre-
dicates.
In many
cases the
alternatives
are, in fact,
exclusive,
but this Is
due to their
natural in-
compati-
biUty.
(see § 81) the predicate is undistributed, and, therefore, indefinite
up to the extent of P (cf. § 72) ; by adding or Q this indefiniteness
of range is increased, but no fundamental difference is made in the
character of the proposition. If the alternation was in the subject
— as, if we should say S or X is P, or 5 or X is P or Q — then
we should change a determinate subject into an indeterminate one,
if the original subject, S, was distributed. Such propositions, how-
ever, are not usually called Disjunctive.
The important question as to the interpretation of Dis-
junctives is whether the alternative form, necessitates that
the several predicates conjoined in it be mutually exclusive
in their application. When it is said S is P or Q, is it
necessarily implied that S cannot be both P and Q ? On this
point there has been great difference of opinion amongst
logicians. It is gi'anted by all that in a great number,
perhaps in the great majority, of cases, the alternatives
do, as a matter of fact, exclude each other. Such are :
' He will either pass or fail ' ; * This book is to be bound
either in calf or in morocco'; 'The rebellion will either
succeed or be crushed' ; 'Any swan is either white or black' ;
' These plays were written either by Shakespeare or by
Bacon.' In all these, and in many similar cases, the accept-
ance of one alternative involves the denial of the other,
and it is argued that whenever this does not appear to be
the case, or is not meant to be the case, it is because of
"our slovenly habits of expression and thought "and that
these are "no real evidence against the exclusive character of
"disjunction" (Bradley, Prin. of Log., p. 124). But, when
such instances are examined more closely there are found to
be two possible explanations of this exclusiveness : — (a) the
terms are, in each case, mutually incompatible in their very
nature, so that both cannot possibly be affirmed in the same
sense of the same subject ; the exclusiveness may be due to
this, or (b) it may be a necessary consequence of the dis-
junctive form itself. As the former explanation is mnni-
festly sufficient in itself, such examples will not prove
whether or not the disjunctive form, as a form, necessarily
involves exclusion. To settle this — which is the point really
DEFINITION AND KINDS OF PROPOSITIONS.
189
in dispute — we must examine some examples in which terms
are disjunctively predicated which are not incompatible with
each other in their very nature. If we still find that the
alternative predicates are mutually exclusive we must regard
this as due to the disjunctive form alone. Let us take such
propositions as ' All candidates must be graduates either of
Cambridge, of Oxford, of Dublin, or of London' ; 'He is
either very timid or very modest ' ; ' He is either a knave or
a fool.' Do we mean, in either case, to deny the possibility
of all the predicates being, at the same time, attributes of
the subject ? May not a candidate be a graduate of more
than one of the universities mentioned ? or would such
plural honours be a bar to his success ? Do we deny that
the person in question can be both timid and modest ? Do
we exclude the possibility that the other is both a knave and
a fool ? In fact, in one sense, may not every knave be said
to be necessarily more or less of a fool, in that honesty is
the highest wisdom ? In each of these examples it seems
certain that no exclusion of one predicate by another is in-
volved. If, then, cases can be found — and, of course, many
more might be instanced — in which the alternative predicates
are not mutually exclusive, it follows that when such ex-
clusiveness does exist it is due to the character of the
alternatives themselves and not to the disjunctive form of
the proposition in which they happen to occur. That form,
as a form, implies no such mutual exclusiveness. If we
wish to show formally that our alternatives are intended to
be exclusive, we can do so by writing S is P or Q, but not both,
which is, of course, a compound proposition [see § 75 (i.) (c)].
By adopting this non-exclusive view of disjunctives, we are,
besides, obeying the valuable logical Law of Parsimony —
that whenever a choice is offered us between a more and a
less determinate meaning, it is safer to choose the latter ;
for we thus avoid the danger of implying in, or inferring
from, any statement more than is justified.
But though the form does not imply mutual exclusiveness yet
there is necessarily some element of difference — that is, of exclusive-
ness— in all alternatives, as otherwise they could not be alternatives
Book II.
Ch. I.
When the
alternatives
are not in-
compatible
they are not
exclusive.
Exclusion is
not, there-
fore, due to
the disjunc-
tive form of
proposition.
In so far as
alternatives
are differ-
ent, they are
exclusive.
100 PROPOSITIONS.
Book II. at all, for they would be identical. In so far as two terms, de-
Ch. I. noting different species of the same genus, differ from each other
they exclude each other. For instance, the being a graduate
of Cambridge is not tlie same as being a graduate of London —
the two agree in the fact of graduation, but differ in the place
where the graduation occurred. As these places are different,
they exclude each other ; that is, one act of graduation could not
take place both at Cambridge and at London. Similarly, the
qualities of timidity and modesty differ from each other in some
points, and in these points of difference are mutually exclusive, or
the two ideas would merge into one. In the same way, the points
in which knavery differs from foolishness are points of exclusion.
So it is always ; in so far as notions differ, they exclude each other;
were it not so they would melt into one, for the mentally indis-
tinguishable is a mental unity. It follows that the logical ideal of
a disjunctive judgment is one in which the alternative predicates
are exhaustive of the denotation of the subject, exclusive of each
other, and co-ordinate species under the subject genus. But our
treatment must cover cases in which neither in thought nor in
language is this ideal realized, and our formal interpretation, there-
fore, of a disjunctive proposition must be that the alternatives are
not necessarily exclusive.
SiimiiKiry. We reach, then, this result : the alternatives in every
disjunctive proposition have something iu common; for
they are always capable of being subsumed under some
wider predicate of the same subject : as species of this
genus they are sometimes, iu their very nature, incompatible
with each other, and are, therefore, exclusive : but, in other
cases they are, in their nature, compatible with each other,
and are then not exclusive, though they can never be
identical : hence the degree of exclusiveness depends on
the nature of the alternatives themselves and not on the
disjunctive form of proposition ; in other words, it is
material, not formal.
80. Relation of Disjunctive to Hypothetical and Cate-
gorical Propositions.
As every disjunctive proposition prescribes an alternative
between a definite number of different predicates, one or
t»EPINITIOK AND KINDS OF PROPOSITIONS.
191
more of which is, therefore, affirmed of the subject, it
follows that, if some of these alternatives are denied, the
others are affirmed ; either categorically — if only one is left,
or disjunctively — if more than one remains. Thus, if we
start with the assertion 5 is either P or Q, and then deny that
S is P, we must necessarily proceed to affirm that S is Q.
Similarly, if the original assertion is 5 is P or Q or R or T,
and this is followed by the denial that S is either P or Q, the
affirmation that S is either R or T is a necessary result. As
this affirmation of one, or more, of the alternatives is an
inference from the denial of the rest of them, it follows that
all Disjunctive Propositions involve a hypothetical judg-
ment of the general form If S is not P it is Q, or. If S is not Q
it is P. These propositions are exactly equivalent to each
other, each being, in fact, the Obverted Contrapositive of
the other (see § 105). But the Disjunctive Judgment makes
explicit a categorical element which is wanting to the hypo-
thetical. Were we confined to the latter, thought would be
condemned to an endless regress. For though If S is M it is P,
gives us in M the ground of P, yet we must go on to
similarly ask for the ground of M. This regress can only
be avoided by assuming that the judgment refers to a more
or less self-contained system. It is such a system that the
disjunctive judgment in its ideal form makes explicit in its
enumeration of the sub-species under the subject genus. It
is in the exhaustive character of this enumeration that the
sufficiency of the hypothetical as a statement of a condition
is found. Hence, we find in the disjunctive the mode of
expressing that systematic connexion which is the only form
in which we can think reality.
Those logicians who adopt the exclusive view of the disjunctive
form deny that its full meaning can be expressed in any one
hypothetical proposition. For, if P and Q are mutually exclusive, it
" follows trom S is either P or Q not only that If S is not P it is Q and
If S is not Q it is P, but also that If S is P it is not Q, and If
S is Q it is not P, and one of these latter forms is required
together with one of the former to express the force of the dis-
junctive, which would then be given either by the pair IfS is not P
Book II.
-sv Ch. I.
As the
affirmation
of one alter-
native in a
disjunctive
follows from
a denial of
all the other
alternatives,
every such
proposition
involves a
hypothetical
of the form
// S is not P
it is Q.
It the alter-
natives were
exclusive, a
disjunctive
could only
be expressed
by:
Jf S {■! not P
it is Q ; and
J/S is P it is
not Q.
192
PROPOSITIONS,
Book II.
Ch. I.
All Disjunc-
tives are
Affirmative.
it is Q, and If S is P it is not Q ; or by the pair If S is not Q it is P,
and 1/ S is Q it is not P. This extra implication which such an
interpretation gives to a disjunctive brings out the force of the
remark made near the end of the discussion of exclusiveness, in the
last section, on the logical Law of Parsimony. The exclusive view
evidently binds us to a greater number of assertions than the non-
exclusive view. As we have adopted the latter we cannot, of course,
regard the inferentials If S is P it is not Q and If S is Q it is not P
as involved in, or as legitimate expressions of any part of the mean-
ing of, the formal disjunctive S is either P or Q.
81. Quality and Quantity of Disjunctive Propositions.
(i.) Quality. It follows from the very nature of Dis-
junctive Propositions (see § 79) that they can only be
affirmative ; for they must give a choice of predicates, one
or other of which must be affirmed of the subject. Proposi-
tions of the form S is neither P nor Q give no such choice, nor
do they increase the scope of the predicate as do propositions
of the form S is either P or Q {see § 79). They are essentially
Compound Categorical Propositions [see § 75 (i.) (&)]. It is
true we can have a disjunctive proposition involving negative
terms— as S is either P or non-Q — but the disjunction is as
affirmative as if both terms were positive.
(ii.) Quantity. The ideal disjunctive judgment is always
both abstract and universal, and expresses relation of con-
tent. But like the Generic Judgment it can be expressed in
terms of denotation, and in this case we get distinctions of
quantity. Thus we get propositions of the form Every S is
either P or Q ; * Every idle man is either incapable of work
or morally blameworthy,' and Some S's are either P or Q ;
' Some laws are either opjjressive or are rendered necessary
by an abnormal state of society.' It is evident, however,
that such particular propositions are of practically no logical
value.
82. Modality of Propositions.
Modality has reference to the degree of certainty, or incertainty,
of a judgment, and is concerned with the various ways in which
differences in this respect are expressed. De Morgan defines a Modal
DEFINITION AND KINDS OP PROPOSITIONS.
193
Proposition as "one in which the affirmation or negation was
"expressed as more or less probable." (Formal Logic, p. 232.)
Some of the scholastic writers on Logic regarded all adverbial
modifications of a proposition as a kind of modaUty. They dis-
tinguished, therefore, between Material Modality — where the modi-
fication belonged either to the subject or to the predicate of the
proposition ; as. He spoke angrily ; and Formal Modality — where
the modification affected the certainty of the relation asserted to
exist between the subject and the predicate. This latter, however,
is the only kind of modification to which the term Modality is
rightly applicable.
Extreme Conceptualists [see § 8 (ii.) (&)]— as Mansel and
Hamilton — refuse to admit the discussion of this subject into Logic
at all, as it is essentially concerned with the matter of the judgment
and not with its mere form. But on the wider view of the science
here adopted {see § 9) it is necessary to examine it, and to estimate
its logical value.
Aristotle divided Modal Propositions into four classes — the
Necessary, the Contingent, the Possible, and the Impossible. This
is, evidently, purely objective ; the subjective aspect of the mind
towards the Necessary and the Impossible is identical — both are
cases of full assurance. It is also difiicult to see how the amount of
belief in a contingent proposition differs from that in a possible
one ; this latter distinction is, in fact, extremely vague. Only by
a reference to the things themselves can it be decided whether the
subject S must necessarily possess the predicate P, whether the two
are absolutely incompatible, or whether their union is more or less
likely. Some scholastic writers, ndeed, reduced the forms of
Modality to two — the Certain and the Possible, Others, influenced
by the analogy with the fourfold scheme of propositions ((/. § 72),
retained the original four, and connected the Necessary with A, the
Impossible with E, the Contingent with I, and the Possible with 0
propositions. The distinction they drew between the two latter
modes was that the Contingent was ' what is, but may not be in the
future,' and the Possible 'what is not, but may be in the future.'
Such a division of Modals, founded as it was upon a purely objec-
tive view of the province of Logic, and utterly artificial as it is, has
little to recommend it. As Dr. Venn says : "A very slight study
" of nature and consequent appreciation of inductive evidence suffice
"to convince us that those uniformities upon which all connexions
LOG. I. 13
Book II.
Ch. I.
Modality is
concerned
with the de-
gree of cer-
tain ty,orun-
certainty of
a judgment.
Aristotle di-
vided Modal
propositions
into :
1. Neces-
sary.
Contin-
gent.
Possible.
Impossi-
ble.
194
PROPOSITIONS.
Book II.
Ch. I.
Kant di-
vided Modal
Judgments
into :
1. Apodeic-
tic.
2. Assertory.
3. Problem-
atic.
The distinc-
tions do not
hold from
the formal
point of
view,
hut they are
important
from the
standpoint
of know-
ledge.
Generic and
Hypothe-
tical Judg-
ments are
apodeictic ;
"of phenomena, whether called necessary or contingent, depend,
"demand extremely profound and extensive enquiry; that they
" admit of no such simple division into clearly marked groups ;
"and that, therefore, the pure logician had better not meddle
" with them " (Logic of Chance, p. 307).
Ka7it regarded Modality from a standpoint essentially different
from that of Aristotle. His view was purely subjective ; he con-
sidered simply the amount of our belief in a judgment. As he dis-
tinguished three degrees of assurance, so he divided Modal Pro-
positions into three classes— the Apodeictic — S must he P ; the
Assertory— S is P; the Problematic— S may be P {cf. % 48). " The
"apodeictic judgment is one which we not only accept, but which
*' we find oiu'selves unable to reverse in thought; the assertory is
" simply accepted ; the problematic is one about which we feel in
" doubt " (Venn, Logic oj Chance, p. 310). If we consider this
distinction simply from the standpoint of formal logic, we cannot
accept it. The apodeictic judgment differs from the assertory only
in the emphasis with which it expresses universal connexion. It
is, therefore, formally nothing more than an assertory judgment —
it only asserts more vigorously. Both judgments claim to be true,
and both express complete belief. For if the belief in the assertory
judgment were not as strong as in the apodeictic, the former would
contain an element of doubt, and would be merely problematic.
But it does not follow that the idea underlying the doctrine of
modality is a useless or even an unimportant one. From the point
of view of knowledge there is undoubtedly a distinction between
truth which is regarded as necessary, and whose overthrow would
affect the whole of our mental construction of the world, and pro-
positions which may be accepted as true, but whose overthrow
would originate no such mental chaos. And this distinction we
find between judgments of connexion of content and judgments
based on mere experience. The former — the Generic and Hypo-
thetical Judgments — express the only interpretation we can give
of certain aspects of reality, or certain elements of our experience,
which is consistent with our conception of the universe as a whole.
Such judgments are in the fullest sense, universal, and as universal,
they are necessary. But judgments of uncontradicted experience
have, as has been already pointed out, no such necessary character
(c/". § 71). Hence, all Generic and Hypothetical Judgments are
apodeictic. Again we have seen that the true ground of the parti-
cular proposition is found in the imperfect determination of the cod-
DEFINITION AND KINDS OF PROPOSITIONS.
195
tent of 5, and that this is most clearly expressed in the Modal
Particulars S rnay be P, and S need not he P, or in the more explicit
forms If S is M it may he P and If S is M it need not be P which
hold the same relation to the universal hypothetical that the modal
categorical particulars do to the generic judgment. Hence all
particular judgments are in their essence problematic ; all truly
universal propositions — i.e., hypothetical and generic judgments —
are apodeictic, and all propositions based on mere uncontradicted
experience are assertory. The whole subject will be clearer after
Induction has been discussed, and we shall then return to it
(see § 160).
Book II.
Ch. I.
Particular
Judgments
are problem
atic ;
Enumora-
tive Judg-
ments (if ex-
perience am
assertory.
13—2
CHAPTER TI.
IMPORT OF CATEGORICAL PROPOSITIONS.
Book II.
Ch. II.
The question
of the im-
port of pro-
positions in-
volves :
(a) Wbat is
related ?
(6) What is
the rela-
tion?
e) Is exist-
ence im-
phed?
83. Predication.
The whole treatment of Logic must depend upon the view
held as to the nature of the predication made in a categorical
proposition, and the consequent import of that proposition
[of- §§ 8 (ii.), 66 ad fin.]. The first point to be settled in
considering this is whether such a proposition expresses a
relation between words only, or between ideas, or between
things. The different answers which have been given to this
question were stated in § 8 (ii.) ; and the view here adopted
was set forth in § 9 — viz., that a proposition interprets an
objective fact, by stating a relation which is apprehended in
thought and expressed by language. It is not necessary to
say more on this fundamental point ; we may pass on to con-
sider two other questions : —
(o) The nature of the relation expressed in predication,
and, as a consequence of this, the aspect in which
the terms should be regarded.
(6) Whether or not a categorical proposition implies the
existence of objects denoted by the terms.
On the first of these subjects several different views have
been held, which will be discussed in the next five sec-
tions ; the question of existence will then be considered in
§89.
IMPORT OF CATEGORICAL PROPOSITIONS.
197
84. The Predicative View.
The predicative view regards the relation expressed be-
tween the terms of a formal categorical proposition as that
between subject and attribute. It makes the element of
denotation in the subject, and that of connotation in the
predicate, the more prominent. The subject is thought as
the name of certain objects, and though it is true they are
indicated indirectly — that is, as members of a class to which
they belong solely because they possess certain attributes
[c/. § 71 (i ) (Z*)]— yet the attention is fixed on them as things,
not on the attributes which their names connote. It is of the
things which possess the attributes that the assertion is made ;
the attributes themselves are not definitely before the mind
at all, but are merely symbolized by the name. Hence it is
that a word which is primarily attributive — such as an adjec-
tive— cannot form the subject of a proposition {cf. § 25). But
in the case of the predicate we are thinking of the attributes
which we affirm of the objects denoted by the Subject ; for
our whole purpose is to predicate such a qualification. We
think, not of two sets of objects which we compare, but of
one set of which we assert an attribute. This is most obvious
when the predicate is a directly attributive word, as when we
say 'AH metals are fusible,' ' The dog is barking'; but it is
equally true when the predicate is a substantive. For instance,
in the proposition ' All the candidates for the appointment
are graduates,' if we examine the meaning we shall find it to
be that the candidates in question possess certain qualifica-
tions which are conveniently summed up in the word
'graduates'; we do not think of graduates as individuals,
but predicate the connotation of the name. And the same
holds in every case ; the predicate asserts a qualification of
the subject, and this qualification consists of the attributes
implied by the predicate.
This is the natural interpretation of a categorical proposi-
tion whose subject is expressed with a sign of quantity,
though it must bo borne in mind that its foundation is to be
found in the Generic Judgment whose essence is that it deals
with content of both subject and predicate. It is also
Book II.
Ch. II.
The Predica-
tive Kiew re-
gards the
relation be-
tween the
terms as
that be-
tween sub
ject and at-
tribute.
The S is,
therefore,
read in de-
notation and
the P in con-
notation.
This inter-
pretation
gives rise to
the four-foH
scheme of
proposi-
tiuiit).
198
PROPOSmONS.
Book II.
Ch. II.
The class
view is that
the S is in-
cluded in
the class de-
noted by
the P.
Both terms
are said to be
read in de-
notation.
Tliis view
b ises kiiow-
Icilge upon
enumera-
tion of in-
stances and
neglects the
unity of
judgment.
fully consistent with the four- fold scheme of propositions ; for,
if the predicate names an attribute and the subject indicates
certain objects, we must either affirm or deny the former of
the latter, and in each case the assertion must be made
either of a definite or of an indefinite number of individuals
{cf. § 72).
*85. The Class-inclusion View.
On the class view the relation between the subject and
predicate is that of inclusion in a class. Both terms are said
to be read in denotation, and the proposition is held to assert
that the objects denoted by the subject are to be found
amongst those denoted by the predicate. Whether the sub-
ject is used collectively or distributively [see § 27 (ii.)] is of
no importance ; in each case it forms part of the predicate.
The predicate, however, is necessarily regarded as a whole
or class — i.e., it is used collectively. ' All owls are birds '
means that each owl — and, therefore, the whole class of owls
regarded collectively — is to be found within the whole class
of birds. This collective interpretation of the predicate is
the only permissible one ; for to take it distributively would
give no real meaning at all ; — each owl is certainly not any
bird. The only possible meaning is that the total class com-
posed of birds contains every individual which can be called
an owl ; or, what is exactly the same thing, that it contains
the whole class of owls. Similarly, a negative j^roposition
means that every individual denoted by the subject is
excluded from the whole class of things denoted by the
predicate, and that the two classes are, therefore, entirely
separate.
With respect to this view it may be pointed out, first, that
though it is, of course, possible to attend to the denotation
of the predicate (c/. § 72), yet in judging it is more natural
and common not to do so. No doubt, as every general term
can be considered both in denotation and connotation, it is
possible so to interpret propositions, and such a mode of
interpretation has been common amongst purely formal
logicians, for it lends itself readily to the exposition of the
IMPORT OP CATEGORICAL PROPOSITIONS.
199
Aclassinter-
pretation of
both S and P
leads to a
five-fold
scheme of
proposi-
tions.
formal aspect of reasoning. But to adopt this interpretation Book II.
as the fundamental import of judgment is to fall into the __ '
error of basing knowledge upon a supposititious possibility of
a complete enumeration of instances, instead of upon an
investigation directed to establish connexion of content.
Moreover, this view of predication neglects the essential
unity of the judgment and regards it as stating a relation
between two independent objects rather than as expressing
an interpretation of one element or aspect of reality.
Further, it must be pointed out that if both subject and
predicate are regarded as classes — and, as was said above, on
this view, the subject may, and the ])redicate rmist, be always
so regarded — then the four-fold scheme of propositions is not
an exhaustive statement of the relations which may exist
between them. We require a five-fold scheme ; for if we
have two classes, S and P, it is evident : —
(a) They may exactly coincide and so be identical
wholes.
(6) S may be included in but not form the whole
of P.
(c) S may include P and not be wholly exhausted.
(d) S and P may partially include and partially exclude
each other.
(e) S and P may wholly exclude each other.
To express these in ordinary language we must give
' some ' the meaning ' some but not all ' [c/. § 71 (ii-)]- We
then have : —
(a) All S is all P.
lb) All S is some P.
(c) Some S is all P.
(d) Some S is some P,
(e) No S is any P.
But, it should be remembered that we have here a state- This scheme
ment of the actual relations which must hold, in fact, relations of
between two classes, not of our knowledge of those rela- tnoWledge^
tions. This scheme, therefore, furnishes us with no means ot them,
200
PROPOSITIONS.
Book II.
Ch. II.
and makes
every propo-
sition singu-
lar.
Hamilton
held that a
proposition
expresses an
equation ;
tliat the
predicate is
always
quantified
in thought;
and that
this quanti-
fication
should bo
expressed.
From the
four forms of
proposition
he obtained
eight.
of expressing the very common state of doubt, when we
know that every 5 is P, but do not know whether or not any
other objects are P as well. Moreover, each of the above
propositions is Singular, as each term is necessarily taken
collectively. For both these reasons the scheme is inappro-
priate to the purposes of logic, and any interpretation of the
proposition which, when strictly carried out, leads to it is
thereby condemned.
*86. Quantification of the Predicate.
As the application of S and P is identical, it follows that,
if both the terms of a proposition are read in denotation,
tlie relation between them is reduced to an equation ; and it
would seem to follow, on this view, that it is necessary to
quantify the predicate. Sir W. Hamilton held this to be
the true relation between the terms of a proposition, and
the only way in which a judgment is really thought. He,
therefore, assumed that the predicate is always quantified in
thought, and hence — as a consequence of his fundamental
postulate, " To state explicitly what is thought implicitly "
{see § 21) — that it should be always quantified, on demand,
in expression. He then took the recognized four forms of
proposition A, I, E, 0, and by making the predicate of each
(1) universal and (2) particular he obtained an eight-fold
scheme. Thus : —
From A
From I
FromE
From 0
( All Sis all P- • - - a/a ' - - TJ ■ - - SuP.
(Alls is some P- - - afi - - - A S a P.
5 Some SisallP - - - ifa - - - Y - - - S 7j P.
\ Some S is some P - - ifi - • - 1 - - - S i P.
No S is any P ana - - ■ B ■ - - S e P.
No S is some P - - - am - - - n - - - S ri P.
S Some S is not any P - ina ---O---S0P.
\ Some S is not some P • ini - - - « - - - S w P.
The symbols afa, etc., were employed by Hamilton. In
them / stands for the affirmative copula, n for the negative
copula, those letters being the first consonants in the words
afjinno and nego; a represents a distributed, and i an undis-
IMPORT OF CATEGORICAL PROPOSITIONS.
201
tributed term. Of course the subject term is always placed
first. The symbols commonly employed, however, are U, A,
y, etc., which were introduced by Archbishop Thomson.
Using these in the way adopted in § 72 we obtain the short
symbolic expressions S u P, S a P, etc.
Hamilton supported his position that the predicate is
thought as quantified by urging that it is often quantified in
expression, either directly — as when we say ' Sunday,
Monday, etc., are all the days of the week ' — or, more fre-
quently, indirectly, by the use of Exclusive and Exceptive
forms of propositions. Thus, he says, ' Of animals man
alone is rational ' means ' Man is all rational animal,' and ' In
man there is nothing great but mind' is equivalent to
• Mind is all humanly great,' that is ' Mind is all that is
great in man.'
From this enlarged scheme of propositions great advan-
tages were said to flow. Amongst the more important results
claimed for it were : — that it made evident that the true rela-
tion between the subject and predicate of a proposition was
an equational one ; that it reduced all forms of the con-
version of propositions to simple conversion ; that it replaced
all the general laws of syllogism by a single canon ; that it
dispensed with Figure in the syllogism, and abrogated all the
special laws of syllogism, and the necessity for Reduction ;
that it increased the number of valid moods to thirty -six ;
that it abolished the Fourth Figure of the syllogism, and
made the order of the premises in the Second and Third
Figures a matter of indifference, and consequently allowed
two conclusions to each syllogism in those figures instead of
one. In fact, the adoption of the quantified predicate was
to revolutionize Logic. Not only has it not done this, but the
whole scheme is now generally and deservedly discredited.
To begin with, it maybe urged against Hamilton's psycho-
logical argument that it is wrong to assert that we implicitly
quantify the predicate in thought. The predicate is re-
garded as an attribute, and is not thought mainly in its
denotation ; still less is it thought as embracing all or some of
its denotation. It is equally wrong to say that the subject
Book H.
Oil. II.
Hamilton
urged that
the predi-
cate is often
quantified,
either di-
rectly or in-
directly.
This scheme
was said to
simplify
logical pro-
cesses.
Thodoctrine
is psycho-
logically
fake,
202
PROPOSITIONS.
Book II.
Ch. 11.
andis worth-
less as an
anal3'sis of
judgments.
Quantifying
the predi-
cate does
not give an
identical
proposition.
Hence,
simple con-
version on
this scheme,
involves an
implicit
specification
of our term 8.
Three views
are put for-
"vard as to
thi- r' eaiiing
if ' some' —
is thought collectively, or that a proposition expresses an
identity of two groups taken as wholes, as this scheme
requires. As the very foundation on which the scheme
rests is, thus, unsound, it naturally follows that the scheme
itself is worthless as an analysis of the forms of judgment.
It may be noted that the supposed psychological founda-
tion of the scheme was always assumed by Hamilton without
the slightest attempt at proof.
In the next place, it follows from the discussion of the
import of the particular proposition that a strictly /ormaZ
statement of identity — that is, a logical equation — cannot be
got from a mere quantified predicate, owing to the in-
definite reference of 'some' [c/. § 71 (ii.)]. To get such
an equation, we must specify, and not simply quantify, the
predicate. For it must always be borne in mind that the
predicate can only be read in denotation by taking it
collectively, as one single group ; and the very essence of
every equational view of the proposition is that each term
is thus understood. If, then, the equational doctrine is to
be strictly adhered to, the simple conversion of each of
these quantified forms, except U and E, involves the im-
plicit reservation that the Some S or Some P denotes the
same group in the converse proposition that it did in the
original one ; a limitation which the mere form of the
proposition does not, of course, indicate. For instance, the
proposition 'All man is some animal' would convert to ' Some
animal is all men,' but this latter form is only true when we
limit * Some animal ' in a way which the simple form of the
proposition does not imply. This formal objection does not
hold when we adopt the predicative view of the import of
propositions, in which they are not regarded as equations.
This leads us to enquire in what sense ' some ' is used in
this new scheme. On this point there is a great indecision
amongst the supporters of the doctrine, and even in the
writings of Hamilton himself. Three views have been put
forward, and it is necessary, therefore, to see to what result
each proposed interpretation of ' some ' will lead us.
First. If ' some ' means some only, then each affirmative
proposition which contains 'some' implies a negative pro-
IMPORT OP CATEGORICAL PROPOSITIONS. 203
position, and vice versd. Thus, from the proposition ' All Book II.
man is some (only) animal,' it must be inferred that ' No _1_ '
man is some (other) animal,' and the former proposition is (i) if 'some
uiuu x>} o^^i^v/ yvy , 11 jneiins some
equally involved in the latter. " This sort of Inference only, the
"Hamilton would call Integration, as its effect is, after gcgeme^il
"determining one part, to reconstitute the whole by bring- redundant,
"ing into view the remaining part" (Kowen, Logic, p. 170).
Hence, the assertion of A involves that of »/, and vice versa, (or A and „
Y and O are
and so with Y and 0. With regard to w "Mr. Johnson pairs whose
"points out that if some means some hut not all, we are led j"^™^!^^
"to the paradoxical conclusion that w is equivalent to U. each other,
77 n • J! j.T_ i 8nd o) IS
" Some but not all S is not some but not all P informs us that equivalent
" certain S's are not to be found amongst a certain portion of *° ^•
" the /"s but that they are to be found amongst the remainder
" of the P's, while the remaining S's are to be found amongst
"the first set of P's. Hence all S is P ; and it follows
"similarly that all P is S. Some but not all S is 7iot some hut
"not all P is therefore equivalent to All S is all P" (Keynes,
Form,. Log., 2nd Ed., p. 299). This may be made clearer by the
aid of symbols. Let X S = Some (only) S, then X S = the rest of
S (X = non-X). Similarly, if V P=some [o7ihj) P, then VP = the
rest of P. Now X S is not V P involves that A" S is V P ; for, if
X S is excluded from only a part of P, it must be included in
the remaining part. And A' S is not V P also implies that XS
isVP; for to say that only a part of S is not found in the class
VP implies that the rest of S is found there. Similarly, VP is XS,
and VP is 'XS. Thus, (X S+XS) = (VP+YP), that is All S is all P.
We are thus reduced to the five forms of proposition — U, A Thus, we get
{or ri), y {or 0), I, E— expressive of the relations of quantity gchlme^
between subject and predicate which we gave in § 85 ad fin. '^^^^^^^{i
Had Hamilton, indeed, started with an analysis of the possible the possible
relations of quantity between two classes, he would have seen ^"lations be-
that they can be only these five, and that an eight- fold scheme t^een two
must, therefore, be redundant. And this redundancy makes
it misleading. For every scheme of the forms of propositions
professes to give nothing but simple, distinct, and irreducible
forms ; if, therefore, some of the forms are not distinct and
not irreducible, the scheme suggests differences in predication
201
PROPOSITIONS.
Book TT.
Ch. 11.
(2) If ' some '
means some
at Uast, we
are expre.-s-
ing, not the
actual rela-
ti ons
between
cl;i8ses, but
our know-
ledge of
those rela-
tions.
On any class
view of pro-
positions
the order of
terms is im-
material ;
hence, Y and
A are really
the same,
and so are rj
and 0.
CO is devoid
of significa-
tion.
We are
again, then,
reduced to
a five-fold
scheme.
where none exist. Further objections to the scheme, grounded
on the fact that some of the new forms of proposition are not
really simple, will be noticed later on.
Second. If ' some ' means some at least, then, in addition to
the objection that one of the terms of an equation cannot be
vague in its application without vitiating the assertion of
identity, it must be maintained that we still do not get an
analysis of the relations of quantity possible between classes ;
for it has been seen that these are only five. But it may be
urged we are here dealing, not with those real objective
relations, but with our knowledge of them, which may be
indeterminate. This certainly puts us on more logical
ground (jcf. § 85 ad fin.). But if the relation between the
terms is merely one of quantity, and they are both classes
regarded collectively, then it is evidently immaterial which
we regard as subject and which as predicate. Still more is
this so if the proposition really states an equation between
the terms. But, in this case, Y and A are not independent
forms ; for both mean that one class forms an indefinite
portion (which may, or may not, be all) of another ; and we
may write the distributed term as the subject, instead of as
the predicate, of Y. Similarly jj and 0 are really the same ;
for each excludes the whole of one class from an indeter-
minate portion of the other. The proposition w ceases to be
significant at all ; for it neither denies nor decides anything.
It denies nothing ; for it is true together with any of the
afiBrmative forms, none of which it contradicts. Even if we
afi&rm All S is all P, yet it remains true that this particular
part of S is not that particular part of P. For example, if
we grant that ' All man is all rational animal,' yet we by no
means deny that Englishmen (who are ' some man ') are
not Frenchmen (who are ' some rational animal '). It decides
nothing ; for it can always be asserted with reference to
any two terms, except they happen to be both singular
names, one of which belongs, and the other does not belong,
to one definite individual ; and in that case, of course, the
'some' is altogether out of place. We are, therefore, again
reduced to five forms, U, A, 0, I, E, though each of these is
IMPORT OF CATEGORICAL PROPOSITIONS.
205
no longer incompatible with each of the others — U, A, and I,
may be true together, and so may 0 and E. This is, of
course, no objection if the propositions are regarded as
simply stating our knowledge of the quantitative relations
between the terms, and not those relations themselves, but
it utterly destroys the position that the proposition is an
equation.
Third. If 'some' means some only va. propositions of the
form A, Y, f] and 0, but some at least in these of the form I
and w, then we have a combination of the above objections.
For A and »/, Y and 0 still form pairs of propositions, the
members of each of which are mutually inferrible from each
other, and w is still entirely without real predicative force.
The using ' some' in two distinct senses in the same scheme
of propositions leads also to the anomalous result that I is
consistent with either U, A, or Y, but that each of these
three is incompatible with both the other two.
In whatever way, then, we interpret ' some,' we find that
an analysis of the forms of categorical proposition will not
give an eight-fold scheme.
We will now examine the new forms — U, Y, f], w — and
enquire whether they are ever used in ordinary speech ; and,
if so, whether they are simple forms of proposition, such as
should alone find a place in a logical analysis of elementary
forms of predication. Dr. Keynes holds that U and Y are
met with in ordinary discourse. He says : " It must be
" admitted that these propositions are met with in ordinary
" discourse. We may not indeed find propositions which
"are actually written in the form All S is all P; but wo
" have to all intents and purposes U, wherever there is an
" unmistakeable affirmation that the subject and the pre-
"dicate of a proposition are co-extensive. Thus, all defini-
" tions are practically U propositions ; so are all affirmative
" propositions of which both the subject and the predicate
" are singular terms" (^Formal Logic, 3rd Ed., p. 176). This
is true, in that the denotation of the predicate in a definition
is undoubtedly identical with that of the subject. But the
main object of definition is not to determine the limits of the
Book II.
Ch. II.
(3) If some'
varies in
meaning,
we have a
combination
of the above
objections.
Definitions
are XJ pro-
positions ;
but to state
identity of
denotation
is not their
main func*
tion.
206
PROPOSITIONS.
Book II.
Ch. II.
The real as-
sertion in-
tended by U
can only be
made by
two A pro-
positions.
Hence, U
proposi-
tions, if ever
made, would
be exponi-
bles ; and
this form is,
therefore,
not simple.
Exclusive
and Excep-
tive Propo-
sitions are
examples of
v»rtich is,
therefore,
not a simple
preposi-
tional form.
The form n,
if ever used,
would also
be exponi-
ble, and,
therefore,
uot simple.
denotation, but to make explicit the connotation. Moreover
the content asserted by the predicate in a definition is
affirmed of every individual denoted by the subject. But
in the U proposition this distributive reference is lost, and
the predication is made of the denotation of the subject as a
whole. A definition is then a U proposition, but tliis is its
least important aspect. The full predication intended, but
not really expressed, by the U form can be made by a double
employment of the A proposition of the traditional logic,
and this mode of expression is not open to the objections
just urged against the U form. Thus, S a P and P a S
together express all that S u P is intended to say. Proposi-
tions such as ' Mercury, Venus, etc., are all the planets,' which
have also been given as examples of U propositions, are not
60 formally ; for there is nothing in ' Mercury, Venus,' etc.,
to show that it is All S. We must interpret it as meaning
* The class composed of Mercury, Venus, etc., is all the
planets,' an awkward form whose full force is given in the
two propositions ' Mercury, Venus, etc., is each a planet ' and
' All the planets are amongst those enumerated.' Thus we
see that the strict U form of proposition is practically never
used ; and, if it were, it ought not to be admitted into a
scheme of simple propositional forms as it would really be
exponible [t/. § 75 (ii.)]-
Exclusive and Exceptive Propositions [see § 75 (ii.) («)
and (6)] are usually given as examples of the Y form. It
may be granted that these propositions can be written in
that form — e.g., ' The virtuous alone are happy ' may be
expressed ' Some virtuous is all happy.' But this does not
make them simple propositional forms ; they are, as we saw
in § 75, compound in their meaning, and may be reduced to
two propositions of the form Some S is P, No non-S is P.
They have, then, no place in a scheme of simple propositional
forms.
The form t] — No S is some P — is never used in ordinary
speech. Dr. Keynes says : "Not S alone is Pis practically ti
" provided we do not regard this proposition as implying
"that any S is certainly P" {op. cit., p, 177). But, if
IMPORT OF CATEGORICAL PROPOSITIONS.
207
* some ' is used in the sense of ' some only,' then S t) P does
imply that All S is some (other) P. And, whichever way we
read ' some' we have here again an exponible proposition ;
and consequently, this form, too, must be excluded from an
analysis of simple forms.
The uselessness of the form <n has been already suflBciently
shown, and it is certain that nobody ever attempts to express
a judgment by means of it.
Hence we cannot agree with those logicians who
advocate the addition of T and rj to the four-fold scheme
in Formal Logic ; for they are not simple prepositional
forms, and are moreover necessarily based on the class
view of predication, against which, as expressive of the
real import of predication, objections have been already
urged. That the four-fold scheme is formally complete
if the predicative view is adopted has already been shown
(see § 84).
We may conclude our criticism in the words of Prof. Adam-
son : " To such a scheme the objections are manifold. It is
" neither coherent in itself, nor expressive of the nature of
" thinking, nor deduced truly from the general principle of
" the Hamiltonian logic. For it ought to have been kept in
" mind that extension is but an aspect of the notion, not a
" separable fact upon which the logical processes of elabora-
" tion are to be directed. It is, moreover, sufficiently clear
" that the relation of whole and part is far from exhausting
"or even adequately representing the relations in which
" things become for intelligence matters of cognition, and
" it is further evident that the procedure by which types of
" judgment are distinguished according to the total or partial
"reference to extension contained in them assumes a stage
"and amount of knowledge which is really the completed
"result of coguition, not that with which it starts, or by
" which it proceeds. . . . Hamilton, it may be added, finds
" it completely impossible to work out a coherent doctrine
"of syllogism from the point of view taken in the treat-
"ment of the judgment" (Article Logic in Ency. Brit.,
9th Ed.).
Book II.
Ch. II.
<t> ia useless.
Hence, none
of the new
forms are
admissible
in an analy-
sis of simple
proposi-
tioual forms.
208
PROPOSITIONS.
Book II.
Ch. II.
Hamilton
read judg-
ments both
in extension
and in com-
prehension.
In extension
the copula
means is con-
tained under;
in compre-
hension it
means con-
tains.
This view re-
quires com-
prehension
to include
all the attri-
butesknowu
to be com-
mon to a
class,
*87. The Comprehensive View.
Sir W. Hamilton held that every jadgment expresses not
only a quantitative relation in extension, or denotation [see
§ 28 (vi.)] between subject and predicate, but also a similar
relation in comprehension ^see § 28 (vi.)]. He says : " We
" may . . . articulately define a judgment or proposition to
" be the product of that act in which we pronounce that of
" two notions thought as subject and predicate, the one does
*' or does not constitute a part of the other, either in the
"quantity of extension, or in the quantity of compre-
" hension " (Lectures on Logic, vol. i., p. 229). As extension
and comprehension vary inversely [cf, § 28 (v.)] the notion
which is smaller in extent is larger in content, and vice versa.
The copula, is, has, therefore, two meanings ; " In the one
" process, that, to wit, in extension, the copula is means is con-
" tained under, whereas in the other, it means comprehends
" in " {ibid., p. 274). For example, ' Mania two-legged' read
in extension means that ' man ' as a class is contained in the
class ' two-legged ' ; read in comprehension it means that the
complex notion ' man ' comprehends, as a part of itself, the
attribute ' two-legged.' Thus, read in extension the predicate
is larger than the subject, but read in comprehension it is
less. From this double meaning of all propositions it follows
that every reasoning must be considered under a double
aspect, and that two kinds of syllogisms are required — the
Extensive and the Comprehensive — the latter being derivable
from the former by changing the meaning of the copula and
transposing the premises.
The part of this doctrine which refers to the extensive
reading of propositions has been examined generally in the
last two sections, and with special reference to Hamilton's
further additions in the last section ; and reasons have been
given for rejecting it. Nor is the Comprehensive view more
tenable. If ' Comprehension' is used correctly, as synonymous
with Connotation [see § 28 (vi.)] then it is false to say that
in every proposition the subject contains the predicate in
comprehension ; of no synthetic judgment (see §§ 40 and 69)
is this true. It is, therefore, necessary, if this view of pro-
IMPORT OF CATEGORICAL PROPOSITIONS.
209
positions is to stand, to make comprehension include ali
attributes known to be common to a class [see § 28 (ii.)] » ^^^,
as a necessary consequence, to make all propositions analytic.
But here we are met with a difficulty. We surely cannot get
out of a notion anything which is not already in it. How,
then, can we express new information ? It is not already
part of the subject-notion, and, on this theorj^, it can never
become a part of it. The fundamental notion underlying
this reading of the terms of a proposition is that a judgment
only expresses a relation of agreement or disagreement
between two concepts. Such a view of judgment is, as we
have pointed out, one-sided and altogether inadequate
{see § 8). It errs in the opposite direction to that which
regards only the denotation of the terms, and which, con-
sequently, is too objective. Both views equally lose si^'it
of the essential unity of the judgment, and regard it as
bringing together elements which were before separate and
unrelated in thought.
* 88. The Attributive or Connotative View.
J. S. Mill held that every proposition whose subject is
not singular is the expression of a relation between attri-
butes. But he did not regard this relation as one of inclu-
sion of the predicate in the subject ; on the contrary he
submits Hamilton's view to a trenchant criticism {see Exam.
of Ham., ch. xviii. and xxii.). Mill consistently maintains
that the attributes implied by a cla'^s-name, and which form
its connotation, are those only which are essential to member-
ship of the class [see § 28 (ii.)] ; and he attaches great im-
portance to the indirectness by which the members of such
a class are indicated by that class-name [cf. § 71 (i.) (S)].
He says : " Though it is true that we naturally 'construe the
" subject of a proposition in its extension,' this extension, or
"in other words, the extent of the class denoted by the name,
"is not(^ apprehended or indicated directly. It is both
" apprehended and indicated solely through the attributes "
{Logic, Bk. I., ch. v., § 4, note). Hence he argues that all
that is really asserted in any proposition whose subject is a
LOG. I. U
Book II.
Ch. II.
and makes
all proposi-
tions arialy-
tic.
Mill hcltl
that every
proposition
which is
not singular
expresses a
relation be-
tween attri
butes.
210
PROPOSITIONS.
Book IL
Ch. II.
Mill is right
in liolding
tlie ultimate
iuij'ort of
judyineiitto
be relation
of content,
but wrong
in deriving
this from
enumera-
tion of
instances.
General Name is that the attributes connoted by the predi-
cate do, or do not, accompany the attributes connoted by the
subject. "Man is mortal" means " Whatever has the attri-
•'butes of man has the attribute of mortality; mortality
" constantly accompanies the attributes of man" {ib/d., § 4).
This is " the formula . . . which is best adapted to ex])ress
'• the import of the proposition as a portion of our theoretical
" knowledge. . . . But when the proposition is considered as
"a memorandum for practical use, v/e shall find a different
"mode of expressing the same meaning better adapted to
*' indicate the office which the proposition performs. ... In
" reference to this purpose, the proposition. All men are
"mortal, means that the attributes of man are evidence of,
" are a mai-k of, mortality ; an indication by Avhich the
" presence of that attribute is made manifest '' {Logic, Bk. I.,
:b.vi.,§5).
This analysis of the ultimate import of a proposition is
right in so far as it regards the denotative proposition a8
only an interpretation of a judgment affirming relation of
content. So far it is in agreement with the position set
forth in § 71. But it is faulty in that it regards connexion
of content as established by simple enumeration of instances,
and justified by mere uncontradicted experience. This weak-
ness is due to Mill's general empiricist position that sensuous
experience is the only possible source of knowledge, a posi-
tion which led him to the doctrine that reality is nothing but
possibility of sensation. The sensuous experience which Mill
puts forward as the only basis of knowledge can, obviously,
never give certainty, or necessity, or universality in the strict
sense of always and everywhere. Even if complete it can
only speak definitely of the past and in terms of weaker or
stronger expectation of the future. Thus, Mill does not
grant that universal judgments express a necessary relation
of content, but only a coexistence hitherto invariable. The
defect of Mill's view is thus fundamental ; he omits the
work of thought in constituting reality, he forgets that
sensuous experience becomes knowledge only when it is
interpreted by being brought under relations conceived by
IMPORT OF CATEGORICAL fROPOSITIONS,
211
the mind, and afterwards proved by experience to be true
of realit}'. It is in this recognition of the constitutive action
of thought that we find necessity. A judgment is necessary
when it expresses the only possible interpretation of the
results of experience, that is, when it harmonizes with the
total and systematic concept of reality.
89. Implication of Existence.
The question we have now to consider is this : Does the
assertion of a categorical proposition necessarily imply that
its terms are the names of really existing things ? This
enquiry has no reference to any particular mode of existence ;
the word is used in its widest sense, and embraces existence
in the spheres of fiction, mythology, and imagination, as well
as in that of physical reality [cf. § 28 (iv.)]. The logical
force of the term is well expressed by Prof. W. James : " In
■' the strict and ultimate sense o£ the word existence, every-
" thing which can be thought of at all exists as some sort of
" object, whether mythical object, individual thinkei''s object,
"or object in outer space and for intelligence at large"
{Mind^ No. LV., p. 331). Now, every logical term represents
something thought of, and, therefore, the mere use of a
term implies the existence of some thing, or things, of
which it is the name. Thus, " A name always refers to
" something. . . . That which is named is recognized as having
'' a significance beyond the infinitesimal moment of the
" present, and beyond the knowledge of the individual. . . .
" It is, in short, characterized as au object of knowledge "
(Bosanquet, Lorjic^ vol. i., pp. 18, 19). As this logical
existence, this mere ' thisness,' is implied by the simple use
of the terms of a proposition, it cannot be specially pre-
dicated, nor is it asserted by the copula. Bat a particular
mode of existence can only be asserted by special predication ;
for it is not involved in the use of either terms or copula
{cf. § 68). Thus, ' The idea of duty exists ' predicates
existence in the realm of men's moral thoughts ; ' Much
misery and crime exist in our large towns ' asserts existence
in the physically real sphere of man's life. These special
11-2
Book II.
Ch. II.
Logical ex-
istence m;y
be in any
sphere.
It coiisist.--
merely in
the apiilio;t-
bility of tht
term to the
thing
named,
and is im-
plied in the
mere use of
a term.
It cannot,
therefore,
be specially
predicated ;
but particu-
lar modes of
existence
must be
specifically
asserted as
predicates.
212
PROPOSITIONS.
Book II.
Ch. II.
The implica-
tion of exist-
ence follows
from the
nature of
the proposi-
tion as well
as from tliat
of the term.
In aflSrma-
tive propo-
sitions the
existence of
the S neces-
sitates thiit
of the P in
the same
sphere, but
not in nega-
tive prupo-
jitiuns.
kinds of existence do not, therefore, touch the question we
are now considering. They only come under our notice aa
predicates, and, therefore, as mere special instances of the
general rule that all predicates and subjects involve the
wider logical existence which consists in the applicability of
a term to the thing named. The implication of existence
follows, therefore, from the very nature of the term. It
is equally necessitated by the nature of the proposition.
Every proposition expresses a relation between a subject
and an attribute {see § 84). But we can only conceive a
subject as a more or less permanent thing capable of re-
ceiving or rejecting more or less transient attributes
(c/. § 68). Hence, both subject and attribute must have,
at least, this much existence — that the latter is capable of
affecting the former, and the former of being affected by
the latter. This minimum of existence is all that can be
implied by all propositions, and it is, therefore, all which
can be regarded as necessarily involved in any. Moreover,
a proposition claims to be true, and, if we disbelieve it, we
can only contradict it by another proposition which advances
the same claim to truth (c/. § 66). But if a proposition is
a mere statement in words, with no corresponding reality
behind, we cannot say, in any intelligible sense, that it is
either true or false. In fact, it ceases to have any real
meaning and becomes a mere sham.
We must conclude, then, that every categorical proposition
— universal or particular, analytic or synthetic — implies,
logically and necessarily, the existence of its terms in some
appropriate sphere. ■5 It now remains to ask whether both
subject and predicate must exist in the same sphere. In
the case of affirmative propositions this must, necessarily,
be the case ; for S a P and S i P assert that in the sphere of
existence in which S holds a place, certain objects — viz.. All
S, and Some S respectively — possess the attribute P. But, in
the case of negative propositions, the predicate does not,
necessarily — though it does usually — exist in the same
sphere as the subject. Now, as the subject is the centre
of attention, and the point at which the judgment touches
IMPORT OF CATEGORICAL PROPOSITIONS.
213
reality, we must regard the sphere of existence to which the
proposition refers as that of the subject. Therefore, we
must say that in a negative judgment the existence of the
predicate in the sphere of the subject is problematic. Of
course, as a term, it exists in some sphere, but not necessarily
in the sphere to which the judgment primarily refers. Thus
in * No woman was hanged in England for theft last year,'
the sphere of the subject — and, consequently, of the whole
judgment — is that of physical existence. But in this sphere
the predicate does not exist at all ; for nobody — man, woman,
or child — received that punishment for theft. The sphere
of existence of the predicate is, therefore, that of imagina-
tion. Similarly, * Some mountains are not fifty thousand
feet high ' does not imply the existence of objects of that
height in the realm of physical things in which the subject
exists. Our final result, therefore, is this : All categoi'ical
propositions necessarily imply the existence of their subjects
in the appropriate sphere ; in affirmative propositions this
involves the existence of the predicate in the same sphere ;
but in negative propositions the predicate does not neces-
sarily exist in that particular sphere, though it does in some
sphere.
This reference to reality is the distinguishing mark of a
categorical as distinguished from a hypothetical judgment.
The latter, as we have seen, involves a supposition as its
very essence, and this supposition may be one which is
actually incapable of phenomenal realization, and yet we
may be sure that the judgment is exactly and universally
true, as it is the result of a valid process of inference, and is
the only possible interpretation of experience (see § 76). But
in the categorical judgment the reference to reality is distinct
and direct. In a singular judgment of direct perception,
such as ' This table is made of oak,' the existence of the
subject seems to be not merely implied but asserted. In
the particular enumerative judgment Some S's are P the
implication of such existence is still very strong, for such
judgments on their face claim to be, and often are, the results
of direct experience. The implication is, undoubtedly,
BnOK II.
Ch. II.
The sphere
of existence
to which the
proposition
refers must
be deter-
mined by
the subject.
The P in a
negative
proposition
need not
exist in that
sphere,
though it
must in
some sphere
Summary.
The refer-
ence to
reality dis-
tinguishes
the categori-
cal from the
hypotheti-
cal judg-
ment.
It is promi-
nent in pro-
portion as
the judg-
ment is
based on di-
rect experi-
ence.
214 PROPOSITIONS.
Book II. weaker in the universal judgment Every S is P, for that, as
Ch^i. ^g Lave seen, is not based upon a completed experience of
instances, but upon an established connexion of content.
But as this connexion can only be established by an analysis
of that mode of reality to which the judgment refers, the
implication of existence, in the appropriate sphere — i.e. in
some mode of reality — is not absent.
CHAPTER TIL
DIAGRAMMATIC REPRESENTATION OF PROPOSITIONS.
90. Nature and Use of Diagrams.
Diagrams are intended to make obvious at a glance the
relations between the terms expressed in a proposition.
That any scheme may do this satisfactorily, it is essential
that :—
(1) The diagrams employed should be self-interpreting
immediately the principle on which they are con-
structed is understood.
(2) Each diagram should be capable of one, and only one,
interpretation ; and, conversely,
(3) Each proposition should be representable by one, and
only one, diagram.
The value of every scheme of diagrams must, therefore,
be estimated by the perfection with which it fulfils these
requirements.
Primarily, the ordinary schemes of diagrammatic repre-
sentation— especially that considered in the next section —
represent the extension, or denotation, of both the terms in a
proposition. It has already been pointed out (see §§ 72 and 85) terms
that it is always possible thus to attend to both the terms in a
formally expressed proposition, though this limited interpre-
tation omits and obscures the most essential aspect of the
unity of judgment. But, in some of the schemes, this refer-
ence to extension is less prominent — aa, for instance, in the
scheme explained in § 92. Even in the first scheme, we may
interpret the circle which represents the predicate as ' cases
Book II.
Ch. III.
Diagrams
Bhoul'l be
self- inter-
preting, and
should cor-
respond ex-
actly with
the element-
ary forms of
proposition.
Diagrams
primarily
represent
the exten-
sion of both
216
PROPOSITIONS.
Book II.
Ch. III.
Mausel held
that Con -
cepts could
not be repre-
sauted by
diagrunis ;
but every
concept has
extension,
and it is this
which the
diagrams
represent.
Diagrams
make imme-
diate infer-
ences more
obvio\is.
in which the attribute P occurs'; when, though we still deal
with extension, we have made the connotative element more
prominent than when we simply say ' the clas3 P.'
Mansel, from the Conceptualist standpoint [see § 8 (ii.) (b)] has
raised a fundamental objection to the employment of diaf^rams in
Logic. He says : " If Logic is exclusively concerned with Thought,
" and Thought is exclusively concerned with Concepts, it is impos-
" sible to approve of a practice, sanctioned by some eminent
" Logicians, of representing the relation of terms in a syllogism by
"that of figures in a diagram." This " is to lose sight of the dis-
" tiactive mark of a concept, that it cannot be presented to the
" sense, and tends to confuse the mental inclusion of one notion in
" the sphere of another, with the local inclusion of a ."mailer portion
" of space in a larger " {Prolegomena Logica, p. 55). Those who do
not agree that " Logic is exclusively concerned with Thought, and
"Thought is exclusively concerned with Concepts" (c/. § 9) will
regaid this objection as based on an inadequate and misleading view
of the science. But even those who do accept the Conceptualist
position must own that every concept has extension ; and it is th's
extension, and not the concept itself, which the ordinary diagrams
aim at representing.
A practical argument for the use of diagrams in Logic is
that they afford aid to the beginner in grasping the exact
scope of a proposition, and the immediate inferences which
can be drawn from it. Even those who are not beginners
often find it well to appeal to diagrams when they are dealing
with a large number of terms, as in the problems with which
Symbolic Logic grapples.
Eul er 's
scheme was
based on the
actual rela-
t i o n s of
classes, each
of which he
represented
by a cii'cie.
91. Eulers Circles.
The best known and most commonly used scheme of
diagrams is that of Euler, a distinguished Swlss mathe-
matician and logician of the eighteenth century. He ex-
pounded it in his Lettres a une princesse d'Allemagne (Lett.
102-5). It is based on the actual relations between two
classes, each of which is represented by a circle. This
necessitates the following five diagrams to express all the
possible relations : —
DIAGRAMMATIC REPRESENTATION OF PROPOSITIONS. 217
Book II.
Ch. III.
This re-
quires five
diagrams,
This scheme admirably satisfies the first criterion of excel-
lence given in the last section — there can be no doubt as to
the information given by each of the above diagrams. But,
as it is founded, not on the predicative view of propositions,
but on an analysis of the possible relations which may subsist
between classes, it is not surprising that the diagrams do not
satisfactorily represent the four-fold scheme of propositions.
They correspond, in fact, to the five elementary forms of
proposition which are necessary to express all possible actual
class relations (see § 85 ad fiit.). Thus : —
I. represents that S and P are coincident — All S is
all P.
II. that S is included in, but does not form the whole of
P — All S is some (only) P.
III. that S includes P, but is not wholly exhausted —
Some (only') S is all P.
IV. that S and P partially include and partiallj^ exclude
each other — Some (only) S is some (only) P.
V. that S and P are nmtually exclusive — No S is any P.
If, however, we try to fit in this scheme of diagrams witli
the ordinary four-fold analysis of simple prepositional forms,
we find that only in the case of E have we an adequate
which cor-
respond to
the element
ary forms of
proposition
expressive of
actual class
relations,
hut do not
satisfactorily
represent
the ordinary
four - fold
scheme.
218
PROPOSITIONS.
Book II.
Ch. III.
A, I, and O
each require
a combina-
tion of these
diagrams to
represent
them ;
four of the
diagrams
are ambigu-
ous in their
reference to
those propo-
Bitious ;
and the same
diagram re-
pres e n ts
botli land O.
Attempts
liave been
made to
avoid tliis
amliiguity
by tiie use
of d..tted
lines.
expression in any one diagram. Bearing in mind the abso-
lute indefiniteness of 'Some' [see § 71 (ii.)], it is plain that
every other form of proposition can be fully represented only
by a combination of diagrams. Thus, for A we require I
and II; for I we need I, II, III, IV; and for 0 we must
have III, IV, V. If, on the other hand, we are given either
of the Figures I, II, III, or IV, we cannot say with certainty
what proposition it is meant to represent. The scheme,
then, is of little value for the representation of the ordinary
forms of proposition ; and when propositions are united into
syllogisms, it becomes so complex as to be practically un-
workable [(/. § 124 (i.)]. Thus, when applied to represent
A, E, I, 0 propositions, the scheme does not satisfy either of
the two last criteria of excellence given in the last section. To
attempt to escape this complexity by representing A by II
alone and both I and 0 by IV alone — as is often done — is
misleading, insufficient, and inaccurate. But even were it not
open to these objections, we should still have an ambiguous
diagram ; for IV would represent indifferently I and 0. To
attempt to avoid this difficulty, as Euler apparently did, by
writing the 5 in the part of the S-circle which is excluded
from the ^-circle (as is done above) when the proposition
intended is 0, and in the part of the diagram common to
both circles when it is I, is not satisfactory ; for this assumes
that we already know what proposition is intended. The
diagram itself still remains ambiguous ; and, if it is given in
the empty and unlettered form, we do not know what
predication is intended.
It has been proposed to avoid this ambiguity by using a
dotted circumference to denote what is indefinite. Thus,
Jevons in his Primer of Logic (pp. 46-7) would represent
I by Fig. VI and 0 by Fig. VII •—
ViL
DIAGRAMMATIC REPRESENTATION OP PROPOSITIONS. 219
This is not satisfactory ; for VI excludes the possibility Book II.
of 5 either coinciding with, or including the whole of, P ; J "
and this latter possibility is equally negatived by VII.
Moreover, the A proposition can still be fully represented
only by the combination of Figs. I and II.
Ueberweg's plan is not open to these objections. He O^ these
{Logic, Eng. trans., pp. 217-218) represents A by Fig. VIII, Ueberwegs
I by Fig. IX, and 0 by Fig. X :- Lc'isfT*
VIIl,
IX.
This gives expression to all possible cases, and we have a
scheme in which each proposition is represented by one, and
only one, diagram, and each diagram can be interpreted by
one, and only one, proposition ; but Fig. X can scarcely be
regarded as sufficiently simple and obvious to be satisfactory.
Fig. V is, of course, still retained to express the E proposi-
tion as it is perfectly unambiguous.
92. Lambert's Scheme.
Lambert's plan is to repz-esent the extension of a term by a hori-
zontal straight line, unbroken when the term is distributed and
dotted when it is undistributed. If, by the force of the proposition
in which it occurs, the extension of a term is partly definite and
partly indefinite — as is the predicate of an afErmative proposition —
the line is partly unbroken and partly dotted. When two terms are
joined in a proposition, the line representing the subject-term is
written a little lower than that which stands for the predicate-term.
In an affirmative proposition, the unbroken part of the hne repre-
senting the subjt ct-term is placed under the unbroken part of that
which indicates the predicate-term ; and in a negative proposition,
the lines are so written that their unbroken parts do not overlap.
The relative lengths of the lines, whether broken or unbroken, is,
Lambert re-
presented
the exten-
sion of terms
by lines,
unbroken if
the turm is
distributed,
and dotted
if it is nil-
diRtribiittd.
220
PROPOSITIONS.
Book IT. in all cases, immaterial. Oa this plan, the elementary forms of
■ categorical proposition can be thus expressed : —
A
£
I
SaP
SeP
S iP
P -
S
P
S
0 - SoP
Thus, the diagram for A sets forth that 5 certainly covers part of
P — the part marked by the continuous line — and may, or may not,
cover the rest, the dots representing our uncertainty.
These dia- This scheme is not quite as self-interpreting as is the one last
w i™h the described, but it has great advantages over that in every other way.
four-fold It does fit in with the ordinary four-fold scheme of propositions.
propositions. Each proposition can be represented by one, and only one, diagram ;
and each diagram refers unmistakably to one, and only one, propo-
sition. It can, for this reason, be employed to represent syllogisms
more readily than can Euler's circles ; except, indeed, in the
modified form which XJeberweg gave them.
The scheme of diagrams given above is a modification of
Lambert's scheme so far as the representation of the I proposition
is concerned. Lambert employed
which is as appropriate to 0 as it is to I. This, probably, accounts
to some extent for the neglect with which his scheme has been
treated by Icigicians.
Dr. Venn's
diagrams are
adapted to
represent
universal
propositions.
93. Dr. Venn's Diagrams.
Dr. Venn, in his Symbolic Logic, explains a very ingenious
plan which he has invented for representing universal proposi-
tions interpreted on the existential or compartmental theory,
that is, as denying or affirming the existence of^the thmgs de-
noted by one or more of the complex terms S P, S P, S P, S P. He
regards an empty diagram as representing no proposition, but as a
DIAGRAMMATIC REPRESENTATION OF PROPOSITIONS. 221
mere framework into which a proposition can be fitted. The frame-
work for any proposition invo'ving two terms is
Book II.
Ch. III.
From the
empty dia-
gram the
compart-
ments are
shaded out
which the
proposition
declares
empty.
We have here four compartments (one being that which lies out-
side both the circles)— S P, S P, S P,SP (where 5 and T denote
non-S and non-P respectively), which corrtspond to the four possible
classes which can be obtained from the combinations of SandPand
their contradictories.
Every universal proposition denies the existence of one or more of
these classes, and this is represented by shading out the compart-
ment. Thus, S a P is indicated by shading out the compartment
5 P (Fig. I), and S e P bv shading out S P (Fig. II) :—
II.
The scheme is adapted to propositions involving more than two The scheme
terms, but becomes cumbrous when the number exceeds five. For is adapted to
1 o • n /-> • propositions
example S ts P or Q is represented in Fig. Ill, and S is both P and Q involving as
in Fig. IV : many as five
terms.
m.
vr.
222
PROPOSITIONS.
Book II.
Ch. in.
It equally well represents a categorical syllogism, in which both
the premises are universal. If, for instance, our premises are M a P
and S a M, the diagram la
but is not
well suited
to represent
particular
proposi-
tious.
which shows at a glance that the relation established between S and
PxsS aP.
But the scheme is not well adapted to particular propositions.
Dr. Venn proposes that a bar should be drawn across the compart-
ments which the particular proposition declares to be saved, but this
is apt to lead to confusion. The scheme, therefore, cannot be used
satisfactorily for all the elementary forms of proposition.
It is pro-
posed to base
a scheme of
diagrams on
the implica-
tions of ex-
istence in a
categorical
proposition.
The possible
classes are to
be written
along a line,
which, when
unbroken,
implies cer-
tainty of ex-
i8tence,and,
when broken,
doubt.
94. Sclieme Proposed.
We would suggest that a satisfactory scheme of diagrams
can be based on the implications of existence contained in a
categorical proposition {see § 89). Every such proposition
combining two terms — S and P — involves reference to the
following classes — the S which is P {SP), the 5 which is not
P (SP), the things beside S which are P (SP). Of those
things outside S which do not possess the attribute P {SP),
the proposition tells us nothing directly, and the existence of
that class is, therefore, always problematic. We propose
that these four classes shall be written along a horizontal
line, v/hich shall be unbroken when it represents a class
whose existence is implied, and dotted when it stands for a
class whose existence, in that sphere to which the propo-
position refers, is doubtful. The omission of the line
representing any one class involving either S or P implies
that its existence is implicitly denied by the proposition.
DIAGRAMMATIC REPRESENTATION OF PROPOSITIONS. 223
This plan would give the following diagrams : —
SP SP tiP
A
Sap;
SeP;
SiP;
SoP:
I-
SP
SP
SP
■]•
s-p
SP
SP
SP
SP
Book II.
Ch. III.
The omission
of a class in-
volving s or
P means de-
nial of exist-
ence.
SP
-I-
SP
■I
The predicates of the negative propositions, E and 0, are
represented by dotted lines in accordance with the view
advocated in § 89 that their existence is not assured in the
same sphere as that of the subject. It will be noticed that a
distributed term is confined to one kind of line — unbroken or
broken, but that an undistributed term contains both (cf.
§ 72). This is natural, as, in an undistributed term a positive
reference is made to some portion of its denotation, but it is
left undetermined what that portion is. The division SP Tiiedianram
may be practically disregarded except for some purposes of ^'■f'Y ^^ '*'^'-
immediate inference, and may, in all other cases, be omitted
from the diagrams ; but it must be remembered that such
omission does not mean that the possibility of that class is
denied, for such denial cannot possibly be made by a
proposition which contains neither S nor P.
In this simplified form the diagrams will be : —
most pur-
pos es by
omitting s p
SP
A
E
I
0
SaP
SeP; I-
SP
-I
SP
SP
I I
SiP; I-
SP
SP
SP
■1
SP SP SF
oP; 1 1 1- - - I
224
PROPOSITIONS.
Book II.
Ch. III.
These dlB'
gT^ms fit in
with the
fourfold
scheme of
propositions.
On the same
plan dis-
junctives
can be repre-
sented, if
they Involve
no more
than three
terms.
Hypotheti-
cals caim t
be repre-
sentud by
diiigvoiiis.
This plan is based on the four-fold scheme of categorical
propositions, with which it thoroughly fits in. Each pro-
position is fully represented by one diagram, and each
diagram can be interpreted by only one proposition. A
glance at any one of the diagrams will show whether, in that
particular case, S is, cr is not, asserted to possess the
attribute P— that is, whether it does, or does not, exist in
the class SP — and whether that statement is, or is not, made
definitely about the whole of S. "&
A disjunctive proposition involving no more than three
terms can be similarly represented. Thus 5 is P or Q is
shown by a dotted line divided into three portions ; for the
existence of neither alternative is assured, though it is
certain that some one, or two, must exist.
SPQ
SPQ
SPQ
If the disjunction embraces more than three terms, however,
it is not possible to represent it simply on this plan. But no
plan — except Dr. Venn's — is at all adapted to the represen-
tation of such disjunctives, and, it may be added, their
representation, if it could be secured, would be of no
practical advantage.
From their very nature Hypothetical Propositions as such
are not capable of diagrammatic representation ; for no
diacrrams can express the relation of dependence which exists
between the consequent and the antecedent (see § 76).
BOOK III.
IMMEDIATE INFERENCES.
CHAPTER I.
GENERAL REMARKS ON IMMEDIATE INFERENCES.
95. Nature of Immediate Inferences.
Inference, or Reasoning, is the deriving of one truth
from others. By this is meant that the new judgment is
accepted as true because, and in so far as, the validity of the
judgments from which it is derived is accepted. Hence,
every inference has a formal and necessary character, and
this is not affected by the truth or falsity of the premises.
The premises may be false and yet the inference may be
formally valid, i.e., valid in the sense of avoiding contradic-
tion within itself. But in the wider sense of validity, in
which the result of the inference must also be consistent
with the whole system of knowledge, the truth of the pre-
mises is, of course, an essential element {cf. § 5). This
aspect of inference will be dealt with when we consider the
doctrine of Induction ; in this and the following Book we
shall be primarily concerned with an analysis of the formal
aspect of the process.
Inference is not a mental process absolutely distinct in its
character from judgment. The essence of the latter is the
explanation of some element of reality by reference of it to
LOG I. 15
Book III.
Ch. I.
Inferenct is
the deriving
of one truth
from others.
It has a
formal as-
pect,
but involves
the deriva-
tion of a new
judgment.
226
IMMEDIATE INFERENCES.
Book III.
Ch. I.
An inference
iucludes
both pre-
mises and
conclusion.
An Immedi-
ate Inference
unfolds the
implications
of a single
judgment.
Many Im-
mediate In-
ferences ai-e
obvious,
some concept already familiar to the mind. In inference
there is the same essential feature, but with this difference,
that the reference is not made immediately, but indirectly
through the medium of some previously accepted truth or
truths. In inference, therefore, we pass beyond the judg-
ment, or judgments, from which we start, and attain a new
point of view ; though, at the same time, the new judgment
thus reached must be a necessary consequence of the data
from which we set out. Inference thus involves both a pro-
cess and its result ; and to each of these the name is sometimes
given. But strictly speaking an inference is the whole mental
construction, and sets forth the connexion between the judg-
ment proved and the evidence which proves it. The judgments
which express the data or evidence are called Premises ; the
judgment derived from them is termed the Conclusion.
Immediate Inference is the process by which the
implications of a single judgment are unfolded. By its
immediateness is not meant that no activity of thought is
required to reach the new judgment — for then it would not
be inference at all — but simply that no datum is necessary
besides the one given judgment. Kant called such inferences
" Syllogisms of the Understanding " to distinguish them
from the " Syllogisms of Reason " in which two premises
are required, and to which the name " Syllogism " is commonly
restricted. They may, however, most appropriately be styled
Literpretative Inferences, as distinguished from mediate in-
ferences obtained from a combination of judgments in which
thought makes a substantial advance to a new truth.
It is often questioned whether Immediate Inferences are
really inferences at all, as no new truth is reached by them.
It may be granted that the majority of them are of compara-
tively small interest, and that the passage of thought from the
premise to the conclusion is a very small and obvious one.
But to object as Mill does that "there is in the conclusion no
" new truth, nothing but what was already asserted in the
" premises, and obvious to whoever apprehends them "
{Logic, II., i., § 2) would be fatal to all inference ; for in
every valid inference the conclusion must be a necessary
GENERAL REMARKS ON IMMEDIATE INFERENCES. 227
consequence of the premises, and, therefore, potentially Book III.
known as soon as these are fully apprehended. The step ^'^^- *•
from premise to conclusion in an Immediate Inference is
small ; but this does not prove that it is no step at all,
or that it is unnecessary to take it. Moreover, the great but their
variety of these steps necessitates a careful and systematic ^^-mdaex-
examination of them; for, without such an investigation, amination
., • 1 1 /. 1 -^ 11 . .... . inorderthat
it IS very doubttul if all the necessary implications contained the full force
in a simple proposition are generally grasped. It is only mefirmay
when we have seen in how many new forms we can be grasped.
express what is virtually contained in any single judgment
that we, as a rule, fully appreciate the meaning of that
judgment.
An examination of the forms of reasoning should begin An exami-
with Immediate Inferences ; for we should know what is reasoning
involved in a single judgment before we go on to enquire should begin
what results will follow from a union of several judgments, diate infer-
ences.
96. Kinds of Immediate Inferences.
There are two main classes of Immediate Inferences : — There are
(i.) The Opposition of Propositions, when, from the given immediate'
truth or falsity of one proposition we infer the truth or inferences:
... . . '■ (a) Oppnsi-
ralsity of other propositions relating to the same matter — Hon of Pro-
that is, having the same subject and predicate. In other fnferences"'^
words, an examination of the opposition of propositions ^^ to truth
means a consideration of the relations as to truth or false- of related
hood which hold between the four forms of propositions, J"'^S'"e'^'^8-
S a P, S e P, S i P, S o P, when 5 and P have the same signifi-
cation in every proposition.
(ii.) Eductions,^ in which, from a given judgment regarded {b)Eductionf,
as true, we derive other judgments which are implied by it ; tionT)f'im-
or, in other words, when we look at the same truth from p^*'^'^ J"'^^-
, . men's.
another point of view, and express the same matter in a
different verbal form.
We shall consider these two kinds of Immediate Inferences
in the next two chapters.
This name is adopted from Miss Jones' Elements of Logic.
15^2
CHAPTER 11.
Book III.
Ch. II.
The Opposi-
tion of' Propo-
sitions meaus
the relation
between any
two proposi-
tions of dif-
ferent form
with identi-
cal 5 and P.
OPPOSITION OF PROPOSITI(;NS.
97. Opposition of Categorical Propositions.
By the Opposition of Propositions is meant the relation
which holds between any two propositions which have
identically the same subject and predicate. Opposed
propositions thus differ in form, but refer to exactly the
same matter ; that is, to the same things, at the same time,
and under the same circumstances. The logical doctrine of
ByOpposition Opposition, therefore, sets forth what implications as to the
thVt'ruthT' ^'""^^ or falsehood of each of the other forms of cate-
gorical propositions are involved in positing (i.e., afHrming as
true), or sublating (i.e., denying the truth of), any one pro-
position. This is, evidently, an entirely technical and
arbitrary use of the v?ord ' opposition.' The natural mean-
ing of the word would be that two opposed propositions
■Opposition' could not both be true together; that is, that opposition
stricted to could exist Only between the pairs of incompatible pro-
positions, A and 0, E and I, A and E. In this sense the
word was originally used. It was, however, found con-
venient to include under the same head the relations between
propositions which are not incompatible, i.e., those between
A and I, E and 0, I and 0. ' Opposition ' thus came to
include the relation between any pair of propositions of
different form referring to the same matter, whether that
relation were one of incompatibility or of compatibility.
When once this technical use of the word ' opposition ' is
clearly understood, it is unlikely to cause any confusion.
As we have universal and particular, affirmative and nega-
falsity, of
one proposi-
tion to the
tru th , o r
falsity, of
the opposed
propositions.
inconsistent
propositions ;
but in-
cludes all re-
lations of
pruptisitions
ditlei ing in
lorm but
identical in
reference.
OPPOSITION OF PROPOSITIONS.
229
tive, propositions, the relations between them will all be in-
cluded under those subsisting between the following pairs : —
(1) A universal and the particular of the same quality :
A and I ; E and 0.
(2) A universal and the particular of opposite quality ;
A and 0 ; E and I.
(3) A universal and the universal of opposite quality ;
A and E.
(4) A particular and the particular of opposite quality ;
I and 0.
This gives us four kinds of opposition, to which the names
(1) Subaltei-nation, (2) Contradiction, (3) Contrariety, and (4)
Sub-contrariety are respectively given. Wo will now examine
these in order.
(i.) Subalternation. Subaltern Opposition exists between a
universal and the particular of the same quality; that is,
between A and I, E and 0. Thus, the propositions differ
in quantity but not in quality. This is one of the technical
kinds of opposition ; for, not only are the two propositions
in subaltern opposition not inconsistent with each other,
but the truth of the universal necessitates that of the
particular. This follows from the Principle of Identity
{see § 17) ; for, by that principle any assertion which is
true of every member of a class must hold of any number
of those individual members, since they must be identical
with some of those included under the distributed term. The
assertion, when made of an indefinite part, simply repeats
an assertion which was contained in the universal pro-
position.
In such a pair of opposites, the universal proposition is
called the Subalternant or Siibalternans, and the particular
the Subalternate or Subaltern. Inference from the former
to the latter is styled Consequentia or Conclusio ad subalter-
natam propositionem ; that from the latter to the former,
Conclusio ad subalternantem.
Hence, the inference of the truth of I from that of A, and
of the truth of 0 from that of E, are ad subalternaiam. The
Book III.
Ch. II.
There are
four kin da of
Opposition :
(1) Subalter-
nation.
(2) Contra-
diction.
(3) Contra-
riety.
(4) Sub-con-
trariety.
Subaltern
Oppo.iitlon
exists be-
tween a uni-
versal and
the particu-
lar of the
same
quality.
By the Prin-
cipleof Iden-
t i t y the
truth of the
particular
follows from
that of the
universal ;
230
IMMEDIATE INFERENCES.
Book III.
Ch. II.
but from the
falsity of the
universal we
cannot infer
either the
falsity or the
truth of the
particular.
The denial
of the par-
ticular in-
volves the
denial of the
universal ;
but the
truth of the
particular
does not in-
volve the
truth of the
universal.
Summary.
assertion of ' All metals are fusible ' involves that of ' Some
metals are fusible'; and, if we posit 'No horses are carni-
vorous ' we equally posit ' Some horses are not carnivorous.'
But, if 5 a P is denied, then this denial holds equally if P
belongs to some only of the S's, or to none of them. Hence,
from the falsity of A we cannot say whether I is true or false.
Similarly, from the denial of E we can neither affirm nor deny
0. For example, if I deny that ' All metals are malleable' 1
do not thereby deny that 'Some metals are malleable.' Neither
do I affirm the latter proposition (though it happens to be
true in fact) ; for, if I did, then the denial of ' All horses are
carnivorous' would involve the assertion of ' Some horses are
carnivorous.' The sublating of the universal leaves us quite
in the dark as to the truth or falsity of its subalternate.
If we now examine the inferences ad suhaltemantem (or
from particular to universal), we find that the denial of the
particular involves the denial of the universal. For what is
not true even in some cases cannot be true in all. The denial
oi S i P means ' There are no such things as some S's which
are P,' and this, evidently, negates the assertion that All S's
are P. Again, if A were true, I must be true by inference ad
stibalfernatam ; and hence, if the falsity of I did not involve
that of A, it would follow that I could be both true and false
at the same time ; which is absurd. The same results hold
with E and 0. Thus, if we deny the truth of ' Some horses
are carnivorous ' we thereby deny that of ' All horses are car-
nivorous '; and if we assert the falsity of ' Some men are not
mortal ' we equally assert that of ' No men are mortal.' But,
to posit the particular cannot justify us in positing the uni-
versal ; for we can never justify an assertion about Every S
by asserting that it holds good with regard to Some S's.
For instance, though it may be true that ' Some men are red-
haired,' it does not follow that all men possess that attribute ;
nor does the truth of ' Some men are not six feet high ' imply
that no men attain that height.
Hence, iVe reach this general result : The truth of the
particular follows from that of the universal, but not vice
versd ; and the falsity of the universal is an inference from
that of the particular, but not vice versd.
OPPOSITION OF PKOPOSITIONS. 231
* These results are illustrated by the diagrams given in Book III.
Ch. ir.
§94.
S P S P S P S P suits can be
[A] I 1 1 [E] 1 — 1 1 1 illustrated
by diagrams.
SP SP SP SP SP SP
[I] I J 1 1 [0] 1 1 1 1
The length of the lines in these diagrams is, of course,
immaterial. There is, therefore, no suggestion that the S is
more extensive in the diagi'ams for I and 0 than in those for
A and E. It is exactly the same in extent ; for the S referred
to must be identical in all cases of opposition. If we now
examine the above diagrams we see that, in each case, the
assertion of the universal includes that of the particular ; for
the unbroken line in the former includes the whole of S, and.
therefore, covers the unbroken line in the latter, which only
necessarily includes a portion of 5. Hence, also, to sublate
the universal would not sublate the particular ; for the denial
that the whole of S should be marked by an unbroken line
does not necessitate the denial that a portion of it may be
correctly so marked. Thus, in A the diagram shows that the
whole of S is included in S f / if the truth of this be denied,
then some, at least, of S must be found in S P, which is shown
to be possible by the diagram for I without necessarily deny-
ing the existence of another part of S in SP. On the other
hand, if we sublate the particular we necessarily sublate the
universal ; for if we strike out from the diagram of the par-
ticular the unbroken line, we have nothing left corresponding
to the unbroken line in the universal, which must, therefore,
be also struck out. But to posit the particular will not posit
the universal ; for the existence of the unbroken line in the
particular will not ensure its existence in the universal, as the
latter covers the whole of S, but the former only necessarily
covers a portion of it.
Similarly, if we use Euler's diagrams [see § 91) we see that A is
represented by Figs. I and II, and I by Figs. I, II, III, IV;
E by Fig. V, and 0 by Figs. Ill, IV, V. Hence, in each case,
232
IMMEDIATE INFERENCES.
Book III.
Ch. II.
C^vtrndirtory
Proposition!
differ both
In quality
a n d i n
quantity.
A and O ;
E and I are
pairs of Con-
tradictories.
By the Prin-
ciple of Con-
tradiction
one of a pair
of Contra-
dictory pro-
positions
must be
false ;
and, by the
Principle of
Excluded
Middle, one
must be
true.
the diagrams representing the universal include some of those whicL
represent the particular. As the particular is true when any one or
more of the diagrams which represent it is secured, it follows that to
posit the universal is to posit the particular. But the sublating of
the universal only removes some of the diagrams which may repre-
sent the particular, and hence does not sublate the particular, which
may still be represented by the remaining diagrams. Similarly, the
truth of the particular is ensured if those diagrams which do not
represent the universal are secured ; and, therefore, to posit the
particular is not to posit the universal. On the other hand, if the
particular is false, all the diagrams which can represent it are
removed, and this removal includes all those which can represent
the universal, which is thereby also declared false.
(ii.) Contradiction. Propositions are contradictory to each
other when they dij^er both in quality and in quantity. Hence,
there are two pairs of contradictories — A and 0 ; E and I
By the Principle of Contradiction {see § 18) both the mem-
bers of such a pair cannot be true together, and by the
Principle of Excluded Middle (see § 19) both cannot be false.
If 'All metals are fusible' is true, it cannot be true that
'Some metals are not fusible'; and, similarly, if 'No lions
are herbivorous ' is a true proposition, then it cannot be true
that 'Some lions are herbivorous.' And, generally, if we
make an assertion about every member of a class, the Prin-
ciple of Contradiction forbids us to deny that assertion about
any member of the same class. Therefore, one of the con-
tradictories in each pair must he false. But, by the Principle
of Excluded Middle, they cannot both be false. For, by
that principle, any given attribute, P, must either belong, or
not belong, to every individual 5. It cannot, therefore, be
false both to make an assertion of Every S and to deny that
same assertion of Some of those S's. Such propositions as
'AH metals are fusible ' and ' Some metals are not fusible '
cannot both be false together.
Or, in detail ; if we deny the truth of Every S is P our denial
holds whether P is denied of the whole of S, or of only part of S.
But the former denial necessarily, by subalternation, includes the
latter, which is, therefore, true in any case. Consequently the
truth of 0 follows from the denial of A. If we deny 0 we really
OPPOSITION OF PROPOSITIONS.
233
assert that 'There are no such things as Some S's which are not f,'
and this is the same as the assertion that All S's are P. If E be
denied, then either all or some S's — in any case the latter — are P,
and, therefore, I is necessarily true. Lastly, if we deny I, we really
say 'There are no such things as Some S's which are P,' and this is
to assert that No S's are P. Therefore, one of the propositions in
each pair of contradictories must be true.
We see, then, that contradictories are incompatible with
regard both to truth and to falsehood. It follows that when
two contradictory propositions are given us we infer, by the
Principle of Contradiction, that one of them is false, and, by
the Principle of Excluded Middle, that one of them is true.
Hence, we can deduce the falsity of one from the truth of the
other, and the truth of one from the falsity of the other. The
relation of contradiction is thus seen to be reciprocal ; the
positing of one proposition and the sublating of its contra-
dictory are assertions of one and the same fact. It will be
seen, as we examine the other forms of Opposition, that in
none of them are the propositions thus mutually inferrible,
and in none of them is there incompatibility with regard to
both truth and falsehood.
Contradiction is, therefore, the most perfect form of
logical opposition.
Whatever we affirm denies something else. The mere
asserting of every S that it is P is, in itself, a denial of any S
whatever that it is not P. To assert, therefore, that Some S's
are not P, in opposition to Every S is P, is the minimum of
denial. It is sufficient to destroy the proposition which it
contradicts, but it does not affirm the falsity of every part of
it. Thus, a pair of contradictory propositions leave no room
for an intermediate supposition ; one or the other must be
accepted as true, as together they exhaust all possible
alternatives.
* These results are illustrated by the diagrams given in
§94.
SP SP SP SP
[A] ^ I 1 1 [E] I 1 1 1
Book III
Ch. II.
Hence, con-
tradictories
are incom-
patible with
regard to
both truth
and false-
hood.
Contradic-
tion is the
only form of
Opposition
in which the
opposed pro-
positions are
mutually in-
ferrible.
Contradic-
tion is the
minimum of
deniaL
Contradic-
tion can be
illustrated
by diagrams.
m
SP
SP SP
[I]
SP
SP
SP
234
IMMEDIATE INFERENCES,
Book III.
Ch. II.
Secondnri/
Coiitraditioii
exists
between
Singular
Pr o p 08 i -
tious.
The absence of S P from the diagram for A showsi that the
existence of anything which is both S and P is denied. This
existence is the very thing which is asserted in the diagram
for 0, which, however, does not deny the existence of the
class posited by A — viz., S P — but merely leaves it a matter
of doubt. The same thing holds in the diagrams for E and
I ; where I posits S P which E sublates, but does not sublate
anything which is posited by E. Hence, one of a pair of
contradictory propositions must be false, but they are in con-
flict with respect to the existence of one class only. In each
case, moreover, the diagrams between them show that no
other assertion can be made concurrently with these two
about any part of S ; for together they give full information
about the S which is P, and the S which is not P, and every 5
must belong to one or other of those two classes. Hence, one
of the contradictories must be true.
Similarly, a reference to Euler's circles (see § 91) will show that
a pair of contradictories between them require the whole series
of diagrams, thus showing that they are together exhaustive of all
possibilities — thus, A requires Figs. I and II, 0 takes up III, IV,
V ; for I we need I, II, III, IV, and E is fully expressed by V — and
no diagram belongs to both members of either pair of contradictories,
thus proving their absolute incompatibility.
Every proposition has a contradictory ; if the proposition
is simple, so is the contradictory, but if the proposition is
compound it can be contradicted in more than one way, and
its full contradictory is, therefore, compound (c/. § 75).
Contradiction is the only kind of opposition which can
subsist between Singular Propositions [see § 71 (i.) (a)] ; for
these can differ only in quality, and, therefore, to ])osit the
one is to sublate the other, and vice versa. This opposition
of singular propositions is frequently called Secondary Contra-
diction.
Contrary (iii.) Contrariety. Contrary Opposition exists hetween a pair
are uuiver- of universal propositions of opposite quality ; that is, between
site quaiit'* -^ ^^^ ^' Thus, contrary propositions differ in quality only,
and not in quantity. By the Principle of Contradiction
OPPOSITION OP PROPOSITIONS.
235
{see § 18) both cannot be true together. For, if two con-
traries were both true, then contradictories would also be
true together. For, by subalternation, the truth of A would
necessitate that of I, and the truth of E would secure that of
0. Hence, A and 0 would be true together, and so would E
and I, But this is impossible ; and, therefore, A and E
cannot be true together. But as a contrary proposition
does not simply deny the truth of the opposed universal as a
whole, but that of every part of it, and thus asserts its entire
falsity, there is a possibility of an intermediate alternative.
Hence, the Principle of Excluded Middle (see § 19) does not
apply, and the propositions may both he false. For, whilst
the negation of a universal allows inference by Contradic-
tion to the truth of the particular of opposed quality, this
latter does not warrant us in deducing the truth of the
universal to which it is subaltern. Though by sublating A
we posit 0, this will not enable us to posit E. Hence, con-
trary propositions are incompatible with regard to truth, but
not with regard to falsitj'. If one is true, the other must be
false, but the falsity of the one does not involve the truth of
the other. It may be equally false that ' All men are red-
haired ' and that ' No men are red-haired ' ; for the one pro-
position does not simply negate the other, but makes the
opposite assertion with an equal degree of generality. It
follows that contrary propositions are not mutually in-
ferrible, and their formal opposition is, therefore, less per-
fect than is that of contradictories, although, of course, they
express a greater degree of material divergence.
From this lesser formal perfection, as well as from the
much greater difficulty of establishing the contrai'y com-
pared with that of merely disproving a given universal
proposition, it follows that contrariety is of much less
formal importance than contradiction. The bringing for-
ward of one single instance which does not agree with a
general proposition is sufficient to disprove it, and the con-
tradiction is secure, as it rests on observed fact. But to
establish — not merely that one 5, or a few S's, but — that
every S disagrees with the general proposition we wish to
Book III.
Ch. II.
By the Prin-
ciple of Con-
tradiction
both cannot
be true,
but, as the
Principle of
Excluded
Middle does
not apply,
both may be
false.
Hence, con-
trary propo-
sitions are
incompa-
tible with
regard to
truth only ;
and are not
mutually in-
feriible.
Contrarictv
is more diffi-
cult to es-
tablish than
is Contra-
diction.
236 IMMEDIATE INFERENCES.
Book III. disprove is a task of much greater difficulty, and the result
Ch. II. ''
secure.
is much less secure against being itself proved false, than is
Kftpjirfl^'^^ the contradictory. For we can scarcely ever be sure that we
have really examined every instance, and one exception is
fatal to our general proposition ; whilst the simple contra-
dictory, being a particular, can only be overthrown by estab-
lishing the opposed general proposition. Thus we see that
contradiction is sufficient for disproof, and is, obviously, a
more secure position to take np than is the assertion of the
contrary. One would deny that ' All men are liars ' with
much greater strength of conviction than one would assert
that 'No men are liars.'
Contrariety * All these points are illustrated by the diagrams given in
trated by 9 94.
diagrams. ^p -^
[A] 1 1 1
SP SP
[E] I ^1 I
The diagram for A sublates S P and posits S P ; that for E
not only posits 5 P which A sublates, but sublates S P, which
A posits. These two propositions are, therefore, seen to be
in conflict with respect to every possible part of S, and they
do not include the possible case that a portion of the class S
may be P and, at the same time, a portion of it not be P ;
that is, that some S\s do, and some do not, possess the attri-
bute P. It follows that both may be false though both cannot
be true.
Similarly, if we illustrate by Euler's circles {see § 91 ) we find that
A requires Figs. I and II, and Eia represented by Fig. V. The two
together, therefore, omit Figs. Ill and IV, thus showing that they
do not together exhaust all possible cases. Their thorough-going
divergence is shown by the fact that, in the diagrams which repre-
sent A, S is entirely contained within P ; and, in the diagram
corresponding to E, is wholly excluded from it.
j'ro''osiii'ons 0^-) Sub-contrarlety. Particular iwopositions stand in
areparticu- suh-contrary opposition to each other ; that is, I and 0 are sub-
entquaiity? contraries. This opposition depends on the Principle of
OPPOSITION OF PROPOSITIONS.
237
Excluded Middle (see § 19) ; for there can be no judgment
intermediate between ' Some are ' and ' Some are not.'
Moreover, to deny the truth of one particular is to assert
that of the universal of opposite quality (by Contradiction),
and from this follows the truth of the particular which is
subaltern to it. Hence, both these propositions cannot be
false. But the Principle of Contradiction (see § 18) does
not apply ; for the ' some ' in the one case is different in its
reference from the ' some ' in the other. Both propositions
may, therefore, be true. The truth, for example, of ' Some
men are red-haired ' does not involve the falsity of ' Some
men are not red-haired ' ; for it is not the same ' some men '
who are referred to in both cases. But the form of the pro-
positions does not show this, since the interpretation of
' some ' must be purely indefinite. Thus, there is no real
contrariety between I and 0, and the name ' Sub-contrary ' ia
entirely arbitrary. This is another instance of the technical
use of the word ' opposition,' as the two propositions are per-
fectly compatible with each other ; both may be, and often
are, true, though both cannot be false. It follows, therefore,
that to sublate the one is to posit the other, but not vice versd.
Hence sub-contrary propositions are inconsistent with regard
to falsity but not with regard to truth.
* This relation is illustrated in the diHgrams given in
§94.
SP SP SP
[I] I I-- 1 1
Book III.
Ch. u.
Both cannot
be false ;
but both
may be true.
Hence, Sub-
contraries
are incon-
sistent only
with regard
to falsity.
These rela-
lations may
be illu 6 ■
trated by
diagrams.
SP
[0]
-I-
SP
SP
Neither diagram denies anything. In each case what is
definitely posited by the one is marked as possible by the
other — the unbroken line in each coincides with a broken
line in the other. Hence, both may be true. But to regard
both as false would be to strike out the unbroken line from
each, and this would involve the entire denial of the existence
of S, which would be inconsistent with the fact that a predi-
cation has been m:ide of it (se § 89).
238
IMMEDIATE INFERENCES.
Book III.
Ch. II.
It has been
argued that,
if ' some '
means 'some
at least ' I
and O may
both be
false ;
but if ' some '
means 'some
only ■ they
must both
be true.
The latter is
tnio, but is
not based on
the logical
meaning of
' some.'
I cannot be
equivalent
to A, and, at
the same
time, OtoE,
when A and
E are both
lalse.
Similarly, if reference is made to Euler's diagrams (see § 91 ), it
is clear that I requires Figs. I, II, III, IV, for its full expression,
and 0 is only fully represented by Figs. Ill, IV, and V ; it follows
that, as both are partly represented by Figs. Ill and IV, they can
be true together ; and, as together they include all the diagrams,
they cannot both be false as that would entirely deny the possibility
of the existence of S, by removing every diagram which can possibly
express a relation in which it stands to P.
It may be well to notice here an argument which has been
advanced against the doctrine of opposition by Mr. Stock. He
says : "If I and 0 were taken as indefinite propositions, meaning
" ' some, if not all,' the truth of I would not exclude the possibility
" of the truth of A, and, similarly, the truth of 0 would not exclude
" the possibility of the truth of E. Now A and E ma.y both be false.
" Therefore I and 0, being possibly equivalent to them, may both be
" false also. In that case the doctrine of contradiction breaks
"down as well. For I and 0 may, on this showing, be false, with-
" out their contradictories E and A being thereby rendered true "
(Deductive Logic, p. 139). But "if I and 0 be taken as strictly
"particular propositions, which exclude the possibility of the
"universal of the same quality being true along with them, we
" ought not merely to say that I and 0 may both be true, but that
" if one be true the other must also be true. For I being true, A is
" false, and therefore 0 is true ; and we may argue similarly from
" the truth of 0 to the truth of I, through the falsity of E. Or— to
" put the same thing in a less abstract form— since the strictly par-
" ticular proposition means ' some, but not all,' it follows that the
" truth of one sub-contrary necessarily carries with it the truth of
"the other" (ibid., p. 140).
The latter part of this argument may be granted at once. It has
been already pointed out that, if ' some ' is used in the sense of
' some only,' each affirmative proposition involves a negative propo-
sition also, and vice versd (see § 86). Hence, each of the propositions
I and 0 involves both I and 0. But this is not the true logical
meaning of ' some ' [cf. § 71 (ii.)]. We may, therefore, leave this
objection as being really beside the mark, and address ourselves to
the former, which takes ' some ' in its ordinary indefinite sense. It
is true that A may be true when I is, and E when 0 is. But, when
it is argued that, because A and E may be false together, therefore
I and 0, being possibly equivalent to them in fact — but not in state-
ment— may both be false also, the question is begged. For, though
OPPOSITION OF PROPOSITIONS.
239
I and 0 are possibly equivalent in fact to A and E in some cases,
they are not possibly so in others ; the very use of the word
'possibly' should call attention to this. And when A and E ar3
both false is just the very case in which I and 0 cannot be both
respectively equal to them. In that case, I maybe equivalent to A,
or 0 to E, but not both I to A and 0 to E. Mr. Stock's argument
really assumes that the ' possibly ' which applies to each separately
applies to both together. Therefore, I and 0 cannot be both false
at once, as long as the purely indefinite character of ' some ' is pre-
served. Hence, the doctrine of sub-contrariety does not break
down ; and, consequently, the argument against the doctrine of
contradiction founded on the assumption that it does, will not hold.
In fact, Mr. Stock's reasoning would prove too much. For, if I and
0 could both be false together, then, by the doctrine of subalterna-
tion, A .and E would, likewise, both be false. Thus, we should reach
the sufficiently absurd result that every judgment which could pos-
sibly be made, either definitely or indefinitely, as to the relation
between a given subject and predicate could be false. But, really,
the doctrine of contradiction needs no proof, as it rests on two of
the fundamental principles of thought — those of Contradiction and
Excluded Middle. The only case in which 0 is equivalent to E and
1 to A, formally and in statement, is when the subject of each pro-
position is regarded as collective, and the propositions are, thus,
really Singular. In that case, contrarietj' and contradiction merge
into one, and the sqxiare of opposition (see § 98) becomes a straight
line, as we are reduced to two alternatives between which there is
no third possibility [cf. § 97 (ii.)]. In no case, therefore, is it true
that " the doctrine of contradiction breaks down."
Book III.
Cb. II.
If I and 0
could b e
both false,
then A and
E would
also be bnth
false ; thus,
no proposi-
tion would
be true.
98. The Square of Opposition.
It has long been traditional in Logic to give, as an aid to
remembering the doctrine of opposition, the accompanying
diagram, called the Square of Opposition.
If this diagram, with the proper positions of the letters
which symbolize the four kinds of propositions, be once
firmly stamped on the mind, but little difficulty will be found
in retaining in the memory the whole theory of opposition.
The universals are placed at the top, the particulars at the
bottom, the affirmatives on the left and the negatives on the
Tlie Sqn.are
of Opposi-
tion is a dia-
gram which
aids the
memory in
retaining
the doctrine
of opposi-
tion.
240
IMMEDIATE INFERENCES.
Book III.
Ch. II.
right. The diagonals, as the longest lines, mark Contradic-
tion, which is the most perfect and thoroughgoing form of
logical opposition [see § 97 (ii.)]. The top line indicates
Contrariety, and the bottom line, parallel to it. Sub-contrariety.
The fact that both are horizontal naturally suggests that
each connects propositions of the same quantity. The per-
Contrarits-
Summary —
pendicular lines appropriately represent Subalternation. As
the diagonals run from the one top corner to the opposite
bottom corner they indicate that contradictory propositions
differ both in quality and quantity. Similarly, the top and
bottom lines suggest a difference in quality only, and the side
lines a difference in quantity only.
99. Summary of Inferences from Opposition.
We will now summarize the inferences which the doctrine
of opposition enables us to draw, when we consider the
results which flow from positing and sublating in turn each
of the four forms of propositions : —
OPPOSITION OF PROPOSITIONS.
241
(1) Posit A. "B J contradiction 0 is sublated.
E is suhlated directly by contrariety to the given
proposition (A), and indirectly by subalternation
from its contradictory (0).
I is posited directly by subalternation to the
given proposition (A), and indirectly by sub-
contrariety to its contradictory (0).
(2) Sublate A. By contradiction 0 is posi/ec?.
E is left douhiful ; for it is neither posited nor
siiblated, either directly by contrariety or in-
directly by subalternation,
I is also left doubtful ; for it is neither posited
nor sublated, either directly by subalternation or
indirectly by sub-contrariety.
(3) Posit E. By contradiction I is snhlated.
A is suhlated, directly by contrariety to the
given proposition (E), and indirectly by subalter-
nation from its contradictory (I).
0 is posited, directly by subalternation to the
given proposition (E), and indirectly by sub-con-
trariety to its contradictory (I).
(4) Sublate E. By contradiction I is posited.
A is left doubtful ; for it is neither posited nor
sublated, either directly by contrariety or in-
directly by subalternation.
0 is also left doubtful ; for it is neither posited
nor sublated, either directly by subalternation or
indirectly by sub-contrariety.
(5) Posit I. By contradiction E is sublated,
A is left doubtful / for it is neither posited nor
sublated, either directly by subalternation from
the given proposition (I), or indirectly by con-
trariety to its contradictory (E).
0 is also left doubtful ; for it is neither posited
nor sublated, either directly by sub-contrariety to
the given proposition (I), or indirectly by sub
alternation to its contradictory (E).
LOO. I. 16
Book III.
Ch. H.
To posit A,
sublates O
and E, aud
posits I.
To sublate
A, posits O,
and leaves E
and I doubt-
ful.
To posit E,
s\;blates I
and A, and
posits O.
To sublate
E, posits I,
and leaves A
and O doubt-
ful.
To posit I,
sublates E,
and leaves A
andO doubt-
ful.
242
IMMEDIATE iNFEKENCfiS.
DOOK III.
Ch. II.
To sublate I,
posits E and
O, and sub-
latcs A.
To posit O,
sublates A,
and leaves I
and E doubt-
ful.
To sublate O,
posits A and
I, and sub-
lates E.
We can draw
most infer-
ences when
a universal
is tnie or a
particular
false.
(6) Sublate I. By contradiction E is posited.
0 is posited, directly bj' sub-contraviety to the
given proposition (I), and indirectly by subalter-
nation to its contradictory (E).
A is sublated, directly by subalternation from
the given proposition (I), and indirectly by con-
trariety to its contradictory (E).
(7) Posit 0. By contradiction A is suhlated.
1 is left doubtful ; for it is neither posited nor
sublated, either directly by sub-contrariety or
indirectly by subalternation.
E is also left doubtful ; for it is neither posited
nor sublated, either directly by subalternation or
indirectly by contrariety.
(H) Sublate 0. By contradiction A ]s jvisiied.
I is piosited, directly by sub-contrariety to the
given proposition (0), and indirectly by sub-
alternation to its contradictory (A).
E is sublated, directly by subalternation from
the given proposition (0), and indirectly by
contrariety to its contradictory (A).
In the above, every result has been given twice over ;
for the positing a universal is the same as sublating the
contradictory particular, and the sublating a universal is
identical with positing its contradictory. Hence (1) and (8),
(2) and (7), (3) and (6), (4) and (5) give exactly the same
inferences. We see from this detailed examination that
from the truth of a universal, or fi-om the falsity of a
particular, we can make definite inferences to the truth or
falsity of each of the three other opposed projiositions. But,
from the falsity of a universal, or the truth of a particular,
the only inference we can make is to the truth or falsity of
the contradictory ; about the other two opposed propositions
we can assert nothing.
It should also be noted that all the above results can be
reached by a consideration of contradiction and subalter-
nation alone, either singly or in combination.
OPPOSITION OP PROPOSITIONS.
243
Tbe following Table exhibits at a glance all the above
results. The kind of opposition through which they are
reached is given by the letter, or letters, in brackets under
each result. C means by contradiction ; S, by subalternation ;
C-y? by Contrariety ; Scy, by sub-contrariety. If any of
these letters is printed in italics it means that the process it
represents is indirect ; that is, the result is obtained, not
immediately from the given proposition, but indirectly
through its contradictory.
Book ITT.
Ch. It.
Table cf in-
f erences
from opposi-
tion.
Given
A
0
E
I
1
A true
false
(C)
false
(Cy.S)
true
(S, Scy)
2
A false
true
(C)
doubtful
doubtful
3
E true
false
(cy, 8)
true
(S, Scy)
false
(C)
4
E false
doubtful
doubtful
true
(C)
5
I true
doubtful
doubtful
false
(C)
6
I false
false
(S, Cy)
true
(Scy, S)
true
(C)
7
0 true
false
(C)
doubtful
doubtful
8
0 false
true
(G)
false
(S, Cy)
true
(Scy, S)
16—2
244
IMMEDIATE INFERKNCrS.
Book III.
Ch. II.
The doctrine
of Opposi-
tiun applies
to proposi-
tions in
■which con-
nexion of
content is
prominent.
lu tlie above treatment the traditional logic has been
followed in considering the propositions under the quantified
form in which denotation is the prominent aspect under
■which the subject is regarded. But the doctrine of opposi-
tion applies in every particular to propositions which make
connexion of content the prominent element and which are
the more fundamental forms of judgment in which the
justification of the denotative proposition must be sought
{cf. § 71). Thus, the Generic Judgment S is P is contra-
dicted by the Modal Particular S need not he P, whilst it has
for its contrary the Generic Judgment S is not P, and for its
subaltern the Modal Particular 5 mai/ be P.
Ilypotheti-
c a I and
Jlodal Par-
ticular Pro-
positions
stand to
each other
in all the
relations of
opposition.
100. Opposition of H3rpothetical Propositions.
The remarks made at the end of the last section apply
equally well to those more definite judgments of connexion of
content which are expressed in hypothetical form. The tine
hypotheticals : If S is M it is P, and Tf S is M it is not P — or
expressed in the more general but less definite symbolism
If A then X, and If A then not X — are universals, and cor-
respond to the A and E categorical forms respectively ;
whilst the more explicit forms of the Modal Particulars :
If S is M it may be P, and If S is M it need not be P — or in
the wider symbolic form, If A then j^ciliaps X, and If A then
not necessarily"^ — correspond to the land 0 categorical forms.
Having thus all the four necessary forms, the whole doctrine
of opposition is applicable.
Similarly with the denotative forms — or conditionals as
we have ventured to call them — which give more concrete
expression to the content of these abstract judgments
{see § 76). Here the four forms are : —
If any S is M that S is always f— corresponding to A.
Jf any S is M that S is never P — „ ,, E.
JfanS is M that S is sometimes P — „ ,, I.
If an S is M that S is sometimes not P „ „ 0
OPPOSITION OF PROPOSITIONS.
245
As the last two forms do not imply more than that S being
P is a possible consequence of its being M, but not that S
actually is P in any one case in which it is M, they may be
often better expressed by : —
If an S is M that S may he P — corresponding to I.
If an S is M that S need not he P — „ „ 0.
* In Suhalternation, the first statement of the particular
conditionals, perhaps, makes the opposition more apparent,
as it is evident that the statement of a universal connection
between S being M and its being P involves the less definite
particular statement. But, it must be remembered that
'sometimes' has here the same absolute indefiniteness as the
ordinary logical ' some.'
* The Contradictory and Contrary of a conditional are
more easily confused than are those of a categorical. At first
sight it might seem that If any S is M that S is P, and If any
S is M that S is not P are contradictories. But they are not,
as they do not exhaust all possible alternatives. A reference
to the table given above will show, indeed, that they are
both universals, and are, therefore, contraries. The true
contradictory of If any S is M that S is P is If an S is M
thai S need not be P, or If an S is M that S is sometimes not P;
as 'If any country is well governed, its people are happy' is
contradicted by * If a country is well governed, its people
need not be happy,' or by ' Though a country is well
governed, its people are sometimes not happy.' These pro-
positions are mutually inferrible, and fulfil all the other
requirements of contradictory opposition [see § 97 (ii.)]-
The contrary of the above proposition would, of course, be
' If any country is well governed, its people are not happy.'
* In Sub-contrariety, the second statement of the particular
conditional is preferable, as not suggesting that S is actually
P in any one case. If it be granted that S being P is a
possible, though not known to be a universal or necessary,
consequence of its being M, then it is evident that the two
propositions If an S is M that S may be P, and If an S is M that
S need not he P may both be true together. But they cannot
Book III.
Ch. II.
•Sometimes'
in the par-
ticular con-
ditional is
as indefinite
as ' some ' in
the particu-
lar categoi i-
cal.
The Contra-
dictory and
Contrary of a
Conditional
are apt to ba
confused.
In Sub-con-
trariety it ie
better to use
those forms
which
merely im-
ply that the
consequent
is a i)ossiblt
result from
the antecft-
dtiuU
246 IMMEDIATE INFERENCES
Book III. both be false ; for to deny that P may be affirmed of S is to
Ch^i. assert that it must not ; and to deny that it need not is to
assert that it must; and these assertions, being contra-
dictories, cannot be true together. Therefore, the original
propositions cannot both be false together.
101. Opposition of Disjunctive Propositions.
The most general symbolic form of the disjunctive propo-
sition— Either Xor Y — is most suitable to those cases in which
the alternative judgments have not the same subject. A dis-
junctive in this form must be regarded as singular, and as,
consequently, only capable of contradiction. The contra-
dictory proposition is Neither X nor Y, and this is not itself
a disjunctive judgment. But the more perfectly stated dis-
junctive judgments, in which several predicates are alterna-
tively affirmed of the same subject, admit of distinctions of
quantity, and propositions of opposite quality can be found
which stand to them in the relations of contradiction and
contrariety. Thus, with the judgment of content, 5 is either
P or Q, the square of opposition can be completed by the
propositions 5 is neither P nor Q (contrary) ; 5 7nay be either P
or Q (subaltern) ; S need not be either P or Q (contradictory).
These distinctions — as in the case of categorical judgments —
stand out yet more clearly in the denotative forms of the
propositions. Here we have the universal affirmative Every
S is either P or Q ; the universal negative No S is either P or Q ;
the particular affirmative Some S's are either P or Q ; and
the particular negative Some S's are neither P nor Q. > But
it will be noticed that none of the negative forms are dis-
junctive propositions. S is neither P nor Q is equally well
expressed in the copulative categorical form S is both P and Q
[cf. § 75 (i.) (a)], and similar propositions express the nega-
tive denotative forms. Hence, the full doctrine of opposition
cannot be said to be applicable to disjunctive j'ropositions.
It may be further pointed out that the particular affirmative
form is of but little value as an expression of knowledge ;
it is far removed from the ideal of a logical disjunctive
judgment, which unfolds the differences within a system, or,
OPPOSITION OF PROPOSITIONS. 247
in other words, predicates a choice between the various Book IIL
species of one and the same genus (c/. § 79). Ch^l.
As a concrete example we may take the proposition ' Every
swan is either white or black.' Its subaltern is ' Some swans
are either white or black'; its contradictory 'Some swans
are neither white nor black,' and its contrary ' No swan is
either white or black.'
CHAPTER TIL
Book III.
Ch. III.
Eductions
are imniodi-
ate iufer-
ences of pro-
positions
whose truth
is iiniihed
by a propo-
sition ac-
cepted as
true ;
i.e., ef the
predications
which can
bj made of
each of the
terms s, S,
P,P.
Each predi-
cation can
be made
with either
a positive or
a negative
predicate.
EDUCTIONS.
102. Chief Eductions of Categorical Propositions.
Eductions are those forms of Immediate Inference by
which, from a given proposition, accepted as true, we
educe other propositions, differing from it in subject, in
predicate, or in both, whose truth is implied by it. Every
Categorical Proposition gives us information of a certain
subject, in terms of a certain predicate. But, each of these
terms has a conceivable negative ; and every categorical pro-
position, therefore, suggests to our minds, directly or
indirectly, four terms — S, P, non-S, non-P. The problem
before us is to enquire what predications about each, or any,
of these possible terms are implied when S and P are con-
nected in any given categorical judgment. In other words,
whether, if we take each of these terms in turn as subject,
the given proposition justifies us in predicating of it any of
the other terms. We need not, of course, consider any forms
of proposition in which the predicate is either the same term
as the subject, or its negative — as S is S, S is non-S, P is not P,
etc. — which are either mere tautology, or are self -contradic-
tory and, therefore, self-destructive (c/. §§ 17 and 18). Our
enquiry is limited to those propositions in which one term ia
S or non-S, and the other P or non-P.
Now, when any one of these four terms is taken as subject,
we have two possible predicates offered to us ; thus, we can
predicate either P or non-P of 5, and either 5 or non-S of P.
This leads us to the kind of Eduction called Ohversion, in
which we retain the same subject but negative the predicate
EDUCTIONS. 249
of the original proposition. Again, if 5 is the subject, and Book III.
P the predicate, of the given proposition, we can form other _; '
propositions whose subjects are respectively, P, non-P, and
non-S, and each of these propositions can take two forms,
one of which is derived from the other by obversion. Thus
we get the following possible modes of inference, most of
which involves a change, not only in the verbal expression
but, in the form of the judgment as thought : —
(1) Obversion — when the subject of the original proposi- The four
tion is unchanged, but the predicate is negatived. Eductions
(2) Conversion — when the subject of the inferred pro- {i)6bversion
position is P, and its predicate S or non-S. ('^^ ^^n'^'
Co) Contraposition — when the subject of the inferred (^)Contro
\ -f -ir •> 'position
proposition is non-P and its predicate S or non-S. (i) inversion
(4) Inversion — when the subject of the inferred propo-
sition is non-S, and its predicate P or non-P.
None of these can be valid inferences from any given pro-
position, unless the inferred proposition is involved in, and
expresses the same truth as, that proposition itself expresses.
We must, therefore, by careful examination, see which of
them are justified by propositions of each of the four
forms, A, E, I, 0.
Each of the inferences (2), (3) and (4) in the above list can Each of the
take two forms, one with a positive, and the other with a has two
negative predicate. Each of these forms is obtainable from of'^^hidris
the other by the process of obversion. As, however, the the obverse
simplest forms are those which have the positive predicates,
the simple names, Conversion, Contraposition, and Inversion, are
applied to the processes by which they are arrived at. Those
propositions themselves are called the Converse, Contrapositive,
and Inverse, of the original proposition ; whilst the corre-
sponding forms with negative predicates are termed the
Obverted Converse, the Obverted Contrapositive, and the
Obverted Inverse, respectively, of that proposition. Thus,
each of these names expresses the relation in which that de-
rived proposition stands to the given one,
250
IMMEDIATE INFERENCES.
Book III.
Ch. III.
Table of
possible
Eductions.
These infer-
ences are
useful as
bringing out
all the impli-
cations of
the original
predication.
If we now use S and P to denote non-S and non-P respec-
tively, we have the following empty schema of possible
Eductions from categorical propositions : —
ui.
IV.
Converses of (1)
Contrapositivesof (1)
Original Proposition
Obverse of (1)
3 Converse of (1)
[n verses of (1)
Ob verted Converse of (1)
S-P
S—P
PS
P—S
Contrapositive of (1)
Obverted Contrapositive of (1)
Inverse of (1)
Obverted Inverse of (1)
PS
PS
S-P
S-P
We have now to see to what extent this empty schema
can be filled out by either of the four kinds of categorical
predication— A, E, I, 0, — when the original proposition itself
is of either of those forms.
Many of the inferred forms are unusual and unnatural
modes of expressing the truth which is stated most simply in
the original proposition. Those of them, too, which contain
negative terms are open, as primary modes of statement, to
the objections made to propositions containing those terms
in § 29. But, when they are regarded as simply secondary
modes of expressing the content of the original judgment,
they are useful ; as they make prominent a fresh side of the
truth there enunciated. And the whole of them together,
by placing that assertion in every possible light, make its
implications much clearer and more definite than a mere con-
sideration of the proposition itself would do.
A-S Obversion and Conversion are the primary modes by
EDUCTIONS.
251
which these eductions are made — for all the other inferences
are obtainable by combinations of these — a detailed consider-
ation of them should precede that of the other forms.
(i.) Ob version is a change in the quality of a predication
made of any given subject, whilst the import of the judg-
ment remains unchanged. The original proposition is
called the Obvertend, and that which is inferred from it is
termed the Obverse.
Whenever we assert anything we, by implication, deny
the opposite. That is, the aflBrmation of any predicate of a
certain subject implies the denial of its negative ; and
the denial of any predicate implies the affirmation of its
negative. The former of these follows from the Principle
of Contradiction — for, if any 5 is f* it cannot be non-P
(see § 18) ; and the latter from that of Excluded Middle
for, if any S is not P it must be non-P (see § 19). All obver-
sions of affirmative propositions, therefore, depend on the
former of these two principles ; and all obversions of nega-
tive propositions on the latter. But, to deny a negative is
to affirm, for two negatives destroy each other ; and to affirm
a negative is to deny ; and, thus, obversion involves no
change of meaning. The matter, therefore, which is ex-
pressed by an affirmative proposition can always be re-ex-
pressed by a negative, and vice versa. This is, however, a
mere change in the mode of expression ; it involves no pro-
cess of thought, and consequently is not a real inference. It
is, however, useful as a first step in contraposition.
From this it follows that the subject of the obverse is the
same as the subject of the obvertend in every respect, as,
otherwise, we should not have a true denial of the opposite
of that obvertend. The quantity of the two propositions is,
therefore, the same. The predicate of the obverse is the
negative of that of the obvertend, and this, to avoid altera-
tion in meaning, necessitates a change in the quality/ of the
proposition. This gives us the one simple rule for obverting
any proposition : —
Negative the predicate and change the quality, hut leave th^
quantity unaltered.
Book IIL
Ch. III.
Obversion
and Conver-
sion are the
primary
modes of
Eduction.
Obversion is
the chang-
ing the
quality of a
proposition,
but neither
its subject
nor its im-
port.
Obvertend —
tlie original
proposition.
Obverse — the
inferred
proposition.
Obversion of
Affirmatives
rests on
Principle of
Contradic-
tion ; of
Negatives
on that of
E.'fcluded
Middle.
Rule for Ob-
version—
Negative
the predi-
cate, change
quality.
252
IMMEDIATE INFERENCES.
Book ITI.
Ch. III.
A and E,
I and O are
pairs of
mutual ob-
Table of Ob-
versions.
Examples of
Obversion.
The formal
negative
term may be
replaced in
an obverse
by a mate-
rial nega-
tive, or pri-
vative,term,
only when
that term is
exactly
equivalent
to the formal
negative.
Apj)lying this rule to the four forms of categorical propo-
sitions, we find that
A ohverts to E, E to A, I to 0, and 0 to I ;
or, expressed symbolically
Original Proposition - -
SaP
SeP
SiP
SoP
Obverse ----..
SeP
Sa'P
SoP
SiP
It must be remembered that obversion is a reciprocal pro-
cess, and, thus, that S a P is as much the obverse of SeP,
as the latter is the obverse of the former.
As material examples we may give the following pairs of
propositions, each member of every pair being the obverse
of the other member : —
(A. All men are mortal.
\ E. No men are not-murtal.
JE. No thoughtful men are superstitious.
I A. All thoughtful men are non-superstitious.
J I. Some men are happy,
\0. Some men are not not-happy.
(0. Some men are not rich.
ll. Some men are not-rich.
* We may often write the obverse in a form more in ac-
cordance with the usages of ordinary speech by using a
material contradictory, or a privative, term [see § 29 (i.) (ii.)],
instead of the formal negative, for the new predicate. But,
unless this term is exactly equivalent in meaning to the
formal negative, we do not make a true obversion by its use.
For instance, in obverting A as given above, we could say
'No men are immortal,' for 'immortal' and 'not-mortal'
exactly correspond. But we could not give ' Some men are
not unhappy ' as the obverse of ' Some men are happy ' ; for
'happy' and 'unhappy' do not exhaust all possibilities, and,
thus, the principle of contradiction does not apply to them.
It is true that this proposition is justified by the given one, for
EDUCTIONS.
253
' not-happy' includes unhappy, as well as all other shades of
departure from ' happy.' But it is not the obverse ; for we
cannot get back from it, by obversion, to our original proposi-
tion. The same holds in the case of all affirmative pro-
positions ; the obverse justifies the denial of all terms which
can be brought under the formal negative. But even this is
not justifiable in the case of the obversion of negative pro-
positions. From ' Some men are not happy ' we cannot con-
clude that ' Some men are unhappy,' for this latter proposition
asserts, not merely the absence of happiness, but the presence
of a certain amount of positive misery [cf. § 29 (i.)]- Still
less can we infer from * Some men are not rich ' that * Some
men ar6 poor ' ; for ' rich ' and ' poor ' are contraries, and
there are many intermediate stages between them, Obversion
is, in short, a formal process ; and, therefore, if we do not
use a formal negative term for our new predicate, we must
make sure that the term we do use is the exact equivalent of
that formal negative.
The results of obversion can be immediately gathered
from an inspection of the diagrams given in § 94, As we
are dealing with propositions involving non-P, it is better
to use the fuller set of diagrams given first in that section : —
Book ITI.
Ch. III.
The obverse
of a proposi-
tion can be
gathered
immediately
from its
diagram.
[A]
[E]
[I]
[0]
_ In the diagram for A, ^only occurs in combination with
S, therefore No S is P. In that for E, S occurs only in the
class SP; hence. All S is P. In that for I, Some S, at least, is
not P, for it is P. Lastly, in that for 0, Soine S is P is given
immediately.
254
IMMEDIATE INFERENCES.
Book III.
Ch. III.
This gives a
fresh illus-
tration of
the implica-
tions of ex-
istence.
Several
other names
have been
given to
obversion.
The so-called
Material
Obversion re-
quires a re-
ference to
the matter
of the
obverse pro-
position,
* The same diagrams also, wben thus applied to obversion,
illustrate afresh the view of existential import adopted
in, § 89. For, in the diagrams for A and I, wherever P
occurs its existence is marked by the dotted line which
implies doubt ; therefore in the negative propositions, S e P
and S 0 P, which are the respective obverses, the existence of
the predicate in the same sphere as the subject is not assured.
But when we affirm S a P and S i P as the respective obverses
of E and 0, our diagrams show us that the existence of these
classes is certain. And this must needs be so ; for, if the
negative propositions, S e P and S o P, do not ensure P in the
same universe as S, then, as, by the principle of Excluded
Middle {see § 19), every S must be either P or P, even if P does
not exist at all in that universe, yet P must.
An examination of Euler's circles (see § 91) will also give the
obverses of A, E, I, 0, though, as each of those propositions, except E,
requires more than one diagram, the results are not so immediately
manifest.
Obversion has been called Permutation (by Fowler, Ray, and
Stock) ; ^quipollence (by Ueberweg, Bowen, and Ray) ; Infinita-
tion (by Bowen) ; Immediate Inference by Privative Conception
(by Jevons) ; Contraversion (by De Morgan) ; and Contraposition
(by Spalding). But Obversion is the most usual name, and is
adopted by the majority of writers either by itself, or (as in the
case of Ueberweg, Ray, Stock, and Jevons) as synonymous with
one of the other names.
* Material Obversion. Professor Bain considers that, in
addition to the formal process we have been considering,
"there are Obverse Inferences justified only on an examina-
" tion of the matter of the proposition. From ' warmth is
"agreeable' we can affirm, by formal obversion, 'warmth is
" not disagreeable, and not indifferent.' We cannot affirm,
"■ without an examination of the subject matter, ' cold is dis-
" agreeable.' . . . Experience teaches us that in an actual
" state of pleasurable warmth, the sudden change to cold is
" also a change to the disagreeable. Whenever an agent is
'•giving us pleasure in act, the abrupt withdrawal of that
I
I
SbtTCTI0N3.
255
" agent is a positive cause of pain. On the faith of this in-
"duction, we can obvert materially a large number of propo-
" sitions regarding pleasure and pain, good and evil " {Ded.
Log., pp. 111-2).
But to call this obversion is unusual. The new proposition
has not the same subject as the old, but a negative of that
subject. It is not derived in any way from the original pro-
position, but, as Prof. Bain himself says, rests on the strength
of an induction quite outside it. A proposition may point
out to us what to examine ; it may suggest a possible result,
and this result may be found to agree with reality. Thus
' warmth is agreeable ' may suggest that ' the opposite of
warmth is the opposite of agreeable,' but we cannot infer the
latter proposition from the former. In fact, it is quite con-
ceivable that two opposite subjects should yet have the same
predicate ; for two opposite states may both be agreeable, or
the reverse. For example, because ' Light is beneficial ' it
does not follow that ' Darkness is harmful,' nor does the
agreeableness of exercise postulate the painfulness of rest.
Book TTT.
Ch. III.
changes the
subject of
the obver-
tend, and is
not an infer-
ence from it.
(ii.) Ocnverslon is the eduction of one proposition
from another by transposing the terms. The original
proposition is called the Converteinl, and that which is
derived from it is named the Converse.
We have, evidently, here a complete alteration of stand-
point, as we have changed the subject or nucleus of our pro-
position. The predication is now made of P in terms of S,
whereas the original proposition contained an assertion about
S in terms of P. Moreover, the truth of the converse follows
directly from that of the convertend. Hence, the process is
a real interpretative infei-ence. Every proposition before
being converted — or, indeed, used in any kind of formal in-
ference— must be reduced to the strict logical form, 5 is P or
S is not P (cf. § 68), and the whole predicate must change
places with the ivhole subject. For instance, the converse of
' Every old man has been a boy ' is not ' Every boy has been
an old man,' but 'Some who have been boys are old men'-
for the original proposition, in its logical form, is 'Every old
Conversion is
the infer-
ence of one
proposition
from
another, by
transposing
the terms.
Convertend —
the original
proposition.
Converse —
the inferred
proposition.
256
IMMEDIATE INFERENCES,
Book III.
Ch. III.
Conversion
does not
change tlie
quality of
the proposi-
tlOQ,
but may
change its
quantity.
Conversion
which is a
valid infer-
ence is called
Illative.
AuUt for
Conversion —
Retain
quality; dis-
tribute no
term not
given as dis-
tributed.
A converts
to I.
This conver-
sion is called
per atcident.
man is a person who has been a boy.' As the converse
simply makes the same assertion as the convertend, looked at,
as it were, from the other side, it is clear that the quality of
both propositions will be the same.
Every act of conversion involves reading the origiual
predicate in its denotation, in order that it may be made a
subject-term. That we really do make this change from a
connotative to a denotative view is shown by the fact that,
if the predicate of the convertend is an adjective — as in
'No crows are white ' — a substantive must be supplied before
we can use that term as the subject of the converse — as 'No
white things are crows.' This involves a consideration of
the distribution of the predicate {see § 72) in order that the
converse may not assert more than is justified by the conver-
tend ; and may necessitate a change of quantity. In other
words, a mere transposition of terms is not always permis-
sible ; we cannot go from ' All cats are animals ' to ' All
animals are cats.' The only conversion we are concerned
with is Illative Convasion ; that is, conversion which is a
valid inference, and in which either both convertend and
converse are true, or both are false. Such conversion must
obey these two rules : —
1. The quality of the proposition must remain unchanged.
2. No term may he distributed in the converse lohich is not
distributed in the convertend.
We must now apply these rules to the conversion of each
kind of categorical proposition.
(a) Conversion of A. In the proposition S a P, whilst S is
distributed, P is not. We cannot, therefore, convert to P a S
for that would break Rule 2 — but we must retain P in its
undistributed condition, and write the converse P i S. Hence
A converts to I, and the conversion involves a change of
quantity from universal to particular. Such conversion was
called by Aristotle Kara nkpoi^ or partitive conversion. This
name has however, given place to the less descriptive one of
conversio per accidens or conversion by limitation. Though the
necessity for this mode of converting A propositions is obvious
EDUCTIONS.
257
enough when the rules for conversion are kept in mind, yet
the improper conversion of A propositions is one of the most
frequent causes of fallacy. Because it is a fairly well estab-
lished fact that all very clever persons have large brains, an
abnormally large cranium is often held to be a sign of great
ability. As lazy persons are often out of work, people jump
to the conclusion that if a man is often out of work he is
necessarily lazy. Since the wages of unskilled labour in
England are low, it is frequently assumed that all badly paid
persons are unskilful. Because all pious people go regularly to
church, regular church-going is commonly regarded as a sure
sign of piety. Such mistakes are continually made, yet they are
on a par with arguing that every animal is a monkey because
every monkey is an animal. No doubt, in some cases — as
tautologous propositions and definitions, or when both sub-
ject and predicate are singular names — the simple converse,
i,e., converse without change of quantity, of A would give a
true proposition. ' Every equiangular triangle is equilateral '
is as true a proposition as is 'Every equilateral triangle is
equiangular.' But its truth has to be established by a sepa-
rate and independent demonstration ; it cannot be inferred
from the latter proposition by conversion. For conversion,
as a formal process of inference, must be applicable to every
proposition of the same kind ; there cannot be two modes of
formally converting A propositions. When the simple con-
verse would be true in fact, it is because of special circum-
stances which do not appear in the statement of the conver-
tend. Hence, as P i S is the only converse which is materially
true in all cases, and is formally true in any, that is the
logical converse oi S a P. For, whilst S a P asserts posi-
tively that the attribute which P denotes is found in every
S, it is not stated whether, or not, it is found in other cases.
This is obvious from an inspection of the diagram for an
A proposition
SP SP
Book III.
Ch. 111.
A is often
improperly
converted.
The simple
converse of
A would
sometimes
be a true
proposition,
but its truth
must be
established
independ-
ently.
That A must
be con verted
per accidens
is evident
from its
diagram.
where it is plain that we cannot say of All P that it is S ; for
the existence of the class SP is shown to be possible. The
LOG. I. 17
268
IMMEDIATE INFERENCES.
Book III.
Ch. III.
E converts
simply to E.
This is obvi-
ous from the
diagram.
figure also shows that the converse is as real a proposition as
the convertend ; if the subject and predicate exist in the
latter, they equally exist in the former.
(6) Conversion of E. An E proposition can be converted
simphj ; that is, without change of quantity. For, S e P
asserts that the attributes connoted by P are found in none
of the objects which 5 denotes, but only in other objects.
Hence, none of the objects in which P is found, and which
are all denoted by P used as a substantive name, possess
the attributes which are connoted by S. The separation
between the things which are 5 and those which possess the
attribute P is total and absolute ; and is, therefore, reci-
procal. Whether we regard it from the side of S or of P,
each individual S differs from each individual P.
Thus, we can convert S e P to P e S. If ' No horses are
carnivorous,' it follows that ' No carnivorous animals are
horses.'
This is plain from the diagram for an E proposition.
The truth of
tlie converse
of E is condi-
tional upon
the exist-
ence of P ; if
stated as a
categorical
proposition
it is, there-
fore, an in-
valid infer-
ence;
SP
SP
"I
-I
* This diagram draws attention to another point — that we
have no assurance of the existence of the subject of the
converse in the universe of discourse fixed by the convertend.
This is a necessary outcome of the view that the predicate of
a negative proposition is not necessarily existent in the
sphere to which that proposition refers (see § 89). But,
if an inference is valid, the inferred proposition must refer
to the same sphere as the original proposition, and must be
true in that sphere if the proposition from which it is
deduced is true. Moreover, its truth must be justified solely
by the given proposition itself, without any information
external to that proposition, or it ceases to be a formal
inference from that proposition. But, in the case of the
conversion of E we cannot be sure that, when the con-
vertend is true, the converse is also true, unless we know
from other and material considerations that the predicate of
EDUCTIONS. 259
the convertend belongs to, and exists in, the same sphere Book III.
as the subject. Of course, in most material examples, both Ch^ii.
the terms of the convertend are known to refer to the same
sphere, and then simple conversion gives a true proposition.
But this is not an inference from the convertend alone, but
from the convertend interpreted in a particular way by
information external to itself. It follows, as the formal
process must apply in all cases and must not travel outside
the given proposition, that the conversion of E is an invalid
process, if the converse is stated as a categorical proposition.
The formally correct statement of the converse is, therefore, its formally
conditional as regards the existence of P — If any P exists, it is statement is
not S. For example, 'No woman is now hanged for theft in /jj^j^^^^'fj
England' converts simply to 'Nobody now hanged for theft
in England is a woman.' But, as a matter of fad, nobody —
man, woman or child — is now hanged in England for that
crime ; though the converse, thus stated, must be regarded
as asserting that some thieves are so punished, for it implies
the existence of its subject in the sphere to which the
convertend belongs — that is, the sphere of actual physical
reality. The true statement would be ' If any person is now
hanged for theft in England, that person is not a woman.'
And, as a statement in this conditional form is the only one
which is true in all cases, it is the only formal inference
which can be drawn, by conversion, from an E proposition.
(c) Conversion of I. As neither term in an I proposition is 1 converts
distributed, it is clear that, by converting it simply, we shall "^^ ^
break neither of the rules of conversion. Thus, SiP converts
to P is, and the proposition remains particular. ' Some herbs
are poisonous ' gives as a converse * Some poisonous things
are herbs.' The 'some' remains, of course, purely indefinite;
and when we speak of the simple conversion of I we do not
mean that ' some ' denotes the same proportion of the total
denotation of the subjects of both convertend and converse.
When the subject of the convertend is a genus of which the
predicate is a species, the simple converse reads somewhat
awkwardly. Thus, ' Some human beings are boys ' converts
2m
IMMEDIATE INFERENCES.
Book III.
Ch. III.
Simple con-
version is re-
ciprocal, but
conversion
per accidens
is not.
The diagram
for I shows
that it is
convertible
simply.
to ' Some boys are human beings,' which, we feel, is not so
definite an assertion as our knowledge of the matter would
warrant us in making. This is particularly noticeable when
we reconvert the converse of an A proposition. The con-
verse of S a P is P i S, and we can only convert this again to
S i P, where the double logical process has led to a loss of
fulness in the statement. For example, 'AH monkeys are
animals ' converts to ' Some animals are monkeys, and the
simple converse of this is ' Some monkeys are animals.' This
shows that conversion per accidens is not a reciprocal process,
as simple conversion is. But, no matter what the I pro-
position is, or whence it is derived, it can, by itself, only
justify us in deducing another I proposition as its converse.
That I is simply convertible is immediately evident on an
inspection of the diagram
SP
SP
SP
I-
-I-
-1
It is plain that the assertion about P is indefinite in the
same way as is the assertion about S. Some P, at least, is 5,
but the diagram can say nothing positive about All P.
0 cannot be
uouverted.
(d) Conversion of 0. As the predicate of an 0 proposition
is distributed, but the subject undistributed (see § 72), we
cannot convert a proposition of that form at all. For, by
Rule 1, S 0 P must convert to a negative proposition with S
for its predicate. This would distribute S ; but Rule 2
forbids this distribution, as S is not distributed in the
convertend. S o P asserts that Some S's have not the
attribute P, but it says nothing about the other possible S's.
Hence, though the Some S^s which form the subject are
entirely separated from all those things which possess the
attribute P, it does not follow that these latter are excluded
from all the S's. It is possible that every P is S, though
there are other instances of S as well (cf. Fig. Ill, § 91 ) which
are not P. For example, ' Some men are not honest ' will
not justify us in inferring that ' Some honest beings are not
men'; nor can we say that some who pass an examinatioa
EDUCTIONS.
261
Book Til
Ch. III.
ThediaErram
for O sliowa
its inconver-
tibility.
do not sit for it, because it is true that some who sit for au
examination do not pass. In many cases, no doubt, the simple
converse of an 0 proposition would be materially true ; thus
' Some men are not black ' and ' Some black things are not
men ' are both true propositions, but neither can be inferred
by formal conversion from the other, for neither statement
is justified by the other.
The inconvertibility of 0 is evident from the diagram
SP SP SP
I '1 1 1
This shows clearly that we can make no definite assertion
about any part of P in terms of 5 ; f or P is entirely repre-
sented by the dotted line which signifies uncertainty.
The doctrine of conversion can aho be traced out from Euler'a
circles (see § 91), though the plurality of diagrams required for
every proposition, except E, makes the process somewhat complex.
To sum up the results we have obtained : —
A converts per accidens ; E and I, simply ; 0, not at nil.
Several logicians have attempted to furnish proofs of the
validity of conversion. These have all taken the indirect form
of a. reductio ad ahsurdum, that is of showing that the assump-
tion of the contradictory of the converse leads to results in-
consistent with the convertend. But as the process is an
immediate application of the formal laws of thought (see
§§ 17-19), it is really a primary one, and as such does not
require proof.
(e) Diverted Conversion. As any categorical proposition By obverting
whatever can be obverted, we can get a new inference from ^e getTnew
the original proposition by obvertinsr the converse, according inference
,,,,.. ,. ° ' ° from a pro-
to the rules given m sub-section (i.). Thus, expressed sym- position,
bolically, we get : —
Table of Cou
■versions.
•Summary.
1.
2.
Original Proposition ...
SaP
S eP
SiP^SoP
Converse of (1)
PiS
Pes
P i S (None)
3.
Obverted Converse of (1 )
Pa's
PaS
PoS
(None)
262 IMMEDIATE INFERENCES,
Book III. As material examples we may give : —
Ch. III.
rOriginal Proposition - A - Every truthful man is trusted.
Examples of J Converse - - - I - Some trusted men are truthful.
onversion. J^Qbverted Converse - 0 - Some trusted men are not untruthful.
/"Original Proposition - E - No cidtivated district is uninhabited.
-j Converse - - - E - No uninhabited district is cultivated.
(^Obverted Converse - A - All uninhabited districts are uncultivated.
{Original Proposition - I - Some British subjects are dishonest.
Converse - - - I - Some dishonest people are British subjects.
Obverted Converse - 0 - So7ne dishonest people are not aliens.
We have purposely chosen examples in which the negative
predicate of the obverted converse can be expressed by a
material negative, or by a privative word. This, of course,
cannot always be done ; and then the derived proposition is
frequently awkward in expression.
conirapnsHion (iii.) Coiitraposition is the inferring, from a given pro-
ring a 'pro^po- position, another proposition whose subject is the contra-
sition with p dictory of the predicate of the original proposition. The
oreu jec . (jgrived proposition is called the Contrapositive ; there is no
Contrapnsiiive corresponding distinctive name for the original proposition.
ferred pro- The contrapositive of any given proposition is most easily
position. arrived at indirectly. It makes a predication about the con-
tradictory of the predicate of the given proposition. Now,
this contradictory appears as the predicate of the obverse of
that proposition. If, then, this obverse can be converted it
gives a proposition of the form required, in which the nega-
tive of the original predicate is the subject, and the subject
of the original proposition is the predicate. Hence, the
Rule for Con- . , ■, °. ^ ,^ ... ■
traposition— Simple rule f or contraposition is : —
First obvert,
then convert. First Obvert, then convert.
This will give, in every case, a proposition differing in
tion^changes 9.uality from the original one ; for obversion changes the
quality, quality, and conversion does not change it back again. But
the quantity remains unchanged, except in the case of the
quantf^^^ contraposition of E ; for, obversion does not change quantity,
unaltered, and, therefore, any change in quantity must be due to the
case of E. subsequent conversion. Now, as A and 0 obvert to £ and I
EDUCTIONS.
263
respectively, and both of these convert simply, the quantity
will remain unaltered. But E obverts to A, which can only
be converted per accidens, and, hence the contrapositive of
the universal negative is a particular affirmative. Thus»
comparing the contraposition with the conversion of universal
propositions in respect to quantity, it is seen that when the
one inference causes a change in quantity, the other does not,
and vice versa. As I obverts to 0, which cannot be con-
verted, there can be no contrapositive of I.
Contraposition is sometimes called Conversion hy Negation,
and, as we see, it can be applied to 0 propositions, and is the
only form of ' conversion ' which can be so applied. But, it
is better not to use ' conversion ' in this sense, as the contra-
positive has not the same subject as the converse, and also
differs from it in quality.
Ohverted Contraposition. — Having obtained the contrapositive
of any proposition we can obvert it, and thus get a proposition
of the same quality as the original one. This Obverted Contra-
positive has for each of its terms the contradictory of a term in
thegiven proposition — its subjectis the negativeof the original
predicate, and its predicate the negative of the original subject.
Some writers have confined the name Contrapositive to this
form. The older logicians all did this, as they held that contra-
position, being a kind of conversion, should not change the
quality of the given proposition. There seems, however, to be
no reason for thus restricting the application of the name.
Both forms are contrapositives, and, when we wish to distin-
guish them, we call the simpler — that is, the one which re-
tains one of the original terms — the contrapositive, whilst
the proposition derived from that by obversion is fitly named
the obverted contrapositive.
We get, then, the following results,expressed symbolically : —
Book ITI.
Ch. III.
1
Original Proposition
SaP
SeP
SiP
SoP
2
[Obverse of (1)] -
ISeP]
[SaP]
[SoP]
[SiP]
3
Contrapositive of (1)
'PeS
~Pi S
(None)
PiS
4
Obverted Contrapositive of (1)
PaS
PoS
(None)
PoS
I cannot be
c o n t r a p o-
sited.
Contraposi-
tion is some-
times called
Conversioniy
Negatiov,
but the
name is not
appropriate.
The Obverted
Contraposi-
tive has
the same
quality as
the original
proposition.
The name
Contraposi-
tive has
sometimes
been need-
lessly con-
fined to this
form.
Table of Con.
trapositives.
Book III.
Ch. III.
Examples of
Contraposi-
tion.
264 IMMEDIATE INFERENCES.
As material examples we may give : —
' Original Proposition - A - Every poison is capable of destroying
[Obverse] - - -[-E]- [No poison is incapable of destroying
life.]
Contrapositive ■ - E - Nothing incapable of destroying life
is poisonous.
Obvd. Contrapositive- A - Everything incapable of destroying
life is non-poisonous.
'Original Proposition
[Obverse] -
Contrapositive -
E - No lazy person is deserving of success.
[A] - [Every lazy person is undeserving of
success.]
I - Some people undeserving of success
are lazy.
Obvd. Contrapositive - 0 - Some people undeserving of success
are not not-lazy.
^Original Proposition - 0 - Some unjust laics are not repealed.
J [Obverse] - - - [I] - [Some unjust laws are unrepealed. ]
j Contrapositive - -I - Some unrepealed laws are unjust.
\.Obvd. Contrapositive - 0 - Some unrepealed laws are not jvst.
We have carefully chosen instances where we can use
terms equivalent in meaning to the formal negatives, in
order that the resultant propositions might not be too far
removed from the usages of ordinary speech. When we
have to use formal negative terms these eductions often
result in strained and unnatural modes of expression. For
example, the obverted contrapositive of ' No plants feed ' if
' Some non-feeders are not non-plants.'
The great value of contraposition is this. The aim of
science is to teach propositions which are in fact reciprocal.
In such propositions the predicate is stated so definitely that
it is strictly characteristic of the subject, that is, it belongs
in exactly that form to nothing else, and the knowledge ex-
pressed by the proposition is, therefore, of the most precise
form attainable. When then S a Z' is established, we want to
know if f a S is also true ; and the readiest way to establish
this is generally to examine cases of S and endeavour to
establish the proposition ^ c P which is the contrapositive of
P a S. The importance of this will appear more clearly in
the discussion of Induction in Book V.
EDUCTIONS. 265
(iv.) Inversion is the inferring, from a given propo- book III.
sition, another proposition whose subject is the con- ch. iii.
tradictory of the subject of the original proposition. inveT^nia
The given proposition is called the Invertend, that which is ^^^ inferring
° , , . . J 1 7- '"'' proposition
inferred from it is termed the inverse. ^^h. s for its
The inverse of any given proposition is most easily subject.
arrived at indirectly, through some of the forms of eduction the given
we have already considered. We can only obtain the con- ^^j^rse— the
tradictory of a term by obverting the proposition of which inferred pro-
o IT IT Dosition
that term forms the predicate. 5 must, therefore, have been ^^ jnvers
made the predicate of a proposition, and then that pro- is most
position must have been obverted for us to get non~S. Two reached in-
eductions — the obverted converse and the obverted contra- directly,
positive — satisfy these conditions. If, then, we can convert
either of these we have an Inverse. Hence the rule for
Inversion is :— f'ti'-/'' ■^"-
version : —
Convert either the Obverted Converse or the Obverted Contra- ^.P.^'^f/*
either the
positive. Obverted
Converse or
In the case of A the obverted converse is P oS [see sub-§ ^'in'traposf-^
(ii.) (e)], and this is inconvertible. But the obverted tive.
contrapositive is Z' a 5 [see sub - § (iii.)], which can be
converted to S i P. As both the terms of this proposition s aP inverts
are contradictories of those which appear in the original converUng
proposition, it is not the simple, but the obverted, inverse, t^e Obvd.
. , , . . . , , Contrap.and
As, however, obversion is a reciprocal process, we can obvert obverting
this io S 0 P which gives the simple inverse. * "^"
In the case of E, the obverted converse is P a S \_see sub-§ s 'J inverts
(ii.) {e)\ which, by conversion, gives the inverse Si P; convertii^
this we can obvert to S o P, which is the obverted *|^® obvd.
' Converse.
inverse.
In the case of I, the obverted converse is P o S [see sub-§ s i p has no
(ii.) (e )], which cannot be converted ; and it has no obverted
contrapositive [see sub-§ (iii.)], and, therefore, it can have
no inverse.
In the case of 0, there is no obverted converse [see sub-§ so phns no
(ii.) ( e )], and the obverted contrapositive is P o S [see
sub-§ (iii.)], which cannot be converted ; 0 has, therefore, no
inverse.
inverse.
Inverse.
266
IMMEDIATE INFERENCES.
Book III. So we may sum up the possible inverses symbolically
C^- "^- thus :—
Table of In-
verses.
The truth of
the inverse
of A is con-
ditional
upon the ex-
istence of S
and "p; that
of the in-
verse of E
upon S and P;
if the infer-
ences are
stated in a
categorical
form, they
are invalid.
1
2
Original Proposition
SaP
Se P
SiP
SoP
[Ob verted Converse of (1)]
[PaS]
3
4
[Obverted Contrapositive of (1)]
[PaS]
Inverse of (1) -
SoP
SiP
(None)
(None)
5
Obverted Inverse of (1)
SiP
SoP
(None) (None)
1
Examples of
Inversion.
N.B. — Only the eduction from which each inverse is immediately obtained
is given. The blank spaces, therefore, simply mean that, if that particular
eduction exists, the inverse cannot be derived from it.
As material examples we may give : —
Original Proposition - A - Every truthful man is trusted.
[Obvd. Contrapositive] - [A]- [Every not-trusted manisuntruthf id.']
Inverse
Obvd. Inverse
^Original Proposition
[Obvd. Converse]
Inverse
Obvd. Inverse
0 - Some untruthful men are not trusted.
■ I - Some untruthful men are not-trusted.
■ E - No unjust act is worthy of praise.
[A]- [Every act worthy of praise is just.]
• I - So7ne just acts are worthy of praise.
• 0 - Some just acts are not umoorthy of
praise.
* As the inverse of A in its affirmative form — i.e. the
obverted inverse — involves both the terms S and P, and as
the inverse of E involves S and P, none of which terms are
guaranteed to exist by the original propositions, it follows
that these eductions must be merely conditional as regards
the existence of both their terms. Thus, the only meaning
of the inverse of A is, If S and P both exist, then some S is P,
and that of "E is, If S and P both exist, then some S is P.
Hence, an inverse from a true proposition is not necessarily
true when stated categorically, but only when stated as con-
ditional upon the existence of both the subject and the
predicate of the inferred proposition, and this conditional
form must be regarded as the true formal inverse.
EDUCTIONS.
267
103. Summary of Chief Eductions. Book III.
In the last section we examined in detail how far, from a - '
given proposition of one of the forms A, E, I, 0, we can |'^u^on8°'
infer propositions in which predications are made about
each of the terms of the original proposition and the con-
tradictories of those terms. In other words, we have inves-
tigated to what extent it is possible to fill up the empty
schema given in the early part of that section. We have
also discussed to what extent the truth of the derived pro-
positions is conditional upon the existence of classes denoted
by certain terms whose existence is not necessarily implied
by the original propositions.
We see that from any universal proposition a predication,
categorical or conditional, can be deduced about each of the
terms 5, S, P, P ; but from a particular we can only infer
predications about S and one of the two terms P and P. It
may be noted that the converses of A are the same as those
of I, and the contrapositives of E the same as those of 0.
These results may be thus tabulated : —
Complete
Table of
Eductionsof
Categorical
Propositions
4-
'I:
A
E
I
0
SoP
. Original Proposition
SaP
SeP
SaP
SiP
2, Obverse of (1) . . .
SeP
SoP
SiP
5 Converse of (1) -
PiS
PeS^
PiS
t Obverted Converse of (1)
PoS
PaS^
PoS
)Contrapositive of (1)
PeS^
PiS
PiS
5 Obverted Contrapositive of (1)
PaS'
PoS
PoS
r Inverse of (1)
SoP
SiP'
SoP'
5,0b verted Inverse of (1)
Sip'
' Conditional upon the existence of p.
* Conditional upon the existence of "p.
3 Conditional upon the exittenoe of ^ and K
' Conditional upon the existence QfS and P.
268
IMMEDIATE INFERENCES.
Book III.
Ch. III.
Some educ-
tions are
material and
no t of
universal
validity.
104. Less Important Eductions.
The eductions we are to deal with in this section are not
purely formal inferences. They do not hold in the case of
all categorical propositions, and their validity, or invalidity,
must, at all times, be decided by a consideration of the matter
of the propositions concerned. They are, thus, not necessary
inferences, and are of considerably less generality, and, con-
sequently, of less importance, than are the eductions we have
already considered in this chapter. They may be classed
under two heads : —
In Inference
by Added De-
terminants
both subject
and predi-
cate are
limited in
the same
way.
This bmita
tion is Deter-
mination,
and the
limiting-
word is a De-
terminant.
The deter,
minant of
the subject
must be
identical
with tliat of
the predi-
cate.
The mean-
ing of a word
being mudi-
fied by the
context,
(i.) Inference by Added Determinants is the deducing,
from a given proposition, another proposition of a nar-
rower application, by limiting both the subject and the
predicate of the original proposition in an identical
manner. Such limitation is called Determination, and is
effected by adding the same qualiiication to each term. Both
subject and predicate are thus made complex (cf. § 74) ; and
each element of the complex term is really a determinant of
the other. But, in speaking of this kind of inference, the
name Determinant is usually restricted to the freshly added
qualification. A Determinant may, therefore, be defined as
a qualification added to a term, tohich, as it does not belong to
that term in its icTiole denotation, limits, or determines, its appli-
cation in this special case \cf. § 74 (ii.)]. Hence, if the same
determinant is added to both subject and predicate, the ex-
tent of each is limited, but each is made more definite, and
the more limited proposition is a true inference from the
wider one. For example, 'AH negroes are men,' therefore
' Every honest negro is an honest man ' ; ' Wrongdoers are
deserving of punishment,' consequently ' Female wrongdoers
are females who deserve punishment ' ; ' Poetry is food for
the imagination,' hence ' Good poetry is good food for the
imagination.' But it must be jyrecisely the same determinant
in each case, and this will not always be secured by using the
same word, for the meaning of words is constantly modified
by the context [c/. §§ 3; 171 (ii.)]- And this modification is
very various, and often of so subtle a character that it escapes
EDUCTIONS.
269
notice. Thus, the employment of this kind of inference is
very liable to lead to fallacy, which must be guarded against
by a careful reference to the special meaning in each case.
If the attributes added as determinants imply any kind of
comparison, the liability to fallacy is enormously increased.
For instance, because it is true that ' An ant is an animal,' it
does not follow that 'A large ant is a large animal,' for
' large ' is a comparative term ; we can only deduce the tau-
tologous proposition that ' A large ant is an animal large for
an ant.' Nor can we infer from 'A bass singer is a man'
that ' A bad bass singer is a bad man,' but only that he is a
man who sings bass badly, which is a very different thing.
If the added attributes imply quite definite qualities, the in-
ferred proposition is more likely to be true, but this greatly
limits the range of this kind of inference. We can infer
from ' A prison is a place of detention ' that ' A stone prison
is a stone place of detention ' ; and from ' A ball is a play-
thing' that 'A leather ball is a leather plaything.' But from
* The unemployed are deserving of help,' we are not likely to
draw the inference that * The unemployed when rioting are
deserving of help in rioting.' In all cases, too, the predicate
must either be a substantive or equivalent in force to a
substantive. From ' The army is worn out by the long
march ' we cannot infer that ' Half the army is hiilf worn
out by the long march ' ; the true inference is ' Half the
army is half the body which is worn out by the long march.'
If the original proposition is negative, then the limiting
the application of subject and predicate makes no difference
in the information conveyed ; for the exclusion is complete
at first, and that wider exclusion necessarily includes the
narrower.
Occasionally a valid inference can be made when the de-
terminants of the subject and predicate are not the same. In
this case, the determinant of the subject is itself the subject
of a proposition of which the determinant of the predicate
is the predicate. Thus, from ' Theft is deserving of pun-
ishment ' and ' Unemployed workmen are poor ' we can infer
' Unemployed workmen who steal are poor men who desarve
Book III.
Ch. III.
this kind of
infereuce is
often falla-
cious;
especially if
the deter-
minant im-
plies compa-
rison.
The predi-
cate must be
a substan-
tive tci'm.
Two propo-
sitions can
be sometimes
combined so
that the
terms of one
are determi-
nants of thosa
of the other.
270
IMMEDIATE INFERENCES.
Book III.
Cb. III.
In Inference
h/ Complex
Conception
the subject
and predi-
cate are made
determinants
of a third
term.
This mode of
inference is
as liable to
fallacy as is
that by
Added De-
terminants.
punishment.' Leibniz thus symbolized such inferences : ' If
A=B and L = M, then A+L = B-\-M,' where = does not
signify equality, but merely denotes the logical copula * is,'
and -|- simply implies the addition of elements to each other
to form a complex term. The formula does not imply that
any two propositions can be thus combined. From ' Lions
are carnivorous ' and ' Oxen are herbivorous,' we cannot de-
duce the statement that ' Lions and oxen are carnivorous and
herbivorous' ; for that would mean that each class of animals
consume both flesh and vegetable food. The combination
can only be made when the terms in the one proposition
limit those in the other [o/. § 74 (ii.)].
(ii.) Inference by Complex Conception is the deducing,
from a given proposition, another proposition of narrower
application by combining both the subject and the predi-
cate of the original proposition with the same name,
whose denotation is thereby limited. This mode of in-
ference is very similar to the last. It differs from it in that,
instead of the original subject and predicate being determined
by the addition made to them, they themselves determine
that added element. Thus, from ' A horse is a quadruped '
we infer ' The head of a horse is the head of a quadruped,'
from ' Arsenic is a poison ' that ' A dose of arsenic is a dose
of poison,' and from ' Poverty is a temptation to crime ' that
' The removal of poverty is the removal of a temptation to
crime.' In these examples, it is the woi'ds ' head,' 'dose,' and
' removal ' which are respectively determined and limited in
their application. The same precautions to avoid fallacy are
necessary in employing this mode of inference as in the case
of Added Determinants. Because ' All judges are lawyers '
it does not follow that ' A majority of judges is a majority
of lawyers,' nor can we infer from 'All great poets are
writers of verse ' that ' A large number of great poets is a
large number of verse-writers ' ; for what would be con-
sidered a large number in the one case would not be so
regarded in the other.
EDUCTIONS. 271
105. Eductions of Hypothetical Propositions. Book III.
Though true hypotheticals are universal, yet we have seen —
that modal particulars take the same general form, and may
be regarded us imperfectly developed hypotheticals {see §§ 76,
100). Embracing these propositions we have forms cor-
responding to each of the four forms of categorical pro-
positions, and the full table of eductions given in § 103 is
applicable to them. These inferences are seen, perhaps,
more clearly when the propositions are not written in the
abstract form directly expressive of connexion of content,
but in the following more concrete and denotative forms
which are justified by, and correspond to them, and which
we have called conditional (of. §§ 76, 100) —
A. If any S is M, then always^ that S is P.
E. If any S is M, then never, that S is P.
I. If an S is M, then sometimes, that S is P.
0. If an S is M, then sometimes not, that S is P.
It must be remembered that 'sometimes' is purely
indefinite, like 'some,' and moreover it does not neces-
sarily imply the actual occurrence of the consequent in any
one instance ; its force is really * it may be,' whilst ' some-
times not ' simply means ' it need not be.'
(i) The eductions from A, expressed symbolically, will
be as follows : —
/Orig. Prop. - A. - If any S is M, then always, that S is P. T-ible of
\ Obverse ■ H - If any S is M, then never, that S is not P. Eductions
from A cou-
i Converse - I - If an S is P, then sometimes, that S is M. ditionala.
Ob. Conv. -0 - If an S is P, then sometimes not, that S is
not M.
/Con trap. - Y. - If any S is not P, then never, that S is M.
\0b. Cotitr. - A - If any S is not P, then always, that Sis not M.
/"Inverse • 0 - If an S is not M, then sometimes not, that S
I is P.
J Ob. Inv. .1 ■ If an S is not M, then sometimes, that S is
l not P.
272
IMMEDIATE INFERENCES.
Book III. As material examples we may give the following, writing
Ch^Il. ^jjg propositions as nearly as possible in the usage of common
Examples of Speech : —
Eductions
from A con-
ditionals.
(Orig. Prop.
-! Obverse
(Converse
Ob. Conv.
{Contrap.
Ob. Contr.
1
Inverse
Ob. Inv.
- A - If any man is Jionest, lie is triisted.
- E - 1/ any man is honest, then never is he not
trusted.
• 1 - If a man is trusted, he is sometimes honest.
- 0 • If a man is trusted, ht is sometimes not dis-
honest.
- "E - If any man is not trusted, he is not honest.
- A - If any man is not trusted, he is dishonest.
- 0 - If a man is not honest, he is sometimes not
trusted.
• 1 - If a man is not honest, he is sometimes dis-
trusted.
Table of fn) The eductions from E may be thus symbolically
Eductions , . j
from E con- Stated : —
ditionals.
rOrig. Prop
\Obverse
/"Converse
\0b. Conv.
f Contrap.
\ Ob. Contr.
[Inverse
Ob. Inv.
'E • If any S is M, then never, that S is P.
A - If any S is M, then always, that S is not P.
E - If any S is P, then never, that S is M.
A - If any S is P, then always, that S is not M,
1
0
If an S is not P, then sometimes, that S is M.
If an S is not P, then sometimes not, that S
is not M.
If an S is not M, then sometimes, that S is P.
If an S is not M, then sometimes not, that S is
not P,
Examples of
Eductions
from E con-
ditionals.
The following are material examples :-
/Orig. Prop,
\^ Obverse
( Converse
\0b. Conv.
'Contrap.
Ob. Contr.
• "E - If any man is happy, he is not vicious.
• A - If any man is happy, he is non-vicious.
- E - 7 /■ any man is vicious, he is not happy.
• A - If any man is vicious, he is not-happy.
- 1 - If a man is not vicious, he is sometimes
happy.
- 0 - If a man is not vicious, he is so7netimes not
not-happy,
• 1 - [f a man is not happy, he is sometimes
vicious.
- 0 - If a man is not happy, he is sometimes not
non-vicious.
EDUCTIONS.
273
(iii)The eductions from I are thus expressed in symbols :— Book IIT
"^ Ch. III.
fOrig. Prop, - I - If anS is M, then sometimes, that S is P.
Obverse - 0 - If an S is M, then sometimes not, that S is EdiJ'ctfoM
"•^^ "• from I con-
- If anS is P, then sometimes, that S is M. ditionals.
• If an S is P, then sometimes not, that S is
not M.
I
0
I
(Converse
Ob. Conv
As material examples may be given : —
{Orig. Prop. - I - If a story is believed, it may be true.
Obverse • 0 - If a story is believed, it need not be untrue.
/Converse - I - If a story is true, it may be believed.
\0b. Conv. • 0 - If a story is true, it need not be disbelieved.
Care must be taken to ensure that the proposition is really
I, and not A in disguise. "Whenever ' sometimes ' implies
actual occurrence, the proposition is really A ; it is only I when
the consequent does not necessarily result at all from the
antecedent. For instance, ' If a man plays recklessly, he
sometimes loses ' is really A ; for it means ' If any man plays
recklessly, it always follows that he has some losses.' Sach
a proposition can, of course, be contraposited and inverted,
processes which the real I propositions cannot undergo.
(iv) The symbolic expressions of the eductions from 0
are
/Orig, Prop.
\ Obverse
{Contrap.
Ob. Contr.
0 - If an S is M, then sometimes not, that S is P.
1 - If an S is M, then sometimes, that S is not P.
I - If an S is not P, then sometimes, that S is M.
0 - IfanS is not P, then sometimes not, that S is
not M.
Ex^amplesof
Eductions
from I con-
ditionals.
Table of
Eductions
from O con-
ditionals.
Or, illustrating by material examples : — Examples oi
Eductions
(Orig. Prop. • 0 - If a man is impulsive, he sometimes is not ^^^^.is!^'
prudent.
Obverse • J - If a man is impulsive, he sometimes is non-
\. prudent.
'Contrap, • 1 - If a man is not prudent, he is sometimes
impulsive.
Ob. Contr, ■ 0 - If a man is not prudent, he is sometimes not
unimpulsive.
LOG. I 18
274
IMMEDIATE INFERENCftg.
Book III.
Ch. III.
The educ-
tions from
disjunctive
propositions
are not dis-
junctive.
Table of
eductions
from a uni-
versal dis-
junctive.
106. Eductions of Disjunctive Propositions.
Eductions can only be drawn from disjunctive propositions
in which alternative predicates are afBrmed of one subject.
They are more clearly seen if we take the denotative forms
of proposition, corresponding to the categorical A and I
(c/l § 81), and the same eductions can be drawn from the
former as from the latter (see § 103). The derived pro-
positions, however, are not themselves disjunctive.
(i.) The symbolic expressions of the eductions from a
universal disjunctive are : —
Examples of
eductions
from a uni-
versal dis-
junctive.
Eductions
from a par-
ticular dis-
junctive.
Every S is either P or Q,
No S is both P and Q.
Some things that are either P or Q are S,
Some things that are either P or Q are not S.
Nothing that is both P and Q is S.
Everything that is both P and Q is S.
I Inverse. Some S's are neither P nor Q.
\ Obv. Inv. Some S's are both P and Q.
As material examples we may give : —
Orig. Prop. Every duty on imports is either protective or a source
of revenue-
No duty on imports fails both to protect native in-
dustries and to be a source of revenue.
Among imposts that either protect native industries or
are sources of revenue are duties on imports.
No impost that fails both to protect native industries
and to be a source of revenue is a duty on imports.
Some imposts which are not duties on imports neither
protect native industries nor increase the revenue.
The obverted forms of the last three eductions can be
easily supplied.
(ii.) The symbolic form of the obverse of the particular
disjunctive Some S's are either P or Q is Some S's are not both P
and Q. Thus, the obverse of ' Some arguments are either
inconclusive or elliptical' is ' Some arguments are not both
conclusive and fully stated.' The forms of the converses
are the same as those from the universal disjunctive.
( Orig. Prop.
( Obverse.
( Converse.
( Ob. Conv.
iOontrap.
Obv. Contr.
Inverse.
Obv. Inv.
Obverse.
Converse.
Contrap.
Inverse.
BOOK IV.
SYLLOGISMS,
CHAPTER I.
GENERAL NATURE OF SYLLOGISM.
107. Definition of Syllogism.
A Syllogism is an inference in which, from two propo-
positions, which contain a common element, and one, at
least, of which is universal, a new proposition is derived,
which is not merely the sum of the two first, and whose
truth follows from theirs as a necessary consequence.
The word Syllogism (Grk. avKKoyiafioQ) seems to have
originally signified ' Computation,' and to have been bor-
rowed by Aristotle from Mathematics. It may, however, be
considered as retaining its strict etymological meaning — ' a
collecting together ' — and as implying that the elements of a
syllogism are thought together. The word thus emphasizes
the fact that a syllogistic inference is one indivisible act of
thought.
As one of the propositions given as data must be universal,
every syllogism is an inference from the general ; in many
cases it is an argument from the general to the particular or
individual. Syllogism is the one means by which a general
principle can be applied to specific instances ; and, in no
Book IY.
Ch. I.
Syllogism —
an infereuce
from two
propositions,
containing a
common
element and
one being
universal, of
a third pro-
position.
Every
syllogism is
an inference
from the
general.
276
SYLLOGISMS.
Book IV.
Ch. I.
The force of
a syllogism
depends
upon the
necessity of
the iuf er-
euce.
The proposi
tions from
which the
inference is
made must
have a
common
element.
case, can the derived proposition be more general than those
from which it is drawn.
The whole force of a syllogism depends upon the necessity
with which the inferred proposition follows from those
given as data, and this necessity must be evident from the
mere form of the argument.
The matter of a syllogism is given in its terms, which vary
according to the subject to which the argument refers. Its
form consists in that relation of the terms by which they
are united in two propositions necessitating a certain con-
clusion. Syllogistic inference is, thus, purely formal, and can,
consequently, be entirely represented by symbols {cf. § 10).
We are concerned in a syllogism, not with the truth or falsity
of either of the individual propositions which compose
it but, simply with the dependence of one of them upon
the other two, so that, if we grant the latter, we, of
necessity, accept the former. The derived proposition,
therefore, propounds no truth which was not contained in
the data. But this is no objection to the syllogism as a
process of inference ; it is, indeed, a necessity if that process
is to be wholly regulated — as we shall show in the next
chapter that it is— by the Laws of Thought {see § 109).
If the given data are objectively true, the proposition
inferred from them must also be true ; but, if the given data
are objectively false, it may accidentally happen that the
derived proposition is true in fact. This is, however, a mere
coincidence ; its truth is known from other sources, and is
not established by the syllogism. For example, from the
data
Lions are herh'worous
Cows are lions
we derive the proposition Cows are herbivorous, which is true,
but whose truth cannot be held to be a consequence of the
" given data, which are both false.
It is essential that the propositions which form the data
should have a common element, as, otherwise, they would
have no bond of connexion with each other, and, conse-
quently, no third proposition could be drawn from their
GENERAL NATURE OP SYLLOGISM.
277
conjunction. But this common element does not appear in
the derived judgment, which is an assertion connecting the
remaining elements of the syllogism.
The Elements of a Syllogism are the propositions and
terms which compose it. ' Terms ' is here used widely to
cover, not only the true terms of categorical propositions, but
also the propositions which form the antecedents and conse-
quents of hypothetical propositions (c/. § 76). The three
propositions which compose a syllogism are called its Proxi-
mate Ai alter, and the terms (in the wide sense just noted)
which are united in those propositions are styled its Remote
Matter. The derived proposition is the Conclusion of the
syllogism, and the two propositions from which it is derived
are the Premises. These names are applicable when the
syllogism is stated in the ordinary and strictly logical form,
in which the premises (proposiiiones prcemissce) precede the
conclusion — as when we say 'Everything which tends to reduce
the supply of any article tends to raise its price ; Protective
Duties tend to reduce the supply of those articles on which
they are imposed ; therefore, Protective Duties tend to raise
the price of those articles on which they are imposed.' But,
when the conclusion is put forward first, as a thesis to be
proved, it was called by the old logicians the Question, and
the propositions which establish it, and which are then intro-
duced by 'because,' or some other causal conjunction, were
termed the Reason. In this form, the syllogism given above
would read — 'Protective Duties tend to raise the price of
those articles on which they are imposed, because they tend
to reduce the supply of those articles ; and everything which
tends to reduce the supply of an article tends to raise its
price.' These latter terms are, however, but little used by
modern writers. The element common to the two premises
is called the Middle Term, and is usually symbolized by M ;
whilst the other two terms are styled the Extremes. Distin-
guishing between the extremes, that which is the predicate of
the conclusion is called the Major Term, and is commonly
expressed by the symbol P ; and that which is the subject of
the conclusion is named the Minor Term, and is generally
Book IV.
Ch. I.
Elements of a
Syllogism —
the proposi-
tions and
terms which
compose it.
Conclusion —
tlie derived
proposition.
Premises —
the proposi-
tions from
which the
inference is
made.
Middle Term
— the element
common to
tlie two pre-
mises— M.
Major Term
—the predi-
cate of the
oonchision-p.
Minor Term
—the sub-
ject of the
conclusion-J
278 SYLLOGISMS.
Book IV. represented by S. The premise in which the major and
^^- ^' middle terms occur is known as the Major Premise ; that in
Major Pre- which the minor and middle terms are found is called the
containing Minor Premise, The order in which the premises are stated
mnor'^'pr ^^' °^ coursc, of no consequence so far as the validity of the
mis«— that argutnent is concerned ; but, as it is customary to state the
s°"nd'JS!'^^ major premise first, that order must be regarded as the
legitimate logical form of a syllogism.
The terms The use of the words Minor, Middle, and Major, to denote
Minor .
Middle, and the terms of a syllogism arose from the consideration of that
primarily^to ^^rm of syllogism in which the conclusion is a universal
the extent of affirmative proposition, and both whose premises are also
a syllogism Universal affirmatives. This syllogism may be symbolized
consisting of jk„
three A pro- J
positions. M a P
S a M
.-.SaP
Here, as the extent of the predicate of an affirmative propo-
sition must be, at least, as great as, and is generally greater
than, that of the subject, it is plain that P must be at least
as wide as, and is probably wider than, M in extent, and simi-
larly with M and S. Hence, the extent of M is, in most cases,
intermediate between that of S and that of P, and, in other
cases, is coincident with that of, at least, one of those terms.
This relation This relation of extent does not hold in all syllogisms and is
doS^'not* iiot essential to the validity of syllogistic apgument. For
hold in all instance
syllogisms, M „ P
and IS not ''' "■ '
essential. M d S
.-. s i y
is a perfectly valid argument, though S is here greater than,
or at least as great as, M in extent. Similarly, when one of
the premises is negative, this relation of extent is not assured.
For example, in
M eP
SaM
.:SeP
the inference ia perfectly just whether P be greater than,
GENERAL NATURE OF SYLLOGISM. 279
equal to, or less than, M in extent ; we cannot tell which is Book IV.
the case, nor is it material, as the total exclusion of P, which ^^'
does not depend on its extent relatively to that of M, is
secured. The names Minor, Middle, and Major, are not,
therefore, appropriate in all cases, if they are regarded as
referring to the extension of the terms ; but they are uni-
versally accepted and recognized, and are as convenient as
any others which could be invented. In another sense, more-
over, the expression 'Middle Term' is quite appropriate, for
that term in every syllogism mediates the conclusion, and is
the middle bond of union connecting the premises.
This terminology of Terms and Premises is primarily The names
applicable to syllogisms which are entirely composed of cate- Middle,
gorical propositions, but it may be transferred, in a large ^rf^a^i ^^^
measure, to those which consist, wholly or in part, of applicable to
hypothetical or disjunctive propositions. This will be dis- coSsS^of
cussed more fully later on (see S 1121. categorical
' V o / propositions.
It has been urged that this naming of the terms and premises is In naming
a vffTepov irporepov, and, therefore, fallacious ; for, it is said, the pre™^ea b
conclusion is assumed in order to name the premises, and, there- reference to
fore, that is first assumed wJiich should only follow from the the conclu-
premises, liut this objection is not valid ; for the reference is not sume only
to any definite proposition {S a P, S e P, etc.) as conclusion, but form'^f*/;
simply to the empty and universal form of proposition S—P. This tl^e conclu
can be assumed, and the naming of the terms and propositions ^''°^'
based on it, without begging the question as to what the conclusion
really is in any syllogism, or whether, indeed, any conclusion at
all can be drawn from any given premises. Such assumption is
necessary to preserve the distinctions of Figure and Mood (see
Ch. Ill), on which so large a part of syllogistic doctrine depends.
The middle term was called, by old writers on Lo»ic, the Othernamea
Argument, as it is what is assumed in order to argue. But that ^>en^to^the
name is now used to denote the whole syllogism, or the process of elements of
inference, which those writers named Argumentation. The major gism!^
premise was frequently termed the Principle ; the reference being
to that most perfect form of syllogism in which the major premise
states a general principle, and the minor— hence called the Eeason—
brings some special case under it. The major was also styled
simply The Proposition, and the minor the Assumption, whilst
as
280
SYLLOGISMS.
Book IV.
Ch. I.
Syllogisms
are of differ-
ent kinds,
according to
the relations
of the pre-
mises.
In a Pure
Syllogism all
the confcti-
tuent propo-
sitions are
of the same
kind, and
maybe Cate-
gorical,
Hypotheti-
cal, or Uis-
iunctive.
In a Mixed
Syllogism
the premises
are of differ-
ent rela-
tions. The
major miy
be hypothe-
tical or dis-
junctive,
and tho
minor cate-
gorical ;
the conclusion was frequently termed the Deduction or CoUeclion.
Mediate, as opposed to Immediate Inference, is frequently called
Discursive, and the process of reasoning is, similarly, termed Dis-
course. Discursive Reasoning is, therefore, that in which an
element is used in the process of inference which does not appear
in the conclusion. The name implies that, as we pass on to the
conclusion, we drop the premises from sight, and retain the state-
ment of the final fact as the one thing we are then concerned with.
108. Kinds of Syllogisms.
As there are different kinds of propositions — Categorical,
Hypothetical, and Disjunctive (see § 67)— all of which can
be used in syllogistic arguments, it follows that syllogisms
can be of different kinds — or relations as it is technically
called (of. § 48).
When both the premises in a syllogism are of the same
character as regards the relation of the terms — categorical,
hypothetical, or disjunctive — the syllogism is said to he Pure,
and the conclusion is, in every case, of the same relation as
the premises. Thus, two categorical premises yield a cate-
gorical conclusion, two hypothetical premises necessitate a
hypothetical conclusion, and from two disjunctive premises
there follows a disjunctive conclusion. There are, there-
fore, three kinds of pure syllogisms — the Categorical, the
Hypothetical and the Disjunctive.
When the premises are propositions of different relations
the syllogism is called Mixed, In the first place, the major
premise may be either hypothetical or disjunctive, and the
minor categorical. A syllogism in which this order was
reversed would be impossible, as the minor premise must
state, in a definite manner, the special case which is to be
brought under the more general statement of the major
premise. This gives two kinds of Mixed Syllogisms — the
Hypothetical, and the Disjunctive. These Hypothetical
Syllogisms are sometimes called Hypothetico-Categorical,
but it is more usual to name a mixed syllogism in accord-
ance with the relation of the major premise. To avoid
confusion, we shall always call syllogisms in which all the
propositions are hypothetical or disjunctive propositions
GENERAL NATURE OF SYLLOGISM.
281
Pure Hypothetical and Pure Disjunctive Syllogisms ; whilst
those with categorical minor premises and conclusions we shall
style Mixed Hypothetical and Mixed Disjunctive Syllogisms,
according to the character of the major premise. In the
second place, the major premise may be hypothetical and
the minor disjunctive. This gives that peculiar form of
mixed syllogism called the Dilemma, in which, according to
the number of terms in the major premise, the conclusion is
either categorical or disjunctive.
We thus get the following table of kinds of syllogisms : —
Syllogisms
1. Pure
2. Mixed
"1
(a.) Categorical.
(6.) Hypothetical.
(c.) Disjunctive.
\a.) Hypothetical.
(h.) Disjunctive,
(c.) Dilemmas.
The distinction between Pure Hypothetical and Pure
Disjunctive Syllogisms on the one hand, and Categorical
Syllogisms on the other is not of as great importance as is
the distinction between hypothetical, disjunctive, and cate-
gorical propositions ; for, in all cases the force of the
syllogism depends on the necessity with which the conclusion
follows from the premises, and the same rules will be found
to apply to all kinds of Pure Syllogism. But the Mixed
Syllogisms require somewhat diiferent treatment.
We shall, in the next three chapters, confine our attention
to Pure Syllogisms, working out the details fully with
categorical syllogisms, and then showing how they can be
applied to pure hypothetical and pure disjunctive syllogisms.
We shall then, in Chapter V, discuss Mixed Syllogisms.
Book IV.
Ch. I.
or, the major
may be hy-
pothetical
and the
miuor dis-
junctive—
this is called
the Dilemma.
Table of
kinds of
syllogisms.
The same
rules apply
to all Pure
Syllogisms,
but Mixed
Syllogisms
require dif-
ferent treat-
ment.
CHAPTER 11.
"Book IV.
Ch. II.
Syllogistic
Reasontng
rests on the
Laws of
Thought—
Aflfirmative
Categorical
Syllogisms
on the
Principle of
Identity ;
Negative
Categorical
Syllogisms
on that of
Contradic-
tion ;
Pure Hypo-
thetical
Syllogisms
on the same
principles,
together
with that of
Sufficient
Reason.
CANONS OP PURE SYLLOGISMS.
109. Basis of Pure Syllogistic Reasoning.
Syllogistic, like aU other purely formal reasoning, rests
ultimately upon the Laws of Thought. The Principle of
Identity {see § 17) is the basis of every affirmative cate-
gorical syllogism, and that of Contradiction (see § 18) of
every negative categorical syllogism. For pure hypothetical
syllogisms an additional reference is required to the Principle
of Sufficient Reason (see § 20).
As both the premises of every syllogism contain the
same middle term (see § 107), each affirmative categorical
premise must state that an element of identity exists be-
tween that term and one of the extremes, and each negative
categorical premise must assert a separation between the
middle term and one of the extremes. If, then, both pre-
mises are affirmative categoricals, the extremes are connected
with each other mediately in so far as each is identical with
the middle term, and identity to the same extent is estab-
lished between them. Or, as Mansel puts it (Prolegomena
Logica, p. 206), " what is given as identical with the whole
" or a part of any concept must be identical with the whole
" or a part of that which is identical with the same concept."
We here use ' Identity' in the sense — explained in § 17 — of
identity amidst diversity, so as to make the principle
cover such statements as S is P. Thus, symbolically, if
S is M, and M is P, then S is P. Of course, if restrictions
of quantity are introduced into the premises, they limit
the identity, and the same limitation must appear in the
CANONS OP PURE SYLLOGISMS.
283
conclusion. If, out of two categorical premises, one is nega-
tive, then, as one extreme is excluded from M, it is excluded
from everything which is identical with M, and, therefore,
from the other extreme ; for the other premise must be
affirmative, and a term cannot at the same time agree with
A/ and with a term which is incompatible with M. Thus,
symbolically, if S is M, and M is not P, then S is not P. These
principles apply equally to pure syllogisms whose premises
are hypothetical propositions. But here the proposition
which forms the 'middle term' of such a syllogism gives the
reason why the proposition which forms the 'minor term' is
the antecedent, whose affirmation is the ground for the
assertion of the proposition which forms the ' major term,'
and is, therefore, the consequent of the conclusion. Thus,
symbolically, from If A, then B ; and if B, then 0, it follows
that If A, then 0 ; the 'Sufficient Reason' being found in
the relation of both these extremes to B.
Book IV.
Ch. II.
110. Axioms of Categorical Syllogisms.
(i.) Axioms applicable to all forms of Categorical
Syllogism.
Instead of appealing directly to the simple statements
of the Laws of Thought, logicians have been accustomed
to give various axioms — which are more or less expansions
of those statements — as the bases of syllogistic reasoning
from categorical propositions. We cannot regard such
axioms, however, as really ultimate ; they are only axioniata
media — or ' middle axioms ' — which, so far as they are not
mere expressions of the simple principles of thought, must
be derived from those principles.
* (a) Whately (Elements of Logic, 5th Ed., pp. 83-4)
gives the following two axioms : —
" 1 . If two terms agree with one and the same third,
" they agree with each other.
" 2. If one term agrees and another disagrees with one
" and the same third, these two disagree with
"each other,"
Logicians
have usually
developed
the Laws of
Thought
into Axioms
of Syllogism,
but none of
these are
ultimate.
Whately's
Axioms.
284
SYLLOGISMS.
Book IV.
Ch. II.
Hamilton's
Axiom.
Thomsou's
Axiom.
None of
these are
unilerivable,
but are based
on the
Principles of
Identity
and Contra-
diction.
They should
bo called
Axioms
rather than
Canons.
* (b) Sir W. Hamilton (Led. on Log., vol. ii., p. 357)
propounded cue such axiom, which he called " the Supreme
Canon of Categorical Syllogisms," in these words :
" In so far as two Notions (notions proper or individuals)
" either both agree, or, one agreeing, the other
" does not agree, with a common third Notion, in
"so far these Notions do or do not agree with
"each other."
* (c) The late Archbisliop Thomson (Laws of Thought,
p. 163) gives a statement of the axiom, which he calls the
' General Canon of Mediate Inference,' and which differs
from Hamilton's in little but form. He says :
" The agreement or disagreement of one conception with
"another is ascertained by a third conception,
" inasmuch as this, wholly or by the same part,
"agrees with both, or with only one of the
" conceptions to be compared."
* A verbal objection may be made to Thomson's state-
ment. For, if, as the words " is ascertained " seem to imply,
two conceptions can only be compared mediately through a
third conception, then all comparison is impossible ; for
neither conception can be compared with this third con-
ception. It would, therefore, be better to read " may be "
for "is." But, putting this on one side as a fault of
expression rather than of meaning, it io evident that the
statements both of Hamilton and of Thomson are — allowing
for the Conceptualist language in which they are expressed
and the Nominalist phraseology of Whately — simply sum-
maries of Whately's two axioms. Their accuracy is un-
doubted, but it is not correct to speak of any such statement
as ' the supreme canon,' if by 'supreme' is meant ultimate or
underivable ; for each is merely a more developed statement
of the Principles of Identity and Contradiction. The word
' Canon ' is, moreover, a not very appropriate name for such
statements. A Canon is a rule, and Hamilton's and Thom-
son's statements are not rules, but axioms, or general prin-
ciples, from which rules may be deduced.
CANONS OF PURE SYLLOGISMS.
285
(ii.) Axioms applicable to only one form of categorical
syllogism.
(«) The Dictum de omni et nullo. The scholastic logiciaus
regarded as the perfect type of categorical syllogism that in
which the middle term is the subject of the major premise
and the predicate of the minor premise — that is, in which
the empty schema is
M P
S M
.-. S P.
Book IV.
Ch.IL
Logicians
commonly
regarded
one type of
syllogism
as funda-
mental ;
tIz.—
M P
S M
All other forms of syllogism can be reduced to this by
applying the various modes of eduction {see §§ 102, 103) to
the pi-emises {see §§ 126-30). The validity of such other
forms can, therefore, be tested, by first reducing them to
this standard form, and then enquiring whether or not they
conform to the general axiom which applies directly to this
form only. These logicians, therefore, gave one axiom aa
the fundamental principle of syllogistic reasoning. This is
the time-honoured Dictum de omni et nullo, which is, per-
haps, most satisfactorily expressed by saying :
Whatever is distributively predicated, whether affirmatively or
negatively, of any class may be predicated in like manner q/
anything which can be asserted to belong to that class.
This axiom is, however, no more fundamental than are
those more generally applicable principles which we have
already examined. Like them, it is simply an expanded
statement of the Principles of Identity and Contradiction ;
for, to predicate anything of a term used distributively is to
make the same predication of each of the constituents of the
denotation of that term.
The Latin form in which the Dictum was commonly given by the
older logicians was Quicquid de omni valet, valet etiam de qui-
busdam et de singulis. Quicquid de nidlo valet, nee de quibusdam
valet, nee de singulis. The common rendering of this into English
was ' Whatever can be affirmed, or denied, of a class may be
'affirmed, or denied, of everything included in the class.' But this
is not satisfactory. The original proposition is not made of a class
and tested
all others by
reducing
them to this
form.
The Dictum
is the Axiom
which applies
to this form.
Statement
of thei>icitt»n.
It is not
fundamental,
but is an ex-
panded state-
ment of the
Principles of
Id entity and
Contradic-
tion.
The ' class '
statement of
the Dictum
is objection-
able;
286
SYLLOGISMS.
Book IV.
Ch. 11.
for the refer-
ence is not
to a term
iised collec-
tively but to
one used dis-
tributively,
and the logi-
cal class is
indefinite
and fixed by
connotation.
The Nota
notce is an
axiom nearly
equivalent
to the Dic-
tum, but
stated in a
counotative
form.
Mill accepted
it as the
practical
form of his
axioms.
as a cfoss— i.e., with a class name used collectively— but of the in-
dividuals which compose the class— i.e., with a class name used dis-
tributively [c/. § 27 (ii.)]. The Dictum itself makes this clear, for
it says de omni, not de cuncto. This rendering of the axiom was
founded upon the class-inclusion view of the import of propositions
[see § 85), and is apt to suggest that the ' class ' whose name forms
the middle term of the syllogism, is a definitely determined and
constituted body of individuals; instead of which, it must be
remembered. Logic regards a class as comprising an indefinite
number of individuals possessing in common certain specified
attributes [c/. § 28 (iv.)].
(b) The Nota notce. The Dictum reads the subject in
each premise— ie., the middle and minor terms— in deno-
tation, though the major term retains its natural connotative
force as a predicate. Those logicians who regarded the
connotation of each term as its most important element
{cf. § 88) framed an axiom corresponding to the Dictum, but
expressing this connotative view. This axiom is Nota notce
est nota rei ipsius. Repugnans notce, repugnat rei ipsi. Mill
adopts this mode of statement as the form of his axioms
best adapted "as an aid for our practical exigencies. . . .
" In this altered phraseology, both these axioms may be
" brought under one general expression ; namely, that what-
" ever has any mark, has that which it is a mark of. Or,
" when the minor premise as well as the major is universal,
" we may state it thus : Whatever is a mark of any mark, is
" a mark of that which this last is a mark of " {Logic, Bk. II,
ch. ii, § 4).
In applying this to negative syllogisms it is necessary
to remember that an attribute may be 'a mark of the
absence of another attribute. It will be noticed that both
statements start with the minor term : the former says — S
has the mark M, which is a mark of P, therefore, S has P:
whilst the latter puts it— S is a mark of M, which is a mark
of P, therefore 5 is a mark of P. The former statement
keeps closer to the nota notce, the latter is a legitimate
generalization of that axiom on the lines on which it is
founded.
CANONS OP PURE SYLLOGISMS.
287
111. General Rules or Canons of Categorical Syllogisms.
(i.) Derivation of Rules from the ' Dictum.' The Dictum
de omni et nullo, as has been said [see §110 (ii.) (a)], is
directly applicable to syllogisms in whose premises the
middle term is the subject of the major, and the predicate
of the minor, premise. To all other forms of syllogism it
applies indirectly through this form. The Dictum may,
therefore, be taken as the axioma medium of all syllogistic
inference (c/. § 109, 110) ; and, consequently, all rules which
govern such inferences must be deducible from it. An
examination of the Dictum will give these in a specific form,
corresponding to its own direct reference to one form only
of syllogism ; but, by a slight generalization they can be
made directly applicable to all forms of syllogism. Such an
examination shows that : —
1. The Dictum speaks of three, and of only three,
terms. There is the * Whatever is predicated ' — which is
the major term; the 'class' of which it is predicated — the
middle term ; and the ' anything asserted to belong to that
class ' — the minor term. This gives the rule that a syllogism
must have three, and only three, terms.
2. Similarly, there are three, but only three, propositions
contemplated by the Dictum. There is that in which the
original predication is made of the ' class ' — the major pre-
mise ; that which declares something ' to belong to that
class ' — the minor premise ; and that in which the original
predication is made of that included something — the con-
clusion. Hence, the rule that a syllogism must consist of three,
and only three, propositions.
3. The Dictum says the original predication is made of
some 'class.' Now this 'class' is, as has just been said
(see 1), the middle term, which is directly regarded by the
Dictum as the subject of the major premise. Thus, the
Dictum tells us that in this form of syllogism the middle
term must be distributed (cf. § 72) in the major premise.
Generalizing this, we get the rule that the middle term must
he distributed in one, at least, of the premises.
Book IV.
Ch. II.
As the Dic-
tum is the
axiom of all
syllogistic
inference,aU
rules which
govern such
inference
are dedu-
cible from it
The Dictum
provides : —
1. That there
be three,
and only
three,
terms.
2. That there
be three,
and only
three, pro-
positions.
3. That the
middle term
be distri-
buted.
288
SYLLOGISMS.
Book IV.
ch. n.
4. That no
term be dis-
tributed in
the conclu-
sion which
is not dis-
tributed in
tlie premises.
5. That one,
at least,
of the pre-
mises be
affinnative.
6. That a
negative
premise
necessitates
a negative
conclusion,
and vice
versd.
4. The Dictum says the original predication may be made
of * anything ' which can be assei'ted to belong to the class ;
therefore, that predication must not be made of a term
more definite than this ' anything.' Hence, if the ' anything '
is undistributed in the premise it must be undistributed in
the conclusion. Similarly, the same predication which is
made of the ' class ' in the major premise can be made of
this ' anything ' in the conclusion ; we are, therefore, not
justified in making a more definite predication, and hence, if
this predication is made by means of an undistributed term
in the predicate of the major premise, it must be made by a
similarly undistributed term in the predicate of the con-
clusion. Generalizing this, we get the rule that no term
may he distributed in the conclusion which is not distributed in
one of the premises.
5. According to the Dictum the minor premise, in the form
of syllogism it directly refers to, must be affirmative, for it
must declare that something can be included in the 'class'
(i.e., in the middle term). This, when generalized, gives the
rule that one, at least, of the premises must be affirmative.
6. The Dictum recognizes the possibility of the original
predication — that is, the major premise in such a syllogism
as it directly applies to — being either affirmative or negative,
and declares that the predication in the conclusion must be
made 'in like manner.' As, according to 5, the minor
premise in such a syllogism is always affirmative, it follows
that when both premises are affirmative the conclusion is
affirmative, and when the major premise is negative, then,
and only then, the conclusion is negative as well. By
generalizing this, we get the rule that a negative premise
necessitates a negative conclusion, and there cannot be a negative
conclusion without a negative premise.
Each rule
applicB di-
rectly to
every form
of syllogism.
(ii.) Examination of the Rules of the Syllogism. We,
thus, get the traditional six general rules, or canons, of the
syllogism. Each of these is directly applicable to every
form of syllogism, and no syllogism is a valid inference in
which any one of these rules is violated. An examination
CANONS OP PURE SYLLOGISMS.
289
of them shows that the first two relate to the nature of Book 17..
a syllogism, the second two to quantity — or distribution of Ch^L
terms, and the last two to quality. They may, therefore, statement
I ,1 -1 cf Rules of
be thus summarized : — SyUogism.
A. Relating to Nature of Syllogism :
I. A syllogism must contain thire, and only three, terms.
II. A syllogism must consist of three, and only three,
projJOSiHons.
B. Relating to Quantity ;
III. The middle term must be distributed in one, at least,
of the p'emises.
IV. No term may be distributed in the conclusion which
is not distributed in a jvemise.
C. Relating to Quality :
V. One, at least, of the premises must be affirmative.
VI. A negative premise necessitates a negative conclusion,
and to prove a negative conclusion requires a negative
premise.
We will now examine each of these rules in detail.
Rules I and II. — These are not rules of syllogistic
inference, but rules for deciding whether or not we have a
syllogism at all. Rule I forbids all ambiguity in the use of
the terms employed in the syllogism ; for, if any term is
used ambiguously, it is really two terms {see § 2G), and so
the argument really contains four, instead of three, terms,
and is not a true syllogism at all, though it may, at first
sight, appear to be one. If there is ambiguity it is most
likely to occur in the case of the middle term, and hence,
Rule III is frequently stated with the additional words
' and must not be ambiguous.' But this is unnecessary, for
Rule I provides against that error, and also against a similar
fault in connexion with either the major or the minor term,
which, if not so common, is equally fatal, when it does occur,
to the validity of the inference.
JOG I 19
Rules I and
II decide
■what is a
syllogisni.
Thefallacy of
Four Terms
ia often
due to am-
biguity iu
one of the
terms,
usually in
the middle
term.
290
BYLLOGISMS.
Book IV. A good example of an ambiguous middle is given by
^^- "• De Morgan {Formal Logic, pp. 241-2)—
" All criminal actions ought to be punished hy law:
" Prosecutions for theft are criminal actions ;
" .•. Prosecutions for theft ought to he punished hij law.
" Here the middle term is doubly ambiguous, both criminal
" and action having different senses in the two premises." If
the middle term is not exactly the same in both premises it
is evident that there is no connecting link between the major
and minor terms. There must be a common element, and
this must be identical in the two premises. Mere resem-
blance, however close, is not enough ; for then S and P might
resemble M in different ways, and so no connexion be estab-
lished between them.
But, as M is connected with P in the one premise, and with S in
the other, it occurs in different contexts, and has different specifica-
tions. We thus have that diversity which is a necessary con-
comitant of identity (c/. § 17). Both this diversity and this identity
are necessary to inference. If M were not identical in both
premises, it could not form a connecting bond between 5 and P ;
and if it had not different aspects in the two premises, it could not
be connected with the ideas S and P, which are different from
each other.
Rules I and
II are in-
volved in the
de6nitiun of
a syllogism.
Again, if S or P is used in a different sense in the conclu-
sion from that which it bears in the premise in which it
occurs, the inference is invalid ; for the premises justify only
the predication of that sarne P which was connected in the
major premise with A/, of that same S which was related in
the minor premise to the same M,
Of Rule II but little need be said. If there are three
terms, two of which are to occur in each proposition, and the
same two in no two propositions, it is evident there must be
three, and only three, propositicns. The very definition of a
syllogism secures this rule directly, and Rule I indirectly : for
two premises with a common term contain evidently three,
CANONS OF PURE SYLLOGISMS.
291
and only three terms, and the conclusion relates the two Book IV,
terms which are not common to the two premises.
Rule III. — The violation of this rule is called the Fallacy
of the Undistrihuted Middle. It is essential that the middle
term should be distributed in one, at least, of the premises, as
only thus can there be any assurance that there exists that
element of identity which is necessary to constitute a bond
of connexion between the extremes. Unless it is certain
that the extremes are related to one and the same part of the
middle term, there can be no inference as to the relation in
which those extremes stand to each other. Now, if only an
indefinite reference is made to the middle term in each pre-
mise, either the same, or an entirely different, part of its
extent may be, in fact, involved in each case. For example,
because All Englishmen are Europeans, and All Frenchmen
are Europeans, it does not follow that All (or any) French-
men are Englishmen. In fact, every possible relation be-
tween 5 and P is consistent -with the two propositions All P
is M and All S is M, where M is an undistributed middle term.
This is seen at a glance by a reference to Euler's diagrams
{see § 91), which give all the possible objective relations of
two classes. Each of those five figures may be entirely
enclosed in a larger circle representing M, and in each
case P a M and S a M will hold true. Thus, it is evident that
from two such propositions no inference whatever can be
drawn as to the relation of S and P. Similar ambiguity will
be found to follow from every other case in which, in a pair
of propositions, M is not once distributed. As, then, we
have no security when M is undistributed that there is any
bond of connexion whatever between S and P, we can draw
no inference concerning the relation of thise two terms-
For, by the Law of Parsimony {cf. § 79), formal inference
must depend upon that bare minimum of assertiou which the
premises must be held to make unconditionally ; and, there-
fore, as it is possible that the same part of M is not referred
to in both premises, we must not assume that it is so referred
to in any particular case. Only, then, by securing the whole
If Nl is un-
distributed
no bond of
connexion
between
S and P is
secured.
hence, there
can be no
inference' m
to their rela-
tion to each
other.
292
SYLLOGISMS.
Book IV.
Ch. II.
t/i need not
be distri-
buted iu
more than
one premise.
of the middle term in one premise can we be certain that
there is an identical element in both premises. And, though
the middle term may be di^tributed in both premises, yet a
single distribution is sufficient to secure this. For, if the
whole extent of M is related to one of the extremes, no
matter what part of M is related to the other extreme, it
must be identical with some at least of the M referred to in
the former case. The real mediation is, of course, through
this common part of A/, whatever its extent may be ; what
that is Formal Logic does not enquire, it deals only with the
definite 'all' and the indefinite ' some,' and rests on the assur-
ance that the former must, of necessity, include the latter.
If we have
an Illicit
Process of
either term
our conclu-
eion is more
deSnitethan
the premises
warrant.
Illicit Major
is only pos-
sible with a
negative
conclusion.
Rule IV. — The violation of this rule is called Illicit Pro-
cess. If the minor term is distributed in the conclusion and
not in the minor premise, we have the Fallacy of Illicit
Process of the Minor Term ; if the same unwarranted treat-
ment is accorded to the major term, it gives rise to the Fallacy
of Illicit Process of the Major Term. As the conclusion must
follow necessarily from the premises, we can never be justi-
fied in making a predication about a definite All S when the
minor premise only refers to Some S. If merely an indefinite
part of S is related to M, that relation can give us no right to
trace, through M, a connexion between P and all S. Because
all criminals are deserving of punishment, and some English-
men are criminals, it does not follow that all Englishmen
are deserving of punishment. The conclusion must be no
more definite than the premises warrant. And the same
holds of the major term. We are justified in relating the
whole of P to S only when the whole of P has been previously
related to M in the major premise. It will be noticed that
we can only use P universally in the conclusion when that
conclusion is negative, for P is always its predicate, and the
predicate of an affirmative proposition is always undistributed
(see § 72). In this case, therefore, one of the premises must
be negative (Rule VI). If that premise be the minor, then
P must be the subject of the major premise, which must be an
A proposition ; but if the negative premise be the major,
CANONS OF PUKE SYLLOGISMS, 293
P may always be its predicate, though it can be its subject only Book IV
when it is universal (cf. § 7"2). In every other case, if P is dis- _1_ •
tributed in the conclusion, we have Illicit Process of theMajor.
P"'or example, if we argue from ' All fishes are oviparous,' and
' No birds are fishes,' to ' No birds are oviparous,' our inference
is invalid. In this case, as in most such cases, the conclusion is
also false in fact. But even were it true in fact it would not
be a valid inference from the premises ; its truth would not be
a result of their truth, but would be an accidental coincidence
and known only through some other source. For instance,
'All fishes are cold-blooded, No whales are fishes, therefore
No whales are cold-blooded ' is exactly as invalid an inference
as the one just considered, though the proposition given as
its ' conclusion ' is objectively true. For there is nothing in
the promises to deny to whales the attribute 'cold-blooded,'
as u ill be seen by substituting the word 'snakes' for 'whales,'
when the 'conclusion' becomes false in fact. Thus, no con-
clusion is justified in which any term is distributed which is
undistributed in the premise in which it occurs. The viola-
tion of this rule may be compared with the simple conversion
of an A proposition, a process which has been already shown
to be illicit [see § 102 (ii.) (o)].
Rule V. — From two negative premises no conclusion can Two negative
be drawn ; for, from mere negation of relation no statement yilid no^con-
of relation can be deduced. It is only when one of the fusion, as no
'' connexion
extremes is connected with the middle term, that we can, isasserte.i
through that connexion, infer its agreement with, or sepa- aneTeUhtr s
ration from, the other extreme. For, if both S and P are ^^ ''•
declared to be separated from M, there is, clearly, no bond
of union to connect them with each other. They may, or
may not, be related in fact ; but whatever relation they hold,
it is impossible to infer it from the negation of relation with
the common element which is contained in the premises.
Compare, for instance, the pairs of negative propositions ; 'No
cows are carnivorous — No sheep are carnivorous ' ; 'No men
are immortal — No negroes are immortal.' In the first case
the minor term is, in fact, wholly excluded from, and, in the
294
(SYLLOGISMS,
Book IV.
Ch. IL
Two negative
premises are
consistent
with every
one of the
possiVile re-
lations be-
tween Sand P.
This holds
if the pre-
mises are
both E ;
or, one E
an.l one O ;
second case, wholly included in, the major term ; but neither
the exclusion nor the inclusion can be deduced from the
premises, which simply separate both the classes represented
, by the major and minor terms from the common attribute
expressed by the middle term ; and which, being identical in
form, must, if they give any conclusion, always give one of
the same form.
It will be profitable to examine this rule more in detail, as its
accuracy has been questioned. There are three possible combina-
tions of negative premises which we will consider separately : —
1. Both premises may be negative universal — i.e., E — proposi-
tions. In this case, both 5 and P are wholly separated from M.
They may, at the same time be (1) wholly separated from each
other, (2) partly coincident and partly separated, or (3) one may be
wholly included in the other. Distinguishing in this last case
between inclusion when the extent of the terms coincide, inclusion
without coincidence of S by P, and of P by S, we have here all the
five po.'sible objective relations between S and P {cf. § 85). This
will be made evident by a reference to Euler's diagrams (see § 91),
which express these relations ; for, to each of the five figures which
represent the relations of S and P we may add a circle M, lying
entirely outside both the circles which represent S and P, and thus
signifying the entire exclusion of both S and P from M. We see,
therefore, that this exclusion is consistent with every conceivable
relation of S and P to each other.
2. One premise may be universal, and the other particular,
negative — i.e., E and 0. In this case M is entirely excluded from
one of the extremes, and partially, at least, from the other. But
the particular never excludes the possibility of the universal being
true in fact [cf. § 71 (ii.)]. Hence, the whole indefiniteness which
obtains when both premises are E propositions remains when one
is E ai'.d the other is 0, and there is added the possibility of other
relations, not between S and P but, between those terms and M.
These may be shown on Euler's diagrams (.see § 91) by drawing
a circle to represent M, which shall intersect one of the circles
representing S and P, and lie entirely outside the other. This
requires seven additional diagrams besides the five which are
necessary when both premises are E. For each of the Figures IV
and V gives two diagrams, according as the S or the P circle is
intersected by the new circle representing M. The indefiniteuess
CANONS OF PURE SYLLOGISMS.
295
of the conclusion is not increased— that is impossible, as there are Book IV.
only five conceivable relations in fact between S and P — but the C!h. II.
indefiniteness of the relations represented by the premises which
should lead to a conclusion is seen to be more than doubled.
3. Both premises ma,y he particular negative — i.e., 0 — proposi- orbotljO;
tions. In this case, each extreme is partially, at least, separated
from M. J Here again, the particular dots not exclude the possibility
that the universal is true in fact. All the former indefiniteness,
then, still remains, and it is still further increased so far as the
relations of Sand Pto M are concerned. For the circle representing
M may now be drawn to intersect both the S and the P circle.
This gives four more diagrams, in addition to the twelve we already
have ; for this double intersection will give a fresh arrangement of
circles in every case except in that of Fig. I, where to intersect
either S or P is, necessarily, to intersect them both, as they are
coincident ; this case was, then, represented in the diagrams
necessary to represent E and 0 premises.
In every case, then, we see that every possible relation of S and no inference
P is consistent with two negative premises ; such premises can, f^^J ^^
therefore, necessitate no conclusion ; that is, no inference can be drawn from
J , . , such pre-
made from them. mises.
* This has been denied by Jevons. He says : " The old
" rule informed us that from two negative premises no
" conclusion could be drawn, but it is a fact that the rule in
'' this bare form does not hold universally true ; and I am
" not aware that any precise explanation has been given of
" the conditions under which it is or is not imperative.
•' Consider the following example :
Jevnns de-
nies this ;
" Whatever is not metallic is not capable of powerful
" magnetic influence,
" Carbon is not metallic.
.'. Carbon is not capable of powerful magnetic influence.
" Here we have two distinctly negative premises, and yet
" they yield a perfectly valid negative conclusion. The
" syllogistic rule is actually falsified in its bare and general
"statement '' {Principles of Science^ 2ud Ed., p. 63).
296
SYLLOGISMS.
Book IV.
Oh. IL
but the
minor pre-
mise in his
example is
really
affirmative.
Mr. Bradley
objects to
expressing
the minor
premise in
the affirma-
tive form,
Expressed symbolically this argument is
No non-M is P
S is not M
.'. S is not P ;
where wo have, apparently, four terms, 5, P, M, and
non-M. But, if we examine the argument more closely, %ve
shall see that what the minor premise really asserts is that S
is included in those things of which P is denied, i.e., the
non-HPs. Hence, the middle term is really non-M, and it is
this which is predicated affirmatively of S in the minor
premise. We reduce the above argument, therefore, to a
simple syllogism by obverting the minor premise [c/ § 102
(i.)], when we get
MeP_
SaM
SeP
which is a perfectly valid syllogism, and in which Rule V is
observed, as the minor premise is affirmative.
Mr. Bradley will not allow this to be any true answer. Referring
to the example just quoted from Jevons, he says : " This argument
" no doubt has quaternio terminorum and is vicious technically,
" but the fact remains that from two denials you somehow have.
" proved a further denial. ' A is not B, what is not B is not C,
"' therefore y1 is notC;' the premises are surely negative to start
" with, and it appears pedantic either to urge on one side that ' A
" ' is non-B ' is simply positive, or on the other that B and non-B
" afiford no junction. If from negative premises I can get my con-
" elusion, it seems idle to object that I have first transformed one
" premise ; for that objection does not show that the premises are
" not negative, and it does not show that I have failed to get my
" conclusion " {Principles of Logic, p. 254), Certainly, the con-
clusion is obtained, and, assuredly, it is a valid inference ; but it is
neither ' pedantic ' nor ' idle ' to urge that one premise has already
been transformed when they are both stated as negative propositions,
and must be reduced back to its natural affirmative form before the
inference can be made. For, as Ueberweg remarks, " this reduc-
CANONS OF PURE SYLLOGISMS.
297
" tion is not an artificial mean, contrived in order to violently
" reconcile an actual exception to a rule falsely considered to be
" universally valid. We only arrive naturally at the conclusion
" when we think the minor premise in the form : S falls under the
"notion of those beings which are not M" (Logic, Eng. trans.,
p. 384). It must be remembered that we can, by obversion,
always reduce an affirmative proposition to a negative, and vice
versd. The decision as to which form is appropriate to any
particular case must be decided by considering whether the judg-
ment to be expressed emphasizes the negafive or the affirmative
element present in every negative proposition (c/. § 70). In this
case, the minor emphasizes the affirmative element.
Ueberweg further points out that this case had not been over-
looked by the older logicians, who explained it in this very way,
and he thinks it "not improbable that the doctrine of qualitative
'* ^quipollence between two judgments [i.e., Obversion] owes its
"origin to the explanation of the syllogistic case" {ibid.). The
authors of the Fort Royal Logic had also considered and solved
the apparent difficulty exactly as we have just done. The ' precise
' explanation ' which Jevons desiderated had, then, been frequently
given. In fact, he had given it himself. In his Elementary
Lessons in Logic (which was published before the second edition
of the Principles of Science) he says : " It must not however be
" supposed that the mere occurrence of a negative particle {not or
*' no) in a proposition renders it negative in the manner con-
" templated by this rule. Thus the argument
" ' What is not compound is an element,
" Gold is not compound ;
" . : Gold is an element,'
''contains negatives in both premises, but is nevertheless valid,
" because the negative in both cases affects the middle term, which
"is really the negative term not-compound^' (p. 134). The rule
then holds without exception that in every syllogism there must be
at least one affirmative premise.
Rule VI. — If one premise is negative, the other must, by
Rule V, be affirmative. Hence, the two extremes are
related to the middle term in opposite waj's. Now, if two
terms agree with each other, they must, necessarily, stand in
the same relation to any third term. If, then, the relations
Book IV.
Ch. II.
but Ueber-
wpg shows
it is the only
natural
reading.
This case
has been
explained
by the older
logicians,
and by
Jevons him.
self.
If the ex-
tremes are
asserted to
stand in op-
posed rela-
tions with til
they cannot
agree with
each other ;
298
SYLLOGISMS.
Book IV.
Ch. IL
hence, a
negative
premise in-
volves a
negative
conclusion,
and a nega-
tive conclu-
sion requires
a negative
premise.
of S and P respectively to M are not the same, but contra-
dictory, relations, S and P cannot agree with each other. We
reach the same result in a slightly different way by con-
sidering that, in so far as anything agrees with A/, it must be
separated from everything from which M is separated. If,
then, one premise declares the agreement of one extreme
with M, and the other premise asserts the incompatibility
with M of the other extreme, those extremes must be inferred
to be incompatible with each other. Hence, a negative
premise involves a negative conclusion. And the converse
of this is also true. The non-agreement of S and P with
each other must follow from the fact that one agrees with,
and the other is separated, wholly or in part, from M. For,
if they both agreed with M, they would agree with each other.
Therefore, a negative conclusion can only be inferred when
one of the premises is negative.
Rules III-
VI are the
only ones
which are
rules of
syllogistic
inference.
These four
are not all
fund a-
mental.
The first
part of Rule
VI can be
deduced
from RuleV.
* (iii.) Simplification of the Rules of the Syllogism.
Our examination of Rules I and II showed that they are not
rules of syllogistic inference at all, but rather preliminary
statements of what a syllogism is. Thus, we have, really,
only four rules of syllogistic inference — two relating to
quantity and two to quality. By a direct application of
these the validity of any argument given in the form of a
syllogism can be tested. They are not, however, equally
fundamental ; for all, except the last part of Rule YI, are
derivable, by a more or less indirect process, either from
Rule III or from Rule IV — these last two being mutually
inferrible.
This has been admirably worked out by Dr. Keynes in his
Formal Logic, and a large part of this subsection is taken from
that work. We w 11 now show in detail bow this reduction in the
number of fundamental rules is effected.
(ft) Tht frst j'art of Ride VI is a corollary from Rule V. — The
fiist part of Rule VI says that a negative premise necessitates
a negative conclusion. Dr. Keynes (Formal Logic, 3rd Edit.,
p. 247) thus shows this to be a deduction from Rule V, which
forbids two negative premises : —
CANONS OF PURE SYLLOGISMS. 299
" If two propositions P and Q together prove a third B, it is plain Book IV.
" that P and the denial of K prove the denial of Q. For P and Q Ch^I.
" cannot be true together without R. Now, if possible, let P (a
" negative) and Q (an affirmative) prove R (an affirmative). Then
" P (a negative) and the denial of R (a negative) prove the denial of
" Q. But two negatives prove notliing."
(b) Rule V is a corollary from Eule III. — Rule V forbids two Rule V
negative premises. This is shown by De Morgan {Formal Logic, kuIq ni" ™
p. 13) to be inferrible from the rule prohibiting undistributed
middle. If the negative premises are both E propoi^itions they may
always be expressed in the form P e M, S e M ; ior if given in any
other form they may be simply converted [see § 102 (ii, ) (b)]. These
by obversion [see § 102 (i.)] give the equivalent propositions
Pa'M
SaM
where, as the middle term, M, is in each premise the predicate of an
affirmative proposition, we have undistributed middle (cf. § 72).
The same proof would hold when one of the premises is pr.rticular,
BO long as M is its predicate. But if one of the premises is particular
and has M for its subject, then, as an 0 proposition cannot be con-
verted [see § 102 (ii.) {d)], we cannot bring the argument into this
form. But, by retaining the original statement of the particular
premise, we can express the apparent syllogism in one of these two
forms
PaM M oP
M oS SaM
in each of which there is no middle term. De Morgan does not
prove this last case quite in this way. He contraposits the particular
premise, which give^
PaM T i M
S i M SaM
but there is nothing gained by doing this, and the former method
has the advantage of simplicity. If both premises are particular,
the invalidity is seen by obverting both when M is predicate of both,
by contrapositing both when M is subject of both, each of which
processes will show undistributed middle ; and by obvtrting the
one which has M for its predicate when M is subject to one and pre-
dicate of the other, which will show four terms.
Dr. Keynes objects to the proof of invalidity by reduction to foui
300 SYLLOGISMS.
Book TV. terms. He says : " This is not satisfactory, since we may often
C^^I- .( exhibit a valid syllogisna in such a form that there appear to b9
"four terms ; e.g.. All M is P, All S is M, may be converted into
"All M isP
" No S is non-M,
" in which there is no middle term " {Formal Logic, pp. 246-7). This
is certainly the case, as we saw in our examination of Rule V in the
last sub-section when we were criticizing Jevons' argument that two
negative premises may yield a valid conclusion. But this objection
does not hold against the reduction of two E propositions to undis-
tributed middle, and as Mr. Keynes goes on to remark, "The
" case in question may ... be disposed of by saying that if we can
" infer nothing from two universal negative premises, d fortiori we
" cannot from two negative premises, one of which is particular"
(ibid.). The only objection which it is possible to bring against the
reduction to undistributed middle is that if M is written as the
subject of the premises, whenever it is possible to do so, we get
a real middle term which is, moreover, distributed. Thus, if the
premises are both E they may be written M e P, M e S, which, by
obversion, give
A/a P
M aS;
but it must be pointed out that, though we have here neither undis-
tributed middle nor four terms yet, M does not mediate a connexion
between S and P, which are the two extremes in the original form
of our premises.
Rul IT ma ^^^ ^"^^ ^^ "^"'^ ^^ inferred from Rule 777.— That illicit process
be inferred indirectly involves undistributed middle is thus shown by Mr.
from Rul9 i<;eynes : " Let P and Q be the premises and R the conclusion of a
" syllogism involving illicit major or minor, a term X which is un
"distributed in P being distributed in R. Then the contradictory
" of R combined with P must prove the contradictory of Q. [For P
" and Q cannot be true together vdthout R.] But any term dis-
" tributed iu a proposition is undistributed in its contradictory. X
" is therefore undistributed in the contradictory of R, and by hypo-
"thesis it is undistributed in P. But X is the middle term of the
" new syllogism, which ia therefore guilty of the fallacy of undia-
" tributed middle" {Formal Logic, 3rd Edit., pp. 247-8).
CANOKS OF PURE SYLLOGISMS.
301
{d) Rule HI may be inferred from JRule I V. — Mr. Keynes remarks Book IV,
that undistributed middle may be deduced from illicit process,
as well as vice versa. This may be thus shown : Let P and Q, be
the premises and E the conclusion of a syllogism involving undis-
tributed middle, and let X be the undistributed middle term. Then
P together with the contradictory of R must prove the contradictory
of Q. For P and Q cannot be true together without E. But any
teim undistributed in a proposition is distributed in its con-
tradictory [cf. §§ 97 (ii), 98]. Therefore X is distributed
in the contradictory of Q. But, by hypothesis, X is undistri-
buted in P, and as it does not appear in E it must be absent
from its contradictory. Hence X is undistributed in the premises
and distributed in the conclusion of the new syllogism, which is,
therefore, guilty of the fallacy of illicit process.
ch. n.
and Rule III
is inferrible
from Rule
IV.
Hence,there
* (e) Results of the Simplification. — There are, thus, ^^.^ ^wo
tive
seen to be only two ultimate rules, one relating to quantity i^lttrnat
and the other to quality. As, however, the two original dependent
rules relating to quantity can each be inferred from the '■^'=^>
other, we may adopt either of these as ultimate. This gives
us two alternative pairs of independent rules. The first
pair is : —
(1) The middle term must be distributed in one, at
least, of the premises.
(2) To prove a negative conclusion requires a negative
premise.
The second, and alternative, pair is : —
(1) No term may be distributed in the conclusion
which is not distributed in a premise.
(2) To prove a negative conclusion requires a negative
premise.
This latter pair has the advantage of exact correspondence
with the rules for Subakernation [see § 07 (i.)] and Con-
version [see § 102 (ii.)] respectively, and of thus showing
the fundamental identity of the mental process in immediate
and in mediate inference.
which are —
(1) Rule III
(2) Rule VI
(second
part),
or: —
(1) Rule IV
(2) Rule VI
(second
part J.
302
SYLLOGISMS.
Book IV.
Ch II.
This simpli-
fication is
not import-
ant, aa Rules
III-VI aro
all required
to directly
test syllo-
gistic argu-
ments.
Three corol-
laries are
deducible
from the
Rules.
Statement
of the Corol-
laries.
Two par-
ticular pre-
mises can-
not distri-
bute enough
terms to
■warrant any
conclusion.
* This simplification is interesting but of no great
practical importance ; for we must still appeal to the last
four rules given in sub-section (ii) as direct tests of the
validity of any argument expressed in the form of a
syllogism. An invalid syllogistic inference need not offend
against either of the two independent rules to which those
four have been reduced, except in a very indirect way,
which is only made apparent by a more or less complex
process of reasoning.
(iv.) Corollaries from the Rules of the Syllogism.
Though the four rules of syllogistic inference [sub-§ (ii.),
(Rules III-VI)] are not equally fundamental, yet all are so
far independent that all are necessary for the immediate
detection of invalidity in syllogistic inference ; and for this
purpose they are sufficient. There are, however, three
corollaries from these rules which, though not abso-
lutely requisite for the detection of syllogistic fallacy,
are useful for that purpose. It was long customary to give
the first two of these as independent rules of syllogism.
They are : —
1. From two 2yarticular premises nothing can he inferred.
2. If one premise is particidar the conclusion must he
particular.
3. From a particidar major and a negative minor nothing
can he inferred.
We will now show in detail how each of these can be
deduced from the rules already given.
Cor. 1. The best proof that two particulars prove nothing
is that given by De Morgan {Formal Logic, p. 14) : — "Since
" both premises are particular in form, the middle term can
" only enter one of them universally by being the predicate
" of a negative proposition ; consequently (Rule V) the other
" premise must be affirmative, and, being particular, neither
" of its terms is universal. Consequently both the terms aa
" to which the conclusion is to be drawn enter partially, and
" the conclusion (Rule IV) can only be a particular affirma-
CANONS OP PURE SYLLOGISMS.
303
" five proposition. But if one of the premises be negative,
" the conclusion must be negative (Rule VI). This contra-
" diction shows that the supposition of particular premises
" producing a legitimate result is inadmissible."
* A somewhat different proof, based on an examination of
each of the possible combinations of particular premises, may
also be given : — In every valid syllogism the premises must
contain one distributed term more than the conclusion. For
if any term is distributed in the conclusion it must also be
distributed in the premise in which it occurs, and, in
addition to this, the middle term must be distributed once,
at least, in the premises. From this it follows that no
conclusion can be drawn from two particular premises. For,
if they are both 0 propositions, by Rule V nothing can be
inferred. If they are both I they contain no distributed
term at all {see § 72) and, thus, break Rule III. If one is
I and the other is 0 then the conclusion must be negative
(Rule VI), and, consequently, it distributes the major term
{see § 72). But I and 0 distribute only one term- between
them, and, therefore, cannot distribute both the major and
the middle terms. Hence, nothing can be inferred, for to
draw a conclusion would be to break either Rule III or
Rule IV.
Cor. 2. That a particular premise necessitates a particular
conclusion may be thus proved : — If both premises are affir-
mative and one particular, they can, between them, distribute
only one term, which must be the middle term (Rule III).
Hence, both the extreme terms are undistributed in the
premises, and, consequently, must be undistributed in the
conclusion (Rule IV) — that is, the conclusion must be
particular affirmative.
If, however, in such a syllogism, one premise is negative,
it distributes its predicate, and the premises, therefore, con-
tain between them two, but only two, distributed terms.
One of these must be the middle term (Rule III). Hence,
only one distributed term can enter the conclusion (R'de IV).
But the conclusion must be negative (Rule VI), and it, there-
fore, distributes the major term, which must, consequently,
Book IV.
ch. n.
One u n 1.
versal and
one particu
lar premise
can only
disti-ibute
enough
terms to
warrant a
particular
conclusion.
304
SYLLOGISMS.
Book IV.
Ch. II.
This corol-
lary can be
proved fium
Cor. 1
A particular
major and a
negative
minor can-
not yield a
conclusion
because the
major term
is undistri-
buted.
All the rules
of syllogism
apply to
pure hypo-
thetical
syllogisms.
be the second distributed term in the premises (Rule IV).
The minor term is, therefore, undistributed in the premises
and must be undistributed in the conclusion — that is, the
conclusion must he particular negative.
As both premises cannot be negative (Rule V), these are
ihe only possible cases.
* A neat proof of this corollary is given by De Morgan
{^Formal Logic, p. 14), vrho thus deduces it from Corol-
lary 1 : — "If two propositions P and Q together prove
" a third R, it is plain that P and the denial of R prove the
" denial of Q. For P and Q cannot be true together without
" R. Now, if possible, let P (a particular) and Q (a universal)
" prove R (a universal). Then P (particular) and the denial
"of R (particular) prove the denial of Q. But two particu-
" lars can prove nothing."
Cor. 3. That nothing can be inferred from a particular
major and a negative minor may be thus proved : — As both
premises cannot be negative the major is affirmative particu-
lar (I), and distributes neither of its terms. The major
term, therefore, cannot be distributed in the conclusion
(Rule IV). But, as one premise is negative, the conclusion
must be negative. Therefore, the major term must be dis-
tributed in the conclusion. This contradiction shows that
no valid inference can be made.
112. Application of the Rules to Pure Hypothetical and
Pure Disjunctive Syllogisms.
(i.) Pure Hypothetical Syllogisms. Since hypothetical
propositions, when we include Modal Particulars under the
name, admit of the same distinctions of quality and quantity
as categorical propositions (see § 78, cf. § 105), there can be
forms of pure hypothetical syllogisms (see § 108) correspond-
ing to every form of categorical syllogism. Hence, all the
rules given in § 111 (ii.) apply to pure hypothetical syllo-
gisms. The denotative, or conditional, forms bear a closer
analogy to the ordinary quantified forms of the categorical
syllogism than do the pure abstract hypothetical forms, and
the application of the rules is more clearly seen when those
CANONS OF PURE SYLLOGISMS.
305
Book TV.
Ch. IL
The ' terms '
are proposi-
tions.
The quan-
tity of the
antecedent
is shown by
the words
'always, 'etc.
The quan-
tity of the
consequent
depends on
the quality
of the pro-
position.
quantified forms are considered. The ' terms ' here are, how-
ever, propositions — the consequent of the conclusion corre-
sponding to the major term of a categorical syllogism, the
antecedent of the conclusion to the minor term, and the
element which appears only in the premises to the middle
term. In considering the distribution of these ' terms ' it
must be remembered that, as 'always,' 'never,' 'sometimes,'
' sometimes not,' in conditional propositions correspond to
'all,' 'no,' 'some,' 'some not,' in categorical propositions,
these words indicate the quantity of the antecedent. The
quantity of the consequent must be determined by the same
rule which decides the quantity of the predicate of a cate-
gorical proposition. That is to say, the consequent of a
negative conditional proposition is distributed, and that of
an affirmative conditional is undistributed (c/. § 72). For
example, If any S is M then always that S is P does not dis-
tribute the proposition that S is P ; for it neither states nor
implies that the only possible condition of S being P is that
it should be M — it is quite possible that S is P under many
other conditions, as when it is /V or Q or X. In short, the
distribution of the ' terms ' in a pure hypothetical syllogism
must not be determined by a reference to those terms by
themselves and out of connexion with their context, any
more than in a categorical syllogism.
(ii.) Pure Disjunctive Syllogisms. Since disjunctive The mies of
propositions are all affirmative [see § 81 (i.)], the syllogistic ?oifaipiy°to
rules (V and VI) relating to quality do not apply. The rule ?^^^ P^^-
for securing the distribution of the middle term (III) can only SynLglsms.
be fulfilled when one of the alternatives in the minor pre-
mise is the negative of one of those in the major premise!
This will be more fully considered in the next chapter
[see § 125 (ii.)].
LOO. I.
20
CHAPTER ITT.
Look IV.
Ch. III.
Figure istle
dispositiou
of W in the
premises.
There are
four
Figures —
I. K—P
S—fH
II.
.S—P
P—M
S—M
III.
.S—P
M—P
M—S
IV.
.S—P
P—M
Mj-S
.S-P
FIGURE AND MOOD.
113. Distinctions of Figure.
"t'igure is the form of a syllogitm as detennined by the
position of the middle term in the two piemises.
If: account is taken of the premises alone, so that it is im-
material which of the extreme terms is the subject, and
which the predicate, of the conclusion, only three figures
are possible. For M must either be (1) subject in one pre-
mise and predicate in the other, (2) predicate in botb, or
(3) subject in both. If, however, it is determined which
term sball be the subject, and which the predicate, of the
conclusion, the distinction of major and minor is introduced
into the premises. The first alternative now becomes two-
fold, according as M is subject in the major and predicate in
the minor premise, or predicate in the major and subject in
the minor.
There are thus four possible Figures of syllogism : —
Fh'st Figure: M is subject in major, and predicate in
minor, premise.
Second Figure : M is predicate in each premise.
Third Figure : M is subject in each premise.
Fourth Figure : M is predicate in major, and subject in
minor, premise.
The empty forms of syllogisms arranged in Figures, and
with the premises written in the usual order of major first
(c/ § 107), are, therefore :—
FIGURE AND MOOD.
307
Fig. I.
A/ P
S A/
Fig. II.
P A/
S M
Fig. III.
M P
M S
Fig.
IV.
P
-M
M
—S
.-. 5—
-P
Book IV.
Ch. III.
Of course, the distinction between the First and Fourth
Figures in no way depends upon the order in which the
premises are written, but upon the distinction between
major and minor premise, which is due to the predetermined
order of the terms in the conclusion (cf. § 107).
114. Axioms and Special Rules of the Four Figures.
Though all syllogistic reasoning is ultimately based upon
the fundamental principles of thought (see § 109, cf. §§ 17-20),
yet separate axiomata media may be given as the more im-
mediate bases of inferences in each of the four figures.
From these axioms special rules may be derived which
directly secure the observance of i!Sie general rules of syllo-
gism {sec § HI) by arguments in the figure to which they
apply.
(i.) The First Figure. Several axioms — which we have
already considered in detail [see § 110 (ii.)]-. have been given
as the foundation of syllogistic inference in this figure, the
most generally received of which is the Dictum de omni et
nullo. As that axiom applies directly to all syllogisms in the
First Figure, the following special rules of that figure may
be immediately gathered from it : —
1. The major premise must he universal.
2. The minor premise must be affirmative.
The derivation of these rules was discussed in § 111 (i.)
3 and 5.
These special rules are merely applications to this par-
ticular form of syllogism of the General Rules discussed
in §111. Thus:—
Special Rule 2.
If the minor premise were negative, the
20-2
Each Figure
has a special
axiom.
The Dictum
de omni is
the axiom
for Fig. T.
Special Rv.Ui
o/Fiy. I.—
1. Major
premise
universal
2. Minor
premise
affirma-
tive.
308
SYLLOGISMS.
Book IV.
Ch. III.
The Dictum
de diverso is
tlie axiom
for Fi^. II.
Si"'<'i'U lollies
of l-'i-g. II —
1. Major
premise
universal.
2. One pre-
mise nega-
tive.
major must be affirmative (Gen. Rule V) ; P would, there-
fore, be undistributed in the major premise, of which it is
the predicate {see § 72). But the conclusion must be nega-
tive (Gen. Rule VI), and P, as its predicate, must be dis-
tributed. But this is forbidden by Gen. Rule IV. Hence,
such a syllogism is impossible ; i.e., the minor premise must
be affirmative.
Special Rule 1. As the minor premise is affirmative it can-
not distribute W, which is its predicate. M must, therefore,
be distributed in the major premise (Gen. Rule III), of
which it is the subject; i.e., the major premise must be
universal.
(ii.) The Second Figure. The axiom on which syllogistic
reasonings in this figure are based is called the Dictum de
diverso. It is most accurately stated by Mansel {Aldrich,
Art. Log. Rud., 3rd Ed., p. 84) in the words : "If a certain
" attribute can be predicated (affirmatively or negatively) of
•' every member of a class, any subject, of which it cannot
" be so predicated, does not belong to the class."
From this axiom the following special rules of the Second
Figure can be immediately derived : —
1. The major premise must he universal.
2. One of the premises must he negative.
The first rule is involved in the words " predicated , , .
" of every member of a class" and the second in the words
" any subject of which it cannot be so predicated." From
this second rule it follows, by General Rule VI, that the
conclusion must be negative. This is also evident from the
words " does not belong to the class " with which the axiom
ends.
These rules are merely applications of the General Rules
of syllogism (see § 111). Thus : —
Special Ride 2. M must be distributed once, at least, in
the premises (Gen. Rule III), and, as M is predicate of both
premises, this can only be secured when one of them is nega-
tive (see § 72).
FIGURE AND MOOD.
309
Special Rule 1. As one premise is negative the conclusion
must be negative (Gen. Rule VI), and distribute its predi-
cate, the major term P {see § 72). P must, therefore, be
distributed in the major premise (Gen. Rule IV), of which
it is the subject ; i.e., the major premise imist be universal.
(iii.) The Third Figure. The axiom which forms the
basis of syllogistic reasoning in the Third Figure is called
the Dictum de exemplo. It may be thus stated : " If any-
" thing which is stated to belong to a certain class is affirmed
" to possess, or to be devoid of, certain attributes, then those
" attributes may be predicated in like manner of some
"members of that class."
An examination of this axiom will immediately make
clear that the special rules of the Third Figure are : —
1. The minor premise must he affirmative.
2, The conclusion viust be particular.
The first is evident from the words " is stated to belong to
" a certain class" being used of the subject of the conclusion,
and the second from the predication in the conclusion being
restricted to ^^ some members of that class."
These rules are merely applications to syllogisms in the
Third Figure of the General Rule given in § 111. Thus: —
Special Rule 1. If the minor premise were negative, the
major premise would necessarily be affirmative (Gen. Rule V).
Then, as P is the predicate of the major premise, it is
undistributed {see § 72). But a negative premise necessitates
a negative conclusion (Gen. Rule VI), and this would dis-
tribute P. But this is impossible (Gen. Rule IV). Hence
the minor premise cannot be negative.
Special Rule 2. As the minor premise is affirmative, and
S is its predicate, S is undistributed {see § 72). Therefore,
S can only be the subject of a particular conclusion (Gen.
Rule IV). Thus, the second special rule is seen to be im-
plied in the first.
Book IV.
Ch. Hi.
The nictum
de exemplo is
the axiom of
Fig. III.
Special Rulet
of Fig. III.—
1. Minor
premise
affirma-
tive.
2. Conclu.
sion par-
ticular.
310
SYLLOGISMS.
Book IV.
ch in.
The Dictum
lie reciproco
is the axiom
of Fig. IV.
Special Ritlet
of Fig. IV.—
1. If major
affirma-
tive, minor
universal.
2. If minor
affirma-
tive, con-
cliLsion
particular.
3. If one
premise
negative, _
major uni-
versal.
(iv.) The Fourth Figure. The axiom of the Fourth
Figure was called by Lambert the Dictum de recijjruco, but
no one statement of it has been generally' agreed upon even
by those logicians who accept that Figure, which many do
not. The following may be suggested : ' Whatever has a
' predicate affirmed, or universally denied, of it, may itself
' be predicated, particularly and with like quality, of any
* thing which is affirmed of that predicate ; and whatever
'has a predicate universally affirmed of it may itself be
'universally denied of any thing which is universally denied
'of that predicate.'
The following special rules of the Fourth Figure may be
deduced from this axiom : —
1. If the major premise is affirmative, the minor must be
universal.
2. If the minor premise is affirmative, the conclusion must
he piarticidar.
3. If either premise is negative, the major 7nust be
universal.
The first rule is explicitly stated in the last clause of the
axiom, where the minor premise is negative. When the minor
premise is affirmative it is implicitly involved in the axiom,
for we can only be sure that anything has really been affirmed
about the predicate of an affirmative proposition (which is un-
distributed) when the affirmation is made of the same term
distributed, so as to secure every part of its denotation.
The second rule is explicitly stated in the first clause of
the axiom, and the third rule is involved in the words
"universally denied" in the first clause, and "universally
" affirmed " in the second.
These special rules are only applications of the General
Rules (see § 1 11) to the Fourth Figure. Thus :—
Special Rule 1. If the major is affirmative, M, which is its
predicate, is undistributed {see § 72). But M must be dis-
tributed in one of the premises (Gen. Rule III). Therefore,
M must be distributed in the minor premise, of which it is
the subject ; i.e., the minor premise must be universal,
FIGURE AND MOOD.
311
Special Rule 2. If the minor premise is affirmative, 5,
which is its predicate, is undistributed (see § 72). It must,
therefore, be undistributed in the conclusion, of which it is
the subject (Gen. Rule IV) ; i.e., the conclusion must be
particular.
Special Rule 3. If either premise is negative, the con-
clusion must be negative (Gen. Rule VI), and distribute P
(see § 72). P must, therefore, be distributed in the major
premise, of which it is the subject (Gen. Rule IV) ; i.e., the
major premise must be universal.
It will be noticed that these rules are all hypothetical.
This is because in the Fourth Figure either premise may
be negative, or both may be affirmative ; so that in this
Figure, we cannot give a general rule as to the quality of
either premise.
From these rules we deduce the following corollary : —
Cor. Neither of the premises can be a particular negative
proposition.
This follows from Rules 1 and 3. For a negative pre-
mise requires the major to be universal (Rule 3). The
major, therefore, cannot be 0. And if the major is affirma-
tive, the minor must be universal (Rule 1), and, conse-
quently, cannot be 0- Hence, neither premise can be
particular negative.
(v.) Classification of Special Rules. The special rules of each
Figure are intended to guard against any infraction of the General
Rules III and IV, i.e. they are rules of quantity. No special rule
is given when the application of one of these general rules is obvious
and immediate. The following table shows which special rule in
each Figure provides against the fallacy of breaking either of tho e
General Eules :
Book IV.
Ch. III.
Iallacy guarded against
Fig. I
Fig. 11
Fig. Ill
Fig. IV
Undistributed Middle
1
2
1
Illicit Major - . . .
2
1
1
3
Illicit Minor ....
2
2
Corollary —
Neither pre-
mise can be
0.
Tho special
rules are
rules of
quantity.
312
STL LOG ISMS.
Book IV.
Ch. in.
General Rule IV is immediately applicable to the minor term of
Figures I and II ; or we may say more specifically : If the minor
premise in Figures I and II is particular, the conclusion must be
particular.
Corollary 1 to the General Rules of Syllogism [see§Hl(iv.)]
secures us against an Undistributed Middle in Figure III.
Syllogisms
in every
figure are
conclusive.
Figures II
and III are
appropriate
to certain
kinds of
arguments.
Fig. I best
shows the
character of
syllogistic
inference.
A, E, I, O,
can all be
proved in
Fig. I ;
A can be
proved in no
other figure.
S and p
occupy same
positions in
premises as
in conclu-
sion.
* 115. Characteristics of each Figure.
The scholastic logicians, following Aristotle, regarded the
first figure as the only perfect and cogent form of syllo-
gistic inference, and asserted that the validity of syllogisms
expressed in any of the other figures could only be made
evident by reducing them to that figure [c/. §§ 110 (ii.) (a) ;
126]. But, as all syllogistic argument is really based on the
fundamental principles of thought (see § 109, cf. §§ 17-20),
such reasoning is perfectly conclusive in every one of the
four figures ; in any one, denial of the conclusion involves a
denial of one of those fundamental principles. Moreover,
there are various kinds of arguments, directed to the estab-
lishment of certain classes of conclusions, which fall into
either the second or the third figure more naturally than
into the first. Each figure, indeed, has its peculiar character-
istics, and, with the exception of the fourth, its appropriate
ere. These we will now briefly examine.
(i.) The First Figure. In the first figure alone the
distinctive character of syllogistic inference — the subsump-
tion of a special case under a general rule — is shown by the
very form of the argument. Moreover, conclusions of each
of the four forms of categorical proposition — A, E, I, 0 —
can be proved in this figure ; and A can be the conclusion in
no other figure (see § 116). It is, thus, the figure in which
deductions from general scientific principles are most fre-
quently expressed ; for deductive science chiefly aims at
establishing universal afiirmative propositions. Further, in
this figure neither of the extreme terms sufEers an inversion
of position ; for the minor term is subject, and the major
term predicate, both in premise and in conclusion. Thus^
FIGURE AND MOOD.
313
that element of each term— whether denotation or conno-
tation— which is predominant in the premises remains pre-
dominant in the conclusion.
On these accounts, and because the Dictum de omni et nullo
applies directly to syllogisms in the first figure only, it was
regarded by Aristotle as the only perfect figure, an opinion
in which he was followed not only by his scholastic disciples
but by such modern logicians of eminence as Sir W.
Hamilton. Its superiority over the other figures, however,
may be granted without rejecting them as worthless.
(ii.) The Second Figure. In this figure negative conclu-
sions only can be proved [see § 114 (ii.)] ; it is, consequently,
most employed in arguments intended to disprove some
assertion. It has been called the Exclusive Figure, because,
by means of a succession of syllogisms in it we can exclude,
one by one, every possible predicate of a subject but one.
Thus ;—
S either is, or is not, A;
But, Every A is X,
and S is not X,
.*. 5 is not A.
If S is not A it either is, or is not, B;
But, Every B is Y,
and S is not Y,
.; S is nut B.
And so on, till we are left with only one possible conclusion
— S is P. Such a process is called abscissio injiniti, because it
is a repeated excision of what the subject is not. The
following, which can be easily thrown into the above form,
may be given as an illustration : — This act either was, or
was not, done deliberately ; but the doer is not a thoughtless
man who would act without deliberation. If it was done
deliberately it either was, or was not, done from a sense of
duty ; but the doer is not a man who would disregard his
conscience. If it was done from a sense of duty, these
painful consequences either were, or were not, foreseen ; but
these consequences always follow such an act, and so could
Book TV.
Ch. III.
Fig. II can
prove nega-
tives ouly.
By a series of
syllogisms
in Fig. II we
can exclude
all possible
predicates
but one —
Abscissio in-
finiii.
314
SYLLOGISMS.
Book IV.
Ch. III.
P undergoes
inversion of
position.
Fig. Ill can
prove parti-
culars only.
S undergoes
inversion of
position.
Fig. IV is of
little im-
portance.
Both S and P
undergo in-
version of
position.
Summary.
not have been unforeseen. Hence, these painful conse-
quences were foreseen, and we must conclude that the man
voluntarily faced personal discomfort from a sense of duty.
The second figure is not so natural as the first, in that
one of the extremes suffers inversion, the major term being
subject in the major premise and predicate in the conclusion.
This involves a change from the denotative to the connota-
tive reading of that term (c/. §§ 68, 84).
(iii.) Tlie Third Figure. In this figure particular con-
clusions only can be proved ; it is, therefore, specially
adapted to the establishment of exceptions to a general rule.
Arguments in which the middle term is singular, or definite
in quantity, in both premises fall naturally into the third
figure, as such terms are the true subjects of the propo-
sitions in which they occur (cf. § 68). In this figure, as in
the second, there is one inversion of position, the minor
term being predicate in its premise, and subject in the
conclusion. This, of course, involves a change from the
counotative to the denotative aspect of that term.
(iv.) The Fourth Figure. But few syllogisms find a
natural expression in this figure, as in it there is a complete
inversion of the order of thought. The minor term is
predicate in its premise and subject in the conclusion, whilst
the major term is subject in its premise and predicate in
the conclusion. Each of the extreme terms, therefore,
appears in a different aspect in the conclusion from that
which it bears in its premise. The chief value of the
fourth figure, indeed, is theoretical ; as it is a possible
arrangement of terms, its recognition as such is necessary
to complete the formal doctrine of figure (c/". § 113).
(v.) Summary. We have now shown that each figure
— with the exception of the fourth — has its appropriate
sphere, though the first is the most natural, as it retains
one order of thought throughout. The special uses of
each are thus expressed by Lambert : " The first figure is
" suited to the discovery or proof of the properties of a
" thing ; the second to the discovery or proof of the dis-
FIGURE AND MOOD.
315
Mood de-
pends on the
quality and
quantity of
premises
and conclu-
sion.
" tinctions between things ; the third to the discovery or Bo-^^k
" proof of instances and exceptions ; the fourth to the *^'j_
" discovery or exclusion of the different species of a genus."
On this last we may observe that the relation of species and
genus would be much more satisfactorily established by a
syllogism in the first figure, in which the name of the
species is the minor, and that of the genus the major, term
than by one in the fourth figure, in which the major term
denotes the species and the minor term the genus.
116. Determination of Valid Moods.
Mood is the form of a syllogism as determined by tlie
quality and quantity of the three constituent propo-
sitions, e.g., A A A, E A E, A 0 0, are different moods of
syllogism. As a mood may be valid in one figure and not in
another, the full description of a syllogism requires the
statement both of its mood and of its figure. We must now
enquire how many such fully specified syllogistic forms are
valid ; i.e., we must determine the number of valid moods
of syllogism, using the word ' mood ' in a narrower sense to
denote this more specific description. Such determination
may be made either directly, by enquiring what premises are
capable of yielding each of the four possible forms of con-
clusiou— A, E, I, 0 ; or indirectly, by examining all possible
combinations of premises and excluding those which offend
against any of the syllogistic rules. The latter mode of
procedure is the more commonly adopted in text-books on
Logic, but it is both less philosophical and less scientific than
the former. We will, therefore, examine the direct methods
only.
(i.) Direct Determination. We may directly determine The number
the number of valid syllogistic forms— or ' moods ' in the moods llu
narrower sense— by appealing immediately to the funda-
mental Principles of Thought which form the ultimate basis
of syllogistic reasoning {see § 109 ; c/. §§ 17 and 18) ; to the
General Rules of Syllogism (see § 111) ; or to the Special
Rules of each Figure (see § 114). We will examine the two
former of these in turn.
In a nar-
rower sense
' Mood '
specifies
both mood
and figure.
be deter-
mined di-
rectly :
316
SYIXO0TSM3.
Book IV.
Ch. III.
(a) By reference
to the Law3 of
Thought.
A can only be
proved in the
mood A A A in
Fig. I.
E can be proved
in :
E A E in Pig. I.
AEEinFig. IV.
(a) By Reference to the Fundamental Principle.^ of Thought.
The conclusion to a categorical syllogism must be of one of
the forms A, E, I, or 0.
(1) To prove A. If P is to be affirmed of every S through
the medium of M, it is evident, by the Principle of Identity,
{see § 17), that P must be affirmed of every M, and that the
connotation of M must be affirmed of every S ; i.e., that
every S must be M, and every M must be P. Thus, the only
premises which yield an A conclusion are M a P, S a M ; and
the syllogism is
M aP
SaM
:S a P
which is the mood A A A in Figure 1.
(2) To prove E. If P is to be denied of every S through
the medium of M, then, by the Principles of Identity and
Contradiction (see §§ 17, 18), M must be affirmed of the
whole extent of the denotation of one of the extremes, and
entirely excluded from the denotation of the other.
Now, M is entirely separated from P when the major
premise is either M e P or P e M. Combining each of these
with SaM, in which M is affirmed of every S, we get
(1)
MeP
SaM
.-.Sep
(2)
PeM
SaM
:SeP
Each of these is of the form E A E, the first in Figure I,
the second in F'igure II.
Similarly, M is entirely separated from S when the minor
premise is either S e M or M e S. Combining each of these
with P a M, in which M is affirmed of every P, we get
(3) (4)
PaM PaM
Se M M eS
SeP
'.Sep
FIGURE AND MOOD. 31?
Each of these is of the form A E E, the first in Figure II, Book IV.
and the second in Figure IV. Ch^il.
There are, thus, four moods in which E can be proved —
one in Figure I, two in Figure II, and one in Figure IV.
(3) To prove I. If P is to be affirmed of an indefinite i can be proved
part of the denotation of S through the medium of A/, then AilinFig. I.
by the Principle of Identity (sfg § 17), an indefinite portion ^^J linFi£rlIL
of the denotation of each of the extremes must agree with AAI'
one and the same portion of the denotation of M. This can aaI)^'^^^' ^^
only be assured when one, at least, of the extremes is
aftirmed of every M, and, at the same time, agreement to a
more or less indefinite extent is predicated between M and
the other extreme.
If P is affirmed of every M, and there is wholly indefinite
agreement between S and M, we have
(1) (2)
M aP M aP
S i M M i S
.: S i P .: S i P
These are both of the form All, the first in Figure I,
and the second in Figure III.
If S is affirmed of every M, and there is agreement, wholly
or partially indefinite, between the denotation of M and that
of P, we have
(3)
(4)
(5)
' (6)
MiP
M aP
PiM
PaM
MaS
MaS
MaS
MaS
.'.SiP .'.SiP .-.SiP .-.SiP
Of these (3) and (5) are in the mood I A I, and (4) and (6)
in A A I ; (3) and (4) are in Figure III, and (5) and (6) in
Figure IV. There are thus seen to be six moods in which I
can be proved— one in Figure I, three in Figure III, and
two in Figure IV.
(4) To prove 0. If P is to be denied of an indefinite
318
SYLLOGISMS.
Book IV.
Ch. in.
O ran be proved
in :
E I O in Fig. I.
i^O}inFig.IL
EAO)
EIO WnFig.IIL
OAOJ
EIO |inF'gIV.
portion of the denotation of S through the medium of M.
then, hj the Principles of Identity and Contradiction (see
§§ 17, 18), either
(rt) P must be denied of certain A/'s which are affirmed to
be S, or '
(/3) M must be both aflSrmed of every P and denied of
some S's.
(a) In the first case either M must be entirely separated
from every P, and agree, in whole or to an indefinite extent,
with some S's/ or P must be denied of an indefinite number
of M's whilst every A/ is aflarmed to be 5. The first condition
is fulfilled when the major premise is either M e P or P e M,
and the minor S i M, M a S, or M i S. Comoining each of
these minors with each of the majors we get
(1)
(2)
(3)
(4)
(5)
(6)
M eP
PeM
M eP
M eP
PeM
P ,' U
S iM
S iM
MaS
M iS
MaS
M iS
SoP
SoP
SoP
SoP
SoP
SoP
Of these (3) and (5) are in the mood EAO, and all the
others in the mood EIO; the first is in Figure I, the
second in Figure II, the third and fourth in Figure III, and
the fifth and sixth in Figure IV. The second condition of
the first case is fulfilled by the syllogism
(7)
M oP
MaS
SoP
which is in the mood 0 A 0 in Figure IIL
(/3) The second case gives the syllogism
(8)
PaM
SoM
SoP
which is in the mood A 0 0 in Figure II.
There are, thus, eight moods in which 0 can be proved ;
flGUEE AND MOOD.
319
one in Figure I, two in Figure II, three in Figure III, and Book IV.
two in Figure IV. ^'Un-
collecting our results it appears that : — Summaiy,
A can be proved in only one mood, and only in
Figure I.
E can be proved in four moods, and in every Figure
except the Third.
I can be proved in six moods, and in every Figure
except the Second.
0 can be pioved in eight moods, and in every Figure.
Thus 0 is seen to be proved in the greatest number of
moods, and A in the smallest. But these propositions are
contradictories, and the establishment of the one dis-
proves the other. Hence, it is often said that A is the most
difficult proposition to establish and the easiest to disprove.
At the same time, it must be remembered that " universal
*'■ affirmative conclusions have the highest scientific value, be-
" cause they advance our knowledge in a positive manner
"and admit of reliable application to the individual. The
" universal negatives come next ; they guarantee only a nega-
" tive but a distinctly definite view. Then come the par-
" ticular affirmatives, which promise a positive advance, but
" leave us helpless in the application to individual cases.
" Lastly, the particular negative conclusions are of the lowest
" value. Particular propositions, however, are by no means
" without scientific meaning. Their special service is to ward
" off false generalizations. The universal negative or affirma*
"tive judgment, falsely held to be true, is proved not true
" by the particular affirmative or negative conclusion, which
" is its contradictory opposite" (Ueberweg, Logic, Eng. trans.,
pp. 436-7).
(6) By Reference to the General Rules of Syllogism {see (J) By refer.
§^ ^ , \ ence to
lliJ- General
Rules of
(1) To prove A. Both premises must be affirmative ^y^^^ffi^™-
(Rule VI) ; and, consequently, distribute only their subjects
320 SYLLOaiSMS.
Book IV. (..ee § 72). S, being distributed in the conclusion, must be
Cii^il. distributed in — i.e., be the subject of — the minor premise
(Rule IV). Tliis leaves M to be distributed in the major
premise (Rule III), of which it is, therefore, the subject.
Thus, we get the syllogism
MaP
SaM
.: S aP
which is of the form A A A in Figure I.
(2) To prove E. One of the premises must be negative
(Rule VI) and one must be affirmative (Rule V). Both
S and P are distributed in the conclusion {see § 72), and must,
consequently, be distributed in the minor and major pre-
mises respectively (Rule IV). M must also be distributed
in one of the premises (Rule III). But the premises can
between them distribute three terms only when both are
universal ; one is, therefore, E, and the other A. In the E
premise, both M and one of the extremes are distributed ;
the other extreme must, therefore, be distributed in— i.e., be
the subject of — the A premise. In the E premise M may be
either the subject or the predicate, as both are distributed.
Hence, we get four possible syllogisms with an E conclusion
(1) (2) (3) (4)
MeP PeM PaM PaM
SaM SaM Se M M e S
.\SeP .:SeP .'. S e P .: S e P
Of these (1) and (2) are in the mood E A E, and (3) and
(4) in A E E. The first is in Figure I, the second and third
in Figure II, and the fourth in Figure IV.
(3) To prove 1. Both premises must be affirmative (Rule
VI). As neither S nor P is distributed in the conclusion, M
is the only term whose distribution is necessary in the pre-
mises. It is immaterial whether P is or is not distributed in
the major premise as this cannot affect the conclusion. But
(2)
(3)
(4)
(5)
(6)
MaP
MiP
MaP
PiM
PaM
M iS
M aS
Mas
M aS
M aS
FIGURE AND MOOD. 321
the distribution of S in the minor premise wo;iIcl permit the Book IV.
conclusion to be A, and, therefore, in premises which can Ch^i
yield only I, S must be undistributed. This leaves us any
combination of affirmative premises in which M is, and 5 is
not, the subject of an A proposition. We thus get
(1)
MaP
SiM
.: SiP .'.SiP .:SiP .-.Sip .: SiP .-. S i P
Of these (1) and (2) are in the mood All, (3) and (5) iu
I A I, and (4) and (6) in A A I. The first is in Figure 1,
the second, third, and fourth in Figure III, the fifth and
sixth in Figure IV.
(4) To prove 0. One premise must be negative (Rule VJ).
P is distributed in the conclusion (see § 72), and must, conse-
quently, be distributed in the major premise (Rule IV).
M must be distributed in the premises (Rule III). If the
major premise is E, both P and M are distributed in it, and
either may be its subject ; no term need be distributed iu the
minor premise, but M may be, i.e., the minor premise is
either 1 ov M a S. If the major premise is 0, P must be its
predicate, and M must be distributed in the affirmative
minor premise, which will he M a S. If the major premise
is A, P must be its subject, and A/ alone should be distributed
in the negative minor premise, which will he S o M. Hence
we get
(1) C-^) (3) (4)
M e P PeM M e P M e P
SiM SiM M aS MIS
So P .-. S 0 P .'.S 0 P . .: So P
(5) (6) (7) (8)
PeM PeM M o P PaM
M a S M iS M aS S o M
.: S 0 P .: S 0 P .-. S o P .: S 0 P
Of these (3) and (5) are iu the mood E A 0, (7) iu 0 A 0,
LOG. I. 21
322
SYLLOGISMS.
Book IV.
ch. in.
Tho names
iu the
muemonic
lines specify
the moods
by indicat-
iug the
quality and
quantity of
the three
propositions
by the
letters a, e,
i, 0.
(8) iu A 0 0, and all the others in E I 0. The first is in
Figure I, and second and eighth in Figure II, the third,
fourth, and seventh in Figure III, and the fifth and sixth
iu Figure IV.
These results are, of course, identical with those we
obtained by the more philosophical method of appealing
directly to first principles.
(ii.) The Mnemonic Lines. Each method of determina-
tion has led us to the result that there are nineteen valid
moods, in the narrower sense of the term — four in Figure I,
four in Figure II, six in Figure III, and five in Figure IV.
It is customary to designate these moods by the names
which compose the following mnemonic lines ; each of
these names containing three vowels, and thus specifying a
mood by indicating the quality and quantity of the constituent
propositions by the usual symbols — A, E, I, 0 ; thus Cesare
denotes the mood E A E in Figure II : —
Barbara, Celdrent, Ddrii, Ferid(\ViQ prioris :
Cesare, Cdmestres, Festlno, Bdrocd, secundaa :
Tertia, Ddrapit, Disdmis, Ddtlu, Felapton,
Bocardo, Ferlson, habet : Quarta insuper addit
Brdmantip, Cdnienes, Dimdris, Fesdpo, Fresison.
These mnemonics are given here for the convenience of
referring to the moods by their ordinary names, but the full
explanation of their import must be deferred till we treat of
Reduction, in the next chapter.
Fundamen-
tal Hj/Uo'
ghm~no
term uu-
necessarily
iistributed.
* 117. Fundamental and Strengthened Syllogisms.
Of the nineteen valid moods of syllogism, fifteen may be
called Fundamental, as in them neither premise is stronger
than is necessary to produce the conclusion ; i.e., neither of
the extreme terms is distributed in the premises without
being distributed in the conclusion, and the middle term is
distributed only once. But there are two moods in Figure
III — Darapti and Felapton — and one in Figure IV — Fesapo,
in which the middle term is distributed in both premises,
FIGURE AND MOOD.
323
and oue mood in Figure IV — Bramantip — in which the
major term is distributed in the major premise, but not in
the conclusion. These are called Strengthened Syllofjisms, as
in Darapti and Felapton either premise, in Fesapo the
minor, and in Bramantip the major, premise may be
weakened to its subaltern particular proposition without
affecting the conclusion, and when this ia done the syllogism
is in one of the fundamental moods.
118. Subaltern Moods or Weakened Syllogisms.
When from given premises a conclusion is deduced which
is weaker than the premises warrant, the syllogism is said to
be Weakened, or to be in a Subaltern Mood. There can, thus,
be a subaltern mood correspuuding to every mood with a
universal conclusion. There are five universal moods (see
§ 116), viz., Barbara and Celarent in Figure I, Cesare and
Camestres in Figure II, and Camenes in Figure IV ; and the
corresponding subalterns may be named Barbari, Celaront,
Cesaro, Camestros, and Camenos. The particular conclusions
drawn in these moods are, no doubt, justified by the premises,
but such weakened conclusions are misleading, as they suggest
that the universal cannot be deduced. As, however, the
predication in the minor premise has been made of all the
denotation of the minor term, nothing can be predicated in
the conclusion of one part of that denotation which cannot be
similarly predicated of any other part, and, consequently, of
the whole. Such syllogisms are, therefore, not admitted
into the list of independent legitimate syllogisms, for their
conclusions can be obtained by subalternation [see § 97 (i.)]
from the conclusion of the corresponding fundamental
syllogism. They are, indeed, practically useless, as only a
part of what really results from the premises is taken.
♦ Each of these subaltern moods is a strengthened
syllogism, except Camenos (A E 0 in Figure IV) ; for in
each of the other four moods the minor premise may be
weakened without affecting the conclusion.
♦ Including subaltern moods, then, there are two strength*
ened syllogisms in each figure : —
Book IV.
Ch. in
strengthened
Syllogism — a
term distri-
buted in the
jj remises
more than is
requii'ed.
Weakened
Syllogism or
Subaltern,
Mood —
particular,
instead of
universal,
conclusion
Such
syllogism'
are super
fiuoua.
324
SYLLoaisirs.
Book IV.
Ch. HI.
In Figure T - - (A A I), (E A 0) ;
In Figure II - (E A 0). (A E 0) ;
]n Figure III - A A I, E A 0 ;
In Figure IV - A A I, E A 0 ;
but those in brackets, being subalterns, are superfluous.
Of course, the other strengthened syllogisms, although in
them a particular conclusion is infeired from two universal
premises, are not subaltern moods ; for the particular con-
clusion is the utmost the premises will allow, their super-
fluous information referring, not to the minor but, either to
the middle or to the major term.
Kig. 1 shows
by its form
the funda-
meutal
nature of
syllogistia
inference
AAA and
EAEarethe
ultimately
distinct
moods.
Moods of
1. Barbara-
Hi a P
San
.: Sa P
119. Valid Moods of the First Figure.
In the First Figure the fundamental nature of syllogistic
reasoning — the application of a general rule to a special
instance — is most clearly seen. The major premise gives the
general principle, whilst the minor premise states the special
case to which that general principle is to be applied
Reasonings of the greatest scientific value are, therefore,
most naturally expressed in this figure. There are, as we
have seen (see § 110) four valid moods in this figure, each of
which has one of the four forms of categorical proposition
for its conclusion. But the difference between a mood with
a particular conclusion and that with the universal conclusion
of the same quality is merely in the degree of definiteness
with which the general principle can be applied to the
objects denoted by the minor term. The forms of argument
which are ultimately distinct are, therefore, two ; one
proving the presence, and the other the absence, of an
attribute in the special case under consideration.
We will now consider each of the four moods in some
detail.
(i.) Barbara. This is the most important of all the forms
of syllogistic inference, and the one most frequently
employed — though often elliptically — not only in all
branches of science but in common life ; for to establish a
universal connexion between subiect and attribute is the
FTQURB AND MOOD.
325
constant effort of thought, and object of research. Its Book IV.
schema is ch^ii.
MaP
SaM
SaP
The necessity of the A conclusion is plainly seen when the
mood is represented by the diagrams suggested in § 94—
MP
MP
SM
SM
•I
Diagram of
Barbara.
Barbara-
From-
Mathema
tics.
The line — whether wholly or only partially unbroken —
which represents the total extent of M being drawn of the same
length in each premise, it is evident that the conclusion must
be Every S is P.
Ashort examination of examplesinsomeof thechief domains Bxamplesof
of thought will establish the great importance of this mood.
All direct mathematical demonstrations of affirmative
theorems are given exclusively in such syllogisms, though
they are frequently abbreviated by the omission of an
explicit statement of the major premise. For example, the
whole argument of the First Proposition in the First Book
of Euclid consists of three syllogisms in Barbara — two
proving the equality of each of the newly constructed lines
with that given, and each assuming as its major premise that
' all lines drawn from the centre of a circle to the circum-
ference are equal to each other'; and one establishing the
equality of the two newly constructed lines with each other,
the implicit major premise being that ' things which are
equal to the same thing are equal to one another.' And so
throughout. " This syllogistic concatenation is the spinal
" cord of mathematical demonstration. The mathematician
"shortens the form of expression, but the syllogistic /orm of
'* thought cannot be removed without destroying the force of
" the demonstration itself " (TJeberweg, Logic, Eng. trans,.
pp. 404-5).
320
SYLLOGISMS.
Book IV.
Ch. III.
Prom
Pliysics.
From
Grammar.
Fiom Law.
lu Physics again the syllogistic form of thought is the
only one by which particular phenomena can be explained ;
and here, again, Barbara is the most important mood. From
the general law of the radiation of heat — that, unless some
medium intervenes, a warm body radiates part of its heat
through the atmosphere to a colder body surrounding it — we
infer that, as the surface of the earth on a clear night is a
warm body under those conditions, it will thus become cooled.
Similarly, "the explanation of the formation of dew rests on
" the syllogism : Every cooling object whose temperature is
" below that of the so-called point of dew, attracts to itself
" out of the atmosphere a part of the watery vapour con-
" tained in it, and causes it to precipitate itself on it ; the
" superficies of the earth, and especially of plants, are colder
" in clear nights than the atmosphere, in consequence of the
" radiation of the heat to the space around ; and, therefore,
" when the cooling exceeds a certain limit, they attract a
" portion of the watery vapour contained in the atmosphere^
"and make it precipitate itself on them" (Ueberweg, ibid.,
p. 406).
In Grammar, again, we see the same syllogistic process.
For example, we have in English Grammar the general rule
that names ending in -y not immediately preceded by a vowel
form the plural by changing the -1/ into -i and adding -e«/
now lady is such a noun ; therefore, the plural of lady is
ladies. In French, a verb expressing doubt is followed bv
the subjunctive mood ; douter is such a verb, therefore,
douter is followed by the subjunctive mood.
All application of Laio is equally syllogistic. The whole
aim of legal procedure is to determine whether or not a
particular case does, or does not, fall under a certain general
rule, and, if it does, what are the resultant consequences.
Thus, in a criminal trial, the law which has been violated
furnishes the major premise, the examination of the acts of
the accused person supplies the minor premise, whilst the
verdict of * Guilty ' or ' Not Guilty ' gives the conclusion
from those premises ; a conclusion to which the sentence of
the judge gives practical effect.
FIGURE AND MOOD.
327
In Medicine the reasonings are equally syllogistic. The
whole of diagnosis is an attempt to subsume a particular
ailment under some general class of disease, of which the
appropriate treatment is more or less known. The diagnosis
is itself syllogistic ; for it is an inference that the case in
question is a certain kind of disease, because it exhibits
certain symptoms which are the marks of that disease.
Thus, the diagnosis gives the appropriate minor premise,
' This is a case of such a disease ' ; the treatment adopted by
the physician is the practical expression of the conclusion
that such and such a remedy will be efficacious. For
example, ' Lupus is cured by Dr. Koch's lymph ; this is a
case of lupus ; therefore, this case is to be treated by Dr.
Koch's lymph.'
Reasonings in Economics are of a like kind. Thus, we
have the general principle — which is itself deduced syllogisti-
cally from the conception of the relations of Supply and
Demand — that, other things being equal, everything which
tends to limit the supply of a commodity tends to raise its
price; but protective duties levied on imports tend to
restrict the supply of the commodities on which they are
imposed ; hence, it follows that such duties have a tendency
to raise the prices of those commodities.
In EtMcs, too, our judgments that such and such conduct
is worthy of praise are the results of a syllogistic process by
which we subsume the conduct in question under a general
rule. For instance, when we praise a particular hero for his
patriotism we do so because we apply to his special case the
general rule that all patriotism is praiseworthy.
The explanation of historical phenomena is another case of
syllogistic inference. Thus, Schiller explains the length and
violence of the Thirty Years' War by bringing it under the
general principle that all religious wars are marked by the
greatest pertinacity and bitterness, because every man takes
one side or the other from personal feelings and not, as in
ordinary wars between nations, simply on account of the
place of his birth. Similarly, the fall of the Roman Empiro
is understood when it is regarded as an instance of the
Book IV.
Ch. J 1 1.
Prom
Medicine.
Prom
Economics,
From
Etbics.
From
History.
•^28
SYLLOGISMS.
Book IV. general law that, as nations adopt luxurious and vicious
ch^i. habits, they lose their pristine vigour, become effeminate,
and fall an easy prey to more hardy barbarians. In like
manner, both experience and reason teach us that nations
which are ground down by oppression will at length burst
out into revolution ; by this general law we may explain
the French Revolution of the Eighteenth Century, and may
even foretell, with more or less assurance, the probable fate
of one or two modern European states.
2. Celarent —
M e P
So M
.-.Sep
Diagram of
Celarent.
Examples of
Cdarent.
(ii.) Celarent. This is the typical mood in which it is
proved that a certain subject does not possess certain attri-
butes. As it is neither of so much importance, nor of so
much interest, to prove what a thing is not as to show what
it is, this mood is not so universally used as Barbara. Its
schema is
MeP
SaM
.\SeP
The necessity of the conclusion is again evident from the
diigram
MP
I-
MP
•I
SM SM
which shows the entire exclusion of S from P.
As examples we may give : — ' What is involuntary is not
to be overcome by punishment ; stupidity is involuntary ;
hence, stupidity cannot be overcome by punishment.'
' Duties on imports levied solely for the purposes of revenue
are not protective ; all English import duties are of this
class ; therefore, no English import duty is protective.'
Ihrii —
Map
S i M
■ Sip
(iii.) Darii. This mood is merely an indefinite form of
the process of reasoning employed in Barbara. Its schema
is
MaP
SiM
.8iP
FIGURE AND MOOD, 329
and it is represented by the diagram Book IV.
f J ° ch. III.
MP MP ^. _,
I Dia^am of
SM SM SM
I
where SM is written under MP so as not to appear to ex-
clude the contingency that any possibly existing SM may
be P. Of course, in these diagrams, the order in which
the lines denoting the various possible classes are written is
of no importance.
As examples of Darii we may give : * Every act which is Examples A
done from a strict sense of duty is formally right ; some acts
which mankind generally condemn are done from such a
motive ; therefore, some acts which are generally condemned
are formally right.' ' All just governments aim at securing
the welfare of their people ; some autocracies have been just ;
therefore, some autocracies have aimed at securing the wel-
fare of the governed.'
* The value of this mood, as of all others in which particu-
lar propositions are proved, is limited, but not destroyed, by
the indefiniteness of the conclusion. For all that is in-
definite is whether or not any S's exist of which the predication
contained in the conclusion cannot be made. The value of the
knowledge that Some S's are P (or, in moods with particular
negative conclusions, that Some S'« are not P) is not to be
denied because the premises leave us without information
concerning other possibly existing S's. The desire for know-
ledge, no doubt, must hold this limited information to be
insufficient ; but it is not insufficient in the sense that our
conclusion has excluded the possibility that the predication
can be made of every 5. So long as the purely indefinite
character of * some ' is borne in mind, there can be no fallacy
in such an inference ; for a particular conclusion does not
imply the sub-contrary proposition ; and the assertion of the
minor premise assumes that some S's are known to be cases
to which the general principle which is the basis of the in-
ference may be applied.
330
SYLLOGISMS.
Book IV.
Ch. III.
4. Ferio—
M e P
S i M
.-.SOP
Diagram of
Ferio.
Examples of
Ferio.
(iv.) Ferio. This mood occupies the same relation to
Celarent as Darii does to Barbara. The remarks made at
the end of the last sub-section are, therefore, applicable here.
Its schema is
M eP
S i M
.•.SoP
and its diagram
MP
MP
SM SM
SM
•which shows the definite exclusion of Some S's from P, but
leaves the existence and relation of any other S's proble-
matic ; so that, if any such exist, they may, or may not,
he P.
As examples of Frrio may be given : — ' No act done from
a right motive is deserving of punishment ; some acts whose
consequences are disastrous are done from a right motive ;
therefore, some acts whose consequences are disastrous are
not deserving of punishment.' * No protective duty is im-
posed for purposes of revenue ; some of the French import
duties are protective ; therefore, some of those duties are
not imposed for purposes of revenue.'
Moods of
Fig. il :
120. Valid Moods of the Second Figure.
In this figure, E and 0 are the only possible conclnsions,
and each of these can be proved in two moods, in one of
which the major, and in the other the minor, premise is
negative. There are, therefore, four valid moods to be con-
sidered.
1. Cesare —
P e M
S a M
.-.Sep
(i.) Cesare. The schema of this mood is
PeM
SaM
SeP
flGURE AND MOOD.
fi.ncl it is represented by the diagram
PM PM
i
SM SM
331
\-
PaM
SeM
.'.SeP
and its diagram
PM
PM
SM
Book IV.
Ch. III.
Piagram of
C'esare.
The following examples may be given from Aristotle's
Nicomachean Ethics (II, 4): — 'The emotions do not make
men either praiseworthy or blameworthy ; the virtues and
vices do this ; therefore, the virtues and vices are not
emotions.' * The affections are not acts of choice ; the
virtues are acts of choice ; therefore, the virtues are not
affections.' ' Opinion is not limited in its range of objects;
moral choice is so limited ; therefore, moral choice is not
opinion ' (Hid., Ill, 4).
(ii,) Camestres. The schema of this mood is
Examples oi
Cesare.
2. Camestrfs-
PaM
SeM
.-.Sep
Diagram of
Camestres.
We may again illustrate from Aristotle's Ethics, ' The
faculties, or capacities for feeling emotions, are natural
gifts ; virtues are not natural gifts ; therefore, virtues are
not faculties ' (II, 4). Or we may say : * All acts which are
fit subjects for moral judgment are deliberate ; no impulsive
act is deliberate ; therefore, no impulsive act is a fit subject
for moral judgment.' The way for the discovery of the
existence, place, and size of the planet Neptune was prepared
by reasoning, which was really in this mood. The astronomer
Leverrier argued that the sum total of the worlds belonging
to our solar s^fstem must determine the orbit of UranuSi
Examples of
Camestres.
882
SYLLOGISMS.
Rook TV. an^> as the known worlds did not fully do this, that, there-
Ch. in. fore, all the worlds of our solar system were not known.
S, Festino —
S i M
.•.SoP
Diagram of
Festino.
(iii.) Festino. This mood is the indefinite form cor-
responding to Cesare. Its schema is
PeM
SiM
,\SoP
and its diagram
I-
PM
SM
Pli
SM SM
Examples of
Festino.
4. Baroco —
PaKI
S 0 M
.-.SoP
Diagram of
Laroco.
\ehicb shows that Some S's do not possess P, bnt leave;^
quite muehuite the relation between P and n.ny other S's
which may possibly exist.
Examples of Fentino are : — 'No truthful man prevaricates ;
some statesmen prevaricate ; therefore, some statesmen are
not truthful.' 'Nowise men are superstitious; some edu-
cated men are superstitious ; therefore, some educated men
are not wise.'
(iv.) Baroco. This mood bears the same relation to
Camestres that Festino does to Cesare. Its schema is
PaM
So M
SoP
and its diagram
PM
PM
J
SM
1—
SM
"1
SM
1
Of course, there is no suggestion that SM — if any such class
exists — corresponds with PM ; the boundaries of the classes
represented by dotted lines are absolutely indefinite.
FIGURE AND MOOD.
333
As examples of Baroco we may give — ' Whatever is true is Book IV.
self -consistent ; some of Hamilton's logical theories are not Ch^ii.
self - consistent ; therefore, some of Hamilton's logical r:x"mpiesoi
theories are not true.' ' All truly moral acts are done from
a right motive ; some acts which benefit others are not done
from such a motive ; therefore, some acts which benefit
others are not truly moral.' ' All moral choice is fixed ou
the possible ; some wishes relate to the impossible ; there-
fore, some wishes are not of the nature of moral choice '
(Aristotle, Ethics, III, 4).
121. Valid Moods of the Third Figure.
In this figure, only I and 0 propositions can be proved, but Mooda of
each can be the conclusion of three moods ; for both pre- ^^^' ^^^ '
mises may be universal, or either may be particular. There
are, therefore, six valid moods to be considered.
(i) Darapti. The schema of this mood is
1. Darapti-
Map
Mas
.-.Sip
MaP
MaS
.-. S i P
and its diagram
MP MP
MS
MS
Diagfram of
DaraptU
where it is evident that the part of S which is M must
correspond with the part of P which is M, but the extent,
and, indeed, the existence, of any other S and P are left
problematic.
An example of Darapti is found in Ai'istotle's Ethics (III, 7),
where he argues that as ' to do or to forbear doing what is
creditable or the contrary is in our own power, and these
respectively constitute the being good or bad, therefore, the
being good or vicious characters is in our own power.'
Another example ia : ' All whales are mammals ; all whales
are water - animals ; therefore, some water - animals are
mammals.'
Kxamples of
DaraptL
334 SYLLOGISMS.
Book IV '^^^ ^^^^ ^^^^ ^^ *'^'^ mood the middle term is distributed in each
Ch. III. premise makes it a peculiarly appropriate form in which to express
those syllogisms which have two singular propositions as premises.
Professor Bain (Deductive Lotjic, p. 159) denies that these aie
genuine syllogistic inferences at all. He takes the examjjle
Socrates fought at Delium,
Socrates was the master of Plato,
.'. The master of Plato fought at Delium ;
and says that " the proposition ' Socrates was the master of Plato,
" and fought at Delium,' compounded out of the two premises, is
" obviously nothing more than a grammatical abbreviation. " The
step to the conclusion " contents itself with reproducing a part of
" the meaning, or saying less than had been previously said . . .
"Now, we never consider that we have made a real inference, a
" step in advance, when we repeat less than we are entitled to say,
" or drop from a complex statement some portion not desired at the
" moment." But the same argument would apply to every syllogism,
and with especial ease to all those in the Third Figure. For, in every
syllogism, the premises can be combined into a single statement,
and the conclusion always says " less than had been previously said."
Indeed, the fact that " we repeat less than we are entitled to say,"
which Professor Bain regards as fatal to the claim of such an argument
to be considered a true syllogism, is the characteristic of all dis-
cursive thought, which is so called for the very reason that it does
leave out of sight the data of which it has made full use, and con-
cerns itself only with the predication which can be made about the
special case it is considering,
2. iHsamis^ ^ii^ Disamis. The arguments in this mood are very
K as similar to those in Darapti. The schema of the mood is
•"• ^ * '' MiP
MaS
SiP
and its diagram is
o*-
MS MS
FIGURE AND MOOD.
335
As examples we may give : * Some pronouns in English are Book IV
inflected; all such pronouns arewords of English origin; there- *-'J^'^
fore, some words of English origin are inflected.' 'Some grati-
fications of appetite are injurious to health ; all such gratifica-
tions are pleasant at the moment ; therefore, some things which
give pleasure at the moment are injurious to health.'
Examples of
Dlsamit.
(iii.) Datisi. This mood, again, is very like the last two.
in fact, as an I proposition can be simply converted [see
§ 102 (ii.) (c)], it is a matter of very small moment whether
an argument is expressed in Disamis or in Datisi. The
schema of the mood is
MaP
M iS
Datisi—
Map
M i S
.Sip
.-. SiP
and it is represented by the diagram
MP
MP
MS
MS
MS
Diagram o£
Datisi.
■I
Examples are ; ' All wars cause much suffering ; some
wars are justifiable ; therefore, some justifiable courses of Kxamplesof
conduct cause much suffering.' ' All diseases entail suffer- o«'»«i-
ing ; some diseases are preventible ; therefore, some pre-
ventible causes of suffering exist.'
(iv.) Felapton. This mood is the negative corresponding
to the affirmative Darapti. Its schema is
M eP
MaS
.-. SoP
and the diagram which represents it is
MP MP
MS
MS
4. Fdapton —
M e P
Mas
.-.SoP
Diagram of
Fdapton.
336
SYLLOGISMS.
Book IV -^^ example is: 'No brave man fears death; all brave
Ch. in. meu fear dishouour ; therefore, some who fear dishonour do
Example of not fear death.'
Felapton.
6. Bncardo —
M o P
Mas
.-.SOP
Din gram of
Bocardo.
Rxample of
Bocardo.
(v.) Bocardo. This mood gives the same conclusion its
its strengthened form, Felapton; it is the negative cor-
responding to the aflSrmative Disamis. Its schema is
M oP
MaS
.: SoP
and its diagram
MP
MP
MS
MP
MS
•I
which makes it plain that the 8 which coincides with MP is
not P.
A good example is given by Ueberweg (Logic, Eng. trans.,
pp. 425-6) : " Some persons accused of witchcraft have not
" believed themselves to be free from the guilt laid to
" their charge ; all those accused of witchcraft were accused
" of a merely feigned crime : hence some who were accused
" of a merely feigned crime have not believed themselves
" free from the guilt laid to their charge."
Feri^o'i)
M e P
M i S
,S o P
Diagram of
Ji'trUon.
(vi.) Ferison. This is Felapton with a weakened minor
premise, and corresponds to the affirmative Datisi. Its
schema is
MeP
Mi8
SoP
and its diagram
M?
MS MS
■■[
^P
MS
f-IGURE AND MOOD.
337
As examples we may give : ' No truly moral act is done g^^^ ^^^
without deliberatiou ; some such acts are followed by painful ch. ill.
consequences ; therefore, some acts whose consequences are 2^.^^^^, ^^
painful are not done without deliberation.' ' No aggressive perison.
war is justifiable; some aggressive wars are successful;
therefore, some successful wars are not justifiable.'
122, Valid Moods of the Fourth Figure.
Comparatively few arguments fall naturally into the Fourth Moods of
Figure. The arguments with A, E, and I conclusions re-
spectively which can be expressed in it generally fall into
Figure I. But if we wish to fix attention on the term
which in the First Figure would be the predicate of the con-
clusion, we throw the argument into the Fourth Figure,
where that term becomes the subject of the conclusion. The
two moods in Figure IV with an 0 conclusion can generally
be expressed, at least as naturally, in the Third Figure.
(i.) Bramantip. The schema of this mood ia
PaM
MaS
Bramtm-
tip —
PaM
Mas
SiP
and it ia represented by the diagram
PM
PM
MS
s \ p
Diagram ol
Bramantip.
MS
As an argument which falls more naturally into this mood
than into Barbara, with the extreme terms transposed in the
conclusion, may be given : ' All the important operations of
nature are common ; things which are common escape our
attention ; therefore, some things which escape our attention
are important operations of nature.' Similarly, from the pre-
mises 'All moderate physical exercise is beneficial to health ;
everything beneficial to health is inculcated by the Moral
Code,' we shall, if our attention is concentrated on moral pre-
LOG. 1. 22
338
SYLLOGISMS.
Book IV.
Ch. III.
Diagram of
Cameiiei.
Examples of
Canunes.
Dimaris —
P i Af
Af as
.'.SiP
Diagram of
Dimaris.
cepts, most naturally conclude that ' Amongst the command.^
of the Moral Code is one which insists on moderate physical
exercise.'
(ii.) Camenes. This mood holds the same relation to
Celarent as Bramantlp does to Barbara. Its schema is
PaM
M eS
SeP
and it is represented by the diagram
PM
PM
MS
I-
MS
In this diagram S is entirely represented by a dotted line,
which implies that its existence is doubtful. The only con-
clusion, therefore, which is formally justified by the premises
is of the conditional form — If any S exists, ii is not P. As
examples may be given : ' All squares are parallelograms ; no
parallelogram is a trapezoid ; therefore, no trapezoid is a
square.' ' All truly brave men prefer death to dishonour ; no
one who prefers death to dishonour is capable of a mean
action ; therefore, no one who is capable of a mean action is
truly brave.' In both these examples we know independently
that S exists.
(iii.) Dimaris. This corresponds with Darii, as the two
preceding moods do with Barbara and Celarent respectively.
Its schema is
PiM
MaS
and its diagram
PM
MS
SiP
PM
PM
MS
FIGURE AND MOOD.
339
An example is ' Some parallelograms are squares ; all squares Book IV.
are regular figures ; therefore, some regular figures are ^^- ^'^•
parallelograms.'
(iv.) Fesapo. The schema of this mood is
PeM
MaS
.'. SoP
and it is represented by the diagram
PM
V -H
MS
V
PM
MS
Example ol
Jiiiiiaris.
4. Fcsapo—
PeM
U as
.-.Sop
Diagram of
Fesapo.
As the minor premise assures us of the existence of M in
that sphere of existence to uhich the syllogism refers, we
may simply convert the major premise to the categorical
proposition M e P [cf. §§ f^D, 102 (ii) (6)], and the syllogism
is then in Felapton in Figure III.
As examples may be given : 'No trades-unionist is employed Examples of
in this factory ; all who are employed here earn good wages ;
therefore, some who earn good wages are not trades- unionists.'
' No inference which falls under Aristotle's definition of
inferences in the First Figure is either of the form Fesapo
or of the form Fresison ; every inference of these forms is
in the Fourth Figure; therefore, some inferences in the
Fourth Figure do not fall under Aristotle's definition of
inferences of the First Figure.'
(v.) Fresison. This mood gives the same conclusion as ^- Fresison-
. PeM
its strengthened form, Fesapo. Its schema is mis
PeM
MiS
SoP
.'.SoP
and it is represented by the diagram
PB
MS
diagram of
Frtsiton.
340
SYLLOGISMS.
Book IV,
Ch. in.
Example of
Fresison.
The ordi-
nary treat-
ment of
syllogism
tacitly
claims ex-
istence for
every term.
If all propo-
sitions im-
ply exist-
ence of S.and
affirmatives
of P, every
mood is
valid except
Camciies, and
its subaltern
Camenos, in
Fig. IV ;
As au example may be given : ' No noble man does mean
actions ; some wbo do mean actions succeed in life ; tberc-
fore, some who succeed in life are not noble.'
123. Syllogisms and Implications of Existence.
The great majority of writers on Logic do not examine how far
the legitimacy of the various syllogistic moods recognized as valid
is dependent upon the implications of existence contained in the
premises. As De Morgan says [Formal Logic, p. Ill) : •' Existence
"as objects, or existence as ideas, is tacitly claimed for the terms of
"every syllogism." But this assumption should not be taken on
trust ; for the inference in any syllogism is formally invalid, if the
conclusion contains an implication of existence which is not present
in the premises.
Our enquiry in § 89 led to the adoption of tlie view that all
propositions imply the existence of their subjects in the appropriate
sphere ; that in afBrmative propositions this involves the existence
of the predicate in the same sphere ; but that in negative proposi-
tions the predicate does not necessarily exist in the same sphere as
the subject, though it does in some sphere. We must, therefore,
see what effect this view will have upon the legitimacy of the
inferences in the moods of syllogism which are generally accepl.id
as valid because they break none of the syllojistic rules.
As one premise in every syllogism is affirmative, the existence of
the middle term is always guaranteed. The extremes are, there-
fore, the only terms we have to consider in this connexion. Now,
the inference contained in any syllogism is valid, if the conclusion
does not imply the existence of any term whose existence is not
guaranteed by the premises. If both premises are affirmative, the
existence of every term is assured ; consequently, all such syllo-
gisms are legitimate. If the conclusion is negative, it implies
the existence of 5, but not that of P. Hence, this conclusion can
legitimately follow from the premises only when they imjily the
existence of S. This imf)lication is present in every case except
when S is the predicate of a negative minor premise. 5 occupies
this position in Camenes alone of the nineteen recognized moods.
It follows that the conclusion of this mood is illegitimate as a
formal inference when stated categorically. The same ojojectiou
necessarily holds against the subaltern mood Camenos, which is the
weakened form of Camenes (c/l § 118). This problematic character
FIGURE AND MOOD.
341
of the conclusion of Camenes is apparent from the diagram for thab Book IV
mood [see § 122 (ii)]. Ch. III.
124. The Representation of Syllogisms by Diagrams.
The main purpose of applying diagrams to the represen-
tation of syllogisms is to make immediately obvious to the
eye, by means of geometrical figures, the relation established
between the extreme terms by the premises, and, thus, to
render easier the apprehension of the conclusion. The
scheme of diagrams adopted in this work has been thus
employed in the consideration of the valid moods of each
figure («ee §§ 119-22). It remains for us to examine how
far the other schemes of diagrams described in §§ 91-3 fulfil
the came nurpose.
* (i.) Euler's Diagrams. The diagrams most commonly
adopted by logicians are the circles of Euler. The
fundamental objections to the application of these dia-
grams to the fourfold scheme of propositions have been
already stated (see § 91). As every proposition — except E —
requires a plurality of diagrams for its complete representa-
tion, it is evident that the combination of the two premises
of a syllogism can only be fully set forth by a series of
diagrams, which must, by its very complexity, go far to pre-
vent that immediate obviousness which is an essential feature
of any diagrams which are to be an aid, and not a hindrance,
in apprehending the result of an argument. Simplicity has
often been attained by representing each proposition by only
one of the diagrams which express it ; but this is erroneous
and misleading. To show what is asserted by the premises,
every case must be set forth, as is done by Ueberweg and
Mr. Keynes. For instance, to represent Barbara by
The mean-
ing of
diagrams
should be
obvious.
Euler's
Circles art
most
commonly
employed,
but they can
only repre-
sent syllog-
isms by com-
jilex combi-
nations of
dia;,'r.'ims.
342
SYLLOGISMS.
Book TV.
Oh. III.
Representa-
tion of
Barbara by
Euler's
circles.
— as is done by Jevons {Frimer of Logic, p. 54) and by Mr.
Stock {Deductive Logic, p. 200) — is to suggest that 'some'
means 'some but not all,' and to ignore its absolute indefinite-
ness. Each premise of Barbara requires two diagrams to
express it ; thus —
M aP
SaM
To represent the conclusion, we must combine each of the
diagrams which express the major premise with each of
those setting forth the minor premise. This gives a com-
bination of four diagrams, and unless they are all considered,
we cannot be sure that the result given by those we have
examined will not be inconsistent with that yielded by those
we have omitted. Thus —
(i) and (a) give
(i) and (&) give
FIGURE AND MOOD.
343
(ii) and (a) give
(ii) and (b) give
Book IV.
Ch. III.
If we omit the consideration of M, the last three diagrams
reduce to one so far as S and P are concerned, and we are
left with the two diagrams which express S a P.
Similarly, if we combine E and A propositions as premises,
we require two diagrams to represent the syllogism, for A can
only be fully expressed by using the two diagrams given
above, and E by diagram V. on p. 217. There are, therefore,
two combinations, and these, moreover, will be lettered and
interpreted differently according as the A proposition is the
major or the minor premise.
If we take a syllogism involving a particular premise, the Representa-
representation becomes still more complex. To take Fcstino, ^^^tino by
for instance, the major premise requires only one diagram, E^ier's
but for the minor four are needed —
PeM
344
SYLLOGISMS.
Book IV.
Ch. in.
SiM
The combination of major and minor in every possible way
yields no less than eight diagrams —
(i) and (a) give
(i) and {b) give
(i) and (c) give
FIGURE AND MOOD.
345
Book IV.
Ch. III.
(i) and (d) give
From this group of figures, we have, by disregarding M,
to find the relation of S and P. On examination we find that
(1), (2), (3), (6) express the relation of entire mutual ex-
clusion between S and P ; that (4) and (7) represent the
partial coincidence and partial exclusion of those terms, and
(5) and (8) give the case in which P is entirely included in,
but does not form the whole of, S. We reach, then, the
three diagrams which express the proposition S o P. Of
these diagrams Mr. Stock (Deductive Logic, p. 202) gives (2),
(4), (6), (7), and omits the others. This writer has repre-
sented every valid mood by Euler's diagrams (ibid., pp. 200-
210), but in no case has he given all the Figures necessary to
a complete statement.
Probably the above examples are sufficient to convince the
reader that, though it may be a useful exercise of ingenuity
thus to represent the different moods of the syllogism yet,
the result will scarcely make the reasoning more immediately
self-evident. Indeed, the chief value of this system of
diagrams is the negative one of showing what premises will
346
8YLL0GTSMS.
BiVOK IV.
Ch. III.
Lambert's
diagrams
are less com-
plex than
Euler's.
Barbara in
Lambert's
diagrams.
Festino in
Lambert's
diagrams.
not yield a valid conclu.sion; when the diagrama are com-
patible with every possible relation between S and P — as in
the case of two negative premises [see §111 (ii)] — we know
that no conclusion can be drawn.
(ii.) Lamlert's Diagrams. Lambert's diagrams (c/. § 02) re-
present syllogisms with much less complexity than Euler's circles.
To take the same moods as examples : —
Dr. Venn's
diagrams
clearly pre-
sent syllog-
isms with
universal
premises.
Barbara
P-
M-
S-
FesHno
P-
M
S
Cetare in
Dr. Venn's
diagrams
But, though simple, these diagrams are apt to be misleading--
that for Barbara, for instance, suggests that S cannot be co-exten-
sive with P, though it does not imply this, as the lengths of the
lines do not indicate the relative extent of the classes they repre-
sent.
(iii.) Dr. Venn's Diagrams. Dr. Venn's system of diagrams, as
has been already stated (see § 93), is well suited to represent
universal, but not particular, propositions. Only the moods with
two universal premises can, therefore, be conveniently represented
in this way ; but for such moods the diagrams are very neat and
clear. As examples we wdl take Cesare and Darapti : —
Cesare
FIGURE AND MOOD.
347
The maior premise — P e M — asserts the non-existence of the Book IV.
r^Vi TTT
class P M (cf. % 89) ; we therefore shade it out in the diagram, "^n— 1_"-
Similarly, the minor premise — SaM — destroys S M, i.e., all of S
which is outside M. We see at a glance that iVo S is P.
Darapti « Darapti in
^ '" Dr. Venn s
diagrauis.
The major premise — M a P — destroys all M which is outside P,
and the minor premise— M a <9 — removes all of M which is outside
S. It is then immediately obvious that Some S is P. The only
way in which particular propositions can be distinguished from
universals is by drawing a line through each compartment to be
saved, and when this is done the conclusion is much less obvious.
Eor example, Festino would be represented by
Festino in
Dr. Venn's
diagrams.
A further objection which lies against the use both of these
diagrams and of those of Euler to represent the syllogism is that
they give no indication of Figure. This objection is of more
weight against Euler's system than against this of Dr. Venn ;
for the former was invented as an illustration of ordinary logic,
but the latter is based on an interpretation of the import of pro-
positions in which the distinction of subject and predicate no
longer exists, and, when that distinction is removed, Figure
necessarily disappears.
348
SYLLOGISMS,
Book IV.
ch. in.
Every mood
of categori-
cal syllo-
gism has its
correspoud-
ing form in
pure hypo-
thetical
syllogisms,
but only
those com-
posed of
universal
propositions
are import-
ant.
Hypotheti-
cal syllo-
gisms can bo
expressed in
categorical
form.
Examples of
p)ire hypo-
thetical syl-
logisms—
125, Figure and Mood in Pure Hypothetical and Dis-
junctive Syllogisms.
(i.) Pure Hypothetical Syllogisms. As hypothetical
propositions — including the modal particular forms — have
the same distinctions of quality and quantity as categorical
propositions {see § 78), it follows that they can be combined
into syllogisms in exactly the same number of ways. There
can, therefore, be forms of pure hypothetical syllogism cor-
responding to every figure and mood of categorical syllogism,
and governed by the same rules [c/". § 112 (i.)]. But, as the
universal hypothetical propositions are the only ones of
much importance [see § 78 (ii.)], it follows that the important
pure hypothetical syllogisms are those composed of such
propositions ; and of these, those which coiTCspond in form
to Barbara are the most useful, and the most frequently
employed. Moreover, as the whole force of syllogistic in-
ference consists in the necessity with which the conclusion
follows from the premises (c/, § 107), and as this necessity
is not affected by the hypothetical or categorical form in
which those premises are expressed, it follows that such
hypothetical premises can always be reduced to the cate-
gorical form without affecting the validity of the inference.
This reduction is most conveniently made when the quantified
— or conditional — forms of the hypothetical are employed,
as they correspond most closely with the quantified form in
which the propositions composing a categorical syllogism are
usually written. Of course, when this is done, though the
inference is equally necessary, the abstract and necessary
character of the conclusion is more or less hidden.
It will be sufficient to give an example of a pure hypo-
thetical syllogism in each figure expressing each of our
propositions in the quantified denotative form.
1. Corre-
pponrllng
to Jiarbara,
Figure I. Corresponding to Barbara we have the form
If any S is X, that S is P,
If any S is M, that S is X,
.♦. If any S is M, that S is P.
FIGURE AND MOOD.
349
As a material example maj be given: 'If any person is
selfish, he is unhappy ; if any child is spoilt, that child ia
selfish ; therefore, if any child is spoilt, he is unhappy.'
Figure II. Corresponding to Cesare is the foim
If any S is P, then never is it X,
If any S is M, then always it is X,
.•. If any S is M, then never is it P,
An example is : 'If any act is done from a sense of duty,
it is never formally wrong ; if any act is done from purely
selfish motives, it is always formally wrong ; therefore, if
any act is done from purely selfish motives, it is not done
from a sense of duty.'
Figure III. Corresponding to Bocardo is the form
If an S is X, then sometimes it is not P,
If any S is X, then always it is M,
.'. If an S is M, then sometimes it is not P.
We may give as an example : ' If a war is just, it is some-
times not successful ; if any war is just, it is always waged
in defence of some right ; therefore, if a war is waged in
defence of some right, it is sometimes not successful.' Here
it is evident nothing is lost by transferring the syllogism to
the categorical form, and saying : ' Some just wars are not
successful ; all just wars are waged in defence of some
right ; therefore, some wars waged in defence of a right are
not successful.' This has exactly the same force as the
conditional form, for the latter does not imply that the want
of success is a necessary consequence of the character of the
war. But, in the examples with universal conclusions, it is
evident there is such a dependence of consequent upon
antecedent, which is lost if the syllogism be transferred to
the categorical form.
Figure IV. Corresponding to Dimaris is the form
If an S is P, it is sometimes X,
If any S is X, it is always M,
.', If an S is M, it is sometimes P.
Eooiv IV
Ch. 111.
2. Cor re.
Bponding
to Cesare.
3. Corre-
sponding
to Bocardo.
i. 0 o r r « -
spending
to DimariA
350
SYLLOGISMS,
Book IV.
Ch. III.
This may be illustrated by : 'If the currency of a country
consists of inconvertible bank notes, it is sometimes depre-
ciated ; if the currency of any country is depreciated, it
causes an artificial inflation of prices ; therefore, if the
currency of a country causes an artificial inflation of prices,
it sometimes consists of inconvertible bank notes.' Here,
again, it is evident that the antecedent does not state the
necessary ground or reason for the consequent, and nothing
is lost by reducing the whole argument to the categorical
form.
Pure diP-
jiiuotive
syllogisms
correspond
to the
affirmative
moods of
categorical
syllogisms.
One of tlie
alternatives
in the minor
must nega-
tive one of
those in the
major.
(ii.) Pure Disjunctive Syllogisms. The possibility of
syllogisms consisting entirely of disjuncti\e propositions
has not been usually considered by logicians. Indeed, it is
only with certain limitations that such syllogisms are pos-
sible at all. They can, to begin with, only be syllogisms
with an affirmative conclusion, as no disjunctive proposition
can be negative [see § 81 (i.)]. Only the affirmative moods
are, therefore, possible, and, of these, that corresponding to
Barbara is the only one of any importance. Further, we
only secure a middle term when one of the alternatives in
the minor premise negatives one of those in the major pre-
mise. From
S is either P or Q
S is either P or R
no conclusion can be drawn, except that S is either P or Q
or R which simply sums up the premises. But from
8 is either P or Q
S is either P or R
we can draw the conclusion S is either Q or R, This will,
])erhaps, be more clearly seen if each premise is expressed as
a hypothetical proposition. "We can write the premises in
the form
If Sis Pit is Q_
If S is Wit is P
HGURE AND MOOD. 351
whence it follows that If S is R it is Q, which expresses the Book TV.
disjunctive S is either Q or R. Such syllogisms are, however, Ch^i.
of infrequent occurrence. As the order of the alternatives
is indifferent it will be seen that distinctions of figure have
bare no proper application.
CHAPTER IV.
I? EDUCTION OF SYLLOGISMS.
^ch'^iv^" 126 Function of Reduction.
jied^on— Reduction is the process by which a given syllogistic
theKgm-e argument is expressed in some other Figure or Mood.
"yiiog^m. ^ Reduction has generally been confined to expressing in the
First Figure arguments given in the other Figures, and
though the process may be applied with equal ease to chang-
Reductinn ing reasoniugs from any one figure to any other which con-
themostim- taius the required conclusion, and even from one mood to
portant. another in the same figure [c/. § 128 (i) (c) (3)], yet these
processes are of no great uti'ity.
There has been a good deal of dispute as to the place
Reduction, in this narrower sense, should fill in syllogistic
theory. The view taken on this point will depend upon the
If the axiom principle adopted as the basis of the syllogism. Aristotle
for Fi^' 1 is and the scholastic logicians, who regarded the First as the
regarded as 7 .
the prin- Only perfect figure, and the dictum de omni et nullo as the
syiiog"lstic basis of all syllogistic inference, taught that reduction to
inference. Figure I is absolutely necessary to establish the validity of
Reductionis ^ „ • , ^ a • Ix. t. a r K unr-x/ m
necessary, any syllogism not expressed in that figure [see § 110 (ii) (a)j.
The same view is held by Kant and all other logicians who
adopt a principle directly applicable to the First Figure as
Reduction i ^^® basis of all syllogistic reasoning [see § 110 (ii)]. On
notneces- the other hand, those who hold, with Lambert, that each
vaiidi'ty.'^if ° figure rests on its own dictum (see § 114) regard reduction
each figure j^g both unnatural and unnecessary. Tbe figure of a syl-
h;is its own . .
axiom, logism, they justly argue, is due to the nature of the pro-
KEULCTION OF SYLLOGISMS. 353
positions which form its premises, so that some arguments Book IV.
fall most naturally into figures other than the First, and to Ch^^.
reduce them to that form is to substitute an awkward and
unnatural expression for a simple and natural one. More-
over, the validity of such arguments is as immediately
obvious as is that of the moods of the First Figure, and,
consequently, Keduction is unnecessary. The view here nor if all
adopted — that all syllogistic reasoning rests ultimately on rest*on ^e
the fundamental principles of thought, of which the dicta of Laws of
the different figures are mere limited expressions («ee §§ 109,
110, 114) — leads to the same conclusions. As these principles
apply equally to syllogisms in all the figures, Reduction, as
a proof of validity, is superfluous. But it does not follow but it showo
that Reduction has no legitimate place in syllogistic theory. ^^^ ^^iwis^
It is true that the reasoning does not become more cogent tic process,
by being reduced to the First Figure, but its distinctive
character is more immediately obvious in that figure than in
any other [see § 115 (i)]. Reduction thus makes evident
the essential unity of all forms of syllogistic inference, and
systematizes the theory of syllogism by showing that all
the various moods are, at bottom, expressions of but one
principle.
127. Explanation of the Mnemonic Lines.
The primary intention of the mnemonic lines given in The primary
§ 116 (ii) is to indicate the processes by which syllogisms of'thT^
in figures other than the first can be reduced to that figure. {^g^J^" to"
This is most ingeniously done by means of the consonants indicate the
employed. For convenience of reference we will here repeat eduction to
the lines :— ^ 'S"^« ^
Barbara, Celarent, Darii, Ferioqne prioris :
Cesare, Camestres, Festino, Baroco [or Faksoko], secundae :
Tertia, Darapti, Disamis, Datisi, Felapton,
Bocardo [or Doksamoski, Ferison, habet : Quarta insuper
addit
Bramantip, Qimenes, Dimaris, Fesapo, Fresison. 03
LOG. 1
354
SYLLOGISMS.
Book IV
Cb. IV
Explanation
of the
mnemonics :
s — simple
conversion.
p — conTcr-
sion per ae-
cideni.
m— transpose
premises.
fc — obversion.
its— contra-
position.
tt — obverted
conversion.
e — indirect
reduction.
Each letter
refers to the
preceding
proposition.
Transposi-
tion of pre-
mises neces-
sitates con-
version of
new conclu-
Bion.
The two additional names, given in square brackets, refer
to the direct process of reduction, whilst Baroco and Bocardo
indicate the indirect process adopted by the scholastic lo-
gicians.
Some writers replace the c in Baroco and Bocardo by Tc,
but this letter is required for the two additional mnemonics
for those moods, and cannot, therefore, be used in the older
ones without confusion, as it would then denote two entirely
different processes.
The initial letters of the moods in the First Figure are
the first four consonants. In the other figures : —
s denotes simple conversion of the preceding proposi-
tion.
p indicates that the preceding proposition is to be con-
verted per accidens.
m signifies metathesis, or transposition, of the pre-
mises.
h denotes obversion of the preceding proposition.
ks indicates obversion followed by conversion — i.«., con-
traposition— of the preceding proposition.
sk signifies that the simple converse of the preceding
proposition is to be obverted.
c shows that the syllogism is to be reduced indirectly
(conversio syllogismi, or change of the syllogism).
When one of these letters occurs in the middle of a word,
one of the premises of the oi-iginal syllogism is to undergo
the process of eduction indicated. Now, when one of the
changes indicated is the transposition of premises, the posi-
tion of the extreme terms is reversed, and the major term of
the original syllogism becomes the minor term of the new.
The conclusion must, therefore, be converted to bring it to
the original form. Thus every word in which m occurs ends
in s, p, or sk, and these letters indicate that the conclusion of
the new syllogism is to be converted. It will be noticed that
no other significant letter ends a word. The only meaningless
letters are thus seen to be r, t, Z, 7i, and h and d when they are
not initial. Several attempts so to change the forms of the
REDUCTION OF SYLLOGISMS. 355
words as to omit meaningless letters, and to employ a dis- Book IV.
tinctive letter for each mood have been made, but none of ^^- ^^•
them is likely to replace the traditional forms.
128. Kinds of Reductions.
It was indicated in the last secton that there are two There are
kinds of Eeduction — Direct and Indirect, the latter being ReducUon—
usually restricted to the moods Baroco and Bocardo.
(i.) Direct or Ostensive Reduction. Reduction is direct i. mrect-
ichen the original conclusion is deduced from premises derived TOnchision
from those given. The original premises are changed by con- 'f deduced
version, transposition, or obversion. premises
(a) Conversion.
(1) The moods Cesare, Festi?io, Datisi, Ferison, and ' sion,
Fresison, are reduced to the First Figure by
simply converting one, or both, of the premises.
For example, Cesare (Fig. II) becomes Celarent
in Figure 1 : —
P e M M e P
SaM SaM
changed by :
(") Conver-
.-.SeP r.SeP
and Fresison (Fig. IV) become Ferio (Fig. I) : —
PeM MeP
MiS SiM
.-.SoP .-.SoP
A comparison of the diagrams of each of these moods {see
§§ 119-22) will show that those in Figure III are absolutely
identical with those in Fig. I, and that the others differ only
in assuring the existence of P ; for the existence of M is in
no case doubtful in the syllogism, as it is implied in the
minor premise.
(2) The moods Darapti and Felapton are reduced by
converting the minor premise per accidens. Thus
DarajJti (Fig. Ill) becomes Darii (Fig. I) : —
MaP MaP
MaS SiM
.'. 8 i P .-. S i P
856 SYLLOGISMS.
Book IV. A comparison of diagrams again shows the equivalence of
Ch^. ttjgge moods as far as the relation of S and P is concerned,
the presence of the possible class SM in the First Figure
being quite immaterial.
(3) Fesapo (Fig. IV) is reduced to Ferio (Fig. I) by the
simple conversion of its major, and the conversion
per accidens of its minor premise : —
PeM MeP
MaS S % M
.:SoP .-.SoP
The diagrams again illustrate the equivalence of the
moods, the guarantee of the existence of P given in that for
Fesapo not affecting the relation of 5 and P.
(5) Transpo. ^j^ Transposition of premises. This, as has been seen,
involves conversion of the new conclusion.
(1) The moods Bramantip, Camenes, and Dimaris, all in
Figure IV, reduce to the First Figure by merely
transposing the premises. Thus Bramantip be-
comes Barbara : —
PaM ^,.,^^ ^__^ MaS
MaS S^XCT PaM
.-.SiP .\PaS
,; (byConv.)SiP
A comparison of diagrams shows that they are identical if
S and P are transposed in the premises — a transposition
necessitated by the change in the order of the premises.
(2) Camestres and Disamis are reduced to the First
Figure by transposing one premise with the
simple converse of the other. Thus, Disamia
(Fig. Ill) becomes Darii (Fig. I) : —
MiP ^-^....^^^.^^ MaS
MaS ..--^^^'""■--^ PiM
,\SiP r.PiS
.«. (by conT.) S i P
REDUCTION OF SYLLOGISMS.
357
The diagrams again show the equivalence of the moods
when S and P are transposed in the premises.
(c) Obversion.
(1) The mnemonic Faksoho indicates that Baroco (Fig.
II) may be reduced to Ferio (Fig. I) by contra-
positing the major premise and obverting the
minor. Thus : —
PaM
SoM
.SoP
M tP_
SiM
SoP
A comparison of the diagrams shows that the SM in that
for Baroco bears the same relation to the P as the SM does in
that for Ferio.
(2) Similarly £)t»^samos^ signifies that Bocardo (Fig. Ill)
may be reduced to Darii (Fig. I) by contrapositing
the major premise and making it the minor, and
then obverting the simple converse of the new
conclusion. Thus : —
M oP
MaS
MaS
PiM
SoP
,:PiS
.'. (by conv.) S i P
.•,(byobv.)So P
A comparison of diagrams shows that P bears the same
relation to S in that for Bocardo, as S does for P in that for
Darii.
(3) By the use of obversion, any mood can be reduced to
a mood of similar quantity, but opposite quality,
in the same figure. For example, Celarent may
be reduced to Barbara (Fig. I) by obverting the
major premise : —
MeP Map
SaM SaM
Book IV.
Ch. IV.
(c) Obver-
sion.
•.SeP
SaP
358
8YLL00TSMS.
Book IV.
Ch. IV.
3. Indirect—
proves a con
elusion to be
legitimate
by showing
that its cou-
Iradictory is
lot.
ilethod of
Indirect
Reduction.
fndlrect
lieduction
uf Baroco.
and Disamis to Bocardo (Fig. Ill) by obverting
the major premise : —
MiP
MaS
MoP
MaS
SiP
.'.SoP
but such reductions serve no useful purpose, as
the difference between affirmation and negation
must always remain fundamental {cf. § 70J.
(ii.) Indirect Reduction. Reduction is indirect tvhcn a
new syllogism is formed which establishes the validity of the
original conclusion by showing the illegitimacy of its Contra-
dictory. This method is also called Reductio ad impossibile,
but that name is not so appropriate as Reductio j^er impossibile
or Reductio ad absurdum. It can be applied to any mood,
though in practice it is usually confined to Baroco and
Bocardo ; and this application is the only one contemplated
in the original mnemonics. The method is founded on the
Principle of Contradiction (s^ee § 18). When a conclusion
is legitimately deduced from two given premises, it is for-
mally true ; when it is not so deduced from them, it is
formally false. In judging of the validity of an inference,
this formal truth, or self-consistency, is all we are concerned
with. Now, if the conclusion is formally false, its contra-
dictory must be formally true (see § 18). If this con-
tradictory is combined with one of the original premises, a
new syllogism is formed whose conclusion will either be
identical with, or will contradict, the remaining original
premise. If it contradicts it, it proves that the contradictory
of the original conclusion was formally false, that is, that
conclusion was formally true. Thus the validity of the
original syllogism is established.
For example, Baroco is proved valid by a syllogism in
Barbara. For if the conclusion, S a P,is formally false, then
its contradictory, S a P, is formally true, i.e., is an inference
from the two premises P a M, S o M. Replacing the premise
followed by c by this contradictory of the original con-
REDUCTION OF SYLLOGISMS. 359
elusion, we get the following ayllogism in Barbara, with P for Book IV.
its middle term : — J — '
PaM PaM
SoM ^ V, SaP
.:SoP -^ > .-.SaM
Thus, if 5 a P is formally true so is 5 a M. But SaM
contradicts SoM which is one of the original premises, and
is, therefore, formally false. Hence, S a P is also formally
false ; i.e., the original conclusion, S o P,\s formally true, and
Baroco is a valid mood.
Similarly with Bocardo. If the conclusion, S o P, is indirect
formally false, its contradictory, S a P, la tormally true, of Bocardo.
Replacing the premise followed by c by this proposition,
we get a syllogism in Barbara, with S for its middle term : —
A/oP ^ ^ SaP
MaS ^^^^ MaS
i-
S o P ^ ■^ .-.MaP
But MaP contradicts the original major premise M o P.
Therefore, M a P is formally false, and this entails the formal
falsity of SaP. Therefore, the original conclusion, S o P,
is formally true, and Bocardo is a valid mood.
This process, which was adopted by the scholastic logi-
cians because of their dislike of negative terms, is, cer-
tainly, very cumbrous, and as both the moods to which it is
commonly applied can be reduced much more simply by the
direct method, it might well be banished from Logic. It
should be noted that this indirect process is not reduction in
the same sense as the direct method is ; in tlie latter, the
new syllogism is the same argument as the old, in the former,
it is an entirely different argument.
129. Reductions and Implications of Existence.
(1) On the view we have adopted (see § 89) that every pro- If every
position implies the existence of its subject, the simple conversion proposition
of E, and, consequently, the contraposition of A, which involves it, existence of
are invahd processes. Therefore, no reduction which involves s,
360 SYLLOGISMS.
Book IV. either of these processes is legitimate, unless the existence of the
Ch. IV. predicate of the E proposition which has to be converted is implied
but nega- ^ t'^e other premise. The simple conversion of E is involved in
tives do not the reduction of the moods Cesare, Camestres, Festino, in Figure
of p, the re- II» and of Camenes, Fesapo, and Fresison in Figure IV. In every
auction of oa^gg jjj ^}jigjj tjjQ ■£ proposition to be converted is a premise, its
Invalid. predicate Is M, whose existence is implied in the other premise. In
Camestres the conclusion of the new syllogism has also to be con-
verted, but its predicate is S, which is the subject of the original
minor premise, and whose existence is, therefore, assured. In
Camenes, however, S is the predicate of the original minor pre-
mise, which is negative ; its existence, therefore, is not implied,
and, consequently, the simple conversion of the new conclusion,
P e S, ia invalid. The reduction of Camenes is, therefore, an
illegitimate process. The contraposition of A is only employed
in the direct reduction of Baroco {Faksoko). Here, the obversion
of the minor premise shows that the existence of M is implied in
that premise, and so justifies the contraposition of the major,
which involves the simple conversion of P e M. The direct reduc-
tion of Baroco is, therefore, legitimate. The indirect reduction of
Baroco and Bocardo is also valid, as, on this view, the doctrine
of contradiction holds good. Our examination, then, confirms the
conclusion we reached in § 123, that, on this theory of the exist-
ential import of proposition, Camenes alone of the recognized
moods is invalid.
130. Reduction of Pure Hypothetical Syllogisms.
*othJticai^' '^^^ validity of the reduction of any syllogism depends
syllogisms Upon the legitimacy of the processes of immediate inference
du^ed^stoi- involved. With hypothetical propositions, including the
lariy to modal particulars, all these processes are valid (see 8 105).
categoncals. j .i - , xi ^- i i, • \ t i
and, therefore, pure hypothetical syllogisms can be reduced
in exactly the same way as categorical syllogisms. For
example, the pure hypothetical syllogism corresponding to
Examples— Cesare (Fig. II) {cf. § 125) is reduced to the form in Figure I
agreeing with Celarent, by simply converting th', major
premise, so that we get : —
BEDITCTION OF SYLLOGISMS. 361
(oonv. of orig. major) If any S is X, then never is it P, Book IV.
If any S is M, then always it is X, J "
.*. If any S is M, then never is it P. iifg with'^^^
Cesare.
The form corresponding to Bocardo (Fig. Ill) (c/. § \2b) form agree-
is directly reduced to that agreeing with Darii by contra- ^Bocwrdo.
positing the major premise and transposing the premises.
The new conclasion has then to be converted, and the
converse obverted. We thus get : —
(orig. minor) If any S is X, then always it is M,
(contrap. of orig. major) If an S is P, then sometimes it is X,
If an S is P, then sometimes it is M ;
,'. (by conv.) If an S is A/, then sometimes it is P,
,: (by obv.) If an S is A/, then sometimes it is not P.
And the form corresponding to Dimaria (Fig. lY) {cf. Form agre»
§ 126) is reduced to that agreeing with Darii by transposing ^samru.
the premises and converting the conclusion. Thus we
get:—
(orig. minor) If any S is X, it is always M,
(orig. major) If an S is P, it is sometimes X,
.'. If an S is P, it is sometimes M ;
.: (by conv.) If an S is A/, it is sometimes P.
CHAPTER Y.
Book IV,
Ch. V.
The hypo-
thetical pre-
mise is the
major, the
categorical
is the minor.
The charac-
ter of syllo-
gistic infer-
ence is more
evident
when the
majorisenu-
merative
than when
it is abstract
in form.
MIXED SYLLOGISMS.
131. Mixed Hypothetical Syllogisms.
When one of the premises of a syllogism is a hypothetical
and the other a categorical, proposition, the former is called
the major, as it furnishes the ground of the inference ;
whilst the latter is the minor, as it states a case in which
the major is applicable. The inference conforms to the
same principles whether the major premise is stated in the
fundamental abstract connotative or in the derived con-
crete enumerative form, which we have called conditional
{see § 76). But in the latter case the fundamental char-
acter of syllogistic inference — the application of a general
principle to a special case — is perhaps more plainly seen
than in the former. For, when the major premise is a con-
ditional proposition, it lays down, in so many words, a
general dependence of one phenomenon upon another,
though it makes no assertion as to whether or not either
of these phenomena occurs in any special instance. The
categorical minor affirms, or denies, the occurrence of one
of these phenomena in some special case, and thus enables
us, by applying the general rule given in the major, to con-
clude as to the occurrence, or non-occurrence, of the other
phenomenon in that same case. When, however, the major
premise is stated in the abstract hypothetical form making
explicit the ground for the connexion of content — // S is M
it is P — then the application to reality is not made through
some particular instance of S, but must be mediated by the
MIXED SYLLOGISMS.
363
ascertained nature of S itself ; in other words the minor
premise must be the generic judgment 5 is A/, and the con-
clusion is the generic judgment S is P.
(i.) Basis of Mixed Syllogistic Reasoning from a
Hypothetical major premise. As the inference in these
syllogisms is as purely formal as when both the premises are
categorical, it must ultimately rest on the fundamental prin-
ciples of thought (see §§ 17-20). There is a very distinct
reference to the Principle of Sufficient Reason (see § 20),
which may indeed be regarded as the ax'ioma medium of such
syllogisms. This principle of thought and necessary postu-
late of knowledge compels us to grant the conclusion which
follows from any data we have accepted. Applied to syllo-
gisms with a hypothetical major premise this means that, if
in the minor we assert the antecedent of the major to be true
in fact, we must accept, as a conclusion, the truth of the
consequent. But a stricter examination shows that this is
an application of the Principle of Identity. On the other
hand, if, in the minor, we deny the consequent of the major,
we must, in the conclusion, reject the antecedent. For, by
the Principle of Excluded Middle, the antecedent must be
either true or false, and, if it were true, the consequent
would be true ; and by the Principle of Contradiction,
neither the antecedent nor the consequent can be both true
and false ; therefore, the denial of the consequent neces-
sitates that of the antecedent.
(ii.) Determination of Valid Moods. It is thus seen
that the assertion of the truth of the antecedent of a hypo-
thetical proposition justifies the assertion of the truth of
the consequent, and the denial of the consequent necessitates
the denial of the antecedent. But the same consequent may
result from more than one antecedent ; and, therefore, the
denial of the given antecedent will not justify the denial of
the consequent, nor will the assertion of the consequent
warrant that of the given antecedent. For example, though
if a man is shot through the heart he dies, yet men also die
from other causes. The denial that he is shot through the
Book IV
Ch. V.
These infer-
ences rest
ultimately
upon the
Laws of
Thought,
with the
Principle of
SuflBcient
Reason as
an axioma
medium.
In fere no*
follows fnjm
affirniHtioa
of A, or de-
nial of 0.
As C may
follow from
other ante-
cedents be-
sides A, no
inference
follows from
denial of A,
or affirma-
tion of 0.
364
SYLLOGISMS.
Book IV.
Oh. V.
heart will not, therefore, warrant the denial of his death ;
nor will the assertion of his death necessitate the statement
that it was due to this particular cause. We may express
symbolically the various antecedents which lead to the same
consequent, using the most general formula of the hypo-
thetical proposition, as in this respect it does not matter
whether the consequent has the same subject as the ante-
cedent or not (c/. § 76) : —
To deny A Is
sualogous
to Illicit
Major, and
to affirm 0
corresponds
to Undis-
tributed
Middle.
JfA,thenG.
If X, then G.
If Y, then G.
If Z, then G.
Here, it is evident that if we deny A, we still leave open
several possibilities of the occurrence of 0, for either X, Y,
or Z, may be true ; and if we assert C, though we, thereby,
assert one of its possible antecedents we cannot tell which
one ; nor have we, indeed, in either case, any security that
all the possible antecedents of C are known to us. If, in-
deed, A is the only possible antecedent of 0, its denial
is a material justification for the rejection of 0, and
the affirmation of 0 is a material warranty for that of A.
But these material conditions do not hold in all cases, and we
are not, therefore, justified in assuming them in any ; in
formal inference we can deal only with that which holds
universally.
Now, as 0 may follow from several other antecedents
besides A, it corresponds to an undistributed term. When,
however, the denial of C is deduced from the denial of A, 0
is used universally in the conclusion. Again, when 0 is
affirmed, it is affirmed in one case only out of several possible
ones ; to posit A as a result of such affirmation of 0 would be
to disregard this. Thus, the fallacy of denying the ante-
cedent is analogous to an illicit process of the major term,
and that of affirming the consequent bears a similar resem-
blance to an undistributed middle. In each, the unwar-
ranted assumption is made, that the major premise embraces
every case in which the consequent can be true.
MIXED SYLLOGISMS.
365
There are thus two, and only two, valid processes of
syllogistic inference from a hypothetical major premise.
They are covered by the canon : —
To posit the antecedent is to posit the consequent ; to sublate
the consequent is to sublate the antecedent.
In the former case the syllogism is said to be Constructive,
or in the Modus Ponens ; in the latter case, Destructive, or in
the Modus Tollens.
When, in such a syllogism, the major premise is a negative
hypothetical, it is more convenient, and equally natural, to
regard the negation as belonging to the consequent [see § 78
(i) ad fin."]. The major may, then, take any one of four
forms, as both the antecedent and the consequent may be
either affirmative or negative. There can, therefore, be four
forms both of the Modus Ponens and of the Modus Tollens.
But it must be remembered that these names have no refer-
ence to the quality either of the minor premise or of the con-
clusion, but simply to whether the minor enables us, in the
conclusion, to posit the consequent, or to deny the ante-
cedent, of the major, whatever that antecedent or consequent
may be. To each of these varieties of the two moods
separate names are given by German logicians. These
names, however, are based on the quality of the minor
premise and the conclusion — ponens marking affirmative,
and tollens negative, quality — and thus the same name may
denote either a Modus Ponens or a Modus Tollens. Still using
the one general formula to denote all forms of hypothetical
propositions, these varieties of the two moods are thus
expressed symbolically : — •
Book IV.
Ch. V.
In Modus
Ponens, by
positing A
we posit 0 ;
in Modus
ToUeiis, by
sublating 0
yro sublate
A;
the former
is a Construc-
tive, the
latter a
Destructive
Syllogism.
Both A and
C may be
either
affirmative
or negative
{A) Modus Ponens.
(1) Modus ponendo ponens.
If A then 0,
A,
.••0.
Different
forms of tha
Modtu
Ponent.
366
SYLLOOISMS.
Book IY.
Ch. V.
Different
forms of the
Modus
TolUm.
(2) Modus ponendo tollena.
If A, then not 0,
A,
.-. Not 0.
(3) Modus tollendo ponens.
If not A, then 0
Not A,
,•. 0.
(4) Modus tollendo tollens.
If not A, then not 0,
Not A,
.-. Notll
(B) Modus Tollens.
(1) Modus tollendo tollens
If A, then 0,
Note,
.'. Not A.
(2) Modus ponendo tollens.
If A, then not C,
C,
.-. Not A.
(3) Modus tollendo j^onens.
If not A, then C,
NotO,
771..
(4) Modus ponendo ponem.
If not A, then not 0,
G,
77K.
Th&Modui The identity of the names of the subordinate moods
t'tTrndut points out that the Modus Ponens and the Modus Tollens are,
Tollens ATQ a,t bottom, identical. On comparing the maiors of the
mutually ' . . , ,
convertible, moods with the same name it is seen that they are the
MIXED SYLLOGISMS.
367
obverted contrapositives of each other, with the antecedent
and consequent transposed. It follows that, if we obvert
the contrapositive of the major of any form of the Modus
Ponens, we shall get the corresponding form of the Modus
Tollens ; and that the latter can be similarly reduced to the
former. For example, if we take the modus poneivdo ^ionens
of the Modus Ponens
If A, then 0,
Book IV
Ch. V.
.*. C,
and obvert the contrapositive of its major, we get
If not C, then not A,
A,
•• 0,
which is the modus ponendo ponens of the Modus Tollens.
Similarly, if we take the modus ponendo tollens of the Modus
Tollens,
If A, then not C,
C,
.'. Not A,
by obverting the contrapositive of its major we get
If C, then not A,
• _C^
.*. Not A,
which is the corresponding form of the Modus Ponens.
It must be borne in mind that as, in a hypothetical propo-
sition when it is stated in the conditional or enumerative
form, the subject of both the antecedent and the consequent
are quantified, the minor may sublate the consequent of the
major by affirming either its contradictory or its contrary ;
in each case, however, we are only justified in asserting the
contradictoi-y of the antecedent of the major as oar conclu-
eion. Thus, from the premises *If all prophets spoke the
truth, some would be believed ; but none are believed ' we
In Modus'
Ponens the
conclusion
must be thj
contradic-
tory of A.
368
SYLLOGISMS.
Book IV. are only justified in inferring that ' some prophets do not
^' ^' speak the truth,' not that ' no prophets do so.'
Examples of (iii.) Exajuples. We wUl now give some material examples
Set leal ^^'^ of the various forms of mixed hypothetieal syllogisms : —
Byllogiaiaa.
(A) Modus Ponens.
(1) Modus ponendo pone7i8. ii any country increases in
wealth, it increases in power ; England is in-
creasing in wealth ; therefore, England is increas-
ing in power.
(2) Modus ponendo tollens. If any import duty is
imposed simply for revenue purposes, that duty
is not protective ; English import duties are
imposed simply for purposes of revenue ; there-
fore, English import duties are not protective.
(3) Modus tollendo ponens. If any swan is not white, it
is black ; Australian swans are not white ; there-
fore, Australian swans are black.
(4) Modus tollendo tollens. If any war is not defensive,
it is not just ; the wars waged by Napoleon the
Great were not defensive ; therefore, those wars
were not just.
iB) Modus Tollens.
(1) Modus tollendo tolleno. If any country is civilized it
has a population amongst whom education ia
general ; the people of Russia are not generally
educated ; therefore, Russia is not a civilized
country.
(2) Modus ponendo tollens. If any social institution is
justifiable, it oppresses no class of the commu-
nity ; slavery does oppress a class ; therefore,
slavery is not a justifiable social institution.
MIXED SVLLOGISMS. 369
(3) Modus tollendo ponens. If any railway is not re- Book IV.
quired in the district through which it runs, it is *^^-
a financial failure ; the great English lines are not
financial failures ; therefore, they are required in
the districts through which they run.
(4) Modus ponendo ponens. If any country has no
capital invested abroad, its imports will not
exceed its exports ; England's imports do exceed
her exports ; therefore, England has capital in-
vested abroad.
A few examples may be added of similar inferences when
the hypothetical major has not been reduced to the funda-
mental form with the same subject to both antecedent and
consequent.
{A) Modus Ponens.
(1) Modus ponendo ponens. If all men are fallible, all
philosophers are fallible ; but all men are fallible ;
therefore, all philosophers are fallible.
(2) Modus ponendo tollens. If all our acts are within
our own control, no vice is involuntary ; all our
acts are within our own control ; therefore, no
vice is involuntary,
(3) Modus tollendo ponens. If vindictiveness is not a
justifiable emotion, all punishment should be
simply preventive ; vindictiveness cannot be jus-
tified ; therefore, all punishment should be simply
preventive.
(4) Modus tollendo tollens. If seeking his own pleasure
is not man's chief end, the Egoist is not truly
moral ; the seeking his own pleasure is not man's
chief end ; therefore, the Egoist is not truly
moral.
liOG. I. 24
370
SYLLOGISMS.
Book IV.
Ch. V.
Mixed hypo-
thetical
syllogisms
can be ex-
pressed as
categoricals;
Modus
Ponens in
Kg. I;
Hodus
ToUens in
Pig. II.
(B) Modus ToUens.
(1) Modus toUendo tollens. If all prophets spoke the
truth, Bome would be believed ; but none are
believed ; therefore, some do not speak the
truth.
(2) Modus ponendo tollens. If some of a man's deliberate
acts are wholly determined by circumstances, he
is not morally responsible for them ; but a man is
morally responsible for all his deliberate acts ;
therefore, no such acts are wholly determined by
circumstances.
(3) Modus tollendo ponens. If no men were mad, lunatic
asylums would be useless ; but tbey are not use-
less ; therefore, some men are mad.
(4) Modus ponendo ponens. If the earth did not rotate
on its axis, there would be no alternation of day
and night ; there is such alternation ; therefore,
the earth does rotate on its axis.
(iv.) Eeduction to Categorical Form. A hypothetical
proposition cannot be satisfactorily reduced to the cate-
gorical form, as it includes an element of doubt as to the
concrete existence of its elements which would disappear
in such reduction (see § 77). But in a mixed hypothetical
syllogism this element of doubt is removed by the categorical
character of both the minor premise and the conclusion.
The mediate inference of the syllogism will, therefore, be
exhibited without material alteration, if we express the
major in the form — The case of A heing true is the case of
0 being true. The minor of the Modus Ponens may then be
written — This is the case of A being true, and that of the
Modus Tollens may take the form — This is the case of 0 being
false. The Modus Ponens of these syllogisms is then seen to
be in the First Figure, and the Modus Tollens in the Second.
Such reduction is, however, awkward ; and its only value is
to give a fresh proof of the fundamental unity of the syllo-
gistic process in whatever form it may be expressed.
MIXED SYLLOGISMS.
371
132. Mixed Disjunctive Syllogisms.
A Mixed Disjunctive Syllogism, in the strict sense of
the term, is one in which the inference is drawn from
the disjunctive form of the major premise.
(i.) Basis of syllogistic inference from a disjunctive
major premise. If two alternatives are given in the major
premise, the denial of one of them in the minor justifies the
assertion of the other in the conclusion. Such an inference
is purely formal, and is, therefore, based on the fundamental
principles of thought (see §§ 17-19). Though the common
formula for a disjunctive proposition ia 5 is either P or Q, yet
even here the alternation is, at bottom, between the two
propositions S is P and S is Q. And an alternation may be
equally well asserted between two propositions with different
subjects, as Either S is P or M is Q. If, then, we denote the
alternative propositions by X and Y, we shall have the
simple formula for disjunctive propositions — Either X or Y,
which is more comprehensive than the customary S is either
P or Q. Now, if the major premise is the disjunctive propo-
sition Either X or Y, we know that one, at least, of these
alternatives must be true, i.e., not X ensures Y. If the
minor premise denies X, it must, by the principle of
Excluded Middle (see § 19), affirm not X, and this, by the
Principle of Identity {see § 17) justifies the affirmation of Y.
But the alternatives may be both negative — Either not X or
not Y, and this may be written Not both X and Y. Here
again, if one of the alternatives is false, the other must be
true ; i.e., X ensures not Y. If, then, the minor posits X, it
must, by the Principle of Contradiction (see § 18), deny Y.
for, in this case, X and Y cannot be true together.
Book IV.
Ch. V.
In a Mixed
Disjunctive
Syllogisia
the infer-
ence is
drawn from
the disjunc-
tion in the
major pre-
mise.
The infer-
ence rests on
the Laws of
Thought.
(ii.) Forms of Mixed Disjunctive Syllogisms. The denial To deny any
of one alternative, then, justifies the affirmation of the other, nltives is^'to
And, if the number of alternatives is greater than two the airmail the
same rule holds — the denial of any number justifies the
372
BYLLOQISMS.
Book IV.
Ch. V.
affirmation of the rest, categorically if only one is left, dis-
junctively if more than one remain. Thus : —
Eit'h''r X or Y or Z,
Neither X nor Y,
and : —
Kvei-y dis-
junction can
be expressed
as two alter-
natives.
The asser-
tion of one
alternative
does not
justify tlie
denial of the
other.
Rule.
All mixed
Disjunctive
Syllogisms
sire in the
Modus tol-
Undo voneiis.
z.
Either XorY or Z,
Not X,
. •. Either Y or Z.
But the number of alternatives may always be expressed
as two by considering, for the moment, two or more of them
as one ; and then both the minor premise and the conclnsion
retain the categorical form. This combination is most natur-
ally effected when the alternative propositions have the same
subject, so that the major premise can be written in the
form Every S is either P or Q or /?, which may be expressed as
Every S which is not P is either Q or R. But such reduction of
the number of alternatives is, of course, only apparent, and
serves no good purpose.
As a disjunctive proposition does not imply that the
alternatives are mutually exclusive {see § 79), we cannot
infer the denial of one of them from the assertion of the
other. Those logicians who hold the opposite view, of
course, assert that this can be done. But, even if the
exclusive view were right, and 5 is either P or Q implied that
8 could not be both P and Q, yet when it is inferred that S is
not Q because it is P, the inference is plainly made from the
categorical proposition. No P is Q, which the disjunctive
major premise is held to imply, instead of from that major
premise itself. Such an argument, therefore, even if valid,
would not be a disjunctive syllogism. We may, then, give
as the canon of syllogistic inferences from a disjunctive
l)roposition : —
To suhlate one member {or more) of any alternation is to posit
the other member or members.
This gives one mood only of mixed Disjunctive Syllo-
gisms, commonly called the Modus tollendo ponens because it
posits one alternative by sublating the other.
MIXED SYLLOGISMS.
373
As, however, both the alternative members may be either
affirmative or negative, this mood may take four forms,
corresponding to the subordinate forms of the two more
fundamental moods of mixed hypothetical syllogisms. Both
minor premise and conclusion, therefore, may be either
affirmative or negative categorical propositions. The forms
are thus expressed symbolically, the first being the stan-
dard :—
(1) Either X or Y,
NotX,
.-. Y.
(2) Either X or not Y,
NotX,
.-. Not Y. '
(3) Either not X or Y,
X,
.-. Y.
(4) Either not X or not Y,
X,
.-. Not Y. '
Boor IV.
Ch. V.
There are
four forms
of this raood.
depending
on the
quality of
the alterna
tivea.
Statement
of these
forius.
(iii.) Reduction of Mixed Disjunctive Syllogisms. As
every disjunctive proposition may be expressed in hypo-
thetical form (see § 80), every disjunctive syllogism may be
expressed as a mixed syllogism with a hypothetical major
premise. When this is doae, the above four forms are seen
to be equivalent to (1) the modus tollendo ponens, (2) the
modus tollendo tollens, (3) the modus ponendo ponens, and
(4) the modus ponendo tollens of the Modus Ponens when the
denial of the first alternative is taken as the antecedent of
the hypothetical major premise, and to the same forms of
the Modus Tollens when the denial of the second alternative
is so taken. As every syllogism in the Modus Ponens is re-
ducible to a categorical syllogism in the First Figure, and
every syllogism in the Modus Tollens to a similar syllogism
in the Second Figure [see § 131 (iv.)], it follows that every
Every
Mixed Dis-
junctive
Syllogism
can be re-
duced to a
mixed hypo
thetical
syOogism ;
and through
these to a
categorical
syllogism in
either Fig. I
or Fisr. II.
374 SYLLOGISMS,
Book IV. disjunctive syllogism can be expressed at will as a categorical
Ch^. Byllogism in either of these figures. This again illustrates
the essential unity of the syllogistic process, though the
reduction has no other value.
Examples. (iv.) Examples. As examples of the four possible forms
of mixed disjunctive syllogisms we may give : —
(1) Every tax which provokes general dissatisfaction is
either onerous in amount, or unjust in its inci-
dence ; the unpopular Poll Tax of Richard II
was not onerous in amount ; therefore, it was
unjust in its incidence.
(2) Any country which maintains a protective tariflE
either intends to subordinate present to future
advantage, or fails to see its own interests clearly ;
America, in maintaining her protective policy,
has no intention of subordinating the interests of
the present to those of the future ; therefore,
she fails to see her own interests clearly.
(3) Every revolution is either unjustifiable, or is pro-
voked by oppression ; the French Revolution of
1789 was justifiable ; therefore, it was provoked
by oppression.
(4) Any penalty which fails to diminish the crime of
which it is the appointed punishment, is either of
insufficient severity, or is sometimes not incurred
by the criminal ; the penalty for murder thus
fails, and being death, is of sufficient severity ;
therefore, its infliction on the culprit is not
certain.
We will add a few examples in which the alternatives in
the major premise have not the same subject : —
(1) Either the ancient Athenians were highly civilized,
or the highest artistic culture is possible amongst
a people of inferior civilization ; but this latter
MIXED SYLLOGISMS.
375
alternative is impossible ; therefore, the ancient Book IV.
Athenians were highly civilized. _1_ '
(2) Either vice is voluntary, or man is not responsible
for his actions ; but man is so responsible ; there-
fore, vice is voluntary.
(3) Either no man should be a slave, or some men are
incapable of virtue ; but no men are incapable
of virtue ; therefore, no man should be a slave.
(4) Either poverty is never due to misfortune, or desert
sometimes goes unrewarded ; but poverty is some-
times due to misfortune ; therefore, desert does
sometimes go unrewarded.
(v.) Disjunctive Syllogisms in the wider sense. Some logicians A syllogism
call every syllogism which contains a disjunctive premise a disjunc- junctive
tive syllogism. They thus obtain such syllogisms in every figure, premise is
T, 1 « •= ./ c jj^^ disiunc-
For example :— tive unless
the argu-
ment de-
pends on the
alternation.
Fig. I
M is either P or Q, etc.,
S is M,
,'. S is either P or Q, etc.
Fig. II
P is either M or N, etc.,
S is neither M or N, etc,
S is not P.
Fig. Ill
M is either P or Q, etc.
M is S
Fig. IV
PisM
M is either S or 1, etc.
.'. Some S is either P or Q, etc. .'.Something which is either S
or T, etc., is P.
But in such syllogisms as these the inference does not, in any
sense, depend npon the disjunction. They are, indeed, merely cate-
gorical syllogisms with one or more complex terms ; but this com-
plexity has no bearing upon the process of inference, which ia
purely categorical. Such syllogisms should not, therefore, be called
Disjunctive.
376
SYLLOGISMS.
Book IV.
Ch. V.
IHlemma — a
syllogism
with a com-
pound hypo-
thetical
major and a
disjunctive
minor.
It gives a
choice of al-
ternatives.
133. Dilemmas.
A Dilemma is a syllogism with a compound hypo-
thetical major premise and a disjunctive minor.
In other words, the major contains a plurality either of
antecedents or of consequents, which are either disjunc-
tively affirmed, or disjunctively denied, in the minor. The
peculiar feature of a dilemmatic argument is the choice of
alternatives which it thus offers ; and, when it is used in
Rhetoric, the aim is to make these alternatives of such a
kind that, whilst one must be accepted, all lead to results
equally disagreeable to an opponent. Hence arose the saying
'to be on the horns of a dilemma.' Strictly speaking, a
Dilemma contains only two alternatives ; if three are offered
we have a Trilemma; if four, a Tetralemma; and if more
than four, a Polylemma. As these more complex forms are
governed by the same principles as the dilemma, it will be
sufficient to consider the latter.
A dilemma
is either
Conatruclive
or Destruc-
tive,
and either
Simple or
Complex.
Four main
forms of
Dilemma :
(i.) Forms of the Dilemma.
(a) Determination of Forms. Like all mixed hypothetical
syllogisms, a dilemma may be either Constructive — when the
antecedents are affirmed ; or Destructive — when the conse-
quents are denied. In the former case, there must, of
necessity, be two antecedents in the major premise, as other-
wise the minor premise could not be disjunctive ; but there may
be either a single consequent — which the conclusion will affirm
in the same form, which is usually the simple categorical ; or
two consequents — when the conclusion will always be disjunc-
tive. In the former case the dilemma is Simple; in the latter
case Complex. Similarly, the major premise of a destructive
dilemma must contain two consequents, which may have
either one or two antecedents, the dilemma being again
Simple or Complex accordingly. We thus get four main
forms of the dilemma, which may be expressed by the
following formulae, in which each letter represents a propo-
sition ;—
MIXED SYLLOGISMS.
377
(1) Simple Constructive.
(a) If either A or B, thc7i 0,
Eithrr A or B,
.-. c. ~
(5) ij^ ciiAer A or B, ^/iew eti^er C or D,
Either A oj- B,
.-. £JiAer 0 or D.
(2) Sirrtple Destructive.
(a) If A, f^era Jo'^ C a77(l D,
Either not C o- not D,
Book IV.
Ch. V.
(1) Simple
Construc-
tive.
(2) Simple
Destruc-
tive.
.♦. Not A.
(5) If both A and B, <^en hoth C an<f D,
Either not C oj- not D,
.', Either not A or not B.
(3) Complex Constructive.
If A, then 0, and if B, then D,
Either A or B,
.-. Either C or D.
(4) Complex Destructive.
If A, then C, anrf ?/ B, then D,
Either not 0 or not D,
.*. Either not A or not B.
The second form of the Simple Constructive dilemma is
simple because the alternative hypotheticals which form the
major premise have only one consequent. The conclusion
is disjunctive because this single consequent is disjunctive in
form. Similarly, the second form of the simple destructive
dilemma is not complex, although it has a disjunctive con-
clusion, for that conclusion is merely the simple denial of
the one single antecedent of the major premise. It thus
appears that these forms are not fundamental, but are only
special cases of somewhat greater complexity of the simple
forms (c/. Keynes, Formal Logic, 3rd Ed., pp. 317-8 notes).
(8) Complex
Construc-
tive.
(4) Complex
Destruc-
tive.
378
SYLLOGISMS.
Book IV.
Ch. V.
Examples of
Dilemmas.
It will be noticed that the major premise of both forms
of the simple destructive dilemma has its consequent copu-
lative, and not disjunctive, in form. The reason is that
when two consequents are alternatives their disjunctive
denial will not justify the denial of the antecedent ; for, if
one of two alternatives is false, the other must be true
{cf. § 79), and the truth of one consequent is all that the
antecedent of such a proposition demands. It is necessary
that hoth the consequents should be connected with the
whole antecedent, in order that the denial of their conjunc-
tion may justify the rejection of the antecedent as a whole.
We will now illustrate each of the above forms.
(1) (a) Simple Constructive. The inhabitants of a besieged
town might express their position in some such dilemma as
this : ' If we hold out, we shall suffer loss by the bombard-
ment destroying our property ; if we surrender, we shall suffer
loss through having to pay the enemy a heavy ransom ; but
we must adopt one or other of these two courses ; there-
fore, whichever way we act, we are bound to suffer loss.-
(6) This form, which is more indefinite than the former,
neither antecedent being limited to one consequent, is much
less frequently employed. As an example of it we may
give : ' If either England is over-populated or its industry
is disorganized, many people must either emigrate or live in
deep poverty ; England at present suffers either from over-
population or from disorganization of industry ; therefore,
many Englishmen must either emigrate or live in deep
poverty.'
(2) (a) Simple Destructive. Euclid's proof of Proposition
VII of the First Book may be exhibited as a dilemma of
this kind : ' If two triangles on the same base, and on the
same side of it, have their conterminous sides equal, then
two angles are both equal and unequal to each other ; but
they are either not equal or not unequal ; therefore, the
existence of two such triangles is impossible.'
Whately {Elements of Logic, 5th Ed., pp. 117-8) gives the
MIXED SYLLOGISMS. 379
following example of such an argument : " If we admit the Book IV.
" popular objections against Political Economy, we must ^^' ^
" admit that it tends to an excessive increase of wealth :
" and also, that it tends to impoverishment ; but it cannot
*' do both of these ; (i.e., either not the one, or, not the other)
" therefore we cannot admit the popular objections, &c."
(5) This form is very seldom used. As an example we
may give : ' If compulsory education is unnecessary and no
legal regulation of the conditions of the labour of children
is justifiable, then all guardians of children both understand
and try to perform their duty to those under their charge ;
but some guardians either do not understand their duty to
their young wards or do not try to perform it ; therefore,
either compulsory education is necessary or some legal
regulation of the conditions of children's labour is justi-
fiable.'
(3) Complex Constructive. A good example of this form
of dilemma is found in the oration of Demosthenes On
the Crown, where he argues : ' If .ffischines joined in the
public rejoicings, he is inconsistent ; if he did not, he is
unpatriotic ; but either he did or he did notj therefore,
he is either inconsistent or unpatriotic'
The following argument is in the same form : * If the
Czar of Russia is aware of the persecutions of the Jews in
his country, he is a tyrant ; if he is not aware of them, he
neglects his duty ; but either he is, or he is not, aware of
them ; therefore, either he is a tyrant or he neglects his duty.*
(4) Complex Destructive. This, again, is not a very com-
mon form. An example is : * If the industry of England is
well organized, there is work for every efficient labourer who
seeks it, and if all labourers are industrious, all will seek
work ; but either some labourers cannot get work or they
will not seek it ; therefore, either the industry of England
is not well organized or some labourers are idle.'
(6) Mutual Convertibility of Forms. Like the simpler
mixed hypothetical syllogisms [see § 131 (ii.) ad Jin.], the
380 SYLLOaiSMS.
Book IV. constructive and destructive dilemmas are, at bottom, iden-
Ch^. tical ; for any form of the one may be converted to the
The Con- corresponding form of the other by obverting the contra-
and De- positive of the major premise. Thus, the complex destructive
structive g^^^j complex constructive dilemmas are, fundamentally, the
forins are ^ ' . •" .
mutually same, and each of the two forms of the simple destructive
byobvertiug Js mutually convertible with the corresponding form of the
the contra- gimple constructive dilemma. In illustration of this it will
positive of "^ .
the major. be sufficient to reduce each of the destructive to a construc-
tive form.
Simple Destructive, (a) By obverting the contrapositive
of the major premise and retaining the original minor, we
get:—
If either not C or not D, tJien not A,
Either not 0 or not D,
.'. Not A ;
which is the simple constructive form with negative, instead
of affirmative, elements.
(h) Similarly, by obverting the contrapositive of the
second form of the simple destructive we get the second form
of the simple constructive, with negative elements : —
If either not C or not D, then either not A or not B,
Eithe)' not 0 or not D,
,*, Either not A or not B.
Complex Destructive. The obverted contrapositive of the
major premise being taken, we get : —
If not 0, then not A, and if not D, then not B,
Either not C or not D,
.*. Either not A or not B ;
which is the complex constructive form with negative
elements.
This convertibility may be illustrated by an example. A
MIXED SYLLOGISMS.
381
man in bad health, and who has no income but his salary,
may argue that his recovery is hopeless, either in the simple
destructive dilemma : ' If I am to regain health, I must both
give up work and live generously ; but I cannot do both of
these (i.e., either I cannot do one, or I cannot do the other) ;
therefore, I cannot regain health'; or in the simple con-
structive : ' If I either continue to work, or live meagrely, I
cannot regain health ; but I must either continue to work or
live meagrely ; therefore, I cannot regain health.'
Book IV.
Ch. V.
(c) Other Views. Very great diversity exists amongst logicians
as to what arguments are, and what are not, properly called
dilemmas, and equally divergent definitions have been given of that
form of reasoning. The forms we have given are the simplest, but
they may be modified by having a hypothetical proposition for the
minor premise, when, of course, the conclusion will also be hypo-
thetical. Or again, the major may be written in the negative form
when, of course, the conclusion will be negative in the constructive
dilemmas, whilst in the destructive dilemmas the minor premise
will be afiirmative.
Several logicians, including Jevons, follow Whately and Mansel in
recognizing only three forms of dilemma — the first form of the simple
constructive, and the complex constructive and complex destruc-
tive. They so define the dilemma as to make it essential that the
hjrpothetical major should have more than 07ie antecedent. Whately
{Elements of Logic, 5th Ed., pp. 117-8) rejects the simple destruc-
tive form, on the ground that the disjunctive denial of several
consequents " comes to the same thing as wholly denying them ;
"since if they be not all true, the o?ie antecedent must equally fall
•'to the ground; and the Syllogism will be equally simple." This
is perfectly true, and, we may add, if it were not, such a form of
reasoning would be absolutely invalid ; for in every destructive
hypothetical syllogism the consequent must be denied. Moreover,
the same argument applies with equal force against the simple con-
structive form, for the disjunctive assertion of the antecedents comes
to the same thing as wholly affirming them ; since if one be true,
the one consequent must equally follow. In fact, as these two forms
are mutually convertible, they must stand or fall together ; and the
simple constructive is the most frequently employed, and the most
generally acknowledged, form of dilemma. Hamilton, indeed,
Logicians
differ as to
the defini-
tion and
forms of the
dUemma.
Whately,
Mansel, and
Jevons, re-
ject the
simple
destructive
form.
But it
stands on
the same
ground as
the simple
construc-
tive.
382 SYLLOGISMS.
Book IV. excludes it when (Lect. on Logic., vol. i., p. 350) he defines a
Ch. V. dilemma as " a syllogism in which the sumption [i.e., major premise]
Hamilton "is at once hypothetical and disjunctive, and the subsumption
includes only " [;.e., minor premise] sublatea the whole disjunction, as a conse-
" quent, so that the antecedent is sublated in the conclusion."
This gives the form
If A, then either C or D,
Neither C nor D,
.-. Not A,
which appears also to be the only one contemplated by Lotze (see
Logic, Eng. trans., vol. i., p. 127, cf. Outlines of Logic, p. 70), and
is recognised by Kant, Ueberweg, Thomson, Bain, and other
logicians. To this Mansel {Aldrich, Art. Log. Bud., 3rd Ed.,
p. 107) objects, on the ground that it is " merely a common dis-
" junctive syllogism"; that is, its major may be expressed in the
form
Either not A, or C or D.
This does not seem conclusive, as any hypothetical proposition can
be similarly reduced to the disjunctive form ; for example the
major premise of the simple constructive dilemma may be ex-
pressed— Either C, or neither A nor B.
But, when we obvert the contrapositive of the major premise of the
above form we get : —
If neither C nor D, then not A,
Neither C nor D,
.-. Not A.
which con- TJ^jg cannot be a dilemma, for it contains no disjunction at all. It
tains no true '
alternative may be expressed : —
°^*^^ Jf both not C and not D, then not A,
Both not C and not D,
.-. Not A ;
which corresponds to the form with aflBrmative elements : —
Jf both A and B, then C,
Both A and B,
in which the absence of any alternative element is still more plainly
seen.
MIXED SYLLOGISMS.
383
XJeberweg {Logic, Eng. trans., p. 455) regards as an essential
feature of a dilemma that " whichever of the members of the dis-
" junction may be true, the same conclusion results." This
excludes all the complex forms, in which the conclusion is disjunc-
tive, and includes the form we have just rejected as well as several
forms in which the major premise is not hypothetical, and which
he also enumerates under the head of " disjunctive inferences in the
"wider sense" [ibid,, p. 456 ; cf. § 148 (v)]. But the element of
doubt marked by the hypothetical character of the major premise
is an indispensable characteristic of a dilemmatic argument, for
without it there can be no real choice of alternatives.
Professor Fowler (Deductive Logic, pp. 114-8) recognizes both
the complex dilemmas and the first form of each of the simple, and
Mr, Stock {Deductive Logic, p. 271) adds to these the second form
of the simple constructive, but does not give the corresponding
form of the simple destructive.
Thomson {Laws of Thought, p. 203) defines a dilemma as "a
"syllogism with a conditional [i.e., hypothetical] premise, in which
" either the antecedent or consequent is disjunctive." He gives
three examples — the simple constructive, the form contemplated
by Hamilton's definition, and the following modification of the
latter : —
If some A is B, either the m that are A, or the n that are A, are B,
But neither the m that are A, nor the n that are A, are 8,
'.A is not B,
where the letters symbolize terms. But this form must be rejected
together with that of which it is a modification. Thomson, in fact,
misses the essential point that the minor premise must be disjunc-
tive, and his definition is much too wide, as it would cover many
forms which certainly have no claim to be called dilemmas
such as : —
// A, then either C or D,
A,
Book IV.
Ch. V.
Ueberweg
excludes
the complex
forms,
and includes
some whose
major is
categovicaL
Fowler and
Stock recog-
nize four
forms.
Thomson's
definition is
too wide.
. •, Either C or D.
Dilemmas,
* (ii.) Reduction of Dilemmas. A dilemma is, formally, mixedhh)o.
only a somewhat elaborate kind of mixed hypothetical syl- thetical syl-
logism, and is, consequently, governed by the same canon as i^ educed"
such inferences, and may be reduced in the same way to the caiToms""
384
SYLLOGISMS.
Book IV.
Ch. V.
It may also
bo thrown
Into a scries
of pure hy-
pothetical
syllogiiiina.
The disjunc-
tive minor
must ex-
haust all
alternatives,
or the
dilemma is
not cogent.
A faulty di-
lemma may
be rebutted
by transpos-
ing the con-
sequents,
and changing
theii' quality.
categorical form [cf. § 131 (ii), (iv)]. Thus, the Simple
Constructive Dilemma may be expressed : —
The case of either A or 3 being true is the case of 0
being true,
This is the case of either A or B being true,
.'. This is the case ofC being true;
and similarly with the other forms.
The formal validity of a dilemma may also be ex-
hibited by resolving it into a series of pure hypothetical
syllogisms, by reducing the disjunctive minor to the hypo-
thetical form (see § 80). For example, the Complex Con-
structive form of Dilemma may be reduced to two such
syllogisms : —
(orig. major) If B, then D,
(from orig. minor) If not A, then B,
.'. If not A, then D ;
from orig. major) But, If not C, then not A,
.*. If not C, then D,
i.e. Either 0 or D.
But such reductions are only interesting as affording a
fresh proof that all syllogistic inference is of essentially the
same character.
(iii.) Rebutting a Dilemma. The conclusiveness of a
dilemma depends upon material, as well as formal, considera-
tions. Not only must the connexion of antecedent and con-
sequent be a real one, but the disjunction in the minor pre-
mise must exhaust every possible alternative. The difficulty
of securing this is the reason dilemmatic arguments are so
often fallacious.
Yery often a faulty dilemma can be rebutted or retorted
by an equally cogent dilemma proving the opposite con-
clusion. In such a case, the consequents of the major change
places, and their quality is changed. Thus
If A, then C, and i/"B, then D,
Either A or B,
.-. Either 0 or D,
may be rebutted by the dilemma
MIXED SYLLOGISMS.
385
If A., then not D, and i/B, then not C,
Ether A or B,
Book IV.
Ch. V.
.'. Either not C or not D.
But the conclusion proved is not really incompatible with
that of the original dilemma, for both can be satisfied by C
and not D or by D and not C being true together. Only the
complex constructive forms of the dilemma lend themselves
to this treatment (though destructive dilemmas can be
reduced to the constructive form and then rebutted), and, of
course, only those in which some flaw exists in the original
argument ; a valid dilemma cannot be rebutted. There are
several classical examples of dilemmas thus rebutted, the
consideration of which will tend to make the subject clear.
An Athenian mother is said to have advised her son not to
enter public life; 'for,' said she, 'if you act justly men
will hate jou, and if you act unjustly the gods will hate
you ; but you must act either justly or unjustly; there-
fore, public life must lead to your being hated.' This
argument he rebutted by the equally cogent dilemma :
' If I act justly the gods will love me, and if I act unjustly
men will love me ; therefore, entering public life will make
me beloved.' But, according to the given premises, a public
man must always be both hated and loved ; the given con-
clusions are not, therefore, incompatible.
More famous is the Litigiosus. Protagoras agreed to train
Euathlus as a lawyer, one-half the fee to be paid at once, and
the other half when Euathlus won his first case. As Euathlus
engaged in no suit, Protagoras sued him, and confronted
him with this dilemma : ' Most foolish young man, if you
lose this suit you must pay me by order of the court, and if
you gain it you must pay me by our contract.' To which
Euathlus retorted : ' Most sapient master, I shall not pay
you ; for if I lose this suit I am free from payment by our
contract, and if I gain it, I am exonerated by the judgment
of the court.' Of this difficulty several solutions have been
offered. The most reasonable seems to be this : As Euathlus
had until then won no case, the condition of the bargain was
LOG. I. 25
Classical
examples of
rebutted
dilemmas —
On public
life.
The Litigi-
osus.
386
SYLLOGISMS.
Book IV.
Ch. V.
The Croco-
iilu*.
not fulfilled, aiui the judges should have decided in his
favour. It was then open to Protagoras to bring a fresh
suit, when the judgment must have goae against Euathlus.
Somewhat oimilar is the Crocodilus. A crocodile had
seized a child, but promised the mother that if she told him
truly whether or not he was going to give it back, he would
restore it. Fearing that if she said he was going to give it
back, he would prove her wrong by devouring it, she
answered, 'You will not give it back'; and argued : 'Now
you must give it back — on the score of our agreement if my
answer is true, and to prevent its becoming true if it is
false.' But the crocodile answered : ' I cannot give it back,
for if I did your answer would become false, and thus 1
should break our agreement ; and even could your answer
be correct I could not give it back, as that would make it
false.' On this Lotze says : ' There is no way out of this
" dilemma ; as a matter of fact however both parties rest
" their cases on unthinkable grounds ; for the answer really
" given can as little be true or untrue independently of the
"actual result as could the answer she might have given, an
"answer which only differs from this in being more
"fortunate" {Lor/ic, Eng. trans., vol. ii., p. 20). For, had
she said * You will give it back,' then its restoration would
both have made her answer true and have fulfilled the
agreement.
1
CHAPTER VI.
ABRIDGED AND CONJOINED SYLLOGISMS.
134. Enthymemes.
An Enthymeme is a syllogism abridged in expression
by the omission of one of the constituent propositions.
The most common form in which syllogistic arguments are
met with is the enthymematic. The tendency of speech is
always to state explicitly no more than is required for
clearness ; and as, in most cases, when two of the con-
stituent propositions of a syllogism are given the third is
sufficiently obvious, it would be an offence against brevity
to express it in ordinary discourse. It is, therefore, but
seldom that fully expressed syllogisms are met with outside
treatises on Logic. Especially in epigrams and other witty
sayings, the enthymematic form is common ; and by means
of it, charges can he insinuated which it would be impolitic
to advance openly. As a good exam])le of this we may refer
to Shakespeare's famous version of Mark Antony's Oration
over Caesar's body. It is almost needless to say that when a
B])eaker or writer draws an inference which he knows to be
fallacious, he naturally adopts this shortened form of state-
ment, as the fallacy is then less likely to be detected than it
would be if the argument were set out at length. Thus, ft
false conclusion may be supported by a perfectly true pro
mise, the implied premise being, of course, false.
Although when one of the premises is omitted from a
syllogism, the resulting enthymeme appears at first sight to
draw the conclusion from only one premise, yet, it must be
Book IV.
Ch. VI.
Enthymeme—
A syllogiam
in which one
of the con-
Btituent pro-
positions U
ouiiited.
Both pre-
mises are
necessary in
thought.
388
SYLL0GIS5IS.
Book IV.
Ch. VI.
Enthymemes
are of three
orders—
1. Major
omitted.
2. Minor
omitted.
3. Conclusioii
omitted.
remembered that it does not really do so. The implied
premise is equally necessary with that which is expressed as
a ground for the conclusion. The abridged form of ex-
pression does not affect the form of the thought, and the
inference is, therefore, fully mediate — not immediate, as it
would be were the conclusion drawn from one premise alone.
Indeed, as the distinction between an enthymeme and a fully
expressed syllogism is, primarily, one of expression, it be-
longs to Rhetoric rather than to Logic.
It is more frequently the case that the omitted proposition
is a premise than that it is the conclusion. Some logicians,
indeed, have so defined enthymemes as to exclude the latter
form altogether. But this limitation cannot be justified, as
the omission of the conclusion is by no means uncommon.
According to which proposition is omitted, enthymemes are,
therefore, of three ordera ;—
First Order — when the major premise is omitted.
Second Order — when the minor premise is omitted.
Third Order — when the conclusion is omitted.
For example, the argument of the fully stated syllogism :
— ' All democratic governments are liable to frequent
changes in foreign policy ; the English government is
democratic ; therefore, the English government is liable to
frequent changes in foreign policy ' — may be expressed by
an enthymeme of each order :—
First Order, ' The English government is liable to fre«
quent changes in foreign policy, because it is democratic'
Second Order. * The English government is liable to
frequent changes in foreign policy, because all democratic
governments are liable to this,'
Third Order. ' All democratic governments are liable to
frequent changes in foreign policy, and the English govern-
ment is democratic'
When an enthymeme is of the first or second order, more
frequently than not the conclusion is stated first, and the
AURIDGED AND CONJOINED SYLLOGISMS. 889
premise given in its sujjport is introduced by some such Book IV.
illative particle as ' because ' or ' since.' Ch^l.
When an enthymeme is of the third order, it is, of course, The figure
immediately obvious to which of the syllogistic figures it ine^e^ofthe
belongs. When it is of either the first or second order, this "'"* oy ,
° -Til ..... . . -econd order
must be determined bj^ the position in the given premise of can be deter.
either the minor or the major term. If the given premise The'^positioc
contains the subject of the conclusion, it is, necessarily, the >' s or p.
minor premise, and the enthymeme is of the first order ; if
it contains the predicate of the conclusion, the enthymeme is
of the second order. Now, if both the given premise and
the conclusion have the same subject, the enthymeme must
be in either the First or the Second Figure ; for in those
figures only is S the subject of the minor premise. Similarly,
if both the propositions have the same predicate, the figure is
either the First or the Third, for in these P is predicate of
the major premise. If the predicate of the conclusion is the
subject of the given premise, the argument belongs either to
Figure II or to Figure IV, in each of which P is the subject
of the major premise. Finally, if the subject of the conclusion
IS the predicate of the given premise, the figure is either the
Third or the Fourth, for in each of these S is the predicate
gf the minor premise.
In all cases of categorical enthymemes, one term is An enthy-
common to both the given propositions. Similarly, if the not be cate
enthymeme is the abridged statement of a pure hypothetical, ^^"caL
or of a pure disjunctive, syllogism, one of the propositions
which, in those syllogisms, take the place of terms will be
found in both the given propositious. For instance, ' If
any child is spoilt, he is unhappy ; because if any child is
spoilt, he is sure to be selfish' is a pure hypothetical enthy-
meme of the first order. But if an enthymeme of the
first order is an abridged statement of a mixed syllogism,
the expressed propositions will contain no common element;
as one of them will be the antecedent, and the other the
consequent, of the hypothetical major. Thus, * There is
alternation of day and night, because the earth rotates on its
axis' is such an enthymeme. But here we cannot limit the
390
SYLLOGISMS.
Book IV,
Ch. VI.
fully expressed syllogism to a single possible form, for the
implied major premise may be 'If the earth rotates on its
axis, there is alternation of day and night,' when the
syllogism is a mixed hypothetical in the modus ponens ; or
' Either the earth does not rotate on its axis, or there is
alternation of day and night,' when the syllogism is dis-
junctive. But if the enthymeme is of either the second or
the third order, there is no such choice.
Sylloffisms
may be con-
nected with
each other.
135. Progressive and Regressive Chains of Reasoning.
The matter with which thought deals forms in itself a
connected whole, and the advance of knowledge continually
makes this connexion more evident to us. We find that
from those ultimate general principles which are so self-
evident that they are called Axioms, we can deduce other
principles of less generality, but which are yet themselves
the immediate ground for others of still less scope ; and
this process may be carried on through several stages. Thus
we get a train of syllogistic reasoning, in which the con-
clusion of each syllogism becomes a premise in that which
follows ; so that the last conclusion — which may be of a very
specialized character — is shown to be ultimately, though
indirectly, dependent upon some axiom of the widest gener-
ality. Such a process, which is very common in Mathematics,
and is constantly employed by Euclid in his direct proofs,
may be thus represented symbolically : —
(1)
^2)
(3)
Y a P (majoi)
X a Y (minor)
.:Xa P
(concl.)
X aP
(major)
MaX
(minor)
,M a P (concl.)
M a P (major)
S a M (minor)
S a P (concl.)
(1)
(2)
(3)
S a Y (minor)
Y a X (njajor)
.:Sa X
(concl.)
SaX
(minor)
XaM
(major)
.'. S a M (concl.)
S a M (minor)
M a P (major)
.*. S a P (conol.)
ABRIDGED AND CONJOINED SYLLOGISMS.
391
In the former case, the conclusion of each syllogism forms
the major premise of that which follows ; in the latter, it
becomes the minor premise, which has, in this case, been
written first in each syllogism to make the connexion more
prominent. Each two members of such a train of syllogisms
are thus connected by a common proposition ; and the syl-
logisms thus related are called respectively Prosyllogism and
Episjjllogism with respect to each other. These terms are,
therefore, purely relative ; and the same syllogism may be at
once a prosyllogism and an episyllcgism, with reference to
different members of the chain of syllogisms in which it
occurs ; as is, in fact, the case with the second syllogism in
each of the above examples. We may, therefore, give the
following definitions : —
A Prosyllogism is a syllogism whose conclusion is a premise
in the syllogism with which it is connected.
An Eplsyllogism is a syllogism one of whose premises is
the conclusion of the syllogism with which it is con-
nected.
In the trains of reasoning we have j ust examined, the pro-
gress of thought has been from prosyllogism to episyllogism.
Such a demonstration is called Progressive, Episyllogistic,
or Synthetic, and it may either consist of categorical or of
hypothetical propositions. A gooJ example of the latter is
given by Ueberweg {Logic, Eng. trans., p. 464) : " If there
is a medium obstructing the motion of the planets, then
the path of the earth cannot be constant nor periodical, but
must always become less : If this be the case, then the
existence of organisms on the earth cannot have been (nor
can remain) eternal. Hence, if there is this medium,
organisms must have at one time come into existence, and
will wholly pass away. If oi'ganisms once existed for the
first time on the earth, they must have arisen out of in-
organic matter. If this is the case, there has been an
" original production (generatio asquivoca). Hence, if this
" obstructive medium exists, there has been an original pro-
" duction."
Book IV.
Ch. VI.
The conclu-
sion of a Pro-
syllogism is
a premise in
an Episytf
logisM.
When the
reasoning
starts with
the prosyllog-
ism,itiscall^
Progressive,
Episyllogistic,
or Synthetic
392
SYLLOGISMS,
Book IV.
Ch. VL
When the
reasoning
starts with
the episyl-
logism, it is
called Re-
gressive, Pro-
§yUogistic,Ot
Analytic.
iBut, instead of starting from au axiom of the widest
generality, in physical science it more frequently happens
that the highest and most general principles are the last to
be discovered. "Certain general propositions are first dis-
'■ covered (as, e.g. the laws of Kepler) under which the in-
" dividual facts are syllogistically subsumed. The highest
" principles are discovered later (e.g. the Newtonian law of
"Gravitation) from which those general propositions are
" necessary deductions '' (Ueberweg, ibid., p. 465). In such a
course of reasoning, thought advances from the episyllogism
to the prosyllogism, going backwards further and further
towards first principles. A demonstration of this kind is,
therefore, called Kegressive, Prosyllogistic, or Analytic.
It may be thus represented symbolically, the episyllogism
heinc stated first, as in such a train it comes first in the order
cf thought : —
(1) S a P (concl.)
'.• M a P (major)
S a M (minor)
(2) M aP (concl.)
'.• X a P (major)
M a X (minor)
(3) XaP (concl.)
'.' Y a P (major)
X a Y (minor)
(1) SaP (concl.)
'.• /;/ a P (major)
5 a M (minor)
(2) M aP (concl.)
•.■XaP (major)
/./ a X (minor)
(3) SaM (concl.)
•.• Y a M (major)
S a Y (minor)
I
Polysyllogism
—a chain of
connected
s llogisms.
In the first example we have a continuous regressive chain,
reaching back to the widest general statement Y a P, and the
major only of each syllogism being established by a pro-
syllogism. In the second, each of the premises in the epi-
syllogism is established by a prosyllogism, of which it forma
the conclusion.
Such a train of reasoning, whether progressive or regres-
sive, is often called a Polysyllogism.
ABRIDGED AND CONJOINED SYLLOGISMA
393
136. Sorites.
A Sorites is a progressive chain of reasoning whose
expression is simplified by the omission of the conclusion
of each of the prosyllogisms.
The Sorites is, thus, a series of enthymemes, of which the
first is of the third order, as both its premises are stated ;
and the last is of either the first or the second order, as one
premise and the conclusion are given (c/. § 134). But each of
the intermediate enthymemes is represented by one premise
alone, as the other premise is the omitted conclusion of the
preceding prosyllogism. In somewhat different words, there-
fore, it may be said that a sorites is a series of enthymemes,
in each of which, except the first, one premise is implied by
a prosyllogism, and the other is explicitly stated. From this
it follows that a full analysis of a sorites resolves it into
a number of separate syllogisms, less by one than the total
number of premises.
(i.) Kinds of Sorites. It was seen in the last section that
the conclusion of a prosyllogism may form either the minor
or the major premise of the episyllogism. There are, conse-
quently, two forms of sorites — the Aristotelian, in which the
suppressed conclusions form the minor premises of the
following episyllogisms ; and the Goclenian, in which they
form the major premises. The symbolic expression of each
may be thus given : —
A ristotelian Sorites — Every S is X
Every X is Y
Evei-y Y is Z
Every Z is P
Every S is P
, . iiiVKi y
Goclenian Sorites — Every Z is P
Every Y is Z
Every X is Y
Every S is X
Book IV.
Ch. VI.
Sorites — a
progressive
cliaiu of
reasoning,
with the
cnolusion
of each pro-
syllogism
omitted.
A Sorites is
a sei-ies of
enthymemes
In the
Aristotelian
Sorites, the
omitted
conclusions
form the
minor pre-
mises of the
succeeding
syllogisms ;
in the
Ooclenian
Sorites they
form the
major.
< *
Every S is P
394
SYLLOGISMS.
Book IV.
Ch. VI.
A soritea
may be
analysed
iuto syllog-
isms, one less
in number
than its pre-
mises.
It will be noticed that in the Aristotelian form the subject
of the conclusion is stated first, and the predicate of the
conclusion occurs in the last premise ; in the Goclenian form
the subject of the conclusion is the subject of the last
premise, and the predicate of the conclusion is found in the
first premise. The latter form, therefore, corresponds the
more closely to the customary order in which the premises
of a syllogism are stated. Both forms contain the same
premises, though their order is reversed. This led Hamilton
to regard the Aristotelian form as expressing a reasoning in
comprehension, and the Goclenian form as expressing one in
extension ; since in the latter we start with the premise
which contains the two terms of widest extension, and in the
former we end with that premise.
If both forms are analysed into their constituent syllogisms,
it will be seen that in the Aristotelian form the omitted
conclusions — which we enclose in square brackets — form the
minor, and in the Goclenian form, the major premises of the
succeeding episyllogisms.
Analysis of Aristotelian Sorites.
(!) Every X is Y (major)
Every S is X (minor)
.'. [Every S is Y] (concl.)
(2) Every Y is Z (major)
[Every S is K] (minor)
.*. [Every S is Z] (concL)
(3) Every Z is P (major)
[Every S is Z] (minor)
.'. Every S is P (concl.)
Analysis of Goclenian Sorites.
(1) Every Z is P (major)
Every Y is Z (minor)
.•. [Every Y is P] (concl.)
ABRIDGED AND CONJOINED SYLLOGISVIS.
395
(2) [Every Y is P] (major)
El^eril X is Y (minor)
.*. [Every X is P] (concl.)
(3) [Every X is P] (major)
Every S is X (minor)
.'. Every S is P (concl.)
It IS evident that the two forms agree in the fact that
each omitted conclusion is a premise of the following
syllogism. Now, this advance from previous to consequent
inferences is the characteristic of progressive reasoning
(cf. § 135) ; it is, therefore, an error to speak of the
Goclenian Sorites, as some logicians have done, as a re-
gressive form of reasoning.
Either form of sorites may be entirely composed of
hypothetical propositions. In the Goclenian Sorites the
last premise may be categorical, and then the concluding
enthymeme is the abridged form of a mixed syllogism, in
which the categorical minor premise either posits the ante-
cedent, or snblates the consequent, of the implied conclusion
of the preceding prosyllogism ; e.g. : —
If C, then D, If C, then D,
If B, then C, IfB,thenG,
IfA,the}iB, If A, then B,
A, Not D,
.•7d! .-. Not A.
lu the Aristotelian Sorites, however, the same result can
only be obtained by adding to the sorites a categorical minor
premise, and then regarding the implied conclusion of the
preceding prosyllogism as the major, instead of the minor
premise of the last episyllogisra. In other words, a mixed
syllogism at the end of a sorites must, in all cases, cor-
respocd to the Goclenian form ; e.g. : —
If A, then B, If A, then B,
If B, then C, if B, then C,
If C, then D, If G, then D,
A ' Not D,
Book IV.
Ch. VI.
Every sorites
is a progres-
sive chain of
reasoning.
A sorites
may consist
of h ypo-
tlietical pro-
positions ;
but only the
last syllo-
gism can be
mixed ;
and that
must be of
the Gocle-
nian form..
D.
•. Not A.
396
SYLLOGISMS.
Book IV. The following example of a categorical Aristotelian
CKj^l. Sorites may be given from Aristotle {Poet, vi) : ' Action is
Examples of that in 'which happiness lies ; what contains happiness is the
end and aim ; the end and aim is what is highest ; there-
fore, action is what is highest.' As an instance of a similar
sorites composed of hypothetical propositions we may give :
' If any man is avaricious, he is intent on increasing his
wealth ; if he is so intent, he is discontented ; if he is
discontented, he is unhappy ; therefore, if any man is
avaricious, that man is unhappy.' In the following the last
syllogism is a mixed hypothetical : ' If the soul thinks, it is
active ; if it is active, it has strength ; if it has strength, it
is a substance ; now the soul thinks ; therefore, the soul is a
substance.' In all these cases, if we reverse the order of
premises, we get a sorites of the Goclenian form.
Rula of
Aristotelian
Sorites —
1. Only last
premise
negative.
2. Only first
premise
particular.
(ii.) Special Rules of the Sorites.
(a) The Aristotelian Sorites. In this form of sorites, the
predicate of the last premise is, in the conclusion, affirmed or
denied of the first subject, through one or more inter-
mediate propositions. Each intermediate term must, there-
fore, be affirmatively predicable of the whole of the preceding
one, or the chain of connexion is broken. This gives us the
two following as
Special Rules of the Aristotelian Sorites :—
1, Only one premise, and that the last, can be negative.
2. Only one premise, and that the first, can be particular.
The necessity of these rules is evident when the sorites is
analysed into its constituent syllogisms.
Bale 1. More than one premise cannot be negative ; for,
as a negative premise in any syllogism necessitates a negative
conclusion (see § 111), if more than one premise in the sorites
were negative, one of the constituent syllogisms would
contain two negative premises.
i
ABRIDGED AND CONJOINED SYLLOGISMS.
397
If any premise in the sorites is negative, the conclusion
must be negative ; therefore, the predicate of the conclusion
must be distributed in the last premise, of which it is the
predicate ; i.e., the last premise must be negative.
Mule 2. As every premise except the last must be affirma-
tive, it is evident that if any, except the first, were particular,
it would involve the fallacy of undistributed middle.
(h) The Goclenian Sorites. In this form of sorites the
predicate of the fii'st premise is, in the conclusion, either
affirmed, or denied, of the subject of the last, through one or
more intermediate propositions. Each intermediate term
must, therefore, be affirmed universally of the succeeding
one, or the necessary connexion will not be secured. We,
thus, get the two following as
Special Rules of the Goclenian Sorites : —
1. Only one premise, and that the first, can be negative.
2. Only one premise, and that the last, can he particular.
A consideration of the constituent syllogisms again showa
the necessity of these rules.
Rule 1. As in the Aristotelian sorites, a plurality of
negative premises would result in one of the syllogisms
containing two negative premises.
If any premise is negative, the conclusion must be nega-
tive ; therefore its predicate must be distributed in the
first premise, of which it is the predicate ; i.e., the first premise
must be negative.
Rule 2. If any premise but the last were particular, the
conclusion of the syllogism in which it occurred would also
be particular, and, as that proposition would be the major
premise of the succeeding syllogism, we should have the
fallacy of undistributed middle.
The above rules assume, in each case, that the sorites is
entirely in the First Figure ; i.e., that each of the constituent
syllogisms is in that figure. We must now enquire whether
this is a necessary limitation of this form of argument.
Book IV.
Ch. VI.
Rulei oj
Goclenian
Sorites- -
1. Only first
premise
negative.
2. Only last
premise
particular.
398
SYLLOGISMS.
Book IV.
Ch. VI.
Hamilton
held that a
forites
could be in
the Second
and Third
Figures, as
well as in
the Fii-8t ;
but the
reasonings
he refers to
are not
Rorites.
Dr. Keynes
shows that
sorites in
Figs. II and
III are pos-
sible.
(iii.) Figure of Sorites. Hamilton held that a sorites is possible
in the Second and Third Figures, as well as in the First. He says
(Lectures on Logic, vol. ii, p. 403) : " In Second and Third Figures,
" there being no subordination of terms, the only Sorites competent
" is that by repetition of the same middle. In First Figure, there is
" a new middle term for every new progress of the Sorites ; in
"Second and Third, only one middle term for any number of
"extremes." He thus indicates such forms as the following : —
Second Figure.
No X is M
Xo Y is M
No Z is M
Every P is M
'No^X or Y or Z is f.
Third Figure.
Every M is X
Every M is Y
Every M is Z
Every M is P
Some X and Y and Z are P.
But neither of these is a chain argument, or sorites, at all.
There is not one conclusion drawn from a succession of premises, all
necessary to its establishment : but as many different conclusions as
there are syllogisms, though they are summed up into one compound
proposition (cf. § 75).
Dr. Keynes, whilst agreeing in the rejection of Hamilton's forms,
yet shows that a sorites is possible in both the Second and Third
Figures (see Formal Logic, 3rd ed., pp. 3.30-2). In the Second
Figure the form would be (the suppressed conclusions being enclosed
in square brackets) : —
Some A is not B,
Every C is B,
[. •. Some A is not C],
Every D is C,
[. •. So7ne A is not £)],
Every E is D,
. '. Some A is not E
Of this Dr. Keynes says : " This is the only resolution of the
" sorites possible unless the order of the premises is transposed, and
" it will be seen that all the resulting syllogisms are in Figure II
" and in the mood Baroco. The sorites may accordingly be said to
" be in the same mood and figure. It is analogous to the Aristotelian
" sorites, the subject of the conclusion appearing in the premise
" stated first, and the suppressed premises being all minors in their
" respective syllogisms" (op. cit., p. 331).
ABRIDGED AND CONJOINED SYLLOGISMS.
399
As a Sorites in the Third Figure Dr. Keynes gives one of Book VT.
the following form (the omitted conclusions being in square Ch. VI.
brackets) : —
Some D
is not E,
Every D
is C,
Some C
is not
f].
Every C
is B,
Some B
is not
E],
Every B
is A,
Some A
is not
E.
" These syllogisms are all in Figure III and in the mood Bocardo ;
"and the sorites itself may be said to be in the same mood and
" figure. It is analogous to the Goclenian sorites, the predicate of
*• the conclusion appearing in the premise stated first, and the sup-
' ' pressed premises being majors in their respective sjdlogisms "
(ibid.).
These examples establish Dr. Keyues' contention, and the
criticisms passed upon it in the first edition of this book must be
withdrawn. The examples on which that criticism was based, and
which were given by Dr. Keynes in the earlier editions of his book,
were susceptible, as he himself admits, of more than one analysis,
and one of those analyses would resolve them into the first figure.
But this is not possible with the examples given above.
The special rules of sorites given in (ii.), of course, do not apply
to sorites in other figures. The occurrence of such sorites is so rare,
and their importance so small, that it is not necessary to here work
out such rules in detail. That task may be left to the ingenuity of
the reader.
(iv.) History of Sorites. The name Sorites is not employed by
Aristotle in the modern sense, though he alludes to such chains of
arguments. It appears to have been first used in this way by the
Stoics, from whom it was adopted by Cicero, though its common
acceptance was much later. Tlie Goclenian Sorites received its
name from Goclenius, who first discussed it in his Isagoge in
Organum Aristotelis (1598).
Ancient writers used the name Sorites — cywpbs, a heap— to denote
a particular kind of fallacy, based on the difficulty of assigning
an exact limit to a notion : — ' Does one grain of corn make a
lieap?' 'No.' 'Do two?' 'No.' 'Do three?' 'No.' Thus the number
may be successively increased by imity, till the person questioned
A chain of
enthymemes
was first
called a
sorites by
the Stoics.
The Gocle-
nian Sorites
was first dis-
cussed by
Goclenius.
In ancient
writers, So-
rites was the
name of a
fallacy.
400
SYLLOOISJIS.
Book VI.
Ch. VI.
has either to contradict himself by affirming that one grain dooa
make a heap of that which before its addition was not a heap,
or to deny the name to a pile of corn of any assignable magnitude,
no matter how enormous it might be. A similar sophism was
that which the old logicians termed Calvus, and which began with
the enquiry whether pulling one hair from a man's head made him
bald. Similarly, it might be asked, ' When does a kitten become a
cat Y Such fallacies really rest on a confusion between the collec-
tive and distributive use of terms {cf. § 171 (v.).
Epicheirema
—a regres-
sive chain of
reasoning
with one
premise of
each pro-
syllogism
omitted.
Symbolic
forms cf epi-
cbeiremas.
137. Epicheiremas.
An Epicheirema is a regressive chain of reasoning
abridged by the omission of one of the premises of each
prosyllogism.
Each prosyllogisin, therefore, appears in the epicheirema
as an enthymeme, though the episyllogism is stated in full.
Each prosyllogism furnishes a reason in support of one of
the premises of the episyllogism, and the whole epicheirema
may be described as a syllogism with a reason given in
support of one or both of its premises. When one premise
only is thus supported, the epicheirema is Single ; when both
are furnished with reasons, it is Double/ and when those
reasons themselves have other reasons attached to them, it is
Complex. The progress of thought in an epicheirema is
from the episyllogism to the prosyllogisms on which it
depends ; from the conclusion to the principles which
support it.
Symbolic examples of the Double Epicheirema are : —
(1)
(2)
Every M is P, because it is X,
Every S is M, because it is K,
Every S is P.
Every M is P, because every A is,
Every S is M, because every B is.
.'. Every S is P.
In the first case the enthymemes expressing the reasons
are both of the first order, the suppressed major premises
ABRIDGED AND CONJOINED SYLLOGISMS.
401
Book VI.
Ch. VI.
being Every X is P, and Everi/ Y is M. In the second case
both the enthymemes are of the second order, the implied
minor premises being Every M is A, and Every S is B. Of
course, both need not be of the same order. If we leave
out one of the reasons in either of the above examples we
have a single epicheirema. A complex Epicheirema would
be:—
Every M is P, because it is X. and every X is K,
Every S is M,
.'. Every S is P.
The full analysis of this would give the first example of a
regressive chain of reasoning given towards the close of
§ 135. Additional examples can be framed by omitting other
premises in the same complex reasoning. Of course, both
premises may be similarly supported by a chain of reasonings,
but the arguments then become very complex.
We will now illustrate what has been said by material Examplesof
examples of the two forms of double epicheirema given nias ^^'^^
above.
(1) 'All unnecessary duties on imports are impolitic,
as they impede the trade of the country ; the
American protective duties are unnecessary, as
they support industries which are quite able to
stand alone ; therefore, the American protective
tariff is impolitic'
(2) * All Malays are cruel, because all savages are ; all
the aboriginal inhabitants of Singapore are
Malays, because all the natives of that part of
Asia are ; therefore, all the natives of Singapore
are cruel.'
LOG. T.
26
CHAPTER VII.
Book IV.
Cb. VII.
Pliilosophers
of the Empi-
ricist School
— as Locke
and Mill —
have asserted
that all infer-
ence is from
particulars to
particulars ;
FUNCTIONS OF THE SYLLOGISM.
138. Universal Element in Deductive Reasoning.
The essential feature of syJlogistic reasoning is the sub-
sumption of a particular case under a general rule ; in other
words, every deductive inference must rest on a universal
element [cf. §§ 107, 115 (i.)]. This necessity has been denied
by philosophers of the empiricist school, who hold that all
knowledge is derived from experience. Thus, Locke (Essay
on the Human Understanding, Bk. IV, Ch. xvii, § 8) says :
" It is fit . . . to take notice of one manifest mistake in the
" rules of syllogism, viz., that no syllogistic reasoning can be
" right and conclusive, but what has at least one general
" proposition in it. As if we could not reason, and have
" knowledge about particulars ; whereas, in truth, the matter
" rightly considered, the immediate object of all our reason-
" ing and knowledge is nothing but particulars." Mill, in
his chapter on "The Functions and Logical Value of the
Syllogism " [Logic, Bk. II, Ch. Ill), adopts and expands the
same view. He says : " All inference is from particulars to
" particulars : General propositions are merely registers of
" such inferences already made, and short formulae for
" making more : The major premise of a syllogism, con-
" sequently, is a formula of this description : and the
" conclusion is not an inference drawnyrow the formula, but
" an inference drawn according to the formula : the real
■' logical antecedent, or premise, being the particular facts
' from which the general proposition was collected by
FUNCTIONS OF THE SYLLOGISM. 403
"induction" (§ 4). "Though there is always a process of Book IV.
" reasoning or inference where a syllogism is used, the ^h. vn.
" syllogism is not a correct analysis of that process of
" reasoning or inference ; which is, on the contrary (when
" not a mere inference from testimony), an inference from
" particulars to particulars ; authorized by a previous in-
" ference from particulars to generals, and substantially the
" same with it ; of the nature, therefore, of Induction. But
" while these conclusions appear to me undeniable, I must
*' yet enter a protest against the doctrine that the
"syllogistic art is useless for the purposes of reasoning,
'' The reasoning lies in the act of generalization, not in
"interpreting the record of that act; but the syllogistic
■' form is an indispensable collateral security for the correct-
" uess of the generalization itself. . . . The value ... of and Miu ro-
•' the syllogistic form, and of the rules for using it correctly, gy[io^*]j^ a,
" does not consist in their being the form and rules according valuable
"to which our reasonings are necessarily, or even usually test^of**"
" made ; but in their furnishing us with a mode in which '■easoninfj.
" those reasonings may always be represented, and which is
" admirably calculated, if they are inconclusive, to bring
" their inconclusiveness to light" {ihid., § 5). The "universal
"type of the reasoning process" is "resolvable in all cases
" into the following elements : Certain individuals have a
" given attribute ; an individual or individuals resemble the
" former in certain other attributes ; therefore they resemble
" them also in the given attribute" {ihid., § 7).
This view of the syllogism was accepted by Sir J,
Herschel, Dr. Whewell, Mr. Bailey, Professor Bain, and
other logicians ; but it has been strongly opposed by Mansel,
De Morgan, Dr. J. Martineau, Dr. Ray, Professor Bowen, and
Sir W. Hamilton, and is in conflict with the traditional logical
doctrine.
In examining this doctrine we will pass over the obvious
contradiction involved in saying that "a^/ inference is from
" particulars to particulars " and yet asserting the possibility
of an " inference from particulars to generals," and ask at
once whether we do really reason from particulars to par-
404
SYLLOGISMS.
Book IV.
Ch. VIL
But the in-
ierence is
really luade
irora a
universal
element
In the par-
ticular.
ticulars at all. The possibility of this has been strenuously
denied by Mr. Bradley, though he is by no means an
upholder of the syllogism. He says : " The thesis io be
" proved is that an inference is made direct from par-
" ticulars, as such, to other particulars. The conclusion
" which is proved is that from experience of particulars
" we somehow get a ])articular concluhion " {Principles of
Lorjic^ pp. 323-4). This conclusion may be granted al
once. We frequently do reason by analogy from our
experience of particulars to another particular instance,
and such reasoning is fairly described in the last sentence
quoted above from Mill, though we must demur to the claim
that it is the " universal type of the reasoning process.''
Such arguments are often fallacious, but even when they are
valid, on what do they really rest ? Surely on a generaliza-
tion. They are, as Mill says, arguments from resemblance.
But, " whenever we reason from resemblance we reason from
" identity, from that which is the same in several particulars
" and is itself not a particular. And is it not obvious that,
" in arguing from particular cases, we leave out some of the
" differences, and that we could not argue if we did not leave
" them out ? Is it not then palpable that, when the diHer-
" ences are, disregarded, the residue is a universal ? Is it
■' not once more clear that, in vicious inferences by analogy,
" the fault can be found in a wrong generalization ?" (Brad-
ley, ibid., p. 326). Mill's error springs from a too material
view of Logic \cf. § S (ii) (c)]. He fixes his attention upon
the whole concrete instance on which the generalization is
founded, and overlooks the fact that it is not from this
instance as a whole — e.e., as particular — that the conclusion
is drawn, but from only some elements of it, and that these
are made the basis of the inference simply because they are
regarded as common to all similar cases — i.e., are universal.
This is evident upon a careful examination of an example he
gives (ibid., § 3) : " It is not only the village matron, who, when
" called to a consultation upon the case of a neighbour's child,
" pronounces on the evil and its remedy simply on the recol-
" lection and authority of what she accounts the similar case
1
FUNCTIONS OF THE SYLLOGISM. 405
" of her Lucy." But why does she account it " a similar Book IV.
"case".^ Is it not because she regards the symptoms " '
observed in both cases as marks of the presence of a certain
disease ? But if so, she is reasoning, not from her Lucy as
an individual but, from the universal connexion between a
certain disease and the symptoms Lucy exhibited in her
sickness ; and thence she infers that tlie remedies which
proved efficacious in that case will prove equally beneficial,
not only in this new case of the neighbour's child but, in all
similar cases which may be brought under her notice.
Thus, even in cases where the inference is apparently
founded on one or more particular experiences, it is really
based on the universal element in which they agree ; and
this may be expressed in a general proposition which forms
the major premise of a syllogism.
* 139. Validity of Syllogistic Reasoning.
Not only has the syllogistic process been asserted to be xhe syiio-
valueless, but its very validity has been frequently denied, g''*™ ^'^^ ^^^'^
fl AHA pt" #*ll
on the ground that it involves the fallacy of petitio prin- to involve a
cipii. Strictly speaking, this should mean that the conclusion ?'?"/><'?'"«-
of every syllogism is itselE assumed as one of the premises ;
but, more loosely, it is held to imply that the premises pre-
suppose the truth of the conclusion, and cannot, therefore,
be used to establish it. This argument was advanced, in the
third century, by Sextus Einpiricus, who said that the major
premise must result from a complete testing of every
instance which can come under it, and that, therefore, to
deduce an individual fact from a general principle is to
argue in a circle. The same argument has been adopted by
the Empiricist school generally. Thus, Mill says : " It must
" be granted that in every syllogism, considered as an argu- JfJitsXs
" nient to prove the conclusion, there is a petitio principii ; '''^^^•
"... that no reasoning from generals to particulars can
" as such, prove anything : since from a general principle we
•' cannot infer any particulars, but those which the principle
26«
4UB
SYLLOGISJIS.
Book IV
Ch. VII.
Answers —
1. Theinajoi
premise is
not a mere
snmmatioi
of insrance
but gener-
ally ex-
presses a
necessary
connexion
of attri-
butes.
2. A minor
as well as a
major pre-
mise is an
essential
part of
every
syllopism.
" itself assumes as known " {Logic, Bk. II, Ch. iii, § 2).
Mill then proceeds to argue that the real inference is from
particulars to particulars, and that the syllogism is merely a
guarantee of the validity of those inferences {cf. § 138).
If a universal proposition be regarded as a mere ' universal
of fact,' or summary of examined particulars, the cogency
of this objection to the syllogism must be granted. Very little
' reflexion is, however, needed to show that the vast majority
of universal propositions are made on the strength of the
examination of a small number of instances ; indeed, no
writer has insisted on this more strongly than Mill himself.
The justification of such general propositions will be the
subject of the next book, in which we shall deal with Induc-
tion ; it is sufficient here to say that their force depends
upon a recognition of the fact that they are ' universals of
reason,' or expressions of necessary connexions of attributes.
The truth of such a proposition is recognized, and even held
to be necessary, before the totality of instances which come
under it have been examined, or are, indeed, known. For
instance, the laws of Kepler are syllogistically applied to all
newly discovered planets and satellites without a doubt of
the accuracy of the conclusion. Similarly, the universal
validity of the law of gravitation was held to be so certain,
that when the observed orbit of the planet Uranus appeared
to violate it, the existence of a disturbing cause was inferred
— an inference which led to the discovery of the planet
Neptune.
Again, a syllogistic inference requires the combination of
both premises, but the objection we are considering involves
the tacit assumption that the minor is unnecessary. " When
"you admitted the major premise," says Mill, "you asserted
"the conclusion" (iftz'c?). But, if so, surely the minor pre-
mise is superfluous. Mill, indeed, denies this, on the ground
that the major premise does not individually specify all it
includes, but only indicates them by marks, and that the
office of the minor premise is to compare any new individual
with the marks. *' But since, by supposition, the new in-
•' dividual has the marks, whether we have ascertained him
FUNCTIONS OF THE SYLLOGISM. 407
'• to have them or not ; if we have affirmed the major Book IV.
" premise, we have asserted him" to have them (Mill, ibUt, Ch^n.
§ 8, note). As, however, this assertion was evidently not
foreseen when the reasoner affirmed the major premise, Mill
has to introduce a novel doctrine of ' unconscious assertion.'
But, as assertion is an act of judgment, it cannot be un-
conscious ; and to say, as Mill does (ibid., § 2, note), that a
person can assert a fact which he does not know is not only
to talk very bad psychology but to fall into an absolute
contradiction in terms. The necessity to a syllogistic in-
ference of the minor premise is, then, a proof that such an
inference is not a petitio principii.
If the syllogism were really open to the charge of petitio 3. Knowledge
principii, it would, of course, follow that no advance could be ^'^^ 'dr'
made in knowledge by its means. But the objection springs syllogistic
from a too objective view of Logic ; from neglecting to 'Argument,
remember the difference between what is in the facts of the j-'^^ '^'.'^'j^®
3xternal world, and what we know to be in them. Inference syllogism ia
cannot, of course, give us more than already exists in the obfecUve*'''*
vorld, but it may help us to see and understand more. It view of
is, indeed, our imperfect knowledge which makes inference "
of new truths possible. Were our knowledge complete, all ^"P^^j^^^g^
truths would lie open before us, and such inference would luakes
be both unnecessary and impossible. For the trutli of the ,,ossibie.
conclusion is, in fact, concomitant with that of the premises
from which we deduce it ; it does not succeed them, thougli
Dur perception of ic may follow our perception of them. For,
though the objectors to the syllogism deny the fact, it is
certainly possible to accept the premises without deducing uispossibie
the conclusion. The shortness of the syllogistic process, t'^accepttiie
and the triteness of the examples of it commonly given in aud not
treatises on Logic, disguise this possibility, and give plausi- conclusion,
bility to the assertion that no advance in knowledge is really
made by syllogism. But, because, as statements of fact, the
premises contain the conclusion, it by no means follows that
"in studying how to draw the conclusion, we [are] studying
" to know what we knew before. All the propositions of
" pure geometry, which multiply so fast that it is only a
408
SYLLOGISMS.
Book IV.
Ch. VII.
This possi-
bility is ex-
emplified
in mathe-
matics.
4. Proof does
not depend
on novelty.
Bumm!\ry.
" small and isolated class even among mathematicians who
" know all that has been done in that science, are certainly
"contained in, that is necessarily deducible from, a very
"few simple notions. But to he known from these pi'emises
" is very different fi'om being known loith them. Another
" form of the assertion is that consequences are virtually
"contained in the premises, or (I suppose) as good as con-
" tained in the premises. Persons not spoiled by sophistry
" will smile when they are told that knowing two straight
" lines cannot enclose a space, the whole is greater than its
" part, etc, — they as good as know that the three inter.sections
" of opposite sides of a hexagon inscribed in a circle must be
" in the same straight line. Many of my readers will learn
" this now for the first time ; it will comfort them much to
" be assured, on many high authorities, that they virtually
" knew it ever since their childhood. They can now ponder
" upon the distinction, as to the state of their own minds,
"between virtual knowledge and absolute ignorance" (De
Morgan, Formal Logic, pp. 44-5).
Nor, indeed, even were this objection true would it be to
the point. It is a psychological, not a logical, objection. A
proof does not cease to be a proof because it is thoroughly
familiar to any individual mind. The conclusions of geometry,
for example, do not cease to be inferences from mathe-
matical axioms and definitions because the process of reason-
ing by which they are reached is understood and remembered.
We may, indeed, look upon a formally stated syllogism as
an analysis of the mode of deductive inference, and as such
an analysis it makes explicit elements which, in the actual
drawing of the inference, may be implicit, and so escape
superficial observation. But, as was shown in the last section,
in all deductive inference there must be an application of a
universal judgment to a particular case ; in other words, the
elements of syllogism must be present though each may not
separately engage attention.
"VVe may, then, sum up our answer to the charge of petit io
principii brought against the syllogism under four heads : —
the major is essentially not a mere summation of observed
FUNCTIONS OF THE SYLLOGISM.
409
instances ; the minor is a necessary part of every syllogism ;
it is possible to accept the premises without drawing the
conclusion, and hence to make progress in knowledge by
means of syllogism ; and the fact of inference depends on
the rigidity of the proof, not on its novelty.
Book IV,
Ch. VII.
140. Limitations of Syllogistic Reasoning.
Having shown the validity and value of the syllogism, we
have now to enquire whether it is the only type of valid
mediate inference. This has been strongly asserted by many
logicians. Thus Whately claims that the syllogism is " the
"form to which all correct reasoning may be ultimately
" reduced" {ElemeiUs of Logic, 5th Ed., pp. 14-5) ; Professor
Bowen asserts that "Reasoning, as such, must always be
"syllogistic" {Logic, p. 353); and Dr. Ray says: "The
" syllogism is the type of all valid reasoning ; for no reason-
"ing will be valid . . . unless it can be thrown into the
" form of a syllogism" {Ded. Log., p. 254).
In opposition to these claims it has been pointed out that
the syllogism deals only with propositions which express the
relation of subject and attribute, and that inferences from
other relations, though they may be perfectly valid, not
only are not made syllogistically but, cannot be satis-
factorily expressed in that form. Such, for example, is the
argumentum a fortiori— A is greater than B,B is greater than C ;
therefore, A is greater than C. Various attempts have been
made to express such arguments syllogistically, the most
successful of which is Hansel's (Art. Log. Rud., 3rd. Ed
p. 198)—
" Whatever is greater than a greater than C is greater thanC;
" A is greater than a greater than C,
" Therefore, A is greater than C."
But the whole argument is really assumed in the major
premise, and the inference is, therefore, invalidated by a
petitio principii ; moreover, B does not appear in the premises,
which cannot, therefore, express the whole argument.
Many logi-
cians have
claimed that
the syllo-
gism is the
only type
of valid
reasoning.
But infer-
ences from
relations
other than
that of sub-
ject and at-
tribute can
not be ex-
pressed syl-
logistically.
410
SYLLOGISMS.
Book IV.
Ch. VIL
A Logic of
Relatives
would deal
with all re-
lations,
but will pro-
bably never
be worked
out,
though at-
tempts to
classify rela-
tions have
been made,
of which Mr.
Bradley's is
the most
successful.
If, then, account is to be taken of all valid inferences, we
need a Logic of Relatives which " shall take account of
" relations generally, instead of those merely which are
" indicated by the ordinary logical copula * is.' " (Venn,
Symbolic Logic, p. 400.) Such a logic has never been worked
out, and, perhaps, never will be ; for, as Dr. Venn says {ihid.,
p. 403) : " the attempt to construct a Logic of Relatives
" seems . . . altogether hopeless owing to the extreme
" vagueness and generality of this conception of a Relation."
Some attempt to classify copulas of relation was, indeed,
made by De Morgan {Syllahus, pp. 30, 31), who divided them
into convertible " in which the copular relation exists between
" two names both ways" and inconvertible^ in which it does
not ; and into transitive " in which the copular relation joins X
"with Z whenever it joins K with K and Y with Z," and
intransitive, in which it does not. As an example of a copula
which is both convertible and transitive, De Morgan gives
* is fastened to.' But the great majority of relations would
be both inconvertible and intransitive, and the classification
cannot be said to have much value. Mr. Bradley {Logic,
pp. 243-4) gives a list of relations, which though it " does not
pretend to be complete " is yet, probably, the best classifica-
tion which has yet been put forward. He calls them
"principles of inference," and enumerates five : —
(1) Synthesis of Subject and Attribute. Under this all
syllogistic inferences can be brought,
"(2) Synthesis of Identity. Where one term has one
" and the same point in common with two or
" more terms, there these others have the same
" point in common as *If >4 is the brother
" of B, and B of C, and C is the sister of D, then A
" is the brother of D.'
"(3) Synthesis of Degree, When one term does, by virtue
" of one and the same point in it, stand in a
" relation of degree with two or more other
" terms, then these others are also related in
" degree . . as ' ^ is hotter than B and B than C,
FUNCTIONS OF THE SYLLOGISM.
411
"therefore \ than C " The argumentum d fortiori Book IV.
cornea under this head. Ch^ll.
" (4 and 5) Syntheses of Time and Space. When one and
"the same term stands to two or more other
" terms in any relation of time or space, there we
" must have a relation of time or space between
" these others. Examples : ' A ia north of B and
" B west of C, therefore C south-east of y4 ' ; M is a
" day before 8, B contemporary with C, therefore
"C a day after A'"
The validity of the arguments in classes (2) to (5) may be
granted at once, as may the fact that they are not syllogistic.
But it must be pointed out that neither are they deductive ;
for in them is no subordination of a special case under a
general principle, but an inference of co-ordination from
particular to particular. No doubt, the validity of the
inferences rests upon material considerations of degree, time,
space, etc., which are universally applicable ; but these con-
siderations stand in the same relation to the special arguments
as the dicta of the four figures do to the syllogisms in those
figures ; and are not, therefore, the implied major premises
of the arguments. The syllogism remain.s, then, as the one
type of deductive reasoning, and should not be discarded on
account of the existence of these other valid inferences,
whose scope is not very great, and whose want of generality
must always make them of but little importance. On the
contrary, as Leibniz says (Noiiv. Ess., iv, 17, § 4): "The
" discovery of the syllogism is one of the most beautiful and
•' greatest ever made by the human mind ; it is a kind of
" universal mathematic whose importance is not sufficiently
' known, and when we know and are able to use it well, we
" may say that it has a kind of infallibility : — nothing can be
" more important than the art of formal argumentation
" according to true logic."
Argumen «
from rela
tions other
than subject
and attri-
bute are not
deductive.
The syllo-
gism is the
one type of
deductive
inference.
END OF VOLUME I.
INDEX.
The Roman numerals (i., ii.) refer to the volumes, the Arabic figures
to the pages.
A dicto s.q., ii. 232 ; 235 ; 243-6
Absolute terms, i. 75-6
Abstract terms, i. 72-5
Accentus, ii. 231 ; 235 ; 252
Accidens, predicable, i. 79 ; 80;
85-6
fallacy, ii. 232 ; 236 ; 255-6
Accidental propositions, i. 88 ;
160-1
Activity of thought, ii. 2-5
Adamson : on quantification of
predicate, i. 207
Added determinants : inference
by, i. 268-70
^quivocatio, ii. 231 ; 235 ; 238-
42.
Affinity in classification, i. 142-3
Affirmative propositions, i. 161
Agreement : Method of, ii. 142 ;
144; 152-3; 157
Alphabetical classification, i. 138
Ambiguities of language, i. 5-9
Amphibolia, ii. 231 ; 235; 250-1
Ampliative propositions, i. 88 ;
1(10-1
Analogous terms, ii. 268-9
Analogy, ii. 64-6; 71-82
confirmation of, ii. 80-2.
fallacies in, ii. 236 ; 267-70
logical character, ii. 74-6
— — relation to enumerative in-
duction, ii. 71-4
LOG. I. II. 1
Analogy : strength of, ii. 76-80
suggestive value, ii. 64-6 ;
73-4; 86
Analysis: metaphysical, i. 126
methodof,ii. 212-4; 219-20
qualitative, ii. 70-4; 121-41
Mill's Methods of, ii.
141-59
quantitative, ii. 160-87
Analytic chains of reasoning,
i. 392; 400-1 ; ii. 212-4; 219-
20
Analytic propositions, i. 88 ;
160-1
Analytical keys, i. 132-3; 139
Analytically-formed definitions,
i. 121
Ants: sense of hearing, ii. 127-
30
Apodeictic judgments, i. 194-5
Application of concepts, i. 64
Applied Logic, i. 20-3
Argon, discovery of, ii. 133-7
nature of, ii. 137-41
'Argument,' i. 279
'Argumentation,' i. 279
Argumentum a fortiori, i. 39 ;
409
ad absurdum, ii. 291
ad haculum, ii. 289
ad hominein, ii. 289
ad ignorantiam, ii. 290
INDEX.
Argumentiim ad popnlum, ii.
289-90
ad veremindiam, ii. 290
Aristotelian sorites, i. 393-7
Aristotle : classification of fal-
lacies, ii. 231-2
doctrine of induction, ii.
32-3
on modality, i. 193-4
on non causa pro causa, ii.
290; 292
sclienie of categories, i.
90-y
scheme of predicables, i.
78-80
view of analogy, ii. 75
Art distinguished from Science,
i. 12
Artificial classification, i. 136-7
Assertory judgments, i. 194-5.
Assumption, i. 279
Atoms : natui-e of, ii. 73-4 ; 98
Attribute terms, i. 73
Attributive view of predication,
i. 209-11
Austin : definition of ' law,' ii.
200
Averages : constancy of, ii. 198-
200
Axioms : character of, ii. 278
mathematical, i. 39 ; ii.
201-5
of categorical syllogism, i.
283-6
rules of, ii. 222
undue assumption of, ii.
236; 278-9
Bacon : classification of fallacies,
ii. 35
doctrine of induction, ii.
34-9
on crucial instances, ii.
102-3
on enumerative induction,
ii. 36
on influence of language on
thought, ii. 239
Bain : on Aristotle's categories,
i. 97-8
on material obversion, i.
254-5
on syllogisms with singular
premises, i. 334
Ballot-box theory of nature, ii.
53-4
Baibara, i. 324-8
Barbari, i, 323
Baroeo, i. 332-3
Basis of division, i. 123
■ of induction, ii. 55-8
■ of syllogistic reasoning, i.
282-3
Baynes : on Aristotle's cate-
gories, i. 9"! -6
Beauty of flowers : origin of, ii.
123-4
Benecke : on connotation, i. 55
Bias : effect on observation, ii.
Ill
Bifid division, i. 130-3
Blind experiment, ii. 118-20
Bocardo, i. 336
Boccaccio : fallacy of roasted
stork, ii. 245
Boole : on universe of discourse,
i. 60
Bosanquet : definition of hypo-
thesis, ii. 62
of induction, ii. 58.
on analogy, ii. 71.
on connotation, i. 56
on ground and cause, ii. 30
on inconceivabilit)' of con-
tradictory, ii. 206
on inseparable association,
ii. 204
on logical existence, i. 211
on nature of experiment,
ii. 116
on negative instances in
analogy, ii. 81
on perception and inference,
ii. 114
on qualitative analysis, ii.
122
INDEX.
Bosanquet : on reality, ii. 3
on relation of deduction and
induction, ii. 61
on uniformity of nature in
substances, ii. 195
on unity of knowledge, ii.
207
on vera causa, ii. 94
Bowen : on integration, i. 203.
on iiniversality of syllogism,
i. 409
Bradley: on inference from nega-
tive premises, i. 296
on inference fi'om par-
ticulars, i. 404
on infinite judgments, i.
162
on laws of thought, i. 32 ;
33; 36
on logic of relatives, i.
410-1
on Mill's inductivemethods,
ii. 146-7; 149; 150
Bramantip, i. 337-8
Breadth of concepts, i. 64
Brown : on causation, ii. 16
on hypotheses, ii. 86-7
Buckle : on constancy of suicide-
rate, ii. 199
Burke : on false analogy between
community and individual, ii.
270
Cairnes : on limiting cases in
definition, i. 113
Camenes, i. 338 ; 340-1
Camenos,!. 323; 340-1
Camestres, i. 331-2
Camestros, i. 323
Canons of pure syllogism : cate-
gorical, i. 287-98
corollaries from, i. 302-4
derivation of, i. 287-8
disjunctive, i. 305
hypothetical, i. 304-5
• simplification of, i. 298-
302
of mixed syllogism, i. 365
Cams : on cause and effect, ii. 26
Categorematic words, i. 42-3
Categories : Aristotle's scheme,
i. 90-9
Descartes' scheme, i. 100
Kant's scheme, i. 103-6
Mill's scheme, i. 101-3
nature of, i. 89
Spinoza's scheme, i. 100
Stoic scheme, i. 100
Thomson's scheme, i. 100
Categorical propositions, i. 13-4 ;
156-80; 196-224; 228-44;
248-70 [see Propositions)
syllogisms, i. 282-347 ; 362-
60; 402-11 {see Syllogisms)
Causa cognoscendi, ii. 29
essendi, ii. 29
imtnanens, ii. 20
transiens, ii. 20
vera, ii. 93-95; 98
Causation, ii. 10-31
a postulate of knowledge,
ii. 11-2
and conservation of energy,
ii. 24
— — axiom of, ii. 25-8
Hume's doctrine, ii. 12-4
Mill's doctrine, ii. 16-9
modern empirical view, ii.
14-6
rational doctrine, ii. 19-25
Cause and ground, ii. 28-30
Causes : final, ii. 30-1
plurality of, ii. 18-9 ; 27
theories of, ii. 52
Celarent, i. 328
Celaront, i. 323
Cesare, i. 330-1
Cesaro, i. 323
Cessante causa cessat effeeUis, ii.
23
Chains of reasoning, i. 390-401
Characteristics of syllogistic
figures, i. 312-5
Circulus in dejinietido, i. 116-7
in demonstrando, ii. 282
Circumstantial evidence, ii. 84-5
3
INDEX.
Classification : affinity in, i.
142-3
alphabetic, i. 138
and unity of nature, ii. 10
artificial, i. 136-7
by series, i. 145-6
by type, i. 144
evolution in, i. 142-3
general, i. 139-44
natural, i. 136-7; 139-44
nature of, i. 134-6
of fallacies, ii. 235-6
Aristotle's, ii. 231-2
Bacon's, ii. 35
Mill's, ii. 233-5
Whately's, ii. 232-3
■ rules of, i. 127-9 ; 137 ;
141-4
special, i. 137-9
Class-inclusion view of predica-
tion, i. 198-200
Class terms, i. 48-51
Clifford : on explanation, ii.
190-1
on Lord Kelvin's hypothesis
as to nature of atoms, ii.
73-4; 98
on proof of hypothesis, ii.
104-5
on scientific thought, ii.
62-3
on theories of light and
matter, ii. 106-7
Co-division, i. 123-4
Co-existence : uniformities of, ii .
47-8
Cognate genus, i. 82
species, i. 82
' Collection ' in syllogism, i. 280
Collective terms, i. 49-60
Colligation of facts, ii. 50
Compartmental view of predica-
tion, i. 220-1
Complete definition, i. 121
Complex conception : inference
by, i. 270
Compositio, ii. 231 ; 235; 246-8
Compound propositions, i. 178-80
Comprehension of concepts, i. 64
Comprehensive view of predica-
tion, i. 208-9
Comte : on object of science, ii.
91
on test of theory, ii. 100
Conception : views as to, i. 16-7
Conceptualism, i. 17
Conceptualist view of Logic, i. 1 7
Conclusio ad subalternantem, i.
229
ad subaltertiatam, i. 229
Conclusion of syllogisms, i. 277
Concomitant variations : method
of, ii. 144; 145-6; 156; 157
Concrete terms, i. 72-4
Conditional propositions, i. 184 ;
244-6; 271-3 {see Hypo-
thetical Propositions)
syllogisms: mixed, i. 362-70
{see Hypothetical Si/Uo-
gisms)
pure, i. 304-5 ; 348-50 ;
360-1 {see Hypothetical Syllo-
gisms)
Connotation, i. 51-7 ; 60-4
difficult)' of assigning, i.
56-7
limits of, i. 54-6
• of abstract terms, i. 74-5
^ of proper names, i. 45-6,
53-4
relation to denotation, i.
60-4
synonj'ms of, i. 64
Connotative view of predication,
i. 209-11
Consequens, ii. 232 ; 236 ; 256-7
Consilience of inductions, ii. 61
Consistency : Axiom of, i. 32-3
Construction of the conception,
ii. 62
Constructive definition, i. 120-1
Content, i. 65
Contingent propositions, i. 193
Continuity : principle of, ii. 208-9
Contradiction of propositions, i.
232-4; 245; 246-7
INDEX.
Contradiction of terms : formal,
i. 07-70
• material, i. 65-7
principle of, i. 33-4
Contraposition of propositions,
i. 262-4: 271-3; 274
fallacies in, ii. 236 ; 257
Contrapositive, i. 262
Contrariety of propositions, i.
234-6; 245; 246
of terms, i. 70-1
Conventional language : charac-
ter of, i. 2-3
• growth of, i. 6-9
Converse, i. 255
Conversio per accidens, i. 256 ; ii.
255
ConTersion of propositions, i.
255-62; 271-3; 274; ii. 63-4
fallacies in, i. 257 ; 260-1 ;
ii. 236 ; 255-7
Convertend, i. 255
Copula, i. 40 ; 157-S
Copulative propositions, i. 178
Gormitiis, ii. 254-5
Corollaries from canons of pure
syllogism, i. 302-4
Correlative terms, i. 76-7
Cosmology, i. 25
Criterion of truth, ii. 205-7
Crocodilus, i. 386.
Crookes, experiments on : argon,
ii. 138; 140
momentum of light, ii. 119
Crucial instances, ii. 102-4
Ctim hoc ergo propter hoc, ii.
274
Darapti, i. 333-4
Darii, i. 328-9
Darwin : observation of orchids,
ii. 113
observations on formation
of vegetable mould, ii.
124-7
on origin of beauty of
flowers, ii. 123
Batisi, i. 335
Davis : on character of proof, ii.
284
on non causa pro causa, ii.
290
on paronyms, ii. 243
on petitio principii in Aris-
totle, ii. 284
on question-begging epi-
thets, ii. 281
Davy : discovery of unsuspected
conditions, ii. 120
' Deduction' in syllogism, i. 280
Deductive reasoning : relation
to inductive, ii. 54-5 ;
60-1
universal element in,
i. 402-6
sciences, ii. 212-3
Definition : analytically-formed,
i. 121
by type, i. 122
circle in, i. 116-7
complete, i. 121
connexion with discovery,
i. 111-2
constructive, i. 120-1
essential, i. 121
fallacies in, i. 114-8; ii.
235 ; 237-8
functions of, i. 107-8
genetic, i. 120-1
ignotum per mqne ignotum,
i. 116
ignotum per ignotius, i. 116
imperfect, i. 121-2
incomplete, i. 121-2
kinds of, i. 118-22
• limits of, i. 110-4
negative, i. 118
nominal, i. 118-20
per genus et differentiam, i.
108-10
perfect, i. 121
predicable of, i. 79
real, i. 118-20
relation to division, i. 125
rules of, i. 114-8 ; ii. 221
substantial, i. 120
INDEX.
Definition: synthetically-formed,
i. 121
too narrow, i. 115
• too wide, i. 115
-utility of, i. 107-8
verbal, i. 118-20
Demonstration : rules of, ii. 222
De Morgan : corollaries from
rules of syllogism, i.
302-3; 304
on ambiguous sentences, ii.
251
on ambiguous terms, ii.
239; 241: 242; 248
on classification of fallacies,
ii. 230
on errors in measurement,
ii. 164-5
on exceptions to a rule, ii,
272
— — on extreme cases, ii. 271
on fallacies a dicto s.q., ii.
244-5 ; 246
on fallacies of accent, ii. 252
on frequency of fallacy, ii.
229-30
on iqnoratio elenchx, ii. 286 ;
287 _
on logic of relatives, i. 410
on logical existence, i. 340
on mathematical inductions,
ii. 46
on method of induction, ii.
59-60
on modality, i. 192-3
on paradox, ii. 228-9
on paralogisms, ii. 228
on scholastic logic, ii. 283
on simplification of rules of
syllogism, i. 299
on squaring the circle, ii.
280
on sympathetic powder, ii.
264
on use of illustrations, ii.
288
on validity of syllogism, i.
407-8
De Morgan : on value of logic, i.
24
on vera causa, ii. 93
Denotation, i. 57-64
Depth of concepts, i. 64
Descartes : rules of method, ii.
214-7
scheme of categories, i, 100
Description, i. 122
Desitive propositions, i. 180
Determinant, i. 268
Determination of magnitude, ii
52; 160-87
of syllogistic moods, i. 315-
22 _
Determinative subordinate
clauses, i. 177
Diagnosis, i. 143-4
Diagrams : Euler's i. 216-9 ;
341-6
Lambert's, i. 219-20 ; 346
proposed scheme, i. 222-4
representation : of proposi-
tions by, i. 215-24
of syllogisms by, i.
341-7
use of, i. 215-6
Venn's, i. 220-2 ; 446-7
Dichotomy: division by, i. 130-3
Dictum de diverso, i. 308
■ de exemplo, i. 309
• de omni et nnllo, 285-6
de reciproco, i. 310
Difference : Method of, ii. 143 ;
144; 154-5; 157-8
Differentia, i. 80 ; 83-4
Digby, Kenelm : sympathetic
powder, ii. 264
Dilemmas : definition of, i. 376
forms of, i. 376-83
rebutting, i. 384-6
reduction of, i. 383-4
Dimaris, i. 338-9
Direct reduction, i. 355-8
Disamis, i. 334-5
' Discourse,' i. 280
universe of, i. 59-60
Discovery; method of, ii. 213- 1
6
INDEX.
Discretive propositions, i. 178-9
Discursive reasoning, i. 280 ; 334
Disjunctive propositions, i. 187-
92; 246-7; 274 {see Fro-
posifions)
syllogisms : mixed, 1. 281 ;
371-5
pure, i. 281 ; 305 ;
350-1
Distinction of meanings of words,
i. 126
Distinctive explanation, i. 122
Distribution of terms, i. 172-3
Dividend, i. 123
Dividing members, i. 123
Divisio : fallacy, ii. 231 ; 235 ;
246-8
nonfaciat saltum, i. 124
Division : basis of, i. 123
bifid, i. 130-3
• character of, i. 123-4
dichotomous, i. 130-3
distinguished from partition
and analysis, i. 126
fallacies in, i. 124 ; 127-9 ;
ii. 235 ; 248
material, or classification,
i. 134-46
material element in, i. 125-6
operations resembling, i.
126
purely formal, i. 133-4
relation to definition, i.
125
rules of, i. 127-9
too narrow, i. 128-9
too wide, i. 129
utility of, i. 125
Divisions of Logic, i. 13-9
Eductions : added determinants,
i. 268-70
complex conception, i. 270
contraposition, i. 262-4 ;
271-3; 274
• conversion, i. 255-62 ; 271-
3; 274
definition of, i. 248
Eductions : inversion, i. 265-6 ;
271-3; 274
kinds of, i. 248-50
obversion, i. 251-5 ; 271-3 ;
274
of categorical propositions,
i. 218-70
of disjunctive propositions,
i. 274
of hypothetical propositions,
i. 271-3
summary of chief, i. 267
Effect : analysis of, ii. 27-8
Elements of syllogism, i. 277-80
Empirical laws, ii. 43-4 ; 197-
200; 272-4
Empiricism, i. 402-6; ii. 1-2;
6-7; 12-9; 33-9; 40-8; 53-4;
55-7
Enthymemes : definition of, i.
387
orders of, i. 387-90
Enumeration of instances, ii.
55-6; 68-70
Enumerative induction : ii. 63 ;
66-71
relation to analogy, ii. 71-4
Epicheiremas : definition of, i.
400
kinds of, i. 400-1
Episyllogism, i. 391
Episyllogistic chains of reason-
ing, i. 390-1 ; 393-400; ii.
212-4; 221-5
Equivocal terms, i. 44-5 ; 126 ;
ii. 238-42
Essential definition, i. 121
propositions, i. 88 ; 160-1
Ether : concept of, ii. 97
I'iuler : diagrammatic representa-
tion of propositions, i.
216-9
diagrammatic representa-
tion of syllogisms, i. 341-6
Evolution in classification, i.
142-3
Example : argument from, ii. 33 ;
76
DTDEX.
Exceptive propositions, i. 178-80
Excluded Middle : principle of,
i. 34-7
Exclusive propositions, i. 179
Existence, implications of : in
predication, i. 211-4
in reduction, i. 359-60
in syllogisms, i. 340-1
Existential view of predication,
i. 220-1
Experience, ii. 2-5
individual, ii. 3-5
universal, ii. 3.
Experiment : and hypotheses, ii.
117
blind, ii. 118-20
■ character of, ii. 114-6
■ function of, ii. 117-20
Mill's Methods of, ii. 141-
59
natural, ii. 116-7
■ necessity of, ii. 114-5
negative, ii. 118-20
relation to observation, ii.
114-7
■ symbolic statement of prob-
lem of , ii. 117-8
Experimental sciences, ii. 212-3
Experimentum crucis, ii. 102-4
Explanation : distinctive, i. 122
• nature of, ii. 188-91
Explicative propositions, i. 88 ;
160-1
subordinate clauses, i. 177
Exponible propositions, i. 179-
80
Exposition : method of, ii. 213-4;
220
Extension of concepts, i. 64
Extremes of syllogism, i. 277
Fact and theory, ii. 2-3; 11 ; 49
Fallacies: a dicto s.q., ii. 232
235; 243-6
■ Accentus, ii. 231; 235
252
- — Accidens, li. 232 ; 236
255-6
Fallacies: ^quivocaiio, ii. 231 ;
235; 238-42
ambiguous middle, i. 289-
90 ; ii. 238
Auiphibolia, ii. 231 ; 235 ;
250-1
change of basis of division,
i. 127; 128; ii. 235; 248
classification of, ii. 235-6
Aristotle's, ii. 231-2
Bacon's, ii. 35
■ ■ Mill's, ii. 233-5
Whately's, ii. 232-3
Compositio, ii. 231 ; 235 ;
246-8
Consequens, ii. 232 ; 236 ;
256-7
contradictory definition, ii.
235; 237-8
Crocodilus, i. 386
Divisio, u.. 231 ;
246-8
false analogy, ii.
267-70
false opposition, ii. 235 ;
254-5
Figura dictionis, ii. 231 ;
235; 242-3
four terms, i. 289-93; ii.
236; 259
Igyioratio elenchi, ii. 232 ;
236; 285-90
- — illicit contraposition, ii.
236; 257
illicit conversion, i. 257 ;
260-1 ; ii. 236 ; 255-7
illicit inversion, ii. 236;
257-8
illicit major, i. 292-3 ;
236; 259
illicit minor, i. 292;
236; 259
in conception, ii. 235 ;
237-48
in deductive inference, i.
289-93; ii. 236; 259-60
in disjunctive propositions,
ii. 235 ; 253
235 ;
236
11.
11.
8
INDEX.
Fallacies : in hypothetical pro-
positions, ii. 235 ; 253
in immediate inference, ii.
235-6 ; 254 8
in inductive inference, ii.
236; 261-77
in judgment, ii. 235 ;
249-53
in method, ii. 236 ; 278-92
Litigiosus, i. 385-6
logical, ii. 232
material, ii. 233
nature of, ii. 227-30
Xon causa pro causa, ii.
232; 236; 290-2
non-consecutive division, i.
124; 127; 129; ii. 235;
248
non -exhaustive division, i.
127; 128-9; ii. 235 ; 248
of assumption of axioms, ii.
236; 278-9
of confusion, ii, 234
of deiinition, i. 114-8; ii.
235; 237-8
of generalization, ii. 234 ;
236; 270-7
• of observation, ii. 234 ; 236 ;
261-7
of ratiocination, ii. 234
• of simple inspection, ii. 234
Petitio principii, ii. 232 ;
236; 279-86
Plures interrogationes, ii.
232; 235; 254-5
self-contradictory judg-
ment, ii. 235 ; 249
Sorites, i. 399-400
undistributed middle, i.
291-2; ii. 236; 259
Faraday : experiments on elec-
trical conduction, ii. 102
experiments on source of
power in voltaic pile, ii.
131-3
Felapton, i. 335-6
Ferio, i. 330
Ferison, i. 336-7
Fesapo, i. 339
Fcstino, i. 332
Figura dictionis, ii. 231; 235;
242-3
Figure : axioms and special rules
of, i. 307-12
characteristics of, i. 312-6
— — distinctions of, i. 306-7
in pure disjunctive syllo-
gisms, i. 350-1
in pure hypothetical syl-
logisms, i. 348-50
of sorites, i. 398-9
value of, i. 312-5
Final causes, ii. 30-1
First Figure : axiom and special
rules of, i. 285-6; 307-8
■ characteristics of, i. 312-3
moods of, i. 324-30
Flamsleed : on value of scientific
instruments, ii. 183
Force of concepts, i. 64
Form : Bacon's doctrine of, ii.
35-6
Formal division, i. 133-4
Logic, i. 20-3
Four terms, fallacy, i. 289-93 ;
ii. 236; 259
Four-fold scheme of propositions,
i. 171-3
Fourth figure : axiom and special
rules of, i. 310-1
characteristics of, i. 314
moods of, i. 337-40
Fowler : on dilemmas, i. 383
on fallacies of observation,
ii. 263
on undue respect for
authority, ii. 273
Fresison, i. 339-40
Fresnel : discovery of circular
polarization, ii. 100-1
Functions of definition, i. 107-8
of language, i. 3-5
of reduction, i. 352-3
Fundamental syllogisms, i. 322
Fundamentum divisionis, i. 123
relationis, i. 77
9
INDEX.
Gfctieral propositions, i. 164-7
terms, i. 48-51
Genei'alization : and resem-
blance, ii. 195-6
■ connection with induction,
ii. 191-2
■ empirical, ii. 197 - 200 ;
272-4
fallacies in, ii. 234 ; 236 ;
270-7
in language, i. 7-8
nature of, ii. 192-4
possibility of error in, ii.
194-5; 270-7
Generic differentia, i. 84
judgment, i. 166 ; 167.
proprium, i. 85
Genetic definition, i. 120-1
Genus, i. 79 ; 80 ; 81-3
Gesture language, i. 2
Goclenian sorites, i. 393 ; 394-
6; 397
GoUancz : on date of Two Gentle-
men of Verona, ii. 85
' Government,' ambiguity of, ii.
240-1
Grammar : relation to Logic, i.
28-9
universal, i. 28
Green, T. H. : on change, ii.
10-11
— — on inconceivability of con-
tradictory, ii. 206
on necessary truths, ii. 205
on reality, ii. 2-3 ; 7-8
on uniformity of nature,
ii. 7; 8-9
on Whewell's doctrine of
induction, ii. 50-1
Grote : on Aristotle's categories,
i. 92-3; 98-9
on Greek geometrical rea-
soning, ii. 256
on non causa pro causa, ii.
291
Ground and cause, ii. 28-30
Growth of language, i. 6-9
Hamilton : arrangement of cate-
gories, i. 99
axiom of induction, ii. 34
axiom of syllogism, i. 284
• on circular reasoning in
Plato, ii. 282
on dilemmas, i. 381-2
on figure of sorites, i. 398
on judgments in compre-
hension, i. 208
on real and verbal defini-
tions, i. 119
postulate of Logic, i. 38-9
quantification of predicate,
i. 200-7
Hegel : on nature of analogy, ii,
76 _
on superficial analogies, ii.
76-7
Herschel : on fallacies of obser-
vation, ii. 266
on fusion of marble, ii.
101-2
on nomenclature, i. 147
on object of induction, ii.63
-on previous knowledge in
observation, ii. 112
■ on simplicity of theories,
ii. 107
on theories of light, ii,
103-4
on vera causa, ii. 93
Hobbes : definition of name, i,
41
on cause, ii. 21
Hoffding : on causation, ii. 26
Momonymia, ii. 238-42
Hume : doctrine of causation, ii,
12-4
Hutton : theory of rock forma-
tion, ii. 101-2
Huxley : on universality of law,
ii. 12
Hypotheses non Jingo, ii. 40 ; 86
Hypothesis : agreement with
fact, ii. 99
conditions of validity, ii.
95-9
10
INDEX.
Hypothesis : definition of, ii.
62
■ descriptive, ii. 88-90
development of, ii. 83-108
establishment of, ii. 104-8
• extension of, ii. 100-2
■ formation of, ii. 83-8
function of, ii. 83-5
kinds of, ii. 88-92
of cause, ii. 90-2
of law, ii. 90-2
rules of formation, ii. 86-7
simplicity of, ii. 107-8
suggestion of, ii. 63-6
working, ii. 88-90
Hypothetical propositions, i . 155 ;
181-7; 244-6; 271-3 {see
Fropositions)
syllogisms : mixed, i. 281 :
362-70 {see Si/Uogisms)
pure, i. 281 ; 304-5 ;
348-50; 360-1 {see
Syllogisms)
Hysteron proteron, ii. 280-2
Identity : principle of, i. 31-3
Ignoratio elenchi, ii. 232 ; 236 ;
285-90
Illative conversion, i. 256
Illicit process, i. 292-3 ; ii. 236 ;
259
Imitative language, i. 2
Immediate inferences : kinds of,
i. 227
• nature of, i. 15 ; 225-7
Imperfect definition, i. 121-2
induction, ii. 33
Impersonal judgment, i. 156-7
Implication of terms, i. 64
Implications of existence : in
predication, i. 211-4
in reduction, i. 359-60
■ in syllogisms, i. 340-1
Import of categorical proposi-
tions, i. 13-4; 17-8; 196-
214
• of disjunctive propositions.
i. 187-90
Import of hypothetical proposi-
tions, i. 181-4
Impossible propositions, i. 193
Inceptive propositions, i. 180
Incompatibility of terms, i. 64-71
Incomplete definition, i. 121-2
Inconceivabilityof contradictory,
ii. 205-6
Indefinite terms, i. 36 ; 68-9
Indesignate propositions, i. 169-
71
Indirect reduction, i. 358-9
Individual terms, i. 45-7
Induction : and Probability, ii,
54
■ Aristotle's doctrine, ii. 32-3
Bacon's doctrine, ii. 34-9
• basis and aim, ii. 55-8
enumerative, ii. 63; 66-71
imperfect, ii. 33
• Jevons' doctrine, ii. 53-5
method of, ii. 58-60
Mill's doctrine, ii. 40-8
Newton's doctrine, ii. 39-40
perfect, ii. 33
postulates of, ii. 1-31
relation to deduction, ii.
34; 37-8 ; 60-1
Hcholastic doctrine, ii. 33-4
■ Whewell's doctrine, ii. 48-
53
Inductive inference : nature of,
i. 15; ii. 55-61
Methods: Mill's, ii. 141-59
Inertia, ii. 145-6
Inference : by added determi-
nants, i. 268-70
by complex conception, i.
270
definition of, i. 14 ; 225
from particulars, i. 402-5 ,
li. 44 ; 192
immediate, i. 15 ; 225-74
kinds of, i. L4-6 ; 19
Infima species, i. 81
Infinite judgments, i. 162-3
terms, i. 36 ; 68-9
Inseparable accidens, i. 86
11
INDEX.
Integration, i. 203
Intension of concepts, i. Gl
Inverse, i. 265
Inversion of propositions, i.
265-6; 271-4
fallacies in, ii. 236; 257-8
Invertend, i. 265
James : on logical existence, i.
211
Jevons : diagrammatic represen-
tation of syllogisms, i.
341-2
doctrine of induction, ii.
53-5
on Bacon's inductive
method, ii. 38
on bias in observation, ii.
Ill
on character of successful
investigator, ii. Ill
on classification by type, i.
144
on diagnosis, i. 143
on ether, ii. 97-8
■ on false analogies, ii. 66
on generalization and ana-
logy, ii. 192-3
on hidden identity of phe-
nomena, ii. 195
on inference from negative
premises, i. 295 ; 297
on limits of accurate mea-
surement, ii. 162
on method of experiment,
ii. 122
on natural experiment, ii.
116
on necessity for experi-
ment^ ii. 114
on negative experiments,
ii. 118-9; 120
on negative observation, ii.
112
on Newton's inductive
method, ii. 40
• on relation of induction to
deduction, ii. 54-6
12
Jevons : on theories of light, ii.
103
on working hypotheses, ii.
88-9
Johnson : on w propositions, i.
203
Joint Method, ii. 143 ; 144-5 ;
155-6
Judgment : generic, i. 166 ; 167
hypothetical, i. 181-7
-impersonal, i. 14; 156-7
infinite, i. 162-3
modal particular, i, 169 ;
186
nature, of, i. 13-4 ; 17-8 ;
154-60; 181-6; 187-92;
196-2U
unity of, i. 14 ; 15-6 ; 159-60
Kant : infinite judgments, i.
162
— — modality, i. 194
on Aristotle's categories, i.
94
scheme of categories, i.
103-6
Kelvin, Lord : theory of nature
of atoms, ii. 73-4 ; 98
Kepler: character of his laws,
ii. 45-6; 49
discovery of laws of plane-
tary motion, ii. 66
scientific caution of, ii. 87-8
Keynes : on figure of sorites, i.
398-9
on quantification of predi-
cate, i. 205 ; 206
on universe of discourse, i.
60.
simplification of rules of
syllogism, i. 298-301
Kinds: natural, i. 83; 136
— — of language, i. 2-3
Knowledge : analysis of, i. 1-2
Leibniz on, ii. 222-4
postulates of, i. 30-9 ; ii.
1-31 [see Laws of
Thought)
INDEX.
Lamb : quotation of pun, ii.
242
Lambert : diagrammatic repre-
sentation of propositions,
i. 219-20
diagrammatic representa-
tion of syllogisms, i. 346
Language : ambiguities of, i.
5-9
conventional, i. 2-3
definition of, i. 2
■ functions of, i. 3-5
generalization in, i. 7-8
growth of, i. 6-9
imitative, i. 2
kinds of, i. 2-3
relation to Logic, i. 1-3
specialization in, i. 8-9
Laurie, H. : canon for method of
difference, ii. 157-8
on Mill's inductive methods,
ii. 148-9.
Law: meanings of term, i. 11 ;
ii. 200
of parsimony, i. 189 ; 291
Laws of Phenomena, ii. 52
of Thought : discussion of,
i. 30-9
relation to : contradic-
tion, i. 232
contrariety, i. 234-5
conversion, i. 256 ; 261
mixed disjunctive syl-
logisms, i. 371
mixed hypothetical syl-
logisms, i. 363
■ mood, i. 315-9
obversion, i. 251
subaltemation, i. 229
sub-contrariety,i. 236-7
syllogism, i. 282-3
Least Squares: Method of, ii.
185-7.
Leibniz: on knowledge, ii.
222-4
on value of syllogism, i. 411
principle of Sufficient
Reason, i. 37
Lewes : on conception, ii. 204
on subjective and objective
Logic, i. 18
Lewis : on disregard of counter-
acting causes, ii. 274-5
on interaction of cause and
effect, ii. 276-7
Light : rival theories of, ii.
103-4
Limitations of syllogism, i. 409-
11
Limiting subordinate clauses, i.
177
Limits of definition, i. 110-4
Linea predicamentalis, i. 81
Litigiosus, i. 385-6
Locke : on nature of inference,
1.402
on origin of causation, ii.
11-2
Logica docens, i. 12
uteris, i. 12
Logic: as science or art, i.
12-3
definition of, i. 10-2
divisions of, i. 13-9
general relation to other
sciences, i. 24-5
material or applied, i. 20-3
• of relatives, i. 410-1
origin of, i. 10
pure or formal, i. 20-3
relation to: Grammar, i.
28-9
Language, i. 1-3
Metaphysics,!. 25-6
Psychology, i. 26-7
Rhetoric, i. 27
Thought, i. 10-2
scope of, i. 17-8 ; 19-20
subject-matter of, i. 1
uses of, i. 13
Lotze : on Aristotle's categories,
i. 94
on basis of analogy, ii. 76
on classification, i. 135-6
on Crocodilus, i. 386
on dilemmas, 1. 382
13
INDEX.
Lotze : on Kant's categories, i.
106
on negative terms, i. 36
on observation and experi ■
ment, ii. 115
on similarity, ii. 75-6
on simplicity of hypotheses,
ii. 107-8
on suggestion of hypotheses,
ii. 65
Lubbock : experiments onhearing
in ants, ii. 127-30
Mach : on cause and effect, ii. 26
on hypotheses, ii. 90-1
— — on mechanical theory of
universe, ii. 209-10
on scientific ideas, ii. 208
on unity of nature, ii, 9-10
Mackenzie ; on fallacies in Mill's
Utilitarianism, ii. 243 ;
247
on false analogy in Plato,
ii. 367-8
on simple observation, ii.
Ill
Magnitude : determination of , ii.
182-7
Major premise, i. 278
term, i. 277-9
illicit process of, i.
292-3 ; ii. 236 ; 259
Mai -observation : fallacies of, ii.
234; 265-7
Malus : discovery of laws of
crystallization, ii. 65
Mansel : on Aristotle's categories,
i. 96-7
on diagrams in logic, i. 216
on dilemmas, i. 382
on Kant's categories, i.
105-6
Material element in division, i.
125-6
obversion, i. 254-5
of thought, i. 1-2
or Applied Logic, i. 20-3
view of Logic, i. 18
Mathematical axioms, i. 39; ii.
201-5
Means ; Method of, ii. 183-5
Measurement : elimination of
error in, ii. 163-5
importance of, ii. 160-1
instruments for, ii. 183
limitations of accuracy, ii.
161-3
Mechanical theory of universe,
ii. 209-10
Mediate inference, i. 15.
Membra dividentia, i. 123
Metaphors : fallacies due to, ii.
268-70
Metaphysical analysis, i. 126
universals, i. 171
Metaphysics, relation to Logic, i.
25-6
Method, i. 15; ii. 211-26
defects of, ii. 225-6
definition of, ii. 211
general rules, ii. 214-8
kinds of, ii. 211-4
of induction, ii. 58-60
scope of, ii. 211
Middle term, i. 277-9
undistributed, i. 291-2 ; ii.
236; 259
Mill, J. S. : axioms of syllogism,
i. 286
classification of fallacies, ii.
233-5
controversy with AVhewell,
ii. 45; 48-51
doctrine of: analogy,u.79-80
causation, ii. 16-9; 43-4
induction, ii. 40-8
syllogism, i. 402-7
uniformity of nature,
ii. 5-8; 42-3
experimentnl methods : aim
of, ii. 151-2
basis of, ii. 147-8
canons of, ii. 141-4
character of, ii. 148-51
-claims made for, ii.
146-7
14
INDEX.
Mill, J. S. : experimental
methods : examples
of, ii. 144-5
• inability to yield proof,
ii. 152-5
real functions of, ii.
156-9
■ fallacies in Utilitarian-
ism, ii. 242-3 ; 247
on ambiguous terms, ii. 240
on Aristotle's categories, i.
94-5
on axioms, ii. 202-5
on conception, i. 16
on conuotative abstract
names, i. 74
— — on education and discontent,
ii. 272-3
on empirical laws, ii. 43 ;
47; 272-3
on experimental sciences, ii.
212
on fallacies of observation,
ii. 261 ; 262 ; 264-5 ; 266
on function of hypotheses,
ii. 83
on generalization and induc-
tion, ii. 192
on ' imperfect ' induction,
ii. 41-2
— - on important attributes, i.
141-2
on inductive method, ii.
83-4
on inference from particu-
lars, i. 402-5 ; ii. 44 ;
192
on Kepler's inductions, ii.
45-6
on mathematical inductions,
ii. 46
on nature of generalization,
ii. 192
on necessary truths, ii.
20-5
on nominal and real defini-
tions, i. 119
on order of nature, ii. 109
Mill, J. S. : on ' perfect ' induc-
tion, ii. 41
on predication, i. 209-10
on scope of Logic, i. 18
on uniformities of co-ex-
istence, ii. 47-8
on validity of syllogism, i.
405-6
on vera causa, ii. 93
scheme of categories, i.
101-3
two theories of inference, ii.
44-8; 83-4
Minor premise, i. 278
term, i. 277-9
ilUct process of, i. 292 ;
ii. 236 ; 259
Mixed syllogisms, i. 362-86 {see
Syllogisms)
Mnemonic lines, i. 322 ; 353-5
Modal particular judgments, i.
169; 186
Modality, i. 192-5
Modtcs poiiens, i. 365-70.
tollens, i. 365-70
' Money,' ambiguity of, ii. 240
Moods: determination of, i. 315-
. 2^ .
in mixed disjunctive syllo-
gisms, i. 371-5
in mixed hypothetical syllo-
gisms, i. 363-70
in pure disjunctive syllo-
gisms, i. 350-1
in pure hypothetical syl-
logisms, i. 348-50
names of, i. 322 ; 365-6
of First Figure, i. 324-30
of Fourth Figure, i. 337-
40
of Second Figure, i. 330-3
of Third Figure, i. 333-7
subaltern, i. 323-4
Moral universals, i. 170-1
Mutatio conclusionis, ii. 285
Name, character of, i. 41-2
definition of, i. 41
15
INDEX.
Natura non agit per saltum, ii.
208
'Nature,' ambiguity of, ii. 241
Natural classification, i. 136-7 ;
139-44.
experiment, ii. 116-7
kinds, i. 83 ; 136
Necessary propositions, i. 193
truths, ii. 200-7
Negation, basis of, i. 162
Negative definitions, i. 118
experiments, ii. 118-20
instances in analogy, ii.
80-2
observation, ii. 112-3
premises, i. 293-7
-propositions, i. 161-3
terms, i. 36 ; 67-9
Neptune, discovery of, ii. 66
Newton: doctrine of induction, ii.
39-40
experiments on laws of
pendulum, ii. 130-1
■ on hypotheses, ii. 39-40
■ rules of philosophizing, ii.
92-5
scientific caution of, ii. 88
Nomenclature, i. 146-50
Nominal definitions, i. 118-20
Nominalism, i. 16 ; 17
Nominalist view of Logic, i. 17
JVoM causa pro catisa, ii. 232 ;
236; 290-2
Non-observation : fallacies of, ii.
234; 262-5_
iS'oM per hoc, ii. 290
Hon propter hoc, ii. 290 ; 291
Nota notes, i. 286
Numerically definite proposi-
tions, i. 173
Objective view of Logic, i. 18
Observation : fallacies in, ii. 234 ;
236; 261-7
nature of, ii. 109-14
■ relation to experiment, ii.
114-7
Obverse, i. 251
Ob version : material, i. 254-5
of propositions, i. 251-5;
271-3; 274
Obvertend, i. 251
Olszewski: experiments on argon.
ii. 138-9
Opposite terms, i. 70-1
Opposition: contradictory, i. 232-
4; 245; 246-7
contrary, i. 234-6 ; 245 ;
246-7
definition of, i. 227 ; 228
• fallacies in, ii. 235 ; 254-5
kinds of, i. 228-9
• of categorical propositions,
i. 228-44
■ of disjunctive propositions,
i. 246-7
of hypothetical propositions,
i. 244-6
square of, i. 239-40
subaltern, i. 229-32; 245;
246-7
sub-contrary, i. 236-9 ; 245-
6
summary of, i. 241-4
Origin of hypotheses, ii. 62-82
of logic, i. 10
Ostensive reduction, i. 355-8
Paradox, ii. 228-9
Paralogism, ii. 228
Parsimony : law of, i. 189 ; 291
Particular propositions, i. 167-9
Partitive conversion, i. 256
Pearson, K. : on causation, ii. 15
Pendulum: laws of, ii. 130-1
Perfect definition, i. 121
induction, ii. 33
Petitio principii, ii. 224 ; 232 ;
236 ; 279-85
and syllogism, i. 405-9
qumsiti, ii. 279
Phantoms of the Cave, ii. 35
— — of the Market-place, ii. 35
■ of the Theatre, ii. 35.
of the Tribe, ii. 35
Phenomena : laws of, ii. 52
16
INDEX.
Physical partition, i. 126
Plurality of causes, ii. 18-9 ; 27
Plurative propositions, i. 174-6
Plures interrogationes, ii. 232 ;
235 ; 254 5
Polylemma, i. 376
Polysyllogism, i. 392
Porphyry : scheme of predicables,
i. 80-8
tree of, i. 86-7 ; 132
Port Royalists : on analysis, ii.
219
on analysis and synthesis,
ii. 212
on Aristotle's categories, i.
93-4
on begging the question,
ii. 281
on faulty sequence in
Euclid, ii. 225-6
on logical sequence, ii. 216
on reductio ad impossibile,
ii. 226
on rules of method, ii. 215
rules of synthesis, ii. 221-2
Positive terms, i. 68
Possible propositions, i. 193
Post hoc ergo propter hoc, ii. 274
Postulates of Knowledge, i. 30-9 ;
ii. 1-31 (see Larvs of Thought)
Practical science, i. 12
Predicables : Aristotle's scheme
of, i. 78-80
definition of, i. 78
Porphyry's scheme of, i. 80-8
Predicamental line, i. 81
Predicaments, i. 89-106 (see
Categories)
Predicate, i. 40 ; 158-60
quantification of, i. 200-7
Predication : attributive view of,
i. 209-11
class-inclusion view of, i.
198-200
compartmental view of, i.
220-1
comprehensive view of, i.
208-9
LOG. I. II. 17
Predication : conceptualisi; view
of, i. 17
connotative view of, i. 209-
11
• existential view of, i. 220-1
implication of existence in,
i. 211-4
material view of, i. 18
meaning of, i. 196
nominalist view of, i. 17
objective view of, i. 18
predicative view of, i. 158-
60; 197-8
quantification of predicate
view of, i. 200-7
Preindesignate propositions, i.
169 71
Premises, i. 14 ; 277-80
Prerogative instances, ii. 37
Principle : definition of, i. 11
' Principle' in syllogism, i. 279
Principles of Thought, i. 30-9
229 ; 232 ; 234-5 ; 236-7 ; 251
256 ; 261 ; 282-3 ; 315-9 ; 363
371 {see Laws of Thought)
Privative terms, i. 70-1
Probability : basis of, ii. 165-70
independence of time, ii.
169-70
of alternative conditions,
ii. 178-80
of compound events, ii.
171-8
of conjunction of inde-
pendent events, ii. 171-4
of dependent events, ii.
174-6
of events which can happen
in a plurality of ways,
ii. 176 8
of recurrence of an event,
ii. 180-2
of simple events, ii. 171
Problematic judgments, i. 194-5
Progressive chains of reasoning,
i. 390-1; 393-400; ii. 212-4;
221-5
Proper names, i. 45-6 ; 53-4
INDEX.
• Preposition ' in syllogism, i.
279
Propositiones prcpmisftce, i. 277
Propositions, categorical : affir-
mative, i. 161
-^ analysis of, i. 156-60
analytic, i. 88 ; 160-1
comijound, i. 178-80
contradiction of, i. 232-
4
contraposition of, i.
262-4
contrariety of, i. 234 6
conversion of, i. 255-62
— copulative, i. 178
desitive, i. 180
discretive, i. 178-9
distribution of terms
in, i. 172-3
-■ eductions of, i. 248-
70
exceptive, i. 179-80
exclusive, i. 179
exponible, i. 179-80
four-fold, scheme of,
i. 171-3
— general, i. 164-7
— Hamilton's scheme of,
i. 200-7
— implications of exis-
tence in, i. 211-4
— ^ — ■ import of, i. 13-4 ; 15 ;
17-8 ; 156-60 ; 196-
214
■ — inceptive, i. 180
— indesignate, i. 169-71
inversion of, i. 265-6
justification of,i. 165-6 ;
167 ; 169
• negative, i. 161-3
numerically definite,
i. 173
obversion of, i. 251-5
opposition of, i. 228-44
particular,i. 163; 167-9
plurative, i. 174-6
preindesignate, i, 169-
71
Propositions, categorical : quality
of, i. 161-3
quantity of, i. 163-76
relation to disjunctive,
i. 190-2
relation to hypothetical,
i. 184-6
— remotive, i. 178
represented by dia-
grams, i. 215-24
singular, i. 163-4
subalternation of, i.
229-32
■ sub-contrariety of, i.
236-9
synthetic, i. 88 ; 160-1
universal, i. 163-7
verbal, i. 88 ; 160-1
with complex terms, i.
176-7
conditional : character of,
i. 184
— eductions of, i. 271-3
opposition of, i. 244-6
definition of, i. 15 ; 154
diagrammatic representa-
tion, i. 215-24
disjunctive : definition of,
i. 187
eductions of, i. 274
— — interpretation of, i.
188-90
misinterpretation of, ii.
235 ; 253
nature of, i. 187-90
opposition of, i. 246-7
quality of, i. 192
quantity of, i. 192
relation to hypothetical
and categorical, i.
190-2
— — hypothetical : definition of,
i. 181
eductions of, i. 271-3
misinterpretation of, ii.
235; 253
nature of, i. 181-4
opposition of, i. 244-6
IS
INDEX.
Propositions, hypothetical:
quality of, i. 186
— quantity of, i. 186-7
relation to categorical,
j. 184-6
relation to disjunctive,
i. 190-2
kinds of, i. 155-6
modal particular, i. 169 ;
186
modality of, i. 192-5
Proprium, i. 79 ; 80 ; 84-5
Prosyllogism, i. 391
Prosyllogistic chains of reason-
ing, i. 392 ; 400-1 : ii. 212-4 ;
219-20
Proximate matter of syllogism,
i. 277
Proximum genus, i. 81
Psychology : relation to Logic,
i. 26-7
' Publish,' ambiguity of, ii. 239
Quadruped, logical, ii. 259
Qualitative analysis: character of,
ii. 121-2
examples of, ii. 122-41
Quality : of categorical proposi-
tions, i, 161-3
- — - of disjunctive propositions,
i. 192
of hypothetical propositions,
i. 186
Quantification of predicate, i.
200-7
Quantity : of categorical proposi-
tions, i. 163-76
of disjunctive propositions,
i. 192
of hypothetical propositions,
i. 186-7
'Question' of syllogism, i, 277
Eamean tree, i. 86-7 ; 132
Ramsay : experiments on argon,
ii. 133-41
Rational theory of universe,
ii. 210
Ray : diagrammatic representa-
tion of syllogisms, i. 343
on universality of syllogism,
i. 409
Rayleigh, Lord : experiments on
argon, ii. 133-41
Real definition, i. 118-20
kinds, i. 83 ; 136
proposition, i. 88 ; 160-1
Realism, i. 16
Reality, i. 1-2 ; ii. 1-5
and thought, ii. 2-3
Empiricist view of, ii. 1-2
'Reason' of syllogism, i. 277;
279
Reasoning, i. 14-6 ; 19
in a circle, ii. 282
Rebutting a dilemma, i. 384-6
Reductio ad impossibile, i. 358 ;
ii. 226 ; 290 ; 292
Reduction : and implications of
existence, i. 359-60
direct, i. 355-8
function of, i. 352-3
indirect, i. 358-9
kinds of, i. 355-9
mnemonics for, i. 353-5
of dilemmas, i. 383-4
of mixed disjunctive syllo-
gisms, i. 373-4
of mixed hypothetical
syllogisms, i. 370
of pure hypothetical syl-
logisms, i. 360-1
ostensive, i. 355-8
Regressive chains of reasoning,
i. 392; 400-1 ; ii. 212-4; 219-
20
104;
Relation of propositions, i.
155-6
Relative terms, i. 75-7
Relatives : Logic of, i. 410-1
Remote matter of syllogisms,
i. 277
Remotive propositions, i. 178
Repugnant terms, i. 71
Resemblance and analogy, ii. 78-
80
19
INDEX.
Residual phenomena, il. G5-6
Residues : Method of, ii. 143 ;
145; 156; 157
Ehetoric : relation to Logic, i.
27
Robertson : on method, ii. 211
Rules : of axioms, ii. 222
of classification, i. 127-9 ;
137; 141-4
of definition, i. 114-8 ; ii.
221
of demonstration, ii. 222
of division, i. 127-9
of method, ii. 214-8 ; 221-5
of mixed syllogism : dis-
junctive, i. 372
hypothetical, i. 365
of pure syllogism : i. 287-98
- — corollaries from, i.
302-4
— derivation of, i. 287-8
simplification of, i. 298-
302
of sorites, i. 396 ; 397
of synthesis, ii. 221-5
Scholastic doctrine of induction,
ii. 33-4
Science : distinguished from art,
i. 12
Scientific instruments, ii. 113-4
nomenclature, i. 146-50
■ terminology, i, 150-3
Scope of concepts, i. 64
of Logic, i. 17-8 ; 19-20
Second Figure : axiom and
special rules of, i. 308-9
characteristics of, i. 313-4
moods of, i. 330-3
Selection of the idea, ii. 52
Separable accidens, i. 86
Sequence in discourse, ii. 216-8
Sidgwick, Prof. H. : on utility
of defining, i. 108
Sigwart : on Bacon's inductive
method, ii. 37
on Mill's doctrine of indue
tion, ii. 48
Sigwart : on IMill's inductive
methods, ii. 153
on statistical uniformities,
ii. 199-200
on uniformity of nature, ii. 7
Simplification of theory, ii. 51 ;
107-8
Singular propositions, i. 163-4
terms, i. 45-7
Smollett : false analogy in, ii
269
Sophism, ii. 228 ; 231-2
Sophiamata extra dictionein, ii.
232
in dictione, ii. 231
Sorites : Aristotelian, i. 393-7
definition of, i. 393
fallacy of, i. 399-400
figure of, i. 398-9
Goclenian, i. 393 ; 394-6 ;
397
• history of, i. 399
kinds of, i. 393-6
rules of, i. 396 ; 397
Specialization in language, i.
Species, i. 80 ; 81-3
Specific differentia, i. 84
projwium, i. 85
8-9
69-
Specification of instances, ii
70
Spencer, H. : fallacies in Educa-
tion, ii. 247 ; 284 ; 285
on scope of Logic, i. 18
Sphere of concepts, i. 64
Spinoza : categories, i. 100
Square of opposition, i. 239-40
Squaring the circle, ii. 280
Statistical uniformities, ii. 198-
200
Stock : diagrammatic representa-
tion of syllogisms, i. 345
dilemmas, i. 383
on opposition, i. 238
Stoddart : on universal grammar,
i. 28
Stoic scheme of categories, i. 100
Strength of analogies, ii. 76-80
Strengthened syllogisms, i. 322-3
20
INDEX,
Subaltern, i. 229
genus, i. 81
moods, i. 323-4
species, i. 81
Suhalteriians, i. 229
Suhalternant, i
Suhalternate, i.
229
229
i. 229-32
Subalternation, i. 229-32 ; 245
246-7
Sub-contrariety, i. 236-9 ; 245-6
Sub-division, i. 124
Subject, i. 40 ; 158-60
Subordinate clauses, i. 177
Substantial definition, i. 120
terms, i. 51 ; 72-3
Sufficient Reason : principle of,
i. 37-8 ; ii. 1 ; 28-30
Suggestion of hypotheses, ii. 63-6
Suicides : constant ratio of, ii.
198-9
Suramum genus, i. 81
Superficial analogies, ii. 76-7
Syllogisms, categorical : and im-
plications of existence,
I. 340-1
axioms of, i. 283-6
basis of, i. 282-3
canons of, i. 287-304
— — ■ determination of moods,
i. 315-22
figures of, i. 306-15
fundamental, i. 322
reduction of, i. 352-60
representation by dia-
grams, i. 341-7
rules of, i. 287-304
strengthened, i. 322-3
weakened, i. 323-4
with singular premises,
i. 334
chains of reasoning, i. 390-
401
definition of, i. 275
dilemmas, i. 376-86 (.see Di-
lemmas)
elements of, i. 277-80
enthymemes, i. 387-90 (see
Enlhymemes)
Syllogisms : epicheiremas, i.
400-1 {see Epicheiremas)
fallacies in, i. 291-3 ; ii.
236; 259-60
form of, i. 276
kinds of, i. 280-1
matter of, i. 276 ; 277
mixed disjunctive : basis of,
i. 371
canon of, i. 372
forms of, i. 371-5
in wider sense, i. 375
reduction of, i. 373-4
mixed hypothetical : basis
of, i. 363
canon of, i. 365
character of, i. 362-3
moods of, i. 363-70
reduction of, i. 370
nature of, i. 275-81
premises of, i. 277-80
pure disjunctive : figures
and moods of, i. 350-1
rules of, i. 305
pure hypothetical : figures
and moods of, i. 348-
50
reduction of, i. 360-1
rules of, i. 304-5
sorites, i. 393-400 (see
Sorites)
terms of, i. 277-80
Syllogistic reasoning: chains of, i.
390-401
limitations of, i. 409-11
universal element in, i.
402-5
validity of, i. 405-9
Syncategorematic words, i. 42-3
Synonyms, i. 6
Synthesis: method of, ii. 212-4;
221-5
of Degree, i. 410-1
of Identity, i. 410
of Space, i. 411
of Subject and Attribute
i. 410
of Time, i. 411
21
INDEX.
Synthetic chains of reasoning,
i. 390-1 ; 393-400 ; ii.
212-4 ; 221-5
propositions, i, 88 ; 160-1
Synthetically-fgrmed definitions,
i. 121
Systematization, ii. 207-10
Swift : pun, ii. 242
Teleological nature of value in
analogy, ii. 76-8
Terminology, i. 150-3
Terms : absolute, i. 75-6
abstract, i. 72-5
analogous, ii. 268-9
class, i. 48-51
collective, i. 49-50
concrete, i. 72-4
• connotation of, i. 51 -7 ; 60-4
contradictory, i. 65-70
contrary, i. 70-1
definition of, i. 15 ; 40-1
denotation of, i. 57-64
distribution of, i. 172-3
divisions of, i. 44
equivocal, i. 44-5 ; ii. 238-
42
■ extreme, i. 277
general, i. 48-50
incompatibility of, i. 64-71
indefinite, i. 36 ; 68-9
individual, i. 45-7
infinite, i. 36 ; 68-9
major, i. 277-9
middle, i. 277-9
- — minor, i. 277-9
negative, i. 36, 67-9
of syllogism, i. 277-80
positive, i. 68
privative, i. 70-1
relative, i. 75-7
repugnant, i. 71
single-worded and
worded, i. 41-2
singular, i. 45-7
substantial, i. 51 ;
univocal, i. 44
Tetralemma, i. 376
many-
72-3
Theories of Causes, ii. 52
Theory: and Fact, ii. 2-3; 11;
49
— — • definition of, ii. 105
simplification of, ii. 51 ;
107-8
Third Figure : axiom and special
rules of, i. 309
characteristics of, i. 314
moods of, i. 333-7
Thomson : axioms of syllogism,
i. 284
on dilemmas, i. 383
scheme of categories, i. 100
Thought : activity of, ii. 3-5
and things, i. 1-2; ii. 2-3;
11 ; 49
form and matter of, i. 21
laws of, i. 30-9 ; 229 ; 232 ;
234-5 ; 236-7 ; 251 ; 256 ;
261 ; 282-3; 315-9; 363;
371 {see Laws of Thought)
validity of, i. 11-2
'lolum divisum, i. 123
Tree of Porphyry, i. 86-7 ; 132
Trilemma, i. 376
Truths : necessary, ii. 200-7
Tti quoque, ii. 289
XJeberweg; axiom of consistency,
i. 32-3
on circular definition, i.
116-7
on crucial instances, ii. 102
on dilemmas, i. 383
on hypotheses, ii. 90
on inference from negative
premises, i. 296-7
on nominal and real defini-
tions, i. 119 ; 120
on relative value of conclu-
sions, i. 319
on syllogistic reasoning : in
mathematics, i. 325
in physics, i. 326
Undistiibuted IVIiddle, i. 291-2 ;
ii. 236 ; 259
Uniformity of nature, ii. 5-9
22
INDEX.
Unity of nature : ii. 5-10 ; 209-10
meaning of, ii. 8-9
origin of, ii. 5-8
scope of, ii. 9-10
Universe : mechanical theory of,
ii. 209-10
of discourse, i. 59-60
Universal element in deductive
reasoning, i. 402-5
grammar, i. 28
propositions, i. 164-7 ;
181-4; 190; 192
Univocal terms, i. 44
Use of definition, i. 107-8
of diagrams, i. 215-6
of division, i. 125
of Logic, i. 13 ; 24
of reduction, i. 352-3
' Utter,' ambiguity of, ii. 239
Validity of syllogism, i. 405-9
of thought, i. 11-2
Value of figure, i. 312-5
Vegetable mould : formation of,
ii. 124-7
Venn ; diagrammatic representa-
tion of propositions,!. 220-2
diagrammatic representa-
tion of syllogisms, i, 346-7
on denotation, i. 58-9
on generalization, ii. 191-2
on logic of relatives, i. 410
on modality, i. 193-4
on universe of discourse, i. 59
Vera causa, ii. 93-5 ; 98
Verbal definition, i. 118-20
language, i. 2-3
proposition, i. 88 ; 160-1
Voltaic pile : source of power in,
ii. 131-3
Wallace : on origin of beauty of
flowers, ii. 123
on varieties of melons, ii. 80
Weakened syllogisms, i. 323-4
Wells : investigations into dew,
ii. 144-5
Whately ; axioms of syllogism,
i. 283
on circular reasoning in
physics, ii. 282
on classification of fallacies,
ii. 230-1 ; 232-3
on dilemmas, i. 381
on universality of syllogism,
i. 409
- — view of analogy, ii. 75
Whewell ; controversy with Mill,
ii. 45; 48-51
doctrine of induction, ii. 48-
53
— on application of theory of
gravitation, ii. 100
on character of scientific
mind, ii. 87
on circular polarization, ii.
100-1
on classification by type, i.
144
on definition, i. 112
on diagnosis, i. 144
on fact and theory, ii. 49
on inaccurate observation,
ii. 85
on Kepler's inductions, ii.
45-6
on Newton's Rules, ii. 95
on nomenclature, i. 146;
148-50
on Ptolemaic hypothesis, ii.
89
on scope of Logic, i. 18
on simplicity of hypothesis,
ii. 108
on suggestion of hypothesis,
ii. 66
— — on tentative hypotheses,ii. 86
on terminology, i. 150-3
on test of hypotheses, ii. 100
on vera caiisa, ii. 94
prediction of absence of tide,
ii. 101
Words : classification of, i. 42-3
World as unity, ii. 4-10 ; 209-10
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