LOGIC
DEDUCTIVE AND INDUCTIVE
BOOKS BY JOHN GRIER HIBBEN, Ph.D.
PUBLISHED BY CHARLES SCRIBNER S SONS
LOGIC, DEDUCTIVE AND INDUCTIVE,
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LOGIC
DEDUCTIVE AND INDUCTIVE
BY
JOHN GRIER HIBBEN, PH.D.
STUART PROFESSOR OF LOGIC IN PRINCETON UNIVERSITY
NEW YORK
CHARLES SCRIBNER S SONS
1906
COPYRIGHT, 1896, 1905, BY
CHARLES SCRIBNER S SONS
XortoooO
J. S. Gushing & Co. Berwick & Smith Co,
Norwood, Mass., U.S.A.
STo
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PKEFACE
THIS book consists of two parts, the Deductive and
the Inductive Logic. The former treats of the general
nature of our thought processes as well as the fundamental
principles and practice of deduction, and is now published
for the first time. The latter is my Inductive Logic which
was published in 1896, now revised and incorporated in
this volume. It has been my endeavor to present in con
nection with the more formal and traditional treatment of
the deductive logic also some considerations which have
been contributed by the discussions of the modern logic
and which find expression in such works as those of Sig-
wart, Lotze, Erdmann, Green, Bosanquet, Venn, and others.
The illustrations and examples contained in the text are
taken as far as possible from the sphere of everyday expe
riences, in order that they may represent modes of actual
reasoning pursued by the common run of mankind. With
this end in view, all the stock examples which have grown
old and infirm in the service of many generations of stu
dents in logic have been omitted. Moreover, the material
as well as the formal significance of the judgments em
ployed in reasoning has been emphasized in order that the
student may come to regard logic as a living process of
thought functioning in a normal and natural manner, and
not as an artificial manipulation of certain dead elements
mechanically adjusted one to another.
The illustrations which appear in the Inductive Logic,
and which are taken from the experiments of Faraday,
Tyndall, Darwin, Pasteur, Lubbock, and others, are quoted
vii
Vlll PREFACE
for the most part at considerable length, not merely be
cause in the concrete case the universal principles of rea
soning and of method are often most forcibly discovered,
but also because the experiments of such pioneers in re
search actually create these methods of investigation, or
at least serve to render them exact and definite.
In Chapter XIV, Part I, "A Generalization of Immediate
Inferences," I have presented some original material, this
being an attempt on my part to summarize all the possible
transformations of any given proposition according to a
scheme suggested by the Aristotelian square of opposition,
and developed along similar lines. In addition to the
general field usually covered by writers on deductive logic,
there is appended a discussion on " Extra-syllogistic Reason
ing," being Chapter XVIII, Part I.
I wish to avail myself of this opportunity to express my
appreciation of the suggestions and help which I received
from my colleagues, Dean Andrew F. West and Professor
Winthrop M. Daniels, in the preparation of the Logical
Exercises which appear at the end of Part II.
J. G. H.
PRINCETON, N.J.,
December 23, 1904.
CONTENTS
PART I
DEDUCTIVE LOGIC
CHAPTER I
PAGE
THE NATURE OF THOUGHT 3
Definition and nature of Logic, 3. Thought as reflection, 4.
The four functions of thought, 4. Concept, judgment, in
ference, 10. Logic as a normative science, 11.
CHAPTER II
THE CONCEPT 13
Relation of identity to diversity in concepts, 13. The
natural history of the concept, 15. Logical and empirical
concepts, 16. Genetic concepts, 22. Thought and lan
guage, 23.
CHAPTER III
THE JUDGMENT ......... 25
The essential nature of judgment, 25. Universal and
singular judgments, 26. Relation of judgment to reality,
27. The element of necessity in judgment, 30. The uni
versal element in judgment, 31. Judgment and language,
33. Subject, predicate, and copula, 33.
CHAPTER IV
THE UNIVERSAL JUDGMENT 36
The categories of Aristotle, 37. Heads of Predicables, 38.
Various types of judgment, 40. Extension and intension,
denotation and connotation, 42.
CONTENTS
CHAPTER V
PAGE
DEFINITION 44
Nature of definition, 44. Real and nominal definition, 44.
Rules of definition, 45. Definition by description, 47.
Definition for purpose of identification, 48. Genetic defi
nition, 48.
CHAPTER VI
DIVISION AND CLASSIFICATION 50
Nature of division, 50. Rules of division, 51. Dichotomy,
61. Contrary and contradictory, 52. Trichotomy, 53.
Empirical and logical divisions, 54. Nature of classifica
tion, 55. Artificial and natural classification, 56. Serial
classification, 58. Effect of the doctrine of evolution on
theory of classification, 59. Classification of the sciences,
61. Classifications of Bacon, Comte, and Spencer, 62.
CHAPTER VII
THE SINGULAR JUDGMENT 67
Its relation to the universal judgment, 67. Impersonal,
perceptive, and demonstrative judgments, 67. Determinate
reference, 68. Indeterminate reference, 69. Judgment
concerning a proper name, 69.
CHAPTER VIII
THE NEGATIVE JUDGMENT 73
Nature of the negative judgment, 73. Its function of
exact determination, 74. Its positive ground, 75. Signifi
cant negation, 75. Implication in negation, 76. Infinite
negation, 77.
CHAPTER IX
THE CATEGORICAL, HYPOTHETICAL, AND DISJUNCTIVE
JUDGMENTS ..... 78
The nature of each, 78. Their relation to universal and
singular judgments, 79. Their relation to the progressive
stages of knowledge, 81. Modality of judgments, 83.
CONTENTS Xl
CHAPTER X
PAOI
THE NATURE OF INFERENCE 85
Logical and psychological elements in inference, 85. Ob
jective and subjective necessity, 87. Data of perception,
88. System as ground of inference, 89. The implicit and
explicit, 92. Inference mediated through the universal, 93.
Conceptual processes, 94. Explanation, 94. Relation of
inference to judgment, 95.
CHAPTER XI
THE LAWS OF THOUGHT 98
The law of identity, 98. The law of contradiction, 100.
The law of excluded middle, 101. The law of sufficient
reason, 102.
CHAPTER XII
IMMEDIATE INFERENCE 103
Immediate inference, a misnomer, 103. The processes of
implication and transformation, 103. The square of opposi
tion, 104. Practical suggestions based on opposition, 108.
CHAPTER XIII
On TRANSFORMATIONS OF JUDGMENT FORMS . . . 110
Conversion, 110. Content and form in conversion, 113.
Obversion, 114. Contraposition, 114.
CHAPTER XIV
A GENERALIZATION OF IMMEDIATE INFERENCES . . 117
Summary of possible transformations, 117. The A square,
118. The E square, 119. The / square, 120. The O
square, 120.
CHAPTER XV
MEDIATE INFERENCE THE SYLLOGISM .... 122
Structure and functions of the syllogism, 122. Distribu
tion of terms, 125. Rules for criticism of validity of syllo-
x ii CONTENTS
PAGE
gisms, 126. Modification of these rules in special cases, 129.
Enthymeme, 130. Prosy 11 ogism, episyllogism, and the
sorites, 132.
CHAPTER XVI
MOOD AND FIGURE . .... 131
The valid moods, 134. Figure, 137. Mnemonic lines,
139. Reduction, 140.
CHAPTER xvn
THE HYPOTHETICAL AND DISJUNCTIVE SYLLOGISMS . . 142
Hypothetical syllogism, 142. Disjunctive syllogism, 145.
Dilemma, 145. Trilemnia, 148.
CHAPTER XVIII
EXTRA-SYLLOGISTIC REASONING . . . 149
Reasoning from particulars to particulars, 149. The typi
cal case a disguised universal, 151. Inference based upon
given relations, 152. Its relation to the underlying system,
154. The logic of relatives, 156.
CHAPTER XIX
FALLACIES 157
Formal fallacies, 157. Material fallacies, 158. Equivo
cation, 158. Amphiboly, 159. Composition, 159. Divi
sion, 160. Accent, 160. Figure of speech, 160. Accident,
161. Converse accident, 161. lynoratio Elenchi, 162. Non
sequitur, 164. Petitio Principii, 164. Non causa pro
cawsa, 165. Many questions, 165.
CONTENTS Xlll
PART II
INDUCTIVE LOGIC
CHAPTER I
PAOK
INDUCTION AND DEDUCTION 169
Various opinions concerning their relative importance,
169. Regarded as different phases of one and same pro
cess, 170. Their relation to the ground of inference as
system, 170. Their relation to the universal, 171. Truth
and fact, 171. Mutual dependence of deduction and induc
tion, 172.
CHAPTER II
THE ESSENTIALS OF INDUCTION 175
The inductive hazard, 175. Basal postulate of induction,
176. Its epistemological nature, 177. Reduction, 177.
Law and rule, 180. Law as a hypothetical universal,
181. Induction in practical affairs of life, 181. Scientific
spirit, 182.
CHAPTER III
TYPES OF INDUCTIVE INFERENCE 183
Method of enunciation, 183. (a) Perfect induction, 184.
(6) Incomplete induction, 186. (c) Probability, 186.
Method of Analogy, 187. Method of Scientific Analysis,
188. The causal element in these various methods, 189.
CHAPTER IV
CAUSATION 195
Phenomenal significance of causal concept, 196. Philo
sophical significance, 197. Logical significance, 198. Origin
of belief in uniformity of nature, 199. Popular and scien
tific idea of cause, 201. Causal analysis, 202. Limitations
of knowledge, 203,
CONTENTS
CHAPTER V
PAGB
THE METHOD OF CAUSAL ANALYSIS AND DETERMINATION 206
Sequence, 206. Concurrence, 207. Coexistence, 208.
Collocations, 209. Transfer of energy, 211. Quantitative
determination, 211. Observation and experiment, 213.
Negative determination, 217. Pseudo-causal connections,
219.
CHAPTER VI
MILL S INDUCTIVE METHODS THE METHOD OF AGREE
MENT 222
The five methods, 222. Agreement, 224. Symbolical
representation, 225. Variation of instances, 227. Obser
vation, 228. Simple enumeration, 228. Sequence and
coexistence, 229. Criticism of this method, 229. Agree
ment as a method of suggestion, 232. Illustrations, 232.
CHAPTER VII
THE METHOD OF DIFFERENCE 236
Its relation to agreement, 236. Its characteristic features,
236. Symbolical representation, 238. Relation to negative
determination, 239. Difference and combinations, 239.
Criticism of the method, 240. Practical difficulties, 242.
Illustrations, 245. Blind experiments, 247.
CHAPTER VIII
THE JOINT METHOD OF AGREEMENT AND DIFFERENCE . 248
Relation to method of difference, 248. Its characteristics
and symbolical representation, 249. Illustrations, 250.
Advantages of this method, 257.
CHAPTER IX
THE METHOD OF CONCOMITANT VARIATIONS . . . 258
Characteristics and symbolical representation, 258. Quan
titative determination, 259. Graphical representation, 260.
Advantages in its psychological impressions, 201. Illustra
tions, 262. Comprehension of unknown forces by this
method, 266. Precautions in using this method, 267.
CONTENTS XV
CHAPTER X
PAQI
THE METHOD OF RESIDUES ....... 271
Characteristics and symbolical representation, 271. A
deductive method, 272. Its function suggestive, 273. Illus
trations, 273. Its practical value, 277.
CHAPTER XI
PREDICTION AND VERIFICATION ...... 278
The inducto-deductive method, 278. Illustrations, 279.
Bacon s anticipations of nature, 283. Scientific thought,
284. Indirect method of prediction, 286. Exceptional
phenomena, 288. Generalization, 289. Mathematical
method, 290.
CHAPTER
HYPOTHESIS .......... 291
Its relation to induction, 291. Illustrations, 292. Func
tion of the imagination in hypothesis, 299. Analysis and
synthesis, 300. Requirements of a logical hypothesis, 301.
Consilience of inductions, 309. Experimentum Crucis, 310.
Mill and Whewell controversy, 312.
CHAPTER XIII
ANALOGY ........... 314
Analogy and induction, 314. Natural kinds, 314. Anal
ogy and classification, 315. Teleological analogy, 317.
Suggestion the chief function of analogy, 323. Require
ments of true analogy, 325. Analogy and probability, 329.
CHAPTER XIV
PROBABILITY .......... 330
Probability and causal determination, 330. Relation to
enumerative induction, 332. Various kinds of inference in
sphere of probability, 333. Coincidence and cause, 345.
Circumstantial evidence, 346. Probability and method of
residues, 350.
xvi CONTENTS
CHAPTER XV
PAG*
EMPIRICAL LAWS 351
Various degrees of probability in inference, 351. Various
kinds of empirical laws, 352. Empirical uniformity result
ing from the method of agreement, 357. Empirical laws
and laws of an ultimate nature, 357.
CHAPTER XVI
INDUCTIVE FALLACIES 359
Errors of perception, 360. Errors of judgment, 362.
Errors of imagination, 366. Errors of the conceptual pro
cesses, 369. The psychological nature of these fallacies, 372.
CHAPTER XVII
THE INDUCTIVE METHODS AS APPLIED TO THE VARIOUS
SCIENCES 374
Method varies with different kinds of phenomena, 374.
Difficulties in method due to complexity of phenomena, 378.
Phenomena of one science interpreted in the light of others,
380. Deductive method of some sciences replaced by the
inductive, 381.
CHAPTER XVIII
HISTORICAL SKETCH OF INDUCTION 385
Socrates, 385. Plato, 385. Aristotle, 386. Roger Bacon,
387. Leonardo da Vinci, 388. Telesius, 389. Campa-
nella, 389. The experimental investigators, 390. Francis
Bacon, 390. Locke, 392. Newton, 393. Herschel, 394.
Whewell, 395. Mill, 396.
LOGICAL EXERCISES 399
INDEX .... , 435
PART I
DEDUCTIVE LOGIC
CHAPTER I
THE NATURE OF THOUGHT
LOGIC is a word derived from the Greek Aoyos, which
means thought or reason ; and in this origin may be found
the essential significance of logic, that it treats of the
nature and of the laws of thought. Before it is possible
to appreciate the characteristic features of the laws of
thought, it is necessary to understand the general nature
of the processes of thought themselves. While the process
of thought is various, its most common and conspicuous
manifestation may be described as that phase of the mind s
activity which regards any specific object which may be
presented to it in the light of the general body of knowl
edge. For example, a person may chance to pick up a stone,
which he holds in his hand for a moment and immediately
throws away. It has been in the focus of his attention for
a fleeting moment only, and has excited no activity of
thought whatsoever. He has observed but has not thought
about it. Suppose, however, it does arrest his attention and
he begins to think about it, what is the nature of this
thinking which goes on in his mind ? If his knowledge of
geology is meagre, the result of the application of it to the
special object of inquiry may be merely the assertion that
the stone which he holds in his hand is some kind of a
fossil. If, however, his knowledge is more extensive and
has grown out of a wide experience, he will be able no
doubt to refer the fossil in question to its proper geological
age, and to give some satisfactory description of the general
nature and habits of that species of animals to which it
4 DEDUCTIVE LOGIC
belongs, thus, in a measure more or less explicit, recon
structing its probable life history. Thinking, therefore,
may be defined in one of its aspects at" least as the process
of interpreting the special by the general, or the new expe
rience by the old.
This definition of thought may be further illustrated by
the word reflection, which is often used as synonymous
with thought. Thus we say that we will reflect about a cer
tain proposition, which is equivalent to saying we will think
about it. The process of reflection is essentially one of
illumination. The very word reflection suggests the light
ray which flashes from one object of vision to another ; so,
also, in a figurative sense, it signifies the illumination which
one object of knowledge sheds upon another. In the reflect
ing mind, the new element of experience, whatever it may
be, is held in the focus of the light rays which converge to
that point from all the surrounding parts of the general body
of knowledge until its essential nature is fully revealed.
In this process which we call thinking, or reflecting, there
are in all four functions involved.
1. The first function of thought consists in the trans
formation of the crude data of knowledge furnished by
the senses into forms of such a nature that they can be
readily used in the various operations of our thinking
processes. The form which thought necessarily assumes
for the prosecution of its own activity is always that of
a universal idea; that is, an idea which possesses a one
ness of meaning but admits of an indefinite variety of
application. The universal is sometimes called a group
idea, or a class idea, by which a number of individuals are
embraced under some one general designation. If our body
of knowledge consisted merely in the total number of par
ticular experiences arranged in the form of a series wherein
each separate term remained distinct and completely uncon
nected with any other term, then obviously new experiences
could be added to, but never could be assimilated with,
THE NATURE OF THOUGHT 5
such a body of knowledge. Indeed, a disconnected array of
isolated experiences would hardly merit the name of knowl
edge at all. On the contrary, the elements constituting the
body of our knowledge must be so related and coordinated
that similar elements fall together in such a manner that
a single thought form shall be able to express them all.
Thus, when a geologist says that a certain stone is a fossi],
he means that in his general body of knowledge he has
framed an idea known by the word symbol " fossil," which
embraces under it innumerable special cases, and that one
of these is the stone in question. Thus objects of per
ception can be grasped by the mind and become definite
objects of thought. This grasp of the mind by which a
number of special cases are held together by a single idea
of a nature so universal as to comprehend them all is known
as the process of conception, and the universal idea itself
which is the result of that process is known as the concept.
This word is from the Latin concapio, to take together. The
corresponding German word is Bey riff, which has the same
root as our English word " grip." In the concept the mind
grasps all the essential features which characterize a given
group or class of objects, and holds them together in such a
manner as to constitute an elemental thought form. The
process of thinking, therefore, is fundamentally a conceptual
process, and this primary function of thought consists in
constructing whatever is given through the processes, j?J
perception into the forms of concepts,
2. ^The second function of thought consists in the reduc
tion of the total mass of concepts to some kind of systematic
order. Every concept as it is formed must be received into
the general body of knowledge and assigned to its proper
place and position. The concepts must be arranged in their
due rank and order according to their natural relations of
coordination or subordination. In order that our concept^
may be used as instruments of knowledge, they must admit
of a constant and consistent reference to the general system
6 DEDUCTIVE LOGIC
of which they form constituent parts. These elemental
forms of thought must have their origin in order and not in
chaos. They must be subject to underlying laws of relation,
and not to accident or caprice. Thus the botanist not only
possesses an idea of the general nature of a certain species
of plant, that is, a concept of it, but he knows also definitely
its particular relation to the classified system of plants as
a whole. He is able therefore to describe the species in
question by the relative position which it occupies in the
system itself. Knowledge of the species is obtained not
merely through an understanding of what it is, but also of
what its proper setting may be.
3. JThe third function of thought consists in referring
whatever may be before the consciousness as the object of
thought to its appropriate concept. Such a reference is a
process of interpretation, and represents the central and
most essential feature of all thinking. This mode of inter
pretation may be brought about in several ways.
(a) In the first place, any one portion of our general
body of knowledge may be interpreted in the light of some
other. Thus by way of interpretation or explanation I may
refer one concept to another concept which embraces it as
a smaller class or group within a larger one j e.g. the Trap-
pists are a Eoman Catholic brotherhood.
(6) Again, the concept lends itself to a further use as an
instrument of knowledge, by revealing the various charac
teristics which constitute its nature according as the trend
of thought at the moment may happen to emphasize one or
another of them. In the ordinary processes of thought we
never use a concept in the totality of its significance. We
attend only to a single phase of the concept s meaning at a
time; and our thought selects always that particular phase
of the meaning which is pertinent to the special object of
thought under consideration. Thus the concept, govern
ment, is an exceedingly complex idea, and may be consid
ered from various points of view, as to its general nature,
THE NATURE OF THOUGHT 7
whether democratic, monarchial, despotic, etc. ; or as to its
special functions, whether that of the judicial, legislative, or
executive. The concept as a complex idea may always be
subjected to a more precise determination by the concentra
tion of thought upon one or more of its special attributes or
relations.
(c) In the third place, a particular experience in the
field of sense perception may be interpreted by referring it
to the appropriate concept of which it may be regarded as a
special case. The knowledge given through the senses is
rendered more definite by this reference of it to a concept
which serves to illumine it.
This process of thought which renders the elements of
consciousness more definite by a reference in any one of the
three ways mentioned above to some interpreting concept
is known as the judgment. There is a universal tendency
of thought to transform every concept into the form of a
judgment, because the very presence of a concept in con
sciousness challenges our thought to express some definite
assertion concerning it, and such an assertion is itself a
judgment. As long as there is sustained interest in any
concept which occupies the focus of attention, there is a
constant play of thought about it ; we turn it over in our
minds ; we examine it on all sides ; we put questions to
ourselves about it; and the result is a series of judgments
as regards its nature and the several relations which it sus
tains to cognate concepts. Thus our general knowledge
serves to illumine the specific portion of it which is the
special object under contemplation. So also when the
object of consciousness is a particular object of sense per
ception, we form a judgment by referring it to its appropri
ate concept. Thus in the judgment, Arsenic is a poison, we
have as it were a cross-section of our knowledge in general-,
but in the judgment, this substance which is in the test-
tube before me contains arsenic, the reference is to a special
object in the field of vision which is interpreted by means
of its appropriate concept.
8 DEDUCTIVE LOGIC
Every new experience which is more than a fleeting im
pression, and which is drawn into the field of our attention,
gives rise to one or more judgments of this latter kind. As
I am writing, I look out from a hillside which commands a
wide prospect; and as I observe the various objects in the
field of vision, my thought immediately refers them to their
appropriate concepts by way of more definite characteriza
tion. There is the winding road through the valley, sepa
rating the green meadow from the wood beyond; in the
meadow cows are grazing; by the side of the road flows a
stream, rushing over its rocky bed arid losing itself in the
dark shadows of the wood ; in the distance are the uplands
again bounded by the horizon line, above which the clouds
are hanging low and threatening. Such a description of the
various objects of perception within a field of vision forms a
series of particular instances referred to their corresponding
concepts. They are simple judgments of identification,
a reference of an object immediately before us to a familiar
idea which through its word symbols satisfactorily describes
it. Such a scene however naturally gives rise to more com
plex ideas, which represent the fourth function of thought.
4. This fourth function of thought consists in the process
of. unfolding whatever may be necessarily implied in our
judgments, but is not explicitly asserted. Thus in the scene
described above, I am able to make certain statements which
are warranted by the facts, but which are not the result of
simple observation. I am led to venture the assertion that
there are trout in the stream before me ; that by reasonable
skill and perseverance a fisherman may hope to fill his creel
there in a few hours ; that the threatening clouds, the east
wind, and sultry atmosphere will bring rain; and that it
would be wise under such conditions to fish worm rather
than fly. Judgments such as these are far more complex
than the simple judgments of identification or recognition.
We may call them judgments of elaboration. What is actu
ally given is combined with our general knowledge in such
THE NATURE OF THOUGHT 9
a manner as to render explicit the full measure of all which
is necessarily implied. Thus the assertion that the stream
contains trout is based upon an experience of many years,
and in this way the past is used as a means of interpreting
the present. In like manner my past experience of atmos
pheric conditions enables me to interpret the present condi
tions as indicating the approach of rain. Moreover, the
dark and stormy day is judged to be more suitable for worm
than fly fishing, because on account of the coming rain the
natural flies will not be on the wing, and the rain itself will
wash from the hillsides and banks into the stream grubs
and worms, which the expectant trout will be in readiness
to take. My general knowledge has enabled me in this
particular case to make statements which go beyond that
which is actually perceived, but which nevertheless I am
constrained to believe true, because necessitated by what is
known. And this is the essential feature of all inference.
But inference is not confined to the interpretation of that
which is given in perception. It may serve also to interpret
any part of our general body of knowledge by any other
part, or by the whole. Thus two judgments of a universal
nature may be brought together in such a manner that their
combination furnishes elements of knowledge which are not
given by either judgment separately. We know, for in
stance, that the sum of the angles formed on the same side
of a straight line at a given point equals two right angles ;
also, that the exterior angle formed by extending a side of a
triangle equals the two opposite and interior angles. These
two judgments, when put together, necessitate the infer
ence that the sum of the angles of a triangle equals two
right angles. Inference therefore is essentially a process
by which our thought combines given elements of knowledge
in such a way that the result contains something which the
given elements in their isolation fail to disclose.
There are some general considerations in reference to these
four functions of thought which should be presented. In the
10 DEDUCTIVE LOGIC
first place, the word function itself is significant. It in
dicates an activity which is dependent upon other activities
correlated with it. Each of the four functions of thought is
closely connected and coordinated with the other functions,
and no one is complete in itself. The concept is an essential
element of the judgment, for the judgment is merely the
concept rendered definite through assertion. Moreover, in
ference is a process which consists essentially in the expan
sion and elaboration of our judgments. Inference is itself a
judgment, only it is a judgment which is reached indirectly.
And in the formation of any judgment it is exceedingly
difficult to eliminate altogether the inferential elements,
inasmuch as every judgment contains more than is actually
given in perception, or in a series of perceptions. The result
rests largely upon that which is necessitated by our general
knowledge, and this is essentially inference.
In these coordinated relations which unite concept, judg
ment, and inference, it is natural to regard judgment as the
central function of thought. From this point of view the
concept may then be defined as the judgment in its potential
form ; that is, the concept contains in an indefinite way all
the possible elements of knowledge which it is the function
of the judgment to make explicit. The inference is the
judgment, as we have seen, exhibited in its relation to other
judgments upon which it depends as the warrant of its
validity. The concept is an abridged form of judgment,
while the inference is an expanded form ; and the unit of
thought, therefore, which lies at the basis of all thought
processes is the judgment. It is to thought what the
element is to chemistry.
Again it is to be observed that in the process of inter
preting a given object of knowledge by means of its corre
sponding thought form, it happens that the object in question,
say a given object in the field of vision, will in some measure
at least modify the thought form to which it is referred.
Thus every new experience is both interpreted by our
THE NATURE OF THOUGHT 11
general body of knowledge, and also in turn widens the range
of that knowledge and changes its nature to a greater or
less extent. Especially is this true concerning any object
of knowledge which is so new as to be wholly unfamiliar.
There is then no appropriate thought form to which we can
refer it. We must so analyze the properties of the object in
question and compare it with other instances of the same gen
eral kind, as to construct a basis for the formation of a con
cept which shall embrace the new order of phenomena under
consideration. Such a new concept has to be fitted into the
main body of concepts, and the process of readjustment
among the old concepts is sometimes a most complex and
difficult one. This is illustrated in a striking manner by
the newly formed concept of radium and the various prop
erties of radio-activity. To receive this new concept into
the main body of concepts requires a readjustment of our
former ideas of matter, conservation of energy, etc., which
is almost revolutionary.
Again, among the philosophical sciences, logic is usually
grouped with ethics and aesthetics under the general class
of the so-called normative sciences. A normative science is
one which refers all its phenomena to some standard, or
norm of value to which they are required to conform. The
standard in ethics is that of the right or the good; in
aesthetics, of beauty ; and in logic, of truth. Truth may
be defined as correspondence with reality. The real is the
world as it is constructed by us in consciousness. It is
coextensive with the whole received body of knowledge.
It is the world which is revealed to us through the senses,
it is true ; but at the last analysis it is that world as we
interpret and understand it.
To say therefore that the logical demand of our concepts
is that they must be true signifies that every concept must
clearly and adequately embody the essential features of all
the particular instances in experience which have formed
the basis of its derivation ; the concept moreover must be
12 DEDUCTIVE LOGIC
capable of a constant reference, that is, it must not contain
any element of variability which prejudices its integrity as
a concept. To say that a judgment must be true signifies
that when the judgment expresses the general relation of
any concepts to each other within the same system, it must
conserve the general order which characterizes the system
as a whole, and all interrelated parts of it ; and when the
judgment is of the form of a particular experience referred
to its appropriate concept, then all such references must be
exact. To say that an inference must be true signifies that
the conclusion reached through the process of inference
must be of such a nature that every element of it will find
complete warrant in that which is adduced as its ground.
The logical standard, therefore, which must be realized in
all cases demands clear and adequate concepts of a constant
meaning arranged in an orderly system, so that every
reference to it of any particular object of thought must be
exact, and every inference based upon it must be valid.
While truth may manifest itself in many ways as clearness,
adequacy, constancy, consistency, exactness, or validity,
nevertheless these are all but various instances of a single
elemental principle which underlies the ultimate standard
of logical thinking.
CHAPTER II
THE CONCEPT
THE concept as a form of thought embraces a number
of phenomena which, however much they may differ, have
nevertheless an underlying unity. The ratio of the elements
of similarity to those of diversity in our concepts is by no
means a constant one, but admits of considerable variation.
1. In the first place, the diversity may be reduced sub
stantially to zero, as for instance in such a concept as that
of a silver dollar. The differences which exist between the
several particular cases of this general concept are so minute
as to be overlooked; the similarity alone attracts the at
tention. Each one is an exact copy of every other, and the
idea of any diversity is here practically eliminated.
In reality, however, no two phenomena are precisely
alike. As Leibniz once remarked, " Xo two leaves on the
same tree are alike." And Plato in the same vein has said
that " If two things were exactly alike, there would not be
two but one." Therefore, while the element of diversity
may be reduced to zero as regards its practical relevancy
and as regards the essential significance of the concept in
question, nevertheless it is always present in some appre
ciable degree and may be discovered to a discriminating
observer.
2. There is a second class of concepts wherein the diver
sity is more apparent, and yet the likeness is quite as
obvious. Thus the concept, dog, embodies all the charac
teristic features of the dog race, and yet is so elastic
and ample an idea as to hold in one and the same mental
grasp such diverse breeds as the mastiff, the bull-dog,
13
14 DEDUCTIVE LOGIC
the French poodle, the greyhound and dachshund. A con
cept such as this is typical of the general run of concepts
which require no unusual penetration to disclose the funda
mental elements of similarity in spite of the wide range of
differences.
3. There is, however, a third class of concepts which
require more than ordinary insight, it may be the insight of
genius, in order to discover the unity which lies hidden be
neath an obscuring mass of manifold differences. It required
the analytic mind of a Newton to grasp under one concept
such diverse phenomena as the fall of a body to the earth
and the moon s revolution about its orbit. In the one case
there is motion in a straight line, in the other the motion
is in the path of an ellipse ; in the one the body actually
falls to the earth, in the other it is forever falling but never
falls. Nevertheless, the two are similar. The course of
the moon may be resolved into two distinct motions ; the one
centripetal, which is a direct falling toward the earth, the
other the centrifugal, which holds the former in check and
modifies the direct fall toward the earth so that the result
is the present elliptical orbit of the moon. Therefore it may
be truly said that the moon is always falling toward the
earth in a manner precisely similar to that of the ordinary
falling body upon the earth s surface. The only difference
is the counter force which is operative in the one case and
not in the other. Such a difference however so obscures
the general features of resemblance in the resulting phe
nomena that a surface observation devoid of any deeper
reflection may pronounce them so different as to possess no
point in common. It is characteristic of the trained mind
that it is able to penetrate beneath the surface and discover
points of similarity which escape the notice of unreflecting
observation. Fortunately for the generality of intelligence,
the phenomena of human experience for the most part fall
together into natural groups whose underlying bond of unity
is perfectly obvious. Nature is so prodigal of her creations
THE CONCEPT 15
that innumerable individuals of the same species are forced
upon our attention. The common events of life repeat them
selves with a regularity which compels the recognition of a
constant and common principle as their basis. Therefore it
becomes a natural habit of mind to see things together by
reason of their common features. The most primitive of all
our judgments, and that which lies at the foundation of all
other judgments, is that which is based upon the recognition
of similarity among phenomena. The concept has its ori-
gin in the recognition of similarity among several percepts. 1
As Schopenhauer has remarked, "We get the stuff and
content of our concepts from observation." In our obser
vation, the various instances of some one general kind of
phenomena fall together in our minds on account of their
similarity. They form a series of similar percepts, each
term of which differs from every other term, and yet all in
a certain sense are alike. The mind grasps the essential
features of similarity, fusing them together according to an
underlying unity which persists in spite of the differences.
The result is the concept. Insuch a process, the mind has
subjected the various percepts to an analysis which sepa
rates whatever is peculiarly individual in each instance
from the elements which are characteristic of the series as
a whole. This is essentially a process of abstraction ; it is
what Aristotle calls d<tu pe<ns. There is also a complement
ary process of synthesis, Trpoa-Oea^ according to Aristotle,
which consists in building up the common elements obtained
by the analysis into a complete whole. The resulting
product is in no sense merely an image in the mind of the
blended percepts, but is essentially an ideal construction of
thought which is sufficiently comprehensive and elastic to
admit of application to all particular cases of it. These
processes of separating and uniting, of tearing down and
building up, of analysis and synthesis, have become so con-
l ln the terminology of psychology, the process of perceiving is called
perception ; the resulting product, however, is k::n\vn as the percept.
16 DEDUCTIVE LOGIC
firmed a mental habit that we are not conscious of them, but
come to regard our concepts in the light of original mental
possessions rather than thought forms which we ourselves
have fashioned out of the various phenomena in experience.
The unconscious blending together of the essential charac
teristics of a group of phenomena forms a concept which is
at first barely more than a general impression, a vague
mental grasp of the kind of objects represented by it. The
mind has not yet worked over its first impressions and has
not formed the crude data of its perception into clear and
adequate concepts. The concept at this preliminary stage
of its evolution is called empirical, signifying that it is the
result of a superficial experience which has been subjected
to no critical analysis whatsoever. The word empirical in
philosophy signifies whatever is the result of experience ; it
has, however, a secondary meaning which implies that the
experience in question is a limited one. It is in this sec
ondary sense that the phrase empirical concept is used.
On the other hand, the logical or scientific concept, as it
is often called, is one which has been formed as the result
of some conscious effort to analyze the various phenomena
which form the basis of the concept in question so as to
obtain a clear and adequate idea of their essential charac
teristics. The logical concept differs from the empirical in
the following particulars :
1. The logical concept is always characterized by a grow
ing loss of particularity. The preliminary rough draft of
our concepts always shows the coloring of the particular
instances whence they have arisen. Our first experiences
are necessarily few in number, and they are not sufficiently
numerous to afford a basis for the elimination of all charac
teristics which are not essential. Certain features which
may be common to a limited number of instances will often
disappear when that number is increased. The disappear
ance of such characteristics or their appearance in a sporadic
manner merely proves that they do not belong to the essence
THE CONCEPT 17
of the concept. It is an evidence of an ignorant or untrained
mind that it associates its concepts with particular experi
ences. Such an intellect we are pleased to call provincial
or insular. The nature of the logical concept is always
indicated by its independence of the special case. Thus
the concept of gravitation is not confined to the earth s
attraction of bodies upon its surface. It rises above such
a particular instance, and presents the idea of universal
attraction, of which the force of gravitation upon the earth
is but a small and insignificant instance.
The objection has been urged by Berkeley for instance
that this growing loss of particularity in the concept in
dicates an increasing indefiniteness, inasmuch as the elimina
tion of one particular attribute after another tends to reduce
the concept itself to a bare form stript of all definite features.
Consequently, it is insisted, our concept of a rose must be one
devoid of any specific color, form, or fragrance, and our con
cept of a dog must be one of no particular breed, habit, or
disposition; concepts, therefore, are but the spectral forms
of real objects. This view, however, is based upon a radical
misunderstanding of the essential nature of a concept. For
the concept is freed from the particular attributes which
characterize the various percepts only in a certain sense.
While these attributes are not preserved as such in the
concept, they are nevertheless conserved. The particular
attribute of color, or of form, or of habit is indeed dropped
out of mind in framing the concept, but there is always a
compensation for the loss of the particular by substituting
in its place the possibility, not only of the attribute in
question, but of all others of the same general kind. Instead
of the particular we have the potential which admits of an
indefinite degree of variation. Thus the concept of a rose
admits of any shade of color whatsoever which is compatible
with the whole range of experience regarding roses. In
this adaptability to all possible varieties of color, the poten
tial color of the concept is vastly richer in content, and far
18 DEDUCTIVE LOGIC
more comprehensive, than the single color of any particular
rose could possibly be. So, also, the concept of a dog is
not confined to a particular breed ; it embraces the potential
of all the possible breeds of dogs. Indeed, the mental
process of constructing a concept may be regarded as that
of transforming the various observed attributes of the same
general order into a potential attribute which is lodged in
our minds as a comprehensive symbol embracing every
possible variety of detail.
This potential variation in any concept should not remain
indefinite and vague, but should have definitely prescribed
limits. Thus the color possibility of the concept of a rose
possesses a very wide range of variation; that of the violet,
however, is narrowly circumscribed. The leaf of a beech
tree shows a definite pattern, which is preserved in the
midst of a variation which is essentially one of size alone,
and that within known and easily recognized limits. In the
leaf of the sassafras tree there is a far wider possibility of
variation. On one and the same tree of this species there are
leaves of three distinct patterns. Whatever may be our con
cept of the sassafras leaf, it must certainly provide for this
characteristic variation of form. Moreover, the range of
variation may itself be subject to a variation under chang
ing circumstances. The leaf of the maple is in its normal
appearance green. It admits of a wide variation of shade but
always within the limits of the one color. In the autumn
tints, however, there is a remarkable expansion of the range
of variation, the green turning into the various shades of
brown, yellow, red, gold, and crimson. The possibility of a
wide range of variation in the attributes of our concepts
render them as thought forms exceedingly elastic in the
processes of thinking, while, on the other hand, the definite
limitation of their possible variation renders them quite as
serviceable for exact reference and determination. There is
thus a double gain both of precision and facility in the
exercise of our thought activity.
THE CONCEPT 19
2. There is a second characteristic of the logical concept
in distinction from the roughly generalized empirical con
cept ; namely, that it is freed from all dependence upon any
mental picture in order to render it clear and intelligible.
This feature of the logical concept grows out of the former
given above, the growing loss of particularity; for the
particular can be represented to thought in the form of a
memory image of the original experience. Not so, however,
with the universal idea which lies at the basis of the concept.
The concept is not a composite picture in the mind of a
series of percepts. As far as a mental picture might sug
gest resemblances, we would naturally classify together the
whale and the fish, or the bat and the bird. Dissociated
however from the representations of the outer appearance
of these animals, the bat, as regards the essential elements
which go to make up the concept, is far more closely allied
to the whale than to the bird. The mental picture is in
deed a help to our thinking, but strong minds must learn to
forego such adventitious aid. The undeveloped mind that
of the savage, or the child is dependent upon pictures,
symbols, or figurative representations. In the process of
education, as the mental activity becomes trained and dis
ciplined, the need of colored chalk, of illustrative diagrams,
and of picture-books, becomes less and less in evidence.
In the evolution of the religious sentiment, this is notice
able in a marked degree. The early religions, notably that
of Judaism, endeavored to convey spiritual truth through
an appeal to the senses mediated by a brilliant symbolism.
This, however, was superceded by that religion which laid
stress upon a worship in spirit and in truth, with no in
direct appeal to the thought through the senses, but by
means of ideas which directly enlightened the eyes of the
understanding. An appreciation of the truth in this wise is
essentially logical inasmuch as the truth appears in a form
which appeals immediately to the reason.
3, A third characteristic of the logical concept is its
20 DEDUCTIVE LOGIC
tendency to progressive differentiation ; that is, a breaking up
into smaller concepts which are more precisely determined,
and more distinctly separated in thought one from the
other. The first rough concepts which are formed embrace
without discrimination all sorts of individual instances which
may happen to present any surface resemblances whatso
ever. The child may have at first but one vague concept
which applies equally well to a cow, a horse, and a mule.
Later in the growth of knowledge, this indefinite concept
breaks up into more definite ones, and the child learns to
discriminate between the cow and the horse, and between
the horse and the mule. Wherever there is knowledge
which is comprehensive and exact, the corresponding con
cepts are nicely differentiated and precisely determined.
For most persons, it is a sufficient identification of a bird
to recognize it as a hawk. But for the ornithologist such
a reference is altogether too general and indefinite. He
wishes to know which one of the several different species
of hawk the particular bird may happen to be. Nothing is
gained however by the mere multiplication of the number
of concepts, unless at the same time we are able to discrimi
nate between them. The discriminating mind is essentially
the logical mind. The means by which our concepts may
receive more precise determination will be discussed later
in the chapter on the negative judgment.
There are two ways by which our concepts may be broken
up so as to give rise to new concepts. The one has already
been mentioned, the analysis of the concept into smaller
and smaller groups, each group, however small, represent
ing a complete whole, or complex of attributes. The other
does not regard a complex of attributes which together
constitute the characteristic features of a distinct species of
plant or animal ; it regards the rather some single attribute
and concentrates the attention upon that. This attribute is
first viewed in all the particular instances where it occurs,
and then fashioned into the form of a concept by considering
THE CONCEPT 21
it, in and by itself, quite apart from any of the instances
which illustrate it. Such a concept is known as an abstract
concept. It is oar idea of a particular quality or attribute
of a thing apart from the thing itself. The concrete con
cept, on the other hand, is our idea of a thing as composed
of a complex of attributes, and none of them separated in
thought from the thing in which they all inhere. Thus we
have the abstract concept of motion as distinct from the
concrete concept of a moving body ; the abstract concept of
a sweet or sour flavor as distinct from the concrete concept
of sugar, or of a lemon, in which the attributes sweet and
sour may find expression. So also we have the abstract
concepts of activities apart from any actors, such as speak
ing, swimming, fighting, etc. There may be abstract con
cepts not merely of attributes and activities, but also of
relations which may exist between different objects of per
ception, or between concepts as the case may be, quite apart
from the objects or concepts thus related, such as the con
cept of cause and effect, of organization, of sequence, or of
coexistence. There may be also abstract concepts involving
a combination of several attributes, and yet held apart from
any definite thing or object of knowledge in which they
inhere, such as the complex concepts of freedom, of philan
thropy, of the good, the true, the beautiful. The possi
bility of the various forms of abstract concepts and of the
resulting combinations which may be made out of the sepa
rated elements is indeed without limit. Our logical faculty
is thus given an indefinite scope. The ability to combine
the given elements of knowledge into new forms gives to
our thought a mighty instrument of discovery and of
progress.
4. There is still another characteristic of the logical in
distinction from the merely empirical concept, a character
istic, however, which is realized only in that higher order
of concepts which merit the designation of scientific. The
nature of these concepts is radically distinct from that of
22 DEDUCTIVE LOGIC
the most accurately formulated concepts of the kind which
has so far been described. Instead of representing a corre
lated nexus of common characteristics which discover them
selves to observation in the several special cases, this new
order of concepts represent rather the fundamental con
structive principle, which both underlies the actual produc
tion of every particular instance, and also serves to preserve
the integrity and constancy of its being as well. Such a prin
ciple may assume various forms. It may express simply
the method of producing the different instances which fall
under a single concept. Thus the concept of a conic section
represents the several sections which may be made of a cone,
according as the angle of the cutting plane is varied. There
will result consequently either a point, straight line, circle,
ellipse, parabola, or hyperbola. A mere observation of the
general features of these lines would never disclose their
common nature. They fall together in one and the same
group on account of their common origin, while a simple
variation in the manner of their production gives rise to a
pronounced differentiation in the results.
Again the constructive principle may represent a summa
tion of all the component elements of the object in ques
tion, with possibly the formula of their relative proportions
added. Thus the concept of sulphuric acid can be repre
sented by the symbol, H 2 S0 4 , an exact statement of the
chemical elements in the proper proportions which consti
tute the essential nature of the compound. A concept of
this kind is very different from that concept of sulphuric
acid which represents its several properties and affinities.
Again, this constructive principle may appear in the
form of a law which is operative in producing and sustain
ing the various organisms which may be referred to it.
Thus the concept of natural selection represents a most
comprehensive law, which explains the origin of new
species in the evolution of natural organisms. Every
species has its own constructive principle, which, if discov
THE CONCEPT 23
ered, would form the truest and most satisfactory concept
of that species. These various forms of the concept repre
senting a constructive principle rather than a mere complex
of common attributes are known by the one name of gen
etic concepts; that is, concepts which refer to the com
mon origin of a class of particular instances, rather than
to the characteristic features of their common nature.
There now remains to be considered a topic of consider
able interest and importance, namely, the relation of the
concept to the word which serves as its symbol. As a
symbol the word does not exist for itself, but only for the
meaning which it represents. A symbol always refers to
something which lies outside of itself, and language is a
system of symbols by means of which thought finds signifX
canT expression^ The word Ao yos has a twofold meaning
m Greek (a) the thought itself and (6) the word or words
which stand for the thought. Aristotle calls the one 6 <TU>
or 6 eV Trj ^v\fj Ao yos, and the other 6 t^w Aoyos, that is,
the one, the inner logic ; the other, the outer logic. Lan
guage, therefore, is the external symbol of the inner thought.
We have seen that the logical concept is characterized by
a freedom from all entanglements with any particular per
cepts, or anything like a picture representation of the
same. In this respect, the word as a symbol forms a most
excellent vehicle for the expression of concepts in their
pure thought significance. For the word is perfectly color
less and is freed from all local or temporal associations.
The growth of language has paralleled in this respect the
growth of thought, inasmuch as there has been a constant
tendency for words to lose whatever original associations
of a particular or pictorial nature they may have had. The
Hebrew word for anger was derived from a root which
signified the boiling over of a pot of water, a suggestive
picture of the heat and energy of passion. This primitive
picture, however, has passed away and only the significant
thought remains. So, also, the word green meant, according
24 DEDUCTIVE LOGIC
to its derivation, the color of growing things, the green
natural objects ; but now its meaning has burst these limita
tions and possesses a far wider scope.
There is also a parallel differentiation of words accom
panying the progressive differentiation of thought. Pro
fessor Max Miiller refers to the fact that the Hawaiians
have only one word to express the various ideas of love,
friendship, gratitude, kindness, and respect. To discrimi
nate between the different shades of meaning which these
several ideas signify, a corresponding variety of verbal
symbols has been found necessary. Thus the inner and the
outer thought have progressed together, and the line of prog
ress is always toward a more complete denniteness of mean
ing, in which the finer distinctions of thought may be felt
and expressed.
There is no doubt that clearness of thought is often
greatly obscured by the medium of language. Words come
to acquire strange twists and turns which are productive of
much misunderstanding and error. It is the office of the
logical mind to determine the meaning of words, and to use
the word which most precisely and adequately expresses the
thought. Obscurity in the use of language, however, may
usually be traced to obscurity in thought. Clear thinking
will always find a medium of clear expression.
CHAPTER III
THE JUDGMENT
THE essential function of the judgment is to give definite-
ne ss to the concept. When the concept appears in thought,
it is never as a complete element in itself, but it is always
as a constitutive element of a judgment. The concept
exercises no more independent a function apart from the
judgment than does the sap separated from the tree whose
entire structure it permeates. If we attempt to hold the
concept in the focus of thought, it will always appear
elusive and indefinite. It becomes definite only as it sug
gests some judgment, or it may be a series of judgments of
which it serves to form the basal element. We may seem
at times to hold the concept before the mind as a naked,
unattached idea ; but it is a barely momentary result which
is reached. The concept maintains such a shadowy form,
only so long as we do not concentrate our thought upon it.
As soon as it becomes in any sense an object of thought, it
challenges some assertion either concerning its nature, or its
relations to other concepts in our general body of knowledge.
If we make a list of various concepts, such as iron, educa
tion, freedom, army, horse, bird, fern, and so on, the eye
rapidly traverses such a list, instantly recognizing the mean
ing of each word, as it occurs, and immediately passing on
to the next. But in such a process we have not really taken
these various concepts into our thought. There has been
merely a series of mental reactions in the recognition of the
meaning of word symbols. Such a recognition is nothing
more than the vague sense of familiarity which the several
words are capable of arousing. If, however, an unfamiliar
25
26 DEDUCTIVE LOGIC
word should appear in such a list, it would immediately
give us pause ; we would begin to think about it, to turn it
over in our minds, endeavoring to discover its general nature
and the relations which it sustains to our other concepts.
This would at once give us a series of judgments. Or if any
one of the suggested words should elicit any special interest
on our part, the various processes of thought would be found
to react upon it in such a manner as again to yield a num
ber of judgments. Thus if we allow our thought to dwell
upon such an idea as that of education with something more
than a mere passing recognition of a familiar word, then we
at once find ourselves constructing some definite assertions
concerning this idea, as to its general nature, its various
forms, its methods and scope, and the fundamental princi
ples which underlie its essential significance. Whenever a
concept swings into the focus of thought, it at once forms a
centre whence radiates a series of judgments. The judg
ment may be defined, therefore, as a concept which is ren
dered definite through some assertion concerning it. The
concept is a potential judgment, or rather it is the potential
of many judgments which are implicitly contained in it.
The judgment is the concept in its unfolded form. The
concept, moreover, may be regarded as an unstable com
pound which through the barest contact of thought sepa
rates into its elemental parts and relations expressed in the
form of judgments.
There are two ways by which we may render any concept
definite through assertion, thus producing two general types
of judgment.
1. The concept may be referred to another concept which
forms an essential element in its constitution, or else to one
which sustains some essential relation to it. Thus we may
have the judgment as follows : The constitution of a nation
embodies the fundamental principles underlying the judi
cial, legislative, and executive functions of government.
In such a judgment we have the interpretation of one
THE JUDGMENT 27
concept by others which enter into its composition as
constitutive elements of it.
We may have also a judgment which expresses relations
between concepts as the following : Liberty is not possible
in a country where there is no respect for law.
In both of these illustrations the judgments relate to our
knowledge in general.
2. A concept, however, may be made definite by a reference
to some particular instance which illustrates it, or a particu
lar instance in turn may be referred to a concept which in
terprets it. Thus the concept of philanthropy may be made
clearer and more definite by a reference to certain specific
persons and their deeds ; and on the other hand a special
case, as a peculiar light in the northern sky, may be explained
by a reference to a familiar concept which serves to interpret
it, such as the aurora borealis. Under this head of a refer
ence of the particular experience to its corresponding univer
sal, we have the great body of our judgments of identification
or recognition, such as, This plant is a fringed orchis,
or That is a red-winged blackbird, or That substance is com
bustible. In these cases, the particular instance is referred
to the appropriate class or group within which it naturally
falls, or to a general attribute which characterizes it.
Of the two forms of judgment, thus outlined, the former
represents some phase of our knowledge in general ; the lat
ter, the application of some phase of our general knowledge
to some special case.
Whether the judgment consists of the characterization of
our knowledge in general, or the interpretation of a particu
lar experience by means of our general knowledge, it remains
true that in either case the judgment itself must rest upon
a sound basis of reality. _ A judgment which cannot show
some basis of reality upon which it rests is a judgment in
name only. It may be a fancy, a dream, a query, a hope, but
it is not a judgment. In this particular respect, the judg
ment may be defined as the reference of a concept to reality.
28 DEDUCTIVE LOGIC
This is more obvious in the second form of judgment, where
there is a reference of a particular experience to a concept,
for in this case the reality which underlies the particular
experience in question furnishes the evident basis of reality
to the judgment itself. Thus if one should say, That is the
wreck of a sailing vessel, then the actual object of perception
evidencing its own reality to the senses is to be regarded as
referred to the general concept, a wreck of a sailing vessel,
which thus identifies and explains it.
It is well to remark in this connection that every object
of perception evidences its own reality to the senses, and
this sense of reality attaching to an object is as definite and
as clear a quality of the object as its form, size, color, or any
of its various properties and activities.
This attribute of reality must be regarded as a simple,
unanalyzable element of consciousness which is immediately
given and attested in the very process of perception itself.
The reference to reality in the first kind of judgment
that is, a judgment which relates to our knowledge in
general is not so patent ; but nevertheless the reality is
present as an unseen but secure foundation for the truth
of the judgment. In this first form of judgment wherein
one universal idea is related to another universal idea,
where do we find any basis of reality ? If one idea can be
explained by another idea, in that very process it would
seem that we had swung clear of reality altogether. This is
not so, however. Take the judgment, The collie is an excel
lent sheep-dog ; here the reference to reality is not direct,
it is true, but it is indirect. These concepts, collie and
sheep-dog, have their origin in a series of perceptual experi
ences, and whatever reality may lie at the basis of the
percepts is conserved in the concepts which emerge from
them.
Conceptual reality is based upon perceptual reality.
The concept is real in the sense that it traces its origin to
the several concrete instances whose reality basis is dis-
THE JUDGMENT 29
closed in perception. When the latter is wanting, the
possibility of the reference of the concept to reality is at
once removed. Thus we may have an assertion which has
the form but lacks the substance of a genuine judgment,
such as the following: The Centaur is an animal which
has the body of a horse with the head and shoulders of a
man. Here is a concept, standing as the subject of a judg
ment, which can lay claim to no perceptual ancestry in our
experience. We can form a clear mental picture of it ; it
can even be rendered intelligible to the understanding of a
child ; but there is no real experience at the root of the idea,
and therefore it is nothing but a spectral form of a judg
ment. Every true concept in distinction from a pseudo-
concept, such as that of a Centaur, or a Jabberwock, and the
like, is referable to some real experience in much the same
way as the genuine dollar note may be referred to the gold
coin which it represents and by which it is redeemable.
The counterfeit note presents the same general appearance
as the genuine, but it has a different history, and must be
traced to another origin which is of such a nature as to
render it base and valueless.
There is, however, a certain phase of reality which char
acterizes many of our judgments, and which underlies the
very processes of judgment themselves, but has no origin in
perception. This is the reality which is discovered in the
very nature of thought itself. It is essentially a thought
reality, and as such we become aware of it quite indepen
dently of any particular experience, and only by it indeed
is our experience rendered intelligible. This kind of reality
is illustrated in those self-evident truths which are the com
mon possession of all rational beings, such as the axioms of
geometry, the principle of universal causation, the appreci
ative judgments of moral worth or aesthetic value. Such
judgments are given as examples of a large group of judg
ments which evidently imply as their basis a form of reality
which cannot be traced to an origin in mere perception.
30 DEDUCTIVE LOGIC
There is, of course, that school in philosophy which denies
the possibility of any reality of this kind, but insists that
all forms of reality whatsoever, when completely analyzed,
will reveal an ultimate origin in experience, and that the
so-called intuitive truths have had their beginnings in con
sciousness at the earliest stages in human development, and
have been transmitted through many generations, attaining
in each generation more complete expression, and more
exact formulation ; consequently for us they appear in
thought as judgments which seem to be self-attested and
to have no origin in our particular experiences. In the
present discussion as to the nature of the reality which
underlies our judgments, this question has no direct bear
ing. We find in our consciousness certain judgments which,
for us, at least, whatever their remote origin may be, ap
pear as intuitive truths evidencing their own reality with a
compulsion of thought quite as irresistible as that which
attests the reality of any object which we may see, or hear,
or touch.
These forms of reality whether attested by perception or
by the necessity of thought itself have this in common,
they present themselves to consciousness in such a manner
that we are constrained to yield them a permanent and
constant recognition. To refuse a place to them in thought
would mean the denial of our intellectual integrity. Reality,
as regards its significance for logic, may be defined as that
which we are constrained to think. The real compels our
thought. The dream or fancy can be dispelled by the wak
ing consciousness or the commanding will. Not so, how
ever, with the real object of perception, or the necessary
implications of thought. The ghost is laid by the reassuring
judgments of common sense, but not the lightning, the
thunder, the storm which have overtaken us. The child s
world is one in which there is no sharp distinction between
fact and fancy ; especially is this true with the child s world
of play. But contact with the world of growing knowledge
THE JUDGMENT 31
brings many disillusions and the reduction of many cher
ished fancies to the impossible and absurd.
It is not strictly a question of logic, this question con
cerning the ultimate nature of reality. It is essentially a
question of metaphysics, for metaphysics has primarily to
do with the ultimate nature of things, of time, of space,
of causation, of God, man, and the world, and therefore of
the ultimate nature of the reality itself which underlies
these various manifestations of it. The special question
which metaphysics puts is this: Does our knowledge of
the world represent it really as it is ? May not reality be
very different from that which it appears to be in my per
ceptions ? Are the sea, the sky, the wood, portrayed in
my thought with an exact correspondence to the reality
which constitutes them what they are ? It is evident that
there is here room for much discussion, for much difference
of opinion, and for much confusion of thought as well. But
logic is satisfied from its point of view, if there is assurance
that in the body of knowledge which represents our world
as we conceive it there are elements which maintain a con
stant character, and that whatever we come to think about
them is due to a necessity which underlies their essential
nature. Logic therefore is not concerned with the ultimate
nature of reality, but it does demand as the basis of all
knowledge certain elements which are grounded in neces
sity and admit of a constant reference in thought.
Moreover, that which appears to the individual as a nec
essary experience, a necessary truth, or a necessary demon
stration receives constantly through intercourse with one s
fellow-men a social confirmation and verification. We find
for the most part that our judgments run parallel to those
of the generality of mankind. What we think, other men
think. What is true for me, I believe is true for you also.
The debatable area of conflicting opinion is to be regarded
as knowledge in the making. Questions which divided
men s minds a generation or two ago are many of them now
32 DEDUCTIVE LOGIC
settled, and the results formulated in universally accepted
judgments. Even where there may remain an outstanding
difference of opinion, there is always some common ground
of necessity which is recognized; and the lack of agree
ment is due to the fact, not that the basis of our knowledge
is uncertain and shifting, but that human judgment is
fallible, owing to the limitations of experience, the want of
insight, the presence of prejudice, or the undue submission
to authority. All these disturbing elements enter into the
processes of thought and cause perturbations of judgment.
In spite however of such disagreements, the presence of
an underlying necessity as the basis of knowledge is at
tested and sealed by the general agreement of our judg
ments with those of our fellows. The communication of
thought, the communal interests and activities, the indus
trial and social faith which is preserved between man and
man, the laws both written and unwritten which command
respect and obedience, the universally recognized standards
of civilized life, all attest a common recognition of one and
the same element of necessity, and a common interpre
tation of the many phases of its manifestation.
The difference between the man who is sane and one who
is not lies in the absence of this social factor of agreeing
judgments. With the insane mind there is a feeling of
necessity which, however, is without foundation. The world
in which he lives and moves and has his being is for him
a necessary world. But in it he dwells alone. No one else
can enter it, or understand his view of it. He believes that
he is Julius Caesar, or Napoleon, and it may be consistently
thinks and acts in that character. But for him there is no
fellowship in thought, for his experiences are not believed
and his judgments stand in conflict with those of all the
rest of mankind.
It is incumbent upon us, therefore, as logical beings, to
make sure of the basis of reality which we believe under
lies our judgments. In the investigation of any subject
THE JUDGMENT 33
concerning which we regard ourselves entitled to a judg
ment, not only should we seek as wide a range of observa
tion as is possible concerning the facts upon which we
found the judgment, but we should acquaint ourselves also
with what other men have thought and have written upon
the subject. This is to be done, not that we may slavishly
acquiesce in their judgments, but that by a critical exami
nation of all that is known and reported we may be the
better able to defend our own position, or the more reason
ably to modify or to abandon it as the case may be.
We come now to discuss the relation of the judgment as
a form of thought to its corresponding expression by means
of language. The judgment expressed in language is known
as a proposition. The grammatical form of a proposition
consists of subject, predicate, and copula. The copula is
some form of the verb " to be " either expressed or implied.
It is always implied in any verb which may appear in a
proposition. Thus the proposition, He rows a boat, is equiv
alent to the proposition, He is rowing a boat ; wherein the
verb " to row " breaks up into the participle of the verb com
bined with the auxiliary verb " to be." This can occur with
any verb whatsoever, and therefore in any verb there is im
plied some form of the verb " to be " ; consequently every
proposition may be regarded as composed of subject, predi
cate, and copula.
The logical function of these three parts of speech needs
some further exposition. In the first place, there is a dis
tinction between logical subject and predicate on the one
hand, and grammatical subject and predicate on the other.
The logical subject of every proposition is some phase of
reality ; the logical predicate is always the significant idea
which the judgment contains applied to this phase of reality
in order to characterize or interpret it. The judgment in
this connection may be defined as the interpretation of some
phase of reality by means of some universal idea. The reality
is the logical subject ; the universal idea interpreting it is the
34 DEDUCTIVE LOGIC
logical predicate. This statement may be illustrated as fol
lows : Let us take a judgment of the type in which a partic
ular experience is interpreted by means of a concept, This
is an excellent essay on the labor question. Here the sub
ject, denoted by the demonstrative adjective "this," refers
directly to a point of the world of reality, evident to the
senses, something visible and tangible ; the predicate is the
complex concept, "an excellent essay on the labor question,"
which is asserted of the subject in question. The thought
form interprets the perceived reality simply. In this type
of judgments, the logical subject and predicate coincide
with the grammatical subject and predicate.
In the other form of judgment which is a characterization
of some phase of our knowledge in general, the logical sub
ject and predicate do not coincide with the grammatical
subject and predicate. For instance, let us consider the
proposition, All permanent reforms emanate from the
people. Here the grammatical subject is "all permanent
reforms " ; the grammatical predicate is that they " emanate
from the people."
Now it is the function of the copula to fuse together the
grammatical subject and predicate into one idea which forms
the heart of the judgment and its real logical predicate.
For while language separates the grammatical subject and
predicate, the two must be conceived as merely parts of one
and the same idea in thought. The grammatical predicate,
in this case the phrase " emanate from the people," is an
essential characteristic of the grammatical subject, " perma
nent reform " ; together they form but a single idea, namely,
a permanent reform emanating from the people. This is
the logical predicate which is affirmed of this particular
phase ot that reality which lies at the basis of every true
judgment, and though not expressed in the grammatical
form of the proposition nevertheless constitutes its logical
subject. The logical significance of this judgment, if ex
plicitly expressed, would be somewhat as follows : The
THE JUDGMENT 35
world of reality as I am constrained to regard it is such
as to necessitate that all permanent reforms should emanate
from the people. And this may serve as a type of all our
universal judgments ; they affirm of some phase of reality
the central idea which constitutes the heart of the judgment
itself. A false judgment contains at its heart a central
idea to which there is no corresponding subject in the real
world of knowledge.
When we come to put into words the single idea which
always lies at the root of the essential unity of the judg
ment, why do we separate this unitary idea into two, the
grammatical subject and the grammatical predicate? The
reason is that while the idea in question represents a single
unified thought, it is nevertheless complex and capable of
an analysis into two component elements, one the grammati
cal subject and the other the grammatical predicate. Judg
ment is a process which consists in relating one phase of
an idea to another phase of the same idea, and in rendering
evident the unity which underlies them. The grammatical
subject forms one of these phases ; the grammatical predi
cate forms the other. The copula serves to bring them to
gether and to affirm their unity. Thus every proposition
is the expression of the complementary processes of analysis
and synthesis ; the analysis is expressed by the grammatical
subject and predicate, the synthesis by the force of the
copula whose function it is to blend the two into one logical
idea which forms the very essence of the judgment itself.
In connection with this discussion of the relation of
language to thought, it would be well to call attention to
the meaning of the word term in logic. A term is any word
or combination of words considered as a part of a proposition,
that is, as subject or predicate. The term, therefore, is the
expression in language of the concept as an integral part
of the judgment.
CHAPTER IV
THE UNIVERSAL JUDGMENT
JUDGMENTS, as we have seen, are of two kinds. The
first represents some one or other of the many phases of
our general knowledge. The second serves to interpret the
special case in the light of general knowledge. The first
type is known as the universal judgment. Our general
body of knowledge is composed of judgments of this kind,
and if they are to prove serviceable in the interpretation of
special cases as they arise, they must together form an
orderly system. The concepts which form the constitutive
elements of these judgments are all interrelated. J|pverv
concept represents a point whence radiate lines of connection
with many other concepts. It is impossible to frame a
judgment which shall contain a concept out of all relation
to other concepts.
If, for instance, we analyze the full significance of the
abstract concept of redness, we at once relate it in our
thought to the general color system, which in turn we refer
to light as its source. The idea of light at once suggests the
ether vibrations which affect the retina of the eye, and this,
in turn, the transmission of the physiological disturbance
which occurs in. the retina to the optic lobes of the brain,
and then the resultant reaction which is attended by the
consciousness of a color sensation. Thus the examination
of any concept will reveal an indefinite number of relations
extending into the general body of our knowledge. Their
formulation gives rise to a series of descriptive judgments.
Our knowledge therefore so far as it is worthy the name
36
THE UNIVERSAL JUDGMENT 37
of knowledge, represents an organized _sy stein of relations^
Moreover, there are certain general principles which underlie
the process of organizing the various elements of knowledge.
These principles pertain to the very nature of thought itself,
and man has universally employed them in constructing
his world of knowledge. These fundamental principles are
called the categories of thought. They indicate the various
possible ways by which conceptual elements are related so
as to form the unitary idea which lies at the basis of the
judgment.
As given by Aristotle, the categories, ten in number, are
as follows :
1. ovaia substance 6. TTOTC time
2. TTOO-OV quantity 7. KCUT&U posture or attitude
3. TTOLOV quality 8. exv having
4. Trpds rt relation 0. TTOICIV acting
5. TTOV place 10. Trdax^v being acted upon
Thus any concept whatever may be regarded from one or
more of these points of view, as to its substance, what it
is ; as to its various attributes, its dimensions and weight ;
the relations which it sustains; its space and time condi
tions ; its relative position as regards its surroundings ; as
to what it may possess ; as to how it acts ; and how it is
acted upon. The list exhausts the possibilities of descrip
tion. The word Karrjyopta, as used by Aristotle, means as
sertion or predication. The table of the categories presents
the possibilities of the various kinds of assertion. We have
seen, moreover, that a judgment is a process essentially of
assertion. The categories therefore give us the possible
varieties of judgment. These categories suggest a cor
responding division of words into the various parts of
speech. The substance corresponds to the noun ; quantity,
quality, and relation to the adjective ; place, time, and pos
ture to the adverb; having, acting, and being acted upon to
the verb. Thus we have outlined the possibilities, not only
38 DEDUCTIVE LOGIC
of thought relations, but of the expression of the same in
language.
There are, moreover, certain considerations in reference to
these categories which enable us to coordinate the various
portions of our knowledge so as to form out of them a system
which shall show unity and order. These considerations
are as follows :
The first category is substance ; the other nine categories
give the various kinds of possible attributes which together
serve to determine the essential nature of any concept, that
is, its substance, this first of the categories. Of these various
attributes, some will be common to a number of concepts.
This will enable us to group similar concepts together.
Other attributes will be unique as regards some particular
concept. They will serve as a distinguishing mark of the
concept in question. Others again appear in certain special
instances of a concept, but not in all. This serves to mark
the distinction between constant and variable attributes, a
distinction which is exceedingly valuable from the stand
point of logic ; for it draws the line between attributes and
relations which have a universal validity and those which
are shifting and uncertain.
The above considerations are formulated under five tech
nical terms, known in logic as the Heads of Predicables ;
that is, the various ways in which a predicate given by any
one of the categories may be affirmed of a subject, oT^Tthe
concept regarded in the light of the first category, substance.
These terms are as follows :
1. Genus.
2. Species.
3. Property.
4. Differentia, or Specific Difference.
5. Accident. 1
1 Aristotle gives but four forms, including "species " under "genus,"
and instead of "differentia," giving "definition."
THE UNIVERSAL JUDGMENT 39
They are the Heads of Predicables, as given by Porphyry
(230-300 A.D.) in his Introduction to Aristotle s Treatise on
the Categories.
Genus and species are relative terms and can best be
defined together. The genus is always a larger class which
embraces two or more smaller classes under it by reason of
their common attributes.
The species is any one of the smaller classes which is
embraced under the genus.
The pro]3erty__is an attribute which pertains to the very
nature of the_concept_itseTir"
The differentia is that particular property which serves to
distinguish a given species from all others belonging to the
same genus.
The accident is an attribute which does not pertain to the
essential nature of the concept, and therefore may be present
or absent without affecting the integrity of the concept in
question.
These distinctions may be illustrated in the following
proposition :
Democracy (species ) is a form of government (genus) in
which the supreme power is vested in the people (differ
entia) ; it is attended by certain dangers due to the dissipa
tion of responsibility (property) ; it is regarded in the
United States by some as a proved success, by others as
still in the experimental stage (accident).
The several species under one genus are called cognate
species.
A generic property is one which grows out of the idea
represented by the genus, and which therefore all cognate
species have in common.
A specific property is one which grows out of the idea
represented by the differentia, and belongs therefore only
to one of a number of cognate species.
Genus and species, being relative terms, a concept may
be regarded as a species relative to a genus which embraces
40 DEDUCTIVE LOGIC
it, but a genus relative to the various species which it
embraces.
There is however the sum mum genus, which can be re
ferred to no larger class, and also the infima species, which
cannot be broken up into any smaller classes.
In the light of these various distinctions, we may group
our judgments in several classes, according to the dif
ferent ways by which the concepts in these judgments are
related.
1. The possibility of referring a species as a subject to
its corresponding genus as a predicate ; e.g. The purple
martin is a swallow.
2. The possibility of referring a genus as a subject to the
various species under it which together form the predicate ;
e.g. The swallow may be a purple martin, a barn swallow, a
cliff swallow, etc.
3. The possibility of describing any species as a subject
by one or more of its properties as a predicate ; e.g. Cast
iron has a specific gravity of 7.20.
The special case of this group is where the property
chosen is the differentia of the species ; e.g. Capital is
wealth which is actually used for producing more wealth.
4. The possibility of describing a concept by its accident ;
e.g. Some animals can swim.
A judgment in this latter form is known as a particular
judgment. It is not a statement in terms of a universal ;
neither indeed can it be as long as the predicate is an ac
cident of the concept which appears as subject.
It must not be overlooked, however, that any predicate
which is an accident may be raised to the higher level of a
property in reference to any concept, provided that concept
is only more specifically limited. Thus if we change the
above proposition by inserting the limiting adjective "web-
footed," the predicate at once becomes the property of the
subject thus limited, and instead of a particular judgment,
as in the former case, we now have the universal judgment,
THE UNIVERSAL JUDGMENT 41
All web-footed animals can swim. In general it may be
said that an accident of any species always becomes the
property of that same species under certain definite restric
tions. Every accident, therefore, is a potential property.
To call any attribute of a species an accident is a confes
sion of ignorance, for if we only know the corresponding
limitation of the species in question, the accident at once is
transformed into a property.
If our knowledge were perfect, we should be able to
explain all accidental variations, even the most minute and
seemingly insignificant. Each so-called accident could then
be regarded as a property and be referred to some constant
element within the nature of the concept itself as its cause.
Every variation in nature, whether of color, or form, or
peculiarities of habit and disposition, has a good and suffi
cient reason why it is what it is and not anything else. To
call such variations mere accidents of a species is of course
a confession of ignorance. This leads us to the fifth pos
sibility of reference.
5. The possibility of referring properties of concepts to
definite conditions as their cause. The causal relation when
expressed or implied in a judgment not only renders that
judgment more definite and consequently serves to perfect
the order of the general body of knowledge, but it also
furnishes the ground for the judgment itself and conse
quently serves to justify it. Take for instance the proposi
tion, A conic section formed by a cutting plane parallel
to the base of a cone is always a circle. Here the circle, as
regards a conic section in general, is an accident, but as
regards a conic section under the condition that the cutting
plane is parallel to the base, it is an essential property.
The condition determines the property, and the two are
related as cause and effect. So, also, to further illustrate
this relation, the freezing or boiling of water may be regarded
as accidents, so far as the concept of water in general is
concerned. They are, however, properties of water when
42 DEDUCTIVE LOGIC
specifically determined by the freezing and boiling condi
tions.
.Knowledge, therefore, which is vague and indefinite, gives
rise to judgments whose predicates are accidents of the
subject concept. Definite knowledge, on the other hand,
always gives rise to judgments whose predicates are prop
erties of the subject concept. The bond of connection or
inherence between any species and its property forms the
ground of the universal judgment.
In the various relations which concepts may sustain to
one another in the general scheme which has been given,
there are, in the main, two points of view from which a
concept may be regarded, giving rise to two different kinds
of judgment. The one point of view is known as that of
extension and the other that of intension. The extension
j3f a concept refers to the range of its application "The
intension refers to the various properties which constitute"
its meaning. The term denotation is used as equivalent^ To
extension; and connotation as equivalent to intension. The
term content is also used in much the same sense as con
notation or intension. By some writers the terms extension
and intension are applied to concepts, while denotation and
connotation are applied to terms, the language symbols of
concepts. In ordinary usage, however, extension and de
notation are used interchangeably; so also intension and
connotation. Two questions naturally arise in reference to
any concept : the first, what is its meaning ? and the sec
ond, to what extent within the range of our knowledge
may it be applied ? It is obvious that these two questions
are mutually dependent. It is impossible of course, to
know the number of special cases to which the concept may
be applied if we know nothing of its distinctive properties ;
and, on the other hand, we can know nothing of the distinc
tive properties unless we possess some knowledge of the
special cases illustrating them.
Tiu- distinction between intension and extension gives
THE UNIVERSAL JUDGMENT 43
rise to two topics known as definition and division. Defi
nition is the process of unfolding the connotation of any
term, and division is the process of unfolding the denotation
of a term ; that is, the former tells what it is, the latter to
what instances it may be applied. These two processes we
will now consider more in detail.
CHAPTER V
DEFINITION
DEFINITION is the process of unfolding the connotation
of a concept. A statement giving the complete connota
tion, however, would be overloaded and would weigh down
our thought and its expression with a superfluous burden.
If a definition serves to locate a concept in its proper region
within the general body of knowledge, and in addition dis
tinguishes it from all other cognate concepts which may
fall within the same general area of thought, then it may
be said to perform its function satisfactorily. The function
of definition is expressed by the following rule ^Definition
consists in referring any concept to its proximate genus, i.e.
the genus immediately above it, and also in giving its
appropriate differentia.
To define means to set limits or bounds. This rule in
dicates two defining circles : the first, the genus, marks the
larger area within whose range the concept belongs; the
second, the differentia, draws a narrower circle which sepa
rates the concept within it from all others which lie within
the outer circle, and yet outside this inner circle of more
exact specification. This method of defining is a procedure
which should always be followed when it is possible. There
are other modes of definition which are less complete, but
which it is sometimes necessary to employ, as will be shown
later. The above method, however, is preferable, as it alone
can give what is known as the essential definition.
A distinction is drawn by some logicians between a real
and a nominal definition. The real definition is regarded as
one which gives the meaning of the concept ; the nominal,
44
DEFINITION 45
as giving the meaning of the term which is the language
symbol of the concept. Some writers, as Sigwart and Mill,
declare that there can be no such thing as a real definition,
inasmuch as the process of defining consists in unfolding
the meaning of words. Definition, from this point of view, is
merely the art of fitting the word to the idea which it repre
sents. It seems to me, however, that the process of defini
tion must primarily refer to the meaning of the thought,
and only in a secondary sense to the meaning of the word
which is the symbol of the thought. For the symbol can
have no meaning, except as it represents some thought
behind it. And, in the second place, to define means to
render definite. Consequently, a definition of terms presup
poses always a preliminary transformation of our ideas from
an indefinite to a definite state of determination. It is
thought determination alone which can afford a basis for
exact verbal definition. To draw a line of distinction be
tween a real and a nominal definition is to misunderstand
the relation which obtains between a symbol and that which
it symbolizes.
There are certain rules which should be observed in
definition :
1 . The term defined should be coextensive with 1 lie defini
tion, neither ^realer nor less. The following is an example
of the violation" aFTTii s rule : Logic is a normative science.
Here the term " normative science " is not coextensive with
" logic," for it includes ethics and aesthetics as well as logic.
2. The definition should not contain any superfluous
material. ^ Take the following definition : An hallucina
tion is a fancied perception (genus) without basis of fact
(differentia), and which indicates an abnormal state of
consciousness. The latter clause, while quite true, is alto
gether superfluous. The definition should be always in as
concise a form as possible.
3. JThe definition should not repeat the term to be defined
either explicitly or implicitly. The violation of this rule Is
46 DEDUCTIVE LOGIC
known as defining in a circle (circulus in definiendo). In an
examination recently given the terms " percept " and " con
cept " were defined as follows : A percept is that which
is perceived. A concept is that which is conceived. These
definitions are incorrect also for another reason, because
they contain no proper genus. Instead of a true genus to
which the term defined is referred there is substituted the
indefinite and unsatisfactory phrase " that which."
Under this head of explicit or implicit repetition of the
term to be defined may be included all synonyms of the
term in question. There is the following remark of Hume
which illustrates this. Speaking of the definition of the
term " efficacy/ he says : " I begin with observing that the
terms of efficacy , agency, power, force, energy, necessity, con
nexion, and productive quality are all nearly synonymous ;
and therefore it is an absurdity to employ any one of
them in defining the rest. By this observation we reject at
once all the vulgar definitions which philosophers have given
of jwwer and efficacy ; and instead of searching for the idea
in these definitions, must look for it in the impressions
from which it is originally derived." l
It sometimes happens that in a compound term the in
cidence of the definition falls only upon one of the elements
which compose the compound. In such a case, the other
element of the compound term may be repeated in the defi
nition. Thus the terms, " vesper-sparrow," " gun-metal,"
" armored cruiser," may be defined by referring each to its
appropriate genus, " sparrow," " metal," " cruiser," and then
giving its corresponding differentia.
4 A definition should never be in obscurer language than
k the term to be defined. The violation of this rule is called
" ignotum per ignotius"
An example of this is the following: A state is an
ethnic unit which lies within a geographical unit.
1 Hume, A Treatise of Human Nature. Edited by Green and Grose,
p. 451.
DEFINITION 4T
Sometimes, however, in defining technical terms it is
necessary to use technical words, and an impression is
given to the uninitiated at least of an obscure definition.
Such a definition is Herbert Spencer s of evolution. " Evo
lution is a continuous change from an indefinite incoherent
homogeneity to a definite coherent heterogeneity through
successive differentiations and integrations." In this defi
nition every term used has a definite connotation with which
every student of the subject has become familiar, and there
fore to such an one this definition is exceedingly luminous.
5. A definition should never contain negative expressions
when it is possible to state it by means of the proper posi
tive terms.
The following is a violation of the rule :
A utilitarian is one who does not believe in an intuitional
basis of morals.
It is always desirable to define any term by what it is
rather than by what it is not.
There are certain terms, however, which by their very
nature admit of a negative definition only. Such terms
are the following, anarchist, blindness, unarmored cruiser,
supernatural, and the like.
There are other forms of definition which are substituted
for the ideal form ger yentis et dijf event mm. Sometimes
they are mere makeshifts at definition, when one is ignorant
of the true genus or differentia; and often for some spe
cial reason they better serve the purpose of a satisfactory
definition.
They are as follows :
1. Definition by description. When the genus or the dif
ferentia is unknown, then the concept may be described by
its various properties. A person thinks that he has dis
covered a new species of plant. He is in doubt as to its
precise differentia. An exact definition is impossible. He
wishes, however, to publish some account of it. The only
course which is possible under the circumstances is to give
48 DEDUCTIVE LOGIC
a complete description of it, especially as regards those prop
erties in which it deviates in any marked degree from the
type. The description may serve as a basis for the discovery
of the real differentia.
It often happens when one begins a new study, and the
material he has to deal with is unfamiliar, that precise defi
nitions are impossible. At this preliminary period descrip
tion must take the place of definition. Later with the
mastery of the subject comes the possibility of framing
satisfactory definitions.
2. Definition for the purpose of identification. Instead of
the differentia" wliich may be a property that is not evident
to a surface observation, there may be substituted in the
definition another property which is readily observable and
which serves as a mark of identification. Thus we may de
fine an acid as a chemical compound which turns blue litmus
red. It is not a definition of an acid, but it is a most
convenient formula of identification. Or we may define
sassafras as a tree of the laurel family whose bark has an
aromatic odor or taste. Such formula are most valuable as
working definitions. Sometimes the property wliich best
serves as a basis for identification is a very insignificant
one. Thus the color markings of birds, such as the white
tail-feather of the vesper-sparrow, may furnish a convenient
and perfectly satisfactory basis for identification. It may
be that the peculiar mode of flight may serve a similar
purpose. Jn all such instances a superficial property^is
substituted for the differentia.
3. The genetic definition, which refers the concept to be
defined to its origin. The genetic definition, in giving the
origin of the concept, furnishes at the same time a method
by which special instances of the concept may be produced,
and made available for observation and experiment. Thus
the genetic definition of sulphuric acid is given by the
formula H 2 SO 4 . Here the compound is defined by the com
ponent elements of which its essential nature consists. The
DEFINITION 49
genetic definition of a certain dye would be in terms of the
formula by means of which the dye may be produced. So
also all recipes, prescriptions, and methods of construction
may be regarded as definitions of this class. Any concrete
instance may be produced at will by following the sugges
tions contained in the definitions. Thus it is a genetic
definition of a right cylinder that it is a solid body con
ceived as generated by the rotation of a rectangle about one
of its sides as an axis. So also the various colors of the
spectrum may be defined in terms of the number of vibra
tions corresponding to each color.
The genetic definition is one which has always a practical
significance inasmuch as it furnishes knowledge in such a
form as to subserve the ends of utility. It not only tells
us the meaning of certain ideas, but it also indicates how
we may apply them in the arts, the sciences, and the practi
cal needs of our lives.
CHAPTER VI
DIVISION AND CLASSIFICATION
DIVISION is a process by which the denotation of a con
cept is exhibited. The result is that form of judgment in
which the subject term represents the concept regarded as a
genus, and the predicate term contains the several species
which fall under it. The process of definition always
underlies that of division, for we must know the differentia
of each species before it is possible to consider it as a dis
tinct group under a given genus. In dividing a concept
into its appropriate species, one may proceed in a number of
different ways according to the point of view he may choose
to take. 3?he point of view determines in every case the
so-called principle of division (ftmdamentum divisionis).
Thus we may divide the general concept, education, ac
cording to the principle of the progressive stages of educa
tion regarded as a process, as primary, secondary, collegiate,
university, and professional ; or the principle chosen may be
that of the general nature of the course of studies pursued,
such as the common school, academic, scientific, technical,
etc. ; or again, the principle of division may be an historical
one, giving the periods of ancient, mediaeval, and modern
education. It is obvious that the principle of division will
vary according to one s special interest or purpose. There is
thus a wide range of possibility as regards the analysis of
our various concepts. There is no beaten road for thought
to travel, but each one may cut out his own path. In the
midst of this variety of choice, however, there are certain
rules which logic imposes upon the free play of thought.
Within the bounds of these restrictions the inventive spirit
50
DIVISION AND CLASSIFICATION 51
may range at will ; but the violation of them brings con
fusion and inconsistency of thought. The rules are :
1. There must be but one principle of division. A
violation of thlarmle, for instance, would be such a divi sion
as that of the concept "education" into primary, secondary,
collegiate, technical, scientific, and professional.
2. The members of a division should be mutually
exclusive; no two members of a division should overlap.
The above example illustrates the violation of this rule also.
The following furnishes another illustration: The division
of the discontented classes in society into socialists, an
archists, nihilists, and populists.
While the violation of the first rule produces overlapping
divisions, nevertheless the same error may be due to other
causes even when the requirements of the first rule are
realized.
.",. The division must be exhaustive. No possibility
should 1)0 overlooked and omitted from the division, Thus
if we divide conduct into two classes, the moral and immoral,
the division is at fault because of its incompleteness. There
is still a third class which is omitted, namely, that of con
duct which is morally indifferent, and concerning which it
is not possible to affirm that it is either moral or immoral.
There is a particular method of division known, as Dichot
omy which provides for an exhaustive division under all
circumstances, It consists in dividing a concept into two
parts, according to the presence or the absence of a
differentiating attribute which is chosen as the principle
of division. This may be illustrated by the so-called "Tree
of Porphyry," which exhibits a continued division of that
most general and all-comprehensive concept, being.
52 DEDUCTIVE LOGIC
Being
I
I I
corporeal incorporeal
I I
animate inanimate
I
I
sensible insensible
I
I I
rational irrational
I
r n
Plato Aristotle and other individuals
Such a division is more curious than satisfactory, for one
of the members in each successive division is left indefinite,
being designated by what it is not, rather than by what it is.
Moreover, if a positive term is substituted for the negative,
and its precise connotation is attempted, it will in all
probability not be a complete opposite of the first term of
the dichotomy. If this is the case, the division itself is not
complete, for the dividing of a concept into two members
which are not exact opposites renders it possible to inter
polate between them one or more possibilities which do not
belong to the one or the other of the extremes. In this connec
tion it is necessary to distinguish between contradictory and
contrary or opposite terms.
jContradictory terms are such that they divide the whole
universe of thought between them and admit of no middle
ground.
Contrary terms stand opposite to each other as extremes,
^but there is a possibility of middle ground between them.
Animate and inanimate are contradictory, bitter and
sweet are contrary terms.
A dichotomous division requires its terms to be related as
contradictories. There is perhaps no error in division which
is more frequent or more insidious than this, of dividing a
DIVISION AND CLASSIFICATION 53
concept into members which sustain contrary rather than
contradictory relations to each other. This is seen particu
larly in debate where an opponent will often confront one
with a choice of alternatives, either this course or that,
when, however, there is a third possibility unnoticed, or
purposely ignored. It is the third possibility which we
should always have in mind, and endeavor to discover
whenever the necessities of a dichotomous division are
forced upon us. There can be no free choice of the mind
unless all possibilities are presented.
On this very account division very often takes a threefold
form, that of Trichotomy ; because when a concept is divided
into two members exhibiting some one or more opposed
characteristics, a third member representing a mediating
position between the two naturally suggests itself. This
form of division which expresses extreme terms in relation
to the middle ground between them has played an important
role in the history of philosophical thought. For instance
Aristotle s theory of morals was based upon the principle
that right conduct always lies between two extremes, neither
of which commends itself to the reason. Thus courage, which
is the mean between cowardice on the one hand and rash
ness on the other, takes rank as a virtue and is freed from
all criticism which is called forth naturally by the extremes.
So, also, according to Aristotle, temperance is the virtue
which avoids the extremes of ascetic abstinence and un
bridled desire.
The trichotomous division is further illustrated in the
dialectical method which grew out of the teaching of Kant,
and which was developed by Fichte and brought to its com
plete expression by Hegel. The meaning of " dialectic " may
be gathered from Plato s usage of the term, which with him
signified the process of argument between two disputants,
who in their controversy for and against a given proposition
render this exceedingly valuable service, namely, that the
course of debate brings to light whatever fundamental
54 DEDUCTIVE LOGIC
elements of truth the opposed positions may have in com
mon. This idea Hegel has applied to the evolution of all
truth which he declares develops progressively through three
stages. The first is the thesis, the primary proposition as
originally affirmed ; the second is the antithesis, the opposed
proposition ; the third is the synthesis, the reconstruction of
these two from a higher point of view which discloses the
unity underlying the two extreme positions. Hegel insists
that a scheme such as this forms a universal programme
according to which the evolution of all thought must pro
ceed.
A distinction is drawn in logic between the so-called
empirical and logical divisions. A logical division is one
which applies the principle of division to any given concept,
and notes all the possible members of the division which
result from such a process. The empirical division is the
result of a critical examination of the logical division to the
end that all members of such a division which cannot be
realized actually in experience may be eliminated. A strictly
logical division may give certain ideal groups which are
rendered impossible actually because of certain necessities
of the concrete situation, or because of the general economy
of nature.
As an illustration of the former, the genus, regular poly
hedron, may be divided according to the number of the
bounding planes. Now applying to the genus the principle
of division which is the number series, and without taking
into consideration any other limiting conditions whatso
ever, we get regular polyhedrons according as their faces
are:
4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, etc.
However, a second question forces itself upon our consid
eration. Are the space conditions such that all of these
supposed regular polyhedrons can be actually constructed ?
DIVISION AND CLASSIFICATION 55
The answer is that only the following are possible, those
having sides as follows :
4, 6, 8, 12, 20.
Thus the formal division has been corrected through an
appeal to the actual conditions which are imposed by the
existent space relations.
Again to illustrate what may be called the limitations due
to the economy of nature, we have the following division
of mankind, according to differences of color : White men,
black, red, yellow, orange, green, etc.
Such a division is the result of applying a color principle
of division in its full rigor and extent to the concept in
question. When, however, we ask in addition the question
as to the prodigality of nature in this respect, we find that
the actual colors found among the various races of man are
limited, and therefore our division must be corrected by
striking out such colors as green, orange, etc., which have
no empirical confirmation in fact.
There is a difference as regards order of procedure between
dividing a concept simply according to the possible varia
tions of some selected property irrespective of any con
sideration of the actual limitations which may occur in
experience, and starting with actual classes as they have
been observed in experience, and grouping them together in
a system as related members of one and the same genus.
This latter process is that of classification which will be
considered next.
Classification is a term which is used for the most part
interchangeably with division, but, as regards strictly logical
usage, classification is a process which is the inverse of
division proper. The problem of classification, therefore,
is that of arranging given classes into a system whose
unity is such that it can be regarded as forming the under
lying ground of the several classes in question. Moreover,
classification proper represents usually a more elaborate
56 DEDUCTIVE LOGIC
scheme than simple division. In classification the process
of division is many times repeated, so that the original genus
not only has its species grouped under it, but each species in
turn may be regarded as a new genus, and its corresponding
species indicated, and so on until a series of injlmce species is
reached.
A classification may be of two kinds, either artificial or
natural. In an artificial classification, the principle of classi
fication selected is some characteristic which is external to
the essential nature of the elements to be classified. In a
natural classification, the principle of classification selected
is a property which forms a constituent part of the essential
nature of the elements to be classified.
1. In an artificial classification the characteristic which is
selected as the basis of the classification is either an accident,
or at least an unimportant property of the elements to be
classified. The consequence is that the various members of
the classification which fall together in the same group
possess in common only this arbitrary or artificial mark
selected as the basis of classification, and are dissimilar in
all other respects.
This kind of a classification is best illustrated by the
alphabetical catalogue of books in a library. The initial
letter of the author is regarded as a differentiating mark. It
brings together in one group an indiscriminate variety of
books which have in common merely the one artificial mark.
Such a classification, however, serves its purpose most satis
factorily in furnishing a convenient key for reference.
An artificial classification generally may be said to per
form some such function as this, namely, of realizing some
definite and specific purpose, and is therefore essentially a
working classification. It must not be thought that an
artificial classification is necessarily an imperfect or unsat
isfactory classification. On the contrary, for the end to
which it is designed, it serves a most useful purpose.
2. A natural classification is based upon one or more
DIVISION AND CLASSIFICATION 57
properties directly connected with the essential nature of
the elements to be classified. In a natural classification the
members which fall together in the same group should not
only agree as regards the common property which is selected
as the basis of the classification, but also as regards a large
number of cognate properties. A property therefore should
be selected as the basis of classification which has the largest
number of correlated properties inseparably connected with
it, so that whenever the given property is present, the cor
related properties will always accompany it. Such a
property is known as a diagnostic property. It is like the
significant symptom which indicates to the physician the
nature of a disease, because the symptom in question always
has a number of other symptoms correlated with it and which
forms therefore a basis of exact diagnosis. A diagnostic
attribute, therefore, will bring together in one and the same
group members of the classification which have in common
not merely a large number of properties, but these proper
ties form a system of correlated and interconnected elements
which together constitute what is known as a natural kind.
In a natural classification, the various members therefore
form these groups of natural kinds, or, as they are sometimes
called, real kinds. In a zoological classification we would
have such natural kinds as vertebrates, mammals, reptiles,
etc. The mammals, for instance, have not merely the dif
ferentiating mark in common, but also a complex system of
correlated properties which are built about the central and
distinguishing property of the kind.
Natural classifications obtain in all the sciences, wherein
the subject-matter is arranged in groups according to a
natural determination of kind. The classifications of ani
mal and plant life are the best illustrations which we have
of natural classification. A natural classification furnishes
an excellent basis for comparative study, for, in the method
of grouping according to kind, resemblances are most easily
observed and significant relations suggested, while at the
58 DEDUCTIVE LOGIC
same time characteristic differences are rendered most promi
nent. It often happens in a natural classification that the
fundamental property chosen as the basis of the classifica
tion, and which is of such a nature as to determine the
essential structure or function of a definite kind, is neces
sarily of such a nature that it is not disclosed to a surface
observation. Thus the classification of birds, for instance,
is based largely upon fundamental differences in anatomical
structure. Birds, not as we see them, but as they are when
stripped of plumage and in their nakedness, are the real
objects of consideration in such a system of classification.
The result is that in the same group there will appear side
by side a number of birds whose surface markings are
exceedingly disparate, such as the blue jay and the crow,
or the English sparrow and the cardinal. It is always a
broadening experience, as regards our habits of thinking,
when we are able to discover some essential similarity at
the basis of a marked surface dissimilarity.
In arranging the various cognate species in any scheme
of classification, they should be arranged in some kind of
order so that the more closely allied species are placed side
by side. It is not only necessary to exhibit the unity under
lying each distinct species, but also the connection which
exists between several species closely related to each other.
This is especially to be desired when several cognate species
together form a series of progressive development. In
such a series, every term representing a distinct species
should occupy a place in the classification which will at
once show its dependence upon the terms preceding it,
and its influence in turn upon the terms which follow it.
Every term thus looks before and after, and the series as a
whole is characterized by an ever increasing complexity of
attributes and functions. This principle of an ordered
series in classification, which the doctrine of evolution has
emphasized, is applicable not merely to the classification of
animal and plant life, but has a far wider sphere of applica-
DIVISION AND CLASSIFICATION 59
tion. Herbert Spencer has taken the theory of biological
evolution, and has applied it with skill and insight to the
various branches of knowledge, as politics, sociology, history,
psychology, ethics, etc., so that as a result the classification
of the subject-matter in these disciplines shows a graded
series of progressive development.
The doctrine of evolution, moreover, has affected the
general theory of classification in the further demand that
the progressive series should exhibit as far as possible the
transition cases between the most closely allied of cognate
species. In the traditional view of classification according to
natural kinds, it was held most stoutly that each member of a
series of cognate species that is, each natural kind must
be regarded as cut off wholly from every other, even from
that with which it is most of kin. It is the ancient doctrine
of the immutability of species. The theory of evolution,
however, insists that the seemingly distinct species shade
off by inappreciable degrees of difference, so that the gap
between any two may be filled up by transition cases show
ing the possibility of a continuous transformation from one
to the other. These transition cases, or missing links, can
not always be supplied in experience ; but the contention of
the evolutionist is that in many cases they have been
supplied, and that if our experience were not so limited,
they could be supplied in many more. There is an illustra
tion, however, which does show a classification in which the
transition cases between groups may be shown perfectly
without any defects due to the limitations of experience.
This illustration is from the sphere of mathematics, and
therefore is relieved of the complexities and consequent
difficulties which obtain in reference to natural phenomena.
We know that the various conic sections may be divided
into the following groups, the point, straight line, circle,
ellipse, parabola, and hyperbola. These are not to be re
garded as distinct classes, each one lying wholly outside
of all the others, but as so related that the circle for instance
60 DEDUCTIVE LOGIC
may be exhibited as the special case of the ellipse, and that
it may be shown how through a continuous transformation
the ellipse may become a circle. In like manner, the
parabola may pass over into the ellipse on the one hand,
or into the hyperbola on the other.
When also limiting cases between species are forthcoming
in biological classification, they serve to form a graduated
series in which the presence of transition cases between
allied groups discloses their underlying unity. The tradi
tional doctrine of the immutability of species breaks down
in the face of such instances. The distinct groups of fishes
and amphibians are differentiated by the presence of gills
in the one and of lungs in the other. In the case of the
so-called group of Dipnoi, the African mud-fish, there were
discovered in one and the same animal both lungs and gills.
It forms, therefore, an intermediate transition type between
the fishes and the amphibians. Moreover, the links which
the existing forms of animal life have not been able to
supply have been found in many cases in the record of
extinct forms preserved in the various geological strata
of the earth s surface.
The unity of widely divergent species is illustrated by
Von Baer s law, that the history of evolution of species in
the race is repeated in miniature in the development
observed in the embryo of each individual. Thus the egg
of a bird in the various stages of transformation passes
through a series of forms, resembling in a rough way it is
true, but still resembling successively a worm, then a fish,
then an amphibian, then a reptile, and finally the full-
formed bird. That all these variations in form are due to
variations in the one constructive basal principle is clearly
seen, inasmuch as the different transformations occur within
the one organism, bounded by the enveloping wall of the
egg. It is the function of classification, therefore, to show
whenever it is possible the unity which underlies its various
groups, and holds them together in a single system through
DIVISION AND CLASSIFICATION 61
bonds not of external relation merely, but of an inner kin
ship.
Every science naturally seeks to arrange its material in
an orderly manner which results in some scheme of classifi
cation. In the sciences such as zoology and botany, the
systems of classification are developed to such an extent of
detail that the intermediate genera and species between the
summum genus and the infima species are specified by a
series of terms which serve to indicate a more and more
elaborate degree of specification. These terms in their
order of specification are as follows : kingdom, group, sphere,
class, order, family, tribe, genus, subdivision, species, variety,
and finally, the separate individuals. These terms may not
all be used in any one system, but they form a kind of
skeleton scheme, any parts of which are available for the
general purposes of classification. It should be remembered
in this connection that the terms genus and species, accord
ing to logical usage, are to be regarded always as relative
terms applicable to any classes whatever, which are sub
ordinated one to the other. Thus the term order is a genus
as regards the family, but species as regards the class.
In a system of classification, the names assigned to
various species are often compound terms made up of the
genus and differentia of the species, e.g. fringed gentian,
red-winged blackbird, smooth-coated collie, etc. The name
not only indicates its place in the general system of classifi
cation, but is at the same time a shorthand expression of its
definition.
Not only has each science a classification of its own
material, but attempts have been made also from time to
time to classify the various sciences in some one general
system which shall show their essential relations and
dependencies. This has proved to be a most engaging
problem to philosophical minds, a problem, however, as
perplexing as it is absorbing. There have been three
attempts in modern times which are of special interest,
62 DEDUCTIVE LOGIC
the classification of the general branches of knowledge by
Bacon, and the classifications of the sciences by Comte and
Spencer.
Bacon s classification of all learning, his so-called " Intel
lectual Globe," is based upon the threefold division of the
mind, memory, imagination, and reason, to which corre
spond the three general divisions of learning, history, poetry,
and philosophy. The classification in its main lines and
without going into all its minute ramifications is shown on
facing page. 1
This classification affords abundant scope for the exercise
of one s critical faculty as regards the validity of the various
divisions which Bacon makes in the course of his analysis
of human learning.
Bacon insisted that every classification of human knowl
edge should exhibit its various members as branches con
nected with a common trunk ; the classification of Comte is
based upon a principle radically different. His purpose is
to show the various sciences in their order of progressive
development. He insists that together they form a series
of increasing complexity in which each science is dependent
upon those before it, and is itself a natural propaedeutic to
those which follow it.
Comte s classification of the sciences proceeds in the follow
ing order : Mathematics, Astronomy, Physics, Chemistry,
Biology, Sociology, the Science of Morals. In order that
the significance of this series may be fully appreciated, the
following passage from Comte is appended :
" In morals we study human nature for the government
of human life. All our real speculations, the most abstract
and the most simple not excepted, necessarily converge toward
this human domain, for indirectly they help us to the knowl
edge of man under his lower aspects, on which the nobler are
dependent. . . . Paramount as the theory of our emotional
nature, studied in itself, must ultimately be, without this
1 Bacon, The Dignity and Advancement of Learning, Book II, etc.
DIVISION AND CLASSIFICATION
63
-HJ
* _i
"
-
_
r
G4 DEDUCTIVE LOGIC
preliminary step it would have no consistence. Morals thus
objectively made dependent on Sociology, the next step is
easy and similiar ; objectively Sociology becomes dependent
on Biology, as our cerebral existence evidently rests on our
purely bodily life. These two steps carry us on to the con
ception of Chemistry as the normal basis of Biology, since
we allow that vitality depends on the general law of the
combination of water. Chemistry again in its turn is ob
jectively subordinate to Physics, by virtue of the influence
which the universal properties of matter must always exer
cise on the specific qualities of the different substances.
Similarly Physics become subordinate to Astronomy when
we recognize the fact that the existence of our terrestrial
environment is carried on in perpetual subjection to the condi
tions of our planet as one of the heavenly bodies. Lastly,
Astronomy is subordinated to Mathematics by virtue of
the evident dependence of the geometrical and mechanical
phenomena of the heavens on the universal laws of number,
extension, and motion." l
Mr. Spencer takes exception to Comte s arrangement of
the sciences in serial order, insisting that such a grouping
of the sciences represents neither their logical dependence
or their historical dependence. In this connection he gives
his definition of a true classification which may be of inter
est to quote here as we have already emphasized the
fundamental principle which lies at its basis. " A true classi
fication," says Mr. Spencer, " includes in each class those
objects which have more characteristics in common with one
another, than any of them have in common with any objects
excluded from the class. Further, the characteristics pos
sessed in common by the colligated objects, and not possessed
by other objects, involve more numerous dependent charac
teristics. There are two sides of the same definition. For
things possessing the greatest number of attributes in com
mon are things that possess in common those essential at-
i Comte, System of Positive Polity, Vol. IV, pp. 161-162.
DIVISION AND CLASSIFICATION
65
tributes on which the rest depeiid; and, conversely, the
possession in common of the essential attributes implies the
possession in common of the greatest number of attributes." l
The classification of Mr. Spencer proceeds upon this
principle with the following result:
Science is
that which treats of
the forms in which
phenomena are
known to us
Abstract Science
that which treats
of the phenom
ena themselves
in their ( Abstract
elements concrete
I Science
in their f Concrete
totalities 1 Science
| Logic
I Mathematics
Mechanics
Physics
Chemistry
etc.
Astronomy
Geology
Biology
Psychology
Sociology
etc. 2
In the above the terms abstract, abstract-concrete, con
crete, need some further explanation in order that one may
understand the sense in which Mr. Spencer uses them. By
abstract sciences he would designate those sciences which
deal with fundamental principles detached from any par
ticular incidents which may illustrate them ; as, for in
stance, the necessary relations which obtain in logic and
mathematics and which maybe proved and formulated quite
apart from any concrete demonstration. By the compound
term abstract-concrete he means those sciences which are
partly concrete inasmuch as they investigate actual phenom
ena themselves, but abstract inasmuch as the phenomena
investigated are only detached portions of more complete
1 Spencer, Essays, Scientific, Political, and Speculative, Vol. II, p. 76.
2 Ibid., Vol. II, p. 78.
66 DEDUCTIVE LOGIC
wholes, as, for instance, the examination in chemistry of the
special properties of oxygen by themselves and apart from
the whole body of chemical phenomena. By the purely con
crete sciences, Mr. Spencer refers to those sciences which
investigate phenomena pertaining to complete aggregates,
and the relation of all separate parts to one combined
whole. Thus, as Mr. Spencer says : " The geologist does not
take for his problem only those irregularities of the earth s
crust that are worked by denudation ; or only those which
igneous action causes. He does not seek simply to under
stand how sedimentary strata were formed; or how faults
were produced; or how moraines originated; or how the
beds of Alpine lakes were scooped out. But taking into
account all agencies cooperating in endless and ever varying
combinations, he aims to interpret the entire structure of
the earth s crust. If he studies separately the actions of
rain, rivers, glaciers, icebergs, tides, waves, volcanoes, earth
quakes, etc., he does so that he may be better able to
comprehend their joint actions as factors in geological
phenomena, the object of his science being to generalize
these phenomena in all their intricate connexions as parts
of one whole. l
These classifications of Bacon, Comte, and Spencer have
been given here somewhat at length inasmuch as they pre
sent an excellent idea of the difficulties attending the classi
fication of such complex phenomena, as well as to furnish
suitable material for the exercise of one s critical faculty in
respect to the measure in which these systems have realized
the rigorous requirements of the laws of classification.
1 Spencer, Essays, Scientific, Political, and Speculative, Vol. II, p. 89.
CHAPTER VII
THE SINGULAR JUDGMENT
THIS type of judgment differs from the universal judg
ment in the essential feature that it refers a single object of
thought to our general body of knowledge which serves to
interpret it, while the universal judgment is concerned solely
with the universal characteristics arid relations which obtain
within the general body of knowledge itself. The singular
judgment deals with special cases in the light of our gen
eral knowledge. The universal judgment deals only with
the various phases of general knowledge in the light which
is reflected from one part to another. The single instance
which forms the subject of the singular judgment may be
actually present in the field of perception, or it may be re
instated in consciousness through the processes of memory.
The change in tense may be regarded as unessential, and
the term perceptive judgment is often used as synonymous
with singular judgment, whether the given perception is in
the past or present time. The so-called narrative judgment
is only the perceptive judgment referred to past time, and
therefore does not constitute a distinct type of judgment.
If the whole field of perception is taken in an indefinite
manner as the object of thought, and no particular part
of it specified for special consideration, then the judgment
which results is known as the impersonal judgment, e.g. It
is raining, it is hot, it is a charming day, etc. The imper
sonal pronoun in such judgments refers to reality which is
present in consciousness in a wholly undifferentiated man
ner. If, however, this indefinite range of reality is more
precisely determined by focussing the consciousness at any
67
68 DEDUCTIVE LOGIC
one particular point in the field of perception, we have as a
result the so-called demonstrative judgments, introduced by
the demonstrative pronoun or adjective, e.g. This is magnetic
ore ; this black sand is magnetic ore. The latter is really a
combination of two judgments, This is black sand, and it
is magnetic. The perceptive judgment always originates at
the focal point of the perceptual processes, just as the uni
versal judgment originates at the focal point of the concep
tual processes. A similar variety of assertion is also possible
in reference to the perceptive or singular judgment as was
found to obtain in reference to the universal judgment.
Thus the single subject in perception, or in memory, may
be rendered definite by referring it to its appropriate genus
or species, or by describing it by its properties, differentia, or
accidents.
There are two functions of the perceptive judgment which
correspond in a general way to the two functions of definition
and division. Corresponding to definition there is the func
tion of determinate reference. And corresponding to division
there is the function of indeterminate reference.
1. By determinate reference is meant the identification
of the single object of perception in question with its ap
propriate genus or species, e.g. That is a fossil of the car
boniferous age. In such a judgment we have satisfactorily
disposed of the single object of perception by referring it to
the general class to which it belongs. It is a process simi
lar to that of definition. Indeed, this judgment may lead
naturally to a definition of the general class to which we refer
the specific object before us; for the question may be put,
What is a fossil of the carboniferous age ? The answer
would be its definition. In every process of referring a
single object of perception to the concept which explains it,
the knowledge of the definition of the concept employed is
always implicit in such a judgment. It is not explicitly
stated, however, unless the terms used need to be further
explained or illustrated. It should be remembered that a
THE SINGULAR JUDGMENT 69
definition proper can never be given of a single object of
perception, but only of the concept of which the object in
question is a special case. The definition of the universal
of course illumines the special case which falls under it.
2. Corresponding to division there is the process of in
determinate reference, wherein the object of perception is
referred to one of several possible concepts as its appropriate
characterization, and the judgment takes the following form,
x is y, z, or iv. Thus, we may say, this test-tube contains
nitric acid, or hydrochloric acid. The genus is known, but
the precise species is undetermined, and a range of possibility
is expressed. The various possibilities which occur in such
a judgment correspond to the several members of a division.
In the process of division a concept is represented in terms
of its possible varieties of manifestation. In the singular
judgment of indeterminate reference, the single object of
thought is referred to the various possibilities which may
be alleged as its explanation. However, in the division of
a concept the various possibilities may all be successively
realized, but in the indefinite reference of a single object of
perception to several possible concepts, one only of the pos
sibilities can be the true reference to the actual exclusion of
all the others. Thus we may have the general statement
that violent death may be due to accident, murder, or suicide.
Here each one of the possibilities mentioned may, in turn,
illustrate by an actual instance the genus concept of violent
death. But if it is affirmed that in any single case before us
a given person s death is due to accident, murder, or suicide,
we have the specific case referred to an indeterminate cause
in such a manner that only one of the possibilities in
question can be true.
There is a form of judgment which seems to be a transi
tion case between the universal and singular judgment. It
is the so-called individual judgment whose subject refers
to an individual designated by a proper name, or in some
manner which emphasizes the individuality of the subject.
70 DEDUCTIVE LOGIC
Such a judgment is in one sense merely a special case of
the singular judgment ; in another, however, it is closely
allied to the universal judgment. For the subject, as an
individual capable of being designated by a proper name,
may be regarded also as the possible subject of an indefinite
variety of judgments. Every personality which is designated
by a proper name is exceedingly complex and admits of an
endless variety of manifestation in thought, word, and deed.
We have in every such personality a unity in the midst of
a diversity of varying states and activities. But this prop
erty of a unity in the midst of diversity expresses the
essential characteristic of the universal, which is capable of
realizing its essential unity in manifold ways. The uni
versal always represents a centre of identical reference so
that the many varieties of its special cases possess in com
mon one and the same underlying principle of being. So
also we trace all acts and moods of an individual to one
source. They vary indefinitely, but he is one. And so the
individual judgment lies midway between the singular and
the universal judgments. If our judgment concerning any
person has been suggested by a definite field of perception
of which the person in question forms the focal point of
attention, then the judgment is merely a special case of the
ordinary perceptive judgment. If however the judgment
concerning a person refers to his general character or
reputation, the judgment partakes more of the nature
of the universal. If I say, General Grant showed consum
mate skill at the siege of Vicksburg, the judgment is of the
nature of the usual singular judgment. If, however, I say,
General Grant was a man who possessed consummate powers
of generalship, then the judgment is more closely related
to the universal judgment, for it holds true of the person
referred to in every conceivable variety of circumstances.
In this connection there is suggested the traditional query
as to whether a proper name has any connotation. It of
course has a denotation in that it denotes or points out the
THE SINGULAR JUDGMENT 71
individual of whom it is the symbol. But does it connote
any complex group of coordinated properties which compose
its essential nature, as in the case of concepts, such as iron,
government, library, and the like. The answer to this ques
tion is evident if we regard the person designated by the
proper name as in a certain sense a universal ; that is, pos
sessing a unity of nature or of character in the midst of the
indefinite variety of phenomena through which the person
in question has become familiar to us. If the proper name
is to us anything more than a name, it must stand for this
central core of character, which is composed of a com
plex of properties. These properties together constitute
the connotation of the proper name. And if there were
not such an association of correlated properties with the
name itself, the name would be a symbol merely of the
actual object of perception, the man to whom we could point
with the finger or designate by a special time or space rela
tion, but the name alone could never call to mind a definite
kind of character, a personality to attract or repel, a memory
to enkindle ardor or inspire devotion. Names which live
and have power over the thoughts of men must themselves
stand for thought. Moreover, certain proper names have
received such definite connotation that they have come to be
used strictly as universals, and without their original deno
tation which was solely individual ; e.rj. A Daniel has come
to judgment. He is a veritable Shy lock, etc.
There are some judgments which seem to lack altogether
any element of universality. Instead of referring the single
instance to its appropriate concept, this kind of judgment
merely relates one single instance to another ; e.rj. This leaf
is similar to that one. New York is north of Philadelphia.
A is greater than B. Even in these illustrations, however,
there is present that which is an essential element in every
judgment, namely, a universal idea. Such a judgment fuses
the two single instances which it contains into an identity
which is constituted by the relation existing between them.
72 DEDUCTIVE LOGIC
This relation itself is the universal. In the illustrations
given, the affirmation of similarity, of relative position, or
of relative magnitude, forms as complete a universal as in
the usual instances of the reference of a single organism to
its appropriate genus.
The relation of the singular judgment to the universal
is a reciprocal one as regards their mutual dependence.
The special case becomes intelligible only as we are able
to view it in the light of our general knowledge, and our
general knowledge has a significance for us only so far as
we understand the special cases which constitute the ground
of our generalizations. As Agassiz once remarked, " A
generalization can never mean more to one than his own
particular experience will admit."
CHAPTER VIII
THE NEGATIVE JUDGMENT
So far in this discussion, judgments of assertion only, or
affirmative judgments, have been under consideration. We
come now to the examination of the negative judgment.
We have seen that the function of the copula in the affirma
tive judgment is to fuse into one the subject and predicate
terms of the universal judgment, and in the singular judg
ment to assert that the given object in the field of percep
tion or in memory is one with the concept to which it is
referred. The process in either case is essentially con
structive.
The negative judgment, on the other hand, holds apart the
subject and predicate terms. It denies the possibility of
explaining the one concept by the other, or of interpreting
the single case by the universal in question. The negative
judgment stands guard over our general body of knowledge,
excluding whatever is altogether false, and also whatever
may be false under certain conditions but may be true
under others. It is thus through the process of the nega
tive judgment that thought becomes discriminating. Our
first judgments upon any unfamiliar subject are most
naturally vague and indefinite. The truth which they
contain is mingled with much that is erroneous. It may be,
as is often the case, that a given object of perception is
recognized as belonging to a certain genus, but we do not
know to which one of several species under this genus it
should be assigned. But as our knowledge grows, the
various special cases become distinct through well-recognized
differences, which, when stated, constitute a series of negative
73
74 DEDUCTIVE LOGIC
judgments. This process of differentiation serves to render
knowledge more exact. This is essentially the method
which Socrates pursued with his pupils, asking of them the
meaning of some idea, such as virtue, or justice, and then
examining the conventional definition given in the light of
certain concrete instances of virtue or of justice which
differed radically from the definition. Accordingly the
definition had to be changed so as to adapt itself to these
negative cases. In this manner vague and general notions
upon which little thought had been bestowed were trans
formed into clear and precise ideas. The old dictum, Omnis
determinate est negatio, expresses the essential function of
the negative judgment as that of exact determination through
the process of negation. This process of negation sets a
limit beyond which a given concept cannot be applied. A
limit thus set serves as a boundary of exact determination.
It marks always a line of distinction between what is and
what is not as regards the essential nature of any concept.
The process of exact determination by means of negation
may be analyzed into its three component stages which form
the programme of all exact thinking :
1. The first rough draft of knowledge, which is neces
sarily vague and indefinite.
2. The critical limitation of this primary assertion by a
number of negative judgments, which show where it breaks
down, where it does not apply, and wherein the unessential
may be eliminated.
3. The reconstruction of the original statement modified
by the necessary restrictions, which the process of negative
criticism has disclosed as essential. The result is knowledge
in exact and definite form.
Thus the beginner in the study of chemistry has a vague
idea of chemical affinity, that certain elements enter into
a number of various combinations to form compounds. But
as his knowledge grows, he finds himself face to face with a
series of negative facts, which must be reckoned with,
THE NEGATIVE JUDGMENT 75
namely, that all elements indiscriminately do not combine
together ; that they which are capable of combining do not do
so in any proportions whatsoever ; that combinations which
are possible under certain temperature conditions are not in
others ; that elements which unite under ordinary circum
stances will not unite in the presence of certain other
elements. Consequently, when the idea of chemical affinity
comes to be restated in the thought of the advanced student
of the subject, it must be necessarily more definite and exact
by reason of these very negative instances which have
emerged in the course of his investigations.
Moreover, every negative judgment which possesses any
value as knowledge must rest upon some positive ground.
Mere denial of itself means nothing. For when pushed for
a reason of our denial, we must be prepared to give some
positive ground for the conviction that is in us. When we
say, It will not rain to-night, our judgment rests upon our
interpretation of the actual weather conditions. We venture
the negative statement because we are positive concerning
the significance of the present atmospheric conditions. And
also, if we should say of a certain friend, He did not do
the mean act of which he is accused, we rest such a denial
upon our knowledge of his character, abundantly tested and
proved by years of close companionship. If a person should
affirm that he does not expect to be conditioned in a certain
examination, and the only ground he could allege for his
belief were merely the indefinite feeling that he would
not fail, such an uncertain foundation would be absolutely
worthless. A definite negation must have the ground of
definite knowledge, or otherwise it has no force.
A distinction moreover is often drawn between signifi
cant and non-significant denial. Significant denial occurs
within the region which lies near the line of differentiation
between affirmation and negation. The non-significant de
nial occurs in the region remotely separated from this line
of differentiation. Thus, to say that a chrysanthemum is
76 DEDUCTIVE LOGIC
not an animal would be a non-significant denial. But to
say that one of the lower orders of animal such as that of
the sea-anemone is not a chrysanthemum would be a sig
nificant denial, because it resembles the chrysanthemum in
external appearance. There are so many marks in common
that one may fail to recognize at the first glance the
differentiating mark which separates the two cases.
Significant denial often carries with it also the implication
that under certain changed conditions the relation or refer
ence which is denied would become true. Thus the state
ment that water does not boil on the top of a mountain at
212, implies that it would boil however at some other tem
perature. If we say that the elements, oxygen and hydrogen,
will not unite in a one-to-one proportion, there is the impli
cation that they will unite in some other proportion. Again
the statement, that a certain man having made such a politi
cal blunder could not be nominated for governor, implies
that had it not been for the political blunder in question, he
might have been nominated for governor. A distinction
however should be drawn between limiting conditions and
conditions whose removal do not alter the force of the
original denial. Thus in the statement, Do not trust the
Greeks bearing gifts, the phrase "bearing gifts " is not a
limiting condition, the removal of which would alter the
statement at all. The meaning is, Do not trust the Greeks
even though they bear gifts ; that is, do not trust them at
all. Likewise the statement, There are no ghosts in mod
ern times, should not be interpreted as meaning that there
were ghosts in ancient times. The nearer incompatible con
cepts approach a limit beyond which denial passes over into
assertion the more significant does the denial become, and
the greater the possible difference of opinion which may
arise in reference to it. It is in the field immediately adja
cent to the limiting cases that dispute arises. When I say
that the American Beauty rose is not yellow, no one disputes
such an assertion ; and, moreover, there is no suggestion in
THE NEGATIVE JUDGMENT 77
this statement as to the real color of the American Beauty.
But if I say that a certain shade of red does not match a
given sample, the denial on my part may provoke a differ
ence of opinion ; and because the range of variation is so
narrow, the implication is that the true color must be very
near the one mentioned and within the region of the various
shades of red.
If denial asserts an incompatibility between concepts
which is absolute, that is, if there is no common point of
similarity at all between them, the judgment is called an
infinite negation. Such judgments being completely with
out significance are always nonsensical ; e.g. A stone has no
conscience. A triangle has no lungs. Between the limit on
the one hand of the infinite negation, and on the other of
the limiting case which separates denial from assertion,
there are all grades of denial possible according to the order
of their growing significance. Near the limit of assertion
denial becomes the subject of dispute and controversy.
Further removed the denial is unquestioned. Further still,
it becomes a truism, a commonplace of knowledge, soon
passing into the region of the grotesquely absurd and mean
ingless. To know just where assertion ends and where
denial begins is characteristic of the exact mind ; to know
just where denial ceases to be significant is characteristic of
the relevant mind.
CHAPTER IX
THE CATEGORICAL, HYPOTHETICAL, AND DISJUNCTIVE
JUDGMENTS
THERE are three forms which our judgments may take,
the categorical, hypothetical, and disjunctive.
The categorical judgment is assertion in its simplestjorm,
unconditioned, unanalyzed, and unexplained ; e.g. That man
is a half-breed ; whales are mammals. It expresses either
a fact, or else a generalization based upon a number of
facts.
The hypothetical judgment is an assertion subject to a
given limitation, or regarded under certain specified condi
tions. It does not refer to a concrete special case, but
rather to the abstract universal relations which form the
ground of all the possible special cases which may be con
ditioned by the relations ; e.g. If in an isosceles triangle a
line is drawn from the apex perpendicular to the base, it
will bisect it; if hydrogen, oxygen, and sulphur unite in
the proportions H-,S0 4 , they will form sulphuric acid.
Our knowledge, it must be remembered, forms a system
of interrelated parts. The hypothetical judgment is con
cerned essentially with the necessary connections which
obtain between these various elements. It asserts the
fundamental relations which exist between any ground and
its consequence. In our body of knowledge regarded as a
system, the hypothetical judgments constitute the basal
lines of construction ; by them part is related to part, and
part to the whole.
.The. disjunctive judgment is an indeterminate assertion
Concerning various possibilities which may exist in refer-
78
VARIETIES OF JUDGMENT FORMS 79
ence to a given subject, and which are of such a nature that
tlu-. establishment of the truth of any one necessarily excludes
the others ; e.g. The invading fleet may attack Newport, Cape
Cod, or Gloucester. One may travel from New York to
Philadelphia by the Reading, or the Pennsylvania rail
roads.
We have divided all judgments into two general types,
the singular judgment and the universal. Of these, the
singular judgment is naturally categorical, for it is an
assertion concerning a fact or a group of facts. If the
categorical is changed in form so as to make it a hypo
thetical, this is done by reason of a universal hypothetical
judgment of which the singular hypothetical judgment in
question is merely a special case, and therefore the hypo
thetical nature is due to the universal relation which is as
sumed as underlying it. Thus in the judgment, If this
substance is an acid, it will turn blue litmus paper red, we
see that the hypothetical relation expressed concerning the
special case is merely a single instance of a relation which
holds universally. It is only in this indirect manner that
a hypothetical judgment can apply to a special case. The
hypothetical is essentially a mode of expressing universal
relations. There are two cases in which the hypothetical
form of judgment is naturally used.
1. When we wish to express the necessary connection of
cause and effect between any given elements in a system
of related parts, e.g. If you double the pressure, you halve
the volume of gases.
2. When we wish to express a more exact differentiation
of our concepts by means of a reference to their specific
differences, e.g. If a triangle has two of its sides equal, it
is an isosceles triangle. The hypothetical form is used also
when the differentiating mark cannot be regarded as of the
essence of the concept in question, and even when it is abso
lutely arbitrary, provided only it serves to point out unmis
takably the concept in question. Thus the signal of Paul
80 DEDUCTIVE LOGIC
Revere was in this form, If the enemy come by land, there
will be a single light in the belfry ; if by sea, two lights.
The essential function of the hypothetical is to show this
relation of dependence of any one element upon another in a
system of interrelated and coordinated parts. The system it
self may be one of nature, or one arbitrarily assumed or agreed
upon by mutual consent, or of common convention. The
main thing is that the system should be of such a nature
as to render the connection which constitutes the hypotheti
cal relation absolutely uniform and necessary.
It is of course possible to change any categorical judg
ment of the universal form into a hypothetical. Thus,
All crows are black, may be put into the form, If there is a
crow, it is black. The hypothetical in this case is however
not the natural form of expression, and the reason is that
in such a judgment the necessary connection of ground and
consequent is not brought to the fore. There must be
in the very constitution of the crow a sufficient ground
for its customary color ; nevertheless its precise nature is
unknown and lies in the background of the simple asser
tion itself. It can be said therefore in general that when
a universal judgment presents an unanalyzed content, it
takes the categorical form ; when however the content is
analyzed so as to exhibit within it the connection of ground
and consequent, then it takes the hypothetical form.
Again, the disjunctive judgment naturally expresses a
universal relation. When it refers, as it often does, to a
special case, the disjunction is really based upon our knowl
edge of general conditions. When we say, for instance,
that a certain line must be equal to, greater than, or less
than some other given line, we do so because we know that
any line whatsoever must be equal to, greater than, or less
than any other given line. So also a physician may pronounce
a suspicious case of sore throat to be either scarlet fever or
diphtheria. His judgment in this case is grounded wholly
upon his knowledge of such cases in general. Therefore,
VARIETIES OF JUDGMENT FORMS 81
although the disjunctive judgment may in form deal with
a single instance, it always contains by implication a refer
ence to the universal conditions which are illustrated in the
special case.
The disjunctive judgment, moreover, contains both a
categorical and a hypothetical element. It is categorical
inasmuch as it asserts a definite area of possibility. It is
hypothetical inasmuch as the possibilities are related in
such a manner that if any one is true, the others are false,
and if any one is false, one of the others must be true.
Such a hypothetical implication renders the disjunctive
judgment significant; otherwise it would be without mean
ing. To illustrate this, let us examine the following dis
junctive judgment, A certain murder was committed by an
enemy or by a burglar. The categorical element in this
assertion limits the possibilities to the two alternatives
mentioned, and excludes suicide or any other possibility.
The hypothetical element lies in the implication that if
either one of the possibilities is proved, it negatives the
other.
Moreover, the categorical, disjunctive, and hypothetical
judgments may be regarded as various stages in the prog
ress of knowledge from that which is indefinite and inde
terminate to that which is definite and determinate.
The categorical judgment represents the primary stage
of vague assertion, wherein the conditions upon which the
asserted fact depends have not been fully analyzed.
The disjunctive is a statement of the various antecedents
which may have given rise to the given fact.
The hypothetical is the critical analysis of these various
antecedents, and the determination of that particular one
which bears an essential and necessary relation to the fact
in question.
All knowledge necessarily begins with a vague assertion.
The very fact that it is a beginning renders the assertion
vague. We hear, for instance, that a man has died suddenly
82 DEDUCTIVE LOGIC
under suspicious circumstances. Our first statement is
merely that a murder has been committed. A closer exam
ination of the surroundings will suggest various possibilities
by way of explanation. We settle finally upon the definite
conviction that the murder was committed by an enemy;
because we know that the dead man had an enemy who
had repeatedly threatened to take his life, and we have
therefore the general hypothetical principle to guide us,
that if a man has an enemy who has repeatedly threatened
to take his life, that man s murder may be presumably
traced to this as its explanation, provided there are no
other guiding indications. Or if the question should be
raised as to which one of several possible species is referred
to in any given instance, then we have a series of significant
hypothetical to assist us in the exact determination. We
may have the disjunctive statement that whales are either
sperm whales or right whales. This is more precisely de
termined in our body of general knowledge by means of
the two hypothetical : if the whale does not have in its
mouth baleen or whalebone, it is a sperm whale ; but if it
has baleen in its mouth, it is a right whale.
The process of the exact determination of a disjunctive
judgment may be effected through a series of negative judg
ments as well as positive. Instead of determining any olie
member of a disjunction positively, by discovering its differ
entia or necessary condition, we may reach a like result by
a process of elimination. If we have given several possible
explanations of a certain situation we may examine each in
turn and prove it to be impossible, and so narrow the range
by successive elimination until one only is left. Negation
becomes especially significant when there are but two
possibilities in reference to any given situation. The elimi
nation of either one leaves the other in full possession of
the field. Thus, if in the case of a murdered man it can be
proved negatively that he never had an enemy, and that
there was no one who would have sought his life through
VARIETIES OF JUDGMENT FORMS 83
hatred or because of an injury received, we are then forced
to the explanation that the man was murdered by a burglar
or some one other than an enemy. This process of elimina
tion by negation is trustworthy so far as we are sure that
the negative judgment is true, and that also we have com
pletely embraced all possibilities in our disjunction.
We have seen that every process of judgment consists in
establishing a unity of some kind among the elements of our
thought. Now this unifying bond in judgment admits of
a certain degree of variability, being more or less definite in
nature. Its degree of variability determines what is known
as the modality of judgments.
If this unifying bond is actual, the judgment is known as
an assertorical judgment. If the judgment expresses a pos
sible relation only, it is a problematical judgment. If the
judgment expresses a necessary relation, that is, where the
unifying bond expresses not merely that which is but that
which must be, the judgment is known as apodeictic.
The categorical judgment naturally takes the assertorical
form, e.g. x is y.
The disjunctive judgment naturally takes the problemati
cal form, e.g. x may be y, or z, or w.
The hypothetical judgment naturally take the apodeictic
form, e.g. If x is y, then z must be w.
There may, however, be a change of modality as regards
any one of the forms of judgment, categorical, disjunctive,
or hypothetical. Thus the categorical judgment will be
found in the various forms as follows : x is y, x may be ?/,
x must be y. The first of these is the natural way of express
ing the categorical ; for the form, x may be y, implies other
possibilities, and at least the negative possibility that x may
not be y. Therefore the problematical mode of the judgment
is to be regarded as implying a disjunctive. Moreover, the
categorical form, x must be y, implies a hypothetical judg
ment as its basis, for the assertion of necessity naturally
implies some knowledge of the fundamental relation of
84 DEDUCTIVE LOGIC
ground and consequent which underlies such necessity.
Thus each phase of modality has its own natural form of
expression; the assertorical expressing itself in the categori
cal judgment, the problematical in the disjunctive, and the
apodeictic in the hypothetical.
CHAPTER X
THE NATURE OF INFERENCE
THE nature of inference may be unfolded in two ways.
We may consider what it is in its outward aspect ; that
is, through its phenomenal manifestation in what it effects ;
or it may be more strictly denned in terms of its warrant
or ground. From the first point of view we examine infer
ence as regards its psychological significance ; that is,
what is inference considered as a psychical experience, its
nature, and characteristics ? But we must consider also the
second question, whether there is any necessity limiting
and determining the subjective experience, which presents
the character of a law having universal validity. What
goes on in the mind during the process of inference ? Also,
what is the rationale of such a process ? These questions
we will examine more closely, in order to show the nature
of inference under the two aspects, the one psychological
and the other logical.
It is a well-recognized fact in psychology that, in our
simplest as well as the more complex perceptions, the inter
pretation of the data of perception always goes beyond the
strict content of the data themselves. We see more than is
given in the field of vision immediately before us. The mind
supplies here and there the necessary parts that are lacking in
the actual elements of perception, and yet which are necessi
tated by the known nature of that which is actually given.
We form our judgment of distance indirectly, and not through
direct observation. So, also, our idea of a third dimension is
acquired by a process, marvellously complex, in which the
data both indicate and yet are transcended by the results.
Whether the nativist or empiricist holds the true position
86 DEDUCTIVE LOGIC
concerning original psychical experience, it still must be
conceded according to either theory that the development
of our perceptions corresponds to a law of growth based
upon accumulated inferences. Inference has been defined
as the indirect reference of a content to reality, and as such
we see the beginnings of inference in the most simple of our
perceptions. Every perception contains a direct reference
to reality, but also something which in a greater or less
degree is Deferred indirectly to reality. The fact that our
knowledge as given in the complete perception contains
more than is actually mediated through the avenues of the
senses is due to the apperceptive processes of consciousness.
Mind is active in perception, and not a mere passive recep
tacle. That which is given, the raw material of the senses,
is elaborated and extended, as it is combined with the
wealth of representative and conceptual material, which
the mind brings to every new perception. To this extent,
at least, the mind possesses a creative function. A certain
appearance of sky, combined with peculiar conditions of
wind and temperature, leads one to assert, with some de
gree of certitude, that it will rain before morning. The
prediction is an inference based upon and growing out of
the actual data of perception, and yet far outrunning them.
We recognize a friend from his step or voice. The mere
perception is only a sound. That it is associated with a
person, and not an animal, or a thing, is an inference ; that
it is the particular person whom we recognize as a friend
and can call by name, even before we turn around to con
firm the opinion by direct testimony of vision, this is a still
further inference. And even when we open our eyes in
simple vision itself, we fill up many a gap in our minds,
and give depth and distance, and interpret the contrasts
of light and shade, and the play of colors, through the
process of inference, although we may not be aware of the
process itself, which is automatically operative through
long-continued habit. When we thus regard inference as
THE NATURE OF INFERENCE 87
a psychological phenomenon, it may be readily explained
by the laws of comparison, association, recognition, generali
zation, etc. And, as such, inference has a subjective force,
at least, and leads to the habit of prediction and expecta
tion. The will, influenced by the resulting belief, leads
to activities consistent with such expectation.
Here, however, the question arises which is urged with
such force by Hume, Is there objective validity as well as
subjective necessity ? This leads to a consideration of
inference, from the second point of view, above mentioned.
We may be constrained to believe certain things concerning
the great world lying beyond the sphere of immediate con
sciousness ; but what warrant have we in so doing, or what
assurance that our conclusions are correct ? May we not be
deceived, after all, and by some psychological trick be led to
regard the phenomena of consciousness as quite otherwise
than that which obtains in reality ? We may have a strong
aversion to sitting down at a table where the number of per
sons will be thirteen. But has the subjective conviction,
that one of the thirteen will die in the course of the year,
any value when we come to refer it to reality, and ask our
selves the nature of the ground upon which the conviction
is based ?
On the other hand however it is quite a different kind of
necessity which constrains us to judge that if a person jumps
off of the roof of a house, he must surely fall to the ground
below. Some grossly superstitious and ignorant people may
believe the former with as obstinate a conviction as the
latter, so that a purely psychological criterion of the
strength of conviction is not at all adequate or satisfactory.
Is there any other criterion ? In what instances does this
subjective constraint proceed from the necessities of reality ?
or, in other words, in what cases are we able to discover a
logically grounded warrant which compels the inference, in
distinction from the mere psychological compulsion which
is occasioned by the psychical tendencies of association and
generalization ?
DEDUCTIVE LOGIC
This leads us to consider the logical, in distinction from
the psychological nature of inference. Inasmuch as the
characteristic feature of inference consists in this, that while
depending upon certain data of perception, it nevertheless
wholly transcends them, the question naturally suggests
itself, whether it is something within the data themselves,
cr without, by virtue of which the mind thus goes bevond
them in the process of inference. If it lies wholly without
the data, it must be something imposed upon them by the
mind, and as such can have only a psychological force and
value. For instance, the belief that if thirteen sit down
together at a table, one will die in the course of the year,
can have only a subjective value and significance. This is
true in all cases where the necessity of conviction finds its
origin in prejudice or in superstition, or it may be in the
force of authority. In all such instances we feel the lack
of a satisfactory logical ground. However, on the other
hand, if the data of consciousness contain within themselves
that which enables us to transcend them at the same time
that we interpret them, there is external validity for our
inference that has a logical worth. This seems at the first
glance to be a paradox. How can any content enable us to
state concerning it more than is contained within it ? The
answer to the seeming paradox is that every concept, and
every perception as well, have both an explicit and implicit
content. We never attain complete vision or perfect appre
hension.
There are, moreover, many points of view, each giving
additional knowledge concerning any phenomenon present
in consciousness. We see, therefore, only in part, and yet
that which is seen contains certain necessary implications
concerning that which is not seen. In the progress of
knowledge, subsequent observations, different points of
view, are ever confirming and amplifying our inferences,
enabling us to perceive immediately what formerly was only
inferred. The process by which the implicit is becoming
THE NATURE OF INFERENCE 89
explicit indicates a necessary relation existing between that
which is known mediately and that which is known imme
diately. Moreover, consciousness has been represented as a
stream, or an intricately interwoven web, something ex
tremely complex. Every part is related both proximately
and remotely. There is no such thing as an isolated per
ception ; every perception has its complex relations and
connections. So also every concept which is formed by
generalization through comparison and abstraction of our
perceptions as interpreted by us, possesses this character
istic of greater or less complexity. In this manner the
world of consciousness is constructed, that is, the world as
it is for us. This forms a complex whole made up of parts,
which in themselves may be regarded as wholes, and yet
which may be still further divided and subdivided.
Such an interrelated whole we may style a system, or, in
other words, a complex whole whose parts are congruently
arranged. The idea of system finds expression in the " Law
of Totality," that our knowledge is capable of arrange
ment in a self-consistent and harmonious system, and which
moreover in its content and form faithfully represents
objective reality. 1 We find, therefore, that in the focus of
consciousness at any one time, whether in the sphere of per
ception or in the region of representative or the conceptual
processes, whatever is given carries with it always certain
implications, and therefore certain necessary relations. This
is specially emphasized in Bosanquet s definition of system :
" System is a group of relations, or properties, or things, so
held together by a common nature that you can judge from
some of them what the others must be." Two facts re
garded as independent and considered separately may give
no information beyond their explicit contents ; but when
conjoined, they imply more than the sum of their parts.
1 Ueberweg, A System of Logic and History of Logical Doctrine, pp.
540 f.
2 Bosanquet, The Essentials of Logic, p. 140.
90 DEDUCTIVE LOGIC
How often two ideas in separate minds yield no result ; but
brought together, they give light. Isolation negatives
inference. To unfold whatever is given in all its manifold
implications is the process of inference. Its warrant lies in
the fundamental postulate of knowledge which we are con
strained to assume ; namely, that our consciousness must be
self-consistent throughout. Whatever is admitted as true
must find a congruent place in the system to which it is
possible to refer it. The necessity of fitting it in its proper
place gives rise to certain implications which necessitate
corresponding relations and attributes. And if it could not
be put into such a place, we would feel that we should have
to surrender the idea of self-consistency in the variously
related elements of our consciousness. The very integrity
of our mental life necessitates this conviction.
Therefore a part being given, we supply in our minds
other parts, or the whole to which the given part must nec
essarily belong. To achieve this, with logical warrant, our
knowledge of the part must be adequate to the extent that
we know that the element under consideration cannot be
complete in itself, but must be supplemented by its appro
priately related elements which with it go to make up the
complete system. We infer the nature of the flower not yet
in bud by the sprouting leaf. The one necessitates the
other by virtue of their common inherence in the same plant
system. We know that figs do not come from thorns nor
grapes from thistles. Columbus, noting the seaweed, and
birds, and the drift of the sea, inferred a shore beyond, to
which he was constrained by the necessities of thought to
refer them. It is said of Cuvier that he was able to re
construct part for part the entire frame and organism of an
animal whose fossil tooth alone formed the original datum.
He knew the system to which it must have belonged and to
which it alone could possibly be referred. An interesting
quotation from Cuvier himself illustrates most appropriately
this function of inference. He says, in his Oasemens Fosailea :
THE NATURE OF INFERENCE 91
" I doubt if any one would have divined, if untaught by
observation, that all ruminants have the foot cleft, and
that they alone have it. I doubt if any one would have
divined that there are frontal horns only in this class ; that
those among them which have sharp canines for the most
part lack horns. However, since these relations are con
stant, they must have some sufficient cause ; but since we
are ignorant of it, we must make good the defect of the
theory by means of observation : it enables us to establish
empirical laws which become almost as certain as rational
laws when they rest on sufficiently repeated observations ;
so that now whoso sees merely the print of a cleft foot may
conclude that the animal that left this impression ruminated,
and this conclusion is as certain as any other in physics or
morals. This footprint alone, then, yields to him who
observes it the form of the teeth, the form of the jaws, the
form of the vertebrae, the form of all the bones of the legs,
of the thighs, of the shoulders, and of the pelvis of the
animal which has passed by." l
In the common conduct of everyday life we infer beyond
the immediate present experience to future happenings and
in a similar manner. My train is half an hour late. I
know I must miss my connections at the station ahead ; for
the train I am hoping to catcli at that place is scheduled to
leave five minutes after the time of arrival of the train I am
now on. The time relations here necessitate my missing
my connections. This is rendered still more certain if they
are rival roads ; on no account will one wait for the other.
Moreover, the train I hope to make is made up and leaves
the station in question, and so I cannot fall back upon the
favoring chance that it also may be detained en route, and
so enable me, after all, to reach it in time. Thus, with
every additional knowledge of the system which forms the
ground of my inference, and the various conditions which
affect it, the validity of my inference is thereby increased.
1 Quoted by Jevons, Principles of Science, 2d ed., p. 683.
92 DEDUCTIVE LOGIC
Inference regarded as the analysis of a system of inter
related parts is illustrated in the following paragraph of
Professor James : " The result of reasoning may be hit upon
by accident. Cats have been known to open doors by pulling
latches, etc. But no cat, if the latch got out of order, could
open the door again, unless some new accident at random
fumbling taught her to associate some new total movement
with the total phenomenon of the closed door. A reasoning
man, however, would open the door by first analyzing the
hindrance. He would ascertain what particular feature of
the door is wrong. The lever, e.g., does not raise the latch
sufficiently from its slot case of insufficient elevation
raise door bodily on hinges ! Or door sticks at top by fric
tion against lintel press it bodily down ! I have a
student s lamp of which the flame vibrates most unpleas
antly unless the collar which bears the chimney be raised
about a sixteenth of an inch. I learned the remedy after
much torment, by accident, and now always keep the collar
up with a small wedge. But my procedure is a mere asso
ciation of two totals, diseased object and remedy. One
learned in pneumatics could have named the cause of the
disease and then inferred the remedy immediately." l
Inference, therefore, may be regarded as a deep penetrat
ing insight. The explicit is that which lies upon the surface,
which the mind immediately grasps, for it lies directly in
the focus of consciousness. Whereas the implicit is beneath
the surface, and is revealed only through a searching analy
sis. This difference may be exhibited through the distinc
tion between the actual and the potential. A child regards
gunpowder merely as a pile of coarse-grained sand. The
man sees what the child sees, but also the existing possibili
ties under certain conditions of explosive force. He appre
hends the potential as well as the actual ; and his inference
as to the possible results is based upon his superior insight.
It is therefore the well-furnished mind which sees things
1 James, Psychology, Vol. II, pp. 339, 340.
THE NATURE OF INFERENCE 93
as most widely related, and discerns the potential as well
as the actual manifestation, which will prove the most
fertile in accurate inference, in prophetic suggestion, and
in inventive resource.
The whole world of reality, as well as that of knowledge,
may be considered as one system, embracing within the
unity of its totality all the various systems with their com
plicated parts. From this point of view everything sustains
relations to everything else in the universe. The original
signification of the term universe is thus emphasized. This
thought, no doubt, Tennyson had in mind in the following
verse :
Flower in the crannied wall,
I pluck you out of the crannies,
I hold you here, root and all, in my "hand,
Little flower but if I could understand
What you are, root and all, and all in all,
I should know what God and man is.
We can, in this connection, best exhibit the precise nature
and function of the universal in inference. The possibility
of unfolding the properties or relations of anything in all
its implications depends upon our knowledge of the univer
sal concept to which the properties or relations in question
are naturally referred. While a singular proposition is the
statement of the mere occurrence of a phenomenon, the
universal always implies a knowledge of the conditions
and relations of the phenomenon. 1 Insight is only possible
where there is a wealth of universal concepts. We see an
animal which we observe to be cloven-footed. We infer
that it also chews its cud. We do not observe this. The
assertion does not arise directly from observed reality, but
indirectly through the generic concept that has grasped to
gether the two attributes, of chewing the cud and cloven
feet as always and necessarily coexisting in one and the
same animal. Inference, in this sense, may be regarded
iSee Green, Philosophical Works, Vol. II, pp. 284, 285.
94 DEDUCTIVE LOGIC
as the indirect reference of knowledge to reality, and this
is always mediated through the universal. The universal
has this characteristic feature, that it preserves an identity
in the midst of manifold differences. The same thought
may be expressed by saying that the universal manifests
a unity in the midst of diversity. However widely different,
in many respects, the animals may appear that chew the
cud, as the cow, deer, sheep, etc., there is always the
constant characteristic that they are cloven-footed.
Such a point of identity furnishes the constant factor
which determines the nature and the validity of the in
ference. Were it not for this conceptual power of the mind,
this ability to grasp phenomena in their universal essence,
and consider them as interrelated and connected, we could
never pass beyond individual and particular experiences
which would form a series of wholly disconnected events.
Knowledge could not then form a self-consistent system,
or inference possess any higher worth than a haphazard
guess. As Green says, " A mere fact/ a fact apart from
relations which are not sensible, would be no fact, would
have no nature, would not admit of anything being known
or said about it." l
Moreover, inference is not merely employed to extend
the field of consciousness in unfolding supplementary ele
ments lying beyond the sphere of direct cognition; the
elements may all be given immediately, and inference em
ployed to discover their connection and interrelations, and by
virtue of what bond they belong in one or the same system.
Inference here functions as explanation. A man is found
dead ; there are many wounds upon his person, and evidences
of a struggle in an out-of-the-way place upon a lonely road.
Such a combination of facts calls for an explanation which
shall be consistent with them. The facts must all be cor
related in a system whose related facts and the unity of
the whole will completely satisfy the mind. The mind
J Green, Philosophical Works, Vol. II, p. 301.
THE NATURE OF INFERENCE 95
is satisfied only when all hang together in what seems the
only possible self-consistent coordinated system. The facts
being given, they must be read backward to their origin.
The other aspect of inference is the reading of facts for
wards, or unfolding them in their necessary consequences.
Inference is the reply to the natural questions of the mind,
whence and whither ? And the process is essentially the
same, whether its peculiar mode consists in the evolution or
the involution of that which is given in consciousness.
Moreover, the mere psychological inference, the subjective
extension of the data of consciousness without any objec
tive ground or warrant, should ever be corrected, or even
at times wholly set aside by means of the truly logical
inference. Where the psychological experience, in tran
scending simple presentation, proceeds upon strictly logical
grounds, and has objective validity as well as subjective
necessity, we possess a warrant of the highest possible
worth.
The relation of the process of inference to that of judg
ment may be expressed in the following definition thatjn-
ference is a judgment plus tin- reason for it. Whenever the
reason for a judgment is obvious, the inferential element
falls into the background. The judgment then appears
merely as a restatement of a well-known truth which no one
would think of gainsaying, or as the result of referring a
familiar object of perception to its generally recognized
concept. But if the averred truth is challenged, or if the
reference of the perceived object is not clear, then in order
to make good the judgment, recourse must be had to some
phase of the inferential process. We have the accepted
judgment that lightning is a form of electrical discharge.
Such a statement commands assent without question. But
when Franklin proved the identity of these two phenomena,
it was by a process of inference in which it was necessary
to establish the common ground of these two phenomena.
If one should point to a bird circling above a field in
96 DEDUCTIVE LOGIC
majestic lines of flight, and say, "That is an eagle," the
observation would probably receive immediate assent. It
would pass then as an obvious judgment of perception.
If, however, the statement should meet with dissent, or an
opposed judgment should be urged that it is a crow, then
the inferential element revealing the necessary ground of
the judgment would at once come to the fore. It would
be possible to point out that the flight of the bird is so
characteristically the flight of an eagle that it could not be
mistaken or confused with that of a crow. It will be readily
seen that the inferential element is contained potentially
in every judgment. A direct assertion, received without
question, is the judgment in its simplest form. An indirect
statement, showing that it must be true because of its nec
essary connection with some other judgment, is an inferred
judgment. In the light of this distinction the difference
between judgment and inference may be defined as fol
lows :
The judgment is a direct reference of a concept to reality.
The inference is an indirect reference of a concept to
reality.
The differentiating line is evidently a variable one. Its
variability depends upon the presence or absence of any
occasion which demands a fuller explication of the ground
of a judgment. As long as the ground is obvious and the
judgment unchallenged, it is not necessary to offer any proof
of it. If however it is necessary for any reason to give an
explicit statement of the ground underlying a judgment,
then at once the inferential element passes from its potential
stage into its developed form as actually expressed. It is
often the opposition of a negative judgment which provokes
the inferential process underlying some positive assertion.
Inference may be deductive or inductive. It is deductive
when the process shows that from a universal prmciple~or~
law there must follow some special case, or some more
special phase of that principle or law. It is inductive when
THE NATURE OF INFERENCE 97
the process shows that a general principle or law must result
from the investigation of special cases.
When we reason that a man s conduct under certain given
circumstances will be honorable or dishonorable, as the case
may be, our inference is based upon our general knowledge
of the man s character, and the inferential process is one
of deduction. When however we reason that a man must
have a certain kind of character in the light of a number of
particular instances which we have observed, our inference
is based upon our interpretation of these special cases as re
vealing an underlying universal nature which we call the
man s character. Such a process is one of induction. 1
Part II, Chapter I, on " Deduction and Induction."
CHAPTER XI
THE LAWS OF THOUGHT
IN order that we may be able to justify our judgments
and relate them to each other and to the main body of our
knowledge, we must recognize certain fundamental and
universal principles known in logic as the laws of thought.
These laws are as follows :
1. The Law of Identity.
2. The Law of Contradiction.
3. The Law of Excluded Middle.
4. The Law of Sufficient Reason.
1. The law of identity requires every concept to repre
sent some phase of reality which remains essentially the
^same. This does not mean an identity which admits of no
variety ; for we have seen that it is of the very nature of
the concept to manifest many shades of difference within
the variety of special cases which illustrate it. It does
mean however that in spite of manifold differences, there
is a central core of essential identity which remains con
stant and unaffected by the various unessential changes.
This law has been formulated in the simple expression
A A. Such an expression is true but meaningless, and
were the law of identity restricted to such an expression of
it, there could be no progress in thought, for every judg
ment would be a mere tautology lacking any significance
whatever. The law would be more exactly formulated by
the expressions A A = A" = A ", etc. ; that is, every vari
ety of A is nevertheless A, or every special case of A is the
same as every other special case of A in spite of all differ-
THE LAWS OF THOUGHT 99
ences. This law therefore is merely the expression of the
unity which is the ground of all our judgments. Inasmuch
as inference has been defined as the reference of a judgment
to its proper ground, then this law, regarded as a law of
inference, demands that such ground must be something
abiding, no matter what variety of form it may assume. If
the ground to which we refer a judgment in the process of
inference is uncertain and shifting, then the inference itself
is invalidated. Every inference therefore requires as its
ground a relation which is constant, that is, identical with
itself.
This abiding ground which gives validity to our infer
ence may be either (1) a single thing or person whose self-
identity is obviously preserved, or (2) it may be a universal
whose very nature is such that it preserves a unity in spite
of the manifold differences in the various instances which
illustrate it. As an example of inference wherein the
identity is that of a single person there is the story of
Thackeray s of the old Abbe, who, one day conversing with
a party of intimate friends, chanced to say, " A priest has
strange experiences; why, my first penitent was a mur
derer." At this moment, the principal nobleman of the
neighborhood enters the room. " Ah, Abbe ! here you are ;
do you know, ladies, I was the Abbe s first penitent, and
I promise you my confession astonished him ! " The two
statements of the Abbe and the nobleman become signifi
cant solely because of their identity of reference to one and
the same individual. 1 Again in the case wherein the identi
cal ground is not an individual but is a universal, a state
ment might be made that a certain cloth will fade. When
asked for a reason, the reply might be, because that cloth
contains a dye which always does fade. It is evident that
the validity of such an inference depends upon the constant
nature of the peculiar kind of dye in question. The show
ing of a universal property of the dye, such as that of fading,
1 This illustration is taken from Bosauquet s Essentials of Logic, p. 140.
100 DEDUCTIVE LOGIC
forms in this case the justifying ground of the inference
that the cloth containing the dye will fade. A true uni
versal assures an identical ground, and therefore the pos
sibility of a constant reference as completely as does a
single individual.
2. The law of contradiction is that judgments which are
opposed to each other (as this is a, and this is riot a ; or a
is 6, a is not b) cannot both be true. The truth of either
one renders the other false. This is essentially the axiom
of consistency. It serves to buttress the law of identity.
The latter demands the preservation of a unity in spite of
differences. The law of contradiction draws a line of limi
tation as a boundary to these differences. Beyond such a
line, the differences contradict the underlying unity which
must be preserved in accordance with the law of identity.
It prevents the reference of incompatible properties to one
and the same subject at the same time and in the same sense.
The law of contradiction applies to judgments which are
opposed in a contrary as well as a contradictory manner.
The contradictory, it will be remembered, is the general
term for the total area of negation lying outside the denn
ing boundary of the positive term to which it is opposed.
The contrary is any special case of the contradictory which
may be designated by a part of the area of total negation.
The judgments a is &, a is not b, are contradictorily opposed.
The judgments a is b, a is c, are contrarily opposed when
ever c is any property incompatible with b. To such judg
ments the law of contradiction also applies ; if it is true that
a is b, then the statement that a is c must be false.
We have seen that a bare denial as in contradictory oppo
sition is not significant, and that significant denial rests
upon the knowledge of some property or relation which is
contrary to the alleged assertion which it opposes. Most
of our denials, therefore, are contrary rather than contradic
tory. Inconsistencies arise in thought more often by the
endeavor to unite properties slightly contrary than those
THE LAWS OF THOUGHT 101
wholly contradictory. Controversies which take the form,
It is, It isn t, and are conducted by continued reiteration
of bare assertion and denial, are always meaningless and
futile. If a statement is made that a certain ore is gold,
we may deny it merely by saying it is not. This is contra
dictory opposition. We may say also, It is iron pyrites,
i.e. a special case of that which is not gold. The denial
is significant and represents contrary opposition. The law
of contradiction applies equally to the two cases. If the
statement, It is gold, is true, then both of the following
statements are negatived: It is not gold; also it is iron
pyrites.
3. The law of excluded middle is, that between two
judgments contradictorily opposed there is no middle or
tliird judgment which is true. Om> or the other of the
Two given judgments must belrue. This law, however,
cToes~nbt apply to judgments which express contrary oppo
sition, for it is of the very nature of contraries that there
is middle ground between the extremes which they repre
sent. Both statements, x is greater than y, x is less than
2/, may be false, because of the middle possibility x=y.
However, contrary statements in the light of special circum
stances which render them an exhaustive disjunction come
under the law of excluded middle, e.g. He had either to
jump from the window, or perish in the flames. The cir
cumstances were such as to leave no other course open. A
contrary relation within a limited universe of discourse
thus ranks as a contradictory relation because the limita
tion of the area of relevant subject-matter cuts out a
middle ground which in an unlimited universe of thought
might otherwise appear. Much of the loose thinking,
especially in untrained and unreflecting minds, arises from
the careless assumption of contradictory alternatives, when
in reality they are merely contrary. The middle ground
is overlooked, and logical confusion inevitably results.
The law of excluded middle always secures an exhaustive
102 DEDUCTIVE LOGIC
disjunction, and therefore renders a negative statement
significant inasmuch as the other and opposed alternative
is then necessarily true.
4. The law of sufficient reason is that every judgment
must be based upon some satisfactory ground which fully
justifies it. This law was first formulated by Leibniz
(1646), and placed by him side by side with the law of
contradiction. It is so intimately associated with the
great philosopher that it would be worth while to have
his own statement of it. " Our intellectual inferences rest
on two great principles: the principle of contradiction, and
the principle of sufficient reason, in virtue of which we
know that no fact can be found real, no proposition true,
without a sufficient reason why it is in this way rather
than in another." This law is essentially the statement of
the fundamental logical basis upon which all inference
rests, namely, that our knowledge forms a system of inter
related and coordinated parts, and that any single element
can be determined only when its relation is known to some
other element or elements upon which it depends. It is
a law which recognizes a reciprocal dependence of part to
part throughout the entire body of knowledge. It is a
corollary of this law that every judgment contains a poten
tial inference ; for every judgment is true in so far as it is
based upon a sufficient ground, and to render explicit the
ground upon which it rests is itself the process of inference.
In these four laws we find that certain logical demands
are made to which all processes of thought must adhere.
The law of identity demands a basis of constant reference }
the law of contradiction, that of consistent treatment ; the
law of excluded middle, that of an exhaustive survey of
possibilities ; and the law of sufficient reason, that of ade
quate explanation. There are many rules which are given
for guidance in the various processes of inference, which,
however, are merely adaptations of some one or other of the
several phases of these four fundamental principles.
CHAPTER XII
IMMEDIATE INFERENCE
IN tne traditional logic the distinction is drawn between
immediate and mediate inference, the former being the di
rect reference of a judgment to its ground, the latter the
indirect reference of a judgment to its ground through the
medium of one or more intervening judgments. Such a dis
tinction, however, will not hold. All inference is indirect.
Indeed inference is defined as the indirect reference of a
concept to reality. The difference between the so-called
immediate and mediate inference is rather one of degree.
In the immediate inference from a given proposition in the
form, All x is y, to the derived proposition, Some x is y, the
process is not as direct as it seems. It assumes, tacitly at
least, another mediating judgment that whatever is true of
a class generically is true of every member of the class,
the old Aristotelian dictum. Such a judgment as this, how
ever, is so obvious that it falls into the background, and
the inference seems to be immediate. Immediate inference,
therefore, may be regarded as an abbreviated form of infer
ence in general. The term " immediate reference," however,
in the history of logic, is not applied to any inference what
ever which employs an obvious mediating judgment, but it
is restricted to certain definite aspects of inference dependent
upon general considerations of a self-evident character.
These considerations give rise to two well-defined types of
immediate inference according as the process is one of
implication or transformation.
1. The process of implication depends upon the funda
mental relations which exist between "all" and "some" and
103
104 DEDUCTIVE LOGIC
between "yes " and "no" ; that is, if we have a judgment, for
instance, in the form of a universal affirmation, all are, what
is implied in reference to the particular affirmation, some
are, or the universal negative, none are, or the particular
negative, some are not? The possible combinations which
we are able to make with the terms, " all," " some," "none,"
" some not," give us four distinct types of judgment which
for convenience of reference are designated by the four
vowels A y E, /, and as follows :
A = The Universal Affirmative ; All x is y.
\E = The Universal Negative ; No x is y.
I = The Particular Affirmative ; Some x is y.
= The Particular Negative ; Some x is not y.
Judgments which differ as universal and particular are
said to differ in quantity ; those which differ as affirmative
and negative are said to differ in quality. It will be seen
that the question of the various implications involved in the
relations which these several kinds of judgment sustain to
one another, is a general question which has to do with the
significance of the forms which all our judgments assume,
whatever may be their content; for any judgment concern
ing any object of knowledge must be put in one or another
of these four forms.
Now if a judgment in any one of these four forms is given
as true, certain necessary implications will follow in refer
ence to the other three. Likewise, if any judgment is.given
as false, certain necessary implications will follow.
In order to exhibit these relations in as clear a manner as
possible, Aristotle devised the scheme of placing the four
kinds of judgment each at a corner of a square, known as the
Aristotelian square, or the square of opposition. The latter
term is misleading, however, as all the relations are not
opposed, but only those obtaining between affirmation and
negation. A better term, which covers all the possible rela
tions, is implication. The judgments are arranged about
IMMEDIATE INFERENCE
105
the square so that the universals are above, the particulars
beneath, the affirmatives at the left, and the negatives at the
right. This arrangement will give us the following :
All X is y
A
THE SQUARE OF ARISTOTLE
Contrary
No x Isy
E
Some x is y
Subcontrary
O
Some x is not y
In the above, the word " some " is to be regarded as equiv
alent to " some at least." In the proposition, Some x is ?/,
there is no indication, as far as the bare form is concerned,
whether it may not also be true that All x is ?/, or, on the
other hand, that Some x is not y. " Some," used in this
sense, is the " some " of preliminary investigation, wherein a
connection has been established between x and y, but the in
vestigation is not fully complete. Upon further research, it
106 DEDUCTIVE LOGIC
may be that exceptions will be found which might render a
generalization impossible, or it may be that the connection
can be so firmly established as to admit of a generalization
as regards its logical force. " Some," in this sense, lies be
tween the terms " all " and " some only," and is equivalent
to " some at least."
Now, as regards the various relations which this diagram
illustrates, there are the following :
1. The subaltern relation between the universal (either
affirmative or negative) and its corresponding particular is
so called because the particular is regarded as being subor
dinated to the universal. The relation between universal
and particular is such that if the universal is true, the par
ticular is true also ; but if the particular is true, the truth
of the universal is left in doubt. The truth of a particular
judgment, as based upon the truth of the corresponding uni
versal, follows from our fundamental law of identity, that
the universal preserves its essential unity in all the particu
lar forms of its manifestation. The indeterminateness of the
universal, when the particular is given as true, is due to
the possibility that the connection expressed by the particu
lar judgment in question may be accidental, and therefore
not a part of the essential content of the species as a whole.
Moreover, if the universal is false, the particular is left in
doubt. It may be true or false, according to the concrete
circumstances in any given case. The reason for this is
that the bare denial of a universal is always ambiguous. It
may be a total denial by confronting it with the opposite
universal, or it may be a partial denial by pointing out
exceptions to it ; which, of course, render the affirmed uni
versality false. But the falsity of a particular renders its
corresponding universal false; for, if the particular state
ment is not true, much less will be the universal, which
embraces the particular under it.
2. The contrary relation between A and E propositions is
such that if either of the related judgments is true, the other
IMMEDIATE INFERENCE 107
must be false, but if either is false, the other is indeter
minate. For it is obvious that between " all " and " none "
there is middle ground, and therefore they are related as con
traries ; and it is the nature of the contrary relation that,
according to the law of contradiction, the truth of one ren
ders the other false ; and, as there is middle ground between
them, the law of excluded middle does not apply, and there
fore the fact that one is false merely leaves the other inde
terminate.
3. The subcontrary relation between /and is the inverse
of the contrary. Here the falsity of either renders the other
true, but the truth of either leaves the other indeterminate.
This is perhaps more difficult to see. It should be remem
bered that " some " = " some at least." Now, if it is false that
Some x is y, it must be true that Some x (at least) is not y,
which latter statement is not incompatible with the fuller
statement that No x is ?/, for it is merely a special case under
it. But if it is true that Some x (at least) is y, we have seen
that by the very significance of " some " thus interpreted,
the question as to whether there may be exceptions ex
pressed in the form Some x is not y is left in doubt.
4. The contradictory relations between A and and
between E and / are such that if either is true, the other is
false, and if either is false, the other is true. This follows
directly from the law of excluded middle. That the propo
sitions, All x is y, and Some x is not y, have no middle
ground between them is evident. It may be put in this
way : if a judgment is always true, it admits of no excep
tions, and if it has exceptions, it is not always true ; if a
judgment is not always true, it must have exceptions, and if
it does not have exceptions, it must be always true.
These relations may be summarized as follows :
1. Given A true, then /is true, the others false.
2. Given E true, then is true, the others false.
3. Given A false, then is true, the others unknown.
108 DEDUCTIVE LOGIC
4. Given E false, then / is true, the others unknown.
5. Given / true, then E is false, the others unknown.
6. Given true, then A is false, the others unknown.
7. Given / false, then A is false, the others true.
8. Given false, then E is false, the others true.
These eight statements may be still further condensed as
follows :
I. Given A or E true, I or false, the corresponding
subaltern is the same, the others opposite.
II. Given A or E false, / or true, the corresponding
contradictory is opposite, the others unknown.
There are two practical suggestions which emerge from
these dry symbols, which may prove not only interesting
but also of some value. (1) The one is that the trend of
logical thought is always from the universal to the particu
lar, from " all " to " some," and that procedure in the oppo
site direction is one of the most fertile sources of error in
thinking. It is the well-known fallacy of hasty generali
zation, namely, the collecting of a few instances of experi
ence and immediately raising them to the rank of a
universal. There is no procedure of thought which needs
to be so carefully safeguarded as that from " some " to
"all." (2) Again there is the principle which I would
call, the economy of refutation. It is this : Whenever in
discussion or debate a universal judgment is advanced, do
not attempt to controvert it by the opposite universal, but
rather by the opposite particular. There will be less diffi
culty in proving a particular, and thus a strategic point of
advantage will be gained. If a proposition is advanced in
the form All x is ?/, to refute it, it is only necessary to prove
essential exceptions in the form of Some x is not y. Thus
in the Harvard-Princeton debate in 1896, the question was,
Resolved that Congress should take measures to retire all
the legal tender notes. Princeton maintained the affirm a-
IMMEDIATE INFERENCE 109
tive. Harvard s attack upon this position was not, as might
have been expected, a universal negative, namely, that no
legal tender notes should be retired by Congress, but a
particular negative, that not all but only some should be
retired. It is a useful rule to remember in debate,
never attempt to prove more than is necessary to overthrow
your opponent s main contention.
CHAPTER XIII
ON TRANSFORMATIONS OF JUDGMENT FORMS
THE different forms of judgment may be subjected to
various changes, some of which give slightly new shades of
meaning, without however altering the logical force of the
judgment itself. The original judgment and its transfor
mation must be logically compatible. This is the criterion
by which all transformations are to be tested. These trans
formations may be produced in various ways: by an
interchange of subject and predicate; by a change in the
quantity or quality l of the judgment ; by the change of a
term to its contradictory; or by certain complex changes
involving all of these.
The interchange of subject and predicate is called the
Conversion of a proposition.
If it is a proposition of the A form, All x is y, its simple
conversion will give All y is x. This, however, alters the
logical force of the original proposition ; for, if we have
given the form All x is ?/, it may be that the predicate y is
the common mark of a number of species besides x, such as
x, z, w, etc. Therefore, y is not a distinctive mark of x at
all, and it does not follow that because All x is y, therefore
All y is x. In the conversion of an A proposition the uni
versal force is lost, and only a particular is possible. Thus
All x is y becomes Some y is x. This is called conversion by
limitation, or conversio per accidens.
With the universal negative, however, No x is y, a simple
conversion is possible, because the negative asserts a com
plete incompatibility of x and y, and such being the case, it
is a matter of indifference whether we say that x cannot be
1 See page 104.
110
TRANSFORMATIONS OF JUDGMENT FORMS 111
fused into any unity with ?/, or that y cannot be fused into
any unity with x. Thus No x is y becomes by conversion
No y is x.
With the particular affirmative form, Some x is y, a simple
conversion into Some y is x is also possible, because if
some x forms a unity with y, some y at least must be pres
ent with x to constitute that unity. Thus Some x is y
becomes, by conversion, Some y is x.
But with the particular negative, Some x is not y, the
simple conversion Some y is not x does not necessarily
follow; for the subject y may represent a species and the
predicate x its corresponding genus. Obviously, Some y
is not x will be false, for the species must fall wholly
within its corresponding genus. Thus if we have a judg
ment of this kind such as, Some reptiles are not snakes,
and convert it, we get Some snakes are not reptiles, which
is obviously false. Thus a particular negative cannot be
converted.
The possibilities of conversion may be summarized as
follows :
Converted
A All x is y / Some y is x
E No x is y E No y is x
I Some x is y / Some y is x
Some x is not y No result
Given
The above are the only transformations which are pos
sible when we regard the form of the propositions merely.
If, however, in addition to their mere formal structure, we
take into consideration their content, that is, the meaning
of the subject and predicate terms and their relation to each
other in any judgment, then a greater range in conversion
is possible.
Thus, in the universal affirmative, if the subject and
predicate are coextensive terms, or if they are coordinate
properties of the one and the same concept, then a simple
conversion without change is possible. Given, All equian-
112 DEDUCTIVE LOGIC
gular triangles are equilateral. By conversion we have All
equilateral triangles are equiangular.
Or if the universal proposition is in the form of a defini
tion, i.e. a concept referred to its genus and differentia,
then simple conversion is possible. Democracy is govern
ment by the people. A government by the people is a
democracy. It is evident that an indefinite reference of a
concept to a class genus merely, or a description of a concept
by one or more of its attributes, will give a proposition
which admits of conversion only by limitation, i.e. change
of " all " to " some " ; but, on the other hand, a definite refer
ence which serves to differentiate the concept in question
admits of simple conversion.
The same observation applies to the conversion of a
hypothetical judgment. Given a judgment of the form,
If x is y, z is iv, it does not follow that if z is w 9 x is y ; for
there may be other antecedents which will give us z is w,
as well as the given one x is y. Thus, given the judgment,
If the democrats win, they must carry New York State,
it does not follow that if they carry New York State, they
will win.
It is the aim of all exact thinking, of all scientific formu
lation especially, to render thought so definite that a simple
conversion is possible. It is not sufficient to refer a species
to a genus, which is a class embracing also many other
species, but to so refer the species in question by means of
its differentiating properties, that the reference will become
distinctive. Moreover, while a given consequent may follow
from many antecedents, it is the aim of exact thinking to
connect certain specific marks which accompany that con
sequent with certain causal conditions present in some one
of the many possible antecedents and not present in the
others. Simple conversion is then of course possible.
Logical error arises when judgments expressing inexact
references are converted simply by unreflecting persons.
As, for instance, when an ignorant foreigner reasons that be-
TRANSFORMATIONS OF JUDGMENT FORMS 113
cause all travellers who give unusually large tips are Ameri
cans, that therefore all Americans will give unusually large
tips. The error is more apt to arise when subject and predi
cate, or antecedent and consequent, approach very near the
boundary of simple conversion, but have not quite reached
it. The margin is so narrow however that it is overlooked,
and error naturally results. Thus, no one would think of
converting the proposition, All United States Senators
are members of Congress, into All members of Congress
are United States Senators, but many might fall into the
fallacy of converting the proposition, All the democrats
in the Senate voted against the bill, into All Senators who
voted against the bill were democrats.
The wider range of conversion which is rendered possible
by the consideration of content in addition to that of form
merely, may also be illustrated in the particular affirmative,
Given, Some x is y ; then, if it is known in addition that y
is a species of x, we may convert the particular into a
universal, and get All y is x as the result. Thus, if we
have given the judgment that Some birds of prey are vultures,
we can convert it so as to obtain All vultures are birds of
prey.
Again, in the particular negative, conversion, which is not
possible by consideration of form alone, becomes possible
if, on examination of content, we know that the predicate
is not a species of the subject. Thus, if we have given Some
birds of prey are not hawks, we can convert it into Some
hawks at least are not birds of prey. But if the predicate is
a species of the subject, conversion is impossible, e.g. Some
governments are not republics. The relation of form to
content is such in general that not merely is it impossible
to interpret the full significance of a proposition without
knowing its content, but also it is impossible to assent to
any formal statement whatsoever unless we know in addition
the significance of the terms used. The proposition, All x
is y, is a mere skeleton form, but in the actual judgments
114 DEDUCTIVE LOGIC
of our thinking x and y are replaced by definite concepts
with a real significance. Our first thought, therefore, is
whether the real concepts which we substitute for x and y
in our symbolic form will admit of a universal affirmative
assertion, or of a universal negative, etc. Form without
content is meaningless ; content without form is confusion.
The one is always a function of the other.
We come now to a second kind of transformation, known
as Obversion. It consists in a change in the quality of a
proposition from affirmative to negative, or from negative to
affirmative, and at the same time a compensating change of
the original predicate to its corresponding contradictory.
If the original proposition is true, a single change of
quality would render the transformed proposition false,
therefore the predicate term is changed by way of com-
pensation, because the reference of any predicate to a sub
ject has the same logical force as that of excluding the
contradictory of that predicate from the same subject.
Given, All such conditions are impossible, by obversion we
have No conditions of such a nature are possible. The
same process holds in the obversion of the other forms
of judgment, and we have the following tabulated sum
mary :
Obverted
All x is y A No x is not-?/ E
No x is y E All a; is not-v A
-Some x is y I Some x isjAfdt-y O
Some x is not y Some x is\iot-y I
Given
The term " uoi-y " is usually expressed by some form of a
negative affix such as impossible, w?icontrollable, etc.
There are several complex transformations formed by
the combined processes of conversion and obversion. Of
these the so-called ^Contraposijive is formed by subject
ing the given proposition to tnr ee transformations, as fol
lows :
TRANSFORMATIONS OF JUDGMENT FORMS 115
1. Obversion.
2. Conversion,
u. Ubversioii.
Given the proposition : All scholarly work is logical,
1. By obversion, No scholarly work is illogical.
2. By conversion, No illogical work is scholarly,
o. By obversion, All illogical work is unscholarly.
In the final proposition, which is the contrapositive, it
will be seen that the subject and predicate of the original
proposition have been interchanged and each replaced by
its corresponding contradictory. The contrapositive may be
defined, therefore, as a transformation which substitutes
for the given terms their corresponding contradictories,
and at the same time interchanges the subject and predi
cate positions. The three processes by which the contra-
positive is formed may be omitted, and the contrapositive
formed directly according to the above definition. The
processes, however, form the proof that this direct transfor
mation is admissible. 1
There is another proof for the contrapositive of a uni
versal affirmative which is as follows : Given, All x is y ;
then All not-?/ is not-x. For what is not-?/ must be either
x or not-x. But if it is x, it is also ?/, according to the
given proposition. This, however, is impossible, for the same
concept cannot be both not-?/ and y. Therefore, the other
alternative must be true, namely, that not-?/ must be not-z,
which was to be proved.
When an A proposition is given, its contrapositive is also
an A proposition. When, however, the given proposition is
of the E form, there is a loss of logical force, and the result
of the three processes is an proposition.
1 Some logicians regard the contrapositive as the result merely of the
two processes, obversion and conversion. This, however, is merely a mat
ter of definition, and no confusion can result, because the additional process
of obyersion simply carries the operation one step farther.
116 DEDUCTIVE LOGIC
Given, No insane persons are responsible, E.
(1) By obversion, All insane persons are irresponsible, A.
(2) By conversion, Some irresponsible persons are in
sane, .7".
(3) By obversion, Some irresponsible persons are not
sane, 0.
In a similar manner it will be readily seen that the
contrapositive of an O proposition is also an proposition.
The / proposition yields no contrapositive, because the first
step of obversion gives an proposition ; the second step of
conversion cannot be applied to an proposition, and con
sequently the process is blocked at this point.
It is well to remember that the contrapositive is formed
by taking contradictories of the original subject and predi
cate ; for, if contraries are taken, the process is rendered
invalid. For instance, if we have given the proposition,
All honest acts are moral, the contrapositive, according to
rule, would seem to be, All immoral acts are dishonest.
This, however, is not true, and the reason is that the terms
" honest" and "dishonest" are not contradictory but contrary,
for between honest and dishonest acts there is the middle
ground corresponding to acts concerning which the ques
tion of honesty is not raised at all.
CHAPTER XIV
A GENERALIZATION OF IMMEDIATE INFERENCES
As the various immediate inferences by opposition have
been generalized in the ancient logical square, the question
suggests itself, cannot a similar method be applied to the
other forms of immediate inference? And the following
is the result of the problem thus proposed.
The possible transformations of a simple proposition may
occur in any of the following ways : by a change of the
quality of a proposition, i.e. change from affirmative to
negative and vice versa; or, by a change of quantity, i.e.
from universal to particular and vice versa; or by a change
of either subject or predicate terms by substituting for them
their respective contradictory terms ; or, by an interchange
of subject and predicate in the proposition. Of these pro
cesses or combinations of them, the ones which are legiti
mate inferences are as follows :
Having given, for example, an A proposition, All x is y,
it is possible to infer :
(1) The converse, Some y is x.
(2) The obverse, No x is not-?/.
(3) Converted obverse, No not-?/ is x.
{4) Contrapositive, All not-?/ is not-z.
(5) Obverted converse, Some y is not not-a?.
(6) Inverse, Some not-a; is not y. 1
(7) Obverted inverse, Some not-x is not-?/.
1 The inverse of a proposition has the same predicate, but for its sub
ject the contradictory of the original subject.
117
118
DEDUCTIVE LOGIC
These transformations may be comprehended in the fol
lowing logical square :
x E not-y
or I
A or
not-x
Here I have placed the terms x 9 y, and their contradic
tions, not-#, not-?/, in the corners of the square so that any
term and its contradiction will be situated diagonally oppo
site. The letter A, E, /, or 0, indicates that the two terms
between which the letter is situated may be formed into a
proposition of the character represented by that letter, and
in every case such a proposition is a legitimate inference
from the original proposition, All x is y. Thus, between
the two upper terms, x and not-y, there are possible two
universal negative propositions, one the converse of the
other :
No x is not-y, E.
No not-?/ is x, E.
Between the two lower terms, two particular negative
propositions :
Some y is not not-z, 0.
Some not-ic is not y, 0.
Between either upper one as subject and corresponding
lower one as predicate there is possible a universal affirma
tive. This gives:
All x is y. A.
All not-?/ is not-cc, A.
Between either lower term and corresponding upper one
there is possible a particular affirmative. This gives :
A GENERALIZATION OF IMMEDIATE INFERENCES 110
Some y is x, I.
Some not-x is not-?/, /.
By comparison of these results with the legitimate infer
ences given at the beginning of this discussion, there will
be seen an exact correspondence. This square, therefore,
summarizes exhaustively all possible legitimate infer
ences.
I would note in passing that of the two inferences of the
form, while one is the converse of the other, still it is
not derived from the other by conversion, which process
is logically inadmissible, but is derived independently:
Some y is not not-a, being the obverted converse, and Some
not-a is not y being the inverse.
Again, when E is the original proposition, the possible
inferences are :
(1) No y is x.
(2) All x is not-!/.
(3) All y is not-x.
(4) Some not-?/ is x.
(5) Some not-?/ is not not-x.
(6) Some not-x is y.
(7) Some not-rc is not not-?/.
All of these are comprehended in the same square as that
indicating the inferences from an A proposition, provided
the positions of y and not-?/ are interchanged. This gives
the following square for inferences from an E proposition :
not-y
not-x
120
DEDUCTIVE LOGIC
This agrees with the fact that an A proposition, All x is
y, becomes by obversion an E proposition, No x is not-?/;
in this transformation it is observed that not-?/ has displaced
y. Such a substitution will affect all inferences from the
original proposition uniformly. With this one change,
therefore, the inferences exhibited by the A square and
the E square coincide throughout.
The / square is the same as the A square, with the
exceptions that the E and A inferences become O and /,
respectively, and that the propositions indicated by the two
horizontal lines of the square are to be formed by reading from
left to right only; also that no inference is possible between
not-x and not-?/, i.e. no contrapositive of an / proposition is
possible. The / square is as follows :
x not"y
y not-x
The possible inferences based upon an J proposition are
indicated in this square, and are as follows :
(1) Some y is x.
(2) Some x is not not-y.
(3) Some y is not not-#.
The square is the same as the / square, provided y and
not-?/ are interchanged as above in the case of the E and A
diagrams. The following is the square :
not-y not*
A GENERALIZATION OF IMMEDIATE INFERENCES 121
The possible inferences based upon an proposition are
indicated in this square, and are as follows :
(1) Some x is not-y.
(2) Some not-y is x.
(3) Some not-;?/ is not not-rc.
There is no relation between y and not-x as a possible form
of inference, inasmuch as the inverse of an proposition is
impossible.
CHAPTER XV
MEDIATE INFERENCE THE SYLLOGISM
TRUE inference always contains an element of mediation.
It is the process of grounding a judgment upon some other
judgment essentially related to it, and which stands as the
warrant of its truth. The reference of a judgment to an
other judgment as its ground implies a knowledge of a
third judgment which expresses a universal and necessary
connection between the two. The complete process of me
diate inference, therefore, consists in exhibiting a judgment
as the necessary result of the combination of two other
judgments. Thus, the judgment that a certain heap of
black sand is magnetic is justified when referred to its
ground, namely, that it attracts iron filings. To complete
the process, however, a third judgment is necessary, which
shall express the constant bond of connection between the
given judgment and its alleged ground, such as the judg
ment that whatever attracts iron is magnetic.
This form which mediate inference naturally takes is the
syllogism, which is a process of combining two judgments so
as to produce a third. The above judgments expressed in
syllogistic form would be :
Whatever attracts iron is a magnet.
This black sand attracts iron.
.*. This black sand is a magnet.
It will be observed that the two judgments which combine
to produce the third have a term in common. This is the
middle term of the syllogism. Moreover, the third judg
ment is formed by eliminating the middle term and taking
122
MEDIATE INFERENCE 123
as its subject and predicate respectively the remaining
term in each of the two given judgments. The subject of
the judgment thus formed is called the minor term of the
syllogism, and the predicate the major term. Minor and
major are applied to these terms because in any judgment
the predicate generally refers to a larger class than the
subject.
Of the two given judgments, the one containing the major
term is called the major premise ; and the one containing
the minor term, the minor premise. The premises take
their names from the major and minor terms, and not the
terms from the premises. In most syllogisms, the major
premise is placed before the minor; but this order is not
essential to the structure of the syllogism, or is it by any
means an invariable practice. The judgment which is
derived from the combination of the two premises is called
the conclusion.
It is the peculiar function of the major premise to ex
hibit some phase of our general knowledge; and of the
minor premise, to exhibit some more particular phase of
our general knowledge, or, as it more frequently occurs,
some special case embodied in a concrete experience. It
, of the two combined, that is, of
the syllogism itself, to apply universal knowledge to a spe
cial ease so as to yield its true interpret ;it ion. The proc.-ss
is one which consists essentially in eliminating the middle
or common term. It is the same process which we find in
algebra. Equations are merely a special case of judgment.
The following is in every respect a true syllogism :
x=y.
= z.
There is, however, a difference between the algebraical
equation and the ordinary logical proposition in this re
spect that in the equation it is a matter of indifference
124 DEDUCTIVE LOGIC
whether we say x = y or y = x ; but the proposition cannot
be converted in this manner without impairing its logical
significance.
Compare the following syllogisms :
(1) All x is y. (2) All y is x. (3) Some x is y.
All z is x. All z is x. All z is x.
.*. All 2 is y. . . All z is y. .*. All z is y.
It is obvious that the first of these syllogisms is valid, the
other two invalid. Moreover, it is evident that the position
of the terms in the syllogism, as well as the kind of proposi
tions employed in its structure, whether A 9 E, I, or 0,
have an essential bearing upon its validity. How this
comes to pass and what criteria may be formulated for
testing the validity of syllogisms will appear in the fol
lowing exposition concerning the so-called distribution of
terms.
A term is said to be distributed when it is used in a uni
versal sense, and undistributed when it is used in a limited
or partial sense. The word distributed is regarded as
synonymous with universal, because it is of the nature of a
universal to distribute or apply the full force of its signifi
cance to every individual case which is subsumed under it.
In the proposition, All the schoolmen were logicians, the
subject is distributed in the connection in which it is used,
so that what is affirmed of the class that they were logicians
can be affirmed of every individual of the class. The term
logicians in this connection is undistributed, because it is
only a part of the class of logicians that can be identified
with the schoolmen.
In respect to the four propositions, A, E, I, 0, the follow
ing are the possibilities as regards distributed and undistrib
uted terms.
1. The universal affirmative distributes the subject but
not the predicate. This will be evident, if the given propo-
MEDIATE INFERENCE 125
sition be converted, for while All x is y, by conversion
Some y is x.
.-. x is seen to be distributed, and y undistributed.
2. The universal negative distributes both subject and
predicate. It is a matter of indifference whether we say No
x is y, or by conversion No y is x. In the one case x is wholly
excluded from y } but that is the same as excluding y wholly
from x.
.-. x is distributed, and y is distributed.
3. The particular affirmative does not distribute either
term. For Some x is y gives by conversion Some y is x.
.-. x is undistributed and y is undistributed.
4. The particular negative does not distribute the sub
ject but does distribute the predicate. This cannot be
shown by converting the given proposition, for the par
ticular negative does not admit of simple conversion. How
ever, given the proposition Some x is not y, it is evident
that the subject, some x, is excluded wholly from ?/, there
fore such exclusion must cut off all of y from that special
some x } which is its subject.
.-. x is undistributed but y is distributed.
The above results may be tabulated as follows, the dis
tributed terms being marked with a \/ and the undistributed
with a .
\f o
All x is y A.
v . v/
No x is y E.
Some x is y I.
o . V/
Some x is not y 0.
In determining whether a term is distributed in any
given proposition, the distribution of the subjects will be
readily recognized because indicated by the qualifying terms,
126 DEDUCTIVE LOGIC
"all," "some," "none," or "some not." The distribution
of the predicates may be recalled by the following gen
eralization which is obvious upon inspection of the above
table.
Affirmative propositions do not distribute their predi
cates.
Negative propositions do distribute their predicates.
In reference to the criticism of any syllogism, there
are two fundamental rules of distribution which must be
observed :
1. The middle term must be distributed at least once.
2. If a term is distributed in the conclusion, it must also
be distributed in its premise.
The middle term must be distributed at least once in
order to provide a common point of connection between the
two premises. For if the middle term is undistributed in
both premises, then the major term is related to a part of
the middle term in the major premise, and the minor term
is related to a jtart of the middle term in the minor premise,
and there is no assurance whatever that these two parts
have anything in common.
Given the premises (1) All x is y,
(2) All z is y,
the following diagrams will represent these relations
respectively.
MEDIATE INFERENCE
127
There is nothing in the above relations,
however, to indicate whether within the
common circle ?/, x and z be wholly apart
as in the following diagram
or whether they have some common
ground as
or whether x falls within z as
or whether z falls within x as
The relation between x and z is left wholly indeterminate
by the given premises. If, however, the middle term is dis
tributed at least once, it serves to bring the two premises
into a logically significant relation freed from all ambiguity.
It is not necessary, however, that it should be distributed
twice ; for the object of its distribution is to connect the
two premises. This connection once effected, it is not neces-
128 DEDUCTIVE LOGIC
sary to secure it again ; if the middle term should happen
to be distributed in both premises, the existing connection
is merely confirmed and in no sense invalidated by such
twofold distribution.
The following syllogism will serve as a concrete illustra
tion of the fallacy of an undistributed middle :
All agnostics repudiate the methods of metaphysical in
quiry.
All materialists repudiate the methods of metaphysical
inquiry.
/. All agnostics are materialists.
This conclusion does not necessarily follow. The middle
term, being in the predicate of an affirmative proposition in
each case, is undistributed.
The second rule that a term distributed in the conclusion
must also be distributed in its premise, is directed against
that illogical procedure from a term used in a partial sense
to the same term used in the universal sense. In the dis
cussion concerning the opposition of propositions, it was
seen that the truth of the particular does not imply the
truth of the universal. It is the same principle which
emerges here. The truth of the universal carries with it,
however, the truth of the particular; therefore, it is per
missible to have a term distributed in the premise and
undistributed in the conclusion. The beginner in logic is
liable to confuse these two modes of procedure ; therefore it
should be especially held in mind that the invalid procedure
is only from a term undistributed in the premise to the same
term distributed in the conclusion, or from the particular
to the universal. As a concrete illustration, take the fol
lowing syllogism in which the distribution of terms is
marked :
/ o
All foreigners who are naturalized may vote.
v f V
No native-born citizens are foreigners who are naturalized.
v/ V
,\ No native-born citizens may vote.
MEDIATE INFERENCE 129
This conclusion is obviously incorrect ;jthe major term
is distributed in the conclusion and undistributed in the
premise. When such invalid procedure is concerned with
the major term, it is called the illicit process of the major
term, or simply illicit major; when it is concerned with the
minor term, it is the illicit process of the minor term, or
illicit minor.
There are several special cases in which the general rules
for distribution must be somewhat modified :
1. The predicate of some affirmative propositions is dis
tributed because of a special significance which it may
possess. While according to form alone it would be undis
tributed, the sense may afford additional information which
justifies its distribution. This is the same principle which
was seen to operate in reference to the conversion of a
universal affirmative proposition, All x is y to All y is x
when x and y are coextensive terms.
Thus the following syllogism is invalid because of an
undistributed middle :
All x is y.
All z is y.
.-. All z is x.
Here the form alone serves as the test of its validity.
But in the filling up of such a form with significant terms,
the meaning may possibly render such a syllogism valid.
Thus,
Every government by the people is a democracy.
The United States is a democracy.
.-.The United States is a government by the people.
The middle term in this syllogism is undistributed as
regards its bare form. As regards the meaning of the terms
the major premise may be converted simply, every democ
racy is a government by the people. The term democracy is
in reality therefore distributed, the subject and predicate
terms of the major premise being coextensive.
130 DEDUCTIVE LOGIC
2. There are certain qualifying words which, while restrict
ing the subject at the same time, distribute the predicate.
In all propositions of this kind the subject is undistributed,
and the predicate is distributed. The qualifying words are
" only," " none but," " alone," and the like. In the proposi
tion, None but members of the union will be employed, the
subject is undistributed, and the predicate distributed ; the
logical force of this proposition will be the more readily
seen if we convert it. It then becomes, All who are em
ployed must be members of the union; in this form, the
subject is distributed, the predicate undistributed, as it is a
universal affirmative.
In the two syllogisms following, the first is valid, the
second is invalid, being a case of undistributed middle :
(1) None but members of the union will be employed.
A certain man was employed.
.. He must have been a member of the union.
(2) None but members of the union will be employed.
A certain man is a member of the union.
.. He must be employed.
In the criticism of the various modes of reasoning atten
tion should be drawn to the fact that we seldom find our
thought expressed in the form of a complete syllogism.
Usually one of the parts of the syllogism is omitted, not,
however, because its force is unessential to the reasoning
process, but because it is so obvious that it is unnecessary to
state it explicitly. This condensed form of the syllogism is
known as the Enthymeme, so called, as its name indicates,
because a part of the syllogism is not expressed but in the
reasoning process is carried along in the mind. The omitted
portion is usually the major premise ; that is, the general
principle of which the course of the reasoning in question
forms the special case. Both the minor premise, or the
conclusion, may also be omitted in the construction of an
enthymeme. There are three kinds of enthymeme:
MEDIATE INFERENCE 131
1. With the major premise omitted.
This enterprise will tend to increase the public wealth,
because it will promote the general happiness of the people.
2. With minor premise omitted.
That expedition is doomed to failure, because no small
body of men insufficiently equipped and cut off from their
base of supplies can ever reduce so strongly fortified a
garrison.
3. With conclusion omitted.
All members of that conference were traitors to their
party. And you were a member of that conference. Noth
ing more need be said.
The enthymeme may be tested as regards its validity by
supplying the omitted part, and then applying the usual
rules of the syllogism. But, inasmuch as the enthymeme
expresses the immediate connection between two judgments,
it may be subjected to direct criticism according to the
following criteria :
If the major premise is omitted, the enthymeme consists
of a special case referred to its ground, This is x because
it is y. The enthymeme is valid, provided the ground as
signed for the special case applies as well to all other cases
of the same kind ; that is, according to the symbols used,
if All y is x.
In the enthymeme, He is a free-trader because he is a
democrat, the connection is a valid one provided all demo-
crats are free-traders.
Again, if the enthymeme has the minor premise omitted,
it may be expressed in symbols, as follows:
A certain thing is x, because All z is x. In such a relation,
the special case must be recognized as a special case of the
universal ; that is, we must know that the thing in question
is z.
For instance, given the enthymeme as follows : That
man is a German, for all the crew are Germans. The
inference based upon the assigned ground is valid, provided
132 DEDUCTIVE LOGIC
we know that the man in question is a member of the crew;
that is, if the single case falls with the area of the universal
which is stated as its ground.
Syllogisms may be combined in various ways into chains
of reasoning. When the conclusion of one syllogism becomes
the premise of a second syllogism, the former is called the
prosyllogism and the latter the episyllogism. When we
combine a number of prosyllogisms and episyllogisms so
that all the conclusions except the last are omitted, the
chain of reasoning is called the Sorites. There are two
forms of the Sorites, known as the Aristotelian and the
Goclenian. 1
These forms may be expressed symbolically as follows :
I II
Aristotelian Sorites Goclenian Sorites
A is B. D is E.
-Bis C. CisD.
CisD. B is C.
D is E. A is B.
.-. A is E. .-. A is E.
It will be seen that the middle terms cancel through
out, and the conclusion is formed from the remaining terms
in the first and last premises. Thus, it may be reasoned
that a certain political boss has caused his chosen man to be
made governor of New York ; for he controls the machine,
and the machine controls the party, and the party controls
the state vote, and the state vote creates the governor. The
Sorites is commonly used to indicate the various links of
cause and effect which may be interpolated between an
effect and a remote cause.
The Sorites often appears in hypothetical form, for the
reason that the causal relation is best expressed by a
hypothetical. In the life of Sir James Fitzjames Stephen,
1 Named from Goclenius, a German logician of the sixteenth century.
MEDIATE INFERENCE 133
the following remark of his tutor appears, which illustrates
the hypothetical form of the Sorites, and at the same time
will serve to show how plausibly a Sorites may express a
subtle fallacy : " If you do not take more pains, how can you
ever expect to write good longs and shorts ? If you do not
write good longs and shorts, how can you ever be a man
of taste ? If you are not a man of taste, how can you ever
hope to be of use in the world?"
CHAPTER XVI
MOOD AND FIGURE
A SYLLOGISM may be constructed by combining in various
ways the four propositions, A, E, 7, and 0. The particular
combination employed in any one syllogism constitutes the
mood of that syllogism. Thus, to refer to a syllogism as hav
ing the mood AAA, means that the premises and conclusion
are all universal affirmative propositions; the mood EAE
means that the major premise is a universal negative, the
minor premise a universal affirmative, and the conclusion a
universal negative. The three letters designating the mood
are to be interpreted in the order of major premise, minor
premise, and conclusion.
The problem which the subject of mood presents is to
find which moods are valid ; for there are sixty-four possi
ble permutations of three propositions out of four, repeti
tions such as AAA being allowed. In order to discriminate
between the valid and invalid moods, the following rules
must be taken to guide us :
1. A particular premise gives a particular conclusion.
2. Two particular premises give no conclusion.
3. A negative premise gives a negative conclusion, and
conversely if the conclusion is negative, one of the premises
must be negative.
4. Two negative premises give no conclusion.
The first and second rules follow from the rules relating
to distribution of terms ; this is obvious upon simple inspec
tion. The third rule as to a negative premise giving a nega
tive conclusion and its converse is based upon the necessary
134
MOOD AND FIGURE 135
relation that if one of the two terms major or minor agrees
with the middle term and the other disagrees, then they
must necessarily disagree with each other ; that is, the con
clusion expressing this disagreement must be in the negative
form. As to the rule that two negatives give no conclusion,
it is evident that when the major and minor terms both are
excluded from all relation to the middle term, no indication
whatever is given as to their relation to each other. Ac
cepting these rules therefore as binding, let us examine their
effect upon the sixty -four possible permutations. This prob
lem we will divide into two parts :
(1) What pairs of premises are valid?
(2) What valid conclusions follow from them ?
First, the major premise may be either A, E, 7, or 0, and
the minor premise may be either A, E, I, or 0. The per
mutations resulting from combining these letters to form
possible pairs of premises are as follows :
AA, AE, AI, AO.
EA, EE, El, EG.
IA, IE, II, 10.
OA, OE, OI, 00.
Of these the following cannot stand as pairs of prem
ises :
EE, because there are two negatives.
EO, because there are two negatives.
77, because there are two particulars.
10, because there are two particulars.
01, because there are two particulars.
OE, because there are two negatives.
00, because there are two negatives, and also two par
ticulars.
Eliminating these tentative forms, there remain the fol
lowing :
136 DEDUCTIVE LOGIC
AA, AE, AI, AO.
EA, EL
IA, IE.
OA.
The second question is, given the above premises, what
conclusions are possible ?
AA will give as a conclusion either A or /; but will not
give E or 0, for a negative conclusion requires one of the
premises to be negative. By inspection, after the same
manner, it will be found that AE will give two conclusions,
E and ; so also EA. The remaining, with the exception
of IE, have each one conclusion, All, AGO, EIO, IAI,
OAO.
The premises IE would seem to require the conclusion
and so form a valid mood IEO. This mood, in fact, squares
with all the special rules which we have formed above to
guide us in discussing this present problem. However, it is
impossible to construct a syllogism in this form which does
not contain an illicit major, for the conclusion, being nega
tive, distributes the major term, and the major premise, being
/, cannot distribute either subject or predicate term. For
example, take the following syllogism :
Some x is y I.
No 2 is x E.
Some z is not y 0.
Y is here distributed in the conclusion, but not in the
premise. The syllogism, therefore, in this form is impossible.
The valid moods which remain after this process of elimi
nation which we have now completed are as follows :
AAA AEE EAE All EIO OAO
(AAI) (AEO) (EAO) AGO IAI
The three in parentheses are called the weak moods of
the syllogism, because the conclusion in each case is really
MOOD AND FIGURE 137
implied in the stronger conclusion immediately above it, and
therefore they do not constitute distinct types. The truth
of A always necessitates the truth of /, and the truth of E
always necessitates the truth of O.
There remain all together only eight distinct types out
of the sixty-four which are valid forms of the syllogism.
There is still a further problem which remains to be
considered, whether all of these moods are valid irrespec
tive of the relative positions of the major, minor, and mid
dle terms in the syllogism. The position of the middle
term in reference to the major and minor term constitutes
what is known as the figure of the syllogism. If we
represent the middle term by 37, the minor term by S, and
the major term by P, the four possible figures are as
follows :
I II III IV
M P. P. M. M. P. P. M.
S. M. S. M. M. S. M. S.
.-. S. P. .-. S. P. .-. S. P. .-. S. P.
A change in the relative position of the terms will of
course affect the matter of their distribution, and therefore
the validity of the various moods in the different figures
will turn upon the question of the distribution of terms.
The two rules for distribution, it will be remembered, are
as follows :
(1) The middle term must be distributed at least once.
(2) If a term is distributed in the conclusion, it must be
distributed also in the premise.
The following are the valid moods in the several figures,
the invalid moods being stricken out, and the number
appended being the number of the rule violated in each
case :
138 DEDUCTIVE LOGIC
Figure I Figure II Figure III Figure IV
AAA ~te&? ^AA 2 (AAI) -AAtf(AAI)
Agfi 2 AEE -^AE? AEE
EAE EAE ^AE- (EAO) ~&AS 2 (EAO)
AH -Att^ AH
-A3& AGO -A0&
EIO EIO EIO EIO
J^f 1 J^t 1 IAI IAI
GAG J&AO* GAG
The first figure is called by Aristotle the perfect figure,
for it alone, he averred, conforms to the fundamental canon
of all reasoning. This canon of Aristotle is called the
Dictum de omni et nullo. It has come down to us from the
mediaeval logicians and is formulated as follows:
Whatever is predicated affirmatively or negatively of a
whole class must be predicated affirmatively or negatively
of everything contained under that class. The affirmative
predication is expressed by the phrase de omni, and the
negative by de nutto. 1
Thus the perfect syllogism is a process of applying
our general knowledge (the major premise) to a special
case (minor premise), the conclusion being the special case
interpreted in the light of our general knowledge.
It will be readily seen, also, upon inspection, that the first
figure is the only one of the four which proves any one of
the four propositions, A } E, I, or 0, as its conclusion.
The second figure proves only negative conclusions. It
is used in proving distinctions between things.
The third figure proves only particular conclusions. The
moods with an I conclusion are useful in proving a rule
by positive instances ; the moods with an O conclusion in
proving exceptions to a rule. It will be noticed that in
the third figure the strong moods AAA and EAE are
1 Aristotle stated it, Whatever is said of the predicate is said of the
subject.
MOOD AND FIGURE 139
invalid, but the weakened niood AAI and EAO are
valid.
The fourth figure was regarded by Aristotle as merely
an awkward variety of the figure, and therefore he ignored
it altogether. His pupils, Theophrastus and Eudemus,
however, added its five moods to Figure I, calling them
indirect moods. The fourth figure is called the Galenian
figure from Claudus Galenus (died about 200 A.D.), who
insisted upon ranking it upon the same footing as the other
three figures. In the fourth figure, also, the weakened
moods take the place of their corresponding stronger
moods, the latter being invalid.
The Latin schoolmen in the thirteenth century invented
a system of mnemonic verses for the purpose of assisting
the memory as regards the valid moods in each figure.
While such a mechanical device is not needed by the student
of logic, it is given a place in the text as a curious bit of
logical history. It furnishes also an excellent illustration
of the scholastic type of mind. The lines are :
Barbara, Celarent, Darn, Ferioque prioris ;
Cesare, Camestres, Festino, Baroko, secundae,
Tertia, Darapti, Disamis, Datisi, Felapton,
Bokardo, Ferison, habet ; quarta insuper addit
Bramantip, Camenes, Dimaris, Fesapo, Fresison.
The words printed in italics are artificial words having no
significance whatsoever. Each word represents a mood, its
three vowels indicating the propositions which it contains.
The words " prioris," " secundae," etc., refer, of course, to the
figure in each case. Thus Barbara signifies AAA of the first
figure; Disamis, IAI of the third figure. Some of the
consonants in these words are also significant, indicating
the method by which the moods in any of the three figures
may be reduced to the form of the first figure. Aristotle
insisted that a mood in any other figure could be tested
as regards its validity only after it had been changed so
as to conform to the "perfect figure." This process b
140 DEDUCTIVE LOGIC
called reduction. The significance of the consonants in
reference to this process is as follows :
In the several words, s indicates that the proposition rep
resented by the preceding vowel is to be converted simply ;
p indicates that the proposition represented by the preced
ing vowel is to be converted per accidens, or by limitation,
that is, changing all to some; m (mutare) indicates that
the propositions which stand as the premises are to be
transposed; k means that an indirect proof is necessary
in order to reduce the mood to the first figure. Moreover,
the initial consonants of the so-called imperfect figures cor
respond with those of the moods in the first figure to which
they can be reduced.
Thus Darapti reduces to Darii :
The mood expressed by Darapti is AAI as in the
following :
All B is A.
All B is C.
.. Some C is A.
The p in Darapti indicates conversion of minor premise
per acddens ; this gives the mood All which is the Darii of
the first figure :
All B is A.
Some (7 is B.
.*. Some C is A.
So also Disamis becomes Darii :
Given the syllogism in the form of Disamis:
Some B is A.
All B is <7.
.-. Some C is A.
Here the first s indicates a simple conversion of the major
premises, the ra a transposition of premises, and the final
MOOD AND FIGURE 141
s a simple conversion of the conclusion, all of which will
result as follows :
All B is C.
Some A is B.
.. Some A is (7.
or Some C is A.
The process of reduction has no practical value whatso
ever ; as a device to arrange the syllogism in proper form for
the testing of its validity, it is wholly unnecessary. Every
syllogism, whether of the first or of the other figures, may
be tested quietly by the application of the rules concerning
the distribution of terms. If the middle term is distributed,
and no illicit process either of the major or the minor term
is involved, the syllogism needs no further justification.
CHAPTER XVII
THE HYPOTHETICAL AND DISJUNCTIVE SYLLOGISMS
THE hypothetical syllogism is a syllogism in which the
major premise is a hypothetical proposition, the minor
premise a categorical, and the conclusion a categorical propo
sition also. The hypothetical proposition is of the general
form, If x is y, then z is iv. The conditional clause is
known as the antecedent, the following clause the conse
quent. ,
Let us examine some hypothetical proposition regarding
it as a major premise, and putting the question as to how
many syllogisms may be constructed by means of introduc
ing various minor premises in connection with it. Let us
take the proposition, If the Japanese are to be victorious
in the war with Kussia, they must take Port Arthur. With
this proposition as a major premise, there are four minor
premises possible according as we affirm or deny the ante
cedent, or affirm or deny the consequent, as follows :
(1) They are victorious.
(2) They are not victorious.
(3) They have taken Port Arthur.
(4) They have not taken Port Arthur.
It will be observed that the first and fourth statements
when taken in connection with the major premise give
definite conclusions.
When we affirm the antecedent, They are victorious, the
conclusion follows necessarily, They must have taken Port
Arthur.
142
THE HYPOTHETICAL SYLLOGISM 143
Similiarly, when we deny the antecedent, They have not
taken Port Arthur, the conclusion follows, They are not
victorious.
Granting the truth of the major premise, these two con
clusions must necessarily follow from the respective minor
premises as above stated.
But when we come to the other two cases, the denial of
the antecedent, or the affirmation of the consequent, the case
is very different. If it is stated that they are not victorious,
it does not follow that they did not take Port Arthur, for
they might take Port Arthur and yet fail of victory for
some other reason. And so also, if it is stated that they
have taken Port Arthur, it cannot be inferred that they
are victorious, for here again some other cause may have
operated to prevent victory.
In general therefore the denial of the antecedent or the
affirmation of the consequent leaves the conclusion indeter
minate ; for, as in the special case cited above, there may be
some other antecedent which may give rise to the conse
quent as well as the particular antecedent connected with
it in the given hypothetical proposition which forms the
major premise. This possibility will always render the infer
ence indeterminate. If however it is known that the
given antecedent is the sole antecedent of the given conse
quent, and therefore every other possibility is eliminated,
then the denial of the antecedent, or the affirmation of the
consequent, will also give a determinate conclusion. This
special case of the hypothetical syllogism may be recognized
by the simple test of conversion. Thus if the hypothetical
major premise can be converted simply, then any one of the
four possible minor premises will yield a definite conclusion.
Thus we have the proposition, If any substance turns blue
litmus paper red, it is an acid. Here antecedent and conse
quent are reciprocally related, so that we can also state the
proposition conversely, If the substance is an acid, it will
turn blue litmus paper red.
144 DEDUCTIVE LOGIC
With such a major premise, any one of four conclusions
may be possible according as the antecedent is affirmed or
denied, or as the consequent is affirmed or denied.
It is possible, moreover, to transform any hypothetical
proposition into a categorical form. Let us take the hypo
thetical, If the patient takes this medicine, he will get well.
The two minor premises which give indeterminate conclu
sions are as follows :
(1) He does not take the medicine.
.-. Conclusion is left in doubt.
(2) He gets well.
.-. Conclusion is left in doubt.
Forming these into categorical syllogisms, we have :
(1) The taking of this medicine will restore health.
The patient does not take the medicine.
.-. He will not be restored.
(2) The taking of this medicine will restore health.
The patient s health is restored.
.-. He has taken the medicine.
By examining these two conclusions, obviously invalid, it
will be seen that the denial of the antecedent in a hypotheti
cal syllogism is equivalent to the illicit process of the major
term in the categorical syllogism, and the affirmation of the
consequent is equivalent to the undistributed middle in the
same. The inferences which are always possible in the hypo
thetical syllogism, the affirmation of the antecedent, or the
denial of the consequent, are designated by the Latin phrases,
modus ponens and modus tollens respectively.
The Disjunctive Syllogism. In this syllogism we have as
major premise a disjunctive proposition of the form, A is
either B or C. There are four possible minor premises,
being the affirmation or the denial of either one of the alter
natives. The conclusions which are possible depend upon
THE DISJUNCTIVE SYLLOGISM 145
the nature of the disjunctive major premise. There are
the following cases :
(1) If the disjunction is a strictly logical one, that is,
the terms mutually exclusive and the disjunction complete, 1
~. then the affirmation of either alternative necessitates as a
conclusion the denial of the other, while the denial of either
one necessitates the affirmation of the other. The former is
called the modus ponendo fallens, i.e. the mood which denies
by affirming ; the latter is called modus tollendo ponens, i.e.
the mood which affirms by denying.
(2) If the disjunctive members are not mutually exclu
sive, the affirmation of the one does not necessarily deny the
other. Thus we might have the disjunctive proposition,
The disease is either pneumonia or typhoid fever. The
assertion that it is pneumonia does not necessarily render
the typhoid fever an impossibility ; for a patient may have
both diseases at the same time.
(3) If the disjunction is not complete, then the denial of
one member of the disjunction does not necessitate the
affirmation of the other, for one or more possibilities not
expressed in the original disjunctive statement must be
reckoned with. For instance, let us take the disjunctive
syllogism, The prices of commodities will be either in
creased or lowered by this law.
They cannot be increased.
.*. They must be lowered.
It may be shown that there is a third possibility, namely, the
law does not affect the prices one way or the other.
The Dilemma. This is a complex syllogism in which
both hypothetical and disjunctive propositions are combined.
The dilemma in its most complete form is constructed as
follows : the major premise consists of two hypothetical
propositions, the minor premise, of a disjunctive ; and the
conclusion, of a disjunctive.
i See p. 51.
146 DEDUCTIVE LOGIC
The minor premise may take either one of two forms. It
may affirm disjunctively the two antecedents contained in
the double hypothetical of the major premise; or it may
deny disjunctively the two consequents contained in the
same. If the former, the dilemma is called constructive;
if the latter, destructive. The symbolic representation of
these two forms may be expressed as follows :
(1) The constructive dilemma.
If A is B, C is D ; if E is F y O is H.
Either A is B, or E is F.
.-. C is D, or G is H.
(2) The destructive dilemma.
If A is B, C is D ; if E is F } G is H.
Either C is not Z>, or G is not H.
.*. Either A is not B, or E is not F.
The above being the complete form of the dilemma, there
may be certain variations introduced, as, for instance, instead
of two consequents there may be only one, or instead of two
antecedents there may be only one. The principle of the
dilemma is, however, not affected by these changes. This
principle is essentially that of presenting two possibilities
with definitely determined consequences, so that a choice
must be made between them which in either case results in
embarrassment, confusion, or contradiction. The following
dilemma, which will serve as a type of dilemmas in general,
illustrates these various features :
If the charges of the Senator from South Carolina are
true, I am unfit to remain a member of the Senate ; and if
they are untrue, the man who made them is unfit to remain
a member of this honorable body.
But they must be true or untrue.
.-. Either the Senator from South Carolina is unfit or I
am unfit to remain a member of this body. 1
1 Extract from a speech of Senator McLaurin in answer to Senator
Tillman s charges.
THE DILEMMA 147
It will be observed that the minor premise of a dilemma
states the possibilities to which a given situation gives rise,
and the major premise states the necessary relations which
these possibilities respectively sustain.
There are two parts of a dilemma where a structural
weakness is apt to occur, which of course affects seriously
the validity of the conclusion. The one weakness is an
absence of necessary sequence between antecedent and
consequent in either one or both of the hypothetical
propositions which form the major premise. The other
is the incompleteness of the disjunctive proposition which
forms the minor premise. If the alternatives are not
mutually exclusive, or if they are not exhaustive, error of
course must result. Sometimes a specious argument in
the form of a dilemma may be suddenly presented by an
opponent in controversy or in debate, and produce a tempo
rary confusion of mind because it is not known just where
the fallacy of the dilemma is concealed. It is well to know
therefore the exact sources whence errors in the dilemma
are apt to proceed.
When, in the major premise of a dilemma, the conse
quents do not invariably follow from the given antecedents,
or when other consequents also may follow which are not
mentioned in the premise, then it is possible to form a
counter dilemma which, starting from the same premises,
reaches an opposite conclusion. Both the original dilemma
and the counter dilemma in such cases are at fault, because
they both start from an inadequately expressed hypotheti
cal relation. An illustration of this is found in the classi
cal incident of the Athenian mother who advised her son
not to enter public life ; " for," said she, " if you act justly,
men will hate you, and if you act unjustly, the gods will
hate you ; but you must act either justly or unjustly ; there
fore public life will result in your being hated." The son,
however, brought in rebuttal an equally plausible state
ment; "If I act justly, the gods will love me; and if I
148 DEDUCTIVE LOGIC
act unjustly, men will love me ; therefore, entering public
life will make me beloved."
Trilemma. There is a still more complex form of the
combined hypothetical and disjunctive propositions which
is known as the trilemma. As the name indicates, the dis
junction in the minor premise consists of three members.
This is illustrated in the following statement regarding the
Louisiana Purchase. It is averred that the sale of Louisiana
to the United States was invalid ; because, if it were French
property, Buonaparte could not constitutionally alienate it
without the consent of the Chambers; if it were Spanish
property, he could not alienate it at all ; if Spain had the
right of reclamation, the sale was worthless.
CHAPTER XVIII
EXTRA-SYLLOGISTIC REASONING
THE syllogism, as we have seen, is a form of inference
which is essentially the interpretation of a special case in
the light of a universal concept to which it can be referred.
The function of the major premise is the statement of the
universal principle or relation which forms the basis of the
inference ; that of the minor, the statement of the connec
tion of the special case under consideration to this universal ;
that of the conclusion the investiture of the special case
with the essential properties which belong to the universal.
Now there are certain forms of reasoning which do not ex
plicitly at least conform to this programme of the syllogism,
and which judged by the formal rules of the syllogism must
be regarded as invalid, but which nevertheless are commonly
employed in our everyday inferences and whose validity is
indisputable.
There is in the first place the so-called reasoning from
" particulars to particulars." John Stuart Mill, as is well
known, attacks the accepted view of the syllogism insisting
that the reasoning process is never based upon a complete
universal, but always starts with particulars and concludes
with particulars. 1
In this connection, he gives the following illustrations :
" 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 recollection and
authority of what she accounts the similar case of her Lucy.
i Mill s Logic, Book II, Chap. Ill, 3.
149
150 DEDUCTIVE LOGIC
We all, when we have no definite maxims to steer by, guide
ourselves in the same way; and if we have an extensive
experience and retain its impressions strongly, we may
acquire in this manner a very considerable power of accu
rate judgment, which we may be utterly incapable of justify
ing or of communicating to others. Among the higher
order of practical intellects, there have been many of
whom it was remarked how admirably they suited their
means to their ends without being able to give any sufficient
reasons for what they did ; and applied, or seemed to apply,
recondite principles which they were wholly unable to state.
This is a natural consequence of having a mind stored with
appropriate particulars, and having been long accustomed
to reason at once from these to fresh particulars, without
practising the habit of stating to oneself or to others the
corresponding general propositions. An old warrior, on a
rapid glance at the outlines of the ground, is able at once
to give the necessary orders for a skilful arrangement of
his troops ; though if he has received little theoretical in
struction, and has seldom been called upon to answer to other
people for his conduct, he may never have had in his mind
a single general theorem respecting the relation between
ground and array. But his experience of encampments,
under circumstances more or less similar, has left a number
of vivid, unexpressed, ungeneralized analogies in his mind,
the most appropriate of which, instantly suggesting itself,
determines him to a judicious arrangement."
Mr. Mill is no doubt quite correct in this outline which
he sketches of common procedure in inference. However,
it cannot be claimed, and Mr. Mill is the last one to claim
it, that every particular instance furnishes sufficient ground
for an inference concerning a similar particular instance.
On the contrary, it is only the particular instance of a cer
tain well-defined kind which can give to such an inference the
proper logical warrant and validity. And this special kind
is one in which the particular instance ranks as a typical
EXTRA-SYLLOGISTIC REASONING 151
case. It stands in one s thought as the representative of the
universal of which it is a special case. In our reasoning we
speak of it in terms of its particularity, but the correspond
ing universal is always in the background of thought, and
it invests the particular case with its essential significance.
The particular is merely a disguised universal. The partic
ular as mere particular is barren of any inferential result.
When however it stands as representative of the universal
of which it is a special case, then it serves as a valid ground
of inference. When the village matron argues from her
own child s case to that of some other child, she has in mind,
dimly it may be, but nevertheless truly, some idea which
embraces her child s case and her neighbor s in one and the
same class. She knows, although it may not be explicit in
her thought, that the cure of the child did not depend upon
any circumstance peculiar to her constitution or nature, but
that the treatment employed possessed some essentially
efficacious tendency of a universal nature.
When the argument is, however, narrowed down to a
single special case, and this is made the basis of an infer
ence to another case which closely resembles it, then we
have inference by analogy. 1 There is a marked difference
between the special case which furnishes ground for infer
ence, because it stands in our minds as a typical case repre
sentative of its appropriate universal, and on the other hand
that special case which does not imply a universal at all,
but immediately suggests some resemblance to a similar
case and thus opens the way for reasoning by analogy.
Analogy, as a form of inference, has attached to it an element
of uncertainty so long as its basis is merely a particular
instance. When that particular instance begins, however,
to assume the characteristics of a typical case, and to direct
the thought to its corresponding universal, then inference
by analogy passes over by insensible degrees to the ordinary
syllogistic inference, or inference by subsumption.
i See p. 186.
152 DEDUCTIVE LOGIC
There is, again, another form of inference, which departs
from the syllogistic type but which nevertheless possesses
undoubted logical validity, such as the following :
A is to the right of B.
B is to the right of C.
.*. A is to the right of C.
Judged strictly by the logical rules of the syllogism, the
above conclusion is invalid, because the given syllogism has
four terms, A, B, the right of B } and the right of C. There
is, therefore, no proper middle term ; for B and to the right
ofB are different and can give no identical point of refer
ence for the two premises. Nevertheless this syllogism
holds. No one would think of denying its validity. How
ever, its form alone does not warrant the conclusion ; for we
may construct a syllogism of the same form whose conclu
sion is invalid. For example, in the following syllogism:
A is a friend of B.
B is a friend of C.
.-. A is a friend of C.
it is obvious that the conclusion does not follow necessarily
from the premises. Again, let us take a concrete example
of a line of argument which appeals to many as quite cogent,
but is nevertheless evidently fallacious, such as the fol
lowing :
Princeton has defeated Yale in base-ball.
Yale has defeated Harvard.
.-. Princeton will defeat Harvard.
We are confronted therefore by this problem :
Given the following syllogism,
A sustains certain relations to B.
B sustains similar relations to (7.
/. A must sustain these same relations to C.
EXTRA-SYLLOGISTIC REASONING 153
What kind of relations are they which necessitate such a
conclusion, and what kind are they which leave the conclu
sion indeterminate ; or, in other words, what are the precise
criteria which will differentiate the truly logical ground
from the illogical as regards the nature of the relations upon
which the inference is based ? The answer is not far to
seek. It lies in the very nature of the syllogistic inference
itself. We have seen that every valid inference must pro
ceed from premises which have as common ground some
identical point of reference. 1 If the premises are not joined
at a common point of articulation, their logical force can
not be combined, and without the premises in combination,
no conclusion follows.
Now, in the syllogism expressing relations of a perfectly
general character as given above, the form alone does not
give this necessary point of common reference. We must
look, therefore, for some direct test as regards the nature
of the relations as there expressed. If the relation which
obtains in the major premise is the same as that which
obtains in the minor premise, then evidently this identity
of relation secures the desired identical point of reference,
and therefore furnishes logical warrant for the derived con
clusion. This identity of the relations obtaining in the major
and minor premises can be established indisputably, however,
only when these relations appear in a system of coordinated
parts, wherein there is such simplicity that the relations
of part to part, throughout the whole extent of the system,
can be definitely and exhaustively comprehended. It is
only simplicity of system that gives necessity of inference.
Otherwise in relations which seem to be identical, there
may lurk some unknown and essential differences. From
the premises that A is a friend of B, and that B is a friend
of C, the conclusion that A is a friend of C does not fol
low because the system in which these relations obtain
is so exceedingly complex as to allow the possibility of a
1 Bosanquet, Essentials of Logic, p. 74 f.
154 DEDUCTIVE LOGIC
very wide divergence between phenomena, which upon the
surface seem quite similar. Not so, however, with the
premises, A is to the right of B and B to the right of C.
The conclusion is left in no doubt, for the very reason that
the given relations emerge in a system so simple that no
new or unknown elements can be conceived as disturbing
factors. Think however of introducing a change into this
simple system. Eegard it no longer as a plane surface, but
as the surface of a sphere. The conclusion from the given
premises does not follow necessarily.
Any system, therefore, which is of such simplicity as to
assure the identity of given relations, will always furnish
a logical ground for inferences of the kind we have been
discussing. Such inference is called inference by construc
tion rather than inference by subsumption. It is inference
by construction because the mind takes the material fur
nished by the premises, and places it where it belongs in
an underlying system which is explicitly or implicitly
assumed. The conclusion follows because the construction
has been made within that system and according to the
possibilities which the nature of that system imposes.
With any other system such a construction would not
have necessitated the same conclusion. The conclusion that
the square on the hypothenuse is equal to the sum of the
squares on the other two sides follows only when we con
ceive our right-angled triangle as constructed upon a plane
surface and not upon a sphere. If you say to me, " You
must be a friend to my friend because you are a friend to
me," my reply would be : " Not necessarily ; for in the vast
system of social relations exposed to the many perturba
tions arising from the qualities and the frailties of human
nature alike, the relation of friend to friend is too com
plex, too subtle, too profound, to furnish any simple and
constant basis of inference. There is here something more
than a matter of mere magnitude and position."
In addition to the examples already given there are many
EXTRA-SYLLOGISTIC REASONING 155
other simple systems, which for the most part grow out of
the fundamental categories of thought, and which provide
a logical ground upon which one may construct these in
ferences of relation.
There is the system which expresses solely the relations
of degree, in which it is possible to construct inferences
such as the following :
A is taller than B.
B is taller than C.
.*. A is taller than C.
There is also the simple time system, giving the infer
ence :
A is older than B.
B is older than C.
.-. A is older than C.
We may have also somewhat more complicated relations
within, however, an exceedingly simple system, as the fol
lowing will show :
A and B, two angles of a plane triangle, equal together 95.
.-. (7, the third angle, must equal 85.
These illustrations might be multiplied. They are, how
ever, sufficient to render clear the criteria regarding
all inference concerning related elements of one and the
same system. Whenever identity of relationship can be
established, a valid inference is possible ; and identity of
relationship can be established only in systems of such
simplicity that no unknown elements which might dis
turb the given relations can be conceived. Our thought
must command the system ; otherwise we are never justi
fied in using that system as a basis of reasoning. It should
be added, however, that the relations given in the premises
may be exceedingly complex, provided only the system in
which they inhere remains so simple that our knowledge
commands it fully. Thus, in geometry, there is the possi-
156 DEDUCTIVE LOGIC
bility of indefinitely complex constructions ; there are many
steps in the reasoning process from the statement of the
theorem to the joyful stage of the Q. E. D. ; nevertheless,
there must remain the constant simple system of space and
magnitude relations which constitutes the ground of it all.
There is no limit to the length of a series which may
express continued relations. We may have a related to 6,
b to c, c to d, d to e, and so on. The relations between
proximate terms will not insure like relations between more
remote terms necessarily. Here again our test comes to the
fore. If in such a series the underlying system is so
simple as to render the various relations identical in kind,
then all terms of the series are brought within a closed cir
cuit, as it were, and we can then pass in thought from the
first to the last term as well as from the first to the second.
There is still another kind of inference which is based
upon the nature of certain given relations and partakes of the
general characteristics of immediate inference. It is this,
that whenever we have given a judgment of the form, a is
related to b, the given relation necessitates a converse rela
tion, b is related to a. The converse relation is not identical,
however, with the given relation, but has an essentially
distinct significance, usually of an opposite nature. For
example, we have given A is the father of B, therefore B
is the son of A ; New York is east of Chicago, therefore
Chicago is west of New York. The following may be urged
as an exception to the statement that the converse relation
differs essentially from the given one : A is a friend of B,
therefore B is a friend of A. However, this is only a seeming
exception, for even in the relation of the most intimate friend
ship conceivable, the attitude, feeling, or disposition of one
party in the friendship is never the same as that of the other.
The precise nature of the converse relation will always de
pend upon the nature of the system in which the given
relation obtains. 1
1 See Russell, The Principles of Mathematics, Chap. IX, on " Relations."
CHAPTER XIX
FALLACIES
FALLACIES or errors in reasoning may be formal or mate
rial. The formal fallacy is one which is due to the struc
ture of the reasoning process itself ; the material fallacy is
due to the thought which underlies the structure. The
formal fallacies have been treated indirectly at least in
reference to the various rules of the syllogism, the violation
of which of course results in a fallacy of this kind. It will
be sufficient at this juncture merely to summarize these
fallacies, the most important of which are as follows :
1. Undistributed middle.
2. Illicit process of the major or minor term.
3. Denying the antecedent, or affirming the consequent
of the hypothetical syllogism.
4. Inadequate disjunction of the several members of the
major premise in the disjunctive syllogism ; that is, when
these members are not exclusive and therefore overlap.
5. The incomplete enumeration of possibilities in the
major premise of the disjunctive syllogism.
The material fallacies may be divided, as did Aristotle,
into two classes, those fallacies which are due to language
(rrapa TTJV Aeiv, or in dictione) ; and those which are due to
certain errors in the content of thought itself (!a> 7-775 A. eu>s,
or extra dictionem).
The fallacies which are due to language arise from the
fact that both in single words and in syntactical forms there
may lurk ambiguities of meaning. Any ambiguity of mean
ing in the course of reasoning violates the fundamental law
157
158 DEDUCTIVE LOGIC
of identity, which demands that a single and constant sig
nificance should attach to all the thought elements which go
to make up the data and the processes of our reasoning.
The fallacies due to language are often referred to as fal
lacies of ambiguity. Their violation of the law of identity
will be seen in the several instances which will be given.
These fallacies are as follows :
1. Equivocation. 4. Division.
2. Amphiboly. 5. Accent.
3. Composition. 6. Figure of Speech.
1. Equivocation. This fallacy consists in using a word
or a phrase which is capable of a double meaning, as, for
example :
I have the right to publish my opinions concerning the
present administration.
What is right for me to do, I ought to do.
.-. I ought to publish my opinions concerning the present
administration.
The ambiguity here, of course, lies in the meaning of the
word right, which in the one premise is to be taken in a
legal sense, and in the other in a moral sense.
This fallacy is in reality a fallacy of four terms ; that is,
in every syllogism there should be only three terms, each
term however being repeated. The law of identity de
mands that in this repetition the integrity of significance,
as regards the repeated term, must be preserved. To intro
duce a term, therefore, which is ambiguous violates this
fundamental principle of thought. The law of identity,
however, it must be remembered, allows a certain margin of
variation in meaning, provided only that the essential sig
nificance of the thought is not impaired. There is often a
difference of opinion as to whether a change in meaning
affects the essential significance of a concept or not. For
instance, let us consider the following syllogism :
Whatever menaces the public interests should be pre
vented by law.
FALLACIES 159
The Great Northern Securities Merger menaces the public
interests.
.-. It should be prevented by law.
Here the question is raised, Does this merger menace the
public interests in the sense that it should be punished by
law ? And that, of course, is the point upon which the
argument turns.
2. Amphiboly. This is a fallacy in which the ambiguity
lies in the syntax of the proposition rather than in the terms
of which it is composed. The following, taken from a notice
in the New York Tribune, will illustrate this :
"To-morrow afternoon, at four o clock, the Rev. J. A.
Francis will deliver the third and last address of a series of
plain talks to young men about their perils at the East 8Gth
St. branch of the Y. M. 0. A." The conclusion is obvious.
The following epitaph, also illustrating this same fallacy,
I discovered several years ago on a tombstone in the old
burying-ground at Concord, Massachusetts :
" Sacred to the memory
of
After living with her husband for fifty-five
years, she departed in the hope of a better life."
3. Composition. This is the fallacy due to the supposi
tion that what may be affirmed of individuals separately
may also be affirmed of them when taken together. It does
not follow, for instance, that because the members of a foot
ball team are all individually excellent players, therefore
the team play will show a similar order of excellence. This
fallacy is also illustrated in the following quotation from
John Stuart Mill:-
" No reason can be given why the general happiness is
desirable except that each person, as far as he believes it
to be attainable, desires his own happiness. . . . Each per
son s happiness is a good to that person, and the general
160 DEDUCTIVE LOGIC
happiness therefore a good to the aggregate of persons."
It does not follow, however, that because each desires his
own happiness, therefore all desire the happiness of the
whole. The root of this fallacy is to be found in the neglect
of the distinction between the distributive and the collective
use of a term. A term is used distributively when it is ap
plicable to each individual of the class separately; but
collectively when it is applicable only to all the individuals
which compose the class when taken together. It is the dif
ference between " all " meaning each one, and " all " meaning
all together.
4. Division. This is the converse of composition, and
consists in affirming of individuals separately what is true
only when they are taken together. It does not follow, for
instance, that because a certain board of directors has the
reputation of being exceedingly conservative, therefore any
individual member of that board is necessarily conservative
also.
5. Accent. This is a fallacy due to the undue accentua
tion of a word or clause in any statement so as to create an
implication which the bare words themselves do not indi
cate, and which, moreover, was not intended by the author
of the words. To quote from the text of an author and to
italicize certain words will often necessitate an interpreta
tion quite foreign to the author s mind. This is often done
with malice aforethought, and is an eminently unfair and
indefensible liberty to take with the thought of others.
6. Figure of Speech. This is a fallacy of using different
parts of speech having a common root as though they had
precisely the same meaning. The fact is, however, that a
noun may have a certain meaning, while an adjective derived
from the same root will have acquired a twist of meaning or
a subsidiary significance which will prevent their being re
garded in the light of interchangeable terms. The following,
also from John Stuart Mill, will illustrate this :
"The only proof capable of being given that an object is
FALLACIES 161
visible is that people actually see it. The only proof that
a sound is audible is that people hear it. ... In like man
ner, I apprehend, the sole evidence it is possible to produce
that anything is desirable is that people do actually desire
it." In this quotation, the relation of the word desirable
to desire is not the same as the other two cited, namely,
the relation of the word visible to the word see, and of
audible to the word hear. Visible means that which can be
seen ; audible means that which can be heard ; but desirable
does not mean that which can be desired, rather, that
which ought to be desired.
We come now to the second division of the material
fallacies, those which are due to inconsistencies of thought
rather than to ambiguities in the expression of the thought.
These fallacies are as follows : -
1. Accident. 5. Petitio Principii.
2. Converse Accident. 6. Non Causa pro Causa.
3. Ignoratio Elenchi. 7. Many questions.
4. Non Sequitur.
1. Accident. This is expressed in the Latin phrase, a
dicto simpliciter ad dictum secundum quid.
This is the fallacy of reasoning from what is true as a
general statement (simpliciter} to the same statement which
is restricted or conditioned in some manner (secundum quid).
The following is this fallacy of accident : -
Strychnine is a deadly poison, and therefore it can never
be used as a medicine.
2. Converse Accident. This is expressed in the Latin
phrase, a dicto secundum quid, ad dictum sintyliciter. This
is the fallacy of reasoning from that which may be true
under certain conditions or limitations, to that which how
ever is not true when these conditions or limitations are
removed. This is illustrated in the following argument
which is very often heard :
Certain men have risen to prominent positions who never
162 DEDUCTIVE LOGIC
had a college education ; therefore a college education is un
necessary to equip a man for his life s work.
In reference to these two fallacies, there is a passage in
Lotze which is of interest, and which is well worth quoting
here.
" Two general modes of fallacious thought are developed
by the habitual commission of these fallacies, and illustrate
them on a grand scale. The first is doctrinairism, the second
narrow-mindedness. The doctrinaire is an idealist who refuses
to see that though ideas may be right in the abstract, yet the
nature of the circumstances under which and of the objects
to which they are to be applied must limit not only their
practicability but even their binding force. The narrow-
minded, on the other hand, can recognize and esteem no
truth and no ideal, even the most universally valid, except
in that special form to which they have become accustomed
within a limited circle of thought and personal observation.
Life is a school which corrects these habits of mind. The
parochially minded man sees things persist in spite of him
self in taking shapes which he considers unprecedented, but
he finds the world somehow survives it, and learns at last
that a system of life may be excellent and precious, but that
it is rash from that to argue that it is the only proper mode
of orderly existence. And the enthusiast for ideals, when
he sees the curtailment which every attempt at realization
inflicts on them, learns the lesson which the disjunctive
theorem might have taught him. Every universal P changes
in the act of being applied from something that held sim~
pliciter into something that holds secundum quid, changes
from P to p, 1 p, 2 or p 3 ; to refuse to accept it in any one of
these, which are its only possible shapes, is to ask that it
be realized under a condition which even logic pronounces
impossible." 1
3. Ignoratio Elenchi. This is a fallacy which consists
primarily in an ignorance of the nature of refutation. To
1 Lotze s Logic, Vol. II, p. 5, Eng. trans.
FALLACIES 163
refute an argument, its logical contradiction must be es
tablished. Any proof which falls short of this fails in its
end. The nature of this fallacy has been enlarged in scope,
so as to comprise any argument whatever which does not
squarely meet the point at issue. It is, in many cases, not
so much the ignorance of the point at issue, as purposely
ignoring the point at issue. It is a natural method of argu
ment when one has a weak case. Any subterfuge which
withdraws attention from the point at issue tends, of course,
to strengthen the weaker side, at least as regards the plausi
bility of its position. Suppose a student should be urged
to spend more time upon his Latin or Greek, and he should
excuse his negligence by insisting that in after life he would
never find any practical use for his classics, this would be
the fallacy of ignoratio elenchi.
There are various ways in which this fallacy may be
illustrated, as follows : -
(a) Argumentum ad hominem. This is the fallacy wherein
the argument is diverted from the merits of the case to the
character or the position of one s opponent.
(b) Argumentum ad populum. This is the fallacy of ap
pealing to the passion or prejudice of an audience, rather
than to their reason. It is essentially the argument of the
demagogue.
(c) Argumentum ad ignorantiam. This fallacy consists in
taking advantage of the ignorance of the person or persons
addressed who, consequently, lack the power of discrimina
tion between the true and the false, the relevant and the
irrelevant.
(d) Argumentum ad verecundiam. This is an appeal to
the sentiment of veneration for authority, instead of an appeal
directly to the reason. The weight of great names is with
some persons the most convincing of all arguments. Logi
cally it is not an argument at all. It may serve to confirm
truth, but it does not establish it.
(e) Argumentum ad baculum. This repudiates all argu-
164 DEDUCTIVE LOGIC
ment and resorts to force in order to establish one s
point.
In distinction from these various kinds of subterfuges to
avoid a direct facing of the question, there is the argumen-
tum ad rem, or the argumentum ad judicum, i.e. arguing
directly to the point at issue. All lines of argument should
converge to this central point.
4. TJie Fallacy of the Consequent, or Non Sequitur. This
fallacy was defined primarily by Aristotle as the formal error
of affirming the consequent. It has received, however, in
the course of time, a far wider application, and has come to
be applied to any loose argument whatever, in which the
conclusion does not seem to follow from the premises. It
is very convenient to have the phrase non sequitur where
with to characterize such arguments.
5. Petitio Prindpii. This is the fallacy of begging the
question. This is an attempt to assume the conclusion
without any attempt whatever to prove it. According to
Aristotle this may take place in five ways :
(1) To assume the point at issue directly. This, however,
cannot be done without resort to some rhetorical device to
conceal the absence of any real proof.
(2) To assume some more general truth which involves
the point at issue.
(3) To assume particular truths which it involves.
(4) To assume the component parts in detail.
(5) To assume some necessary consequence of the point in
question.
As an illustration of begging the question, take the fol
lowing extract from a speech of a member of the House of
Commons: "The bill before the House is well calculated
to elevate the character of education in this country, for the
general standard of instruction in all our schools will be
raised by it."
Galileo accuses Aristotle of having committed this fallacy
in his argument that " the nature of heavy things is to tend
FALLACIES
165
toward the centre of the universe, and of light things to fly
from it; therefore, the centre of the earth is the centre of
the universe."
There is a special form of this fallacy known as arguing
in a circle, circulus in probando. This is an attempt to
prove a conclusion to follow from a premise, when in truth
the premise itself depends upon the truth of the conclusion
as its ground. This is illustrated in the following statement
taken from a letter written to one of our daily journals
quite recently: "The left-handed man lacks will power,
for, if not, he wouldn t be left handed."
6. Non Causa pro Causa. This is the fallacy of regarding
as a cause that which is not a cause. It is due to the lack
of discrimination between a mere coincidence and a vent
able cause. There is no fallacy, perhaps, which is so subtle
as this one, and none which is more common. As an exam
ple of this fallacy, we may cite the exploded hypothesis
of a mesmeric fluid to account for the various well-known
phenomena of hypnotism ; also the statement that nature
abhors a vacuum to account for the rise of water in a
pump ; or the belief that any unusual appearance among
the heavenly bodies, as that of a comet, is to be interpreted
as a portent of disaster. Many of our common superstitions
may be traced to this fallacy. Moreover, inasmuch as the
causal relation naturally manifests itself in the form of a
sequence, there is a special case of this fallacy which con
sists in the confusion of mere sequence with a causal con
nection ; this is called the fallacy of post hoc ergopropter hoc.
This is illustrated in the belief which many entertain, that
when thirteen sit down together at a common board, one
of the number will surely die within the year ; or in the
tendency so often observable to attribute the financial pros
perity or distress of the country to some legislative measure
recently enacted.
7. TJie Fallacy of Many Questions. A better name for
this would be the Fallacy of a Double Question, for it con-
166 DEDUCTIVE LOGIC
sists in asking a question which is in the form of a single
question, but which should have been put in the form of two
separate questions. The question which is asked assumes
that another question has been already asked and answered.
This fallacy usually takes the form of asking a question
about an assumed fact whereas the fact is itself in dispute.
Thus the question, How much do you pay a certain member
of your athletic team for his services, presupposes of course
that some amount is certainly paid.
The following anecdote which appeared recently in one of
our daily papers also illustrates this fallacy :
"Charles Bradlaugh, the English free-thinker, once en
gaged in a discussion with a dissenting minister. He
insisted that the minister should answer a question by a
simple Yes or No, without any circumlocution, assert
ing that every question could be replied to in that manner.
" The reverend gentleman rose, and said, Mr. Bradlaugh,
will you allow me to ask you a question on those terms ?
" Certainly/ said Bradlaugh.
" Then, may I ask, have you given up beating your
wife ? "
This completes the table of fallacies usually given in
treatises on logic. All the general types of fallacies are
comprehended in it. There are fallacies, however, which do
not distinctively fall under any one type, but are so subtly
complex as to involve the errors of many. There are, again,
others which arise out of special circumstances, and cannot
be classified under any of the types mentioned. They, how
ever, readily disclose themselves to the open mind which is
freed from sophistry and prejudice.
PART II
INDUCTIVE LOGIC
CHAPTER I
INDUCTION AND DEDUCTION
THERE have been divergent tendencies in the history of
logic, to make either deduction or induction alone the whole
of logical procedure in the process of inference. The fact
that the Aristotelian logic, which is essentially deductive,
has been for centuries exclusively associated with logic as
a whole, has left the impression upon many minds that it
is the beginning and end of the logical encyclopaedia. On
the other hand, John Stuart Mill and his followers have
attempted to analyze the syllogism so as to prove its es
sentially inductive character; and they maintain that all
reasoning is inductive. This is the position in the main of
Bacon, Locke, and Bain. Locke, for instance, insists that
the syllogism is of less value than external and internal
experience, induction, and common sense. 1
So also, in a similar vein, Schleiermacher says : " The
syllogistic procedure is of no value for the real construc
tion of judgments, for the substituted judgments can only
be higher and lower; nothing is expressed in the conclu
sion but the relation of two terms to each other, which
have a common member, and are not without, but within,
each other. Advance in thinking, a new cognition, cannot
originate by the syllogism ; it is merely the reflection upon
the way in which we have attained, or could attain, to a
judgment, the conclusion ; no new insight is ever reached." <
The two opposed views thus indicated do not necessitate
conflicting or mutually exclusive processes. It is better to
1 Essay on Human Understanding, Book IV, p. 7.
2 See Ueberweg, System of Logic, etc., p. 345.
169
170 INDUCTIVE LOGIC
regard them, not as radically different types of inference,
but rather as different phases of one and the same inferential
process. We have seen that inference consists in interpreting
the implications of the system to which the given in con
sciousness belongs. In the light of this definition we can
best indicate the relative functions of induction and deduc
tion in the process of inference. When the system can be
considered as a whole, and is apprehended in its entirety,
then it may become the ground upon which the inference is
based, resulting in unfolding the necessary nature or relations
of any of the parts considered in themselves, or in reference
to the system as a whole. The procedure in such a case
is from the nature of the whole system, to the nature of
the several parts, and their existent relations, and this is
deductive in its essential features.
On the other hand, when we know the various parts, and
proceed from them as data to construct the system which
their known nature and relations necessitate, it is induction,
or procedure from elementary parts to the whole thus neces
sitated. From a knowledge of the planetary system we
can infer the necessary positions of sun, moon, and earth
at any required time, as, for instance, in the calculation of
an eclipse. This is deduction. But when we begin with
investigating the several movements of the different planets,
and from them infer the necessary nature of the system of
which they are parts, we have the process of induction.
Such processes we see must be complementary, and mutu
ally dependent. As Lavater says, " He only sees well who
sees the whole in the parts, and the parts in the whole."
Moreover, the distinction between deduction and induc
tion may be shown through their respective relations to the
universal, which we have seen is the ground of inference.
The question whose answer leads to the deductive process
in reasoning, is, What does the universal necessitate ? In
induction, the question which starts the investigation is,
Into what system may the given material properties or
INDUCTION AND DEDUCTION 171
relations be constructed so as to reach a universal concept
that will be consistent with itself and with the whole of
knowledge which forms the world of consciousness ? In
this there is an analytical discrimination of the essential
from the accidental elements, and the gathering together of
the former into the complex whole which is the universal.
Induction, therefore, is inference viewed from the side of
the differences; deduction is inference viewed from that
of the universal. For instance, we may investigate the
characteristic features of a diamond, and find that a certain
specific gravity, 3.53 as compared with water, is a con
stant and determining attribute, and as such must be in
corporated as an essential element of the general concept
diamond. We can then form the universal judgment, What
ever stones possess this specific gravity are diamonds. Their
differences, regarding size, brilliancy, etc., may all be set
aside as accidental, but the one constant determining fea
ture indicates a oneness in which they all agree.
And so with the other essential attributes. After pos
sessing such knowledge gained inductively, we may use it
practically in a deductive manner ; and it is so used in
discriminating between true and imitation stones, as de
scribed in the following process : " Diamonds, rubies, and
sapphires are now tested by floating to prove their genuine
ness. The liquid used has five times the density of water,
and is composed of double nitrate of silver and thallium.
The tests are rapidly made, as all stones of the same nature
have the same specific gravity, while none of the bogus ones
have the same weight as those they are made to imitate."
Another view of the relation of induction to deduction
may be gained by calling attention to the difference of sig
nificance between the terms, a "truth" and a " fact." A fact
carries with it only the special and individual character of
the particular occurrence in which it is manifested. A
truth, however, is always universal in its very nature, ad
mitting of universal application, and capable of illustration
172 INDUCTIVE LOGIC
in an indefinite number of different facts which embody its
essence. In deduction we have given some truth of uni
versal nature that leads to individual facts that may be
subsumed under it. In induction, we interpret a fact or a
number of facts in the light of their universal implication,
on the ground that there can be no such thing as an isolated
fact, but every fact must have some relation to a universal
to which it must be referred.
While considering the distinctions between induction and
deduction, we must not overlook their mutual dependence.
We cannot proceed in deduction irrespective of induction,
because the universal upon which the deductive process is
based arises in the majority of cases from a previous induc
tion. It is true that the universal term may be in a propo
sition that is known a priori) as the axioms of geometry and
certain space and time postulates ; but a very small propor
tion of major premises can be said to have such an origin,
and their resulting conclusions have very slight material
significance. Deduction that reaches other than purely ab
stract and formal conclusions must rest upon induction for
the material to form its premises. We find this even in
the technical construction of the syllogism, where, for in
stance, the question of the distribution of the terms is
raised. We may insist that a certain middle term is dis
tributed, as it is the subject of an universal affirmative
proposition; but then the further question naturally sug
gests itself, How do we know that the proposition in ques
tion is really a universal ? Its material significance alone
tells us that we may write it as an A or / proposition, as
the case may be. The matter is a function of the form,
and the form a function of the matter. They cannot be
separated, in fact, unless we conceive reasoning as a purely
formal process of determining a conclusion, irrespective of
the truth or falsity of the premises. If we regard the
premises as given, and we accept them with unquestioning
credence, the deduction is purely formal; so, also, if the
INDUCTION AND DEDUCTION 173
various terms are expressed by letters A, B, C, etc., and
devoid of any material significance. Any process of rea
soning based upon a slavish acceptance of premises can only
reach artificial and even false results. In the actual experi
ences of life our premises are not made for us. They must
be constructed by us through our interpretation of reality.
Disregard of this has brought formal logic into much disre
pute, and it has often degenerated into the barren discussion
of logical puzzles and quibbles. Grant a person any prem
ises he may choose to assume, irrespective of an inductive
test of their validity, he can prove black white and white
black.
On the other hand, induction is dependent upon deduction ;
for we cannot reason from particular instances to a universal
proposition, unless we assume as basis of the whole induc
tive process some postulate which has real universal signifi
cance. Otherwise, we reach only a high degree of probability,
but not necessity; a rude generalization, but not univer
sality. When we assert some such general statement as
this, that arsenic always acts as a poison, we have based
the universal character of the proposition upon an under
lying postulate that is understood even though it is not
expressed, such as the uniformity of nature, that under
identical conditions we always look for identical effects.
This will be discussed later more in detail ; it is referred to
at this point merely to illustrate the deductive basis of
induction. Bradley insists that there can be no such thing
as induction, because it always rests upon an implied uni
versal which gives to the process as a whole a deductive
character. 1 His criticism has the force only of proving that
induction cannot be independent of deduction. This depend
ence does not, however, necessarily vitiate the integrity of
induction as a mode of the inferential process. Lotze has
placed special emphasis upon this dependemce of induction
upon deduction. He says: " It is the custom in our day to
1 Bradley, Principles of Logic, p. 342.
174 INDUCTIVE LOGIC
collect into one body the numerous operations which assist
us in ascending from particulars to generals, or to call this
inductive logic, and to set it against the deductive or demon
strative logic along with much disparagement of the latter.
Such disparagement rests on a mistake. The inductive
methods, it is certain, are the most effectual helps to the
attainment of new truth, but it is no less certain that they
rest entirely on the results of deductive logic." !
Moreover, in induction the results obtained and formulated
in general propositions may be extended and often modified
by a deduction which is based upon them as major premises ;
for the deduction thus proceeding from them reveals new
instances which conform or perhaps modify the simple
inductive results themselves. What is popularly called a
hasty generalization, if made a major premise of a syllo
gism, will often lead us astray through the deductions
drawn from it. As soon as we are aware of this, we return
to question the validity of the generalization, whose weak
ness is not appreciated until thus tested and revealed. Thus
deduction serves to extend and correct the results of induc
tion, and at the same time it itself is dependent upon the
results of inductive generalization for the material to form
its premises. We come to see therefore how intimately
associated these two processes are in actual reasoning. For
convenience of illustrating their individual characteristics,
they may be considered as separate, and each investigated
as an independent mode of inference. But they are in reality
mutually related and dependent, and are always found mani
festing their functions together. In any course of reasoning
concerning the conduct of our everyday affairs, or in sci
entific investigation, anywhere, indeed, outside of the
artificial examples of logical text-books, we reason both
inductively and deductively in one complex process.
1 Lotze, Logic, p. 288, See also Bosanquet, Logic, Vol. II, p. 119.
CHAPTER II
THE ESSENTIALS OF INDUCTION
WE now proceed to a more precise determination of the
nature of induction. Its point of view in all reasoning has
reference to concrete instances. They are the data, and
from them general propositions are to result. The pro
cedure is from given facts to laws which are the ground and
explanation of these facts. We are here however at once
struck with the evident break in the course of our reasoning.
Procedure from the particular to the universal cannot be a
continuous process. There is a gap somewhere. The con
clusion contains more than the premises. In deduction, we
are proceeding from the greater to the less, and we experi
ence no violation of our logical sense; but at once we
appreciate the difficulty which attends the reverse process
from the less to the greater. 1 Here we soon reach a point
where we pass beyond the sphere of our experience to the
generalization which necessarily embraces far more than
our experience. This is the so-called inductive leap ; or it
is sometimes referred to as the inductive hazard. But is this
a leap in the dark a wild guess concerning all that lies
beyond the sensuous sphere of our immediate experience ?
This would be the case, were we compelled to use the mere
data of experience as sole ground for our inferences. John
Stuart Mill insists that nothing whatever is given in con
sciousness but particular sensations, and these are but sub
jective states of feeling, and with no assurance of any
definite correspondence with the external world. With such
purely empirical data it is impossible to proceed to truths of
1 See p. 107.
175
176 INDUCTIVE LOGIC
universal validity. It is necessary to postulate some uni
versal truth which the mind through strictly a priori con
siderations is constrained to formulate, and which will serve
to bridge the gulf between the particular and the universal.
This postulate has been variously expressed by different
authors, yet with substantially the same significance in all.
In the older logic, it is put under the convenient formula of
the uniformity of nature ; that is, that beyond the sphere
of experience, phenomena will behave in the same manner
under like conditions, as in the sphere of immediate obser
vation and experiment. In the modern logic this is some
what differently expressed. The phrase "uniformity of
nature," being somewhat indefinite and implying a point
of view purely objective, is not used. Modern writers have
omitted it largely from their terminology. Lotze says:
"The logical idea upon which induction rests is by no
means merely probable, but certain and irrefragable. It
consists in the conviction based upon the principle of
identity, that every determinate phenomenon M can depend
upon only one determinate condition, and accordingly that,
where under apparently different circumstances or in differ
ent subjects P, S, T, U, the same M occurs, there must
inevitably be in them some common element 2 which is the
true identical condition of M, or the true subject of M." l
We have a somewhat similar description of the basis of the
inductive process given by Sigwart: "The logical justification
of the inductive process rests upon the fact that it is an inevi
table postulate of our effort after knowledge that the given is
necessary, and can be known as proceeding from its grounds
according to universal laws." 2 Bosanquet considers as the
basis of inductive inference that which he calls the postu
late of knowledge, that " the universe is a rational system,
taking < rational to mean not only of such a nature that it can
be known by intelligence, but further of such a nature that
it can be known and handled by our intelligence. 7 3
l Lotze, Logic, p. 102. 2 Sigwart, Logic (Eng. trans.), Vol. II, p. 289.
8 Bosanquet, The Essentials of Logic, p. 166.
ESSENTIALS OF INDUCTION 177
I have quoted these passages from Lotze, Bosanquet, and
Sigwart, that we may appreciate the modern tendency to
derive the inductive postulate from an epistemological
source; namely, that our knowledge must be consistent
throughout with itself, part to part, and parts to whole, and
that the world for us is the world as constructed by our
knowledge. Whatever is given in consciousness must be
long therefore in the one place where it appropriately and
necessarily belongs. Here also there must be a place for
everything, and everything in its place. There must be a
uniformity of consciousness ; that is, the primary postulate
and the uniformity of nature is secondary to this, and
implied in it. This postulate may also be expressed as
follows: What is once true is always true. Here "true" is
used in the sense of the universal significance of a fact.
Whenever a concrete instance is present in consciousness, its
existence must be considered as necessitated by some ante
cedent which can satisfactorily account for it, and which
can at the same time be appropriately adjusted to the whole
of our knowledge in interpreting it. Bosanquet says that
" ideally speaking, every concrete real totality can be
analyzed into a complex of necessary relations." 1 These
necessary relations of course have a universal significance,
and therefore in every concrete instance, if we can rightly
interpret it, we may discern the universal element which is
contained in it, and gives it a place and meaning in the
world as cognized by us. Nature, after all, is only another
word for the world as we know it.
There is a sense in which induction may be regarded as
the inverse process of deduction. In deduction the problem
is concerned with the question, What does the universal
necessitate? In induction, the instance is given, and the
problem is, What universal can be discovered which could
give rise to the instance in question? This view of induc
tion is especially associated with the name of Jevons, whose
1 Bosanquet, Logic, Vol. II, p. 82.
178 INDUCTIVE LOGIC
inductive system is described as the inverse of deduction.
He calls it the deciphering of the hidden meaning of natural
phenomena. 1 The name commonly used to designate this
view of induction is that of "reduction," originally sug
gested by Duhamel. 3 This process was known to the old
logicians, who called it " Method," to denote the process of
hunting for middle terms by the aid of which a given con
clusion could be proved. 3 Like all inverse processes, it is
by itself an indeterminate one.
Given, All A is B, and
All B is (7,
we infer by the direct process of deduction that
All A is C.
But in the indirect or inverse process we have given A is
C, and the problem, to find a middle term which necessi
tates such a conclusion, is an indeterminate one. There
may be a number of middle terms. This is analogous in
one respect to the method of integral calculus ; while dif
ferentiation leads to a definite result, the inverse process of
integration leads to an indeterminate result. So also we
multiply two numbers, producing one determinate result;
but inversely, when we have given a certain number, and
ask what factors multiplied together could produce this
number, we may reach several different solutions. The
answer is indeterminate. Professor Jevons, in his scheme
of inductive inference, falls back upon probability to indi
cate which of several possibilities is the most likely one in
the given case. 4 But before the inverse operation can result
in determinate results, the given terms such as A and C
must be subjected to some analysis in order that their
material signification may give suggestion as to the nature
1 Jevons, Principles of Science, p. 124.
2 Duhamel, Afethodes, Vol. I, p. 24.
8 Venn, Empirical Logic, p. 361.
4 Jevons, Principles of Science, p. 219.
ESSENTIALS OF INDUCTION 179
of the middle term. For instance, a man is found dead,
washed ashore by the tide ; the natural supposition would
be that he met his death by drowning. And yet it might
possibly happen that the man died through injuries inflicted
by blows, or by poison, or heart failure. The attendant cir
cumstances and bodily indications must suggest the most
probable cause to account for the given effect. Venn criti
cises Jevons view of induction, that is, making it the
inverse process of deduction, on the ground that it is purely
a formal process, and therefore can lead only to indetermi
nate results. 1
It is always possible, however, to make some analysis of
the material significance of the data, as has been above indi
cated, which relieves the purely formal processes from the
indefiniteness of the results. Bosanquet criticises Jevons
theory of inductive inference, in that the hypothesis pro
posed to account for the given in reality can at best be
only highly probable. 2 However, Venn, Lotze, Bosanquet,
Sigvvart, all allow a place to the inverse function of all
inductive reasoning; their contention, however, is this, that
it does not funish an adequate account of the whole matter. 3
It is interesting to note that Whewell s theory of induc
tion corresponds in the main to this idea of reduction, or
inverse process. He finds in induction a twofold operation
of the mind, consisting in the colligation of facts and the
explication of conceptions. By the colligation of facts he
refers to that insight which is able to see the connections
and relations which necessarily exist between the different
phenomena present in consciousness ; and by explication of
conceptions he refers to the appropriate fitting in of these
related facts to some conception of the mind which most
readily accounts for them. 4 Such a process is merely the
l Empirical Logic, p. 359. 2 Bosanquet, Logic, Vol. II, p. 175.
Venn, 361 ; Bosanquet, Vol. II, p. 175 ; Sigwart, Vol. II, p. 203, 289.
Lotze, Outlines of Logic, p. 93.
4 Whewell, Philosophy of the Inductive Sciences, pp. 172, 202.
180 INDUCTIVE LOGIC
reading of given facts backward to their origin, or substan
tially an inverse process, where the procedure is from the
given concrete to the explanation of the same in terms of the
universal to which it can be most appropriately referred.
So also Mill s account of procedure by hypothesis presents
characteristics similar to this process of reduction.
The end of induction is to discover a law having objective
validity and universal application. There is a distinction
which must be noticed and clearly kept in mind; namely,
the distinction between a law and a rule. Induction seeks
a law, and not a rule. A law expresses the essential and
universal relations subsisting between given phenomena,
eliminating entirely all accidental and local coloring. A
law has objective validity, and preserves a constant nature.
There can be only one law in reference to one and the same
connection of facts. A rule however is subjective, dealing
with the individual s attitude to phenomena, rather than
an explanation of the essential features of the phenomena
themselves. It often is determined in the concrete by that
which is external, local, and accidental. There may be
many rules, varying with many minds and many tastes.
Fundamental and universal laws of political economy be
come maxims and rules in different communities. The laws
of morality, universal and immutable, become rules of con
duct in individual experience, admitting of wide difference
of opinion and diversity of application. 1 In the processes
of induction, therefore, the law is the desideratum, and not
the rule.
Law however is used rather loosely in our ordinary ter
minology. Law as used in jurisprudence has a meaning
quite different from law as used in physical science. And
so, also, the laws of biology, the laws of political economy,
the laws of ethics, are referred to with different shades of
meaning in each sphere. However ambiguous may be the
significance of " law " in ordinary thought and usage, never-
1 Lotze, Logic, p. 335.
ESSENTIALS OF INDUCTION 181
theless in induction it has a constant and a simple signifi
cance, which, if carefully adhered to, will avoid confusion
and obscurity as well, in our inferential processes and
results. Law in induction is always in the form of an
hypothetical universal :
If A is, B is.
It does not assert what has happened, but what should hap
pen under certain conditions. Given the antecedent A, a
certain determinate consequent B is always necessitated.
The relation is constant and invariable, and therefore has a
universal significance.
Induction holds a peculiar and important place in our
everyday life, because it has to do with the analytical
treatment of instances as they appear in experience. The
large part of our conscious thinking has to do with the con
crete, the raw material of experience ; this, induction alone
can handle. Leonardo da Vinci s maxim was " to begin with
experience and by means of it to direct the reason." Thus
the superstructure of knowledge is raised day by day. The
given is continually being interpreted and referred to its
appropriate place, as the stones of the quarry are hewn and
fitted into their proper position in the building for which they
have been designed. There are certain individual experi
ences which it is impossible to determine through our syllo
gistic forms. They cannot be judged deductively. There
is no general category under which they can be subsumed.
They may be formally illogical if thus expressed, and yet
admit of direct investigation and experiment in the induc
tive manner, for the purpose of disclosing the law under
lying them and as yet unknown.
It often happens that through indifference or indolence
we are content to refer many phenomena to long-established
and convenient categories, which, if investigated indepen
dently, we would find could not possibly be so treated. The
1 Ueberweg, Logic, p. 42.
182 INDUCTIVE LOGIC
convenient pigeon hole, because near at hand, receives much
that does not properly belong there. It is the office of
induction to investigate anew the old material, and then
to reclassify in accordance with the revised generalizations
which such investigations may necessitate.
The procedure by induction is in keeping with the scien
tific spirit of the day, to interpret the phenomena of
nature as given, and not to anticipate nature through pre
conceptions, and wrest fact in order to fit theory. It comes
to the sources in nature with empty vessels, to draw and
carry away that which nature alone can give.
CHAPTER III
TYPES OF INDUCTIVE INFERENCE
THE process of induction, as we have seen, is a procedure
from given instances to the discovery of the law which
underlies them, and which is the ground of the connection
of the various attributes and relations that unite in the one
concrete whole. Viewed from the standpoint of the direc
tion of the process, we have found that it is always toward
some general expression of individual experiences, and in
this respect it is the inverse of deduction, which proceeds
from the general to the particulars which are embraced in it.
There is however another and important point of view that
should not be overlooked. We have to consider the mode
of the process as well as its direction ; not merely the result
to be attained, but also the peculiar manner of realizing the
same must be considered. Difference in method here gives
rise to various kinds of inductive inference. The end pro
posed in all is to generalize our experiences as they occur
in the concrete and particular. When I find a given phe
nomenon, A, given in consciousness, and characterized by
several distinctive features among which I note specially
the mark B, the question at once most naturally suggests
itself, Is there a reasonable expectation that I shall always
find B as an inseparable accompaniment of A, so that I can
assert confidently that whenever A is found, B also will be
found? There are three ways of satisfying ourselves as to
the existence of any constant rather than coincidental con
nection between antecedent and consequent, as A and B.
These give rise to three different methods of inductive
research, and they are as follows :
183
184 INDUCTIVE LOGIC
I. The Method of Enumeration.
II. The Method of Analogy.
III. The Method of Scientific Analysis, or search after
causal connection.
Failure to distinguish between the three methods has
given rise to confusion in the definition of and correspond
ing reference to inductive inference ; some authors use
induction in one, and some in another of these senses. It
is necessary to discriminate carefully, and to maintain a
strict consistency in the usage of the terms as defined.
I. The Method of Enumeration. We observe the various
instances in which certain attributes, as A and jB, are con
joined in our experience. We count them in the sense of
noting to what extent they accumulate without noticing
any exception to what seems at least an invariable connection.
We do not necessarily count by precise enumeration reaching
a numerically definite result. We notice merely to what
extent the observed instances of like nature accumulate;
that is, whether a few, a considerable number, or a very
large number. The mere number of instances produces a
certain psychological impression, whatever may be their
logical force. This is brought about through the laws of
association, and creates an expectation of a continuous
repetition of the experience in question. This arises from
a natural tendency of the mind to generalize. We observe
that crows are black ; and the increasing number of confirm
ing instances goes far to establish a connection between the
crow and its color which seems to have universal validity.
The enumeration of instances may lead us to any one of
three results :
1. We may meet with no exception whatsoever, until the
scope of observation completely embraces the sum of all
possible instances. This is complete enumeration, and
when enumeration reaches this limit, it passes over into
deductive reasoning, by virtue of the logical canon that
TYPES OF INDUCTIVE INFERENCE 185
whatever is true of the parts is true of the whole distribu-
tively ; that is, provided the summation of the parts has
been an exhaustive one. We assert that all the sheep of a
given flock are white ; for we have observed each separately,
and no one has been missed in the count. So, also, the
judgment that all planets move around the sun, resulting
from an enumeration of the planets one by one. It is
possible also to have a perfect induction with an infinite
enumeration of parts. This is possible in two cases, as
pointed out by Beneke. 1 First, when the parts are con
nected together continuously in space. This occurs in
geometrical demonstration when the inference, based upon
the simple figure it refers to, is extended to all figures falling
under the like definition. And second, when the parts are
not continuously connected, if it can be proved syllogistically
that what is true of a definite ?ith part, must also be true
for the (n + l)th part.
Perfect induction also embraces arithmetical method and
computation. Here the whole, which is the sum of the
facts in each case, is a totality or universal whose differences,
which are all separate and distinguishable, are yet homo
geneous and equal. 2 There is no qualitative differentiation
of parts, only a quantitative one. The total is the sum of
the units, and each unit is like every other one. If we
have one hundred units making a totality, the one that may
be the twenty-seventh is precisely like the sixty-seventh.
It is a case where each one counts for one and no one for
more than one, in an absolutely literal sense.
It has been urged against perfect induction that it affords
no new information, and, therefore, its results are not
valuable. However, the summation of particulars in abbre
viated forms is always an advantage. It is a labor-saving
process to the mind. It enables the mind to retain a large
number of facts by throwing them into one and the same
1 Quoted by Ueberweg, Logic, p. 482.
2 Bosanquet, Logic, Vol. II, p. 54.
186 INDUCTIVE LOGIC
category ; and it facilitates arithmetical processes by conven
ient comprehending of units within a totality.
2. The second result that is possible, is that, in counting
instances, our enumeration should prove incomplete. From
the necessities of the case, we are often not able to observe
the entire sphere of possible occurrences and cover the whole
ground. It may be that beyond the sphere of our expe
rience, the constant connection between certain phenomena
may be disturbed by the appearance of some variable factor
of which we have been wholly ignorant. It is the possibili
ties beyond the sphere of observation which render uncertain
the results of our count. We are sure as far as we have
observed ; but we have not gone far enough perhaps. Such
results, formulated in general propositions, are termed
empirical laws ; that is, generalizations from an experience
necessarily limited.
3. We have still a third case ; where in our enumeration
of positive instances we meet with exceptions to a greater
or less extent. Here we cannot even sum up the actual
experience in terms of a generalization. There are out
standing exceptions which will invalidate it. We must,
therefore, fall back upon the theory of probability and the
calculation of chances, presuming that, in general, we will
meet with the same proportion of exceptions to positive
instances in the future, that we have already observed in
the past. So we make, in our minds at least, comparative
tables of positive cases over against exceptions, and reach a
summary of the result in the form of a ratio, whose numera
tor will be the number of positive cases observed, and the
denominator the total number of instances including positive
instances and the corresponding exceptions. We observe
that some cryptogamous plants possess a purely cellular
structure ; others, however, do not, being partially vascular.
The probability that a new cryptogam will be cellular can
be estimated only on the ground of the comparative number
of known cryptogams which are cellular, as over against
TYPES OF INDUCTIVE INFERENCE 187
the total number of cryptogams, both cellular and vascular,
previously observed. 1
II. TJie Method of Analogy. Here, also, we start with
the experience that A is characterized by the mark B.
But there is additional knowledge of which we may avail
ourselves in the generalization of some past experience
already effected, such as the following : that A very closely
resembles C, in that the two have many properties or attri
butes in common. The inference by analogy is that C also,
as well as A, will have the mark B. It may be that we
cannot examine C in a number of various instances to see in
how many the mark B occurs. Our only resource is the
inference which is based upon the known resemblances, or
analogies. This kind of inference, for example, was em
ployed by Sir Isaac Newton in a very interesting manner.
He had observed that certain "fat, sulphureous, unctious
bodies," such as camphor, oils, spirit of turpentine, amber,
etc., have refractive powers two or three times greater than
might be anticipated from their densities. He noticed also
the unusually high refractive index of diamond, and from
this resemblance, based upon similarity in reference to one
attribute only, he inferred that diamond also would prove
to be combustible. His prediction in this regard was veri
fied by the Florentine Academicians in 1694. 2 Brewster
made a striking comment upon Newton s inference, to the
effect that if Newton had drawn a like analogy in refer
ence to greenockite and octahedrite as he did concerning
diamond, inasmuch as they, too, have a very high refrac
tive index, he would have been wholly incorrect. This is
an indication of the fact that argument by analogy is not
conclusive.
Bosanquet has very strikingly expressed the essence of
the analogical method in saying that " in analogy we weigh
the instances rather than count them." 3 The distinction
1 Jevons, Principles of Science, pp. 146, 147. 2 Ibid. p. 527.
8 Bosanquet, The Essentials of Logic, p. 155.
188 INDUCTIVE LOGIC
between analogy and enumeration of instances lies in this,
that in the former we count similar attributes in the con
tents of two instances, and balance them against the dis
similar or unknown. In induction by enumeration we count
similar instances, considering them in their totality without
examination and comparison of their respective attributes.
III. TJie Method of Scientific Analysis. The instance in
question, A, which is characterized by the mark B, is sub
jected to a vigorous analytical examination, to show that A
and B are related through a causal connection. This analy
sis is effected either through a minute observation or by
means of exact experiment. The end to be attained by such
analysis is to separate a complex phenomenon into its
several elements, by which process a causal connection may
be revealed, whose very existence is disguised by the com
plexity of the phenomenon. For instance, the phenomenon
of death following the taking of arsenic is an event so com
plex as to evade a precise determination of the causal rela
tion. When analyzed into simpler elements, it is found
that the immediate effect of arsenic upon the bodily tissues
is to harden them so as to prevent their normal functioning.
This is the causal ground of the death due to arsenic.
Moreover, this analytic process, which may be appropriately
called a material one, is supplemented by a formal process
of negation ; that is, an instance in which the suspected
causal element is absent in the complex phenomenon under
investigation, and the related effect, before observed, now
no longer appears. This formal process acts as a check,
and as a verification as well, of the material analysis of the
phenomenon. For example, an antidote, as sesquioxide of
iron, being administered, no death from arsenic occurs ; and
it is also observed that no hardening of the tissues has
resulted, therefore the former result, hardening of tissues
producing death, has been thus corroborated negatively by
the fact that where no hardening of tissues has resulted,
death does not follow.
TYPES OF INDUCTIVE INFERENCE 189
We see at once the advantage of such a method over that
of counting all instances where taking of arsenic has caused
death. The latter is a phenomenally adjudged result; the
former penetrates with analytic insight to the ground of the
phenomenon itself. Thus one instance, if its parts and
their manifold relations are adequately comprehended, may
suffice for a universal conclusion based upon it. It is true,
however, as remarked by Bosanquet, that " number of ob
servations does, as a rule, assist analysis and contribute to
eliminating error. Scientific analysis as such, however,
does not deal with instances, but only with contents." l
In cases where the phenomenon does not reveal its com
ponent elements under observation, and it is impossible to
subject it to experiment, the most likely cause of the effect
in question is tentatively judged to be the real cause, until
it can be verified in reality. This is procedure by hypothe
sis, and is always resorted to as preliminary to a subsequent
experiment which is its test, or else in lieu of such an ex
periment when it is by the nature of the case precluded.
It is a form of ideal analysis. The experiment is constructed
mentally. The phenomenon is separated into what we would
reasonably imagine its simpler elements would be. We are
constrained to believe that if the hypothetical antecedent
existed, it would be adequate to produce the effect. Al
though rising in the sphere of the imagination, it is that
with which the mind is, for the time at least, satisfied as
an explanation of the facts which demand some cause to
account for them. Kegarding induction as a process of
reduction, hypothesis is the assumed universal or middle
term, which will necessitate the phenomenon under investi
gation as its logical conclusion.
We will now proceed to a further examination of these
methods, considered both singly and together.
1. They all proceed upon the supposition that what is
given in consciousness has some necessary ground for its
1 Bosanquet, Logic, Vol. II, p. 118.
190 INDUCTIVE LOGIC
being. In enumerative induction, there is some causal con
nection presupposed, yet in a very general and indefinite
manner, and accompanied by no analysis of the various
concepts either by a systematic observation or experiment.
It is a vague sense of uniformity, which, when observed
for many times, we feel will continue indefinitely. That
which has happened often and not contradicted carries
with it a certain convincing power by dint of bare repeti
tion, especially to persons of narrow experience, and un
accustomed to discriminating observation. Ueberweg has
made the following comment in reference to the so-called
imperfect induction. "The conclusion is made universal
with more or less probability, and the blank which remains
over in the given relations of spheres is legitimately filled
up partly on the universal presupposition of a causal-nexus
in the objects of knowledge, partly on the particular pre
supposition that in the case presented such a causal-nexus
exists as connects the subject and predicate of the conclu
sion. The degree of probability of the inductive inference
depends in each case on the admissibility of this last presup
position, and the various inductive operations, the extension
of the sphere of observation, the simplification of the ob
served conditions by successive exhaustion of the unessen
tial, etc., all tend to secure its admissibility." l
Analogy likewise proceeds upon the assumption of an
underlying cause among the observed phenomena, and this
is more definitely in the foreground throughout the process
than in that of induction by enumeration. Analogy is based
upon the postulate that similar phenomena have similar
causes ; the greater the agreement of the various attributes
of the different phenomena compared, the greater will be
the resultant probability that causes capable of producing
them as effects will be similar. The similarity of the light
ning flash to the electric spark suggested to Benjamin Frank
lin the possibility that they were due to a like origin, and
1 Ueberweg, Logic, pp. 483 f.
TYPES OF INDUCTIVE INFERENCE 191
by experiment his analogical reasoning was actually con
firmed, as is well known. Upon the theory that the world
as it exists for us in knowledge forms a system, to some
place in which every phenomenon given in experience must
be appropriately and necessarily referred, it follows there
fore that a simple experience devoid of any complexity of
parts may fit into several possible places in our world of
consciousness, and remain so far forth indeterminate. How
ever, a complex phenomenon, with many parts intricately
connected, will fit into one unique place only in the system
to which it must be referred. It is like a key that will fit
into only one lock. The presumption therefore is that any
other phenomenon which resembles the first through much
of its entire content, part for part, attribute for attribute,
will also resemble it further as regards other attributes not
yet examined, so that it will likewise fit into the peculiar
place in the system of knowledge to which the first has been
found to belong. There is always a strong probability that
agreement in spheres of great complexity is not a mere
coincidence, but the result of a causal relation. One charac
teristic of a system, which we have found to be the ground
of inference generally, is the coordination of like things
under one concept. Analogy therefore is based upon the
view of causal connections within the system which com
prises the world as given in consciousness.
In the third method, the causal relation is more promi
nent still, and the search for it characterizes the procedure
employed. That which in the other methods may exist
merely as a vague impression is here formulated and made
the direct and sole object of research.
2. The three methods in the order here presented show
an increasing prominence given to the causal connection in
the phenomena of experience. And therefore they possess
a relatively increasing scientific value. As the first has
only indirect reference to the causal connection of its facts,
it is the least trustworthy and has no claim as a scientific
192 INDUCTIVE LOGIC
method. It breaks down as soon as an exception is noted ;
and is even weakened by the fact that it is constantly men
aced by the possibility at least of the appearance of an ex
ception. " How do we know," says Green, " that the instances,
with the examination of which we are always dispensing on
the strength of the rule (that is, our generalization), might
not be just what would invalidate it, if they were exam
ined ? " * We may arrive at the conclusion, based upon our
observation and consequent record, that all sheep are white,
and yet black sheep do occur, in every flock, as the prov
erb has it. According to Aristotle, the proposition that
all swans are white, was a perfectly general one, and yet in
recent times black swans have been discovered in Australia.
Bacon s criticism upon this method has become classic :
" Inductio quae proeedit per enumerationem simplicem, res
puerilis est et precario concludit et periculo exponitur ab
instantia contradictoria et plerumque secundum pauciora
quam par est et exiis tantummodo quae presto sunt pro-
nunciat." 2
The validity of this method of procedure depends largely
upon the probability of our meeting and noticing exceptions
were they to occur. As Lotze puts it : "A man who never
observes a place of public resort but once in every seven
days, and that on a Sunday afternoon, has no right to sup
pose, because it is crowded then, that it is as crowded on a
week-day." * He is here in no position to note the excep
tions even should they occur.
Analogy, unless confirmed by experiment, or upon the
ground of resemblance established by a verifiable hypothesis,
has no claim to be considered as a scientific method. There
may be false analogies depending upon surface resemblances.
A child might conclude that oil would put out fire because
it so closely resembles water, which he knows can extinguish
the flames. The difference between essential and accidental
1 Green, Philosophical Works, Vol. II, p. 282.
2 Novum Organum, i. 105. 8 Lotze, Logic, p. 343.
TYPES OF INDUCTIVE INFERENCE 193
agreement between phenomena can be revealed only when
the underlying cause is ascertained.
The third method alone has scientific worth. True in
duction must be a continued search to discover a causal
relation.
~3. The two first processes fulfil their functions largely
as tentative and suggestive methods. In enumeration of
instances, we are often led to note resemblances which
become the basis of analogy. And analogy suggests, in
turn, hypothesis which is capable of verification through
subsequent experiment.
The question may be put, " Which of the three processes
is induction proper ? " The fact is that it may involve all
three, but it is not complete until it reaches the third, the
experimental method. Analogy is especially fertile in sug
gestion. Scientific minds most carefully trained and versed
in scientific methods of research are often most keen in
noting resemblances, and detecting analogies which become
the basis of their experiments. Newton possessed that rare
insight which, in spite of the manifest dissimilarity of the
two phenomena, could yet discern an essential likeness be
tween the fall of an apple and the gravitating force of the
moon toward the earth.
4. It is also to be observed that the choice of method will
depend largely upon mental habit. Some minds naturally
or by special training and surroundings are given to experi
ment. They have a testing facility and inventive capacity.
Others naturally are susceptible in an unusual degree to
contrasts and resemblances. Others again are accustomed
to accurate observation that is ever pushing beyond and
seeking to extend its sphere. Thus we have a natural divi
sion of these methods according to psychical proclivities.
The choice of method is often conditioned by the force of
circumstances. Experiment is not always possible. Are all
crows black ? There is no connection between the general
organism of the crow and its color that has thus far been
194 INDUCTIVE LOGIC
revealed through analysis or experiment. The only recourse
is to number instances over the widest possible field. We
say, moreover, that Mars may be inhabited ; for it has an
atmosphere similar to the earth and therefore capable
of sustaining life. Analogy is the only guide in such a case,
and it is impossible to verify it either by observation or
experiment.
5. All the methods tend to one end, that of effecting a
generalization of experience. The generalization may be
either a numerically general one, or one expressed in terms
of a generic concept.
(1) The former consists in the extension of several in
stances to their repetition under like conditions.
(2) The second consists in the extension of several in
stances to all cognate species under the same genus.
Examples of these two kinds of generalization are as
follows: The general proposition that all sulphur is com
bustible is of the former kind ; all instances are substantially
of the same nature, and do not differ as distinguishable
species under the same genus, but rather a repetition of
like phenomena. The general concept in the above propo
sition is of the nature of an infima species. On the other
hand, the proposition that all mammals are vertebrates, has
the subject-term in form of a generic concept. Many spe
cies, differing widely among themselves, may be embraced
under it. 1
l Sigwart, Logic, Vol. II, pp. 310, 311.
CHAPTER IV
CAUSATION
WE have seen that induction as a truly scientific method
consists in the analytical determination of the relations of
cause to effect in any complex phenomenon, accompanied
by a generalization of the result obtained. The final out
come of such a process is a universal concept which em
bodies a law, expressed in terms of a constant connection
between antecedent and consequent. As Green has said,
" The essence of induction consists in the discovery of the
causes of phenomena."/ A causal view of the universe
gives rise to logical concepts, whereas a mythological view
of the universe, as in ancient times, resulted in mere empiri
cal concepts, which gave no assurance either of stability or
invariability. It will be necessary therefore to determine
more precisely the logical significance of the causal idea,
which seems to underlie all inductive inference. This is no
easy task. According to Clifford, "cause" has sixty-four
meanings in Plato, and forty-eight in Aristotle. 2
The causal idea has sometimes found expression in the
phrase, the uniformity of nature, or it is often referred to
as the doctrine of universal causation. These two phrases
are often used interchangeably ; this gives rise to confusion
of thought, for their meanings are quite distinct.
Uniformity of nature, strictly interpreted, means that like
antecedents, under precisely the same conditions, will be
followed by like effects ; this idea expresses one phase of
causation, viz. its invariability.
1 Green, Philosophical Works, Vol. I, p. 284.
2 Clifford, Lectures and Essays, Vol. I, p. 149.
195
196 INDUCTIVE LOGIC
The doctrine of universal causation, however, expresses
the impossibility of phenomena rising spontaneously, with
out an antecedent, or antecedents, sufficient rationally to ac
count for them. The two ideas lie at the root of the causal
idea. As Tennyson has put it :
For nothing is that errs from Law.
Some confusion has also arisen from the failure to discrimi
nate precisely between the philosophical and the purely
logical questions relative to the general subject of causa
tion. Causation may be viewed from three different points
of view :
1. What it is phenomenally, that is, as regards its physi
cal aspects.
2. What it is essentially, as regards its real nature. This
is a metaphysical question.
3. What it is in respect to its characteristic attribute of
invariability. This is a purely logical question.
(1) As to the first, what is causation phenomenally?
What is its purely physical significance ? Investigations
in this line have led to the doctrine of the conservation of
energy. This is substantially the assertion that, in every
event, no new energy is called forth which did not exist
before potentially at least, nor can any energy be ultimately
lost; nothing new is created, there is only a change or
transfer from one state or condition to another. Moreover,
the sum total of energy in the universe is a constant quan
tity ; it can neither be added to, nor subtracted from. There
is an excellent illustration of this theory in the admirable
chapter on " Conservation of Energy " by Professor Tait.
I give it somewhat in full : " I allow an electric current to
pass through a galvanic battery, and there is for the moment
a certain quantity of zinc consumed, or, as we may put it, a
certain quantity of potential energy in the battery has been
converted into the kinetic energy of a current of electricity.
That current of electricity passes round some yards of cop-
CAUSATION 197
per wire, coiled round a bar of iron or a number of fine iron
wires which are standing vertically inside this apparatus.
The moment the current passes, these iron wires are con
verted into magnets, but, in consequence of the conservation
of energy, while this is going on they weaken the current.
The current of electricity becomes weaker in the act of
making the magnet, but the moment the magnet springs
into existence, it again is weakened, because, from the
necessities of its position, its mere coining into existence
necessitates the passage of a new current of electricity in
another coil of wire which surrounds this externally, and
finally this last current we can use to produce heat, or light,
or sound." 1 In this cycle of changes we see how closely
connected even disparate phenomena are, and how the ap
pearance of energy in any one definite state is dependent
upon its previous existence in some other state. The
doctrine of conservation of energy, we shall see later on,
may be suggestive as to the nature of the analytical treat
ment of cause and effect.
(2) The philosophical question as to the inner nature of
causation met with one answer generally until the time of
Hume; namely, that the idea of cause signified that the
antecedent was efficient in producing the corresponding
consequent, implying the transfer of power sufficient to
bring about the effect. Hume, however, contended that in the
greatest possible extent of our knowledge, all that we cer
tainly know is this, that one event follows another. We have
no ground for an assertion concerning the manner in which
the sequence is effected, nor for assuming any real tie be
tween them. Hume insisted that phenomena were conjoined,
but never connected. 2 His opponents, as Kant and others,
deny him, however, his fundamental position, that the
origin of the causal concept comes from experience alone.
They urged that it has an a priori origin, a concept simple
1 Tait, Recent Advances in Physical Science, pp. 76, 77.
2 Huine, Essay on Idea of Necessary Causation.
198 INDUCTIVE LOGIC
and unanalyzable, given through intuitive insight ; developed
in the sphere of experience, but not dependent upon expe
rience for its warrant. It is an interesting fact that the
idea of the conservation of energy developed subsequent
to Hume s time. It seems to give evidence which Hume
insisted was not and could not be forthcoming; namely,
concerning the idea of the antecedent as an efficient power.
Through the modern doctrine, the impression of a transfer
of real power is produced, though its mode and manner still
remain a mystery.
(3) The logical aspect concerns not the phenomenal
manifestation of cause and effect, nor their inner nature,
but rather the element of invariability in causation. Two
questions here suggest themselves : First, Is invariability a
fact, a constant element in causation ? Second, How do
we account for its existence ? The first only has truly
logical significance. The invariability of causation, that
like antecedents under precisely the same conditions pro
duce like effects, alone makes induction possible. Mill says
that it is the belief in the uniformity of nature which stands
as the ultimate major premise in every process of induction.
Hume accepted it, and based inferences upon it, and never
challenged it as a working basis as regards the affairs of
everyday life. He acknowledged the element of invaria
bility, and only denied the bond of connection. This ele
ment has peculiar logical significance : without it, it would
be impossible to extend our knowledge beyond the seen and
the heard, indeed that which is seen and heard would then
have no meaning, and no basis for their interpretation and
appreciation. Being assumed, however, as a logical postu
late, we have a basis for induction, a constant to be sought
for and to be depended upon, in explanation of the past and
in prediction of the future.
When we come to the second question, which is essen
tially a genetic one, how the belief in the uniformity of
nature arose, we find two classes which answer respectively
CAUSATION 199
that the belief arose a priori, and on the other hand, from
experience simply. The former is the opinion especially
associated with the Scottish School of philosophy. Hume
holds that it proceeds from a psychological law of custom
or habit, an unbroken line of mental associations induc
ing a belief within, concerning the uniformity of nature
without. Mill has also a like empirical basis for a belief
in the uniformity of nature ; he holds that having observed
uniformity in many experiences, in fact never contradicted,
we generalize so as to cover a sphere beyond our experience.
Moreover, we possess the consensus of testimony, coexten
sive with the history of humanity, of the indefinitely wide
extent of the sphere of causation, and the accompanying
characteristic of uniformity. His position is fortified by
the fact that in the process of incomplete induction, its
probability is strengthened where there has been exception
ally abundant scope for observation, so that there is the
overwhelming conviction that if there had been a time or
place where the law would prove untrue, it would have been
noticed. Instead of universal causation, Mill and his fol
lowers make a more cautious statement, causation as
coextensive with the sum total of human experience. This
is abundantly adequate to embrace all possible circum
stances of practical inference. The immensely high degree
of probability engenders a subjective certitude which in
everyday conduct of affairs, and even in the more exact
requirements of scientific investigation, is never questioned.
Preyer has given an interesting account of the extremely
early appearance of the appreciation of the causal relation
in the case of his child, " who, at the three hundred nine
teenth day of its life, struck several times with a spoon
upon a plate. It happened accidentally, while he was doing
this, that he touched the plate with the hand that was free ;
the sound was dulled, and the child noticed the difference.
He now took the spoon in the other hand, struck with it on
the plate and dulled the sound again, and so on. In the
200 INDUCTIVE LOGIC
evening the experiment was renewed with a like result.
Evidently the function of causality had emerged in some
strength, for it prompted the experiment. The cause of the
dulling of the sound by the hand was it in the hand or in
the plate ? The other hand had the same dulling effect, so
the cause was not lodged with the one hand. Pretty nearly
in this fashion the child must have interpreted his sound-
impression, and this at a time when he did not know a single
word of his later language." l
The theoretical soundness of Mill s speculations, however,
has a flaw, although the practical results may not be thereby
invalidated. The inductive process, which is supposed to
be a truly scientific method, and superior to induction by
simple enumeration must, according to Mill, at the last
analysis, rest upon a principle which is itself based upon an
incomplete induction. A very fair and searching criticism
of Mill is that of Venn s in his Empirical Logic. 2 Whately
insists that the whole question concerning the nature of our
belief in uniformity is irrelevant, as it is a purely psycho
logical and not a logical one. Mansel holds a mediating
position in insisting that the idea of universal causation is
intuitive, while that of uniformity is necessarily empirical.
Sigwart has very trenchantly criticised Mill in that " taking
away with one hand what he gives with the other, he shows
in the uncertainty of his views the helplessness of pure
empiricism, the impossibility of erecting an edifice of uni
versal propositions on the sand-heap of shifting and isolated
facts, or, more accurately, sensations; the endeavor to ex
tract any necessity from a mere sum of facts must be fruit
less. The only true point in the whole treatment is one in
which Mill as a logician gets the better of Mill as an
empiricist; namely, that every inductive inference contains
a universal principle ; that if it is to be an inference and
not merely an association of only subjective validity, the
1 Preyer, The Senses and the Will, pp. 87, 88.
2 Venn, Empirical Logic, p. 130.
CAUSATION 201
transition from the empirically universal judgment All
known A s are B to the unconditionally universal All that
is A is , can only be made by means of a universal major
premise, and that only upon condition of this being true
are we justified in inferring from the particular known A s
to the still unknown A s." l
The whole tendency of the modern logic is to base the
causal postulate upon a ground which is epistemological ;
namely, inasmuch as our knowledge is one and self-con
sistent throughout all its separate elements, there must be
a corresponding invariability in the phenomena themselves,
as there is in the system of knowledge which results from
the interpretation of these phenomena. This is the general
view of Sigwart, Bosanquet, Lotze, and Green. 2
This view may be considered also as an expression of the
Law of Sufficient Reason; namely, that there is an inherent
characteristic of intelligence which demands that every
element of consciousness must be referred to some other
element for its explanation, and that it is only when the
logical connection of ideas corresponds to a real causal con
nection, that the mind discovers a reason for its several
experiences which is satisfying. It has been said by Ueber-
weg, as giving expression to this view : " The external in
variable connection among sense phenomena is, with logical
correctness, explained by an inner conformability to law,
according to the analogy of the causal connection perceived
in ourselves between volition and its actual accomplishment." 3
There is a distinction that is of importance to note be
tween the popular and the scientific idea of cause. The
former is the outcome of the supposition that whatever
immediately precedes the effect has evidently produced it,
and that this is sufficient wholly to account for it. Such
1 Sigwart, Logic, Vol. II, p. 303.
*Ibid., Vol. II, pp. 119, 120 ; Bosanquet, Logic, Vol. II, pp. 220, 251:
Lotze, Logic, p. 68; Green, Philosophical Works, Vol. II, p. 286.
a Ueberweg, Logic, pp. 281, 282.
202 INDUCTIVE LOGIC
an idea of causes leads, at the best, but to a loose and super
ficial determination of the relation between any antecedent
and its consequent, and there is the danger, moreover, of a
hasty inference which results in the fallacy of post hoc ergo
propter hoc. In order to attain a true view of causation, we
must especially attend to the extreme complexity of the
causal connection. There is no such thing as a simple
cause followed by a simple effect. The cause is always a
combination of several elements, circumstances, and condi
tions ; the effect is always manifold. This characteristic
has been admirably presented in Mill s chapter on the
"Plurality of Causes and the Intermixture of Effects." 1 It
is well known that the variation in the height of a barometer
is due partly to the variation of the atmospheric pressure,
and partly to the variation of the expansion of the mercu
rial column due to heat. In exact determination, some
experiment or calculation must precede, before there can be
a discrimination between the elements of the joint effect.
And so also, a number of circumstances may combine to
restore an invalid to health, no one of which alone being
capable of effecting his recovery.
The cause of any phenomenon has been defined by Mill,
as also by Brown and Herschel, as the sum total of all its
antecedents. This statement has been criticised, inasmuch
as the sum total of all antecedents is indeterminate, and
that there is no end to the possible ramifications in all
directions which an exhaustive analysis of any complex
cause will yield. However, the problem is one of reduction
to simplest possible terms within the range of our powers of
observation and experiment. There is much in the sum
total of all the antecedents of any given effect which is
irrelevant. It is the peculiar function of logical analysis to
discriminate between the relevant and irrelevant. The
temperature of the laboratory will not affect, one way or
the other, experiments with falling bodies ; but will essen-
i Mill, Logic, Book III, Chap. X.
CAUSATION 203
tially influence certain chemical experiments, and must
enter as one of the determining factors in the sum total of
antecedents. It may be that certain elements of a complex
whole may seem to us ultimate and unanalyzable, and yet be
themselves systems of more or less complexity. There is
always a limit to analysis, both experimental and mental.
The analysis is to extend to the ultimate parts as far as
possible. It is not an exact process, but a process which
tends to exactness to the extent which the scope of finite
intelligence will permit. The reason is not at fault so much
as the natural limitations of observation and experimental
analysis. The end of our research in causal analysis is to
discover an invariable relation that can be expressed in the
form of an hypothetical universal, If A, then B.
In order to effect this, the complex A must be separated
into its parts, a, b, c, etc., and the effective, and necessary,
and indispensable element producing B must be determined.
Suppose it proves to be a, it may be possible to subject this
to further analysis, and to reduce it to simpler elements, such
as x, y, z, etc., and x be found as the significant element of
the real cause. Each analysis determines a narrower and
still narrower sphere within which the cause lies. A man
is shot. We say the bullet killed him ; then the driving
force behind the bullet; then the explosive power of the
gunpowder ; this in turn was occasioned by the combined
chemical and mechanical energy of its ingredients whereby
a solid is transformed into a gaseous substance many times
its original bulk.
Sooner or later we must reach the end of our analysis,
and the investigation be necessarily checked. No explana
tion is ultimate ; we only transfer our point of view from a
less to a more familiar sphere of interpretation. We do not
feel the need of explaining the very familiar ; though the
most familiar is hardest satisfactorily to explain, because
there is nothing simpler in whose terms we may paraphrase
it. We feel this in giving a definition of terms whose
204 INDUCTIVE LOGIC
meaning we best know, and which we most frequently use.
Mr. Barrett, a former assistant at the Royal Institution,
said of Faraday : " I well remember one day when Mr.
Faraday was by my side, I happened to be steadying, by
means of a magnet, the motion of a magnetic needle under
a glass shade. Mr. Faraday suddenly looked most impres
sively and earnestly, as he said : How wonderful and
mysterious is that power you have there ! The more I think
over it, the less I seem to know. 7 And yet, he who said
this knew more of it than any living man." 1
Although our knowledge is limited as in all cases of
causation however simple, nevertheless, as far as it goes,
the several elements are related logically, that is, necessarily
and universally. We may only know in part, but still we
know, and the world, as interpreted for us in knowledge, is
a world of invariable sequences. The process of inductive
analysis, therefore, consists in reducing a complex antecedent
to its ultimate parts, in order to reveal the element or ele
ments in it which may have caused the given effect. It some
times happens that different elements in an antecedent may
be considered severally as the cause, according to the psycho
logical point of view as regards the interests of the investiga
tor. It is not always that a scientific determination of the
cause is required ; it may be that all that is desired is a
knowledge of that part of the antecedent which is most
closely and prominently connected with the event in
question. An inquiry may be started in reference to the
cause of an epidemic in a community. One may discover
the cause in the carelessness of sanitary engineers ; another
may say the cause lies in the poor construction of the
sewerage ; another says that the cause of the epidemic is a
certain kind of bacilli. Each one is looking at the chain of
events related as cause and effect; but they all look at
different links of the same chain. One element, therefore,
of a complex antecedent may be brought into more or less
1 Gladstone, Michael Faraday, p. 180.
CAUSATION 205
prominence as the efficient element of the cause, according
as the point of view is shifted. If, in the search for the
cause of phenomena, the sum total of antecedents were
always given exhaustively, the explanation might become
so loaded down with details as to burden the mind, and
confuse rather than clear the understanding.
CHAPTER V
THE METHOD OF CAUSAL ANALYSIS AND
DETERMINATION
IT will be well to consider the various cases which will
confront us in seeking to analyze a complex antecedent for
the purpose of discovering its cause.
1. There are instances where cause and effect appear in
evident sequence. There is an antecedent which is fol
lowed by a consequent. If A happens, then B will happen.
Instances of this kind most readily yield themselves to the
process of analysis, because a change in any given phenom
enon is occasioned by the efficiency of the antecedent which
may be observed in connection with the change itself. It
is easier to note active than passive relations, the dynamic
rather than the static. The attention is attracted and held
by change. The bird flying across our path is observed,
and the one perched upon the tree near at hand, however
conspicuous may be its position, is passed by without any
notice taken of it. It is easier to connect the moisture of
the grass with falling rain, than when the same is occa
sioned by the dew. In one case, the causal relation is ex
hibited in operation ; in the other, the connection is veiled.
We find the grass wet ; what preceded it we are not able to
see. There are several instances of sequence among ob
served phenomena which must be carefully discriminated
in order to avoid confusion of thought. They are as
follows :
(1) When we have A followed by B, and A ceases wholly
while B endures for an appreciable time afterwards, or it"
may be permanently. A billiard ball strikes another, the
206
CAUSAL ANALYSIS AND DETERMINATION 207
second goes on by virtue of the newly acquired energy
transferred by impact from the first, which, however, stops
altogether. I throw a ball which lodges on the top of a
building; the effect produced lasts permanently, for the
ball has gained a gravity potential due to the energy im
parted to it by the initial throwing. The old formula,
therefore, does not always hold: "Cessante causa cessat
effectus."
(2) Cases where A ceases, and thereupon B immediately
ceases also, If we cut off the supply of gas which feeds
a flame, the flame at once disappears. There are cases,
however, when an appreciable time must elapse in order
that the transferred energy in the effect may be dissipated.
When we shut our eyes the stimulus causing the percep
tion is cut off, and the perception at once is at an end ;
however, there are cases where the stimulus being very
strong, after-images are induced which remain for some
time in the dark field after the eyes are closed.
(3) Cases where the antecedent is wholly inadequate to
produce the effect, but whose function is merely to liberate
potential energy already stored, and waiting an occasion
for its active manifestation. A slight blow upon a piece
of dynamite causes an explosion wholly disproportionate
to the striking force employed. As is well known, heat is
often an exciting cause of chemical action. In such cases
the real cause is more or less concealed, while that which
is apparent upon the surface is not a cause so much as an
occasion of the phenomenon in question. I touch the pen
dulum and a clock starts and so continues for many hours ;
the swinging pendulum, however, is only the occasion of
liberating the potential energy of the wound-up spring,
and thence the power which runs the clock, pendulum,
wheels, hands, and all.
2. We have also instances not so much of sequence as
of_ concurrence. The planets revolve around the central
sun; here the cause is constant, attended by constant
208 INDUCTIVE LOGIC
effect. The machine never runs down, nor has to be
wound up.
3. Again there are instances of coexistence. These are
more difficult to analyze, for the phenomena do not here
appear as antecedent and consequent in the midst of chang
ing conditions and circumstances. We have coexistence
of two kinds :
(1) Coexisting attributes in one and the same organism.
They are always found together. They form one generic
concept and are called by one name. Cows have horns,
cloven feet, are ruminant, etc. Dogs have their distinct
and constant characteristics. The orange has its correla
tion of color, taste, smell. And so we have the so-called
"natural kinds," i.e. organisms presenting an unique and
characteristic appearance, differentiated thereby from all
others. There are also certain correlations of growth which
present a constant relation between certain attributes, as
the fact, however we may explain it, that cats with blue
eyes are invariably deaf. There are, moreover, illustrations
of the same in an inorganic sphere, as the law which con
nects the atomic weight of substances and their specific
heat by an inverse proportion; or that other law which
obtains between the specific gravity of substances in the
gaseous state, and their atomic weights, they being either
equal or the one a multiple of the other. In many cases,
the bare fact of coexistence must be accepted without being
able to explain the causal ground of it. The several ele
ments present a constant association, and that is all that
can be said about it. In other cases, however, a cause may
be found, for instance, as regards the correlation of warm
blooded animals always possessing lungs ; the connection
between respiration and the generation of heat is found to
depend upon chemical action as its causal basis.
(2) A relation of statics rather than dynamics, as, for
instance, a pillar supporting a roof or arch is said to be the.
cause, in the sense of the sustaining cause, of the super-
CAUSAL ANALYSIS AND DETERMINATION 209
structure. So also the cohesive force which holds together
the particles of a stone. In such cases the energy inherent
in the cause is of the nature of a stress and strain.
4. Under this head are embraced the phenomena of vital
growth or development. These are the most difficult of all
the causal problems to determine ; for it is required to dis
cover the inner necessity of essence, and how the succeeding
stages of development unfold through the play of the cen
tral forces inherent in the very nature and being of the
organism itself. Mill is content with classifying organisms
as different natural kinds, and he is not concerned with the
reason why there should be such and such kinds, nor does
he attempt to discover any law concerning these natural
correlations and the mode of their growth. In inductive
analysis, our concepts must not merely grasp what the natu
ral kinds are, but also what has determined them to be what
they are. Darwin puts special emphasis upon the environ
ment as affecting changes in organisms and producing dif
ferentiating modifications among species. This, however,
must be considered not as sole factor, but one which is com
bined with inner needs and necessities. Moreover, Darwin
has drawn attention to the fact that individual differences
need scientific explanation as well as the common attributes,
as, for instance, why some sheep are black, and why some
pigeons are fantailed and others are not. In all such con
siderations we must not lose sight of the fact that there are
two determining factors, the inner necessity of develop
ment, and the external necessity of causality, as organisms
are acted upon by their environment. 1
5. Cases of collocation where no one element of the cause
is efficient, but together they all combine to produce the
effect. In searching for the cause, we must not only find a
certain amount of energy capable of producing the effect,
but we must also discover what peculiar arrangement of the
elements concerned must exist before the energy in question
l Sigwart, Logic, Vol. II, pp. 322, 330, 331.
210 INDUCTIVE LOGIC
can become operative. Chalmers says that "the existing
collocations of the material world are as important as the
laws which the objects obey, that many overlook this dis
tinction and forget that mere laws without collocations
would have afforded no security against a turbid and dis
orderly chaos." * We would naturally say that the sole
cause of water boiling at 212 is the enveloping heat ; it has,
however, been observed that on top of Mont Blanc, water
boils at 180 instead of 212. This indicates that, in addi
tion to the fire, the atmospheric pressure is an element in
the cause, very easily overlooked. Charcoal and diamond
are of the same substance ; a difference only in the arrange
ment of the molecules results in such radically different
combinations. There are, in the main, three special kinds
of collocations, as follows :
(1) Cases of modifying circumstance. A strong wind
blows down a tree ; this would not have occurred had not
the tree been hollow. The hollowness of the tree is here a
cooperative circumstance that is combined with the efficient
cause, the force of the wind. An instance where arrange
ment of the elements concerned rather than their efficient
energies is productive of the effect, is that of capillarity, the
rising of liquid in a tube of exceedingly small bore. Here
form is more essential to the effect than the expenditure of
any visible energy.
(2) Cases in which certain negative conditions prevent
the realization of the effect. The plants and shrubs die in
a long drouth, because it does not rain. A train collides
with another, because the red signal was not exposed as it
should have been. A match will ignite gunpowder gener
ally, but it fails to do so should the powder prove to be
wet.
(3) There are also cases of counteracting causes, where
the effect of cause A is not realized, as cause B neutralizes
the force of cause A ; as when an anchored boat will not
1 Quoted by Jevons, Principles of Science, p. 740.
CAUSAL ANALYSIS AND DETERMINATION 211
respond to the pull of the oar. Sometimes the cause is not
wholly counteracted, or it may be the counteracting cause
more than holds the positive cause in check, and is itself
operative. The rise of a balloon in the air is due to the fact
that the force of gravity is more than overbalanced by the
expansive force of the gas within the balloon ; one force pull
ing downwards, the other bearing up, and the latter pre
vailing.
Mechanical forces acting in combination admit of a reso
lution of their joint effect according to the theory of the
parallelogram of forces. Chemical and vital forces cannot
be treated in such a way at all. From the character of the
elementary forces in mechanics, one can calculate their com
bination. In chemistry, however, when the elements are
given, the resulting compound cannot be thus determined.
So, also, in vital and mental phenomena, the necessarily com
plex nature of the elements involved prevents not only
prediction of resulting combinations, but even adequate
explanation of that which may be immediately given in
consciousness.
It is necessary, in connection with these various instances
of causal relations, to understand the different modes of the
transfer of energy, which are as follows :
(1) Molar or mechanical, as in the case of a billiard-ball
transferring its energy to another through impact.
(2) Molecular, as heat, chemical and electrical and mag
netic forces, light, etc. One passes into another, as chemical
force producing electric, electric producing magnetic, or
producing heat and light.
(3) Cases where mechanical force becomes molecular, as
friction inducing heat; or cases where molecular becomes
mechanical, as heat transferred into the driving power of an
engine, or electricity applied as a motor. A precise deter
mination of equivalents can be made between molar and
molecular energy; as, for example, it has been found that
it takes the same amount of energy to raise 772 pounds a
212 INDUCTIVE LOGIC
distance of one foot that it does to raise the temperature of
one pound of water 1 F. ; or the heat requisite to boil a
gallon of freezing water would lift 1,389,600 pounds through
a distance of one foot.
As a consequence of the doctrine of the transfer of energy,
a causal law can be so stated as to express the fact that
variations in the antecedents will call for the corresponding
variations in the effect, as, for instance, such a law as the
following: " Resistance in a wire of constant section and
material is directly proportional to the length and inversely
proportional to the area of the cross-section." * The neglect
of quantitative determination of the proportionate variations
of the antecedent and consequent was a glaring defect in the
inductive systems both of Mill and of Bacon.
Through the representation of the various stages of such
variation, it is also possible to establish the upper and lower
limits beyond which the cause does not produce the corre
sponding effect ; as in Weber s law concerning the relation
of stimulus to sensation, that stimulus must increase geo
metrically in order that the sensations increase arithmeti
cally. There is an upper and lower limit beyond which this
proportion does not hold.
The doctrine of conservation of energy creates the im
pression of continuous change in causation, in which the
effect unfolds out of the cause. We do not think of phenom
ena under this aspect as discrete events. More than ever,
in the light of modern science, does the old saying obtain,
"Natura non facit saltum." We no longer look for catas
trophic results in nature, but regard causation as a con
tinuous transfer of potential energy into kinetic or actual
energy.
We come now to the consideration of the method by
which the causal analysis is mediated. This is effected
through observation and experiment. Observation is some
thing more than mere looking at phenomena : it means con-
l Jenkin, Electricitij and Magnetism ^ p. 83,
CAUSAL ANALYSIS AND DETERMINATION 213
centration of attention for the purpose of research ; it means
discriminating insight, an appreciation of likeness and
difference ; it means a penetration beneath surface appear
ances, and an apprehension of the essential features of the
objects of perception. Experiment consists in modifying
the elements which form the complex antecedent in order to
observe the resultant effect upon the corresponding conse
quent. Forces may be added or subtracted ; their intensity
may be varied, increased, or decreased ; the circumstances
or conditions may be altered. Herschel speaks of observa
tion and experiment, as passive and active observation
respectively. When we interfere to change the course of
nature, or to bring natural forces within the range of our
""observation, we are experimenting. Observation is prelimi
nary to experiment, and suggests the lines along which
experiment should proceed. An observation that sees the
parts in the whole and the whole in the parts, is in itself
an analysis of a phenomenon, in course of which process
causal relations must be disclosed. The scientific spirit
demands absolute veracity in observation. One ought not
to be blind to facts even though they tend to contradict
preconceived theories. Bacon has observed that " men mark
when they hit, never mark when they miss." We must
strive against a natural tendency to see things as we would
have them, and not as they strictly are.
We must also carefully distinguish between observed
facts, and inferences which we instinctively draw from
these facts. Observation is preliminary to an inductive
inference, therefore it must not itself involve an inference,
or we should be arguing in a circle. An interesting illus
tration of the difference between observation and inference
based upon it, is narrated in the life of Faraday: "An
artist was once maintaining that in natural appearances and
in pictures, up and down, and high and low, were fixed in
dubitable realities; but Faraday told him that they were
merely conventional acceptations, based on standards often
214 INDUCTIVE LOGIC
arbitrary. The disputant could not be convinced that ideas
which he had hitherto never doubted, had such shifting
foundations. Well, said Faraday, hold a walking-stick
between your chin and great toe ; look along it and say
which is the upper end. The experiment was tried, and
the artist found his idea of perspective at complete variance
with his sense of reality; either end of the stick might be
called upper, pictorially it was one, physically it was the
other." 1
This indicates how readily our inferences and observations
blend, and how difficult it is to separate them in conscious
ness. De Morgan has pointed out that there are four ways
of one event seeming to follow another, or to be connected
with it, without really being so :
(1) Instead of A causing B, our perception of A may
cause B. A man dies on a certain day which he has always
regarded as his last through his own fears concerning it.
(2) The event A may make our perception of B follow,
which otherwise would happen without being perceived.
It was thought that more comets appeared in hot than cold
summers ; no account, however, was taken of the fact that
hot summers would be comparatively cloudless, and afford
better opportunities for the discovery of comets.
(3) Our perception of A may make our perception of B
follow. This is illustrated by the fallacy of the moon s
influence in the dissipation of the clouds. When the sky is
densely clouded, the moon would not be visible at all ; it
would be necessary for us to see the full moon in order that
our attention should be strongly drawn to the fact, and this
would happen most often on those nights when the sky is
cloudless.
(4) B is really the antecedent event, but our perception
of A, which is a consequence of B, may be necessary to
bring about our perception of B. Upward and downward
currents are continually circulating in the lowest stratum of
1 Gladstone, Michael Faraday, pp. 165, 166.
CAUSAL ANALYSIS AND DETERMINATION 215
the atmosphere ; but there is no evidence of this, until we
perceive cumulous clouds, which are the consequence of such
currents. 1
There are certain natural limitations to observation, as
things too minute to be seen, too swift to be carefully exam
ined; there are sounds which some ears can detect, while
others cannot, and shades that some eyes cannot discriminate.
There are effects proceeding from certain causes that are so
slight that we fail to observe them, and yet erroneously infer
that they do not exist. Professor Tyndall has given a strik
ing illustration of the difference of auditory power in two
individuals ; he says : " In crossing the Wengern Alp in com
pany with a friend, the grass at each side of the path swarmed
with insects which to me rent the air with their shrill chirrup
ing. My friend heard nothing of this, the insect music lying
quite beyond his limit of audition." : Much has been done
by inventive skill to increase our powers of observation, and
at the same time to render them more accurate, as the tele
scope, microscope, the vernier for precise measurement of
minute differences of magnitude, the chronograph for time
measurements, self-registering thermometers, the thermopile,
galvanometers, etc. One of the chief problems of scientific
method is to overcome natural limitations of observation
through mechanical devices.
Observations on a large scale and over a considerable
period of time must sometimes be taken in order to disclose
tendencies as seen only in the average or the mean of the
observed results. Thus meteorological, vital statistics, and
others of a like kind, must extend over a large area, and
embrace a large number of instances in order to reach
results of any value. It is known that Tycho Brahe made
an immense number of most exact records of the positions of
the heavenly bodies with the aid of the best of astronomical
1 Quoted by Jevons, Principles of Science, pp. 409-411.
2 Tyndall, On Sound, pp. 73, 74.
216 INDUCTIVE LOGIC
instruments, and these records afterwards became the foun
dation of Kepler s laws and of modern astronomy. 1
The faculty for accurate observation can be increased by
acquiring the habit of examining carefully everything within
the field of vision. We fail to see many things because we
fall into the easy way of passing them by without noting
their presence or appreciating their significance. It was
said of Charles Darwin by his son that " he wished to learn
as much as possible from every experiment, so that he did not
confine himself to observing the single point to which the
experiment was directed, and his power of seeing a number
of other things was wonderful." 2 The open-eyed vision is
the prime requisite for scientific investigation.
The limitations of observation naturally lead to experi
ment, whose special function is to so modify phenomena as
to bring a suspected causal element more prominently into
notice. This can be done by intensifying the force in ques
tion, or by neutralizing all other elements in combination
with it, so that the sole effect of this force in actual opera
tion can be observed. When the cause is not a simple ele
ment, but a combination, then the problem is to vary the
conditions so that but one possible combination can be opera
tive alone, and note the corresponding effect. Given a
certain number of elements, the number of possible com
binations is mathematically determinate, and can be tried
seriatim until all possibilities are exhausted. Venn has
given a long and interesting illustration of this in his Em
pirical Logic? All combinations need not be tried, how
ever; for many will be seen to be either impossible or
irrelevant. The aim is to obtain an antecedent which shall
consist either of a simple element, or a combination such
that with its presence the effect in question is present also,
but with its disappearance the effect is wanting.
1 Gore, The Art of Scientific Discovery, p. 316.
2 Life and Letters of Charles Darwin, Vol. I, p. 122.
pp. 402 ff .
CAUSAL ANALYSIS AND DETERMINATION 217
It is not sufficient to note merely the presence of an ante
cedent connected with a corresponding consequent; scien
tific determination consists also in proving the absence of
the suspected cause in cases where the given effect is not
present. This is called determination by negation. A
proposition which is held affirmatively has only a vague
significance; it must be determined within definite limits
assigned to it by virtue of what it is not. Defining means
to set limits to a term ; these limits grow out of the nature
of the thing itself. The negative judgment marks a transi
tion always from that which is indefinite and incoherent
to that which is definite and coherent. 1
This may be illustrated in the concrete, when in dissec
tion one is tracing a nerve; it is followed throughout its
course by a series of negative judgments though they be
unexpressed : This is not a nerve, but an artery ; this is not
a nerve, but a vein ; this is not a nerve, but a filament, or
shred of muscle, etc. So we rise through negative discrim
ination to a clear apprehension of an object under investi
gation. The original proposition must be readjusted with
every new negative determination. It sometimes happens
that the original proposition is completely negatived by the
negative determination, sometimes again it is confirmed.
A proposition that has not been worked over through such
a process has no real logical worth or scientific value. There
fore in the analysis of phenomena when the suspected cause
and effect are combined in a proposition, it can at first be
held only tentatively. It must be confirmed negatively, or
else readjusted to conform to the negative requirements.
Suppose we have given that A is followed by B as far as we
have been able to observe. We may proceed by experiment
to multiply instances of A s connection with B, but still the
causal relation is not absolutely proved. We must go on
to show that in all cases of noWL there is not-J5, or in all
cases of not-B there is noWl. Negative experiment pro-
i See p. 74.
218 INDUCTIVE LOGIC
duces the contrapositive, or the converse contrapositive, of
the proposition under investigation, which deductively neces
sitates the validity of the original proposition.
This is substantially Mill s method of difference, that if
an instance in which the phenomenon under investigation
occurs, and an instance in which it does not occur, have
every circumstance save one in common, and that one oc
curring only in the former, the circumstance in which alone
the two instances differ is the effect or cause or a necessary
part of the cause of the phenomenon. This method will
be described later; it is the main inductive method, the
others being largely modifications of it. A negative in
stance which is established concerning relations of not-^4
and not-B is absolutely conclusive, inasmuch as not-^1 is
the contradictory of A, and not-B is the contradictory of B.
They are mutually exclusive. No other possibility can be
forthcoming, and the experimental analysis is exhaustive.
Professor Tyndall gives the following account of an experi
ment to determine the cause of resonance. " I hold a vibrat
ing tuning-fork over a glass jar eighteen inches deep ; but
you fail to hear the sound of the fork. Preserving the fork
in its position, I pour water with the least possible noise
into the jar. The column of air underneath the fork be
comes shorter as the water rises. The sound augments in
intensity, and when the water reaches a certain level, it
bursts forth with extraordinary power. I continue to pour
in water, the sound sinks, and becomes finally as inaudible
as at first." l
From this it is inferred that a certain column of water of
definite height is necessary to the production of the sound,
for above and below the limits no sound is heard. This
experiment also indicates that which is most important in
causal determination, a variation in cause accompanied by
a variation in effect, as also a maximum and minimum as
regards the intensity of the sound. Experiment proceeds
1 Tyndall, On Sound, p. 172.
CAUSAL ANALYSIS AND DETERMINATION 219
upon the supposition of the measurableness of phenomena,
and seeks numerically expressible results in this regard.
For instance, by different experiments, Tyndall proved that
the length of the column of air which resounds to the fork
in a maximum degree of intensity is equal to one-fourth of
the length of the wave produced by the fork. 1
The negative determination of a suspected connection
of cause and effect must be precise in order to establish
the causal relation with that degree of accuracy which is
demanded in a truly logical and scientific method. Upon
this point, Bosanquet has a very suggestive passage : " The
essence of significant negation consists in correcting and
confirming our judgment of the nature of a positive phe
nomenon by showing that just when its condition ceases,
just then something else begins. The Just-not is the im
portant point, and this is only given by a positive negation
within a definite system. You want to explain or define the
case in which A becomes B. You want observation of not-12,
but almost the whole world is formally or barely not-J5,
so that you are lost in chaos. What you must do is to find
the point within A where A.\ which is B, passes into A 2
which is (7, and that will give you the just-not-B which is
the valuable negative instance." 2 For example, in Professor
Tyndall s experiment, the significant negative instance was
this, when the water in the tube reached just that height
when for the first time during the experiment no sound was
audible. The discriminating observation that can mark and
measure the precise point of transition from sound to no
sound, has determined accurately the conditions necessary
to produce the sound, and precisely define their limita
tions.
In all observation and experiment, the following possi
bilities should be kept before the mind in order to avoid a
hasty conclusion in reference to a seeming causal connec-
1 Tyndall, On Sound, p. 174.
2 Bosanquet, The Essentials of Logic, p. 134.
220 INDUCTIVE LOGIC
tion. We may think that we have discovered the relation
that if there is A, then there must be B, and the one there
fore the cause of the other, but it may happen that
1. Both A and B are effects of another cause and are
thereby related coordinately in reference to it.
2. A may be merely a liberating circumstance, or an inva
riable accompaniment of B.
3. A may not be the cause of B, but only an element of a
complex collocation which is the cause of B.
4. Each separate instance of B may so differ as to
respond to the action of A in a manner different from the
others.
5. A may be related to B in a system of such a nature
that the system in continuously developing new effects
causes J5, as the introduction of medicine into an organism
whose forces are themselves effecting a healing process.
6. It is often very difficult to tell whether A is the cause
of B, or B the cause of A, as in districts where drunkenness
and poverty are prevalent, or cases of moral and intellectual
feebleness. Which is the cause ? and which the effect ? In
many cases such as these, the forces react upon each other,
the effect tending to increase the intensity of the cause.
7. The connection of A and B may be one of mere coin
cidence, and not of a causal nature whatsoever. Newton
was much impressed with the apparent connection between
the seven intervals of the octave, and the fact that the
colors of the spectrum divide into a like series of seven
intervals. And yet there is no causal connection that can
be proved to exist between the two.
The more we dwell upon these various possibilities, the
more are we impressed with the extreme complexity in
which the relation of cause and effect is involved. The
investigator must bring to his research the spirit of patience
and perseverance, as well as a clear vision and discriminat
ing insight. Sir John Lubbock, in his observations upon
the habits of ants, says that at one time he watched an ant
CAUSAL ANALYSIS AND DETERMINATION 221
from six in the morning until a quarter to ten at night, as
she worked without intermission during all that time. 1 It
is to such patient investigators that nature reveals her
secrets.
i Sir John Lubbock, Scientific Lectures, p. 73.
CHAPTER VI
MILL S INDUCTIVE METHODS THE METHOD OF
AGREEMENT
THERE are certain specific methods by which a supposed
relation of cause and effect may be tested. Before applying
any method however to concrete instances, there is naturally
in mind some suspected causal relation which is the result
of one or both of the two preliminary inductive processes.
As we have seen, these primary processes in inductive in
quiry are induction by simple enumeration, and induction
by analogy. By the enumeration of the special cases in
which we have found a significant coexistence or sequence,
a causal relation is suggested as a possible or probable ex
planation. By analogy also a causal relation is suggested
on the basis that a given phenomenon which in essential
particulars resembles another phenomenon whose cause or
effect is already known will, in all probability, have a like
cause or effect. Enumeration and analogy thus suggest a
probable explanation which is not as yet proved, but which
ranks as a tentative hypothesis. The natural history, there
fore, of the final product of the inductive process recognizes
the initial stages of enumeration and analogy leading to
some preliminary hypothesis, which is to be tested by one
or more of the specific methods of scientific investigation.
These methods have been formulated by John Stuart Mill
and are especially associated with his name. They are as
follows :
1. The Method of Agreement.
2. The Method of Difference.
3. The Joint Method of Agreement and Difference.
222
MILL S INDUCTIVE METHODS 223
4. The Method of Concomitant Variations.
5. The Method of Residues.
The method of agreement consists in inferring the exist
ence of a causal relation, when in a number of varying
instances it is observed that the supposed cause is always
accompanied by the phenomenon in question, as correspond
ing effect.
The method of difference is the comparing of an instance
where the supposed cause is present, accompanied by the
corresponding effect, with an instance having precisely the
same setting, but where the supposed cause is withdrawn,
the effect also disappearing ; the inference of a causal rela
tion is then permissible.
The joint method of agreement and difference is the com
paring of instances where the supposed cause is present,
with similar instances where it is absent ; if the correspond
ing effect is present in the former, and absent in the latter,
group of instances, a causal relation may be inferred. This
differs from the method of difference, that in the latter the
same instance, now with, and again without, the presence of
the suspected cause, is the subject of observation ; in the
joint method it is a number of instances with, compared
with a number of similar instances without, the presence of
the supposed cause.
The method of concomitant variations consists in so modi
fying any given phenomenon that the supposed cause will
vary in intensity ; then a corresponding variation in the
accompanying effect is evidence of a causal relation.
The method of residues consists in the analysis of a given
complex phenomenon, in which all elements save one of the
antecedent are known to be related severally in a causal
manner to all elements save one of the consequent; then
the residual element of the one may be regarded as the
cause of the residual element of the other.
These methods, it is true, deal only with concrete
224 INDUCTIVE LOGIC
instances; but, in so far as these instances discover an
underlying causal connection, they thereby furnish sufficient
ground for a complete generalization, and warrant the induc
tive procedure from special cases to the universal.
We will now examine these methods more in detail. The
brief outline above is intended merely to give a general idea
of the methods, that it may lead to a better understanding of
the more exact statement of their nature and characteristics.
The Method of Agreement. The more precise statement
of this method is given in the first canon of Mill, which is
substantially as follows :
If two or more instances of the phenomenon under investi
gation have only one circumstance in common, the circum
stance in which alone all the instances agree is the probable
cause (or effect) of the given phenomenon, or sustains some
causal relation to it.
The above is based upon the causal axiom that the constant
elements which emerge in any given series of similar phe
nomena are to be considered as connected in some manner
with the cause of the phenomena; but that the variable
elements are not connected with the phenomena in any
causal manner whatsoever.
The method of agreement is illustrated in the investiga
tion of the very common phenomenon of the transformation
of substances from the solid to the liquid state. What is
the one circumstance which is always present when we con
sider the melting of such widely different substances as
butter, ice, lead, iron, etc. ? In all instances, to whatsoever
extent they may be multiplied, of the change from solid to
liquid states, heat has been observed to be present, and
is thereby indicated as the likely cause of the phenomenon
in question. The method may be represented through the
use of symbols which, according to Mill, are capital letters
to denote antecedents, and smaller letters to denote corre
sponding consequents. Let the following be a number
of different instances with the antecedents and con-
THE METHOD OF AGREEMENT 225
sequents arranged in order, and represented as above in
dicated :
ABC abc.
ADE ctde.
AMN amn.
etc. etc.
By inspection of such a table of instances thus analyzed,
and symbolically represented, it will be readily seen that A
is the only element common to all the antecedents, while
a is the only one common to all the consequents. The in
ference, therefore, is that A is the cause of a. It has been
objected to this system of representation that it artificially
arranges the elements of antecedent and consequent, as
though there were a number of distinct cause-elements, each
connected with a correspondingly distinct effect-element, and
it produces the impression that it is quite an easy matter
to see how these causal pairs are thus separately related. 1
As nature presents her phenomena to us, however, there is
such complexity throughout, that the analysis cannot readily
distribute part to part in appropriate causal relations. To
avoid such an error in notation, I have adopted the follow
ing symbols, which will be used hereafter to describe the
various methods. Let us take C as the letter to represent
the supposed causal element, and S, the entire setting of
accompanying circumstances; let e denote the corresponding
effect, and s the sum total of the attendant consequences.
The causal relation will be then indicated, according to the
method of agreement, as follows :
S + C s + e.
S + C s + e.
S" + C s" + e.
Here the setting changes throughout, as indicated by S,
S , S", etc., but C remains constant in the antecedents ; also
1 Venn, Empirical Logic, p. 411.
226 INDUCTIVE LOGIC
the corresponding setting in the consequents changes, as in
dicated by s, s , s", etc., but e remains constant throughout.
Such a notation does not attempt to represent just which
parts of S cause corresponding parts of s, nor by what ele
ments precisely S differs from S and S", etc. It does rep
resent, however, the difference between the variable and
constant elements of the table of instances which are ar
ranged for comparison, and this is the key to disclose the
causal relation.
As an example of this method, let us take the physical
law that different bodies tend at the same time to absorb
and to emit the same waves of light. It is known that every
substance in burning gives its own lines in the spectrum
analysis, sodium, for instance, producing a very bright line
in the yellow portion of the spectrum in a definite locality
(Line D, of Fraunhofer). If now, instead of burning sodium,
we interpose the vapor of sodium in the path of the ray
which should give a continuous spectrum, the phenomenon
is completely reversed; at the exact point where there was
a bright line in the spectrum, a dark line now appears.
Thus the vapor of sodium, acting as a screen, absorbs the
rays which it emits when it acts as the luminous source.
A similar effect is observed in the case of vapors of iodine,
of strontium, of iron, etc. ; it may therefore be regarded as a
phenomenon, admitting of generalization by induction. 1 This
is according to the method of agreement; and we may make
the following representation :
Vapor of sodium acting as a screen = S -f- C.
Vapor of iodine acting as a screen = S + C.
Vapor of iron acting as a screen = S" -f (7.
Vapor of strontium acting as a screen = S " + C.
etc. etc.
i Saigey, The Unity of Natural Phenomena, pp. 94, 95.
THE METHOD OF AGREEMENT 227
The corresponding consequents are : -
Reversing bright sodium line to dark = s + e.
Reversing bright iodine line to dark = s -\- e.
Reversing bright iron line to dark = s" + e.
Reversing bright strontium line to dark = s 1 " -f e.
etc. etc.
Therefore we have :
S + C . . ,
. . . s 4- e.
S + C . . . .
... +e.
S" + C . . . .
. . . s" -f e.
/S " 4- C . . . .
t . . s " _f- e .
etc.
etc.
In this the constant C of the antecedents is the vapor of
any substance acting as a screen ; the constant e is the
reversal in each case of the bright line of the substance in
the spectrum to the corresponding dark line of the same.
From this it is inferred that the vapor of any substance acting
as a screen absorbs exactly those rays which it emits when
it acts as the luminous source.
It is of great importance that the instances selected for
observation or experiment be as varied as possible, so that
widely differing phenomena may be gathered together.
Then if running through them all there is one common
element observed among the antecedents, and one common
element among the consequents, the greater the variation
among the instances the more pronounced will be the signifi
cance of the constant elements. In the illustration given
the substances which are so different as iron, strontium.
sodium, iodine, etc., preclude the possibility of the resultant
phenomenon described being due to the peculiar properties
of any one metal, or group of metals. So many phenomena
and so different in kind are taken as to eliminate the
peculiarities attached to any one in particular. In this re
spect the method is one of elimination. By varying the
228 INDUCTIVE LOGIC
instances the non-essential is eliminated, and the essential,
which remains as the element common to all, is thereby
emphasized, and differentiated from all attendant circum
stances.
.This method also is one of discrimination, of discerning
the constant element under the various changing forms
which it can assume, and as such it is similar to the logical
process of the formation of a concept. The concept is the
grasping of the universal element which is present through
the particular and concrete manifestations of the same.
Through them all there is the like common element which
is the basis of the concept itself. So out of many particular
instances the mind grasps the elements which are common
to all, and considers them as related in a constant and
therefore causal manner, which has in itself the character
of a universal concept and so admits of being formulated in
the form of a law universal, which is the end of all induc
tion.
This method, moreover, is peculiarly adapted to observa
tion, the collating of a number of instances, rather than to
experiment. Instances cannot always be manufactured, and
so it may be beyond the power of experiment to reproduce
them. They can, however, always be the objects of research,
and as such fall naturally into the field of observation.
The question may properly be asked at this point,
How does this method differ from that of induction by
simple enumeration ? The latter we have seen is never
satisfactory because the enumeration cannot be complete,
and may be contradicted by an enlarged experience. This
method however is superior in that it provides for more
than simple enumeration of instances in which the phenome
non in question has occurred ; there must be a corresponding
analysis of the instances, accompanied by a discriminating
insight to distinguish the essential from the unessential.
Number of instances increases the probability that the
variable elements have been eliminated, and enables the
THE METHOD OF AGREEMENT 229
mind to concentrate upon the constant elements that remain
and are thereby disclosed.
This method primarily admits of application to instances
where a sequence is observable ; that is, where antecedent
can be distinguished from consequent by an appreciable
time element. It is however possible to apply this method
to the investigation of coexistences, where it may show that
either the coexisting elements are related as cause and
effect, or that in some causal manner they are the correlated
effect of some cause sufficient to account for them both.
Many instances may be adduced of the prevalence of
poverty and crime associated together. This may indicate
a causal relation between them, and yet a sequence cannot
be observed of sufficient definiteness to indicate which is
the cause, and which the effect. The problem is thus left
indeterminate, with the suggestion of some other cause
which may possibly account for them both. All that the
method of agreement can attain, is by collecting a number
of instances of diverse nature to indicate that in some way
at least poverty and crime are connected by causal ties. The
constant coexistence of attributes in one individual admits
of a similar treatment and similar results. The fact of the
high coloring of male butterflies in a large number of
instances, in reference to a variety of species, indicates a
constant relation between the fact of its being a male
and the possession of brilliant coloring. This inseparable
association indicates a causal relation, which, however,
cannot be more precisely determined by this method. The
full explanation of the phenomenon requires some supple
mentary hypothesis depending upon conditions not disclosed
by this method, an hypothesis such that the high coloring has
the special function of attracting the female butterfly and
has been intensified and developed by natural selection.
The method of agreement is open to criticism at several
points, and yet it must be at the beginning understood that
this method does not rank as a final method. We shall
230 INDUCTIVE LOGIC
soon see that in many cases it needs to be supplemented by
the method of difference, in order either to confirm or to
disprove its tentative results. The chief criticisms that have
been made of this method may be summed up as follows :
1. The cause indicated by the method of agreement is
not thereby proved to be the sole cause of the phenomenon
in question. We may gather together a number of varied
instances where an extensive failure of crops in the summer
has caused hard times during the winter following. And
yet there may be, and as a fact there are, many other causes
which engender periods of industrial depression. We may
say, therefore, that this method is capable of establishing,
tentatively at least, a universal proposition of the form,
All x is y ; it does not, however, attempt to give any indica
tion, one way or the other, regarding the validity of the con
verse, All y is x. Knowing the limitations of a method does
not by any means destroy its legitimacy as a method ; it rather
increases its efficiency within its proper sphere, by the more
exact knowledge as to the precise extent of that sphere itself.
2. It is urged that while it is possible to recognize in
most, if not in all, cases, the common element in the several
effects of similar phenomena, it is not so easy a matter to
separate the common element in the corresponding antece
dents by the simple method of agreement alone. For
instance, in Bacon s illustration of the investigation of the
cause of heat, he cites such disparate phenomena as the
sun s rays, friction, combustion, etc. The element of heat
is readily discernible through them all; but what is the
common element which operates as cause in each case ?
There is the difficulty. Sigwart illustrates this in the case
of the phenomenon of death. The effect can be easily de
tected as similar throughout, but in all the antecedents the
only property common to them all is life, and, therefore, we
are led into the fallacy of attributing to life the cause of
death. 1 We must therefore acknowledge that some phe-
l Sigwart, Logic, Vol. II, p. 341.
THE METHOD OF AGREEMENT 231
nomena may occur in such a variety and such a number of
manifestations as to disguise the nature of the cause under
the mask of a generality too indefinite to be recognized. In
all such instances, the method of agreement must avail itself
of suggestions received from some other source, as to the
nature of the common element in the antecedents. Or, some
minor circumstances attending the effect may indicate more
precisely the nature of the cause, as, for instance, the pecul
iar symptoms associated with death by drowning, which dif
ferentiate it from death due to any other cause.
3. The common element in the antecedents may prove to
be an unessential accompaniment of all the instances exam
ined. Its presence, therefore, may have nothing whatso
ever to do with the observed effects. A number of different
medicines, for example, may produce a certain effect alike
in all instances. The only common element that can be
detected in the various medicines examined may be the
alcohol which is used as the common vehicle of the different
drugs, and yet its effect may be entirely inert as regards the
medicinal qualities in question. The common element really
efficient may be overlooked, and another common element
which is easily discernible may nevertheless remain wholly
inoperative. This difficulty may be overcome by a more
thorough analysis of the phenomena observed, which may
be attained by a judicious variation of the instances, so as
to reveal, in turn, the precise effect of the various simple
elements which together constitute the complex whole of
the phenomenon in question. The defects of the method
in this respect are, in a word, the defects of induction by
simple enumeration.
4. The cause may be present in all the antecedents, and,
notwithstanding the corresponding effect not appear, and
this, not because the two are not related in a causal manner,
but because the cause is neutralized by the associated ele
ments which appear in combination with it in the various
antecedents. For instance, diphtheria germs are the cause
232 INDUCTIVE LOGIC
of diphtheria, and have been found accompanying this
disease in all cases which have been observed. And yet
their presence is often noted when the disease itself does
not develop. The tendency existing is counteracted by the
condition of the organism at the time, so that the dread
bacilli are inoperative and therefore harmless. As we have
seen before, the presence of the effect necessitates the pres
ence of the corresponding cause; but by no means is it
always true that the presence of the cause necessitates the
effect. The cause always produces the tendency at least,
which however may be neutralized.
5. This method is often applied in a very careless way to
the observations of persons who do not possess the power
of accurate discrimination, and therefore observed coinci
dences are hastily assumed to be particular instances of an
universal law. Such procedure leads to superstition and
prejudice. It not only warps the judgment, owing to its
illogical nature, but it also affects indirectly the man s
moral view, as it implies a weakness in character as well
as in mind. This criticism refers however to the abuse
rather than the legitimate use of this method under such
restrictions as have been already indicated.
The chief function of this method is that of suggestion.
It indicates often only the possibility of the existence of a
causal relation ; in other cases it leads to an inference of
high probability. In all cases however it marks merely the
preliminary steps of an investigation which should be fol
lowed up by painstaking experiment. As it is the method
of observation chiefly, it is natural that it should precede
experiment ; for it is only by reflection upon our observa
tions that we discover the nature and relations of phenomena,
which serve as data for subsequent experiment.
I have selected several illustrations to indicate the various
fields of research in which this method of agreement has led
to satisfactory results.
The first refers to the relation between the occurrence of
THE METHOD OF AGREEMENT 233
financial crises and the prevalence of over-production. Guyot,
in his Principles of Social Economy, gives the following in
stances : An enormous consumption of capital in the United
States in the seventies, for the construction of railroads, was
followed by unusual commercial depression. Then the like
outlay in India for railway construction by means of loans
and taxes which absorbed the whole circulating capital of
the Indian population was followed by a devastating fam
ine and general commercial paralysis. Again in Germany
there was an enormous consumption of capital in forts and
armaments and general military equipment, bringing on the
crisis of 1876-1879. England at the same time was unduly
supplying circulating capital to the United States, Egypt,
and her colonies, and a financial crisis was the result.
Through all these varying instances and others of a like
nature which might be added, the constant relation of over-
consumption in the antecedents to the industrial depression
evident in the effect indicates the one to be the cause of the
other, either in whole or in part.
Again it is narrated in Brewster s Treatise on Optics that
he accidently took an impression from a piece of mother-of-
pearl in a cement of resin and beeswax, and, finding the
colors repeated upon the surface of the wax. he proceeded
to take other impressions in balsam, fusible metal, lead,
gum arabic, isinglass, etc., and always found the iridescent
colors the same. His inference was that the form of the
surface is the real cause of such color effects. 1 The com
mon element which appears in all the antecedents is evidently
the same form impressed upon each, which was originally
received from the mother-of-pearl. The cause is moreover
independent of the nature of the substance in each case
which received the impression upon its surface, because
such a variety of substances was chosen as to eliminate the
individual nature of each as an influencing factor in the
result. In this experiment we see the advantage of varying
1 Quoted by Jevons, Principles of Science, p. 419.
234 INDUCTIVE LOGIC
the instances as* far as possible for this very purpose of
eliminating all irrelevant elements. Similar experiments
have proved like results in reference to the colors exhibited
by thin plates and films. Here the rings and lines of color
have been found to be nearly the same whatever may be
the nature of the substance. A slight variation in color is
due to the refractive index of the intervening substance.
With this exception, the nature of the substance is not
operative in producing the color effect, but the form alone.
The celebrated scientist, Pasteur, in the year 1868 was
carrying on his investigations as to the cause of the blight
then devastating the silkworms of France. One of his ex
periments consisted in selecting thirty perfectly healthy
worms from moths that were entirely free from the cor
puscles, which latter are the germs of disease, or at that
time suspected to be the germs of disease. Then, rubbing
a small corpusculous worm in water, he smeared the mix
ture over the mulberry leaves. Assuring himself that the
leaves had been eaten, he watched the consequences day by
day. One after the other the worms languished ; all showed
evidences of being the prey of the corpusculous matter, and
finally, within one month s time, all died. Pasteur s infer
ence naturally was that the corpuscles had produced the death.
Of course his results were not founded upon this experiment
alone, but other experiments, carried on in many different
ways, served to corroborate the causal relation which the
experiment just described had suggested as at least highly
probable.
In medicine also the method of agreement is often used
with effect. Certain drugs are administered in a number of
cases and the results noted. A uniform effect consequent
upon the administration of a given drug indicates a causal
connection capable of generalization. Not only are subjects
in disease, but also in health, selected, and the effects upon
both the normal and morbid natures compared. Thus a
variation in instances is secured. If a number of different
THE METHOD OF AGREEMENT 235
drugs produce like effects, the question at once suggests
itself, What is the property common to them all ? The
method of agreement often gives some indication of this,
when the elimination of the inert properties can be accom
plished through a sufficient variation of instances. The
difficulty lies, however, in this very thing, to so vary the in
stances as to disclose the efficient element present in them all.
Various medicines present a complex nature of such a char
acter that it is extremely difficult to discriminate the precise
effects which the several component parts individually
exercise.
The method of agreement is also used, perhaps uncon
sciously, in the conduct of the everyday affairs of life.
Whenever different phenomena in our experience present
certain characteristics of a constant nature, we are at once
led to suspect a causal connection, and to start upon a more
searching investigation of the same. Too often however
the supplementary investigation is omitted, and the mind
rests content with a few surface resemblances that lead to
a hasty generalization without being more precisely and
adequately determined.
CHAPTER VII
THE METHOD OF DIFFERENCE
THE method of agreement, as we have seen, presents a
causal relation as a suggestion, admitting of a high degree
of probability it may be, but requiring to be tested by some
more scientific method. This is accomplished by the method
of difference. Here a phenomenon is observed, in which
the supposed cause-element and effect-element appear ; then
while all other circumstances and conditions remain unal
tered, the supposed cause-element is withdrawn, or its force
adequately eliminated ; the immediate disappearance of the
supposed effect-element, consequent upon this, indicates a
causal relation between the two. Or the experiment may
be made in a different manner, but to the same end, that
is, a phenomenon may be characterized by the absence of
both cause-element and effect-element; then, if the intro
duction of the cause-element does not disturb the phenomenon
in question, except immediately to produce the effect-ele
ment, the inference may be drawn that the one is the
veritable cause of the other.
Canon of the Method of Difference. If an instance in
which the phenomenon under investigation occurs, and an
instance in which it does not occur, have every circumstance
save one in common, that one occurring only in the former;
the circumstance in which alone the two instances differ is
the effect, or it may be the cause, or a necessary part of the
cause, of the phenomenon.
This method admits of manifold illustration in our every
day inferences. A person is asleep in the room with us,
and we hear the loud noise of a slamming door, and observe
236
THE METHOD OF DIFFERENCE 237
the person at once awakening with a start and exclamation.
We have no hesitancy in ascribing the awakening to the
noise immediately preceding it. We observe again some
one receiving a letter or telegram, and immediately upon
opening it the face grows white with anxiety and fear, the
hands tremble, and there are shown general symptoms of
perturbation. The message received, we say, has caused
the mental shock and physical accompaniments.
Or, taking a simple experiment in quite another sphere,
it was observed by Boyle, in 1670, that an extract of litmus
was immediately turned red by the introduction of an acid.
This subsequently became a test for the presence of acids,
the inference being that an acid has this capacity of chang
ing the litmus to a red color from its original blue.
Professor Tyndall describes an experiment to prove that
waves of ether issuing from a strong source, such as the sun
or electric light, are competent to shake asunder the atoms
of gaseous molecules, such as those of the sulphur and oxy
gen which constitute the molecule of sulphurous acid, lie
enclosed the substance in a vessel, placing it in a dark room,
and sending through it a powerful beam of light. At first
nothing was seen ; the vessel containing the gas seemed as
empty as a vacuum. Soon, along the track of the beam, a
beautiful sky-blue color was observed, due to the liberated
particles of sulphur. For a time the blue grew more intense ;
it then became whitish ; and from a whitish-blue it passed
to a more or less perfect white. Continuing the action, the
tube became filled with a dense cloud of sulphur particles
which, by the application of proper means, could be rendered
visible. 1 In this series of continuous changes, we find the
one antecedent giving the causal impulse to be the beam of
light. It was the one element introduced which started
the several changes leading to the appearance of the sulphur.
The one, therefore, is to be regarded as the cause of the
other.
i Tyndall, Use and Limit of the Imagination in Science, p. 33.
238 INDUCTIVE LOGIC
It is possible to represent this method by means of sym
bols in a manner similar to that of the method of agreement.
Let C be the supposed cause and e the effect corresponding,
while S and s denote the setting of antecedent and conse
quent respectively. We have, therefore, the following :
S + C s + e.
Then, withdrawing (7, we have the absence of e.
S s.
The inference then is that C is the cause of e. Or, we may
have given
S s.
Then if, adding (7, we find that e also appears, represented
by
S + C s -f- e,
we infer that (7 and e have a causal connection.
In the method of agreement, a number of instances are
taken agreeing only in the possession of two circumstances,
the cause and effect elements common to them all. -In
this method, only two instances are taken, and they must
be precisely alike, with the one exception, the presence
of two circumstances in one, that is, the cause and the effect
elements, and the absence of the same in the other. In the
method of agreement, we compare the various phenomena
to note wherein they agree; in the method of difference,
we compare the two phenomena to note wherein they differ.
The logical axiom underlying the two methods is sub
stantially one and the same, differing only in its special
adaptation in each case. The former method rests on the
assumption, which must be accepted as a fundamental postu
late, that whatever can be eliminated from the various
instances is not connected with the phenomenon under in
vestigation in any causal manner ; and the method of differ-
THE METHOD OF DIFFERENCE 239
once is based on tho postulate tliat whatever cannot
natecTIs connected with the phenomenon by a causal law.
"The method of difference is evidently the method by
negation, which has already been indicated as the truly
"scientific process in induction. It is also preeminently the
method of experiment rather than observation ; for the with
drawal or introduction of forces can only be accomplished
at will when we bring the phenomena under experimental
control. At times nature herself may perform the experi
ment for us, and we stand simply as observers to note the
results. This is especially the case in the catastrophic
phenomena, such as volcanic eruption, earthquakes, etc.
Generally speaking, however, the method of difference is the
process of man s manipulation to secure purposed results in
which a causal relation is disclosed.
A question naturally suggests itself, What is there to
determine the precise mode of experiment ? We may have
given a concrete whole of extreme complexity. In our ex
periment, which element shall we proceed to eliminate, in
order to note the result? An answer may be given us
through suggestions received from the results of enumera
tion, analogy, or the method of agreement. If it is not
possible to avail one s self of this contribution from another
sphere of investigation, then the complex whole must be
broken up, as far as possible, into its simplest component
parts, and one after another these parts, singly, then in
pairs, and all other possible combinations, caused to be
withdrawn, or their force neutralized, and the results in
each case noted, as to whether the effect under investigation
disappears. The exhaustion of all possible combinations
must yield some definite result. Suppose, for instance,
there is a complex antecedent involving four separable ele
ments, as A, B, C, D. Withdraw severally A, B, C, and D,
noting results ; then withdraw, in turn, AB, AC, AD, BC,
BD, CD, that is, the possible combinations of four elements
taken two at a time ; then withdraw AB<1, then BCD, ABD }
240 INDUCTIVE LOGIC
and A CD, that is, combinations of four elements taken three
at a time. 1 By such a process there will be disclosed
whether one element alone or whether a combination of two
or more have produced the effect under investigation. The
practical difficulty in separating the elements of a complex
whole, and withdrawing the several combinations from the
whole, renders this process in many cases impossible. The
cause, however, is generally suspected. It may be suggested,
as we have seen, by the method of agreement, by analogy,
or by that insight which at once declares certain combina
tions to be impossible and others irrelevant. There is
generally a mental experiment in which the judgment
rejects unlikely combinations, thus narrowing the field of
investigation and furnishing a tentative hypothesis as a
preliminary to the experiments proper.
The method of difference is open to various criticisms \
the most important are the following :
1. In applying this method, we may be easily misled, in
supposing our two instances are precisely alike with the
one exception of the presence or absence of the supposed
cause, but in reality the instance may differ radically, and
yet we may be unable to detect this. A patient may have
medicine administered to him, and begin at once rapidly to
recover, and yet the very taking of the medicine in itself
may have made such a mental impression inducing confi
dence and hope that the real cause of the recovery may be
due wholly to this mental reaction. Persons taking pills
composed of inert substances have often given evidence of
bodily effects wholly impossible to trace to the medicine
itself. And yet this criticism is one of caution rather than
of censure; for the defects are but difficulties which ex
treme care and insight may overcome.
2. It has been objected that this method may point out
the cause in the concrete instance before the experimenter,
i This process has been illustrated and criticised at length in a striking
manner by Venn, Empirical Logic, pp. 401 ff.
THE METHOD OF DIFFERENCE 241
but that this furnishes no basis whatsoever for a wider
generalization that the effect in question is always produced
by this cause. Sigwart has illustrated this objection by the
instances in which typhus fever has been traced to the
drinking of impure water. 1 The causal relation may be
fully established in the cases investigated, but the universal
proposition does not follow that wherever typhus fever
appears, impure water has been drunk. This objection
applies especially to cases of extreme complexity, where
proximate causes alone can be discovered, and their ultimate
nature which may appear in various forms is not revealed ;
for instance, the impure water is not in itself the ultimate
cause of the typhus fever. It contains the poison germs,
the real cause ; they may be introduced into the system in
some other way. Care therefore should be taken to reveal
the cause in and by itself, and not the cause of the cause.
The objection, therefore, may be in a measure overcome.
To effect a generalization of logical validity, it is necessary
to supplement the method of difference by hypothesis and
subsequent verification, which will be described later on.
3. This method may lead to error in cases where the sup
posed causal element is regarded as the cause in its entirety,
when it jam reality but a part of the cause. If one should
plant seed in a garden and water only one-half of the plot,
and it should follow that only the watered part brought
forth the leaf and flower, then an inference according to the
method of difference might be drawn that the water caused
the sprouting of the young plants. And yet it must be re
garded simply as contributory to the real cause. Such a
difficulty may be obviated by a careful discrimination in the
analysis of the phenomenon investigated.
4. Sometimes a liberating cause may be revealed by a
strict interpretation of the method of difference, when the
real cause is more obscure, and may be overlooked. A stone
may strike a can of dynamite, and the explosion which
1 Sigwart, Logic, Vol. II, p. 420.
242 INDUCTIVE LOGIC
occurs may be traced to the impact of the stone. It is the
one element of difference introduced in the sphere of the
observed phenomena, with the consequent result. The
force existing as a potential is naturally obscure, and apt to
elude observation. Therefore, whenever a cause disclosed
by the method of difference seems to be out of all propor
tion to the effect, it at once suggests the probability that a
potential force not discerned by our powers of observation
has been the real cause, and the former a conditioning cause
merely. Another illustration of this is the experiment of
Priestley, which led to his discovery of oxygen in 1774.
He placed some oxide of mercury upon the top of quick
silver in an inverted glass tube filled with that metal and
standing in mercury ; he then heated the oxide by means of
a glass lens and the sun s rays, and obtained a gas, which
he called " nitrous air," afterwards designated as oxygen.
The heat in this case was the sole element of difference
between the two instances, one in which there was no gas,
and the second after application of the heat, when the gas
was present. Here the heat must be regarded as the liber
ating and not in any sense the producing cause. Again, as
Lotze says, " the fact that with the destruction of a single
part of the brain a definite psychical function ceases, is no
proof that just this single part was the organ which alone
produced that function." 1
In addition to the difficulties attending this method, which
have been enumerated and which have to do with the logi
cal theory of the method, there are also difficulties of a prac
tical nature which arise in the actual application of this
method in experimental inquiry. They are as follows :
1. Care must be taken that, in the two phenomena com
pared, with and without the supposed cause, there shall not
be an interval of time elapsing, in which period some other
cause might be introduced unknown to the investigator, and
yet capable of producing the result, or else of neutralizing
1 Lotze, Logic, p. 322.
THE METHOD OF DIFFERENCE 243
some force that is present and itself capable of producing
the result. For instance, if ;i cKerbical compound b- l.-l t
for an appnviubl* time, we may notice certain changes ami
be able to assert positively that no new element has been
introduced, and yet the action of the air may in itself have
been sufficient to work these changes. When the two phe
nomena to be compared can be presented for inspection
simultaneously, this difficulty is obviated. This is illus
trated in an experiment devised to exhibit the presence of
light effects in the spectrum beyond the violet rays ; that is,
beyond the place where the spectrum seems to end. A
sheet of paper is taken, the lower part of which is moistened
with a solution of sulphate of quinine, while the upper part
remains dry. Let the image of the solar ray fall upon this
sheet ; the spectrum preserves at the top of the sheet in the
dry portion of the paper its ordinary appearance, while in
the moistened portion a brilliant phosphorescence appears
beyond the region of the violet rays. Here the dry and wet
portions are simultaneously presented, and there is but one
point of difference between the two. The inference, there
fore, is readily drawn that the solution of sulphate of quinine
is a substance sensitive to the ultra-violet portion of the sun s
rays, the phosphorescence being the effect of these rays upon
the solution.
2. Extreme care must be taken that, in the withdrawing
of any element in the course of the experiment, no other
element is inadvertently introduced, and that, in adding
any element, no existing element or combination of elements
is destroyed, or their effect neutralized. Mr. Venn has ad
mirably illustrated this difficulty, and I give the following
quotation in full from him : " We suppose that when we
have put a weight into one pan of a pair of scales we have
done nothing more than this, or can at any rate by due cau
tion succeed in doing nothing more. But if we exact the
utmost rigidity of conditions, we easily see that we have
done a great deal more. Our bodies are heavy, and there-
244 INDUCTIVE LOGIC
fore the mere approach to the machine has altered the mag
nitude and direction of the resultant attraction upon the
scales. Our bodies are presumably warmer than the sur
rounding air ; accordingly, we warm and therefore lighten
the air in which the scales hang, and if the two scales and
their contents are not of the same volume, we at once alter
their weight as measured in the air. Our breath produces
disturbing currents of air. Our approach affects the sur
face of the non-rigid floor or ground on which the scales
stand, and produces another source of disturbance, and so
on through the whole range of the physical forces." *
In the Report of the British Association, 1881, an account
is given of Professor G. H. Darwin s experiments to meas
ure the lunar disturbance of gravity at the Cavendish Lab
oratory by means of an extremely delicate pendulum. It
was found that approaching the pendulum in order to ob
serve its reading, the surface level of the stone floor on
which the instrument stood was deflected by the weight of
the observer. As he stood to take the reading, the shifting
of his weight from one leg to the other was perceptible ; so
it became necessary to observe the reading by a telescope
from a distance, or to adopt some similar plan. 2
Faraday was able at will to produce or remove a magnetic
force, through the revealed properties of the electromagnet.
Many of his experiments would have been impossible if it
had been necessary to remove a cumbersome magnet and
reinstate it again and again in his experiments. The elec
tromagnet however could produce or destroy the presence
of magnetic force without any incidental perturbations.
Thus Faraday was enabled to prove the rotation of circu
larly polarized light by the fact that certain light ceased to
be visible when the electric current of the magnet was cut
off, and instantly reappeared when the current was reestab
lished. Faraday says of the experiment himself: "These
1 Venn, Empirical Logic, p. 416.
2 Quoted by Venn in Empirical Logic, p. 419.
THE METHOD OF DIFFERENCE 245
phenomena could be reversed at pleasure, and at any instant
of time, and upon any occasion, showing a perfect depend
ence of cause and effect." 1
3. In some cases it is impossible to remove an element
whichos_aiiposed to be the cause of an effect under investi
gation. Its removal might cause the destruction or the im
pairing of the whole phenomenon. The force therefore
that cannot be eliminated must be neutralized by an equal
and opposing force. For instance, the force of gravity can
not be eliminated ; it must therefore be counterbalanced by
some device of the investigator. In chemistry the removal
of an element from a compound may be impossible without
destroying utterly the compound itself ; in such a case also
a neutralizing agent must be introduced. Darwin wished to
prove that the odor of flowers is attractive to insects irre
spective of the attraction of color. He therefore covered
certain flowers with a muslin net, and still the insects were
attracted to the flowers although the color was thus con
cealed. 2
The following illustrations may serve further to exhibit
the various features of the method of difference :
Mr. Eobert Mallet gives the following interesting account
of his visit to Faraday : " It must be now eighteen years
ago when I paid him a visit, and brought some slips of
flexible and tough Muntz s yellow metal, to show him the
instantaneous change to complete brittleness with rigidity
produced by dipping into pernitrate of mercury solution.
He got the solution and I showed him the facts ; he
obviously did not doubt what he saw me do before and
close to him ; but a sort of experimental instinct seemed to
require he should try it himself. So he took one of the
slips, bent it forward and backward, dipped it, and broke it
up into short bits between his own fingers. He had not
before spoken. Then he said, Yes, it is pliable, and it does
1 Experimental Researches in Electricity, Vol. Ill, p. 4.
2 Darwin, Cross and Self Fertilization, p. 374.
246 INDUCTIVE LOGIC
become instantly brittle. " 1 Here the experiment with and
without the significant antecedent and consequent indicates
the causal relation, especially as the instantaneous effect
precludes the possibility of the operation of any other
cause.
Another experiment of Faraday s is that of his investiga
tion of the behavior of Lycopodium powder on a vibrating
plate. It had been observed that the minute particles of
the powder collected together at the points of greatest motion,
whereas sand and all heavy particles collected at the nodes,
where the motion was least. It occurred to Faraday to
try the experiment in the exhausted receiver of an air-
pump, and it was then found that the light powder behaved
exactly like heavy powder. The inference was that the
presence of air was the condition of importance, because it
was thrown into eddies by the motion of the plate, and
carried the Lycopodium powder to the points of greatest
agitation. Sand was too heavy to be carried by the air. 2
Sir John Lubbock gives an account of experiments per
formed upon insects to prove that the sense of smell is in
some way connected with their antennae. One experiment
was performed by Forel, who removed the wings from some
blue-bottle flies and placed them near a decaying mole.
They immediately walked to it, and began licking it and
laying eggs. He then took them away, and removed the
antennae, all other circumstances remaining the same as
before, after which, even when placed close to the mole,
they did not appear to perceive it. Another experiment
similar to this was tried by Plateau, who put some food of
which cockroaches are fond on a table and surrounded it
with a low circular wall of cardboard. He then put some
cockroaches on the table ; they evidently scented the food,
and made straight for it. He then removed their antennae,
after which, as long as they could not see the food, they
1 Gladstone, Michael Faraday, p. 175.
2 Jevons, Principles of Science, p. 419.
THE METHOD OF DIFFERENCE 247
failed to find it, even though they wandered about quite
close to it. 1
Another experiment is that of Graber to prove the sense
of hearing in insects. He placed some water-boatmen
(Corixa) in a deep jar full of water, at the bottom of which
was a layer of mud. He dropped a stone on the mud, but
the beetles, which were reposing quietly on some weeds,
took no notice. He then put a piece of glass 011 the mud,
and dropped a stone on to it, thus making a noise, though
the disturbance of the water was the same as when the stone
was dropped on the mud. The water-boatmen, however,
then at once took flight. 2
An illustration of the method of difference occurs in the
so-called blind experiments, which are of ten made in chemistry
especially. As Professor Jevons has described such an
experiment: "Suppose, for instance, a chemist places a
certain suspected substance in Marsh s test apparatus and
finds that it gives a small deposit of metallic arsenic, he
cannot be sure that the arsenic really proceeds from the
suspected substance; the impurity of the zinc or sulphuric
acid may have been the cause of its appearance. It is
therefore the practice of chemists to make what they call
blind experiments, that is, to try whether arsenic appears
in the absence of the suspected substance. The same pre
caution ought to be taken in all important analytical
operations. Indeed it is not merely a precaution, it is
an essential part of any experiment. If the blind trial be
not made, the chemist merely assumes that he knows what
would happen." 3
1 Lubbock, On the Senses, Instincts, and Intelligence of Animals, p. 45.
2 Ibid., p. 75. 8 Jevons, Principles of Science, p. 433.
CHAPTER VIII
THE JOINT METHOD OF AGREEMENT AND DIFFERENCE
IT has already been shown that the method of difference
is sometimes not available, inasmuch as it may be neither
possible nor practicable to remove from, the phenomenon to
be investigated the suspected causal element without destroy
ing the phenomenon itself. Sometimes, too, it is impossible
even to neutralize the effect of the causal element if it is
allowed to remain as an integral part of the phenomenon.
This is especially the case in all vital phenomena, and also
in many chemical phenomena. Therefore another method
is resorted to, which is known as the joint method of agree
ment and difference. Inasmuch as the suspected causal ele
ment cannot be removed, we must select another phenomenon
as much like the former as possible, which is however
characterized by the absence of the causal element. By the
simple method of difference, two instances only need be
compared, the one with and the other without the causal
element, but agreeing precisely in every other particular.
In the joint method, the instances with and without the
causal element differ, it may be, in several particulars. A
number of varying instances must therefore be selected so
as to eliminate the possibility of any of these differing char
acteristics being the cause in question. Therefore two sets
of instances are collected and compared. The one set com
prises all the positive instances having the presence of the
supposed causal element, and the second set consists of the
negative instances having the supposed causal element ab
sent altogether. The validity of the method depends upon
the similarity of the two sets of instances. As the similarity
248
AGREEMENT AND DIFFERENCE 249
increases, the method approximates to the simple method
of difference.
The Canon of the Joint Method. It several instances in
which the phenomenon occurs have only one circumstance
in common, while several instances in which it does not
occur have nothing in common save the absence of that cir
cumstance ; the circumstance in which alone the two sets of
instances differ, is the effect, or cause, or a necessary part
of the cause, of the phenomenon.
The symbolical representation of this method may be ex
hibited as follows, using a similar notation to that employed
in the previous methods :
I. Table of positive instances.
S + C s +e.
S + C s +e.
S" + C s" +e.
S " + C *" + *
etc. etc.
II. Table of negative instances.
etc. etc.
In the two sets of instances, the following conditions
must be observed in order to render the method valid :
1. S + C, S + C, S 11 + C, S " + C, etc.,
must be so varied that they reveal but one constant element,
common to them all, as C. It may be that S will resemble
S 1 in more marks than the one, namely (7, and this may be
true of any two or more instances ; however, taken all to
gether, they must possess but the one common element C.
2. In the same way S t may resemble S n in more marks
than merely the absence of C and so for any two or more
250 INDUCTIVE LOGIC
instances in the series S t , S n , S in , etc. However, the one
characteristic common to them all must be the absence of C.
3. If in the instances chosen an element is common to all
in addition to (7, or in the second set its absence, then addi
tional instances must be added to the tables both positive
and negative in order to secure this all-important condition
of elimination through suitable variation.
4. Moreover, the two series, positive and negative, must
have their settings similar. S t , S tl , S in , etc., must resemble
S , S", S ", etc. ; otherwise the negative instances would not
be significant. 1 They must be chosen from the same sphere
as the positive, in order that they may be similar. It is
possible to multiply negative instances ad infinitum, which,
however, would furnish no ground for any inference, be
cause they would be wholly irrelevant to the problem under
investigation.
5. If S t is so similar to S as to be identical with it, and
also s, passes over into s ; then we have the method of
difference in its pure form:
S +C ........ s e.
Here the setting, instead of being similar in the two cases,
is the same in each.
The following is an experiment of Sir John Lubbock s
concerning the sense of smell in insects, which I have chosen
as illustrating this method of inductive research. He took
a large ant and tethered her on a board by a thread. When
she was quite still, he brought a tuning-fork into close
proximity to her antennae, but she was not disturbed in the
least. He then approached the feather of a pen very quietly,
so as almost to touch first one and then the other of the
antennae, which, however, did not move. He then dipped
the pen in the essence of musk and did the same; the
antenna was slowly retracted and drawn quite back. He
i See p. 75.
AGREEMENT AND DIFFERENCE 251
then repeated the same with the other antenna, and with
like result. Care was taken throughout not to touch the
antennte. Lubbock then repeated the experiment with a
number of different ants, and using various substances.
The results in all cases were the same, and the inference
was naturally drawn that the antennae possessed the sense
of smell. In these experiments various substances were
taken having nothing in common save the odor of musk
that had been placed upon them.
In some cases it is not possible to discover positive in
stances in which the only common element is the suspected
cause. In such cases the method is not conclusive in its
results, although it may attain a high degree of probability,
if all the common elements save the suspected cause-element
are known to be irrelevant, or can in any other way be
proved to have no influence whatsoever upon the result.
For instance, an illustration is often given of this method,
which fails in the manner just described. A man is attempt
ing to discover whether a particular article of food disagrees
with him. He notices several occasions, a large number if
you please, when he has eaten this particular kind of food,
and has soon after experienced distress. These are the posi
tive instances. This peculiar distress has never been ex
perienced when he has abstained from the food in question.
The inference is that this food has caused the distress. In
the various instances, however, the sole element in common
is not merely the taking or not taking the food. The per
son s whole bodily organism is common to all the instances.
Within it, unforeseen complications, independent of this
article of food, might have caused the trouble. In such
cases a large number of instances must be resorted to in
order to render the possibility of a coincidence out of the
question.
So also in such cases as the treatment of any given disease
in a hospital. An experiment may be tried in the treat
ment, say, of typhoid fever. One ward may be subjected
252 INDUCTIVE LOGIC
to a particular kind of treatment, and another ward not sub
jected to that treatment. If recovery is hastened in the one
and retarded in the other case, an inference may be drawn
as to efficacy of this treatment. In these instances again,
while they are all different patients, still the nursing, sur
roundings, etc., are common to them all. It must be shown
that these are present both in the negative and positive in
stances, and equally capable of accomplishing the effect if
they had been real causes. They may therefore be elimi
nated in comparing the two sets of instances, because com
mon both to the negative and positive cases. In this also
resort must be had to the number of instances in order to
eliminate chance coincidences. The presence of common
elements in excess of the common causal element may be
represented according to the symbolical notation of the joint
method, by the introduction of another symbol x. Let x
stand for that which is common to all instances in addition
to the common element C. We then have :
I. Set of positive instances.
S + C + x s + e.
S 1 + C + * s 1 +e.
S + c + x s" +e.
" + C + x s" + e.
etc. etc.
II. Set of negative instances.
&n ~^~ x s ir
tS ul -f- x s llt .
etc. etc.
We observe x in all instances both positive and negative.
Being present when the effect occurs and when it does not,
indifferently, we can at once infer that x is not the whole
cause of e. However, it may have united with C in the first
AGREEMENT AND DIFFERENCE 253
set of instances to produce the effect e, so that C without #,
or some part or parts of x, could not alone produce the effect
e. In all such cases the exact force of x must be estimated
in some other way. If x is extremely complex, or subject
to change from forces emanating from within itself, as in
the case of organic phenomena, then it becomes extremely
difficult to determine x; and consequently the method of
agreement and difference does not yield as exact results.
As long as the force of x remains unknown, it becomes the
source of possible disturbance, which may wholly vitiate
the results attained.
Mr. Darwin, in his experiments upon cross and self fer
tilization in the vegetable kingdom, placed a net about one
hundred flower heads, thus protecting them from the bees
and from any chance of fertilization by means of the pollen
conveyed to them by the bees. He at the same time placed
one hundred other flower heads of the same variety of plant
where they would be exposed to the bees, and, as he observed,
were repeatedly visited by them. Here we have the two
sets of instances, in one the flowers accessible to the bees,
and in the other, not accessible. He obtained the following
result. The protected flowers failed to yield a single seed.
The others produced 68 grains weight of seed, which he
estimated as numbering 2720 seeds. Cross-fertilization as
the cause in this case is thus proved. The common element
in all these instances, however, is not merely the presence
in one case and the absence in the other of the bees ; there
is also the element of the common plant structure running
through all of the two hundred instances. This element is,
however, of such an um 7 arying nature in all the instances,
and the number observed so many as to eliminate the possi
bility of any given plant structure possessing unobserved
peculiarities sufficient to produce the result in question. It
may therefore be considered as an inert element as regards
the effects noticed in the one and absent in the other set of
instances.
254 INDUCTIVE LOGIC
Sir John Lubbock, in his researches concerning the dif
ferent functions of the two kinds of eyes in insects, illus
trates the joint method in its general features. The two
kinds of eyes are the large compound eyes, situated one on
each side of the head, and the ocelli, or small eyes, of which
there are generally three, arranged in a triangle between the
other two. He wished to determine the precise function of
the small eyes, the ocelli ; and he has gathered together the
following facts. Plateau has shown that caterpillars, which
possess ocelli, but no compound eyes, are very short-sighted,
not seeing above one to two centimetres. He has also
proved by experiments that spiders, which have ocelli but
no compound eyes, are very short-sighted ; they were easily
deceived by artificial flies of most inartistic construction,
and even hunting spiders could not see beyond ten centi
metres (four inches). Lubbock experimented on this point
with a female spider, which, after laying her eggs, had
rolled them into a ball, and had enveloped the whole with a
silken bag which she carried about with her. Having cap
tured the female and having taken the bag of eggs from her,
he placed it on a table about two inches in front of her.
She evidently did not see it. He then pushed it gradually
towards her, but she took no notice till it nearly touched
her, when she eagerly seized it. He then took it away a
second time, and put it in the middle of the table, which
was two feet four inches by one foot four, and had nothing
else on it. The spider wandered about for an hour and
fifty minutes before she found it, apparently by accident.
He then took it away again and put it down as before, when
she wandered about for an hour without finding it. Like
experiments were tried with other spiders and with the
same results. Plateau also experimented with scorpions
which had ocelli and no compound eyes. They appeared
scarcely to see beyond their own pincers. Moreover, the
ocelli are especially developed in insects, such as ants, bees,
and wasps, which live partly in the open light and partly in
AGREEMENT AND DIFFERENCE 255
the dark recesses of nests. Again, the night-flying moths
all possess ocelli. On the other hand, however, they are
entirely absent in all butterflies, with but one exception,
according to Scudder, namely, the genus Pamphila. Forel
varnished the compound eyes of various insects which
had ocelli as well. The latter however he allowed to
remain in their natural state. Inasmuch as their habits
of flight required powers of vision in these insects extend
ing to a considerable distance, it happened that when placed
on the ground they made no attempt to rise ; while, if
thrown into the air, they flew first in one direction and then
in another, striking against any object that came in their
way, and being apparently quite unable to guide themselves.
They flew repeatedly against a wall, falling to the ground,
and unable to alight against it, as they did so cleverly when
they had their compound eyes to guide them. All these
instances, taken together in their positive and negative
aspects, led Sir John Lubbock to infer that the ocelli were
useful in dark places and for near vision, while the com
pound eyes were for the light and more distant vision. 1
Another illustration of this method may be found in
Darwin s account of the extreme tameness of the birds in
the Galapagos and Falkland islands. I quote some extracts
from his narrative, in which it will be seen that Darwin s
inferences follow from his comparison of the positive and
negative instances before him. He says : " This tameness
of disposition is common to all the terrestrial species of
these islands in the Galapagos Archipelago ; namely, to the
mocking-thrushes, the finches, wrens, tyrant flycatchers, the
dove, and carrion-buzzard. All of them often approached
sufficiently near to be killed with a switch, and sometimes,
as I myself tried, with a cap or hat. A gun is here almost
superfluous ; for, with the muzzle, I pushed a hawk off the
branch of a tree. In Charles Island, which had been colo
nized about six years, I saw a boy sitting by a well with a
1 Lubbock, On the Senses, Instinct, and Intelligence of Animals, pp. 175 ff.
256 INDUCTIVE LOGIC
switch in his hand, with which he killed the doves and
finches as they came to drink. He had already procured a
little heap of them for his dinner ; and he said that he had
constantly been in the habit of waiting by this well for the
same purpose. The Falkland Islands offer instances of
birds with a similar disposition. The snipe, upland and
lowland goose, thrush bunting, and even some true hawks,
are more or less tame. The black-necked swan is here wild,
and it was impossible to kill it. It however is a bird of
passage, which probably brought with it the wisdom learned
in foreign countries.
" From these several facts, we may, I think, conclude that
the wildness of birds with regard to man is a particular
instinct directed against him and not dependent on any
general degree of caution arising from other sources of
danger ; secondly, that it is not acquired by individual birds
in a short time, even when much persecuted, but that in the
course of successive generations it becomes hereditary.
With domesticated animals we are accustomed to see new
mental habits or instincts acquired and rendered hereditary,
but with animals in a state of nature it must always be
most difficult to discover instances of acquired hereditary
knowledge. In regard to the wildness of birds towards
man, there is no way of accounting for it except as an in
herited habit ; comparatively few young birds, in any one
year, have been injured by man in England, yet almost all,
even nestlings, are afraid of him ; many individuals, on the
other hand, both at Galapagos and at the Falklands, have
been pursued and injured by him, but yet have not learned
a salutary dread of him." l
I have given this quotation somewhat at length in order
to show the method of a great investigator in the realm of
nature ; and that it may be seen how naturally he falls into
the method of comparing positive and negative sets of in
stances relative to the object of research. The animal and
i Darwin, Voyage of a Naturalist, Vol. II, pp. 172 f .
AGREEMENT AND DIFFERENCE 257
vegetable kingdoms are especialty adapted to the applica
tion of this joint method, and therefore it is in biology that
it is most frequently employed and where it has yielded the
most fertile results.
The advantage of the joint method over the simple
method of agreement is that it largely eliminates the possi
bility of there being any other cause of the given phenome
non than the one disclosed by the operation of this method.
The method of agreement, as we have seen, often fails of a
definite result owing to the plurality of causes. The joint
method tends to indicate the one and only cause, and when
the instances are rigorously selected according to the condi
tions of the canon, there is a high degree of probability that
the sole cause is discovered. Mr c Mill at this point claims
too much for the method in insisting that it gives a certainty
regarding the sole cause, when the requirements are perfectly
realized. It is impossible to realize the requirements per
fectly. In selecting the negative instances, we are never
sure that we have compassed the entire sphere of significant
negative instances. We may, however, attain results highly
probable in this regard, though they may not reach an abso
lute certainty. Such a statement is more moderate in its
expression, and practically it assures as satisfactory results.
CHAPTER IX
THE METHOD OF CONCOMITANT VARIATIONS
THE method of concomitant variations is a process of
determining a causal relation when, as an element in an
antecedent varies in intensity greater or less, there is
observed a corresponding or concomitant variation in the
consequent.
Canon of the Method of Concomitant Variations. What
ever phenomenon varies in any manner, whenever another
phenomenon varies in some particular, is either a cause or
an effect of that phenomenon, or is connected with it through
some fact of causation.
The latter clause of this canon provides for that circum
stance in which the varying elements may both be con-
comitant effects of a common cause. When we are assured
of the absence of any possible common cause to which we
can assign the two phenomena as effects, then they must
be related between themselves as cause and effect. A
simple illustration of this method is the rise of the mercury
in the thermometer owing to the increase of heat ; its fall,
whenever there is decrease of heat. One varies as the
other concomitantly, and we infer a causal relation that
we at once proceed to generalize without hesitation.
The symbolical representation of this method is as
follows :
S + C dC ..... s + e de.
etc. etc.
Then C is the cause of e.
258
CONCOMITANT VARIATIONS 259
I have used dC and de to denote the increments or
decrements of the cause and effect respectively. This
method is used generally when the method of difference is
impossible, owing to the fact that the supposed causal
element cannot be made to vanish wholly. In all such
cases a variation of the element is resorted to, and the
corresponding result observed. Heat is relative and not
absolute, as also the height of mercury in the tube. The
relation is determined, therefore, by variations, greater and
less. This method is also used to supplement the results
of other methods by which a causal relation has been
determined, but not in exact quantitative terms. It may
be known that a certain phenomenon C is always the cause
of a certain effect e, and the method of concomitant vari
ations will then be of use in determining precisely how
much of a variation in C will cause a specified variation
in e. A law finds scientific expression only when stated in
terms of exact quantitative relation between variations in
antecedent and consequent. We may express the law of
universal attraction in a vague way that bodies always
attract each other, and the greater attraction when the
bodies are nearer together, and the larger they are. But
this statement needs to be recast in terms exhibiting the
precise quantitative variation, bodies attract each other
directly as the product of their masses, and inversely as the
square of their distance. It is evident that the special
function of this method of concomitant variations consists
in^just this quantitative determination. In one respect,
therefore, it may be regarded as a substitute for the method
of difference, and in another way as a supplement to the
method of difference in leading to quantitatively determi
nate results.
The quantitative variation between antecedent and con
sequent may be either direct or inverse variation. The
former is when one increases as the other increases, or
when one decreases as the other decreases. The inverse is
260 INDUCTIVE LOGIC
when one decreases as the other increases, or vice versa.
This may be expressed symbolically
S + C dC . . . . s -f e T de.
We have, for instance, Boyle s law as regards the variation
of volume of gases according to the pressure; that is,
when we double the pressure, we halve the volume. This
may be proved experimentally. The method also was
used by Ricardo to prove his law that the rate of profits
varies in inverse ratio to the rate of wages. We have also
the tendency observed in respect to increase of crimes
when there is decrease of opportunities for labor.
The expression of a law in terms of the quantitative
relation between antecedent and consequent may be facili
tated by a graphic representation of the same, through
corresponding abscissae and ordinates. The varying ante
cedents, for instance, may be laid off on the axis of X,
and each several consequent represented by the corre
sponding ordinate. The resulting curve thus obtained
will represent the law of their mutual relation. If the
equation of the curve can be determined, it will represent
the mathematically exact expression of the law in ques
tion. If this is not possible, it may prove at least sug
gestive of the law which otherwise might have remained
concealed. This graphical method is especially useful in
dealing with physical phenomena. "If the abscissae rep
resent intervals of time, and the ordinates corresponding
heights of the barometer, we may construct curves which
show at a glance the dependence of barometric pressure
upon the time of day. Such curves may be accurately
drawn by photographic processes on a sheet of sensitive
paper placed behind the mercurial column, and made to
move past it with a uniform horizontal velocity by clock
work. A similar process is applied to the temperature and
CONCOMITANT VARIATIONS 261
electricity of the atmosphere, and to the components of ter
restrial magnetism." l
The method of concomitant variation has the advantage
of the psychological impression which it makes. The
mind is more susceptible to the perception of variation in
forces where the change is apparent to the senses, than to
the perception of a constant force, whose constant character
thereby conceals its nature and function from the senses.
Synchronous changes attract the attention, and admit of
ready comparison, as we follow out the variations from
point to point. We may ring a bell in a vacuum, and detect
no sound whatsoever, and then allow the air to enter gradu
ally. We notice that as the air enters more and more freely,
the sound grows louder and louder. The relation of cause
itnd ct iV ct ii thus demonstrated t<> tin- senses in tin- nmst.
vivid manner possible. The variations are exhibited side
b^ sideband thus, presented together in their concomitant
relation, produce the deeper impression.
This method is of special advantage in all experiments
where the intensity of the forces can be varied at will and
the consequent effects exhibited in some palpable manner.
The determination of the heat rays in the solar spectrum is
accomplished by this method. The spectrum may be
received upon a plate pierced with a narrow slit, through
which the rays can act upon a thermo-electric pile, which
will indicate by deflections of a needle the varying intensity
of the heat in the several rays of the spectrum. Now,
move the slit through the whole extent of the spectrum,
beginning with the violet portion. While in the violet, the
indigo, the blue, and even the green, the needle of the ther-
moscopic apparatus will be deflected but slightly, it will
indicate an amount of heat increasing as the slit crosses the
yellow, next the orange, then the red ; and then beyond the
red, and entering the dark portion of the spectrum, we find
the greatest deflection of all. The maximum of heat is
1 Thomson and Tait, Elements of Natural Philosophy, Vol. I, p. 119.
262 INDUCTIVE LOGIC
therefore in a region beyond the observation of the senses
when unaided by experimental device ; and yet revealed
conclusively by this method. 1
Professor Tyndall performed a very interesting experi
ment to prove that the cloud of darkness surrounding flames
of great heat was due to the fact that the heat consumed
the floating motes in the air which serve to scatter the light
which is visible only when thus diffused. The phenomenon
which he endeavored to explain was somewhat as follows :
Beneath a beam of electric light, a red-hot poker was placed,
and from it black wreaths as of smoke were seen to ascend.
A large hydrogen flame being employed, it produced whirl
ing masses of darkness far more copiously than the poker.
Of this Professor Tyndall remarked : " Smoke was out of
the question ; what then was the blackness ? It was simply
that of stellar space ; that is to say, blackness resulting from
the absence from the track of the beam of all matter com
petent to scatter its light. When the flame was placed
below the beam, the floating matter was destroyed in situ ;
and the air freed from this matter rose into the beam, jostled
aside the illuminated particles, and substituted for their
light the darkness due to its own perfect transparency.
Nothing could more forcibly illustrate the invisibility of
the agent which renders all things visible. The beam
crossed, unseen, the black chasm formed by the transpar
ent air, while at both sides of the gap the thick-strewn par
ticles shone out like a luminous solid under the powerful
illumination." Such being the phenomenon and Professor
Tyndall s explanation, it will be seen that he proceeded
according to the method of concomitant variations in the
following experiment of many which he performed to sub
stantiate this theory :
A platinum tube with its plug of platinum gauze was
connected with an experimental tube, through which a pow-
1 Saigey, The Unity of Natural Phenomena, p. 61.
* Tyndall, Fragments of Science, p. 280.
CONCOMITANT VARIATIONS 263
erful beam could be sent from an electric lamp placed at its
end. The platinum tube was heated till it glowed feebly
but distinctly in the dark. The experimental tube was
then exhausted, and filled with air that had passed through
the red-hot tube. A considerable amount of floating matter
which had escaped combustion was revealed by the electric
beam.
Then the tube was raised to a brighter redness and the
air permitted to pass slowly through it. Though diminished
in quantity, a certain amount of floating matter passed into
the exhausted experimental tube.
The platinum tube was rendered still hotter; a barely
perceptible trace of the floating matter now passed through
it. The experiment was repeated, with the difference that
the air was sent more slowly through the red-hot tube. The
floating matter was totally destroyed. The platinum tube
was now lowered until it bordered upon a visible red heat.
The air, sent through it still more slowly than in the last
experiment, carried with it a cloud of floating matter. Pro
fessor Tyndall s commentary upon this experiment is as
follows: "If, then, the suspended matter is destroyed by
a bright red heat, much more is it destroyed by a flame
whose temperature is vastly higher than any employed
in this experiment. So that the blackness introduced
into a luminous beam where a flame is placed beneath it
is due, as stated, to the destruction of the suspended
matter." l
Professor Tyndall also supplemented this experiment by
one which was according to the joint method of agreement
and difference. He prepared oxygen so as to exclude all
floating particles, and found that when blown into the beam,
darkness was produced ; also that hydrogen, nitrogen, car
bonic acid, and coal-gas, when prepared in a similar way,
each produce darkness when poured or blown into the beam.
These instances, combined with various positive instances of
l Tyndall, Fragments of Science, pp. 283, 284.
264 INDUCTIVE LOGIC
illumination of mote-strewn currents of air, illustrate the
method of agreement and difference.
An additional experiment, according to the method of
difference, was as follows : Professor Tyndall placed an
ordinary glass shade in the air with its mouth downward.
This permitted the track of the beam to be seen crossing it.
Letting coal-gas, or hydrogen, enter the shade by a tube
reaching to its top, the gas gradually filled the shade from
the top downward. As soon as it occupied the space crossed
by the beam, the luminous track was instantly abolished.
Lifting the shade so as to bring the common boundary of
gas and air above the beam, the track flashed forth. After
the shade was full, he inverted it ; thereupon the gas passed
upward like a black smoke among the illuminated particles. 1
The method of concomitant variations is not only capable
of illustration by laboratory methods and devices ; it finds
abundant illustration as well in the realm of nature, where
observation alone becomes the instrument of investigation
and where experiment is impossible or impracticable. Lyell,
in his Principles of Geology, gives a very interesting account
of the alternate elevation and subsidence of the temple of
Jupiter Serapis, at Pozzuoli, on the Bay of Naples.* It is
situated in proximity to several volcanoes, Vesuvius, however,
being at some distance. It has been observed that there is
a certain connection between each era of upheaval, and a
local development of volcanic heat ; and on the other hand,
between each era of depression, and the local quiescent con
dition of volcanic phenomena. Before the Christian era,
when Ischia was in a state of eruption, and Avernus and
other points in the Phlegrsean fields were celebrated for their
volcanic character, it was observed that at that time the
ground on which the temple stood was several feet above
water. Vesuvius was then regarded as a spent volcano.
After the Christian era, Vesuvius became active and then
1 Tyndall, Fragments of Science, pp. 284, 285.
2 Chapter XXX.
CONCOMITANT VARIATIONS 265
scarcely a single eruption occurred in Ischia or around the
Bay of Baiae. It was observed that at that time the temple
was sinking. Vesuvius then became quiet for five centuries
preceding the eruption of 1631, and during that period the
Solfatara was in eruption in 1198, Ischia in 1302, and Monte
Nuovo was formed in 1538. Then the foundations of the
temple were observed to be rising again. Vesuvius became
active after that, and has continued so ever since, and during
this time the temple has been subsiding. The inference is
that as the subterranean heat increases, and lava forms
without obtaining an easy vent like that afforded by Vesu
vius, the surface is elevated, but when the rocks below are
cooling and contracting, the pent-up fire being withdrawn
in the eruption of the great Vesuvius, then there is a cor
responding subsidence.
The observation of concomitant variations is furthermore
illustrated in Darwin s researches concerning the formation
of coral reefs, as regards the question which some natural
ists have raised as to which part of the coral reef is most
favorable to the growth of coral. 1 He adduces the follow
ing facts, most of which are the direct result of his observa
tions : " The great mounds of living Porites and of Millepora
round Keeling atoll occur exclusively on the extreme verge
of the reef, which is washed by a constant succession of
breakers. At the Marshall Islands the larger kinds of
coral which form rocks measuring several fathoms in thick
ness prefer the most violent surf. The outer margin of the
Maldiva atolls consists of living corals, and here the surf is
so tremendous that even large ships have been thrown, by
a single heave of the sea, high and dry on the reef, all on
board thus escaping with their lives. In the Red Sea the
strongest corals live on the outer reefs and appear to love
the surf. From these facts it is certain that the strongest
and most massive corals flourish where most exposed. The
less perfect state of the reef of most atolls on the leeward
i Darwin, Coral Reefs, pp. 87 f.
266 INDUCTIVE LOGIC
and less exposed side, compared with its state to the wind
ward, and the analogous case of the greater number of
breaches on the rear sides of those atolls in the Maldiva
Archipelago, which afford some protection to each other,
are obviously explained by this circumstance." There
seems to be here a combination of the method of agreement
with that of concomitant variations. And such a combina
tion may be employed to advantage in cases where the phe
nomena under investigation show forces under varying
degrees of intensity ; the causal relation is more apparent,
and the possibility of fortuitous coincidence is largely elimi
nated if a number of instances can be collected in which the
forces manifest themselves in varying degrees. Accumula
tion of instances, showing concomitant variations in the
forces observed, corresponds to the actual variations which
in an experiment are effected by the investigator himself.
In such observed instances, however, we cannot always have
before us the variations expressed continuously. There
are evident gaps that must be interpolated mentally. In
experiment however of whatever nature, the degrees of
intensity can be exhibited continuously, one degree merg
ing into another through inappreciable increments. There
is thus a gradation which has no gaps to be filled, and the
psychological impression is thereby heightened.
By the method of concomitant variations it is possible also
to represent to the mind the magnitude of an unknown force,
or un observable force, by a comparison with the intensity
of a known force which lies within the sphere of observa
tion. For instance, Mr. Darwin gives an interesting account
in his narrative of the finding near the shores of the Plata
a group of vitrified siliceous tubes which had been formed
by lightning entering loose sand. The internal surface of
such tubes is completely vitrified, glossy, and smooth, and
the tubes themselves are generally compressed, and have
deep longitudinal furrows so as closely to resemble a
shrivelled vegetable stalk, or the bark of an elm or cork
CONCOMITANT VARIATIONS 2G7
tree. Their circumference is about two inches, but in some
fragments which are cylindrical and without any furrows,
it is as much as four inches. Judging from the uncom
pressed fragments, the measure or bore of the lightning
proved to be about one inch and a quarter. In contrast
with the force of lightning as thus revealed in its effects,
Mr. Darwin cites some experiments performed in Paris by
an artificial force of great magnitude indeed and yet with
results that seem insignificantly small in comparison. He
says : " At Paris, M. Hatchette and M. Beudant succeeded
in making tubes in most respects similar to these fulgurites
by passing very strong shocks of galvanism through finely
powdered glass : they failed, however, both with powdered
felspar and quartz. One tube, formed with pounded glass,
was very near an inch long, namely, .982, and had an inter
nal diameter of .019 of an inch. When we hear that the
strongest battery in Paris was used, and that its power on
a substance of such easy fusibility as glass was to form
tubes so diminutive, we must feel greatly astonished at the
force of a shock of lightning, which, striking the sand in
several places, has formed cylinders in one instance at least
thirty feet long, and having an internal bore, where not
compressed, of full an inch and a half ; and this in a mate
rial so extraordinarily refractory as quartz ! " l
The method of concomitant variations may be used in
regard to phenomena whose nature is such as seemingly to
indicate a constant law of variation, and yet inferences
based thereupon lead to false results. It is therefore well
to note some of these instances by way of general precaution
in applying this method.
1. It does not necessarily follow that having observed
two forces varying in a constant ratio through several con
comitant modifications, the same ratio will be preserved
indefinitely through all subsequent changes. Water con
tracts as it is cooling. Suppose we begin to note this con-
1 Darwin, Voyage of a Naturalist, Vol. I, pp. 76 f.
268 INDUCTIVE LOGIC
tinned contracting of water from 100 F. to 90 ; we
naturally expect to find it continuing through 90 to 80.
And as we observe, we find our expectations confirmed.
And so on through to 40, we find that water continues to
contract. It is, therefore, most natural for us to expect to
find water contracting at 39. But just at this point in the
series, there is a break in the continuity of variation; at
39 water begins to expand and so continues until it passes
into the solid form at the freezing-point. The same also is
illustrated in Weber s law, already mentioned, which ex
presses the quantitative relation between the stimulus and
the corresponding sensation. The law is that the force of
the stimulus must increase geometrically, in order that the
intensity of the sensation should increase arithmetically.
This law, however, breaks down towards the upper or
lower limits, with a stimulus of slight degree of intensity
and with one of extreme intensity. We find also an in
crease of temperature as we proceed towards the centre of the
earth of about one degree to every fifty-three feet of descent.
This by no means warrants us in inferring that this ratio
continues constant to the very centre itself. In certain
phenomena, moreover, there are natural limits, as in sound,
for example, where the pitch rises as the number of vibra
tions increases. At a certain point, varying according to
different individuals, increase of vibrations gives no result
ing sound whatsoever ; and so there is a lower limit, vibra
tions may decrease to a point beyond which no sound is
heard.
An illustration of this fallacy, though not strictly of the
method of concomitant variations, is given by Jevons. He
takes the following series of prime numbers : 41, 43, 47, 53,
61, 71, 83, 97, 113, 131, etc. It will be seen that they all
agree in being values of the general expression x 2 + x -f- 41,
where we put for x the successive values of 0, 1, 2, 3, 4, etc.
For instance, let # = in x 2 + ic + 41, we get 41 ; let x = 1
in the same, we get 43 ; when x 2, we get 47 j and so on.
CONCOMITANT VARIATIONS 269
It seems as though we could keep this up indefinitely, pro
ducing an increasing series, always of prime numbers. It
is found, however, that if we take x = 40, in the formula
x * + x + 41, we shall have 40 X 40 + 40 + 41, which equals
1681, and this number is the square of 41 and therefore not
a prime number. At this point the law breaks down. 1
In the sphere of political economy also we might be led
into an easy yet false inference. Suppose a certain farm
yield 500 bushels of corn with a given amount of expendi
ture and labor. We might think that if we double the
expenditure and labor, we will also be able to double the
results, and obtain a yield of 1000 bushels as over against
the 500 of the previous year. Here, however, what is
known as the law of decreasing returns obtains ; to double
the product it may be necessary to triple or quadruple
the labor and expense. Thus in the production of any
plot of land there is a point of equilibrium, which marks
an impassable limit, not of course a limit which could
not be passed if it were wished, but one that no one
wishes to pass, because there is nothing to be gained by so
doing." 2
To know that such false inferences are at least possible
in the application of this method of concomitant variations
to the unknown regions beyond our experience, may serve
at least to keep us on guard under similar circumstances.
2. There are certain phenomena moreover in which an
increased intensity of the force in question may give rise to
incidental effects which tend to neutralize the chief effect to
be attained. For instance, an overdose of arsenic causes
violent contractions of the stomach so that its contents are
immediately ejected, and thus the system is relieved of the
noxious substance.
3. Two elements in a given phenomenon may vary to
gether constantly and yet they may not be related at all as
i Jevons, Principles of Science, p. 230.
a Gide, Political Economy, p. 325.
270 INDUCTIVE LOGIC
cause and effect, but appear as coincidental effects of one
and the same cause. It has been observed that the occur
rence of the aurora borealis has been accompanied by pro
nounced magnetic disturbances. It, however, cannot be
inferred that the former has been the cause of the latter ;
they are probably the varied effects of some widely operat
ing magnetic force.
The precaution above mentioned has already been referred
to as provided for in the canon of this method which states
that the observed concomitant variation may indicate not
always a direct causal element between the two varying
elements, but that they are at least connected with the phe
nomenon under investigation through some fact of causation.
CHAPTER X
THE METHOD OF RESIDUES
THE method of residues consists in the analysis of a given
phenomenon based upon previous inductions, through which
it has been determined that certain elements in the antece
dent have caused certain elements in the consequent ; the
inference is then drawn, that the remaining elements of the
antecedent are necessarily the cause of the remainder of
the consequent. It is a method of elimination of the known
relations so as to simplify the complex character of the phe-
"nomehoh and disclose the relations that are unknown in the
light of a causal connection which we are constrained to
believe must obtain.
The Canon of the Method of Residues. Subduct from
any^ phenomenon such part as is known by previous induc
tions to be the effect of certain antecedents, and the residue
of the phenomenon is the effect of the remaining antecedents.
The symbolical representation is as follows :
Given S + C s + e.
If it is known that there exists already the causal relation
S *,
we may then infer that C is the cause of e. In this, C may
be simple or complex ; if it is simple, the causal relation
established is expressed in its simplest terms and is there
fore a determinate result. If however the residue C is
complex, it must be reduced by experimental analysis to its
simplest elements, and their relation to the elements into
which e can be analyzed further determined.
271
272 INDUCTIVE LOGIC
The most striking illustration of this method, and one
of the most brilliant achievements of science as well, is the
discovery of the planet Neptune by Adams and Le Verrier,
working on the problem independently and reaching the
same result. These astronomers had observed certain per
turbations in the planet Uranus. It did not keep in its
proper orbit as determined by their mathematical calcula
tions based upon the presence of the known stellar bodies.
It behaved as though beyond its orbit was an outer planet,
whose presence alone could account for the observed per
turbations. Adams and Le Verrier then proceeded to calcu
late the exact position of such a disturbing body as determined
by the nature and magnitude of the perturbations of Uranus.
The telescope was then pointed to the exact point in the
heavens, as thus indicated, and the planet Neptune was
revealed to the eye according to the determination of far-
reaching prophecy, which confidently asserted that it must
be there.
The method of residues is really a deductive method
based upon the law of sufficient reason ; so many elements
on the one hand producing so many elements on the other^
if, then, a part of the former can be checked off as cause of
a part of the latter, then the remainder on the one hand
must be the cause of the remainder on the other. This is
pure deduction. For we ask, Why are we constrained to
account for the remainder on one side by the remainder
on the other ? The only possible answer is that it must be
accounted for within the system to which it is referred ;
and but one part therein is left which can possibly account
for it, because all the others are specifically determined in
the known effects which they have produced. This method,
however, has a proper place among the inductive methods,
inasmuch as it is based on previous inductions, and leads
to investigations that can be prosecuted only by the various
inductive processes of experiment.
When the residue of the antecedent is a simple element,
THE METHOD OF RESIDUES 273
and no other possible causal element can lie concealed from
our observation, then the inference is simple and conclusive.
A difficulty, however, may present itself, owing to the fact
that the residual element is apt to be complex and leave the
phenomenon still indeterminate, or there may be a lurking
element unnoticed by us which is the real cause in question.
The function of this method is, therefore, largely suggestive.
It says the effect is not wholly accounted for by the known
causal elements ; there is a residue unaccounted for, and its
cause_ is to be sought in the residue of the antecedent, and
if it io thought that the whole of the antecedent is compre
hended, the question is started, May there not be unobserved
circumstances of the antecedent that further experiment
will be calculated to reveal ? In many cases, therefore, this
method must be supplemented by some other experimental
method in order to secure more precise determination, gen
erally the method of difference. It often happens in inves
tigations in chemistry, astronomy, and physics, that the
actual phenomena vary in greater or less degree from their
expected behavior according to established theory. This
must lead either to a reconstruction of theory, or to a search
for some unobserved force sufficient to account for the dis
crepancy. Herschel was the first to point out the signifi
cance of such discrepancies in scientific research, and he
called them residual phenomena.
An illustration of such a situation and the solution of the
problem thus presented is that of Sir Humphry Davy s ex
periments upon the decomposition of water by galvanism.
" He found that besides the two components of water, oxy
gen and hydrogen, an acid and alkali were developed at the
two opposite poles of the machine. As the theory of the
analysis of water did not give reason to expect these products,
they were a residual phenomenon, the cause of which was still
to be found. The insight of Davy conjectured that there
might be some hidden cause of this portion of the effect ; the
glass containing the water might suffer partial decomposition,
274 INDUCTIVE LOGIC
or some foreign matter might be mingled with the water, and
the acid and alkali be disengaged from it, so that the water
would have no share in their production. Assuming this,
he proceeded to try whether the total removal of the cause
would destroy the effect produced. By the substitution of
gold vessels for the glass, without any change in the effect,
he at once determined that the glass was not the cause.
Employing distilled water, he found a marked diminution
of the quantity of acid and alkali evolved ; yet there was
enough to show that the cause, whatever it was, was still in
operation. The impurity of the water, then, was not the
sole, but a concurrent, cause. He now conceived that the
perspiration from the hands touching the instruments might
affect the case, as it would contain common salt, and an
acid and alkali would result from its decomposition under
the agency of electricity. By carefully avoiding such con
tact, he reduced the quantity of the products still further,
until no more than slight traces of them were perceptible.
What remained of the effect might be traceable to impuri
ties of the atmosphere decomposed by contact with the
electrical apparatus. An experiment determined this ; the
machine was placed under an exhausted receiver, and when
thus secured from atmospheric influence, it no longer evolved
the acid and alkali." 1
By means of the suggestions incident upon this method,
Bunsen, in 1860, discovered two new alkaline metals, cae
sium and rubidium. He was examining alkalies produced
by the evaporation of mineral water from Durkheim. The
flame of these salts was examined by the spectroscope.
Bunsen discovered several bright lines which he had never
noticed before, and which he knew could not be produced
by potash or soda, whose corresponding lines were in close
proximity. He then subjected the mixture to a searching
analysis and succeeded in obtaining two new alkaline sub
stances. When he had separated them, he then tested them
1 Gore, The Art of Scientific Discovery, pp. 432, 433.
THE METHOD OF RESIDUES 275
by the method of difference, by which he found that
they were capable of producing the lines at first no
ticed ; but when withdrawn, the lines instantaneously dis
appeared.
Thomson and Tait, in their Elements of Natural Philoso
phy, have the following reference and illustration of this
method. "When, in an experiment, all known causes being
allowed for, there remain unexplained effects (excessively
slight it may be), these must be carefully investigated, and
every conceivable variation of arrangement of apparatus,
etc., tried ; until, if possible, we manage so to exaggerate
the residual phenomenon as to be able to detect its cause.
It is here perhaps that in the present state of science we
may most reasonably look for extensions of our knowledge ;
at all events, we are warranted by the recent history of
natural philosophy in so doing. Thus, to take only a very
few instances, and to say nothing of the discovery of elec
tricity and magnetism by the ancients, the peculiar smell
observed in a room in which an electrical machine is kept in
action was long ago observed, but called the smell of elec
tricity, and thus left unexplained. The sagacity of Schon-
bein led to the discovery that this is due to the formation
of ozone, a most extraordinary body, of enormous chemical
energies ; whose nature is still uncertain, though the atten
tion of chemists has for years been directed to it." 1
Another illustration of this method is seen in the com
parison of the observed and calculated positions of Encke s
comet. It was found that the comet returned a little sooner
than it should have done, the period regularly decreasing
from 1212.79 days, between 1786 and 1789, to 1210.44
between 1855 and 1858. The inference has been that there
is a resisting medium, as the ether, filling the space through
which the comet passes. What the resisting medium is,
and its nature, is of course a matter of conjecture as far 
