Elements of inductive logic

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INDUCTIVE LOGIC

BY

NOAH K. DAVIS, Ph.D., LL.D.

PROFESSOR OF MORAL PHILOSOPHY IN THE UNIVERSITY OP VIRGINIA

AND AUTHOR OF "THE THEORY OP THOUGHT"

*' ELEMENTS OF DEDUCTIVE LOGIC " ETC.

"Ot< scientice fundamentum eat, Stori fastigium

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NEW YORK

HARPER & BROTHERS PUBLISHERS

1895

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Copyright, 1895, by Harper & Brothers.
All rights reserved.

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PEEFACE

In preparing the present treatise, I have kept in view
the need of collegians and of graduate students in the
universities, and endeavored to furnish them with a satis-
factory hand-book on Induction. The few pages in popu-
lar treatises on Deductive Logic usually allotted to this
co-ordinate branch being utterly inadequate and dispro-
portionate, and thereby greatly underrating its extent and
importance, should be replaced by a separate treatise com-
prehending at least the essential elements of Induction,
and opening the way for its full investigation and applica-
tion. In the hope of supplying this want, I offer to stu-
dents well advanced in the schools the work in hand.

Special students engaged in the pursuit of physical
science, who have not enjoyed a full course in Logic, need
a compact hand-book on Induction, in order to gain a
clearer insight into the principles of the methods they are
employing, and thus to avoid a waste of energy, and the
discouragement of blunders in the dark. To this class of
students, also, and to the general reader who desires a
clearer knowledge of his own mental processes and of
those of the scientist skilled in the discovery of truth, my
work is hopefully addressed.

With these ends in view, I have earnestly tried, first of
all, to be true in matter, then clear and distinct in its treat-
ment. Whoever is acquainted with the literature of the

IV PREFACE

subject will recognize my helps, and will, at the same
time, accord to me some fair measure of independence. A
profusion of illustration has been used, drawn largely from
the humbler departments of knowledge, yet in many cases
taken from the physical sciences, not for display, but for
service, avoiding recondite examples, the purpose being to
teach, not physics, but Logic.

The text in the larger type is for the tyro. The many
marginal notes, which have been added with much pains,
are for the scholarly reader who desires further information.
The abundant references to authorities not only indicate
my own sources, but will serve to direct those interested
to wider fields. As some acquaintance with Deduction
is prerequisite to the understanding of Induction, I have
ventured to make references to my " Elements of Deductive
Logic," the companion of the present work, also a few to
" The Theory of Thought," and to my " Elements of Psy-
chology." I ask indulgence for these references, trusting
that the bad taste will be neutralized by their helpfulness
to those who may have the books at hand.

To Professor Collins Denny, of Vanderbilt University
I am gratefully indebted for encouragement, and for very
many valuable suggestions.

Noah K. Davis.

University of Virginia.

CONTENTS

I.— DEFINITION

Page

§ 1. Logic defined and divided 1

§ 2. In both branches a science of forms 1

§ 3. Induction distinguished from deduction and defined. . 4

§ 4. Induction synthetic in extension and intension 7

§ 5. Analytic judgments distinguished 8

§ 6. Induction a generalization from experience 10

§ 7. Pure truths distinguished from empirical 11

§ 8. Induction a generalization beyond experience 14

§ 9. Summary or closed generalization distinguished 15

§ 10. Identification to establish a minor distinguished 16

§ 11. Search after causal relation distinguished 19

§ 12. The definition adequate and real 21

II.— PRINCIPLES

§ 13. Additional principles requisite for induction 22

§ 14. General meaning of cause and condition 22

§ 15. No simple cause or effect. Preventive cause 24

§ 16. Theoretic view. Definitions of cause and effect 25

§ 17. Recent scientific view of causation 27

§ 18. The principle or axiom of change 29

§ 19. The first principle or axiom of uniformity 31

§ 20. Plurality of effects, its maxim. Joint effects 33

§ 21. The second principle or axiom of uniformity 35

§ 22. Plurality of causes, its maxim. Resultant motion 37

§ 23. Uniformity of nature. The axioms compared 39

VI CONTENTS

III.— PROCESS

Page

§ 24. An inductive inference exemplified 41

§ 25. Its conformity to the definition and axioms 41

§ 26. Its immediate character. Formulas 43

§ 27. Aristotle's inductive syllogism examined 44

§ 28. Hamilton's inductive syllogism criticised 46

§ 29. Whately's and Mill's syllogism criticised 47

§ 30. General objections to the syllogistic view 48

§ 31. The function and application of forms 50

§ 32. Induction immediate. Preparatory process 51

IV.-OBSERVATION

§ 33. Phenomena of coexistence and of succession 54

§ 34. Observation illustrated. Its two modes 55

§ 35. Simple observation. Its application 57

§ 36. Experimental observation. Its prerogatives 59

v.— ENUMERATION

§ 37. Description. Two kinds of enumeration 62

§ 38. Canon and formula of enumeration of cases 63

§ 39. The justification of this form of induction 64

§ 40. Its practical and scientific value 66

§41. Analogy distinguished from metaphor, and described.. 67

§ 42. Canon and formula of enumeration of marks 69

§ 43. Justification and limitation of analogy. Examples. . . 71

§ 44. Its practical and scientific value 73

VI.— PROBABILITY

§ 45. Certainty discriminated. Range of probability 76

§ 46. Practical importance of probable estimates 78

§ 47. Significance of exceptional cases 80

§ 48. Chance occun*ence and concurrence 82

§ 49. Calculation of chance. Two special cases 84

§ 50. Separation of casual from causal phenomena. Canon . 88

§ 51. The elimination of chance concurrences 91

§ 52. The general valuation of probabilities 94

§ 53. Their numerical valuation. Statistics 98

CONTENTS Vll

VII.— DIFFERENCE

Page

§ 54. Scientific or perfect induction. Canon 102

§ 55. Methods of determining causal relations 103

§ 56. The Method of Difference. Canon and formula 105

§ 57. Examples of the method from simple observation 107

§ 58. Examples from experimental observation. Tests 109

§ 59. Formulas of induction and deduction Ill

§ 60. The Method of Residue. Canon and formula 112

§ 61. Examples of discovery by this method 114

VIII.— AGREEMENT

§ 62. The Method of Agreement. Canon and formula 116

§ 63. Examples of the application of this method 118

§ 64. General precautions relative to the methods 120

§ 65. Imperfection of the method of agreement ":. . 122

^ 66. Its results only probable. Its scientific value 123

§ 67. The Method of Double Agreement, Canon and for-
mula 125

§ 68. Illustration of its application. Its prerogatives 127

§ 69. A standard example, the research on dew 128

/ / • -
IX.-CONCOMITANCE

§ 70. Method of Concomitant Variations. Canon and for-
mula 130

^ 71. Illustration of \ts application and insufficiency 132

§ 72. Examples of direct and inverse concomitance 133

§ 73. Measurement of quantity, the mark of advanced sci-
ence 135

§ 74. The service of this method in developing a science. . . 137
§ 75. Three limitations to a mathematical induction 138

X.— DEDUCTION

§ 76. Deductions subsequent to induction. Discovery 141

§ 77. Deductions precedent. Two classes of effects 146

§ 78. The Method of Deduction. Canon and formula 148

§ 79. Three stages in the procedure. Example 151

Vlll CONTENTS

XL— HYPOTHESIS

Page

§ 80. The universal use of supposition or hypothesis 155

§ 81. Supposition involved in all the methods of science. . 158

§ 82. Formal use of hypothesis in the deductive method . . 160

§ 83. Definition of scientific hypothesis 162

§84. Hypothesis of cause with known law. Vera causa.. 162

§ 85. Hypothesis of law with known cause. Other forms. 165

§ 86. Rival hypotheses. Instantice crucis 168

§ 87. Verification alone not proof. Power of prediction . . 169

§ 88. Proof of an hypothesis, two steps. Illustrated 171

§ 89. Example of the use of this method by Newton 174

XII.— NATURAL LAW

§ 90. General definition of law 177

§ 91. Formal and material law 178

§ 92. Moral and natural law 179

§ 93. Distribution of natural law 182

§ 94. Empirical laws of coexistence 183

§ 95. Empirical laws of succession 185

§ 96. Rational derivative laws. Examples 187

§ 97. Explanation in its philosophical sense 190

I 98. Laws of Nature. Examples 193

§ 99. Inductive sciences becoming deductive 197

§ 100. The number of the ultimate Laws of Nature 199

Index 201

ELEMENTS OF
INDUCTIVE LOGIC

I.— DEFINITION

§ 1. Logic is the science of the necessary
forms of thought. This is the definition of pure
logic as distinguished from modified and from ap-
plied logic, and from what is called material logic.
Pure logic, or simply logic, is divided primarily into
Deductive Logic and Inductive Logic. The specific
difference between these will come to light as we
proceed. The latter only is the subject of the pres-
ent treatise-.

§ 2. In undertaking to expound the theory of in-
duction, it is important to state and insist at the
outset that the limitation to forms of thought is as
proper to this branch of logic as it is to deduction.
A number of writers on logic take a contrary view,
holding that Deductive Logic is formal, Inductive
Logic material ; the one having to do subjectively

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with the laws of thought, the other objectively with
the laws of things ; the one being the logic of con-
sistency, the other the logic of truth, especially of
science ; the one being a jprioriy the other a jposteri-
ori in the sense that it considers the character of in-
dividual things or their classes, and thence rises by
induction to their laws. These striking antitheses
are not justifiable. It is impossible to treat of any
matter unless in some form ; the laws of thought
accord with and lead to the laws of things, as every
natural realist maintains ; each branch must require
self-consistency and be truth-giving, else it is wortli-
less ; and the theory of induction, as well as that of
deduction, is aprior% since it likewise demonstrates
its canons, starting from axiomatic principles (§ J^).^

There are other writers on logic who take an ex-
treme view, holding that both Deductive and Induc-
tive Logic are material, that logic in general is an
empirical and not a formal science, having to do with
things and laws of things rather than with forms and
laws of thought.'

^ The reference is to the companion treatise entitled " Elements of
Deductive Logic " (Harper & Brothers). See a discussion of the several
terms of the definition of Logic in Chapter I. of its Introduction.
References to that treatise are in Italics. Figures in Roman type (as,
§ 25) relate to the present treatise.

^ Writers adhering to the school of material logicians, if it may be
so called, usually claim Mr. J. S. Mill as its founder. Prominent
among them is Mr. Venn, notably in his " Empirical Logic." Elsewhere
he says: "With what may be called the Material view of Logic, as
opposed to the Formal or Conceptualist — with that which regards it
as taking cognizance of laws of Things and not of the laws of our own
minds in thinking about things — I am in entire accordance." — Logic

DEFINITION 3

But logic, throughout both deduction and induc-
tion, treats only of form, regardless of matter. To
consider the matter of thought in either branch
would, as Aristotle says, require omniscience, for sci-
ences are possibly infinite ; but the forms of thought
being few can be comprised in a single treatise, and
being the same for all varieties of matter, they alone
need to be studied in their abstract generality in or-
der to discover the necessary processes by which J
truth is attained (§ 5). This, then, is the sole prov- /./(^■^^<^
ince of logic : To unfold the formal principles and ,'
deduce from them tlie formal laws by which we think
material things and their laws.*

of Chance^ Preface, p. x. Afterward (ch. x., § 2) he quotes with
approbation Mr. Mill's saying that the conceptualist view is " one of
the most fatal errors ever introduced into the philosophy of Logic." —
Mill, Logic^ p. 74 (Harper's ed.). For a detailed exposition of Mr.
Mill's views, see his " Examination of Hamilton's Philosophy," ch. xx.
Mr. Venn, notwithstanding his emphatic endorsement of Mr. Mill, gives
us an elaborate work on *' Symbolic Logic," which is necessarily and
essentially formal throughout ; and Mr. Mill in one place very truly says :
" The business of Inductive Logic is to provide rules and models (such
as the Syllogism and its rules are for ratiocination) to which, if induc-
tive arguments conform, those arguments are conclusive, and not other-
wise."— Logic^ p. 308. Moreover, both Mr. Venn and Mr. Mill in all
their logical writings are constantly, and though inconsistently yet hap-
pily, occupied with an exposition of the forms of thought, illustrated
by material examples. Otherwise, indeed, these writings would not
be merely on Logic, but de omnibus rebus et quibusdam aim. The
obscurity which seems to cling so strangely and persistently in these
latter days to the Aristotelic distinction between the form and the
matter of thought, is very remarkable.

^ As Logic treats of the forms of thought, so Grammar treats of
the forms of speech, and Rhetoric of the forms of style. See Hamil-
ton, Discussions, article Logic, p. 139 (Harper's ed.).

4 ELEMENTS OF INDUCTIVE LOGIC

It should henceforth be clearly and constantly
noted that the various technical terms used in treat-
ing induction are names of forms, or second inten-
tions. Some of these are : judgment d^ndi proposition,
genus and species, inference, syllogism, phenomenon,
circumstance, instance or case, cause and effect, ante-
cedent and consequent, experience, observation, gener-
alization, uniformity, law. As elsewhere, however,
we shall here also freely use material examples and
illustrations in first intentions or names of things,
that the reader, while never failing to distinguish the
form from the matter, may be enabled to grasp more
firmly the form by means of concrete matter em-
bodying it (§ 6),

§ 3. Judgments are primarily of two kinds, intui-
tions and inferences (§ 77). Intuitions are self-evi-
dent, necessary judgments, and are divided into em-
pirical and pure. In these all knowledge has its
beginning ; they determine all other judgments. In-
ferences are enunciations in which from something
laid down and admitted, something distinct from
what is laid down follows of necessity.* To infer,

* It has been questioned whether this Aristotelic definition of Syl-
logism, "Analyt. Prior," i., 1, will, as to its last term "of necessity,"
apply to inductive, as it unquestionably does to deductive, inference.
Alexander of Aphrodisias, the Exegete (200 a.d.), in his "Schol. ad
Topica," p. 253, intimates that Aristotle included necessary sequence
in this definition for the specific purpose of distinguishing deduction
by syllogism from induction, a sequence that is not necessary. This
view has been generally adopted. But necessity here means only that
one cannot grant the premises and deny the conclusion without contra-

DEFINITION 6

then, is to derive a judgment from one or more pre-
mised judgments. Inferences also are of two kinds,
deductive and inductive.

Deductive inferences are judgments having a gen-
erality equal to or less than the premises from which
they are deriveid. We may proceed deductively from
all to all^ from some to som£, and from all to some,
but not from some to all (§ 79). Except in quantita-
tive cases, which compare masses, deduction is the de-
termination or specification of a class notion ; it de-
scends the logical scale (§ ^^), and thus is a priori.
It does not generalize, but specifies by inference
from intuitions or from inductions.

Inductive inference, on the contrary, by virtue of
principles to be presently discussed, ascends the log-
ical scale ; it generalizes, proceeding from the par-
ticular or the less general to the universal, from some
to all^ and thus, in the application of its demon-
strated theorems or canons, it is a posteriori. The
inference from some to all completes the possible
procedures, since every judgment concerns either all
or some of its subject (§ 62y

dieting axiomatic truth ; and it would seem, when the inductive prem-
ise expresses a causal relation perfectly ascertained (and the theory
presumes perfection), that the induction of a universal follows of ne-
cessity, in the sense stated. Hence we have ventured to use this
definition as a definition of inference in general, including deductive
inference (both immediate and mediate) and inductive inference.

^ " Induction is inferring a proposition from premises less general
than itself, and Ratiocination [Deduction] is inferring a proposition
from premises equally or more general." — Mill, Logic, p. 125.

It is worth noting that the names Deduction and Induction happily
express by their etymology (Lat. de-ducere and in-ducere) the inverse

t

6 ELEMENTS OF INDUCTIVE LOGIC

This division of the genus inference into deduction
and induction, differentiating the latter as a general-
ization, prepares us for an exact and full definition,
thus: Induction is an immediate synthetic
inference generalizing from and beyond
experience.*

correlation of the processes. The one is to lead or draw from estab-
lished generalities new particulars, the other is to lead or draw in un-
observed particulars under new generalities. In both the procedure
is from the known to the previously unknown.

r ' '* Induction is the process from particulars to universals." — Aris-
tbtle, Topica, i., 12-

"Inductionem enim censemus eam esse demonstrandi forraam,
quoe sensum tuetur et naturam premit et operibus imminet ac fere im-
miscetur. ... At secundum nos, axiomata [propositiones] continentur
et gradatim excitantur, ut nonnisi postremo loco ad generalissiraa ve-
niatur." — Bacon, Instauratio Magna, Dist. Op., p. 3.

" Induction is a kind of argument which infers, respecting a whole
class, what has been ascertained respecting one or more individuals
\ii that class." — Whately, Logic, Index.

Induction is "a formal illation of the universal from the individ-
ual, as legitimated solely by the laws of thought, and abstract from
the conditions of this or that particular matter." — Hamilton, Discus-
sions, p. 157.

"When, having discovered by observation and comparison that
certain objects agree in certain respects, we generalize the qualities in
which they coincide, that is, when from a certain number of individ-
ual instances we infer a general law, we perform what is called an
act of Induction." — Hamilton, Metaphysics, p. 72.

** Induction is usually defined to be the process of drawing a gen-
eral law from a sufficient number of particular cases; Deduction is the
converse process of proving that some property belongs to a particular
case, from the consideration that it comes under a general law." —
Thomson, Outline of the Laws of Thought, § 113.

" Induction is a term applied to describe the process of a true Col-
ligation of Facts by means of an exact and appropriate Conception."
— Whewell, Novum Organon Benovatum, bk. ii., aphorism 13.

DEFINITION 7

§ 4. In the definition are collected a number of
terms needful to further discriminations. That the
inference is immediate will be clearly established in
a subsequent discussion (§§ 24, 32).

That the inference is synthetic is evident, since it
concludes more than the content of its premise.
When from Some nien are mortals, we infer All
men are mortals, the subject is augmented, enlarged
from the narrow Some men of whom we know, to
the wide universal All men; thus adding to the
general class notion something not already contained
in iti Besides, the content of the predicate is aug-
mented ; for in the premise Some men are mortals,

" Induction may be defined, the operation of discovering and prov-
ing general propositions." — Mill, Logic^ p. 208.

( " Induction is that operation of the mind by which we infer that
What we know to be true in a particular case or cases will be true In
all cases which resemble the former in certain assignable respects.
In other words, Induction is the process by which we conclude that
what is true of certain individuals of a class is true of the whole class,
or that what is true at certain times will be true in similar circum-
stances at all times." — Mill, Logic^ p. 210.

" Induction may be summarily defined as Generalization from Ex-
perience. It consists in inferring from some individual instances in
which a phenomenon is observed to occur that it occurs in all in-
stances of a certain class ; namely, in all which resemble the former,
in what are regarded as the material circumstances." — W\\\ Logic^
p. 223.

" Induction is the generalization of conjoined properties, on the ob-
servation of individual instances." — Bain, Logic, Int., § 54.

" Induction is the arriving at General Propositions, by means of
Observation or fact." — Bain, Logic, bk. iii., ch. i., § 1.

In a careful search through Mr. Venn's "Empirical Logic," I was
unable to find that he anywhere ventures upon a succinct definition of
induction. But see his discussion, ch. xiv.

8 ELEMENTS OF INDUCTIVE LOGIC

the class notion mortals is only said to contain some
men, whereas in the conchision All Tnen are mortals
the notion mortals contains under it all men. This
adding to both subject and predicate is a double syn-
thesis.

Changing the form from extension to intension
(§ W) the synthesis remains. From Some men are
mortal, we infer All men are mortal. Here the
mark mortal, which in the premise is attributed to
some men only, is in the conclusion attributed to all
men. The content of the all men is thereby en-
larged by a synthesis of the mark mortal; and the
mark itself is synthetically enlarged from its narrow
attribution to some men, to its wide attribution to all
men. Thus in this view also we find a double syn-
thesis/

§ 5. The term synthetic in the definition clears
induction of a large and important class of judg-
ments which are not synthetic, but analytic. When
a predicate belongs to a subject as something which
is already though covertly contained in it, the judg-
ment is analytic ; as, Man is an animal. Matter is
extended, Birds are oviparous. Table-salt is a chlo-
ride. Such a predicate adds nothing to the con-
ception of the subject, but merely unfolds a constit-
uent mark, essential and original, which is thought
already though confusedly in the subject. This

' There seems to have been a great deal of confusion on this very
simple matter. See, for explanations, Hamilton, Metaphysics^ p. 72 ;
and Logic, p. 337. Cf. Venn, Empirical Logic, p. 366 sq.

DEFINITION 9

form of predication, then, is analytic, a judgment of
partial identity affirming of a subject a portion of its
essence.*

Moreover, every logical definition, being a full ex-
plication of the original and essential marks of the
definitum, is an analytic judgment ; as, Man is a ra-
tional animal^ Matter is extended sitbstance, A hird
is a feathered oviparous winged hiped^ Table-salt is
sodium chloride^ also the general definition of Logic,
and that of Induction now under discussion. All such
judgments being analytic, must be set apart from in-
ductions. Likewise must be set apart all derivatives
from analytic judgments ; as, Man is sentient, Mat-
ter is divisible, Birds incubate, Table-salt is binary.
These are not inductions. Their generality arises,
not from induction, but from the original forming of
a class notion and its definition (§§ 16, 35).

An important consequence of the foregoing dis-
tinction is that induction is always and only of log-
ical accidents ; for the essence of a subject is attained
by its analysis, which essence being predicated yields
an analytic judgment. It is often difficult to distin-
guish between essence and accident, a mark supposed
to be the one sometimes turning out on closer in-
spection to be the other. "We may be practically
embarrassed by this difficulty ; still it is clear, that
induction is of accidents only, not of essence.

We have already discriminated deduction from m-

* A statement of the Kantian distribution of judgments into analyt-
ic and synthetic will be found in " The Theory of Thought " (Harper &
Bros.), p. 93.

10 ELEMENTS OF INDUCTIVE LOGIC

duction ; we may take now another view of the dis-
tinction. While induction is synthetic, deduction is
analytic, since it concludes only a part of the content
of its premises. Under All men are mortals sub-
sume All kings are men, concluding All kings are
mortals. The conclusion is of narrower generality
than the major premise. The process analyzes or
resolves the notion men into its constituents kings
and non-kings, and concludes concerning the former
only. Deduction, therefore, is analytic, and thus is
essentially distinct from and logically opposed to in-
) duction.

r

§ 6. The definition limits the inductive inference
still further to generalization from experience.

A practical acquaintance with any particular mat-
ter by simple observation or by experiment is an ex-
perience. Truth thus known is called empirical
truth. The generalization of induction, since its
ground is experience, is likewise called empirical.
But let it be noted that an experience is always and
only of a particular individual fact or truth ; there is
no experience of a general fact or truth, this being a
product of thinking. An inference a posteriori or
from experience is not necessarily true, nor has it in-
dependent universality, since it is conditioned on the
existing order of things. Moreover, every experi-
ence is attended by some uncertainty, for the closest
observation of the simplest fact is liable at least to
what is called an error of sense, and so far is doubt-
ful. This possibility injects a corresponding uncer-

DEFINITION 1 1

tainty into the inferred generality. Hence the ab-
sence of strict certainty, necessity, and universality is
characteristic of empirical knowledge, that derived
from experience.*

§ 7. To empirical truth evolved by induction is
opposed pure truth, that is, truth not derived from
experience, but given in intuition.' Pure intuitions
in the forms of purely intellectual or non-sensuous
ideas and principles are characterized by strict cer-
tainty, necessity, and universality. Such are the ideas
of space^ of time, oi caxisation, of right ^ such are
the principles or axioms of pure mathematics, as
Two intersecting straight lines cannot enclose an

^ Empirical, from kfiTTEipia ; experience, from experiri. Empirical
knowledge, the knowledge of experience, is the knowledge that a
thing is, yvojaiQ oti lari. Speculative or philosophical knowledge, the
knowledge of ratiocination, is the knowledge why or how a thing is,
yvioaiQ dioTi tan. Se'e motto on the title-page, taken from Trendelen-
burg, "Elem. Log. Arist." The distinction, in these terms, is made by
Aristotle in many places, e.g., yap to fikv oti twv aiffOTjTiKwv eiSkvai^
TO de SioTi TU)v ixaOijfiaTiKojVf etc. — Afial. Post., i. 13. Themistius, his
paraphrast, says : Sid tov (rrjfidov fiiv wg to on, did Oarkpov de tjg to
dion. See Grote, Aristotle^ ch. vii. p. 322. For empirical, see Ham-
ilton, Metaphysics, lee. iii.

- Empirical intuitions, and the inferences from them characterizing
mediate perceptions, are discussed in my " Elements of Psychology,"
§§ 81, 96, 157. For pure intuitions, see Id. §§113, 124. Also the
foot-note in this volume, p. 30. A full consideration of pure truths
belongs to philosophy, hot to logic. Differing views are held as to
their origin and the ground of their undisputed universality. These
views do not specially concern us here. Logic needs only to distin-
guish pure truths from unquestionable inductions, in order to set them,
with their direct consequences, clearly apart.

12 ELEMENTS OF INDUCTIVE LOGIC

area,*, also those of logic, as the primary laws (§ 7)";
also those of ethics, as Trespass is wrong. Every
rule derived from experience has actual, or possible,
or at least conceivable, exceptions ; but a rule intui-
tively discerned by pure reason has universal, un-
limited universality, has no exception in all the uni-
verse of things. An exception is impossible even
in thought.

Let it be remarked that, while pure truth is gen-
eral in the highest sense, its generality is attained
neither by class generalization nor by induction, but
by intuition. Wlien upon an empirical occasion such
truth is intellectively discerned, it is at once, without
any logical process beyond abstraction, seen by the
pure intellect or reason to be strictly universal.
Hence it is not the result of inductive inference, nor
indeed of any kind of inference.

Also we remark that pure principles are synthetic,
since the predicate adds something to the subject
not already contained in it. But they are not, like
inductions, synthetic a posteriori, but are synthetic
a priori, a profound distinction referable to the exer-
cise of pure reason.'

* The phrases a priori and a posteriori were used by the schoolmen
in a sense derived from Aristotle, the former to denote an inference
from cause to effect, the latter to denote an inference from effect to
cause. More commonly now, in Logic, they are used to distinguish
between the deduction of a special from general truth, and the induc-
tion of a general truth from observed facts. In Philosophy, knowledge
a priori, according to Kant, is that which is independent of all expe-
rience and logically prior to it; knowledge a posteriori is that acquired
by observation of facts, and therefore dependent on and logically pes-

i

EFINITION 13

By the foregoing criteria pure intuitive truth is
tiistinguished, and should never be confused with the
empirical generalities obtained by induction. The
importance of this clearance cannot be overestimated.
Its difficulty is enhanced by the fact that both pure
and empirical truths, though so widely distinct in
origin and character, are constantly and intimately
connected, and are therefore especially liable to con-
fusion. In the present treatise we shall be largely
concerned with both kinds ; for logic in general con-
sists essentially of pure truths with deductions from
them of formal rules and canons, and incidentally
makes application of these to matter, evolving mate-
rial and empirical truths.^

terior to experience. The one is knowledge of pure, the other of
empirical, truth. See Critique of Pure Reason, Int. § 1 ; and Hamil-
ton, Logic, p. 385 Am. ed.

^ Sciences which, like Logic, originate in and develop from pure
truths or axioms, are strictly demonstrative and exclusively deductive
(§ 108). Thus Inductive Logic is, as to its formal system, deductive.
Also let it be noted that Pure Mathematics is exclusively deductive.
This is sufficiently obvious, since the formal conclusions it deduces,
being already completely general, cannot be further generalized (§ 130).
It does not admit of any inductive inference. Many logicians have
maintained the contrary, holding that the law of a series, such as
Newton's binomial theorem, is obtained by induction, by generalizing
from a few particulars. But upon consideration it will be seen that
the law in each case is a deduction from the more general principles
of multipUcation'as applied in permutation and combination. The
given members of the series are subsumed, and the law deduced. Be-
sides the already complete generality of all the propositions involved
in the process, we point out that the inference to the law is not a syn-
thetic, but an analytic, process. For example, take the simple series
2, 4, 6, 8 — - -. Its law is: The n'*» term=2n. This is discovered
by an analysis of the given terms, and adds nothing to what is given.

14 ELEMENTS OF INDUCTIVE LOGIC

§ 8. The definition limits the inductive inference
finally to generalization beyond experience.

Induction centres in experience, but it makes a cir-
cuit of untried regions, and, in accord with the ety-
mology of the word, leads in or brings within its
scope a vast assemblage of truths otherwise unknown.
It goes far beyond experience, and by synthesis adds
unobserved facts to our knowledge, very often ascer-
taining with scientific accuracy facts that are perma-
nently beyond the reach of possible observation.
Moreover, it exhausts the field by its comprehensive
all. This excursive and inclusive clean sweep is an
especial characteristic of induction.

An important consequence of the extension of the
inductive inference beyond experience is its liability
to include, in the unexplored region, exceptional
facts. It is true that in an inductive sequence ground-
ed on thoroughly ascertained causal connection, seem-
ing exceptions must be attributed to unknown coun-
teracting causes, and hence are not truly exceptions.
Yet in our ignorance of the possibilities in the outer
region, we are not, even in such case, strictly certain,
and must admit the notion of possible, or at least
conceivable, exceptions. For example, if from obser-
vation of many cases it is inferred that All crows
are hlach^ color being usually considered an unessen-
tial mark or accident, it may be objected that albi-
nos have been seen. So also Every oak-tree hears
acorns is uncertain, for it may be that some are
sterile. To the rule Alkalies have metallic hases an
exception turns up in ammonia. If from finding

DEFINITION 15

table-salt, saltpetre, and others, to be soluble I induc-
tively infer All alkaline salts are soluble^ perhaps no
exception could be named ; but if I generalize more
widely io' All salts are soluble^ my inference is falsi-
fied by sulphate of baryta, and many others.

In a previous section (§ 6) an element of uncer-
tainty, due to what are called errors of sense, was
pointed out. We now find another^ due to general-
izing beyond experience. Since exceptions may act-
ually or at least conceivably occur, it follows that the
empirical universality attained by induction is to
some extent a precarious, a hazardous universality.
This hazard is a derived characteristic of induction.

§ 9. A generalization beyond experience is logical-
ly opposed to a generalization within experience.
Having examined certain individuals of a class, we
may sum up our observations in a single general state-
ment. Thus I may know by direct observation, and
say of my friends, that A few are wealthy, Many are
prosjperotts, Most are industrious. These statements
concerning some at least, perhaps all, being partial,
are termed approximate generalizations. Thorough
observation of each friend may justify my saying,
All are honest, None are covetous. These state-
ments, being total, are termed complete generaliza-
tions, or simply generalizations. In like manner, by
an examination of each one separately, I may ascer-
tain that Every member of my class ofjpupils is stu-
dious, or that Each of the apostles was inspired, or
i\\2i\j All the known planets shine by reflected light

16 ELEMENTS OF mDUCTIVE LOGIC

Likewise, when it is seen that A straight line cannot
intersect a circle in more than two jpoints, and that
this is true also in case of an ellipse, of a parabola,
and of an hyperbola, then, there being no others, we
may lay it down as a universal property of conic
curves. This last example illustrates the modifica-
tion that two or more general truths or laws may
often be reduced to one comprehensive statement
whose extension is no greater than that of its com-
bined components.

This process is truly a generalization, a classifica-
tion, and of great value in condensing expression ;
but it is not an induction, for it does not surpass the
limits of experience. Yet it has been called an in-
duction, and even a perfect and the only perfect induc-
tion.^ But indeed it is not an inference of any kind,
for nothing distinct from what is laid down follows.
Evidently it is merely a summation of the known par-
ticulars, a colligation of the observed facts, an abridg-
ment of their statement by uniting them under one
term. To distinguish it from induction, it may be
called a summary or closed generalization, or, more
widely, a colligation.

§ 10. We have now explained with illustrations
most of the limiting terms in the definition of induc-

* See the subsequent § 2Y. " It is in the transition from some ob-
served particulars to the totality of particulars that the real inductive
inference consists ; not in the transition from the totality to the class-
term which denotes that totality, and connotes its determining common
attribute."— Grote, Aristotle, ch. vL, p. 278.

DEFINITION 17

tion. Also we have indicated and illustrated several
forms of thought excluded by those limitations.

Two formal processes, in addition to those already
examined, each of which results in a new truth, but
neither of which is a generalization, frequently oc-
cur, and cause confusion, inasmuch as they are com-
monly regarded and treated of as inductive proc-
esses. For the sake of clearness, it is needful that
these also should be here examined, in order to be at
once distinguished from induction, and relegated to
their rightful places.

One may be illustrated thus : A ship follows an
unknown coast. After some days the sailor, having
watched the coast and finding himself again at the
starting-point, says : It is an island. Here is cer-
tainly a discovery of a new fact, assigning this land
to the familiar sub -class island. Throughout the
process there is no generalization whatever ; hence it
is neither in part nor in whole an induction. Either
it is merely a gathering up and piecing together in
one the facts of a series of observations, which is
only another sort of colligation, one without gener-
alization, or, what is better, it is a discovery of iden-
tity establishing a minor premise.

This last phrase requires some explanation. The
sailor knows : A land soon sailed about is an island.
He discovers : This land is land soon sailed about /
which discovery merely identifies this land with the
notion of land soon sailed about, thereby estab-
lishing a minor premise, and enabling him to con-
clude : This land is an island. Here is both dis-

18 ELEMENTS OF INDUCTIVE LOGIC

covery and deductive proof. No generalization, and
therefore no induction, is involved/

In like manner, Kepler, having noted several points
in the planet's path, and finding the curve connecting
them to be elliptical, determined the orbit of Mars to
be an ellipse. It had long been known that this orbit
is a curve returning into itself. As a geometer Kep-
ler knew also that a curve returning into itself, with
such and such properties, is an ellipse. He identi-
fied the orbit of Mars, besides being a curve return-
ing into itself, as having such and such properties.
By this identification, he established a minor prem-
ise, and concluded the orbit of Mars is an ellipse.
Afterward Kepler made the induction, known as his
second law, that All planetary orbits are ellipses.^

Similar instances of enlarged discovery by identi-
fication abound. When, after the induction of the
laws of magnetism, other metals besides iron, as nick-
el, cobalt, manganese, chromium, were discovered to
be magnetic, the magnetic laws were at once trans-
ferred deductively to these metals. Franklin, by use
of a kite, identified lightning with electricity. It fol-
lowed that whatever was inductively true of the one
was true of the other.

' Mr. Mill calls this process a description. See his Logic, p. 213
sq., Am. ed. ; and the criticism of Dr. Whewell, Philosophy of Discov-
ery^ ch. xxii., § ii., 15 sq. See also Bain, Logic^ p. 235 sq., Am. ed.

2 Kepler's Laws of the Planetary Orbits are as follow :

Ist. The radii vectores describe equal areas in equal times.

2d, The orbits are ellipses, with the Sun in one of the foci.

3d. The squares of the periodic times are as the cubes of the mean
distances.

DEFINITION 19

Questions of identity to establish a minor premise
are necessarily a part of scientific research, but they
should not be confused, as they often are, with a pre-
cedent process of inductive generalization establish-
ing a major premise or a general law, nor with a sub-
sequent induction to which they may give rise.

§ 11. The other process needing to be distinguished
from induction resembles the preceding in being a
deduction lying between prior and subsequent induc-
tions ; it differs from the preceding in that it is an
inquiry, not into identity, but into causal relation.
Such investigation involves no generalization, and is
often carried on with no present thought of exten-
sion beyond the individual case in question.

For example, a coroner's inquest is held to deter-
mine the cause of a death. All the immediate cir-
cumstances are minutely ascertained, expert medical
testimony taken, and all collateral facts set down in
detail. Then, subsuming the facts under long-settled
and well-known principles and rules, deductions are
made, perhaps quite a series, and the cause of the
death finally concluded to be this or that. There is
no generalization, no induction whatever ; and the
important fact of the cause in this particular case is
ascertained and stated without any intent or thought
of extending the conclusion by induction to all sim-
ilar cases. The procedure indicated is from effect
to cause. The reverse may occur. Thus legislators,
having fixed a certain tax, watch for its effect upon
industry.

20 ELEMENTS OF INDUCTIVE LOQIO

The discovery of the planet Neptune by Leverrier
and by Adams is a notable example. Perturbations
having been observed in the orbital motion of Ura-
nus, each of these astronomers posited hypothetically
an exterior planet as the disturbing cause. Then
by calculation they assigned the place where finally
the telescope revealed its presence. Throughout this
process, in order to deduce the result, they used gen-
eral principles of mathematics, and mechanical and
astronomical inductions already established ; but they
did not make any induction during the process, nor
did they, like Kepler, follow it by any inductive gen-
eralization.

It is important that the formal procedure here ex-
emplified be clearly and emphatically set apart, es-
pecially because, being a necessary preparation for
scientific induction, the two are very liable to be con-
founded, and are actually so confounded by most log-
ical authorities. Preparation for induction is in some
cases the observation of only a single fact. For ex-
ample : This lodestone attracts this hit of iron, iN'ow,
if the statement be unquestionably true, we may pro-
ceed at once to the induction, and say universally :
Lodestone attracts iron. The example is crude, but
even in it we may clearly distinguish the preparatory
fact from the subsequent induction.

When the matter is more complex many observa-
tions of similar cases may be requisite, accompanied
perhaps by much experimental investigation involv-
ing numerous deductions, before it is fully estab-
lished that a certain phenomenon in each of the cases

DEFINITION 21

is nnqnestionably the cause or the effect of another.
Then, but not until then, are we properly prepared
to make a scientific induction from the experience
of these particular cases to all cases of the same class
lying beyond experience.

Thus David Wells made many observations, with
reasonings therefrom, and many careful experiments
on the deposit of dew under various circumstances,
before he could justly conclude that this phenome-
non in these cases was the effect of a reduction of
the temperature of the bedewed surface below a cer-
tain point. When this was established, he was then
prepared to make the induction of the universal law
known as the Wells theory of dew.

The preparation for induction, so far as it involves
inference, is deductive, and should not be confused
with the subsequent induction. In the progress of
the present treatise there will be frequent occasion
to remark this distinction.

§ 12. The various distinctions and eliminations pro-
posed in the foregoing sections are all in accord with
the stated definition of induction. This will be al-
lowed. But perhaps the definition itself may be
questioned. It may be deemed too narrow, or arbi-
trary, or merely nominal, not real (§ 39). In reply
we can only offer the development of the subject in
the following treatise. The definition we have given
will, in its numerous and varied applications, be found
adequately comprehensive, yet sharply distinctive, of
a real mental process of the highest import.

II.— PKmCIPLES

§ 13. It is suflBcientlj evident that the Primary
Laws of Thought cannot be superseded (§ 7 sq.).
Their necessity is universal, holding in induction,
and throughout its collateral processes. But it is also
clear that under these laws alone the inference of all
from some is illicit (§ 79). Hence this very impor-
tant inference becomes legitimate only in view of
certain principles of similar origin and authority con-
joined with the primary laws. Such principles are
evolved from the intuitive fact of causation, the root
of all induction, and that which gives it validity.
They are called the Principles of Induction, or the
Laws of Causation, and are applicable to changes or
events that are purely physical, and to human affairs.

§ 14. A preliminary examination of the notion of
causation is needful. In general, a cause is what de-
termines a change or event. Strictly taken, a change
is a ceasing to be ; an event, a beginning to be ; but
we shall use these terms indifferently. The cause
determines, without possible alternative, that the
event shall be just what it becomes. The cause is
antecedent, the event or effect consequent. When
the stroke of a hammer breaks a stone, the antece-

PRINCIPLES ^ 23

dent blow is the cause, the consequent breaking is
the event determined or the effect produced. A
cause thus producing an effect is called an efficient
cause, to distinguish it from other senses of the word
cause which is used commonly and in this treatise
without qualification t6 signify efficient cause/

^ Aristotle, in "Analyt. Post," II. xi,, and "Meta." I. iii., distinguishes
four kinds of cause, airia, as follow :

1st. The formal cause, to t'i rjv elvai, is the form, idea, archetype, or
TrapdSeiyfia of a thing. The plan of a building in the mind of the
architect is its formal cause.

2d. The material cause, ri vXrj vTroKEifievri, is the matter subjected to
the form. The wood, stone, and iron used in a building constitute its
material cause.

3d. The efficient cause, r) ri rrpioTov sKivrjffty is the proximate mover
producing change. The workmen who erect a building are its efficient
cause.

4th. The final cause, to Tivog tvtKa, is that for the sake of which
the thing is done. The purpose or end for which a building is erected
is its final cause.

The final cause is prior, he says, in the order of nature, but posterior
in the order of time or generation. The efficient cause is prior in time
or generation. The formal and the material causes are each simul-
taneous with its effect, neither prior nor posterior.

But it would, perhaps, be more accurate to say that every cause is
simultaneous with its effect. For cause and effect are correlatives —
neither can exist without the other; they exist only as they coexist.
A cause cannot be so named, except by anticipation, until there is an
effect; nor an effect, except by reference to what has already occurred,
after the change or event has taken place. Their order of succession
is logical, not temporal. Cessante causa cessat et effectus was a scholas-
tic dogma. Mr. Mill speaks doubtfully and rather confusedly. — Logic^
p. 247 sq. Cf. Hobbes, Elementa Philosophica, ch. ix.

The schoolmen made the important subdivision of efficient cause,
cama efflciens^ into the simply genetic causa essendi, a cause of being,
and causa cognoscendi, a cause of knowing, a reason (§ 110). Aristotle
uses aiTia in this latter sense even when treating of induction,

24 ELEMENTS OF INDUCTIVE LOGIC

A condition, in general, is an antecedent that must
be in order that something else maj be. A causal
condition is, specifically, an antecedent determining
the event (§ 110). A merely temporal antecedent is
followed by a subsequent ; a causal antecedent by a
consequent. Mere succession in time, however in-
variable, does not imply causation. IS'ight is fol-
lowed by day, but is not its cause. Day is not con-
ditioned on night, but on a rising sun, and this, then,
is the cause of day, or its determining condition.
There may be a quasi-sequence in time, as when in-
oculation is followed by small - pox ; or none, as
when by expenditure of energy a cannon-ball in-
stantly shatters a wall.

§ 15. In the foregoing example of a hammer and a
stone a single antecedent is named as the cause, and
single consequent as the effect. This is the usage of
common speech. Such a selection from several ante-
cedents or consequents of some one as the cause or the
effect is often quite arbitrary. When a stone falls to
the ground, the cause may be said to be the earth, or
gravity, or the weight of the stone, or the stone it-
self. We may say the explosion destroyed the maga-
zine, or that it shook the land, or was heard miles
away. The selection is perhaps influenced by con-
comitant thoughts, or determined by some special

kTrayoyi], in " Analyt. Prior," II., xxiii., which has occasioned much
confusion in the views of his interpreters. In the present treatise the
unquaUfied word cauae must be understood to mean, as it does in mod-
ern usage, cama efficiens essendi.

PRINCIPLES ' 25

interest. Most frequently that antecedent which,
added to those already assembled, completes the col-
location requisite to produce the change is called the
cause ; as, A sjpark caused the explosion / or. An east
wind produced the rain / or, Malaria induced the
fever. But it is evident that in these cases, and like-
wise in all cases, neither the cause nor the effect is
single and simple. There must be a conjunction of
at least two things to produce a change in either,
and both are thereby changed. There is always
more than one causal condition or antecedent, and
more than one determined consequent.

Even a purely negative fact is often spoken of as a
cause ; as. The calces were hurned because of Alfred's
inattention. Obviously this is unscientific. More
properly the thought is that an event occurred when
a preventing cause was withdrawn. The phrase, pre-
venting cause, is a convenient designation of any
member of a given collocation of antecedents whose
presence hinders change ; as in the examples : A
friction match does not ignite hecause it is wet';
A scotched' wheel does not revolve ; An anmsthetic
prevents pain / an antiseptic, decay. But the no-
tion of a preventive cause is negative, and inaccurate,
for in strictness a cause is essentially positive.

§ 16. In seeking, then, a full knowledge of the
cause or the effect of a phenomenon, all positive cir-
cumstances are to be inspected ; and, having eliminat-
ed those that are immaterial, i. e., not concerned in
the case, we enumerate the rest, recognizing as the

26 ELEMENTS OF INDUCTIVE LOGIC

cause all conditioning antecedents, and as the effect
all conditioned consequents, and omitting to state
only those that are quite obvious. For instance, a
cause is a hammer in motion and a whole stone ; its
effect, a hammer at rest and a broken stone.' It may-
be very difficult or even quite impracticable to enu-
merate completely the antecedents concerned in pro-
ducing an effect, or the consequents of their inter-
action, but nothing short of this can be accepted as
entire theoretical accuracy, though, indeed, all induc-
tive sciences have to be content with merely approx-
imate statements. Thus, popularly speaking, the
cause of vision is light entering the eye ; but a sci-
entific statement would include the optical action of
the lenses of the eye, the physiology of its coats, and
of the nerves and brain, together with the connec-
tion between a special activity of the brain and a
state of mind, a sense-perception. Still the enu-
meration would be only approximate. To state even
approximately the effect in vision would require a
much more subtile analysis. The theoretic ideal re-

^ The notion of cause and effect is confused in many minds with
the notion of agent and patient, whereas the two notions are very dif-
ferent. The latter distinction, that of agent and patient, occurs only
among the antecedents or causes of an event ; as, the hammer strikes,
the stone is struck. The manifestly active member is regarded as
the agent, the apparently quiescent member as the recipient or patient
affected. Still this is arbitrary. We may.say, the stone resists, the
hammer is resisted. The distinction, except when referable to Will
as a determining antecedent, depends merely on the point of view,
and hence, though often convenient, is unessential. See Mill, Logic,
p. 242.

PRINCIPLES 27

quires an exhaustive statement, towards which ideal
our practice strives.

These considerations explain and will justify the
following correlative definitions :

A cause is the aggregate of all the posi-
tively conditioning antecedents of an event.

An effect is the aggregate of all the posi-
tively conditioned consequents in an event. '

§ IT. In the notion of a cause as an efiicient
agent is implied the notion of a force producing the
effect, and this force is properly and scientifically
regarded as the cause. The aggregate of the ante-
cedents is the source of the force, or, more strictly,
the force is manifested by an aggregate of antece-
dents of which it is the property or function. Ex-
amples are, gravity, cohesion, muscular elfort, etc.

Kecent physics, while it regards force as the ever-
present agent of physical change, represents all phys-
ical changes or events" as consisting in a transferring
with often a transforming of energy. Some of the

* Mr. Mill's definition of cause has been widely discussed and ap-
proved. He says: *'The cause of a phenomenon is the antecedent,
or the concurrence of antecedents, on which it is invariably and un-
conditionally consequent." — Logic^ p. 245. Mr. Venn says ; " This
view of causation is very generally accepted in science and in the
logical treatises on Inductive Philosophy, if indeed it may not be
termed the popular view." He then makes some critical remarks. —
Logic of Chance^ ch. ix. We have ventured to propose a modified
statement, because the important terms invariably and unconditionally
are negative, and because the former superfluously implies uniformity
(§ 19). Cf. Hobbes, Mementa Philosophica, ch. ix.

28 ELEMENTS OF INDUCTIVE LOGIC

principal forms of energy which are capable of mut-
ual transformation are mechanical, thermal (heat and
light), electrical, chemical, and neural energy.

It has been proved in many cases, by accurate
measurements of the work done within a given sys-
tem or aggregate of things, that the quantity of en-
ergy therein transferred or transformed or both is
constant. There is neither gain nor loss. Hence it
is inductively inferred that, while in the internal
changes of a group there may be alteration in the
forms of energy, there is no alteration of its quan-
tity. This is the Law of Conservation of Energy
(§ 98 n.). It affirms that, just as the quantity of mat-
ter in the universe is unalterable, so the quantity of
energy is unalterable; though, indeed, these state-
ments are identical, matter being known only by the
manifestation of energy.

The law of conservation is supplemented by the
important distinction between kinetic or actual and
potential energy. In gunpowder is stored up a vast
amount of potential energy which is set free or be-
comes kinetic by virtue of the kinetic energy of a
spark. It is the sum of the kinetic and potential
energies that is constant, while in almost every
change there is a passing more or less complete of
one into the other.

In this modified and refined view we define thus :

Causation is the transfer, with more or
less transformation, of a definite amount of
energy, measured by the amount of work
done, and effecting a new distribution.

PRINCIPLES 29

Physical science of to-day is largely occupied with
the measurement of passing energy in various cases,
with the determination of the quantity rather than
the kind of causes and their correlative effects. But
in all of these investigations, under modified doc-
trines and varied terminology, the logical processes
are formally identical, and there is no need to alter
the view of causation presented in the previous sec-
tions in order to unfold the fundamental processes of
thought involved in physical research.

§ 18. It has already been said that from the intui-
tive fact of causation are evolved the special Princi-
ples of Induction, or Laws of Causation (§ 13). They
are primarily two, the first in logical order being the
Principle or Axiom of Change, as follows :

Every change (or event) has a cause.

This axiom, by virtue of its predominating pure
element, causation, has philosophical necessity (§ 5),
and is strictly universal (§ 7). The bare possibility
of a single exception is utterly inconceivable.^
There lurks an essential self-contradiction in the
phrase. An uncaused event (§ 9). The word chance,
when used in that sense, has no meaning whatever ;
there is no possible notion, and no possible fact cor-

1 The principle is intuitively true, though not altogether pure. The
notion of cause is strictly pure, but the notion of change (or event) is
empirical — that is, it can be had only from experience. See Kant, C.
P. R., Int., § 1. Change, referred to the consciousness of the ob-
server, is the very essence of experience, and is the occasion of the
pure intellectual intuition of causation. See Psychology^ §§ 114, 126.

30 ELEMENTS OF INDUCTIVE LOGIC

responding to it (§ 48). Whenever any change is ex-
perienced, the pure intellect or reason intuitively dis-
cerns that it must have a cause, an efficient deter-
mining cause/ What is the cause may be in most
cases very questionable, but that there is a determin-
ing cause in each and every case is strictly unques-
tionable, or rather is clearly and truly discernible.
The axiom is not merely a law of thought, but is
also a law of things, not merely a logical subjective
necessity, but a real objective necessity in nature,

^ This doctrine of the origin of the present and of other axioms is
according to the intuitional philosophy. The opposed empirical phi-
losophy teaches that all axioms are themselves inductions from ex-
perience, inductions .of widest and unexceptional generality. The
question is discussed in my " Elements of Psychology," § 124 sq. See
also above, § V, and below, § 19 note. Dr. Whewell in his " Philosophy
of Discovery," ch. xxii., severely criticises Mr. Mill's "Logic," and in
§ 71 very aptly says that axioms " may be much better described as con-
ditions of experience than as results of experience." For illustra-
tion of our view : A whole is equal to the sum of its parts is the axio-
matic basis of chemical quantitative analysis ; but should we make an
induction from the myriads of analyses that have been published, the
inference would be : A whole is never equal to, but ever less than, the
sum of its parts.

But as already observed, § V, note, the question of the origin of
axioms is philosophical, not logical. It might be entirely disregarded
in this treatise, since all logicians, empiricists as well as intuitionists,
accept them as irrefragable and unexceptionable, and therefore a safe
and sufficient basis of logical doctrine and scientific proof.

Let us, however, instance their catholicity. So firm is the deep
though obscure conviction in every mind that Bvery change is caused,
that when a change (event) occurs with no assignable causal ante-
cedents, men are prone to invent a cause, a groundless hypothesis ; and
so it comes that in ignorance, in the absence of any apparent natural
cause, one supernatural is often posited ; hence false spiritualism, and,
in general, superstition.

PRINCIPLES 31

holding true throughout the universe, in all space
everywhere, in all time, past, present, and future/

The axiom may be stated : If change is^ cause is;
hence (§ 119)^ If cause is not, change is not. This
form is illustrated by the first law of motion, which
afiirms that a body in motion, if not acted on by
some disturbing cause, will continue to move with
uniform velocity and in the same direction forever.''

§ 19. The second of the two Laws of Causation is
the Principle or Axiom of Uniformity. It is sub-
divided into two axioms, the first of which is as fol-
lows :

Like causes have like effects."

The word like here is to be very strictly construed.
It means more than general resemblance, or striking

^ Burgersdyck says very neatly: Quicquid fit ah alio fit, nihil fit a
seipso.

2 Newton's Three Laws of Motion, " Principia," Introduction, are as
follow :

1st. Every body perseveres in its state of rest, or of uniform motion
in a right line, unless it is compelled to change its state by forces im-
pressed upon it.

2d. Change of motion is proportional to the motive force impressed,
and is made in the right Hne in which that force is impressed.

8d. Reaction is always contrary and equal to action ; or, the actions
of two bodies upon each other are always equal, and directed to con-
trary parts.

^ It might be very correctly stated : Like causes produce, or deter-
mine, or enforce, like effects. But it is needless for logical purposes
to insist on the bond of efficiency. Mr. Mill, following the doctrine of
Hume, and in entire consistency with his own empirical philosophy,
says : " The notion of causation is deemed, by the schools of meta-
physics most in vogue at the present moment, to imply a mysterious

32 ELEMENTS OF INDUCTIVE LOGIC

similarity. It is not merely that observation, even
the most skilful and minute, cannot distinguish cer-
tain cases by any other particular than place or time,

and most powerful tie, such as cannot, or at least does not, exist be-
tween any physical fact and that other physical fact on which it is
invariably consequent, and which is popularly termed its cause ; and
thence is deduced the supposed necessity of ascending higher, into
the essence and inherent constitution of things, to find the true cause,
the cause which is not only followed by, but actually produces, the
effect. No such necessity exists for the purposes of the present in-
quiry, nor will any such doctrine be found in the following pages." —
Logic, p. 236. Nevertheless he frequently speaks of causes as pro-
ducing their effects, and uses the word force a hundred times " in the
following pages." How could he do otherwise, while, apart from
metaphysics, recent physics is almost wholly occupied with the doc-
trines of force and energy ? Again, he says : " The causes with which
I concern myself are not efficient, but physical causes." — Ibid. Why
then should he ever use the word effect i

Mr. Mill posits this first Axiom of Uniformity as the " Ground of
Induction." — Logic, title of ch. iii., bk. iii. In the first section of
the chapter (p. 225) he says: "I regard it as itself a generalization
from experience." That is to say: Induction is grounded on the
axiom of uniformity, and the axiom of uniformity is grounded on in-
duction. This vicious circle he labors, in ch. xxi., with all his great
acumen, to justify, and finds in simple enumeration, avowedly the weak-
est form of induction, which " in science carries us but a little way,"
the source and strength of the ultimate Axiom of Uniformity. See
below, § 40, note. This remarkable attitude of the eminent logician is
a necessary consequence of his underlying philosophy, and is a suicidal
reductio ad absurdum of empiricism.

It is with much hesitation and sincere regret that these points are
noted. Such is my high esteem of Mr. Mill as an acute, comprehen-
sive, and profound thinker, that I do not differ from him when I can
help it. Happily the exceptions taken relate to his philosophical prin-
ciple, rather than to his logical doctrine, and do not materially affect
the latter. The world of science is profoundly indebted to him for
the clearest exposition that has been made since Aristotle of its logical
methods. Bacon pointed out the way, Mill laid it open.

PKINCIPLES 33

but that the cases really and strictly do not at all dif-
fer in any other particular.*

It is evident, upon clear reflection, that this axiom
has the same origin and character as the axiom of
change ; that, when rigidly construed, it is necessarily
and universally true, without possible exception in
nature or in thought.

§ 20. It very often happens, however, that various
phenomena are due to indistinguishable causes. A
certain medicine in one case cures, in another kills.
A chemist in one case obtains crystals of a salt from
its solution, in another he fails. Clouds apparently
alike emit at one time lightning, at another rain, at
another hail, at another snow. Heat softens iron and
liardens clay, it warms to life and scorches to death,
it causes chemical composition and decomposition, it
melts ice, then contracts the water, then expands it,
then turns it to vapor. Electricity is likewise sup-
posed to do of itself a great variety of things.

This mode of statement arises from imperfect ob-
servation, or from an interest that assigns to some
single antecedent a predominance, as though it alone
were the cause (§ 15). In every such instance, how-
ever, there is an incomplete estimate of the causal

^ It should be remarked that the word like or similar is sometimes
replaced by the word same, this word being often used to express, not
strict identity, but the close similarity in things that are distinguish-
able only numerically, only by place or time. Place and time are real
conditions, but not causal conditions, of an event (§ 110\ and hence
are not to be reckoned among its causal antecedents.
3

34 ELEMENTS OF INDUCTIVE LOGIC

conditions, and every clear-thinking scientist knows,
with a strict certainty admitting of no hesitation or
question, that any variation whatever in the conse-
quents is due to some difference in the antecedents,
though he be unable to discern or demonstrate any
difference. This he knows by virtue of the principle
of uniformity. Even the careless observer of ordi-
nary events regulates his thoughts and actions, though
obscurely and confusedly, by the same principle.

Yet, as a concession to an interest, or more fre-
quently to a specific ignorance incident to the practi-
cal impossibility of making a complete analysis and
estimate of the antecedents, a doctrine of so-called
Plurality of Effects is allowed, as expressed in the
Maxim : Regard indistinguishable causes as having
apj>arently a variety of effects.

Every substance has a variety of properties, and
substances are distinguished from each other by their
different properties. A property is the capability
of a body to produce a specific effect. Every body,
then, is a cause producing, according to the forego-
ing maxim, a variety of effects. Thus the sun de-
flects the course of the planets, and emits light and
heat, because of its attractive, luminiferous, and cal-
orific properties. The earth has attractive and mag-
netic properties. Steel is hard, heavy, lustrous, and
elastic. But it is evident that no body manifests a
property except in combination with some other
thing. Its color, for example, becomes manifest only
in its combination with light and vision. This class
of cases, then, does not differ from that already de-

PRINCIPLES 35

scribed. Different antecedents only are followed by
different consequents, "Whenever all the causal an-
tecedents are alike, the consequents are alike.

Again, it is usual to speak of different or even op-
posed phenomena, when invariably coexistent, as the
effects of a common cause. Since doubly refracting
substances always exhibit periodical colors on expos-
ure to polarized light, these diverse phenomena have
been attributed to a hypothetical common cause,
'^he aurora is invariably accompanied by magnetic
disturbance, hence doubtless a common though un-
known cause. There is a simultaneous rise of tides
on opposite sides of the earth, of which phenomena
the moon is known to be the common cause. Such
joint effects, whether in the same or in different de-
grees of descent from the cause, are said to be cau-
sally connected, or related through some fact of cau-
sation. This mode of representation is convenient,
and in accord with the maxim. But the axiom holds
good ; for the common cause of different phenomena
invariably coexistent means only that amidst their
distinctly various antecedents some one at least is
common.

§ 21. The second axiom of uniformity reverses the
first, and is its complement, as follows :

Like effects have like causes.

The same strict construction is to be put on the
terms of this axiom as on those of its fellow. It has
the same intuitive origin, the same necessary and
universal character. That it is an axiom at all has

36 ELEMENTS OF INDUCTIVE LOGIC

rarely if ever been recognized by logicians of any
school. Yet many of the refinements of recent sci-
ence not only proceed upon it, but would be impos-
sible without itj and it is high time it should take its
place in logic. For when all the antecedents and all
the consequents are taken into account, either of
these groups equally and absolutely implies the other.
From a complete knowledge of one the other may in-
fallibly be inferred. Logically, the past is just as truly
contained in its future as the future in its past.^

^ The second axiom of uniformity is formally involved in Newton's
famous "Regulae Philosophandi," introducing bk. iii. of the "Prin-
cipia." The first two of the four Rules, with his comments, are :

1st. No more causes of natural things should be admitted than such
as are both true (verce) and sufficient to explain their phenomena.

Accordingly philosophers say : Nature does nothing in vain, and it
is vain to do by many what can be done by fewer. For nature is sim-
ple, and does not luxuriate in superfluous causes of things.

2d. And therefore {ideoque) of natural effects of the same kind the
same causes are to be assigned, as far as possible {guatenus fieri potest).

As, respiration in man and in beast ; descent of stones in Europe
and in America ; light in culinary fire and in the sun ; reflection of
light in the earth and in the planets.

To these comments of Newton we venture to add the remark that
the illation {ideoqtie) of the second rule from the first is to be construed,
not as a deduction, but as an implication. See § 78, and " Theory of
Thought," p. 103. Also we remark that both rules are likewise im-
plied in the Law of Parcimony, sometimes called Occam's razor, to
which Newton probably had reference in his first comment. See
Psychology^ § 83, note. Also Aristotle says : 6 Qthg kuI r) <pvaiQ ovd^v
fidrnv TToiovmv. — De Coelo, i., 4. Dr. Whewell, in "Philosophy of Dis-
covery," ch. xviii., § 5 sq., descants at some length on these rules.

By virtue of the axiom implied in the rules, that like effects have
like causes, Newton identified celestial with terrestrial gravity. In-
deed, he laid down the Rules in anticipation and justification of the
proof which follows in bk. iii. Also Franklin's identification of light-

PRINCIPLES 37

§ 22. It very often happens, however, that vari-
ous phenomena give rise to indistinguishable effects.
Our powers of observation, even when highly skilled
and aided by the best microscopes and instruments
of precision, are very limited, and in general can
distinguish only the grosser elements of causes or of
effects. Hence it is rarely possible to pronounce
two events strictly alike. Moreover, from the gross-
er elements of an effect, some one is usually selected,
because of its special interest, and treated as though
it alone were the effect, all other consequents being
disregarded. These considerations explain the prac-
tice, even in scientific treatises, of viewing similar
effects as the products of dissimilar causes. It is
clearly a fiction, and in strictness an impossibility.
Yet, in concession to this mode of speecli, which is
convenient and advantageous when not misleading,
a doctrine of so-called Plurality of Causes is admitted,
as expressed in the Maxim: Regard indistinguish-
able effects as having apparently a variety of causes.

Accordingly it is allowed that a man's death is

ning with electricity is by virtue of this axiom. Also the Law of the
Conservation of Energy finds its basis therein (§ 17).

But illustrations from physical science are needless when we con-
sider that our sensations are effects by which we identify or recognize
substances which affect us by their properties. How do I recognize
my friend ? The like effect on me of a presence I attribute to a like
cause. I identify a given substance as gold, only because its effect on
me is like to that produced by gold. I distinguish gold and silver by
their unlike effects. It is clear, then, that this axiom lies in the very
foundation of all knowledge. See, on Genesis of Mediate Perceptions,
Psychology, §§ 158, 159.

38 ELEMENTS OF INDUCTIVE LOGIC

caused in one case by a bullet entering tlie brain, in
another by a knife cutting the heart, in another by a
fever ravaging the intestines, and so on, it being im-
possible to enumerate the various causes of death.
There is no objection to such expressions, if we are
not misled by them. Let it be noted that death is a
purely negative and abstract notion, whereas we are
dealing with positive and concrete phenomena. In
the first case cited, the causal antecedents are a man
and an entering bullet ; the effected consequents are
a corpse and a torn brain ; and so on. It is evident,
even in this gross view, that any variation in a total
cause gives rise to a variation in its total effect. We
allow the useful fiction of a plurality of causes, but
hold, in strict construction, to rigid invariability, to
uniformity.

Another standard example is heat. It is pro-
duced by combustion, by friction, by compression,
by electricity, etc. It would be easy to show that
heat also is only one fact in an aggregate of conse-
quents varying in each case. But it is better, per-
haps, to say that in each case there is a transfer of
energy, effecting a new distribution, partly in the
form of heat (§ 17). As to the sense - perception of
heat, or of white, there is some one condition or set of
conditions which is present in every case, and whose
presence always produces in us that sense-perception.*

* For the usual view of the doctrine of Plurality of Causes, not rec-
ognizing the second axiom of uniformity, see Mill, Logic, bk. iii.,
ch. X. ; followed by Bain, bk. iii., ch. viii. Mr. Venn's view is not
unlike that of our text. See his Empirical Logic, pp. 62, 88. On

PEINCIPLES 39

Another example, one not so readily reduced, is
from the composition of motion. If a ball receive
two simultaneous impacts differing in direction and
intensity, motion is imparted to it, manifest by its
passing along a certain line with a certain velocity.
J^ow the number of impacts which will produce
precisely this effect, also their possible variations in
direction, or in intensity, is infinite. Here, then, it
seems we have an infinite plurality of causes deter-
mining an identical effect ; for, by the second law
of motion, a universal law of nature, the resultant in
all cases must be the same.' This appears to be a
demonstration of plurality of causes ; that its maxim
is rather a principle, falsifying the second axiom of
uniformity. But the resultant motion of the ball is
only one fact among others, the only one patent to ob-
servation perhaps, but not standing alone. Could
we estimate the stress of each impact on the ball,
and the consequence to its interior, together with
the arrest of the impelling agents, evidently we
should find that the aggregate of consequents varies
with every variation in the cause.

§ 23. The two axioms of uniformity express all
that is properly meant by the familiar phrase : Uni-
formity of Nature^ which is sometimes more widely

page 421 he says: "The doctrine of Plurality of Causes is a promi-
nent one in Mill's scheme, and he even attaches too great importance
to it by regarding the plurality rather as formulated by nature than as
arising merely out of practical convenience and convention."

* Newton's Corollary I, from the Laws of Motion. See p. 31, note.

40 ELEMENTS OF INDUCTIVE LOGIC

and thereby erroneously construed.' They clearly
hold good in theoretical strictness, and should regu-
late observation and inference " as far as possible.""
The maxims of plurality are practically admissible
only as a guard against errors arising from defective
or interested observation. In this respect they ren-
der important service, especially in those ordinary
concerns of life wherein only some part of a cause
or of an effect needs consideration.

When the axioms of uniformity are compared, it
will be seen that each might be stated more fully,
thus : Only like causes have like effects^ and Only
like effects have like causes. The first of these com-
pound statements implies: Unlike causes have un-
like effects ; the second implies : Unlike effects have
unlike causes (§ 71). Hence, also, if either is, the
other is ; and if either is not, the other is not ; the
form being conditio sine qua non (§ 119).

^ The phrase obviously requires limitation. No two leaves of the
forest are alike, no two human faces are alike, one star differeth from
another star in glory. So far from being uniform, unanalyzed nature
presents an infinite variety. Likewise, the statement that the course
of nature is uniform, taken in an unlimited sense, is not true. The
events of each day are unlike those of any previous day, and no one
expects history to repeat itself. But amidst this infinite variety, we
discern certain uniformities conforming to the principle that like
causes have like effects, and the reverse, which uniformities reduced
to general expression are termed laws. In this very important sense,
but in no other, is the constitution and course of nature uniform.

2 This phrase in the second Rule, § 21, note, seems to refer to the
very general impracticability of making an exhaustive estimate of the
causal conditions of a given effect.

III.— PEOCESS A

\
§ 24. Sitting by my anthracite fire, I thrust the
poker between the bars of the grate, and after a
while, on drawing it out, see that it is red-hot; it
shines in the dark. A pyrometer at hand shows
that it has reached 1000° F. Here is an experience,
specifically an observation by trial or experiment,
with quantitative measurement. The result is: This
hody of iron heated to 1000° F. has hecome lumi-
nous or glows. It states a particular, individual fact
respecting this piece of iron at this time and place,
and in the present circumstances.

Then from this single fact I infer immediately
the universal proposition : Any and every hody of
iron, at any time and any where, heated to 1000° F.,
lecomes self-luminous. This immediate inference is
an induction.

§ 25. Let us note, in the first place, that the fore-
going inference conforms strictly to other terms in
the definition of induction (§ 3). It is synthetic,
since the predicate adds to the general notion hody
of iron, something not already contained in it. It
obviously generalizes both from experience and be-
yond experience. The basis from which it proceeds

42 "^ ELEMENTS OF INDUCTIVE LOGIC

is ray experimental observation of a fact. It sur-
passes all experience by bringing in or inducting
under a universal statement every strictly similar
fact occurring anywhere in the earth, in the planets,
in the stellar spaces, at any time in the unlimited
past, present, or future/

Secondly, it is in accord with the principle or axi-
om of change (§ 18). A change is observed in the
iron from dull cold to bright hot. By the axiom,
there must be a cause for this changed state in which
we take an especial interest. We observe the ag-
gregate of the positively conditioning antecedents,
finding it, in the rough, to be burning coal and dull
cold iron. This, then, is the cause, having for its
consequents burnt coal and bright hot iron, the ag-
gregate effect (§ 16). A more refined view regards
the aggregate of antecedents as the present source of
a force or cause, determining, in this case, a transfer
to the iron of thermal energy of sufficient intensity
to affect vision (§ 17).

The causal relation being experimentally and def-
initely ascertained, we note, thirdly, that the infer-
ence conforms to the first axiom of uniformity (§ 19).
It assumes that like causes may occur or have occurred
at other times and places, and concludes that in all

^ The notion, not infrequent, that induction bears some special re-
lation to the future, needs correction. Time, in its modifications of
present, past, and future, is not an element in the inference ; nor is
place, near or remote. We do not infer from now to then, nor from
here to there, but from facts observed to facts unobserved, regardless
of time or place. See p. 83, note. On time in judgment see § 60.

PROCESS 43

such cases like effects must follow. This inductive
step is fully authorized by the axiom. The axiom
itself is merely and strictly formal ; the material case
conforms to it, and so is justified. It should be re-
marked, however, that tliere is a varying degree of
hazard in drawing the conclasion, arising not from
the principles involved, but from the uncertainty,
always greater or less, respecting the observed facts
and their causal relation, no empirical matter ever
attaining the strict certainty of intuitive truth (§ 8).
In the example, the quantity and shape of the iron
are disregarded, being considered immaterial circum-
stances; but, this one experiment being taken as the
sole ground, it might fairly be questioned whether
the like effect would follow in a spherical ton of iron,
and so further experiments be prerequisite to the
general conclusion.

§ 26. In examining the inductive process, it is very
important to observe that the inference is immediate, v
This is true of every proper induction. There is no . '
middle term, one cannot be, for both terms of the con-
clusion occur in the same premise ; hence, no syllo- ' \i
gism ; the step is strictly and exclusively immediate. )^.

Havinor established the causal relation between a n "

1 '^
phenomenon and some circumstance, we proceed, in '

conformity with one or the other of the axioms.

One of the axioms is: Zilce causes have like effects.

The corresponding formula is simply as follows:

In this case A causes a y

.'. In all cases A causes a.

44 ELEMENTS OF INDUCTIVE LOGIC

The other axiom is : Like effects have like causes.
The corresponding formula is simply as follows:

In this case a is the effect of A ;
,'. In all cases a is the effect of A.

(I The inductive process herein formulated needs to
be especially remarked, explained, justified, and em-
phasized ; not only because its immediacy is an inva-
riable characteristic, but also because many eminent
logicians and their disciples are at fault on this im-
portant poiiit, holding that induction is essentially a
mediate process, and reducible to the formal syllo-
gism. It seems hard to avoid confusing the induc-
tive inference and other inferences often associated
with it, and to see clearly that it is simple, plain,
direct, and immediate.'

§ 27. It is usual to quote Aristotle in support of
the view that the inductive process is a mediate in-
ference, a syllogism. He has the following form :

X, Y, ZareB;

X, Y, Zareall A;
.-. All A are B.

This he calls a syllogism, using the word generically,

* Dr. Whewell's view is not clear, but it seems consonant with our
own on this point. He says : " The process of induction includes a
mysterious step, by which we pass from particulars to generals, of
which step the reason always seems to be inadequately rendered by
any words we can use ; and this step to most minds is not demonstra-
tive, as to few is it given to perform it on a great scale." — Philosophy
of Discovery, ch. xxii., § 66.

PROCESS 45

in its etymological sense; specifically, a rhetorical
syllogism.' In the same passage he says induction
is contrary to syllogism, meaning logical syllogism.
That the form is not a logical syllogism is evident ;
for the second proposition is one of entire identity ;
there are, then, but two terras in all ; and hence the
question is begged (§ HG). Aristotle's example is :

Man, horse, mule, etc., are long-lived ;
Man, horse, mule, etc., are acholous (or r) ;
.'.' All acholous (or r) animals are long-lived.

He adds : We must conceive that r consists of a col-^
lection of all the particular cases. This, he says, is
induction. His followers, and many logicians of to-
day, call it a perfect, and the only perfect, induction.
Bat the process is from all to all ; and that ambigu-
ously, the first all being cumular, the second distrib-
utive (§§ ^^ 7Jt). Moreover, not generalizing beyond
experience, the process is a closed generalization, a
mere summary, a colligation, and:' therefore not at all
an induction in the modern or Baconian sense (§ 9).
Aristotle nowhere treats of induction in the latter
sense. It was reserved for Bacon to found this com-
plementary branch of logic.'*

* Syllogism, avv-Xkynv, to collect together ; like conclusion, con-clu-
dere, to shut up together. — Theory of Thought^ p. 130. Aristotle
speaks of a conclusion as " a perfect syllogism of the extremes." The
above form he calls 6 k^ eTrayojyrJQ (rvWoyiafiog. — Prior Analyt.^ ii,,
23. For the word tTrdyojyfj {tTTi-dyeiv, to lead or bring upon), see Thom-
son, Outline, etc., § 113, note. It means here an accumulation, a sum-
mation, a colligation (§ 9). Cicero, " De Inv.," fairly translates it by
inductio, but it is quite different from the modern induction.

2 Aristotle says distinctly : We believe everything either through

46 ELEMENTS OF INDUCTIVE LOGIC

§ 28. Another 'syllogistic form, laid down as that
of the mediate process essential in all induction, is
exemplified thus :

This, that, and the other magnet attract iron ,
This, that, and the other magnet represent all magnets ;
.*. All magnets attract iron.*

The correct conclusion in Darajpti as authorized by
the premises is the following :

.*. Some things that represent all magnets attract iron.

syllogism or from induction — ciiravra yap TriffTSvofiev rf Sid avWo'
yifffiov ri £? hirayuiyiiQ. — Prior Analyt., ii., 23. Here as well as in
other passages he notes the two processes as entirely distinct. But
he forgets or relinquishes this, when he presents to us, in the same
chapter, the inductive process as a variety of the syllogism. His view-
has been much discussed. Dr. Whewell, " Phil, of Disc," Appendix
D, examines it at length, and concludes : " Induction from a compara-
tively small number of particular cases to a general law stands in op-
position to the syllogism. . . . Induction is inconclusive as reasoning.
It is not reasoning ; it is another way of getting at truth. ... As true
inductive propositions cannot be logically demonstrated by syllogistic
rules, so they cannot be discovered by any rule." Mr. Grote, "Aris-
totle," ch. vi., p. 268 sq., also discusses the matter at length, and con-
cludes : " We thus see that this very peculiar syllogism is (as indeed
Aristotle himself remarks) the opposite or antithesis of a genuine syl-
logism. It has no proper middle term ; the conclusion in which it
results is [identical with] the first or major proposition, the character-
istic feature of which it is to be immediate, or not demonstrated
through a middle term " (p. 273) " These chapters respecting induc-
tion and example are among the most obscure and perplexing in the
Aristotelian Analytica. The attempt to throw both into the syllogistic
form is alike complicated and unfortunate; moreover, the reasoning
has hitherto been imperfectly apprehended " (p. 275).

* This form is given by Hamilton in his " Metaphysics," p. 72 ; and
more fully developed and defended in his " Logic," § Ixii. See also his
Discussions, p. 156 sq.

PEOCESS 47

But this is not at all what is proposed to be proved.
The conclusion sought and stated is not sjllogisti-
cally authorized, the term all magnets not occurring
in either premise ; and hence that conclusion is ir-
relevant (§ HJi). We remark, however, that the
given form, though not a true syllogism of any
kind, yet involves an immediate inductive inference
from the first premise directly to the conclusion.
The intermediate proposition is superfluous, being
merely a statement in concrete terms of the axiom
of uniformity authorizing the induction of all. Its
omission does not reduce the form to an enthymeme,
for not this material proposition, but only the for-
mal axiom, is in mind.

§ 29. Yery eminent authorities unite in proposing
the following as a type of the inductive syllogism :

Whatever is true of John, Peter, etc., is true of all man-
kind;
Mortality is true of John, Peter, etc. ;
.'.Mortality is true of all mankind.^

1 This form is given by Whately in his " Logic," bk. iv., ch. i., § 1.
It i quoted and approved by Mill, " Logic," p. 225. On the previous
page he says : " As Whately remarks, every induction is a syllogism
with the major premise suppressed ; or (as I prefer expressing it)
every induction may be thrown into the form of a syllogism, by sup-
plying a major premise." On this Grote comments thus : " Even with
this modified phraseology, I cannot admit the propriety of throwing
Induction into syllogistic forms of argument. By doing this we efface
the special character of Induction, as the jump from particular cases,
more or fewer, to an universal proposition comprising them and an
indefinite number of others besides. To state this in forms which

48 ELEMENTS OF INDrCTIVE LOGIC

This truly is a syllogism. But is it an induction ?
l^ot at all. The inference is not from some to alL
The first proposition, whose pre-designation is What-
ever, is by far the widest of the three, and the de-
duction from it is a faultless Barbara, But, say its
advocates, the major premise is an induction ; mean-
ing that it is obtained by induction. Granted ; but
that does not affect the character of the inference
before us, which is undeniably a strictly deductive
syllogism, proceeding from the more to the less gen-
eral. When we ask these advocates: How do you
get this major, their reply is, tliat it is the conclu-
sion of a prior and wider syllogism, whose major
premise is obtained in like manner, and so on, until
we reach the axiom of uniformity, which is the ulti-
mate or primary major premise of the series.' That
is to say, induction is deduction from the axiom of
uniformity.

§ 30. To the doctrine that induction is a mediate
X procedure from an axiomatic premise, we object :

First, the doctrine is confusing. It denies any
specific diflierence between the processes of deduc-
tion and induction. Tiiat they are supposed to arise

imply that it is a necessary step, involving nothing more than the in-
terpretation of a higher universal proposition, appears to me unphilo-
sophical." — Aristotle^ eh. vi., p. 280.

It is curious to note that Mr. Mill, our highest authority in Logic,
holds: 1st. That a syllogism does not and cannot prove anything;
2d. That induction alone is proof; 3d. That induction proceeds by
syllogism. In his Logic, cf. bk. ii., ch. 3, with bk. iii., chs. 1 and 3.

I Mill, Logic, p. 225.

PROCESS 49

from different ultimate premises, their several ax-
ioms, is not a logical difference, and does not justify
that distinction of the methods warmly insisted on
by the advocates of this view, and conventionally
established and recognized as the Aristotelian and
Baconian methods.

Secondly, itis.-«4matuxal.. Logic is in no respect
an invention, but only a distinct statement of the
principles and a development of the formal processes
by which the human intellect actually discovers and
establishes truth, whether of commonplace matter
or of recondite science (§ 3). Now, does the vulgar
or the child mind come to know that Water quenches
fire by a deduction, through a series of syllogisms,
from the principle of uniformity as a primary major
premise? It would give the skilled sophist some
trouble to construct the series, if it be practicable at
all ; and the supposition that ignorant and stupid
people, to whom this and like truths are perfectly
well known, have acquired the knowledge by an in-
tricate syllogistic process, however obscurely per-
formed, is incredible.

JEhirdLy^ it is unnecessary. The doctrine may be
replaced by one much simpler, which substitute will
be confirmed by a little introspection. As the hum-
blest intellect knows, obscurely it may be, yet with
a clearness sufficient for practical application, that
Part of ajpart is part of the whole (§§ 93, 131), and
thereby is able to appreciate the cogency of a simple
syllogism conformed to this formula, so it knows,
with similar obscurity perhaps, that Like causes have

50 ELEMENTS OF INDUCTIVE LOGIC

UTce effects, and a single observation of water quench-
ing lire suffices to establish the general conclusion
immediately drawn according to this formula.

Fourthly, it is not true, which is shown in the sec-
tion following.

§ 31. Let us remark, more particularly than here-
tofore (§ 2), upon the function of a form. Turning
to Pure Mathematics, a science of forms, we note
such familiar instances as 1 : 2 = 3 : 6 ; {x+y) (x—y)
z=zx^+y^; Circ. = 2 ir B, These are formulas of quan-
titative identity, of equality. They are not attained
by generalization of matter, but in entire abstraction
of matter; they have no material content. They
are not generic of things, and do not serve as prem-
ises in material reasonings. They furnish abstract
forms in which concrete matter may be cast. To
apply mathematical forms to matter is the special
function of Applied Mathematics. To make an ap-
plication of a formula is not to draw an inference
from it, but merely to supply it with a content of
suitable matter. Material inferences are not made
from, but according to, a formula.

Precisely the same is true of logic. The Aris-
totelic Dicta express merely syllogistic form (§ 93),
Every material syllogism concludes, in Fig. 1, not
from, but in accord with, these canons. They are
often spoken of as the ultimate major premises in
material reasonings. This is an error. They serve
only to express the abstract form in which sound
reasoning concerning things proceeds.

PROCESS 51

What in this respect is true of deduction is true
also of induction. In its formal character induction
evolves abstract canons and formulas, deduced from
axiomatic principles, which canons and formulas in
ordinary affairs and in the inductive sciences are
supplied with concrete, material facts and things/
In all cases the inductive inference is made, not
from a form, but according to or in conformity with
a form. The notion that the axiom of uniformity
is the ultimate premise in induction is false and con-
fusing. A material conclusion comes always and
only from one or more material premises, in con-
fonnity with certain established abstract forms.

§ 32, We maintain, then, that the inductive proc- \
ess consists wholly and exclusively in a direct im-
mediate inference authorized by the principle of uni-
formity, an inference so simple that in making it a
formal fallacy is well-nigh impossible. This infer-
ence is not reducible to syllogistic form. The at-
tempt results either in a violation of syllogistic law,
and thus is false reasoning, or it presents forced

^ The canons of causation, to be hereafter discussed (§ 55 sq.), are
likewise deduced from the axioms previously laid down, and are
merely abstract formulas of causal relations.

It is remarkable that there should be need to explain the rela-
tion of form to matter, v/hen these words are in familiar and accurate
use in every-day life, even children and the vulgar using them in cor-
rect distinction. E. g. The form of the oration was good, the matter
poor. But logicians, whose specialty it is to mark and regulate the
distinction, often either ignore or reject it ; hence the need of exposi-
tion and insistence. — Cf. § 2 and § ^ and Theory of Thought, p. 5.

62 ELEMENTS OF INDUCTIVE LOGIC

forms, quite unnatural and therefore untrue. iFor,
we repeat, it is the general function of Icjotc to
evolve the forms according to which actual tbniking
is naturally and rightly accomplished ; and its spe-
cial function is to state and demonstrate these forms
clearly and distinctly, so as to dissipate the obscurity
in which they usually lie, even in minds otherwise
highly instructed.

Why, it may fairly be asked, if the process of in-
duction is so simple and infallible, should there be
an elaborate treatise on the subject. Were the for-
mal procedure alone to be considered, we might stop
at this point, having discussed the definition of in-
duction, its principles, and the character of the proc-
ess. Still the ground and manner of its material
applications would need discussion, especially when
taking the form of laws, more especially laws of
nature.

But there is much else to be expounded. The
preparation and complete establishment of the prem-
ise from which to infer inductively is a process of
the highest importance, and often of very great dif-
ficulty. This process also must therefore be sharply
defined, hedged in by rules or methods, and provided
with canons and formulas; thereby constituting a
large and the most intricate part of inductive logic.
This preparation for the induction, so far as it in-
volves inference, is itself strictly deductive, and re-
sults in establishing a causal relation between par-
ticular phenomena. When accomplished, then the
inductive generalization, a single simple step, takes

PROCESS ' 53

place. Its result, a universal truth, perhaps a law,
affords a settled major premise from which deduc-
tions may be made by subsuming special or particu-
lar cases, thus enlarging science in its details.

For instance, Newton made long and laborious de-\
ductions from observations, and thereby settled the \
particular fact that the earth and moon attract each \
other directly as the mass of each and inversely as 1
the distance squared. This preparation accom- /
plished, he then, according to logical order, inferred/
inductively the universal law of gravitation] I Sub-
sequently modern astronomy has been de\^eloped
chiefly by deductions from this law. \ Thus the in-
termediate inductive step, while all -important, is i
single, very simple logically, being immediate, and is
justified by the principle of uniformity.

IV.— OBSERVATION

§ 33. The ground of induction, furnishing matter
to be formalized, is experience (§ 6).' The object
known in experience is a phenomenon. A phenom-
enon is whatever appears, presented either to the
external or to the internal senses. It is the undeter-
mined object of empirical intuition. There are two
great classes of phenomena, those of coexistence and
those of succession.

Phenomena of coexistence are exemplified in the
figure of a body, and in the comparative figures of
separate bodies. Such relations are conditioned on
space and the geometrical properties of space alone;
for, being characterized by simultaneity, they are in-
dependent of time. Thus, that a sphere is two-thirds
of a cylinder whose height is equal to the diameter
of each, is uniformly true in all cases without regard
to time. Very many phenomena of coexistence are
referable to causative antecedents ; as, high tide on
opposite sides of the earth. Other coexisting phe-
nomena are not so referable, but are ultimate ; as,
the ultimate properties of substances. Water has

* " So that the art and practic part of life
Must be the mistress to his theoric."

— K, Em. V.y act L, sc. 1.

OBSERVATION 55

many such properties which always coexist, so that
when we recognize it by some of them, we are sure
of the presence of all.

Ultimate uniformities of coexistence, not being
referable directly or remotely to causation, are not
subject to the principles and methods of induction.
They are, however, subject to observation and classi-
fication, thus forming the basis of kinds, especially
of natural kinds, and thus coming, not under causal
law, but under definition. Hence the various kinds
of rocks and minerals, and of chemical compounds,
expressed by a general name and its definition ; as.
Marble is Grystalline carbonate of lime; also the
natural kinds of plants and animals. These have
ultimate coexisting properties which science obtains
by analysis, and recognizes as constituting the origi-
nal nature of the things, and furnishing the basis of
generalization and classification (§ 5).

Phenomena of succession are conditioned on time,
and are subject to the laws of causation and induc-
tion. Such phenomena are by far the more numer-
ous and important. On a knowledge of them and
their laws is founded every scientific explanation of
past events, also every reasonable anticipation of
future events, and whatever power we possess of in-
fluencing these to our advantage. They constitute
the chief subject of our subsequent inquiries.

§ 34. Attention to a phenomenon and to its at-
tendant phenomena or circumstances is observation.
This implies more or less mental analysis of a whole

66 ELEMENTS OF INDUCTIVE LOGIC

into its constituent parts, and tlieir classification.
When we open our eyes on a landscape, there is an
experience of vision ; what is seen is a whole, whose
parts are quickly distinguished and classified as
mountains, streams, forests, buildings, and so on.
These are kinds of things. They are observed as
coexisting phenomena. A storm arises, clouds gath-
er, rain pours, lightning flaslies, thunder rolls, the
bolt has riven an oak or fired a dwelling, a whirl-
wind threatens yet greater ruin. These are classi-
fied as kinds of events. They are observed as suc-
cessive phenomena, and are recognized as causally
related. Thus observation is discriminating atten-
tion.

Observation has two modes distinguished as sim-
ple observation and experiment. The former takes
place when a phenomenon happens to fall under
notice, or when, without acting upon it, we seek and
find one suited to our ends. The latter takes place
when by an artificial arrangement we produce a suit-
able instance, and by this action bring it under our
observation. The distinction is clear, but does not
imply a logical difference. The character and value
of a fact is not at all affected by the way it is ascer-
tained. The practical difference, however, is im-
portant, and requires consideration.^

^ The verb to experiment always implies activity, while to expeHence
suggests rather a passive, receptive state, and thus is more nearly
allied to simple observation. Both words are from the same deponent
verb experiri, to try or to be tried. We try an experiment (Ger. Ver-
such), we undergo an experience {Erfahrung). But experience and

OBSERVATION 57

§ 35. A phenomenon being given, its cause or its
effect is to be ascertained, as a preb'minary to induc-
tion. In other words, before a scientific induction
can be made, a problem is to be solved : either given
an effect to find its cause, or given a cause to find
its effect. Observation, therefore, must extend be-
yond tlie given phenomenon to its circumstances,
eliminating those that are immaterial (§ 16), and dis-
tributing the remainder as antecedents and conse-
quents. To do this thoroughly and accurately is
often diflScult, requiring great care and skill.^

When an effect is given to find its cause, only sim-
ple observation is applicable. Seeing that a bit of
silver chloride has turned from white to black, and
inquiring the cause of this change, we are limited to
simple observation of the circumstances. We can-
observation are here used synonymously, and as generic of simple ob-
servation and experiment. All imply voluntary attention, which is
essentially active. While perception, taken strictly, is passive, there
is no passive observation. — See Psychology^ § 82 sq., and § 99.

^ The elimination of immaterial circumstances will be considered
subsequently (§ 51). No useful rule can be given for the distribu-
tion of the antecedents and consequents. In general, what has been,
changes, and ceases to be, we should reckon an antecedent \ what was
not, eventuates, and begins to be, we place among the consequents.
" It happens sometimes that when a relation of causation is established
between two facts, it is hard to decide which, in the given case, is the
cause and which the effect, because they act and react upon each
other, each phenomenon being in turn cause and effect. Thus, habits
of industry may produce wealth, while the acquisition of wealth may
promote industry. As Plato remarks, education improves nature, and
nature facilitates education. National character, again, is both effect
and cause ; it reacts on the circumstances from which it arises." —
Lewis, Methods of Politics, i., p. 3Y5.

58 ELEMENTS OF INDUCTIVE LOGIC

not .take an effect and try what will cause it, we can-
not reverse the order of nature. We can only make
note of the circumstantial and substantial antece-
dents, knowing that tlie cause is in them. We may
further observe the phenomenon amid various cir-
cumstances, and so be able to eliminate many that
are not causal conditions, and thus reach a conclusion
more or less probable. In this way we might find
that the like change in several specimens of silver
chloride is probably due to light.

If it happen that two cases occur wherein all an-
tecedent circumstances are strictly alike except that
in one case an antecedent is present with the phe-
nomenon in question and in the other both are ab-
sent, then we have scientific proof that this antece-
dent is the cause of the phenomenon. If two quite
similar bits of silver chloride are observed under
quite similar circumstances except that one is ex-
posed to light and the other not, then the fact that
only the former has turned black is proof that light
is the cause. It is rare, however, that simple obser-
vation is so happy as to find two such cases, or even a
series of cases varying sufficiently to satisfy the de-
mands of scientific proof, nature being constituted
on quite a different plan from that of facilitating our
inquiries.

When a cause is given to find its effect, experi-
mental observation is often applicable ; but there are
many cases, indeed whole sciences, whose matter is
of such sort as to be mostly, if not wholly, out of
the reach of experiment. The mental sciences ad-

OBSERVATION 59

mit it but sparsely, though recently some progress
has been made in experimental psychology. An-
thropology, zoology, geology, and astronomy are sci-
ences whose ground is almost exclusively simple ob-
servation. Thus it is that, in looking from effect to
cause, we are in all cases limited to simple observa-
tion by the nature of the relation; and in looking
from cause to eifect, we are in many cases limited to
simple observation by the nature of the matter.

§ 36. Experimental observation, though applicable
only to the problem of given a cause to find its
effect, and in this natural order only to matter that
can be handled, is nevertheless an extension of obser-
vation, since it multiplies the facts. Also it is a
means of more exact scientific knowledge. Indeed,
simple observation, which when alone hardly yields
sure knowledge of causal relations, has its best re-
sults in furnishing ground for supposition, and in
suggesting intelligent experiment. When in its ex-
ercise we have found reason to suppose that the
blackening of silver chloride is the effect of light, we
have recourse to experiment, reversing the order,
and testing the influence of light upon the salt. We
hereby extend observation, and determine with pre-
cision the cause of the phenomenon.

In the previous section it is indicated that the ob-
servation of a phenomenon in various circumstances
leads approximately to the determination of its cause
or of its effect, by means of the successive elimina-
tion of circumstances that are immaterial. Now,

60 ELEMENTS OF INDUCTIVE LOGIC

it is a prerogative of experiment to vary the circum-
stances at will, and thus intelligently to produce pre-
cisely the sort of variation that conduces most defi-
nitely to the determination we seek, a variation
which perhaps nature does not furnish at all. In
order to know which of the two principal compo-
nents of air, oxygen and nitrogen, supports combus-
tion and respiration, we separate them, thus bring-
ing them into states not found in nature ; then
testing one and then the other with burning and
breathing things, we ascertain, by this especial varia-
tion of circumstances, that oxygen is the effective
component.*

Moreover, when we can produce a phenomenon
artificially, it may be isolated, or at least produced
amid circumstances which are well known, and
hence not liable to be confused with it. For the
study of magnetism, a house is built apart, with no

' One hundred years before Bacon's time, Leonardo da Vinci, the
painter and scientist, wrote: "Theory is the general, Experiments are
the soldiers. . . . We must consult Experience, and vary the cireum-
stances till we have drawn from them general rules ; for it is she who
furnishes true rules. But of what use, you ask, are these rules?
I reply that they direct us in the researches of Nature and the opera-
tions of Art; they prevent our imposing upon ourselves and others.
. . . Nature begins from the Reason and ends in Experience ; but, for
all that, we must take the opposite course: begin from the Experi-
ment and try to discover the Reason." — Venturi, Msai. Bacon em-
phasized " the prerogative of experiment," and urged compliance with
Nature ; '■'■Natura non aliter quam parendo vincitur.^^ Coleridge is more
inquisitorial To experiment is " to bind down material matter under
the inquisition of reason, and force from her, as by torture, unequiv-
ocal answers to prepared and preconceived questions." — Friend.

OBSERVATION 61

iron in its construction, so as to avoid local disturb-
ance. Instead of simply observing electricity in
thunder-clouds, we evolve it in a room by means of
contrivances that are sufficiently understood, such as
the Holtz machine, a voltaic battery, or a dynamo.
A hospital is the best place for studying disease, for
the surroundings and treatment of patients are
largely under control of the physician. When the
phenomenon in question is thus insulated, we proceed
to test it by introducing some well-defined circum-
stance, and noting the consequents. A chemist, hav-
ing obtained apart a new element or compound, ap-
plies various well-known reagents in succession, and
observes what unions or disunions take place. Such
practical isolation of a phenomenon, and the testing
it with various familiar and modifying circumstan-
ces, thus determining definitely its causal relations,
is perhaps the most important prerogative of experi-
mental observation.

v.— ENUMERATIOlSr

§ 37. The first form of induction to be considered
is one that is in very common use and highly im-
portant; natural and right under certain provisos
and limitations, but to be distinguished from sci-
entifically prepared procedure. It is described by
Bacon as Inductio per enumerationem simplicem, ubi
non rejperitur instantia contradictoria. This, he
says, is the only mode of induction that was known
to the ancients, or indeed prior to his time.

It is of two kinds. One arises from a simple
enumeration of cases that resemble each other in a
given mark or marks; the other, from a simple
enumeration of marks in which given cases resem-
ble each other. The one is an inference from a
count of similar instances or cases ; the other is an
inference from a count of similar qualities or marks.
The one deals with matter relatively to its extension ;
the other relatively to its intension (§ W). The
former is called enumeration of cases; the latter,
analogy.^ Of these in their order.

' Analogy is not recognized by Bacon as a kind of simple enumera-
tion. It is usually treated by subsequent logicians as a distinct mode of
inference, sometimes as hardly inductive. The logical place and relation
here assigned will be justified by its treatment in the sequel (§ 41 sq.).

ENUMERATION 63

§ 38. The frequent recurrence of an observed fact
gives rise in the mind to expectation of its renewal.
This bj the laws of suggestion.' Indeed, a single
fact strongly impressed does likewise ; as, A hurnt
child dreads fire. The irrational brute mind seems
to act in this respect like the human mind. But the
human mind reaches a higher plane when, having
observed certain repetitions, it concludes inductively
a general truth. That All crows are hlaclc will be
stoutly maintained by a boor, and not without rea-
son, he never having seen a contrary case. We have
no other proof that All men are mortal, and the ad-
mitted certainty that you and I shall die is a deduc-
tion from this inductive generalization based on enu-
meration.

The formal procedure of induction by an enumer-
ation of cases may be expressed in the following
Canon: If many instances agree in having
two marks in common, then all instances
having one have also the other mark.

The process is forniulated and exemplified thus:

\i A, B, C, D, E are observed to have each

the marks m and n, we make the induction that all
cases having m have also n. Then X, being seen to
have ^ is deductively inferred to have an unseen n.
Newton observed that many highly refractive (m)
substances, as oils and resins, are combustible {n).
Through inferring it of all, he reached the further
conclusion that the diamond, being highly refractive,

^ For Laws of Suggestion, see Psychology^ § 172 sq.

64 ELEMENTS OF INDUCTIVE LOGIC

is also combustible, which was afterwards verified
(see § 47). The inductive step takes this form :

Some specific cases agree in an accidental mark ;
.'. All such cases agree in this accidental mark.

In making the induction the mark generalized is
regarded as a logical accident ; as. All metals are lus-
trous (§ 5). But further investigation may conclude
it to be essential; as, Cows ruminate, All animals
have a nervous system / in which case the mark is
transferred to the definition of .the kind.

§ 39. What justifies this form of induction ? ^
There is in the human mind a natural and strong
tendency to generalize from observed repetitions,
but the only principle that will justify a generaliza-
tion beyond experience is the principle of uniform-
ity. This is the basis of every induction. All men
know that like causes have like effects, with the in-
verse, and though very few may have thought it in
abstract form, still when a uniformity is observed,
when two or more facts frequently and invariably
concur, these are at once suspected to be causally
related, either as one determining tlie other, or as
coexisting parts of a consequent. Which or what is
the cause may be quite unknown and unquestioned,
but an obscure surmise, more or less reliable, that
the concurrence is due to causation, is the authority
for the induction.

If I see a number of men in succession rush by
my window up the street, instinctively I wonder

ENUMERATION 65

what is the matter. I cannot, perhaps, even guess.
Nevertheless, I expect the next one that passes will
follow the others ; having deduced his case from the
induction that, for some cause or other, everybody is
running up the street. On cold, clear nights in the
north the aurora has frequently appeared, hence at,
such times Northerners watch for it. So we all ex-
pect meteors in November, and confidently predict
the zodiacal light in February. Should a comet ap-
pear with a coma not turned from the sun, astrono-
mers would be more bewildered than ever. In case
of a hemorrhage of the lungs, the doctor promptly
administers a dose of common salt. Why? All
that he knows, or that any one knows, is that this
old woman's remedy has often been efficacious.
The inference to all^ which is the inductive step, is
sometimes so obscurely and quickly passed through
that it escapes the attention even of logical ana-
lysts, and the whole process seems to be a direct infer-
ence from particular to particular ; as when a village
matron says : " This physic cured my Susan, there-
fore it will cure your Lucy." But there is surely
an intermediate universal.^

General rules, sometimes called laws, obtained by
induction from a mere enumeration of cases, are
known as empirical rules or laws; for, the causes
being unknown, at least in their modus operandi,
the induction is made solely on the experience of
the cases, without investigating the surmised causes,

* For the contrary view, see Mill, Logic, p. 141 sq.
5

66 ELEMENTS OF INDUCTIVE LOGIO

and so no explanation by reference to more general
laws is assignable. The practice of medicine is very
largely thus empirical (§ 95).

§ 40. What is the value of such imperfect induc-
tion V In the practical affairs of every -day life its
value is inestimable. General truth that has been
or can be scientifically determined is insufficient for
the needs of the scientist, and is unknown to the
vulgar; hence in the vast majority of even most
important concerns we are obliged to use induc-
tion from enumeration of cases as the only avail-
able means to guide expectation and provisory con-
duct. The hazard that attends it is often great;
still, by an extensive multiplication of facts, no ex-
ception occurring, we are able to infer reliable rules.
Not one person in thousands has any other reason
for believing that the sun will rise to-morrow, that
the moon will change, that the seasons will come
and go in fixed order, that industry secures reward,
that no one is content, that money purchases goods,
that physic cures, that water quenches thirst, or even
that he himself can walk and talk.

While this form of induction can never furnish
scientific proof of a universal proposition, but at
best only yields high probability, yet, even in its less

* The custom is to call the Aristotelic procedure discussed in § 27
perfect induction, though truly it is not induction at all, and all induc-
tion proper imperfect induction. I prefer to call induction by enu-
meration imperfect induction, and induction by methods yet to be
expounded perfect induction.

ENUMERATION 67

conclusive instances, it has great scientific value in
serving constantly to suggest causal relations, thus ^
pointing the way to investigation by the sure meth-
ods which are to be hereafter discussed. Thus the
diurnal ebb and flow of the tide, observed for ages,
led at last to the investigation that proved the moon
to be the cause/

§ 41. To induction by a simple enumeration of
cases corresponds induction by a simple enumera-
tion of marks, or analogy.

^ It is quite evident that the mode of induction before us rarely
gives rise to satisfactory knowledge. "Popular notions," says Mr.
Mill, " are usually founded on induction by simple enumeration ; in
science it carries us but a little way. We are forced to begin with it;
we must often rely upon it provisionally, in the absence of means of
more searching investigation ; but for the accurate study of nature, we
require a surer and a more potent instrument." — Logic, p. 227. It ia
surprising, after so excellent a statement, to find this highest authority
holding and laboring to prove that the " ground of induction " is ground-
ed on enumeration. See supra, § 19, note. In this palpable diallelon
(§ 1^6) he is followed, as in other respects, by Mr. Bain. — Logic, bk. iii.,
ch. xi., § 13. Mr. Venn admits the logical fault, but comes to the res-
cue with a psychological justification. — Logic of Chance, ch. x., § 14.

Bacon strongly condemns induction by simple enumeration as un-
scientific, as '"'■mera pilpatioy He says: "Inductio quae procedit per
enumerationem simplicem res puerilis est, et precario concludit, et
periculo exponitur ab instantia contradictoria, et plerumque secundum
pauciora quam par est, et ex his tantummodo quae praesto sunt, pro-
nunciat. At Inductio quae ad inventionem et demonstrationem Scien-
tiarum et Artium erit utilis Naturam separare debet, per rejecticmes
et exclusiones debitas ; ac deinde, post negativas tot quot sufficiunt,
super affirmativas concludere." — Nov. Org., bk. I., aph. 105. Cf. aph.
25 and aph. 69. Mr. Mill quotes approvingly the same passage " as
a final condemnation of this rude and slovenly mode of generaliza-
tion."— Logic, p. 549.

68 ELEMENTS OF INDUCTIVE LOGIC

Analogy is liable to be confused with metaphor.
The latter, taken in a wide sense, is a rhetorical
form wherein, because of some resemblance between
two things, the marks of one are transferred to the
other. Because they are alike in courage, we say;
Achilles is a lion. So also: There is a tide in the
affairs of 7nen, etc; Age is the evening of life;
Gratitude is the inemory of the heart; A ship
ploughs the sea ; James, Cephas, and John were pil-
lars of the church. Such similitudes are used to
adorn and to illustrate, but are inconsequent, and
give rise to a fallacy (§ HD). Analogy likewise is
founded on resemblance, and the name is often very
loosely applied to any and all similes. But as a
logical form, analogy is restricted to such resem-
blances as are consequent, furnishing ground for
logical proof.*

According to its early definition, analogy is an
equality of relations. For example: As is a father
to his children, so is a ruler to his subjects. Here
we have stated, in an equality or identity of re-
lations, the paternal theory of government, from
which may be deduced the duties of citizens." But
it is now usual and better to extend the logical

* Metaphor (from fura-^speiv, to transfer) is a mental transference
of marks. Analogy is not a transference of marks, but because some
marks are observed to be inherent, other marks are inferred to be in-
herent, which is not transference, but inference.

2 With Aristotle analogy is iaSrrjQ Xdywv, an equality of relations.
His example is : wg yap sv trw/iart o\pmg, sv \pvxv vovg. — Fth. Nic.^
I., vi., 12. Formally this is a proportion, an equality of ratios. In
mathematics the term analogy is still used in this restricted sense.

ENUMERATION 69

meaning of analogy to any resemblance, not merely
of relations, but of things and classes of things, that
justifies an inference of further resemblance.

§ 42. Accordingly we have already defined in-
duction by analogy as an inference from a simple
enumeration of marks in which given cases resem-
ble each other (§ 37). A sportsman has found trout
in a deep pool of a clear brook. On coming to an-
other pool, very similar in many observed respects,
he makes the induction by analogy that it is sim-
ilar in yet other respects ; thence he deduces the
probable presence of trout; and casting in his line,
proceeds to verify the case. Solid metal is marked
by a peculiar lustre ; hydrogen has many metallic
qualities ; hence, through an induction by analogy,
it is highly probable that, should hydrogen be so-
lidified, it would exhibit metallic lustre.

The formal procedure of induction by an enumer-
ation of marks may be expressed in the following
Canon: If two instances agree in having
many marks in common, then all marks
in the one are also in the other instance.

The process is formulated and exemplified thus :
If A and A^ are observed to have many marks
in common, we pass by analogy beyond this ex-
perience, and infer all their marks to be common ;
then, having noted that A has a mark m not seen in
A\ we deduce its presence there. The evidence
that brutes are consciously intelligent is analogical.
There being very many physical points notably

70 ELEMENTS OF mDUCTIVE LOGIC

common to man and brnte, the induction is to all
points, making allowance for differences of degree;
and thence is deduced, what cannot be directly ob-
served, the conscious intelligence of the brute. It
is usual to say that brutes show signs of conscious
intelligence by certain actions; but these actions are
merely transient marks obviously common, which
are accepted as analogical signs in the brute of a
deeper mark beyond observation. Let it be noted
that the complete analogical argument here illus-
trated consists of two steps, an induction followed
by a deduction. The first step being obscure, is
usually overlooked, and hence the inference seems
to pass immediately from particular to particular.

When the two cases under consideration are of
the same kind, an essential mark belonging to the
definition of the kind evidently cannot be made the
occasion of analogical inference. Only marks con-
sidered accidental are inferable by analogy. The
inductive step takes this form :

Some accidental marks agree in two specific cases ;
.*. All accidental marks agree in these two cases.

This may yield a fuller knowledge of the essence.
Some, many, accidental marks are common to oaks
and pines. Then, from an induction of all, we con-
clude that, since oaks are observed to be dicoty-
ledonous, pines are so likewise. This, verified by
observation, has been adopted as a generic mark.

Generally the same result may be obtained by
either mode of enumeration. The conclusion that

ENUMERATION 71

I am mortal may be had thus: My neighbor and I
being much alike, and he dying, then I too shall die.
So also Newton's inference that the diamond is
combustible (§ 38) ma}^ be represented analogically.
This might be expected from the striking similarity
of these two forms of induction, and from the con-
vertibility of extension and intension to which they
severally correspond (§ 37).

§ 43. The justification of an inference by analogy,
like that from an enumeration of cases, lies in the
principle of uniformity. The sole support of the
induction is the knowledge or surmise, however ob-
scure, that the marks observed to coexist in the one
ease are causally connected, and hence may be in-
ferred to coexist in the analogous case. Hence, if the
observed property or mark m be known to be un-
connected causally with any of the properties of A
in which A' resembles A^ there is no basis for ana-
logical inference. On the other hand, if the mark
m be known to be connected causally with some one
of these properties of A^ the imperfect induction by
analogy is superseded by a perfect induction (§ 40 n.).
"We must be measurably assured that m is connected
causally with some of the resembling properties with-
out knowing with which it is so connected.

It is evident that if the induction from some cor-
respondences in the two cases to all were fully author-
ized, the result would be one of entire identity ; also
that cases are hardly ever so thoroughly assimilated.
The all of the induction, therefore, can be taken only

72 ELEMENTS OF INDUCTIVE LOGIC

in a loose and doubtful sense even when no con-
traries are observed, and the deduction from it is at
best doubtful. If, along with an observed commu-
nity of many marks, there is also an observed dispar-
ity of others, these as against those proportionally
diminish the probability of the inference. When
the differences balance the resemblances, analogy
affords no presumption.

There are striking points of community between
the senses of smell and taste, and also of hearing and
seeing, which have led by analogy to a fuller knowl-
edge of them.^ Sodium and potassium have many
points of agreement and few of difference ; there is,
therefore, considerable probability that a newly ob-
served quality of one has its counterpart in the
other; or, since qualities are causes, that an effect due
to sodium might also arise from potassium, such as
the rapid decomposition of water at ordinary tem-
perature. An instance may thus have the mark of
being a cause or an effect.

Plato's Republic, whose constitution is modelled
by that of the individual man, is a brilliant ideal ;
but to infer from three leading functions of mind
that there should be three classes of citizens in the
state is inept, for these are not counterparts. Yet,
when we observe that pure reason is legislative,
thought judicial, and will executive, and thus dis-
cover in human nature the approved functions of
departments of state, the resemblances are sufl5cient

* These analogies are more fully stated in " Psyctwlogy, §§ 9, 20.

ENUMERATION 73

to justify an inference by analogy to others that are
derivative.

A famous analogical argument is, that, since the
earth and the moon have many points of resemblance,
and the earth is peopled, therefore the moon also is
peopled. To this it is properly objected, first, that
being peopled cannot be surmised as the effect even
remotely of any or all the enumerated resemblances;
secondly, that the points of difference are much
more numerous and weighty than the resemblances,
and therefore the presumption is decidedly to the
contrary. If we substitute Mars for the moon, the
resemblances are increased and the differences di-
minished, but still the argument fails. A better ana-
logical inference is that the stars, like the sun, are
attended by planets.*

§ 44. Analogy renders good service in practical
concerns by furnishing useful hints that sometimes
ripen into maxims or rules of life. It helps to a
good guess, is an index to truth. The balsam of
Peru, besides many other properties, is medicinal ; the
balsam of Tolu agrees in many of those properties,
and presumably may replace the other in pharmacy.
But such inferences standing alone are very hazard-
ous. The order of plants SolanacecB is defined by
many common points. It includes the tomato, po-
tato, and egg-plant, which are wholesome food. To

' See the anonymous essay, usually attributed to Dr. Whewell, en-
titled " Of the Plurality of Worlds."

74 ELEMENTS OF INDUCTIYE LOGIC

infer this of all other species would be perilous, for
the order includes the thorn-apple, tobacco, and also
belladonna or deadly-nightshade, a virulent poison.

To establish any scientific doctrine whatever anal-
ogy of itself is quite insufficient. The brilliant trea-
tise entitled " Natural Law in the Spiritual World," '
whose argument in support of its leading doctrine,
indicated in the title, is only and can only be from
analogies, has not widened the domain of science
or increased its treasures. Still, the process has
scientific value. It may often profitably be used to
confirm a truth otherwise ascertained, and thus be-
come ancillary to science. It is useful too in ten-
tative or provisional classifications, as those of the
Linnsean botanical system. But its principal service
is to suggest lines of research by certain conclusive
methods to be considered hereafter. The points of
community between hearing and seeing suggested
to Huyghens and to Young, that, as hearing is the
effect of external vibrations of an elastic medium, so
seeing might perhaps have a similar cause. Thus by
analogy originated the hypothesis of an undulating
luminiferous ether.

Analogy has also a negative but great scientific
value in meeting objections, and thus is a useful de-
fensive instrument. The argument of the masterly
treatise entitled "The Analogy of Keligion to the
Constitution and Course of Nature" ^ shows that the
difficulties in religion, natural and revealed, have the

* By Henry Drummond, F.R.G S. ^ By Bishop Joseph Butler.

ENUMERATION 75

same relation to their respective systems that the
difficulties in the course of nature have to the entire
system of nature. If, then, the latter be admitted
to proceed from a divine Author, the difficulties in
the former are not a valid objection to a like origin.
In this statement the analogy is represented ex-
pressly as an equality of relations (§41). It may be
stated. also thus: Nature and religion are largely
analogous — that is, have many likenesses, even as to
difficulties; if, notwithstanding these, a divine Au-
thor is attributed to the former. He cannot, because
of them, be consistently denied to the latter. It is
not intended to prove the divine origin of religion,
but indirectly to confirm proper proofs by showing
that the difficulties in religion, being like those ad-
mitted by the deist to exist in nature, cannot be
offered by him as an objection to its divine origin.
The procedure is evidently ad hominem (§ 108).

VI.— PEOBABILITY

§ 45. It has several times been stated that enumer-
ation furnishes only probable evidence. Let us now
examine the meaning of probability, and consider
its bearing.

To probability is opposed certainty. Only intui-
tion and demonstration, as in pure mathematics, are
attended by pure or strict certainty. Demonstration
starts with and results in certainty, for its ultimate
premises are intuitive necessary principles, and it
carries their strict certainty into its conclusions. The
process is always a deduction, for it proceeds from
the strictly universal to the less general. Both de-
ductive and inductive logic, like pure mathematics,
deduce their formal theorems or canons from intuitive
necessary principles; the process is demonstrative,
the results strictly certain, admitting no degrees.^

But the application of the theorems of logic to em-
pirical matter involves, as in applied mathematics,
the essential uncertainties of experience (§ 8), as well
as those arising from the imperfect fulfilment of the
theoretic conditions. It is clear that any uncertainty
in the premises is followed by an equal uncertainty

* On the feeling of certainty, see Psychology^ §§ 69, 118, 227.

PROBABILITY 11

in the conclusion (§ 91). Hence in the employment
of induction especially, since the material application
of its formal theorems depends wholly on experience,
strict demonstrative certainty is unattainable.

Probable evidence is distinguished from demon-
strative by admitting degrees from the lowest pre-
sumption upward, but not reaching strict certainty.
That the tide ebbs and flows to-day aftords a slight
presumption that it will do so to-morrow; and the
evidence gathers force with each added observation,
until the observations of ages, no exception occurring,
afford by enumeration alone inductive proof of high
order, giving strong assurance that it will do so again,
but not giving certainty in its strict sense. Events
falling within this wide range are regarded as merely
more or less probable.

Having set probability apart from strict certainty,
let us narrow its range by a further distinction. Be-
sides strict or pure certainty we recognize physical
and moral certainty, the former relating to natural,
the latter to human, events.^ These together may be
called empirical certainty. When an order of facts
has been proved by a rigorous application of the de-
terminative methods yet to be discussed, it is scien-
tifically ascertained, and is said to be physically or
morally certain as the case may be — that is, empirically
certain, and not merely probable. Here, then, is the

' Moral certainty, an objectionable phrase, usually quite indefinite,
but too well established to be changed or rejected. The meaning to
which we here limit it is justified by the etymology of moral, from Lat.
moSy moris, manner, custom, habit, conduct.

^8 ELEMENTS OF HTDTJCTIVE LOGIC

upper limit of probability. Its extent is from the
lowest presumption having any, the slightest, evi-
dence in its favor, up to the physical or moral cer-
tainty, the empirical certainty, of scientific truth.
That the sun will rise to-morrow is not strictly cer-
tain, but is physically certain. That, when the sun-
set sky is red, the morrow will be clear, is not a
scientifically ascertained sequence, but has at best
only some degree of probability.

§ 46. Comparatively few phenomena in nature, still
fewer in human affairs, present themselves in a form
suited to close scientific investigation. By far the
greater number, often those of the highest practical
moment, are out of reach of scientific treatment, and
our knowledge of them and our conclusions from
them are uncertain. These fall within the wide
range lying between bare conjecture and empirical
certainty, the range of probability. In such matters
we are dependent on imperfect unscientific induc-
tion, merely approximate generalization, such as is
yielded by enumeration. This probable evidence, in
its very nature, affords but an imperfect kind of in-
formation, yet on a vast multitude of occasions we
have no other resource in guiding our conduct. The
ability to judge fairly of probabilities distinguishes
the man of wdde experience, close observation, and
practical sagacity. When pronouncing what is likely
to be true — that is, like in evidence or circumstances
to some known truth or true event — he rarely errs.

" It is observation that produces, in numberless

PKOB ABILITY 79

daily instances, a presumption, opinion, or full con-
viction that such an event has or will come to pass ;
according as the observation is that the like event
has sometimes, most commonly, or always, so far as
our observation reaches, come to pass at like dis-
tances of time or place, or upon like occasions.
Hence arises the belief that a child, if it live twen-
ty years, will grow up to the stature and strength of
a man ; that food will contribute to the preservation
of its life, and the want of it for such a number of
days be its sure destruction. So likewise the rule
and measure of our hopes and fears concerning the
success of our pursuits, our expectations that others
will act so and so in such circumstances, and our
judgment that such and such actions proceed from
such principles — all these rely upon our having ob-
served the like to what we hope, fear, expect, judge.
And thus it is that to us probability is the very guide
of life." '

" Even when science has really determined the
universal laws of any phenomenon, not only are
these laws generally too much encumbered with con-
ditions to be adapted to every-day use, but the cases
which present themselves in life are too complicated,
and our decisions require to be taken too rapidly, to
admit of waiting till the existence of a phenomenon
can be proved by what have been scientifically ascer-
tained to be the universal marks of it. To be inde-
cisive and reluctant to act, because we have not evi-

^ Butler, Analogy, Int.

80 ELEMENTS OF INDUCTIVE LOGIC

dence of a perfectly conclusive character to act on,
is a defect. If we would succeed in action, we must
judge by indications which, though they do not gen-
erally mislead us, yet sometimes do, and we must
make up, as far as possible, for the incomplete con-
clusiveness of any one indication, by obtaining others
to corroborate it. The principles of induction ap-
plicable to approximate generalization are therefore
not a less important subject of inquiry than the rules
for the investigation of universal truths." *

§ 47. The hazardous validity of the canons of enu-
meration is conditioned on there being no known ex-
ceptions, instantia contradictoria (§ 37). In an appli-
cation to a material case, though no exception may
have been observed, and though we may feel assured
from the extent of the observations that if there
were an exception we should have met with it, still,
since we can never be positive of this, it follows that
a universal by enumeration is never more than prob-
able. We surmise, and perhaps strongly suspect, the
observed uniformity to be due to causation wherein
a real exception is impossible ; as. Horses eat grass,
Cows chew the cud, Birds lay eggs ; ^ but when quite
ignorant of the determining causes, though feeling

» Mill, Logic, p. 41 Y.

2 Such invariable attributes sometimes come to be regarded as es-
sential marks of natural kinds, and then are posited as generic defin-
ing qualities ; as, graminivorom, ruminant, oviparous. In such case
to say, for example, that All birds lag eggs is merely to refer to the
definition, and is not an induction.

PROBABILITY 81

sure of their existence, we can do no more than vent-
ure a highly probable universal proposition.

The saying that a real exception to a causal uni-
formity is impossible is simply a varied statement of
the irrefragable principle of uniformity, and when
a real exception occurs we know at once that the
phenomena in question are not causally related. Be-
fore giving up our probable universal, however, we
should be very sure the exception is real, and not
merely apparent (§ 8). Merely apparent exceptions
frequently occur, due to the presence of some counter-
acting circumstance, some modifying or preventive
cause ; as, when gunpowder fails to explode, being
damp (§ 15). Exceptions of this sort do not invali-
date the induction, its universality being always
under the general condition : Provided there he no
preventing cause. We do not lose faith in a medici-
nal specific because it sometimes fails to cure. But
any exception rightly checks expectation. We hesi-
tate, and recognize the hazard of procedure.

But when a real exception has been detected, this
observation of a contrary forbids the induction of a
universal proposition. The best we can say is Some
(a few, or many, or most, but not all) A's are B / as,
A few springs are silicious ; Many strata arefossil-
iferous ; Most clays are ferruginous. Such incom-
plete uniformities of coexistence are not, cannot be,
cases of causation, and hardly rise to the dignity of em-
pirical maxims, much less of laws. The predicate is
contingent, the coincidence fortuitous. An approxi-
mate generalization of this sort positing Most are, or

82 ELEMENTS OF INDUCTIVE LOGIC

Most are not, obviously requires a comparative knowl-
edge of the total, the observed cases being a majority.
The assertion when limited to these observed cases
is not an induction, but merely a partial colligation
(§ 9), and affords no ground for even a probable in-
ference to unobserved cases. We may only say that
perhaps, perchance, possibly, others correspond.
Newton inferred from oils, resins, etc., the invariable
concurrence of high refrangibility with combusti-
bility, and thence deductively predicted the combusti-
bility of the diamond (§ 38). This haply proved true.
But, as Brewster remarks, had he known the high re-
fractive power of the minerals greenockite and octo-
hedrite, and made the prediction of them, it would
have failed, they being real exceptions invalidating
the induction, and showing the concurrence to be by
chance. Facts that thus concur by chance do not
come within the range of probability indicated in
§ 45, but lie below in a logical region which we shall
now examine, preparatory to a rise from it through
probability into empirical certainty.

§ 48. It has already been said that a chance or for-
tuitous event, a pure accident, a hap, a casualty, in
the sense of an uncaused event, is impossible in fact,
or even in thought (§ 18). There is no such thing
as chance, in antithesis to cause or law, in the whole
realm of being. So taken, the word has no meaning
whatever.

Every event is the effect of causes, and might be
predicted from a knowledge of them. The turning

PROBABILITY 83

up of a particular card is a causal consequence of the
way the pack is handled, and of the place of that
card in the pack ; this last is a consequence of the
way the cards were shuffled ; and so on. When a
leaf, loosened from its stem, falls to the ground, its
final position is strictly determined by causes operat-
ing chiefly during its descent through the resisting
air. Every natural event is physically necessary, but
not physically certain, for there are many that, in
our ignorance, we can neither predict nor explain.*
Such wholly uncertain events are called casual, or
are said to occur by chance. The word chance is
thus used as a common name for the unknown cause
of any single occurrence ; as, The tree fell hy chance
due north. To say, then, that any one phenomenon
is produced by chance is merely a conventional mode
of expressing our ignorance of its cause, and in this
sense the word has no place in logic.'* i

^ The statement that each natural event is physically necessary
means that it is causally determined to be just what it is, without pos-
sible alternative. This is quite apart from our knowledge and belief
respecting it. Physical certainty, as described in § 45, has reference
to knowledge and belief. Certainty and uncertainty are primarily
states of mind, and are attributed secondarily as marks to a recognized
relation among objective facts, when the relation so far as known is
such as to produce some degree of one or the other mental state in the
observer. It is evident that a complete knowledge of the real relation
involving physical necessity would be attended by strict certainty, and
that any inferior degree of certainty is due to a corresponding measure
of ignorance. See the references in § 45, note ; and Whately, I/ygic,
Appendix I., iv. ; also Thomson, Outline^ etc., § 122.

"^ Says Aristotle : SokeI jxtv alria -q rvxVi adrjXov St dv9p(x}mvy
Siavoig.. — Physica, ii., 4.

84 ELEMENTS OF INDUCTIVE LOGIC

But when two or more phenomena or events, that
are in no way related through causation, coexist or suc-
ceed one another, they are said to concur by chance.
In this sense we shall find use for the word. Examples
of such concurrence are : We met hy chance^ and, The
night of CromweWs death, a violent storm hroJce over
London. Also, We chanced to arrive an hour apart;
and, The appearance of the great coTnet of 1861 was
followed hy war. Some such casual coincidences
may recur again and again ; as, Many great hattles
have happened on Sunday. Chance in this sense
may be defined as the possibility of an event, and the
problem of chance is to estimate the value of this
possibility in terms expressing the likelihood of its
recurrence.

§ 49. The logical doctrine of chance, then, pro-
poses to estimate the relative value of a chance. The
clearest illustrations, perhaps, are drawn from games
of chance. In these the probabilities are artificially
balanced ; in other words, there is no probability
either way. Take a toss of a penny. Head or tail ?
It must be one or the other, but it is impossible to
predict which, since there is no ground for proba-
bility in favor of the occurrence of either.' Still,

* The terms chance and probability are very often used synony-
mously ; as, by Laplace in his " Essai Philosophique sur les Probabili-
t^s," and by Mr. Venn in his " Logic of Chance, an Essay on the The-
ory of Probability." In the present treatise we prefer to distinguish
them. Probable cases are those that have some evidence, more or
less, in their favor. The probabilities may be either for or against an

PROBABILITY 85

we are sure that, in the long run of many throws,
the number of heads and tails will be about equal.
'No specific experience seems prerequisite to this as-
surance.

How are we assured, without trial, that the chance
between the two is even ? According to the axio-
matic principle of Sufficient Reason, nothing comes
to pass without a reason why it should occur in that
way, rather than in another.* But, in the case sup-
posed, we are acquainted with the causes at work suf-
ficiently to know that there is nothing, no constant
cause, giving a bias in the long run to either face of
the penny ; that is, there is no cause furnishing a
sufficient reason for inequality. Therefore, inequal-
ity will not come to pass ; or, in the long-run, equal-
ity of heads and tails is reasonably expected.

event — that is, an event is probable or improbable according to the
evidence of causal connection or repugnance. Chance is not a species,
but a pure negation of probability, occupying the ind liferent mean be-
tween the probable and improbable. It is strict uncertainty.

^ Leibnitz, who introduced this principle into logic, says in a letter
to Dr. Clarke: " In order to proceed from mathematics to natural phi-
losophy, another principle is requisite (as I have observed in my *The-
odicsea '). I mean the principle of the sufficient reason ; or, in other
words, that nothing happens without a reason why it should be so,
rather than otherwise. And, accordingly, Archimedes was obliged, in
his book ' De Equilibrio,' to take for granted that if there be a bal-
ance, in which everything is alike on both sides, and if equal weights
are hung on the two ends of that balance, the whole will be at rest.
It is because no reason can be given why one side should weigh down
rather than the other." The reference is to Theod.^ i., § 44. See Mr.
Venn's modified view, Lagic of Chance^ ch. iv., § 8 sq. Evidently the
principle of Sufficient Reason is merely an imperfect statement of the
Laws of Causation, § 18 sq.

86 ELEMENTS OF INDUCTIVE LOQIO

Upon this a priori reasoning, whose subsumption,
however, is empirical, is based the doctrine of the
calculation of chance. " The calculation in general
consists in reducing all events of the same kind to a
certain number of cases equally possible, that is, such
that we are equally undecided as to their existence ;
and in determining the number of these cases which
are favorable to the event of which the chance is
sought. The ratio of that number to the number of
all the possible cases is the measure of the chance ;
which is thus a fraction, having for its numerator
the number of cases favorable to the event, and
for its denominator the number of all the cases
which are possible." ^ We will consider two spe-
cies.

First. — When the uncertain events are taken sever-
ally, the chance of recurrence is expressed hy the num-
ber of cases favorable to it, divided by the whole num-
ber of possible cases. In tossing a penny 2000 times,
we reasonably expect each face to recur about 1000
times. In every single toss, each of the two possible
cases is equally possible — that is, equally uncertain.
The chance, then, of either face recurring is -J. So
likewise in case of drawing a ball from a bag contain-
ing an equal number of black and white balls, or, in
general, in casting equal lots in any manner.' A die,

' Laplace, Essai sur les Frobabilites, p. 7.

2 The remarkable parity of male and female births, statistically as-
certained, fixes the chance of each at ^. The parity of male and fe-
male deaths is an obvious deduction from the parity of births. The

PROBABILITY 87

having six faces, the chance of an ace, or any other
number, is ^', which is only a mode of saying that in
many throws, for instance 600, the ace would recur
about 100 times. Also in each throw the chance
against an ace is |^. If there be in a lottery wheel
^ve prizes in every hundred lots, then the chance of
drawing a prize is .05, or ^^^ ; and the chance of
drawing a blank is .95, or |^.

Second. — When the uncertain events are taken to-
gether^ the chance of their concurrence recurring is
thejproduct of the separate chances. When a pair of
dice is thrown, the chance of an ace with each die
being ^, the chance of double aces is \ x |-=^V? which
is also the chance of an ace twice in succession with a
single die. The chance of cutting a coat-card of the
twelve in the pack of fifty-two is |^f or -^^ ; hence, of
doing so twice in succession, t¥><T3=tIt- ^^^ *^^^
first of three urns contain two black and four white
balls, and the others six white balls each. What is
the chance of drawing a black ball ? The chance of
the drawer taking the first urn is \. In it the black
balls are f of its whole number of balls. Hence the
chance of a black ball is | xf = -|^. Syllogistically :
A is \ C; B is ^ A; .-. B is \ C. Note that if the
eighteen balls were in one urn, the chance would be
the same.

Mathematicians have greatly extended these prin-
ciples, and added others, making application to a

census bulletin of April 27, 1894, shows that in the U. S. males con-
stitute about 51 per cent, of the population.

88 ELEMENTS OF INDUCTIVE LOGIC

great variety of cases, and have thus elaborated a
logico-mathematical system known as the Theory of
Chance/ Its practical applications, however, are not
considerable, nor does its study seem to cultivate sa-
gacity in the estimate of that probability which is
" the very guide of life." ^ We have touched upon
only the simplest elements, and these merely with
a view to their immediate bearing on the general
theory of probability.

§ 50. To set apart casual from causal coincidence,
we need a canon for guidance, since the distinction
is important and often difficult to mark. Absolute
frequency of concurrence will not suffice. Some
events that invariably concur are merely casual ; as,
every change of fortune in one's life concurs with
some change in the position of the planets, but we
no longer believe in planetary influence. On the
other hand, some events that only occasionally con-
cur may be causally connected, the failures being
due to unobserved counteracting circumstances ; as,
rain only sometimes concurs with an east wind.

^ The " Essai " of Laplace, quoted above, and that of Quetelet,
"Sur les Probabilitfes," are the standard authorities.

Besides works already referred to, the " Formal Logic " of Pro-
fessor De Morgan should be named ; also Quetelet's *' Essai de Phy-
sique Sociale," and his " Anthropomfetrie."

•J " Never did I know," says Bulwer, " a man who was an habitual
gambler otherwise than notably inaccurate in his calculations of prob-
abilities in the ordinary affairs of life. Is it that such a man has be-
come so .chronic a drunkard of hope that he sees double every chance
in his favor ?"— What Will He Bo with It ? oh. x.

PROBABILITY 89

From a fact as indefinite as this last example noth-
ing can be inferred. Let us suppose, however, that
rain concurs about as often with east wind as with
any other ; then it is presumably a chance concur-
rence. But if rain concurs more frequently with
east wind than with any other, this indicates that one
can under certain circumstances cause the other, or
somethinor cause both. If the concurrence is less
frequent, this indicates that one, or some cause of
one, can counteract the other. The form of this
procedure, distinguishing casual from causal phenom-
ena, is expressed in the following

Canon: Estimate the positive frequency
of each of the phenomena, and hovr great
frequency of coincidence would take place,
if there were neither connection nor re-
pugnance. Then, if the facts correspond,
the coincidence is presumably casual. If
there be greater frequency, there is pre-
sumably causal connection ; if less, causal
repugnance.

To estimate the positive frequency of a phenome-
non we strike an average on an extended series of
observations. This fixes the ratio between its occur-
rence and its failure to appear. Also it eliminates
mistakes of the senses, accidents, and all errors that
do not arise from some permanent bias. Suppose
we thus ascertain that the phenomenon a occurs once
for two instances of the general circumstances, and
that h occurs once for three. These are their posi-
tive frequency.

90 ELEMENTS OF INDUCTIVE LOGIC

IN'ow, if a and h be independent the average fre-
quency of their coincidence will be once in two times
three, or six, instances ; and hence, if the observed
coincidences be to the instances as one to six, the co-
incidence is presumably by chance (§ 49).

But if the observed coincidence is more frequent
than one time in six, there is presumably some cause
tending to produce it ; if less, some cause tending to
prevent it. The probability of concurrence will in-
crease or diminish with this greater or less frequency.

If, in a certain locality, during the spring months,
it shall have been observed for a number of years
that rain (a) occurs as often as every other day, also
that an east wind (J) occurs as often as every third
day, and that they concur on the average once in six
days, then there is presumably no causal relation be-
tween them — it is a chance concurrence. But if the
observed concurrence be more frequent or less fre-
quent," it is evidence of causal relation.

To vary the illustration : If, in another locality,
fair weather should occur twenty times as many days
in the year as not, and westerly winds three times
as often as not, then, were there no connection or re-
pugnance, fair weather in the long run would concur
with westerly wind five times in seven ; for ff X f = 4-
Now, if the actual concurrence be six in seven, it is
probable that one tends to produce the other, or that
there is some common producing cause; if four in
seven, that one tends to prevent the other, or that
there is some occasional preventing cause.

The principle applies to an enumeration of marks

PEOB ABILITY 91

or analogy. When the coincident marks in two cases
are greater or less in number than chance would af-
ford, we infer that they are causally related, and make
a probable induction respecting unobserved marks.
The East Indian and the English languages have
more comnlon points of syntactical construction, and
similar names for the same things, than chance will
account for, which analogy indicates a common ori-
gin. This renders it probable that other similar feat-
ures are discoverable, so that the existence of some
peculiarity in the one justifies a search for its ana-
logue in the other. The differences between English
and Arabic are greater than chance would yield; hence
a probable repugnancy in fundamental construction,
and an expectant search for still other divergences.

§ 51. The principle involved in the foregoing
canon furnishes ground for the elimination of chance.

In the first place it distinguishes a series of concur-
ring phenomena having real exceptions from such as
have only apparent exceptions (§ 47). As the former
does not justify induction, it is, when exposed, set
aside as the result of chance — eliminated as unfitted
for inductive investigation. That many great battles
have happened on the Sabbath day is an historical fact
from which nothing can be inferred ; for a count would
doubtless find them to be one-seventh of all — a mere
chance, yet striking coincidence.

In the next place the canon helps us in complex
cases to distinguish and eliminate the chance accom-
paniments of a phenomenon undergoing investiga-

92 ELEMENTS OF INDUCTIVE LOGIC

tion. Every phenomenon occurs to observation atnid
circumstances that are immaterial — that is, having no
causative relation to the case. Many of these are
eliminated by the plainest common-sense ; as, in a
chemical experiment in the wet way, it is immaterial
whether the containing vessel be glass of porcelain.
Many are eliminated by isolating the phenomenon as
far as possible, and producing it experimentally amid
well-known circumstances (§ 36). Still some usually
persist whose presence, though invariable, has no bear-
ing on the case, and whose absence would not modify
it. The relative position of the planets was believed
by the alchemists to exert an important influence on
experimental combinations. The chemist of to-day
is sometimes embarrassed by persistent accompani-
ments which are really chance concurrences, and so
need not be regarded. Observation of the phenom-
enon in various situations, artificially varying the cir-
cumstances when practicable, is a means by which,
according to the canon before us, immaterial, chance
circumstances may be detected, and then eliminated
from consideration. This process is especially im-
portant as preliminary to a search for the cause or
effect of a phenomenon by the scientific methods to
be considered subsequently.

An obvious example of the elimination of casual
circumstances is the common-sense explanation of
the progress of the seasons. The fluctuations of
temperature from day to day due to meteorological
change are chance accompaniments, which, being
eliminated, leave the corresponding progress of. the

PROBABILITY 93

sun from solstice to solstice as the one determining
or causal antecedent. The sure profits of a faro-
bank, having a capital too large to be broken by a
run of bad luck, are explained in like manner ; for,
eliminating the chance elements, there remains, in
the very constitution of the game, a small but per-
manent advantage in favor of the banker, which in
the long-run insures his winnings. In all so-called
games of chance which nevertheless involve skill,
as whist, success in the long-run falls to the skilful
players.^

An elimination of the chance elements of a com-
plex phenomenon occasionally discovers small and
hence unsuspected though permanent causes. A
series of throws will detect a loaded die by the turn-
ing up of a certain face oftener than chance will ac-
count for. The slightly more than chance errors of
an instrument of precision indicate some minute per-
manent bias, for which, when determined, allowance
must be made. In this way the obscure diurnal va-
riation of the barometer was discovered. An elimi-
nation of its grosser meteorological fluctuations from
many daily observations, brought it to light and
measurement.'

' Judge Gaynor, now of the Supreme Court of New York, decided
(1894) that horse-raciug is not a lottery within the legal definition any
more than in common speech. The opinion says : " A lottery depends
on a lot or a chance, such as the casting of lots, the throwing of dice,
or the turning of a wheel. In a race the horse-owners pay a sum, not
to win a larger sum by lot or chance, but in order to enter into the
contest of skill, endurance, and speed upon which the stake depends."

2 Even with the best instruments of precision, strict accuracy can-

94 ELEMENTS OF INDUCTIVE LOGIC

§ 52. The preceding considerations prepare us to
examine more definitely the valuation of probabili-
ties. It has already been stated that when a uni-
formity is noted by the enumeration of only a few
instances, there is a slight presumption in favor of
an inductive universal ; and that as observations vary-
ing in circumstances multiply, no contrary case oc-
curring, the probability increases until it reaches the
highest degree, bordering on physical or moral cer-
tainty (§ 45). A deduction from a universal by enu-
meration, subsuming some particular unobserved
instance, is attended by all the hazard involved in
the universal ; and if the particular differs consider-
ably from the observed instances in its circumstan-
ces, the deduction, even from a highly probable

not be expected in a single observation. Therefore it is usual to make
a large number of observations, and, by an application of the Method
of Least Squares, to approximate very closely and surely the true
value. For the best instruments of precision are subject to varia-
tions. Heat, with its irregular warping influence, draughts of air, dust
and consequent friction, distortion by strains, and the slow uneven
contraction of metal which continues long after casting — all these
cause deviations. Moreover, every instrument is liable to some per-
manent bias, due to imperfect construction, which vitiates results, and
therefore must be ascertained and eliminated from each observation.

Another form of permanent bias lies in the observer, some mental
disposition inclining him constantly to perceive in a case more or per-
haps less than is real. Add to this the special action of his muscles
and nerve currents. Allowance must be made, especially in minute
observations on quantity, for the personal bias of each observer. Its
value is expressed in what is called his " personal equation," which
phrase has become familiar to us in connection with astronomical ob-
servations. It is ascertained only by comparing the results obtained
by various observers of tlie same or similar phenomena.

PROBABILITY 95

universal, becomes so precarious as to liave little
value.

It has also been stated that the discovery of a real
exception invalidates the universal (§ 47). The Zulu
of a century ago believed no doubt that All men are
black. To All swans are white there are unaccount-
able exceptions. The satellites of Uranus and Nep-
tune retrograde, and so invalidate All members of the
solar system, move eastward. Such are cases of over-
hasty generalization, a fault of every day and every
hour, acknowledged by the Psalmist in " I said in my
liaste. All men are liars.^^

Exceptions not known to be real, and hence pre-
sumably only apparent, do not invalidate the uni-
versal, since it is conditioned on the proviso that no
interference or prevention takes place. A modifying
or disturbing cause or force may be always present,
and in some cases prevail, becoming a preventive
cause. That all terrestrial bodies fall to the ground
from a given height with like velocity is not invali-
dated by the retarding effect of the ever-present air,
nor falsified in the case of an ascending balloon.
The expression is rendered unexceptionable and still
more general by saying that all bodies tend so to fall.
Thus a tendency, even if never realized, may be
recognized as universal. The generalities of Me-
chanics are rendered more exact by expressing them
in terms of tendency to motion, or pressure.^

* " The habit of neglecting this necessary element in the precise ex-
pression of the laws of nature has given birth to the popular prejudice

96 ELEMENTS OF INDUCTIVE LOGIC

But instances more or less liable to frustration by
unrecognized interferences yield only a questionable
universal, under which the subsumption of an unob-
served particular differing much in its circumstances
is precarious, and the conclusion only more or less
probable. The Greek church has flourished chiefly
among the Slavonic races, the Roman among the
Latin, the Protestant among the Teutonic. Hence
an affinity may be presumed between these several
forms and the character of the races. Change of
time, place, or circumstances, as lapse of centuries,
emigration, political revolution, often breaks this uni-
formity. It is at best an empirical generalization,
whose application to unobserved cases yields only a
low degree of probability.

Approximate generalizations that are not mere
colligations of observed cases (§ 47), but are induc-
tions proper, extending beyond experience, are usu-
ally expressed by Most are or Most are not, or
their equivalents ; as. Most Judges are incorrujpti-
hle. Otherwise we say that the proposition is true
in general, or generally, which in usage implies that
exceptions are recognized at least as possible; as,
It seems to be generally true that Every man has
his jprice, that The wealthy are more virtuous than

that all general truths have exceptions ; and much unmerited distrust
has thence accrued to the conclusions of science, when they have been
submitted to the judgment of minds insufficiently disciplined and cul-
tivated. The rough generalizations suggested by common observation
usually have exceptions ; but principles of science, or, in other words,
laws of causation, have not." — Mill, Logic^ p. 319.

PBOBABiLirr 97

the indigent^ that Punishment deters from crime,
A statement of provisos, when complete, converts
the very general into a universal proposition ; as, An
absolute sovereign will abuse his power, unless his
position depend on the good-will of his subjects, or
unless he have great rectitude and resolution, or un-
less he be guided by a minister having these quali-
ties. So also. Honesty is the best policy , provided it
squares with current opinions, promotes public in-
terest, and is displayed to view. The value of the
probability involved in such generalities cannot be
exactly, numerically estimated. It taxes the sagacity
of the experienced observer to judge their worth in
general statement, and in application to special or
particular cases. They abound in practical' affairs,
and are largely the guide of public and private con-
duct.'

It should be noted that induction by enumeration
very often arises from groups of instances, extends
to similar groups, and thus becomes more reliable,
attaining a higher degree of probability. Thus, if in
many observed groups containing A^s, most A''s are
^, then in all groups containing A^s, most A^s are J3,
If in various counties of Virginia most farms grow

^ The form of the argument isxlfx is, y is; but yis; .'. x (prob-
ably, presumably) i?. This is recognized as a fallacy when the rela-
tion is that of reason and consequent (§§ 91^ 119, I4S) ; but when, as
here, the condition is causal (§ 110\ it affords a probability, a pre-
sumption in favor of the conclusion. For the allowed plurality of
causes (§ 22), which investigation reduces, alone forbids the sine qua
non reading : Only ifxis,y is, which would yield .*. x is. See § 69.
7

98 ELEMENTS OF INDUCTIYE LOGIC

tobacco, then in all counties most farms grow it ; or,
simply, most farms in Virginia grow tobacco. The
inference from observed groups to similar unob-
served groups is more probable than inference to
individuals.

§ 53. An indefinite judgment of probability is
frequently expressed definitely, borrowing the lan-
guage of chance, in the form of a ratio; as. It is
ten to one that a drunkard cannot he reformed ;
and. Not more than one person in a hundred forms
independent opinions in politics or religion^ Such
statements are inaccurate, but, making an approach
toward a measure of probability, are significant of
degree. The statement that As liTcely as not he will
consent is an inference from some one's character to
his conduct as wholly uncertain. A turf-gambler
will bet two or more to one on his favorite racer,
according to his judgment of the ratio of probabil-
ities.

An accurate numerical expression of probability,
like that of chance, is practicable in many instances
both of natural phenomena and of human affairs,

J " What Hobbes says of Charles II. —

'Nam tunc adolescens
Credidit ille, quibus credidit ante Pater '—

is true of the vast majority of men even in the most enlightened
countries. Hence a strong probability that any given individual has
never exercised any independent judgment in politics or in religion.
A hundred to one is a safe estimate of such a probability." — Bain,
Logic, bk. iil, ch. xiv.

PROBABILITY 99

with the modification that the positive frequency of
the phenomenon can very rarely if ever be known a
priori (§ 49), but must be ascertained by observations
reduced to actual count. For example, all the met-
als are white, including shades of gray, except two,
copper and gold. As chance will not account for
this, we presume there is some modifying cause in
the atomic constitution of these exceptions which
determines the difference. Now, since there are
fifty known metals, the probability that hydrogen,
when liquefied, will be white is as 50 to 2. In gen-
eral, then, if we know the exact proportion of in-
stances in an approximate generalization, we can state
numerically the degree of probability of an inference
from it. If there be no exceptions to a well-ascer-
tained uniformity, the probability is at its maximum.
An actual count, extensive and exhaustive, thus
enables us to express probabilities with scientific
precision. Herein lies the inestimable value of sta-
tistics. Statistical estimates and investigations, with
a view to setting up an inductive universal, or at
least a general rule, successfully strive by what is
improperly called a wide induction of facts, prop-
erly a wide enumeration of cases, to approximate the
certainties of exact science. Our decennial census
makes a wide count of very many matters relative
to the lives, property, resources, and occupations of
the people. These are reduced by the Census Bu-
reau, averages struck, and ratios obtained, which,
through induction, justify inferences of great value,
especially to the immediate future.

100 ELEMENTS OF INDUCTIVE LOGIC

For illustration, let us suppose that in a given
county the average number of annual deaths in ten
years is found to be two per cent, of the population ;
then we may confidently infer that in the next dec-
ade a like per centum of mortality will prevail, pro-
vided the population, mode of living, etc., are not
materially changed. This inference is from one
temporal group to another. It would be equally
competent to infer the same per centum of mortal-
ity of an adjoining analogous county for either dec-
ade. We remark that the inference is indifferent
as to order of time, since it would be true likewise
of the previous decade, but that it is greatly weakened
if applied to a case differing considerably in time,
place, or circumstances. Also we remark that, while
this inference from group to group, temporal or
spatial, may reach the highest probability, it fur-
nishes no ground for inference respecting the life of
any individual member of a group.

Such statistics as to term of life, loss by fire, ship-
wreck, and the like, furnish a safe basis on which to
calculate the value of risks, and so justify the in-
vestment of large capital in the business of insur-
ance. For example, the American Tables of Mor-
tality show the results of wide and accurate statistical
observation. Among other averages they give the
expectancy — that is, the probability — of life for differ-
ent ages. A healthful man at 20 years of age has
an expectancy of 42 years more ; at 30, of 35 years ;
at 40, of 28 ; at 50, of 21 ; at 60, of 14 ; at 70, of 8.
The rates charged by a life-insurance office for a

PROBABILITY. ; ; ^''•;'J V,,' '\"t;o>V:

policy of $1000 increase as the expectancy decreases.
It is quite obvious, yet needing to be stated, that
the probabilities of life thus estimated are of no as-
surance to the individual person insured, but only to
the office insuring. The inference from the large
group statistically estimated as to mortality to the
large group the office has in hand holds good, those
who die short of expectancy being balanced on the
average by those who live beyond it, and by this
means the office knows in advance with high prob-
ability the amount from year to year of its disburse-
ments, and rates its charges to correspond.

YII.— DIFFERENCE

§ 54. In view of the foregoing discussion of in-
duction by enumeration it is plain that, were there
no surer canons, the prospect of attaining scientific
truth of unquestionable universality would be hope-
less. The radical defect of enumeration is that in
this preparation for induction there is only a sur-
mise that a determining cause exists, not an ascer-
tained knowledge of the actual determining cause.
Consequently, by conforming to its stated canons,
we reach only a tentative, somewhat probable, but
still, except in the rarer cases, hazardous, generality.
Induction grounded on enumeration is truly induc-
tion, but imperfect, always falling short of empirical
certainty (§ 45).

A knowledge of the cause or effect of a phenom-
enon is scientific knowledge, as stated in the ancient
aphorism : ScientA est rerum cognoscere causas.
Such knowledge is a sure foundation for induction,
and prerequisite to' perfect induction characterized
by empirical certainty. In undertaking now an ex-
amination of the several methods by which this pre-
liminary knowledge is sought, it will be well at the
outset, for the sake of clearness, to express formally
the governing principle of the induction which it

DIFFERENCE ^ 103

conditions. This principle is derived directly from
the prinaarj Laws of Causation, being, indeed,
merely a slight modification of the Axioms of Uni-
formity (§§ 19, 21). It may properly be termed the
General Canon of Perfect Induction, reading thus :
Canon: A cause and its effect being known,
from all like causes like effects are inferable,
and from all like effects like causes are in-
ferable.

Hence it is evident that, in logical order, before
the induction takes place, a preparatory problem is
to be solved : either, a particular cause being given,
to find its effect ; or, a particular effect being given,
to find its cause. When this is done, the induction,
expressed in a strictly universal proposition, is, ac-
cording to the canon, immediately inferred.

§ 55. The several methods of solving the prepara-
tory problem constitute one of the chief considera-
tions of inductive logic, and their application is the
chief difficulty in scientific investigation, the sub-
sequent inductive step itself being an immedi-
ate inference of the simplest character (§ 26).
They are quite commonly called " inductive meth-
ods," though not themselves inductive, but merely
preparations for induction, methods for ascertaining
causal relations between phenomena. To their ex-
position we are now about to proceed. It will be
found to consist in the proof, statement, and illus-
trative application of several Canons of Causation,
or canons of methods for the determination of causal

104 ELEMEI^ OF IimUCTIVE LOGIC

relations, canons which express merely the forms of
thought to which actual processes must conform.
These, like the canon of induction just stated, are
evolved a priori, are derived deductively from the
Laws of Causation. They should not be mistaken
for canons of induction, since they are strictly and
solely tfee formal processes by which a particular
fact of causation may be ascertained, formulating
only a sound and scientific preparation for subse-
quent inductive procedure.

The methods are primarily two — the Method of
Difference, and the Method of Agreement — each
having subordinates. Both accomplish their ends
by a partial elimination of circumstances, in order
to detect which particular circumstances are con-
cerned in the causation. In the Method of Differ-
ence, whatever circumstance cannot be absent with-
out the absence also of the phenomenon under
investigation, is causally connected with that phe-
nomenon ; in the Method of Agreement, whatever
circumstance can be absent without the absence also
of the phenomenon under investigation, is not causal-
ly connected with that phenomenon. These maxims
are obviously derived from the Axiom of Change
(§ 18), which furnishes the basis of the methods.*

^ The methods of scientific investigation now before us are all essen-
tially methods of elimination, and thus conform to Bacon's aphorism
that induction proceeds " by due rejections and conclusions," — Nov.
Org.^ i., 106, already quoted in § 40, note. This process Bacon con-
trasts with the method of " simple enumeration," and justly claims to
be the first to make it prominent ; but his " Prerogatives of Instances,"
id., bk. ii., hardly anticipate the present methods.

DIFFERENCE 105

§ 56. The most important, direct, and simple
method for determining the causal relation between
phenomena is the Method of Difference. It is

Newton's four "Kules for Philosophizing" (§ 21, note) are quite
different from these methods, and have special reference to his own
procedure in the " Principia."

Sir John Herschel, in his " Discourse on the Study of Natural Phi-
losophy," gives, in § 145, five " general rules for guiding and facili-
tating our search, among a great mass of assembled facts, for their
common cause." From the rules he deduces nine " propositions read-
ily applicable to particular cases." Four of these (2, T, 8, 9) are
the four methods, though lacking the prominence given them by Mr.
Mill as the sole and sufficient methods of logical proof. By Her-
schel the four propositions indicated, together with the others, are ex-
pounded as aids to discovery ; the notion that they constitute a system
of logical proof does not seem to have occurred to him. Of his ad-
mirable " Discourse " Mr. Mill says : " It is a work replete with happily
selected exemplifications of inductive processes from almost every
department of physical science, and in which alone, of all books which
I have met with, the four methods of induction are distinctly recog-
nized, though not so clearly characterized and defined, nor their cor-
relation so fully shown, as has appeared to me desirable." — Logic^
p. 297.

Science in all its branches is deeply indebted to Mr. Mill for the first
clear and distinct statement of its logical methods, and the importance
now universally attributed to them is mainly due to his influence. It
was the distinction of his " System of Logic " to draw a clear and
broad line between the Art of Discovery and the Science of Proof.
The latter is Logic. It is concerned mainly with methods of proving
propositions, and only in an incidental way does it aid in suggesting
them. He says : " The business of Inductive Logic is to provide rules
and models (such as the syllogism and its rules are for ratiocination)
to which, if the inductive arguments conform, those arguments are
conclusive, and not otherwise. This is what the four methods profess
to be, and what I believe they are universally considered to be by ex-
perimental philosophers, who had practised all of them long before any
one sought to reduce the practice to theory." — Logic^ p. 308.

The Canons of Causation, as we have designated them, of the pres-

106 ELEMENTS OF INDUCTIVE LOGIC

based, as just stated, on the Axiom of Change, from
which are deduced the following special maxims :

1st. When a consequent appears or disappears, and
with it an antecedent, the latter is the cause or a
part of the cause of the former.

2d. When an antecedent cannot be introduced or
excluded without adding or losing a consequent, the
latter is the effect of the former.

These deductions are comprised in the following
Canon of Difference : If an instance w'herein
a phenomenon occurs, and another wherein
it does not occur, have every circumstance
in common save one in the former, this is
wholly or partly the cause of the phenom-
enon, or its effect.'

A symbolical formula of this canon is as follows :

ABC B C

y z X X y

The larger letters represent particular causal an-
tecedents, or simply causes; the smaller letters, par-
ticular consequents or effects. Each of the larger
letters usually stands for a collocation of distinguish-
able but co-operating factors; each of the smaller,

ent treatise, are the " Four Methods of Experimental Inquiry " drawn
from Mill, " Logic," bk. iii., ch. viii. In transcribing them, we have
ventured to rearrange them and to make some verbal changes in the
interest of logical order, brevity, and precision.

* It should be noted that in this, and in the subsequent Canons, an
instance or case is an observed total analyzed into antecedents and
consequents (§ 35); some one or a group of these is the phenomenon
under investigation, and the rest are its circumstances.

DIFFERENCE 107

for a collective fact. The two groups represent two
instances or cases, one instance affirmative and one
negative of As. If ^ be the particular phenom-
enon under investigation, the fact that it disappears
in the second instance along with A proves that A,
either alone or together with some other antecedent,
is its cause. If A be the phenomenon under in-
vestigation, the fact that it cannot be absent, as in
the second instance, without the loss of s, proves
that 3 is its eifect.

Such is a formal statement in theoretical strictness
of the method of difference, a process of elimina-
tion. It should be observed that, although in its
practical applications only approximate results can
be abtained, yet it is the most rigorous proof of par-
ticular causes or effects that is possible, and when its
theoretic conditions are fairly fulfilled its results are
empirically certain, falling little short of strict dem-
onstration, and thereby furnish a safe premise for
induction (§ 26).

§ 57. Material examples in general accord with
this formal method lie on every hand. It is unwit-
tingly used daily and hourly even by the most
thoughtless and ignorant persons. We cite several
common-sense cases.

I see rain (s) falling, and a cloud {A) in the sky ;
the rain disappears, and with it the cloud ; I infer
this cloud to be the cause, at least in part, of that
rain.^

^ It may be well to recall just here our doctrine on the function and

108 ELEMENTS OF INDUCTIVE LOGIC

A sound {2) strikes mj ear (a?), and I see a swing-
ing {A) bell {B) ; the sound ceases, and with it the
swinging; I infer that the swinging was partly the
cause of the sound. There is here no induction ; but
I might inductively infer, All swinging hells always
produce sound.

If I find my dog shot through the heart, I know,
by the method of difference, it was this that killed
him; for he was alive just now, and all circum-
stances are the same except the wound. Again no
induction ; one might follow, but would be super-
fluous.

A scientific and more recondite example is as fol-
lows: When looking in a spectroscope at the spec-
trum of incandescent sodium {A) and calcium chlo-
rides, I see a very bright yellow line (s) ; just now,
when looking at the spectrum of incandescent cal-
cium chloride, this yellow line was absent ; I infer
that in the present case the incandescent sodium is
the cause of the bright yellow line. Then may fol-
low an induction of all such cases.

The foregoing are inferences from effect to cause.
An inference in the reverse order is : I observe a
shower of hail {A\ and, on going to my conservatory.

application of form, § 31. The canons now before us are merely
formal statements, without any material content. They do not serve
as premises from which material conclusions are inferred, but in their
application the provided abstract form is merely supplied or filled in
with given matter. Thus the forms instance^ phenomenon^ circum-
stance, are simply embodied, in the above example, by weather^ rain,
cUyud.

DIFFEKENCE 109

find the glass broken {£) ; 1 infer, all other circum-
stances being unchanged, that the breakage is the
effect of the hail. Also, just before the hail, I ob-
served a cold nor'wester set in, and infer that the
hail was its effect.

So a pilot, noting that during a thunder-storm the
needle was disturbed, and that during a storm with-
out lightning it was not disturbed, concludes the dis-
turbance to have been effected by the lightning.*

§ 58. The illustrations thus far given are cases of
simple observation, and to this we are limited when
an effect is given to find its cause (§ 35) ; but when
a cause is given to find its effect, we may have re-
course also to experimental observation. Simple
observation of nature often fails to discover, amid
her vast complications, the second case requisite to
fulfil exactly the conditions of proof by this method,
but w^hen we have an approximation indicating the
causal relation, or a suggestion of it from some other
quarter, we may, if the matter be subject to hand-
ling, apply the test of experiment (§ 36).

The conclusion of the pilot, stated above, may be

^ The method of difference is applicable also to inquiry concerning
preventive cause (§§ 15, 47). A patient has intermittent fever {z). If
in the interval he be brought under the influence of quinine {A\ the
fever does not reappear, the quinine acting as a preventive cause,
though we are puzzled to know how. Here £ C is followed by y z x,
and A B Chj y x; that is, in the presence of A, z disappears ; hence
A counteracts B C so far as to prevent the effect z. So, also, as the
old wives tell us, a silver spoon (A) in a common tumbler will prevent
its breaking (z) when it is filled with hot water. This, too, puzzles us.

110 ELEMENTS OF INDUCTIVE LOGIC

tested, inverting the order of proof, and verified
thus: Place a copper wire near and parallel to a
magnetic needle, the latter is not disturbed ; elec-
trify the wire, instantly the needle is disturbed, tak-
ing position at right-angles to the wire; therefore
this disturbance is effected by the electricity. Note
that this test verifies, not the particular conclusion
of the pilot, but an obscure induction from it, that
Electricity deflects the needle.

The previous spectroscopic instance may be tested
and verified, reversing its order of thought, by this
experiment: Produce the spectrum by an incandes-
cent platinum wire, the bright line does not appear;
having touched a pellet of sodium with the point of
the wire, produce the spectrum, instantly the bright
yellow line flashes across ; therefore it is the effect of
the incandescent sodium.

Again, wishing to ascertain which of the two chief
components of air supports breathing life, we put a
mouse in an open jar, and then fill the jar by dis-
placement with pure nitrogen ; the mouse soon dies ;
therefore nitrogen is azotic, and it is the oxygen of
the air that supports life.

Thus the method of difference is pre-eminently a
method of experiment, and the most potent means of
scientific investigation. To it the student of nature
always preferably resorts in cases where its application
is possible. Perhaps nine-tenths of the experimental
research in the chemical, physical, physiological, and
other scientific laboratories, as well as the testing of
ordinary matters, is by the method of difference.

DIFFERENCE 111

Let it be remarked that the foregoing examples of
the method are not inductions. So far as they in-
volve inference, it is deductive. The result in each
case is merely that a certain particular fact is the
cause, or the effect, of a certain other particular fact.
The method of difference only prepares this ground
for the induction of a universal according to the
general canon of induction (§ 54). An inductive
inference is then competent, and is so simple and
direct that thought almost instinctively makes it,
indeed running constantly before the proof of the
particular with an anticipating generalization. It
requires, therefore, some attentive discrimination,
rarely exercised on this point even by logicians, to
avoid confusing the preparatory process with the
logically subsequent induction.

§ 59. Recurring to the first example in § 57, we
inductively infer. Every like cloud always causes
rain. Here rain^ a generalized effect, is attributed
to cloud as its generalized cause. The statement is
in the form of a causal, categorical, universal propo-
sition. Letting A stand for any generalized cause,
and ;§; for its generalized effect, we have : —

If A is, then ^ is ; and

If » is, then A is. — Canon, § 54.

These may be combined in the compound form : —

Only if A is, then % is.

This implies, not merely that if either is, the other
is, but also that if either is not, the other is not.

112 ELEMENTS OF INDUCTIVE LOGIC

Hence affirming either affirms the other, and deny-
ing either denies the other. Such is the character
of the causal conditional, causa essendi^ as distin-
guished from the logical conditional, causa cogno-
scendi (§ 110). Formally and theoretically it is rig-
idly conditio sine qua non.

Now suppose that, having obtained inductively a
universal, some new particular phenomenon of like
sort is observed, then it may be subsumed, and an
unobserved fact deduced, as follows : —

Only if A is, then z is ;

But A is ;

.*. z is

But z is ;

.♦. A is. PONENS.

E. g. When I see just such a cloud in the distance,
I conclude it is raining over there ; or, when at night
I hear the rain on my roof, I conclude there is a
rain-cloud above. Again : —

Only if A is, then Z is ;

But A is not ;
/. z is not.

But z is not ;

.-. A is not. — ToLLENS.

E. g. If there be no such cloud, there is no rain ;
or if there be no rain, there is no such cloud. Other
forms of the so-called conditional syllogism may be
used in these deductions (§ 119 sq.).

§ 60. A modification of the foregoing method of
difference is the Method of Kesidue. After the
principal causes of a complex phenomenon have
been severally ascertained, there often remains a
portion unaccounted for. Sometimes this is so slight

DIFFERENCE 113

as to be overlooked, or else supposed to be due to
errors of observation. But alert scientists have
learned to scrutinize with profit what others neglect.
Indeed, some very important discoveries have re-
sulted from the study of an apparently trifling resi-
due. Separating it from the cognate effects, inquiry
is made for a corresponding surplus in the antece-
dents, which has either been disregarded, or is as yet
unknown, and this, when found, is rightly posited as
the cause of the residuum. The formal process in
such case is expressed succinctly in the following
Canon or Eesidue : Subduct from a complex
instance the consequents of ascertained an-
tecedents, and the residue is the effect of the
remaining antecedents.

The method may be formulated as follows:

ABC

But ^

and C

.'. B C

y z X

X

y

X y

Here the complex instance has yielded to investi-
gation that X is caused by B, and y by C. On sub-
ducting X and y from the total consequents, a resid-
ual phenomenon, s, perhaps quite inconspicuous, is
discovered. This, then, is the effect of A, the re-
maining antecedents.

Note that the two instances, one aflSrraative and
one negative of A ^, characteristic of the method of
difference, appear in the formula. This negative in-
stance, however, is not obtained by direct observa-
tion, but is deduced from the effects which B and G
produce separately. Still the method is as cogent as

114 ELEMENTS OF INDUCTIVE LOGIC

the method of difference itself, provided the prem-
ises B X and C y oi its specific deduction are ob-
tained by that method, and that A is the only agent
to which z can be referred. Otherwise further proof
is requisite.

§ 61. For example : Arfwedson, in 1818, on ana-
lyzing a portion of a certain mineral {A B C\ whose
total weight (y z x) he ascertained, found the weight
{x) of the contained magnesia (B), and the weight {y)
of other components (C). Subducting these weights
{x and y) from the total, a residue (s) was observed.
Searching the mineral for its cause, he discovered a
substance (A), previously unknown, and named it
lithia. In like manner were discovered iodine,
bromine, selenium, and several new metals accom-
panying platinum.

The discrepancy between the observed and calcu-
lated times of eclipses of Jupiter's satellites was a
residue accounted for by the difference of times req-
uisite for the passage of light, previously supposed
to be instantaneous, over his greater and less dis-
tances from us, and on this basis Roemer calculated
its velocity.

The perturbation of the planets was a residue
which led astronomers to extend the law of gravita-
tion from the central body, to which alone it was
at first supposed to be applicable, inductively to all
bodies in the universe.

The geologists who posit early cataclysmic causes
allege in support of their view that, after the effect

DIFFERENCE 115

of all ordinary causes has been allowed for, there is a
large residue of facts proving the existence in geo-
logic eras either of other forces, or of like forces
greatly intensified.

Whoever claims that there is a fundamental differ-
ence in the intellectual capacities of the sexes should
show that, after subtracting from the known differ-
ences all that can be attributed to differences of
physical organization and to the influence of envi-
ronment, there is a residue which can be attributed
only to an ulterior distinction.

YIII.— AGEEEMENT

§ 62. It has already been said that the ordinary
course of nature or of affairs rarely presents cases
fulfilling the requirements of the method of differ-
ence. Moreover, it often happens that these require-
ments cannot be fulfilled by experimental contriv-
ance with sufficiently rigorous accuracy. In such
cases an alternative mode of discovering the cause
of a given effect, or the effect of a given cause, is
afforded by the Method of Agreement. This method
follows the maxim that whatever circumstance can
be absent from a case without the absence also of the
phenomenon under investigation, is not causally con-
nected with that phenomenon (§ 55). It is based on
the Axiom of Change, from which are deduced the
following special maxims :

1st. When a consequent disappears without the
disappearance of a given antecedent, the latter is not
the sole cause of the former.

2d. When an antecedent disappears without the
disappearance of a given consequent, the latter is not
the effect of the former.

3d. The antecedent and consequent, which togeth-
er are constant during the successive disappearance
of each of the others,, are related as cause and effect.

AGREEMENT 117

These deductions are comprised in the following
Canon of Agreement : If instances wherein a
phenomenon occurs have only one circum-
stance in common, this is its cause, or its
effect.

A symbolical formula of this canon is as follows :

ABC A :^ D - ACE

y z^x s^ y z x z v

If z be the particular phenomenon under investiga-
tion, the fact that the three instances containing it
have only one circumstance in common, the ante-
cedent A^ is evidence that A is the cause of z.
Conversely, if A be under investigation, the com-
mon consequent z is its effect.

In the application of this method the instances are
studiously varied so as to eliminate in turn the sev-
eral chance or immaterial circumstances attending
the phenomenon (§ 51). We must follow the Ba-
conian rule of "varying the circumstances"; for a
repetition of strictly similar cases, however numer-
ous, proves nothing, there being no elimination.
Only dissimilar cases eliminate, and so afford proof ;
hence these should be multiplied as far as needful.*

^ The enormous extent to which experiments are sometimes carried
in order to establish causal connection finds illustration in physiolog-
ical investigation by vivisection. M. Paul Bert describes a series of
experiments extending to No. 286. Flourens states that Mcagendie
used 4000 dogs in an effort to prove Sir Charles Bell's theory of the
motor and sensor functions of the nerves, and, having failed, used 4000
more to disprove it ; but that he himself had proved Bell to be right
by the vivisection of 1000 more.

118 ELEMENTS OF INDUCTIVE LOGIC

The questionable possibilities will thus be gradually
reduced in number, and, if the means of elimination
be complete, the inquiry terminates in fixing upon
some one circumstance that has never been absent
when the phenomenon is present.

§ 63. IlTewton observed bright prismatic colors {z)
displayed in white light on a film {A) of a liquid
soap-bubble; like colors in white light on a film of
solid mica; like colors in white light on a film of
air between glass plates. The only common circum-
stance appearing to be white light on a film, he pos-
ited this as the cause of the prismatic colors, now
more fully explained by the interference of light.

Conversely, cause being given to find its effect, if
in several instances an alkali and oil {A) unite, e. g.
potash and tallow, soda and suet, lime and olive oil,
a common circumstance is soap {z) ; this, then, is the
effect of the common antecedent.

From each of these particular determinations an
induction is now competent, thus : Any transparent
film in white light exhihits jprismatio colors; and
Any alkali and oil uniting produce a soap.

Some other examples will be helpful. We observe
in many cases the conversion of solids into liquids,
and these into gases. The bodies so converted have
a great variety of properties. One circumstance
common to the cases is the increase of heat. The
elimination of other circumstances being complete,
this antecedent is rightly assigned as the cause of the
change.

AGREEMENT 119

Brewster proved that the iridescence of nacre is
not due to the nature of the substance^, but of the
surface. Taking an impression of it in wax, he found
on this different substance a like iridescence. It is
now a familiar fact that the surface of glass or metal,
when finely grooved, becomes iridescent.

If a certain occupation or mode of living is found
to be usually attended by a particular disease, it is
reasonably suspected to be the cause of the disease ;
and the exceptional cases wherein the disease does not
occur are suspected to involve a preventive cause
(§§15,47, 56 n.).

Whenever I eat a particular kind of fruit, what-
ever else I may eat or drink, however various my
general state of health, the temperature of the air,
the season, the climate, and divers other surround-
ings, I am taken ill, and rightly consider the eaten
fruit the very probable cause.

A certain plant grows luxuriantly on a certain soil.
If wide observation eliminates very generally the
other circumstances, it is correct to conclude that
soil to be the cause of the remarkable luxuriance of
that plant.

If trade languishes or flourishes under a high tar-
iff, and if it be ascertained that those countries where
the one effect is observed agree throughout in no
other material respect except the tariff, or if this is
observed of different decades in the same country,
the high tariff may be posited as the cause.

Thus it is by the method of agreement primarily
and chiefly that we discern the cause of disease, of

120 ELEMENTS OF INDUCTIVE LOGIC

political revolution, of national characteristics, of
inodificatioos in animal and vegetable physiology, of
the order of geological strata, of changes in language ;
likewise, the effect of storm, of sunshine, and of
snow, of good and bad legislation, of this or that
method of teaching, of one's habits of life, of aesthetic
culture on morals, etc. In short, there is hardly any
department of knowledge wherein the method is not
in constant use.

§ 64. Some general remarks will now be appropri-
ate. The determination of natural kinds, and in
general of phenomena of ultimate coexistence, is by
virtue of similarit}^ or agreement (§ 33). The meth-
ods of induction by enumeration are also founded on
agreement of cases or of marks (§ 37). But the
methods now under consideration are not methods
of induction, but of inquiry into particular cases
of causation. Also they do not apply to ultimately
coexisting phenomena, but only to phenomena of
succession, and in these only to cases of causal succes-
sion.

It is not always easy to determine whether or not
successive phenomena are causally connected. Mere
succession in time is insufficient (§ 14:). The trans-
ference of energy is perhaps the ultimate test, but it
is rarely applicable (§ 17). We must rely mainly on
similar experiences to help us at the outset in distin-
guishing cases of causation, in separating the causal
antecedents from the causal consequents, and in as-
certaining the several components of each (§ 35).

AGREEMENT 121

In studying a case we disregard the immaterial or
chance circumstances. Most instances agree in a
number of these. The objects are subject to gravity,
immersed in air, exposed to light, etc. Unless these
can be supposed to affect the case, they are not taken
into account. But no circumstance should be hastily
rejected. Light was hardly esteemed an agent until
it was detected blackening salts of silver (§ 35) ; now
it is recognized as widely effective in chemical
changes and vital processes.

In the distribution of the antecedents and conse-
quents, as well as in their subdivision, care is requi-
site. Disturbances of the magnetic needle are coin-
cident, more often than chance will account for, with
changes on the disk of the sun, and with auroral dis-
plays. Hence one of these has been mistaken for
the cause of the others, when in fact they are all
properly parts of the effect of some widely prevail-
ing common cause.

Among the unquestionable antecedents occur real
conditions, which should be distinguished from the
causal conditions (§ 110). Thus joints are a condi-
tion, not a cause, of walking. So also molecular mo-
bility is not a cause, but a condition, of crystalliza-
tion. Again, there are certain doubly refracting (z)
substances, Iceland spar being one, having a great
variety of color, weight, hardness, form, and compo-
sition, which qualities, then, are immaterial circum-
stances ; but solidity, transparency, and in general a
crystalline structure, are invariable and essential an-
tecedents, yet not causal conditions {A\ but simply

122 ELEMENTS OF INDUCTIVE LOGIC

real conditions. Su,cli substances exhibit periodical
colors on exposure to polarized light, which is a spe-
cial real, not causal, condition of the periodical colors.
The discrimination and elimination of the real condi-
tions are requisite to avoid misleading confusion.

§ 65. E-eturning to the specific consideration of
the method of agreement, we note that after all the
foregoing general precautions have been observed,
still it is seldom that we have a series of cases either
so simple or so complete as the theoretical formula
indicates. Usually there is a complex tale of many
antecedents and consequents, and it is hard to get
the variety of instances requisite to eliminate all save
one of the important circumstances attending the
phenomenon in question.

Another imperfection in the practical application
of this method is called its characteristic imperfec-
tion, since it is not attributable to the other meth-
ods. An effect given to find its cause is often
due to an apparently possible plurality of causes
(§ 22). Recurring to the formula (§ 62), it appears,
unless the analysis has been thorough, that z may
have been in the first instance the effect of B^ or of
(7, in the second of Z>, in the third of E. Suppose
two distinct drugs, each curative of a certain disease,
and each mixed with an inert drug; applying the
method of agreement we might unguardedly infer
the cure to be the effect of the latter.

This difficulty is wholly due to imperfect analysis
of facts and factors, and not to any inherent imper-

AGREEMENT 123

fection in the theory. But the best analysis even of
the simpler cases is always so far short of perfection
that we must admit in practice the maxim of plural-
ity of causes and regard the colligation (§ 9) of re-
sults as uncertain.

A multiplication of various instances increases the
presumption that A is the cause of ^. The error of
ascribing the cure to the inert drug would hardly
survive even a few cases. Adverting to the first ex-
ample (§ 63), the possibility that the prismatic colors
are the effect in the first instance of the dissolved
soap, in the second of the alumina in the mica, in
the third of the nitrogen of the air, would soon dis-
appear under additional instances, provided the ob-
servations are made amid various circumstances, and
the colligated conclusion, that in each instance the
colors are the effect of white light on a transparent
film constantly present, would soon become a very
strong probability, the uncertainty arising from a
possible plurality of causes being thereby practically
eliminated.

It should be noted that the maxim of a plurality
of effects (§ 20) is likewise to be recognized, and the
uncertainty thence arising to be similarly reduced by
multiplied eliminations. Thus, heat (J.) boils water
(a?), melts metal (^), stimulates growth (5), etc. Elim-
ination of the differences in these effects discovers a
common fact (z) in a specific molecular change.

§ ^^. It is now sufficiently manifest that' in prac-
tice a causal connection between a phenomenon and

124 ELEMENTS OF INDUCTIVE LOGIC

a circumstance cannot be rigidly proved by the
method of agreement. A very high degree of prob-
ability may sometimes be attained, and with this,
when other methods are inapplicable to the case, we
have to be content. In most cases the probability is
of lower degree, varying in value with the multiplic-
ity of differing instances. A rule for estimating this
value is as follows : " Given an effect to be accounted
for, and there being several causes that might have
produced it, but of whose presence in the particular
case nothing is known ; the probability that the ef-
fect was produced by any of these causes is as the
antecedent probability of the cause, multiplied by
the probability that the cause, if it existed, would
have produced the given effect." ^

It is also obvious that the method of agreement is
a method of simple observation rather than of ex-
periment. When the effect of a given cause is
sought, experimental tests are often applicable with
advantage (§ 36). When the cause of a given effect
is sought, simple observation may give rise only to a
suspicion or surmise of the cause ; then, reversing
the order, the suspected cause may often be tried to
see whether z will come of it, which is experimental
observation again. But perhaps yet more often the

' This rule is given by Laplace as the " Sixth Principle," in his
" Essai Philosophique sur les ProbabilitSs," and is described by him
as the " fundamental principle of that branch of the Analysis of Proba-
bilities which consists in ascending from events to their causes." An
excellent exposition, which we have not space to quote, will be found
in Mill, Logic, p. 385 sq., reproduced by Bain, Logic^ bk. iiL, ch. ix., § 13.

AGREEMENT 125

matter is out of reach of handling, and tlien we are
limited to simple observation in both orders of in-
quiry.

Yery generally investigation begins with simple
observation by the method of agreement. Recourse
is had to experiment if practicable, and the intelli-
gent inquirer will never lose an opportunity of re-
sorting to the more cogent method of difference.
Perhaps the chief value of the method of agreement
in scientific pursuit is that it suggests lines of exper-
iment, and the application of other methods yielding
empirical certainty. In itself it is tentative rather
than probative, resulting merely in a greater or less
probability that in the observed cases A is the cause
of 2. Formulae of the induction may be stated thus :

If A is, then probably ^ is ; and
If i^ is, then probably A is.

§ 67. An important modification of the foregoing
method is the Method of Double Agreement. It
consists in applying agreement, first to a series of
cases wherein a certain circumstance attends a phe-
nomenon, and then to a series within the same gen-
eral sphere of circumstances, as nearly similar to the
other as possible, except that the phenomenon in
question and the attendant circumstance are absent.
A comparison of the positive with the negative se-
ries greatly strengthens the inference that the phe-
nomenon and the circumstance are causally con-
nected. There is first an agreement in presence, and
then an agreement in absence, which double agree-

126 ELEMENTS OF INDUCTIVE LOGIC

ment conjoined makes an approach to the conchi-
siveness of the method of difference.

The argument is: Since the positive cases agree
with each other in nothing throughout except in the
presence of the given phenomenon and a circum-
stance, then bj the single method of agreement it is
probable that these are causally connected. More-
over, since the negative cases agree with each oth-
er in nothing throughout except in the absence of
the given phenomenon and that circumstance, this,
considered apart, likewise renders their connection
probable. Therefore, a fortiori^ the two inferences
being conjoined, the connection is still more prob-
able.

The method is stated succinctly in the following
Canon of Double Agreement : If instances
wherein a phenomenon occurs have only
one circumstance in common, and others in
which it does not occur have nothing in
common save the absence of the circum-
stance, this wholly or partly is the cause of
the phenomenon, or its effect.

A symbolical formula of this canon is as follows :

ABC A B D A C F

a h c a h d a c f

B F CD D F

h f c d d f

Let it be observed that no negative instance differs
from any positive instance merely in the absence of
-4, a. If one did, it would satisfy the requisites of

AGREEMENT 127

the simpler, more cogent, and therefore preferable
method of difference, and this would supersede the
other.

§ 68. In correspondence with the formula, sup-
pose a south wind, Auster, from over a marsh, to be
attended by ague in three several instances. In the
first, the weather is Bleak and blighting, also Cloudy
and cold, but not Damp or Foul. In the second, it
is Bleak and blighting. Damp and dewy, but not
Cloudy or Foul. In the third, it is Cloudy and cold,
Foul and foggy, but not Bleak or Damp. The meth-
od of agreement concludes from these cases agree-
ing in presence that probably the ague was in each
caused by Auster charged with malaria from the
marsh.

Again, suppose in the same locality another trio of
winds not Austral and not attended by ague, but
each of the other circumstances appearing in turn in
one or two instances, yet no one in all three. The
method of agreement infers negatively from these
negative cases agreeing in absence that these winds
not Austral did not cause ague.

Now this negative inference greatly strengthens
the prior conclusion that in those cases the ague was
caused by the malarial Auster. For, imagine a se-
ries of negative cases exhaustive of the important
circumstances associated in any instance with A, a.
This series alone would furnish full proof of their
causal connection, as follows : Generalizing from a
colligation of the negative cases, we have —

128 ELEMENTS OF INDUCTIVE LOGIC

If A is not, then a is not ; and v. v. ;
But in a certain case a is ;
.*. In that particular case ^ is ; or v. v. — Tollens.

Practically, however, we can never obtain an ex-
haustive negative series, hence the conclusion is only
probable. But this probability, corroborated by that
arising from the affirmative series, yields a conclu-
sion a fortiori.

In another important respect the prior conclusion
is strengthened still more by the negative series. It
excludes the supposition of a plurality of causes.
For, since the negative series comprises, theoretically
at least, all the antecedents of the affirmative series
except A, without the occurrence of a among its con-
sequents, it follows that none of those antecedents is
a cause of a. Thus the characteristic imperfection
of the method of agreement does not invalidate this
modified method, which therefore is the more cb-
gent, and approaches, though it never reaches, the
demonstrative force of the method of difference.

§ 69. A standard illustration of the method of
double agreement is the research of Wells into the
cause of dew. " It appears that the instances in
which much dew is deposited, which are very vari-
ous, agree in this, and, so far as we are able to ob-
serve, in this only, that they either radiate heat rap-
idly or conduct it slowly ; qualities between which
there is no other circumstance of agreement than
that, by virtue of either, the body tends to lose heat
from the surface more rapidly than it can be re-

AGREEMENT 129

stored from within. The instances, on the contrary,
in which no dew, or but a small quantity of it, is
formed, and which are also extremely various, agree
(as far as we can observe) in nothing except in not
having this same property. We seem, therefore, to
have detected the characteristic difference between
the substances on which dew is produced and those
on which it is not produced. And thus have been
realized the requisitions of what we have termed the
Indirect Method of Difference." This, however, is
not the whole of the research. By the application
of other methods, proof is accumulated, and the the-
ory fully established.*

^ See § 11, last paragraph but one. Mr. Fowler, in his Inductive
Logic, p. 134, note, says : " Dr. "Wells's " Memoir on the Theory of Dew"
is very brief, and deserves to be carefully read by every student of
scientific method. Sir John Herschel, in his " Discourse," etc., § 168,
speaks of the speculation as * one of the most beautiful specimens of
inductive experimental inquiry, lying within a moderate compass,' that
is known to him. Cf. id., p. 155. Our quotation is from Mill, Logic^
p. 299, which is borrowed from Herschel, as above.

IX.— COl^COMITAITCE

§ 70. The Method of Concomitant Yariations,
which may be construed as a modification either of
the method of difference or of the method of agree-
ment, remains to be 'considered.^ There is a large
and important class of cases from which it is imprac-
ticable to eliminate entirely an agent and its conse-
quent. To these cases, therefore, neither of the fore-
going methods, without modification, is applicable.
For instance, the oscillations of a pendulum near a
mountain are .disturbed ; we take it far away, and

* The method has regard to concomitant changes in the degree of a
given phenomenon and a circumstance. Observation having noted a
gain, or a loss, of quantity in an antecedent and consequent, this gain
or loss itself may be taken as a phenomenon and circumstance in
which alone this instance differs from another ; thus fulfilling the con-
ditions of the method of difference. For example, two observations of
a thermometer may discover no difference except a gain of height along
with a gain of heat. Or, a series of observations, noting a gain or a
loss in each of several instances, may be compared as to this point, in
which alone they agree ; thus fulfilling the conditions of the method
of agreement. For example, observations on mercury, iron, water,
and marble, at ordinary temperatures, may agree alone in a loss of
bulk along with a loss of heat. The methods, therefore, are primarily
two (§ 55).

It will be better, however, to disregard this reduction, and treat the
method of concomitant variations as an independent original method.

CONCOMITANCE 131

the disturbance ceases ; this proves, by the method
of difference, that the mountain was the cause of the
disturbance. But we cannot take it away from the
earth, and by tlie same method ascertain the cause of
the oscillations. Nor can we apply the method of
agreement; for, though the earth, a permanent cause,
is always present, so also is the sun, which, by this
method alone, might with equal reason be posited as
the agent. It is evident that some other method
of discovering causal relations is needed. Now, a
pendulum oscillates about a vertical through its
point of suspension, a vertical whose direction in
space varies concomitantly with the earth's motion;
therefore the oscillations of the pendulum about the
varying vertical, and the moving earth, are causally
related.

In general, it follows from the axiom of change
(§ 18), that any modified cause, which, indeed, is a dif-
ferent cause, is followed by a modified effect ; and
any modification of an effect is due to some modi-
fication of its cause. Hence, limiting the view to
progressive changes attending each other, we have the
Canon or Concomitant Variations : If a phenom-
enon varies in any manner whenever a cir-
cumstance varies in some particular man-
ner, they are causally connected.

Only the general fact of a causal connection can
be determined by this method alone. Whether the
phenomenon is specifically the cause or the effect of
its circumstance, or whether they both are not rather
the joint effect {x oc x') of some common cause, must

132 . ELEMENTS OF INDUCTIVE LOGIC

be ascertained by trying whether we can produce one
set of variations, or find one produced, by means of
the other. If so, the relation is that of cause and ef-
fect, and may be symbolically formulated thus :

ABC

X z y

Here B with z, and C with y, remain constant, while A
varies with x, ^

§ 71. It is impracticable to deprive a body, a bar
of iron for instance, entirely of its heat. "We can-
not, therefore, so vary the circumstances as to com-
ply with this requisite of the preceding methods, and
thus discover what effect is due to the heat. But
we can observe a rise of temperature in the bar, and
note that the only concurring modification is an in-
crease of bulk, especially of its length. We con-
clude, by the method of concomitant variations, that
its heat and its length are causally connected.

We find, upon trial, that by adding or withdraw-
ing heat we can increase or diminish its length.
Hence these are not the joint effect (a? oc x') of some
common cause, but are related as cause and effect
{A oc x).

Which is cause of the other? When we increase
the heat, the length increases ; but when we increase
the length by simple traction, the heat does not in-
crease accordingly. When we increase the bulk of
some bodies, as air, the temperature, on the contrary,

CONCOMITANCE 133

falls. Therefore the varying heat is the cause of the
varying length.

This relation being thus particularly ascertained,
we are authorized by the principle of uniformity
(§ 19) to infer immediately and inductively the gen-
eral law that heat expands iron, or metals, or bodies.

For further illustration : Sitting in my study, I
find myself growing too warm, and observe the ther-
mometer on my table rising. Hence these concomi-
tantly varying phenomena are causally related. But
how ? The present method, alone applied, does not
determine. I suspect, however, from previous expe-
riences, that they are the joint effect of a common
cause. On closing the hot-air register the observed
variations cease, proving my surmise to be correct.

§ 72. We cite some examples of direct concomi-
tance : On the earth there is no instance of motion
persisting indefinitely, and hence the ancients held,
by induction from enumeration, that all bodies nat-
urally tend to a state of rest. In proportion, how-
ever, as the known obstructions to motion, such as
friction, resistance of the air, etc., are abated, the
motion is less and less retarded ; as in Borda's ex-
periment with the pendulum in a vacuum, the fric-
tion at the point of suspension being minimized,
the swing continued more than thirty hours. Now,
comparing a whole series of cases, from speedy loss
of motion to prolonged continuance, we observe that
there is a strict concomitance between the degree of
obstruction and the retardation. Therefore, it is in-

134 ELEMENTS OF INDUCTIVE LOGIC

ferred, if obstruction were wholly removed, the mo-
tion would be uniform and perpetual. This proof
is given by ]N"ewton in support of his induction of
the first law of motion (§ 18 n.).

Again, we find that all the variations in the posi-
tion of the moon are attended by corresponding tidal
variations, which is the first step of the process con-
cluding the moon to be the cause determining the
tides.

The science of Geology abounds in illustrations.
Since the agents with which it is concerned, land
and water, subsidence and elevation, denudation and
deposition, are constantly present and acting on the
earth's surface, it being therefore impossible to elim-
inate entirely the influence of any one, the geologist,
in preparing for an induction explanatory of events
long past, is limited very closely to this method.

Also the psycho-physiologist, in seeking to -Q.X the
relations between mental powers and cerebral devel-
opment, also between sensations and neural excitants,
since they are inseparable from mind and body at
large, has small resource at the outset beyond their
concomitant variations.

We cite, also, some examples of inverse concomi-
tance: The apparent size of an object diminishes as
the square of its distance increases. Gravity, which
varies directly as the mass, varies inversely as the
distance squared.

The tendency to chemical action between two sub-
stances increases as their cohesion diminishes, being
much greater between liquids than between solids.

CONCOMITANCE 135

Mariotte's law, the volume of a gas is in inverse
ratio to the pressure, is an induction from observed
and measured concomitant variations.

The greater the elevation of the land, the lower
the temperature of the climate, and the more scanty
the vegetation.

The statistics of crime reveal its general causes.
When we find crimes diminishing according as hab-
its of sobriety and industry have increased, according
to the multiplication of the means of detection and
the more rigorous infliction of penalties, we may
presume their causal connection with circumstances
that do not admit the method of difference.

§ Y3. An important feature of the method still
remains to be considered. It will be suitably pref-
aced by a few general remarks.

The profound and thorough-going distinction be-
tween quality and quantity has been emphatically
noted (§§ 23, 2^, W5 sq), A change in a thing that
leaves it the same thing which it was — that is, one
which does not alter its essence, and so does not
amount to a change of kind — is merely an accident,
often a change in some respect of degree, of quan-
tity. Sciences are at first merely qualitative, classi-
fying their objects, and treating of their several
kinds, but they seek to become also quantitative by
measurement of degrees. When they have passed
into this latter stage they are more highly esteemed,
for then the principles of pure mathematics, the ab-
stract science of quantity, can be applied to their

136 ELEMENTS OF INDUCTIVE LOGIC

concrete facts, and the knowledge becomes more
complete and exact. Astronomy is an illustrious
example of a science founded on observation and a
few broad inductions, and then developed to extraor-
dinary dimensions, and attaining many new and valu-
able results by the application of mathematics.

The several methods of discovering the cause or
the effect of a given phenomenon, which we have
discussed, afford opportunities for passing to a meas-
urement of its quantity which the scientific inves-
tigator is eager to use. The qualitative analysis of
the chemical laboratory, proceeding mostly and w^hen-
ever possible by the method of difference, would be
comparatively poor in results were it not followed
by quantitative analysis. Indeed, alchemy became
chemistry just when the balance was introduced for
quantitative estimates. In the earlier part of this
century most of the phenomena of electricity and
magnetism were known and classified merely as
facts; now they can for the most part be measured
and calculated. The attempt is making to subject
even mental phenomena to measurement, and by the
determination of their relative quantities to raise
psychology to the rank of an exact science. The
effort to bring logic under the dominion of mathe-
matics has been noticed (§ 7^). The result is a
purely artificial structure, as truly so as the calculus
of a fourth dimension, or the geometry of curved
space — ingenious and curious, but without any cor-
responding reality. Such speculation is practically
useless and misleading, and is mentioned here merely

CONCOMITANCE 137

to indicate the strong tendency of scientists to apply
measurement and mathematical form to all branches
of knowledge.

§ 74. We have examined applications of the meth-
od of concomitant variations to cases that cannot be
resolved by the other methods. But it has very
important applications in connection with these.
Especially is it of inestimable importance in deter-
mining comparative quantities. After a causal rela-
tion has been ascertained by other methods, this one
is often applied in determining the ratio of the cause
and effect. Eecurring to a previous example (§61),
when by the method of residue it was definitely as-
certained that the passage of light requires time,
then the variations of the time concomitant with
those of the distance furnished Roemer with data for
calculating its velocit3\

But apart from other methods, this one often
leads to an important measure of quantity. The
velocity of a body falling freely varies concomitantly
with the distance fallen. This is an easy observa-
tion. The exact ratio of the increase of distance
and the increase of velocity is not so readily ascer-
tained, but Atwood's machine determines it to be as

1, 2, 3 to 1, 3, 5 . It also determines the

absolute quantity of fall from rest in the first sec-
ond to be 16.08 feet. From these data can be cal-
culated its fall during any subsequent second, and
its acquired velocity at any point of its fall.

The respective action of the sun and moon in pro-

138 ELEMENTS OF INDUCTIVE LOGIC

ducing the tides may be estimated quantitatively
from the varying positions of those two bodies.

Tiiese examples are sufficient to indicate the im-
portant part the method of concomitant variations
plays in the progress of a science, especially in facil-
itating its passage into an advanced stage, and its
further development under the sway of mathematics.

§ 75. In making a quantitative induction from
measured variations — that is, in applying mathemat-
ical results deduced from observed cases to cases be-
yond experience — provision is to be had on at least
three points.

First, we should know the absolute quantities of
both A and a?, as well as their relative variation.
For, if we cannot fix the total quantity of each, we
cannot fix a thorough-going ratio. Not only must^
and X, or x and x\ vary concomitantly — they must
also vanish together. Because heat expands a body,
we cannot infer that the distance between its par-
ticles is due wholly to heat, so that, if all heat were
withdrawn, they would be in contact ; for we do not
know the amount of heat in a body,^ or the dis-
tance between its particles, and hence cannot know
whether the two would vanish simultaneously. But
in the case of a falling body, cited above, we have
the absolute zero both of the distance fallen, the
starting-point, and of the velocity, the state of rest

^ The thermal zero has not been observed, but by calculation has
been fixed at -273° C, or -459° F.

CONCOMITANCE 139

from which it falls, and are consequently justified in
fixing their ratio.

Second, in general we cannot be sure that beyond
the limit of observation there may not develop some
modifying agent, latent in the observed circum-
stances, which will falsify our induction. The induc-
tion that heat expands bodies (§ 71) is subject, even
in this inexact form of statement, to a number of ex-
ceptions. Yet more emerge when the degree of ex-
pansion and contraction is measured and inductive-
ly posited. Indeed, the contrary sometimes occurs.
Water at ordinary temperatures expands as it warms,
and contracts as it cools, but when cooled below 39°
it begins and continues to expand until it becomes
ice at 32°, which is supposed by Grove to be due to
the setting in of crystallization. .

Third, when the observed variations are within
narrow limits, a very small error in the estimate
may, beyond those limits, enlarge in geometrical
ratio. This occasion for uncertainty, unlike the pre-
ceding, is peculiar to the method of concomitant vari-
ations. It is very hazardous, for example, to extend
an ascertained ratio of expansion and temperature—
that is, the numerical coeflScient of expansion — far
beyond the limits of observation. By being thus
extended the early formulas for the elasticity of
steam have led to disaster.^ So we can be sure of

^ " The formulge," says Sir John Herschel, " Discourse," etc., § 18*7,
" which have been empirically deduced for the elasticity of steam (till
very recently), and those for the resistance of fluids, and other similar
subjects, have almost invariably failed to support the theoretical

140 ELEMENTS OF INDUCTIVE LOGIC

our induction only when it does not greatly exceed
the extreme limits that have been subjected to obser-
vation and measurement.

structures which have been erected on them." Mr. Mill adds:
". . . when relied on beyond the limits of the observations from which
they were deduced." — Logic^ p. 291.

X.— DEDUCTION

§ 76. In the syllogism a general proposition is
premised, from which is inferred a conclusion of
equal or less generality, or a particular individual
fact (§ 3).

The general proposition may be an intuitive pri-
mary axiom, or an inference from axioms. In either
of these cases the process is wholly deductive and
strictly demonstrative or apodictic, as in pure math-
ematics, and in the logic of forms. With it the
present treatise has no concern save to point out
that the formal theorems of induction, and of its
preparatory steps, are deductions from axioms.

Otherwise the general proposition premised is an
induction, from which a deduction is made by sub-
suming some subsidiary truth. The great body of
reasoning in the so-called inductive sciences, and in
the practical affairs of life, is of this character.
Hence a treatise on logic limited to a discussion of
the Aristotelic deductive processes is essentially in-
complete; and, on the other hand, the notion, which
has widely prevailed, that induction is capable of
advantageously superseding deduction, and alone is
worthy of consideration, arises from an entire mis-
conception of the nature and several ends of the two

142 ELEMENTS OF INDUCTIVE LOGIC

processes, and of their essentially complementary re-
lation. *

' The Logic of Aristotle received the title opyavov, not from him-
self, but from his followers. It is clear that he did not regard it as an
organon, an aid or instrument of discovery, but as a propoedeutic. —
See Meta.^ iv., 3 (1005 b. 4). The title came into general use in the
fifteenth century. — See St. Hilaire, De la Logique d^Aristote, torn, i.,
p. 19. Bacon's second book of the " Instauratio Magna" is entitled
"Novum Organum " (1620), and is evidently intended to elaborate an
instrument of discovery. Dr. Whewell, dissatisfied with its methods,
gives us his " Novum Organon Kenovatum."

The designation " Organon " has led to much error. For two cen-
turies after Bacon it was commonly held that his was a new method,
superseding the effete method of Aristotle. But in the last half-cen-
tury a better understanding has come to prevail. Deduction and In-
duction together constitute Logic, and Logic in both branches is merely
" an analysis and systematic exposition of what we are all doing from
morning till night, and continue to do even in our dreams " (Macau-
lay, £Jssay on Bacon). In support of our view of the relation of De-
duction and Induction, we quote the foHowing authorities :

Aristotle says ; " All learning is derived from things previously
known, as we also stated in the Analytics ; and is derived partly from
induction [di tTraywy^g], and partly from syllogism. Now, induction
is the origin of the universal ; but a syllogism is deduced from uni-
versals. There are, therefore, some principles from which the syl-
logism is deduced, which are not themselves syllogistically established;
they are therefore established by induction." — Mc. Eth., vi., 3 (3) ;
cf. ihid., vi., 8 (9); Meta.^ i., 1 , Post. Anal, ii,, 19. Also see Grote,
Aristotle, ch. vi,, p. 276 sq.

Sir John Herschel says : " It is to our immortal countryman, Bacon,
that we owe the broad announcement of this grand and fertile prin-
ciple, and the development of the idea that the whole of natural phi-
losophy consists entirely of a series of inductive generalizations, com-
mencing with the most circumstantially stated particulars, and carried
up to universal laws or axioms, which comprehend in their statements
every subordinate degree of generality ; and of a corresponding series
of inverted reasoning from generals to particulars, by which these
axioms are traced back to their remotest consequences, and all par-

DEDUCTION 143

In the preceding exposition of the several meth-
ods of observation and experiment by which we
contrive to distinguish among a mass of coexistent
phenomena the particular effect due to a given cause,

ticular propositions deduced from them ; as well those by whose imme-
diate consideration we rose to their discovery, as those of which we
had no previous knowledge." — Discourse^ etc.^ ch. iii., § 96. This
passage, which Dr. Whewell prefixes as a motto to his " Nov. Org.
Renov.," reminds us that Buckle, in his " Essay on Induction," says
that Induction is inference from a reality to an idea, and Deduction is
inference from an idea to a reality.

Sir William Hamilton says r ** The deductive and inductive processes
are elements of Logic equally essential. Each requires the other.
The former is only possible through the latter ; the latter is valuable
only as realizing the possibility of the former. As our knowledge
commences with the apprehension of singulars, every class or universal
whole is consequently only a knowledge at second hand. Deductive
reasoning is thus not an original and independent process. The uni-
versal major proposition, out of which it develops the conclusion, is
itself [if not an axiom] necessarily the conclusion of a foregone induc-
tion, and mediately [?] or immediately, an inference, a collection, from
Individual objects of perception or self-consciousness. Logic, there-
fore, as a definite and self-sufficient science, must equally vindicate the
formal purity of the synthetic illation by which it ascends to its
wholes, as of the analytic illation by which it re-descends to their
parts. — Discussions, p. 160 (Harper's ed.). See, also, id., p. 15*7 sq.
Cf. Metaphysics, Lee. vi.

Mr. J. S. Mill says : " We shall, conformably to usage, consider the
name Induction as belonging to the process of establishing the gen-
eral proposition, and the remaining operation we shall call by its usual
name, Deduction. And we shall consider every process, by which any-
thing is inferred respecting an unobserved case, as consisting of an
Induction followed by a Deduction ; because, although the process
need not necessarily be carried on in this form, it is always suscep-
tible of the form, and must be thrown into it when assurance of scien-
tific accuracy is needed and desired," — Logic, p. 154. Cf. Venn, Mn-
pirical Logic, ch. xiv., p. 363 sq.

144 ELEMENTS OF INDUCTIVE LOGIC

or the particular cause which gave birth to a given
effect, it has been repeatedly indicated that, the rela-
tion being first definitely ascertained in a particular
case or cases, the axioms of uniformity authorize a
generalization extending to unknown cases — that is,
an induction of all possible like cases under a uni-
versal proposition or law. Also it has been stated
that such propositions serve as major premises, from
which to make deductions (§§ 32, 59). This is spe-
cifically proof, often resulting in discovery. A new
case being brought under the general proposition,
and a conclusion proved respecting it, this conclu-
sion, if previously unknown, is a discovery.

The induction All matter gravitates has been made,
we will suppose for illustration, from observations
on solids and liquids. Now do gases gravitate ? We
have only to establish All gases are matter^ in order
to deduce All gases gravitate^ or have weight. This,
in form, is not a mere find, but a scientific investi-
gation and discovery.

Again, suppose we have All celestial objects show-
ing a proper motion among the stars, and shining
with reflected light, are planets of the solar system.
We descry a telescopic object, seen by refiection, and
having a proper motion, and discover it to be a
planet. If, furthermore, its path is found to lie be-
tween the orbits of Mars and Jupiter, we have dis-
covered another one of the many asteroids.

The research into the cause of dew (§ 69) led to
the establishment of an inductive generalization,
from which deductions were made to eases thereto-

DEDUCTION 145

fore unexplained, thus resulting in a discovery of
the true cause of certain phenomena, such as the
" sweating " of a pitcher of iced water.

Note that the minor in the first example is a uni-
versal. It is not, however, an induction, but merely
the result of identification under definition (§ 10).
Matter is defined as extended and impenetrable,
which, being found true of gases, gives the proposi-
tion Gases are matter. Questions of identity to es-
tablish a minor are a necessary part of research, but
should not be mistaken for inductive inquiries estab-
lishing a major. Are alloys definite chemical com-
pounds, or mere mixtures, is a question of identity
under definition.

When a deduction to an unobserved fact has been
made, it remains to verify the conclusion. This is
to seek for and observe a particular instance, either
one occurring naturally, or one produced artificially.
Having inferred that Gases gravitate, we exhaust a
vessel of its air, and find that it loses weight. By
the method of difference we rightly judge the weight
lost to be that of the withdrawn air. This verifies
our inference, and also strengthens the premised in-
duction.^

Deduction thus normally subsequent to induction
often leads to further induction, as in the method of
residue (§ 60), and in other preparatory processes.

^ It will be seen, by the foregoing exposition of the general rela-
tions of induction and deduction, that, in logical order, the order of
thought and investigation, induction comes first. In didactic order,
deduction usually comes first.
10

146 ELEMENTS OF INDUCTIVE LOGIC

But deduction has a specific application in the inves-
tigation of certain causal relations which calls for
detailed consideration.

§ Y7. There are two kinds of effect which must be
set clearly apart. The distinction is very important,
and runs deep, being due to the ultimate nature of
things. An effect of one kind has properties quite
different from those of the effect of any of its an-
tecedents operating apart from the others. Thus,
oxygen and hydrogen unite to form water ; but in
water not a trace of the effective properties of either
factor is discernible. Hence it is impossible to de-
duce from such factors the consequent of their con-
joint action. To ascertain it an observation of the
product is requisite, which observation may often be
verified, not merely by direct experiment, but by
an inverse process of analyzing the product into its
originating components. This kind of effect is aptly
termed heterogeneous or heteropathic, the conjoint
effect differing in kind from those separately pro-
duced. It is also called chemical, because the clear-
est and most abundant examples are to be found in
chemical actions ; as, the taste of sugar of lead is
wholly unlike that of acetic acid or any other of its
components, and the color of blue vitriol is nei-
ther that of copper nor of sulphuric acid. There is, in
short, a change of properties so nearly complete that
the effect cannot be predicted from the given cause,
nor indeed the cause from the given effect.^ It is to

^ To this change of properties weight at least has been accounted an

DEDUCTION- 147

the resolution of tin's class of cases that the fore-
going methods are especially adapted.

An effect of the other kind has properties quite
similar to those of the effects of its antecedents oper-
ating separately. When two simultaneous impulses,
which may differ in direction and intensity, impart
motion to a body, the resultant motion is an effect
quite similar to the effects which the impulses acting
successively would produce, and the terminal result is
identical.' Hence it is possible to deduce from sucti
factors the consequent of their conjoint action with-
out observing it. The inference may often be veri-
fied by direct observation of a case, but not by any
reverting analysis of the product, such analysis being
impracticable. This kind of effect is termed homo-
geneous, as of like kind to those separately pro-
duced. It is also called mechanical, since its clear-
est and most abundant examples are to be found in
mechanics, both terrestrial and celestial. In gen-
eral, it is a composition of forces or causes, giving
an intermixture of effects, a homogeneous result not
susceptible of analysis into its originating compo-
nents.'*

exception; for the weight of any composite substance whatever is
always precisely the sum of the weights of its components. This it is
that has made the science of Chemistry possible (§ 18, note, and § 73).
But weight is not truly an exception to the foregoing statements, since
it is not properly a chemical but a mechanical property, not a molec-
ular but a molar activity.

' See Newton's second law of motion, § 18, note.

2 This composition of causes or intermixture of effects is liable to
be confused with plurality of causes (§ 22). In both a number of

148 ELEMENTS OF INDUCTIVE LOGIC

Cases of a homogeneous intermixture of effects
are very much more common than those of the other
class. Indeed, they abound on every hand, and in all
departments of knowledge. A lake is fed by rains
and rivers, but no examination of the lake will tell
how much is due to each. "Wind often concurs with
tide to make high water. The moon's orbit is a re-
sultant of attracting and tangential forces, centripe-
tal and centrifugal. A good crop is a single effect;
the agency, multiple. An invalid plies all means to
regain health ; many influences combine, but the
effect is indivisible. A voluntary effort is the off-
spring of many feelings. The rise and fall of prices,
the general prosperity of a country, and the increase
of population seldom depend on a single cause, yet
the effect is homogeneous.

§ 78. Let us examine these two classes of causal
relation, first with reference to the problem given
cause to find effect, reserving the inverse for treat-
ment in the next following chapter.

As to the class marked by a heterogeneous effect,
since we can infer nothing from the properties of
the antecedents respecting the character of the con-
sequent, we are shut up to the methods of investiga-

antecedents is involved ; but in the latter there is a plurality of distinct
causes, to either of which the effect may apparently be due, and we
are at loss to fix on the true one ; whereas in the former there is a
plurahty of co-operating antecedents, each of which, producing a spe-
cial effect when alone, produces the same when acting conjointly with
the others, and we are at loss to assign to each its due share.

DEDUCTION 149

tion which have already been discussed. These are
based on simple observation of facts or on experi-
ment, and the procedure is a ^posteriori by elimina-
tion.

As to the class marked by a homogeneous effect,
that is, a composition of causes yielding an inter-
mixture of effects, since the consequent is not sus-
ceptible of analysis into its actual constituents, none
of the foregoing methods is competent to cope with
it ; for those methods, proceeding essentially by elim-
ination, require, in order to this, an analysis, a dis-
crimination of the constituent facts of both antece-
dent and consequent. This, as to the consequent,
being impracticable, the preceding methods fail. If
A B 6^ are followed, not by y s x, but by a ; and if
B G still produce a, nothing is eliminated from the
consequent, and no point is gained.^

We are obliged, therefore, in case of a homoge-
neous effect, to seek some other method of investi-
gation. The homogeneity of the effect furnishes
ground for an inference from the effects of the

1 In some exceptional cases, however, the preceding methods yield
results. If A and a vary together, they are causally connected ; and
if with the total disappearance of A there is a loss of Ja, this proves
by the methods of concomitance and difference that A causes Ja. If,
as the weather becomes warmer, one's appetite diminishes, he may be
pretty sure that the appetite is affected by the season, though other
facts co-operate. Dr. Parkes ascertained that a muscle grows during
exercise, and loses bulk during rest ^ but there are other causes of its
growth. If a floating glass globe loses -^ of its displacement on be-
ing exhausted of air, this is proof that the weight of the contained air
caused that much of the displacement.

150 ' ELEMENTS OF INDUCTIVE LOGIC

causes acting apart to the effect of their conjoint
action. This presupposes the ascertainment, by some
of the preceding methods, of the particular effect of
each of the given causes, and generally an induction
of the law according to which each cause operates.
Then we proceed a jpriori to deduce their conjoint
effect, either from the inductions themselves or from
their several consequences. Thus we have a new
distinct method, which, since it proceeds by deduc-
tion, is called the Deductive Method.^

The problem to be solved by the deductive method
is, to find the composite effect from the laws of the
several composing causes. The logical form of the
procedure is concisely expressed in the following
Canon of Deduction : If from the several laws
of a plurality of co-operating antecedents a
composite consequent be deduced, this will
be the conjoint effect of the antecedents.

The method may be formally illustrated as fol-
lows : Let X be the unknown total. JS^ow —
If from A can be inferred -J- x,
and from ^ " " " . f x,
and from (7 " " " ^ x,

and from i> >' " '' —\x,

then their algebraic sum is the conjoint effect x.

For an example of a particular case, suppose we
wish to find the velocity of a train of cars at the foot
of a grade. If we can ascertain that the initial pro-

^ The name is not felicitous, seeing that it is not sharply distinc-
tive, and hence tends to confusion ; but, having been generally adopted
by logical writers, it is here retained.

DEDUCTION 151

pulsion causes a velocity of 10 feet a second, the pull
of the engine while running down 40, gravity 30,
and that friction causes a retardation of 20, then the
sum of the several velocities thus ascertained is its
final velocity. If, now, we actually measure the final
velocity and find it the same as that calculated, our
estimate is thereby verified. Theoretically this is a
very simple case, practically it would be difiicult.^
Yet this method is the sole one applicable to it, and
to a great variety of cases, many of great intricacy ;
nevertheless it has often led to very brilliant results.

§ 79. The deductive method, including its prep-
paration and confirmation, may be viewed as con-
sisting of three several stages.

1st. Induction.' This, the causes having been
separately investigated, makes induction of their
several laws. Many celestial phenomena remained
unexplained until the mechanical laws of certain
causes, especially the laws of motion (§ 18 n.), were
ascertained and furnished a basis for explanation.

^ Often forces are in equilibrium, as in mechanical action and reac-
tion producing rest (§ 18, note, 3d law). If A produces a, and B pro-
duces —a, the causes neutralizing each other as to any perceptible
change, we may have no suspicion that they are in operation at all.
Thus an equal balance at rest gives no sign of the downward forces in
play. Rest is the effect produced, and the forces must be described, in
terms of pressure, by their tendency to produce motion (§ 52).

2 The first stage is called induction because there must be an induc-
tion as the basis of the whole. In many particular investigations the
place of the induction may be supplied by a prior deduction, but the
ultimate major premise of the prior deduction must have been obtained
by induction.

152 ELEMENTS OF INDUCTIVE LOGIC

If the subject be a social phenomenon, the premises
prerequisite to its determination are certain laws of
human action, and certain properties of outward
things by which the conduct of men in society is
influenced. Thus certain political and social ante-
cedents are regarded as explanatory of the French
Kevolution.

2d. Deduction. This infers from the laws of the
causes their combined effect. If the cases subsumed
be general, the conclusions will be general. If they
be particular, so will the conclusions be ; as, the pre-
dicted positions of the planets, found in the nautical
almanac. When the terms of the premises have
been subjected to quantitative measurement (§ 73),
the deduction becomes a process of mathematical
calculation. To determine the path of a projectile,
a cannon-ball for instance, the causes which affect its
range and velocity must first be known and meas-
ured ; as, the force of the powder, the action of
gravity, the angle of elevation, the resistance of the
air, the force and direction of the wind. The laws
of these being given, and particular cases subsumed,
still it is a very difficult mathematical problem so to
combine the results as to deduce the effect of their
collective action.

3d. Verification. This tests the conclusion by
comparing it with actual fact. If these agree, the
conclusion is confirmed. The function of verifica-
tion is not proof, but merely the confirmation of
proof. Still its value is inestimable, and it cannot
be dispensed with. In numerous and important cases

DEDUCTION 153

the agencies are so many and various, often more
or less counteracting one another, that we can hardly
ever be sure that we have taken all into account, or
have estimated rightly those that we know. More-
over, when these conditions are fairly fulfilled, to
make the computation in any but very simple cases
transcends our calculus. "Witness the unsolved prob-
lem of three gravitating bodies. Save in rare in-
stances our results are at best only approximations.
To warrant reliance on the conclusion, it must be
found to accord with a direct observation of the in-
ferred facts, or with an empirical generalization of
them. Should a discrepancy between the inference
and the observation appear, it will lead to a correc-
tion of error, or be indicative of some unnoticed
residue, which may lead to additional discovery
(§ 60).

In Newton's procedure that establishes the iden-
tity of terrestrial gravity with the force that deflects
the moon's motion, or, in other words, that proves
the attraction of the earth to be the cause of the de-
flection, all three of the foregoing stages occur.^

1 The statement that follows is quoted from Mill, Logic, p. 350. It
should be noted that the order of procedure indicated here, and indeed
throughout this treatise, is the logical order. The historical order —
that is, the actual order ofthe thoughts of an investigator — is very va-
rious, anticipating, reverting, passing to and fro over the whole ground;
dwelling now on this point, now on that, overleaping necessary means;
returning to finish the unfinished, making excursions into collateral
regions, etc., so that it would perhaps be impossible for him to record
his actual procedure. But the logical order of statement links all in
a continuous chain of reason and consequent, and may be regarded as
a corrected restatement of the process.

154 ELEMENTS OF INDUCTIVE LOGIC

" First, it is proved from the moon's motions that
the earth attracts her with a force varying as the in-
verse square of the distance. This, though partly
dependent on prior deductions, corresponds to the
first or purely inductive step, the ascertainment of
the law of the cause (§ 89).

" Secondly, from this law, and from the knowl-
edge previously obtained of the moon's mean dis-
tance from the earth, and of the actual amount of
her deflection from the tangent, it is ascertained with
what rapidity the earth's attraction would cause the
moon to fall, if she were no farther off, and no more
acted upon by extraneous forces, than terrestrial
bodies are. That is the second step, the ratiocina-
tion.

" Finally, this calculated velocity being compared
with the observed velocity with which all heavy
bodies fall, by mere gravity, towards the surface of
the earth (§ 74), the two quantities are found to
agree." The proof is thus perfected, the identity
established, the cause of the deflection ascertained
with physical certainty to be the attraction of the
earth. The logical process is complete in all its parts.

XI.— HYPOTHESIS

§ 80. When a novel phenomenon occurs which
cannot at once be referred to its kind or otherwise
explained, we are perplexed and dissatisfied. This
prompts us to assign it provisionally to some known
class or cause to which we suppose it may be-
long. If the matter be trifling, we are usually
satisfied by a guess, and dismiss it. If it be im-
portant, we follow the clew implied in the guess,
and investigate the case until perhaps a plausible
supposition is reached. Closer investigation may
lead to knowledge, but very often we cannot get
beyond a suspicion, a good guess, a fair conjecture,
a reasonable supposition, or at best a probable as-
sumption.

Whoever will attentively consider his own mental
operations will find that almost always they thus
consist at the outset of suppositions, that these guide
his inquiries, and that very often he is unable to
pass beyond to positive knowledge, but must rest
content with probability. He will find, not only
that his thoughts are constantly employed with sup-
positions, and that they comprise the great body of
his most mature reflections, but also that without
the aid of these as percursors it would hardly be pos-

156 ELEMENTS OF INDUCTIVE LOGIC

sible to attain any satisfactory knowledge of any-
thing whatever.

A supposition or hypotliesis has the form of a
representative idea — a mental image of what is at
least logically possible. The making it is the work
chiefly of the reflective or the practical imagination,
the thinking faculty co-operating and restraining.^
That the earth is even now a sphere of molten fluid
intensely hot, enclosed by a thin crust comparable to
an egg-shell, is an hypothesis that required a bold
imagination to frame, and requiring, we may add, a
like imagination to comprehend. A special vigor of
this faculty, disciplined by thought, is characteristic
of discoverers in science and of inventors in the arts.
By it they make tentative excursions into unexplored
regions, increasing and utilizing knowledge.

The methodical use of suppositions in trifles is
precisely the same as in the noblest sciences. One
cannot hear a knock at his door, or see a flash, or
smell an odor, or feel a pain, without instantly, al-
most instinctively, making a supposition to explain
it. Questions in common talk conform to supposi-
tions in mind. Tares appear among the wheat; good
seed was sown; whence come the tares? An enemy
hath done this. The plausible supposition may be
rendered highly probable by circumstantial evidence,
as the courts call it, against the accused, who, while
enjoying the presumption of innocence, is tried on

1 These mental relations are more fully stated with illustrations in
Psychology, §§ 200, 202, 214.

HYPOTHESIS 157

the supposition of guilt. This, unless established by-
direct evidence, remains a supposition — that is, an un-
proved proposition — only becoming more or less prob-
able according to the circumstances. Yet, if it be
shown that no other supposition can be maintained,
this is proof, legal and logical.^ We have passed
from trifles into serious matter. Kow, if we turn to
the great sciences that solve the mysteries of nature,
or to theology that tells us of God, we shall find the
same logical principles and processes, the same use of
conjecture, supposition, and hypothesis, in the course
through which the loftiest truth is attained. It is
the province of logic in general to disclose and for-
mulate the natural processes of thinking, and in par-
ticular to unfold in this place the important part
played by hypothesis."*

^ " Let any one watch the manner in which he himself unravels a
complicated mass of evidence ; let him observe how, for instance, he
elicits the true history of any occurrence from the involved statements
of one or of many witnesses ; he will find that he does not take all the
items of evidence into his mind at once, and attempt to weave them
together; he extemporizes, from a few of the particulars, a first rude
theory [supposition, hypothesis] of the mode in which the facts took
place, and then looks at the other statements one by one, to try whether
they can be reconciled with that provisional theory [hypothesis], or
what alterations or additions it requires to make it square with them.
In this way we arrive, by means of hypotheses, at conclusions not hy-
pothetical."— Mill, Logic^ p. 354.

^ A thesis (Gr.) is a proposition posited (Lat.); an hypothesis is
one supposited or supposed. The words hypothesis and supposition
have thus a like etymology, and are synonyms. The latter in usage
is applied more freely to commonplace and transient notions ; the for-
mer to such as are scientific and settled, and so has rather more dig-
nity.

158 ELEMENTS OF mDUCnVE LOGIC

§ 81. The various methods of investigating to
ascertain the causal relation between a given phe-
nomenon and its circumstances involve necessarily
a constant use of suppositions. "When an effect is
given to find its cause, we are limited to simple
observation, and seek for a natural occurrence of
an instance wherein the antecedents can be noted.
When one has been found, the first step is to reject
the immaterial circumstances, and then to distrib-
ute the remainder into antecedents and consequents.
I^ow, it is quite obvious that even this much cannot
be done, unless there be in the mind of the observer
some idea, however vague and unsettled, of the cause
he is seeking, some suggestion from experience of
analogous cases, some clew, some index, some sur-
mise, conjecture, supposition, to guide him in a ten-
tative application of one or another of the methods
of investigation. This is essential to any intelligent
observation, which otherwise would be no more than
the stupid gazing of a boor. The supposition, aris-
ing perhaps in a very loose and uncertain way, may
soon prove quite erroneous and be rejected, whereby
a negative point is gained. Another takes *its place,
and investigation is renewed, guided constantly by
a supposition. Illustrations of this mode of research
are seen in the various hypotheses on the nature of
comets and nebulae. Others may be taken from va-
rious literary hypotheses which have laid claim to
acceptance ; as, the hypothesis of Wolf respecting
the origin of the Homeric poems ; that of IS'iebuhr,
deriving the stories of early Kome from lost ballads

HYPOTHESIS 159

or epics ; those of Eicliliorn, Marsh, and others con-
cerning the origin of the text of the Gospels ; the
many concerning the authorship of tlie (Economics
attributed to Aristotle, and of the Letters of Junius.
In such cases suppositions are made, and then sup-
ported by circumstantial evidence. The form of
logical procedure in the grave matter of scripture
exegesis, or, generally, in the interpretation of lan-
guage, is quite similar.^

It is equally obvious that all experimental obser-
vation is likewise dependent on supposition. A mere
trial of possible combinations to see what will come
of them, without the further suggestions of a sug-
gested supposition, can elicit nothing, save by chance.
Indeed, that cannot properly be called an experiment
which does not proceed upon some tolerably well
defined hypothesis. Cavendish, suspecting that water
is not an element, was led by positive supposition to
burn hydrogen with oxygen, and thus discovered its
composition. Davy, conjecturing the alkalies to be
metallic oxides, and following a clew suggested by
analogy, proved it on decomposing them in a gal-
vanic circuit. Franklin sailed his kite on a surmise
of the identity of lightning with the electricity of his
machine. Bacon stuffed a dressed fowl with snow,
to test his supposition that cold would keep meat
sweet. Columbus sailed westward on the hypothesis

^ A striking example of the application of the hypothetical deduc-
tive method to interpretation is the deciphering, by Champollion, in
1822, of the famous Rosetta stone, whereby the Egyptian alphabet
was discovered.

160 ELEMENTS OF INDUCTIVE LOGIC

that the earth is round, and hence that he could thus
reach the Indies. Socialists attempt revolution,
and legislators enact tentative laws on hypothetical
grounds. Whenever anybody tries to do any new-
thing with the least modicum of intelligence, he is
trying to realize a suppositive idea, and no scientific
procedure of any sort is possible unless in accord
with a preconceived hypothesis.

§ 82. A more formal use of hypothesis — one more
generally recognized by logicians and scientists — is
now to be examined at some length.

In the previous chapter, of which this is a con-
tinuation, it was shown that the law of each of sev-
eral given causes, which together produce a homo-
geneous effect, being inductively ascertained, we
may, in simple cases, deduce their united effect,
and then verify this result by comparing it with
observed fact. We thus determine the effect to be
expected in certain cases wherein the several co-op-
erating causes intermixing their effects are known.
The logical order of the whole procedure is, first
induction, then deduction, then verification. This
is the deductive method as applied to solve the prob-
lem : Given a certain composition of causes, to find
what homogeneous effect will follow (§ 78). We have
now to show how the deductive method is applied to
solve the inverse problem : Given a homogeneous
effect, to find the cause or causes producing it, or
their laws.

It is quite evident that, owing to the homogeneity

HYPOTHESIS 161

of a mechanical effect, an analysis of it into its com-
ponents is impracticable, and therefore no direct ap-
plication of this or any of the preceding methods will
solve the present problem. The obstacle may also
be explained as due to the quasi-principle of a plu-
rality of causes (§ 22). Referring to the illustration
drawn from the composition of motion, it is evident
that, given the motion of a body, no practicable anal-
ysis of this effect, which is but a part, though the
chief one, of the consequents, will enable us to deter-
mine what force or forces were its actual cause ; since
there is an infinite number of combinations of im-
pulses, varying in intensity and direction, which might
have produced precisely this partial effect. To find
liypothetically, for instance, what impelling force or
forces, with their point of application, direction, and
intensity, might have produced the existing projec-
tile and rotary motion of the earth, is an easy prob-
lem ; but to ascertain what combination of impulses,
if any, did actually produce it, is impossible from
any data we possess.

The difficulty is not always so absolutely insuper-
able. There are many and very important cases, in
which an indirect application of the deductive method
attains, by the aid of hypothesis, results of inesti-
mable scientific value. It consists in substituting
for the induction of the first stage of the direct de-
ductive method (§ 79), an hypothesis of the cause or
of its law, and then proceeding as before, the stages
now being, first hypothesis, then deduction, then

verification. This modified application of the de-
11

162 ELEMENTS OF INDUCTIVE LOGIC

ductive method, we shall now examine more particu-
larly.^

§ 83. Let us first define the term. In its most
general sense, an hypothesis is an unproved, and may
be an unprovable, proposition. More specifically,
it is a proposition laid down, without evidence or
with insufficient evidence, from which to draw con-
clusions relative to facts, under the notion that, if
the conclusions are in accord with known facts, the
hypothesis either is, or is likely to be, true.'* In
undertaking to explain the formal use of scientific
hypothesis, we venture this yet more restricted
definition : A scientific hypothesis is an ideal
assumption of a cause or law.

§ 84. For many cases of a mechanical effect whose
cause is unknown, a known cause, with its known
law, is hypothetically posited ; from this supposition
a deduction is made, and its conclusion verified by
observation. Thus it has frequently occurred to a
scattered cluster of powder-magazines that when
one is exploded the others immediately explode.
How shall we account for or what causes this uni-

* Comte puts the process in a sentence, saying: "Some fact is as
yet little understood, or some law is unknown ; we frame on the sub-
ject an hypothesis as accordant as possible with the whole of the data
already possessed ; and the science, being thus enabled to move for-
ward freely, always ends by leading to new consequences capable of
observation, which either confirm or refute, unequivocally, the first
supposition." — Philoaophie Positive, torn, ii,, p. 434.

'-^ See Mill, Logic, p. 249 ; and Bain, Logic^ bk. iii., ch. 13.

HYPOTHESIS 163

formitj? The hypothesis has been assumed that
aerial vibrations, whose mode of motion and of com-
municating motion are well known, are the cause.
JSTow, if in general aerial vibrations can cause explo-
sion, it is deductively inferred that intense explo-
sives, as cordite, nitrogen iodide, or fulminate of
mercury, shall readily be exploded by the vibrations
which a similar explosion produces, or even by a
musical note. Experiments have verified this con-
clusion, thus rendering the hypothesis probable.

Let it be remarked that in this case the cause hy-
pothetically posited is a vera causa — that is, one
known in other connections to be a cause.^ So we
may assume the cause of an epidemic to be excessive
heat, or bad drainage, or imported bacteria, each be-
ing a vera causa^ and push the inquiry accordingly.
The glacial hypothesis, accounting for the character
and distribution of erratic boulders, assigns the ob-
served action of glaciers and ice-floes as the cause;
and the science of geology in general, finding in the
earth's crust strata and masses of rock quite similar
to observed deposits from water and products of vol-
canic fire, assumes these vercB causm as explanatory
of those ancient formations.

* The phrase vera causa is taken from Newton^s first Rule of Phi-
losophizing (§ 21 n.). His meaning seems plain enough when we con-
sider that he was proposing gravity, a cause known to operate near
the earth, as the cause of planetary motions, to take the place of the
ideal vortices of Descartes, in the theory of celestial mechanics. Still
the phrase has been much discussed, and variously interpreted. See
Herschel, Discourse, etc., § 137 sq. ; Whewell, Phil, of Dis., ch. xriii.,
§ 7 sq. ; Mill, Lo^ic, p. 353.

164 ELEMENTS OF INDUCTIVE LOGIC

It is not, however, as has been claimed, essential
in scientific investigation that the cause assumed
shall be a vera causa, but such assumption brings
the case nearer to and facilitates complete proof, and
hence is the most promising form of this general
mode of inquiry/ There is, indeed, no other limit
to hypothesis than that of imagination ; but natural
science admits only such hypothetical agencies as are
allied, at least by analogy, with known causes and
laws in nature. The assumption of a supernatural
cause to account for a natural event is unscientific,
and characteristic of superstition ; as, to attribute an
epidemic to the ill-will of a witch, or table-rappings
to spirits.

In illustration of a cause wholly hypothetical— that
is, one not a vera causa, but invented and supposed —
we cite the undulatory hypothesis of light originally
proposed by Huyghens. This assumes space to be
filled with an ether whose vibrations, according with
the known laws of vibration in elastic fluids, account
for many of the phenomena of light. The suppo-
sition has given unity to the science of light, and
served as an excellent working hypothesis ; but inde-
pendent evidence of the real existence of such an
ether is still lacking, though it has been earnestly
sought, especially in watching for a retardation of
the motion of comets attributable only to a resisting

1 " Any hypothesis which has so much plausibility as to explain a
'considerable number of facts, helps us to digest these facts in proper
order, to bring new ones to light, and make experimenta ci'ucis for the
sake of future inquiries."— Hartley, 06s. on Man, vol. i., p. 16.

HYPOTHESIS 165

medium. The assumed cause, then, is not a vera
causa, and until it be proved to be one, it is not
strictly proper to speak even of this most admirable
and truly scientific hypothesis as a theory/

§ 85. Instead of a hypothetical cause acting accord-
ing to known law, there maybe posited a known cause
acting according to hypothetical law. For instance,
the kinetic hypothesis of gases assumes that their
mechanical properties are due to a peculiar mode of
activity of the molecules. This activity of the known
cause is supposed to be in accordance with the laws
of motion, inertia and others, which, since they are
known to be true laws in other relations, vercB leges,
correspond in this hypothesis to the verm causm in
those just discussed. The gaseous molecules are rep-
resented as constantly moving with great velocity,
those of hydrogen at zero having a rate of one and
one-seventh miles a second; also as colliding with
each other, and impinging on the sides of a contain-
ing vessel, which expenditure of vis viva is the press-
ure of the gas. As the temperature rises, the mole-
cules move faster, strike harder and often er, and the
pressure is greater. It is their great and inces-
sant molecular activity that causes the expansion and
diffusion of gases, to which is due the uniformity

^ The terms theory and hypothesis should not be used indifferently.
Hypothesis is the more general term including mere conjecture. The-
ory is hypothesis of only the highest order, grounded on a vera causa,
and systematically elaborated. Moreover, after complete proof, the
theorem, though no longer hypothetical, is still called a theory.

166 ELEMENTS OF INDUCTIVE LOGIC

of our mixed atmosphere. This hypothesis of a
peculiar mode or law of activity has been devel-
oped mathematically, and deductions made from it
have been verified. It is accepted by many phys-
icists.

There are also instances in science wherein hy-
pothesis has respect both to the cause and to its law.
The development hypothesis, proposing to account
for the origin of species, was announced, in crude
form, five centuries before the Christian era, and has
never been entirely abandoned. What Mr. Darwin
did for it was to amplify and perfect the hypothesis
of the causes, environment, use and disuse, and hered-
ity, showing that they are verm causes^ then to postu-
late the law of natural selection or the survival of the
fittest, and show it to be a vera lex under which fair-
ly permanent changes of type in both fauna and flora
are actually effected, this being also a verification.
Still, his famous speculation, serving as an excellent
guide of work, and remodelling all branches of nat-
ural history, remains an hypothesis ; it is not a logi-
cally established theory.*

^ See Darwin, " Origin of Species," especially ch. iv. The following
statements, from favoring authorities, are weighty and significant :

" Mr. Darwin's remarkable speculation on the Origin of Species is
another unimpeachable example of a legitimate hypothesis. What
he terms * natural selection ' is not only a vera causa [lex ?], but one
proved to be capable of producing effects of the same kind with those
which the hypothesis ascribes to it ; the question of possibility is en-
tirely otte of degree. It is unreasonable to accuse Mr. Darwin (as has
been done) of violating the rules of Induction. The rules of Induction
are concerned with the conditions of Proof. Mr. Darwin has never

HYPOTHESIS 167

Other examples of approved doctrine wholly hypo-
thetical are Dalton's atomic hypothesis, so prominent
in chemistry, and Boscovich's hypothesis of the ulti-
mate mechanical constitution of matter, which holds
its place in physics. Such double assumptions of
both cause and law must be classed as representative
fictions until discovery take them out of this cate-
gory. Though mere speculations, yet they have sci-
entific value, in promoting unity of conception and
suggesting lines of fruitful investigation.

Finally, an hypothesis may be made respecting the
law of an eifect, the cause and its law being unknown
and unsought. Thus Kepler made and rejected, be-
cause unverifiable, nineteen hypotheses respecting the
orbit of Mars, before he supposed it to be an ellipse,

pretended that his doctrine was proved. He was not bound by the
rules of Induction, but by those of Hypothesis. And these last have
seldom been more completely fulfilled. He has opened a path of in-
quiry full of promise, the results of which none can foresee. And is
it not a wonderful feat of scientific knowledge and ingenuity to have
rendered so bold a suggestion — which the first impulse of every one
was to reject at once — admissible and discussible even as a conjecture ?"
— Mill, Logic, p. 355, note.

"It must suffice to enunciate the belief that Life under all its
forms has arisen by a progressive, unbroken evolution ; and through
the instrumentality of what we call natural causes. That this is an
hypothesis, I readily admit. That it may never be anything more,
seems probable. That even in its most defensible shape there are
serious difficulties in its way, I cheerfully acknowledge. . . . Save for
those who still adhere to the Hebrew myth, or to the doctrine of spe-
cial creations derived from it, there is no alternative but this hypothe-
sis or no hypothesis. For myself, finding that there is no positive
evidence of special creations, and that there is some positive evidence
of evolution, I adopt the hypothesis until better instructed." — Herbert
Spencer, Principles of Psychology ^ § 208, note (2d ed., 1870).

168 ELEMENTS OF INDUCTIVE LOGIC

and found this verifiable (§10). Of like sort is his
hypothesis of the law of refraction of light.

§ 86. An hypothesis, whatever approbation it may
enjoy, if it be found irreconcilable by any modifi-
cation with an observed fact — facts being stubborn
things^must be abandoned. That heat is a mode of
molecular motion was once, but is no longer, an ap-
proved doctrine of physics. The system of cycles
and epicycles, proposed by Tycho Brahe to account
for the celestial motions, fell away as soon as the
relative distances of the planets was measured. Fre-
quently, in the history of science, two or more hy-
potheses, each having its advocates, have been pro-
posed to explain the same class of phenomena. Thus,
in electricity, Franklin's hypothesis of one fluid was
opposed by Symmes's hypothesis of two fluids ; both
are now rejected as failing to accord with the facts.

A fact that decides between two rival hypotheses
was called by Bacon an instantia crucis, a crucial
instance.* When the Copernican system opposed
the Ptolemaic, it triumphed by the instantia crucis
of aberration of light, a fact incompatible with the
earth's being at rest. Foucault's pendulum experi-
ment also is crucial against its immobility. Rival
hypotheses of light mark the «arly history of that sci-
ence. Newton's emission hypothesis supposes light to

^ It is the fourteenth of his Prerogatives of Instances, introduced
thus : " Inter praerogativas instantiarum ponemus, loco decimo quarto,
instanfias crucis ; translato vocabulo a crucibus, quae erectae in biviis,
indicant et signant viarum separationes."— iVov. Org.^ bk. ii., aph. 36.

HYPOTHESIS 169

consist of minute actual particles emitted with great
velocity from luminous bodies. The undulatory hy-
pothesis of Hnyghens, already cited (§ 84), supposes
liofht to consist in the vibrations of an elastic luminif-
erous ether filling space. The absence of mechanical
energy from rays of light, the most delicate experi-
ments failing to discover any vis viva in the con-
centrated solar beam, is a negative instantia crueis
against the emission hypothesis ; one positive is that,
by this hypothesis, the velocity of light on passing
into a denser medium should increase, whereas it was
shown by Fizeau to diminish, being in inverse ratio
to the refractive indices. Moreover, Fresnel showed
that the phenomena of diffraction and of thin plates
are inconsistent with this hypothesis, but clearly
explicable on the other.^ These crucial instances
overthrew the Newtonian hypothesis, and that of
Huyghens has ever since been unrivalled. But let
not the disproof of one be mistaken for proof of the
other. In general, that an hypothesis has no rival,
and is not likely to have one, though it strengthen
presumption, is not proof.

§ 87. A special function of verification, then, is to
establish a crucial instance which will discredit a
rival hypothesis. When this is done, it makes a deep
impression, and strengthens the erroneous notion, so
common even among scientific thinkers, that verifi-
cation somehow is proof. We have already stated

^ See Fresnel's view more fully detailed in Herschel, DiscoursCy etc.y
§ 218. See, also, Ganot, Mments de Physique, §§ 429, 551.

170 ELEMENTS OF INDUCTIVE LOGIC

that its general function is to confirm hypothesis, to
heighten its probability (§ 79). When verifications
are numerous and unexpected, and conform to the
hypothesis with mathematical precision, afid espe-
cially when defeating all proposed rivals, they almost
irresistibly convince. Although any mere hypoth-
esis may at least conceivably be replaced by some
other one not yet devised, it is only the strong and
clear mind that can successfully resist being misled
by such verifications into a confidence proper to em-
pirical certainty alone (§ 45). Mere verifications can
never amount to strict proof.^ Of this much only
may we be sure — if the hypothesis of a cause, as the
luminiferous ether, be at all tenable, then its laws
and the laws of the real cause, whatever it may be,
are at least partially identical. But this identity of
law does not prove identity of cause, for agencies
quite distinct may have identical law ; thus the in-
tensity of all radiants — light, heat, gravity, and oth-
ers— varies inversely as the square of the distance.

The power of predicting entirely new phenomena
has been regarded as a specific mark of the truth of
an hypothesis.'* For instance, it being known that

* An exception, however, should perhaps be taken in case of an hy-
pothesis relative to a single fact, or a group of facts having known
limits. Thus, from the hypothesis that the world is round was in-
ferred, it may be circumnavigated ; which was first fully verified by
the Vittoria, one of Magellan's ships, in 1519-21. Cf. the discovery
of the planet Neptune, § 11.

2 So Dr. Whewell seems to think. See Phil. o/Dis., ch. xxii., § 61.
A very striking case is the prediction, resulting from mathematical
deduction, by Sir William R. Hamilton, verified by Dr. Humphrey

HYPOTHESIS 171

two aerial sound waves may so interfere with one
another as to produce silence, analogy suggested that,
if the undulatory hypothesis of light be true, two
rays may so encounter as to neutralize each other,
and produce darkness ; which prediction was fulfilled
on experiment. This, construed as an argument in
proof of the hypothesis, is plainly fallacia conse-
quentis (§ H5). A fact thus obtained is only one
more added to those already found to accord with
the hypothesis. If the law of the propagation of
light agrees with that of elastic fluids in a number
of known particulars, we may expect it to agree in
others. That a fact was predicted does not in the
least affect its character or bearing. The fulfilment
of such a prophecy merely adds the weight of another
verifying fact to a still unproved assumption. Let
us remember that Newton formed an hypothesis
from which he predicted the combustibility of the
diamond ; which prediction has proved true, yet the
hypothesis has proved false (§ 47).

§ 88. If, then, verification cannot accomplish log-
ical proof, by what process shall it be attained ? The
form is quite simple.

First, the hypothesis in question must be shown
competent to explain all the facts of that class to
which it is applied ; that is, it must lead deductively to
those facts, which deduction is tested by verification.'

Ward of Dublin, of the refraction of a single ray of light, under special
conditions, into a conical pencil.

^ The undulatory hypothesis of light fails even here. It gives no

172 ELEMENTS OF INDUCTIVE LOGIC

Second, it must be shown that no other hypoth-
esis can explain all the facts; in other words, that
any other hypothesis will lead to sonae false result.'

When these two steps have been taken, the proof
is complete, passing beyond the highest probability
that can be attained by the first step alone, and be-
coming physical or moral certainty — that is, empirical
certainty (§ 45). The thesis is no longer an hypoth-
esis, an unproved proposition, but has become a
proved proposition, an established theory.

It is worthy of remark that both parts of this proc-
ess are often recognized in vulgar speech as requi-
site to constitute proof. When a supposition is pro-
posed to account for some commonplace affair, and
questioned, the proposer is apt to say something like
this: It explains the whole matter ^ and the thing
canH he explained in any other way, or, no other
explanation will do. The objector may perhaps re-
ply: It seems to me some other explanation tnight
he found, or is possible, which also implies that estab-
lishing the negative is essential to proof. So in the
courts. Circumstantial evidence of guilt, which in-
deed may be completely refuted by an alihi, a fact
irreconcilable with the supposition, is accumulated
until, in the opinion of both judge and jury, this
and no other supposition can possibly explain the
facts; which result in ordinary cases will justify

satisfactory account of the reflection of light, of the composite char-
acter of white light, of the colors of objects, of the absorption of light,
or of its chemical and vital influences.

^ This is essentially the argumentum ad impossibile (§ IDS').

HYPOTHESIS 173

condemnation, the indictment becoming morally cer-
tain. If the defendant can maintain some other
plausible supposition, doubt remains, and he is enti-
tled to the benefit of the doubt — that is, his guilt is
not proved.

A little consideration will discover that this process
is the rigorous method of difference, the two steps
just described fulfilling its condition of aflSrmative
and negative instances (§ 56). For example, it has
been observed by Hyene of France, and Bizzolero of
Italy, that in every case of the blood of consump-
tives examined there is present a third corpuscle on
which the also ever-present consumption bacillus ap-
parently feeds. The hypothesis is that their coex-
istence, A^ is the cause of the disease, s, which is
thereby explained. Allowing that the observed facts
support, as above stated, this hypothesis, we have
the affirmative instance, ABC with y zx^ the added
letters representing other physical circumstances in
a case. The numerous confirmatory observations,
therefore, by the method of agreement alone, render
the hypothetic causal relation highly probable.

]^ow, the supposition that the third corpuscle alone
may be the cause is precluded by the observation
that its presence is consistent with health. The sup-
position that the bacillus alone is the cause remains.
Dr. Watkins of New York city resolved to test this
last supposition in his own person. Having ascertained
that the third corpuscle was not present in his blood,
he caused himself to be inoculated with the cultus
of tubercule bacilli. The ninety days, during which

174 ELEMENTS OF INDUCTIVE LOGIC

symptoms of consumption or tuberculosis should ap-
pear, passed away without the sign. Thus was sup-
plied the negative instance, B C with x y^ required
by the method of difference ; both the combination
of the corpuscle with the bacillus, A, and the disease,
z, being absent. Therefore, by this cogent method,
the combination is proved to be the cause, no other
hypothesis will answer, and the one laid down be-
comes a fairly estsiblished theory, which may lead to
very important therapeutic results.

§ 89. The discussion may fitly close with a cita-
tion of a standard example. It is Newton's use of
the hypothetical form of the deductive method to
determine the primary laws of the orbital motion of
the planets.*

First, he assumed that the force which constantly
deflects a planet from a rectilinear course, making it
describe a curve around the sun, tends directly toward
the sun. Then he proved deductively that, if it do
so, the radius vector of its orbit shall describe equal
areas in equal times. This was verified by being
identical with Kepler's first law, already empirically
ascertained (§ 10 n.). Newton then proved that if
the force acted in any other direction whatever, the
radius vector would not describe equal areas in equal
times, which consequent is false to fact. This latter
step completes the proof of the first assumption.

^ In the following statement we follow pretty closely the excellent
analysis of Mr. Mill, Logic, p. 351.

HYPOTHESIS 175

For, let A he Si force acting centrally \ A B C^ the
planets and a central force; B C^ the planets apart
from a central force. Now the planets and a central
force produce 2, areas proportional to the times, with
X 2/, effects other than z ; the planets apart from a
central force produce x y only. Hence it is rigor-
ously proved by the method of difference that A^
a force acting centrally, is the causal law of s, areas
as the times.

Second, having thus determined the direction of
the deflecting force, Newton proceeded in like man-
ner to ascertain the law of quantitative variation of
that force. He assumed that the force varies in-
versely as the square of the distance. From this he
deduced Kepler's second and third laws, which veri-
fied the hypothesis. He then proved that any other
law of variation would give results inconsistent with
Kepler's laws already known to be true. This com-
pletes the proof of the second assumption.

Newton then used these conclusions as premises
under which, by the direct deductive method, the
motion of the moon was brought as a special or par-
ticular case, and terrestrial gravity proved to be its
cause. This argument is detailed in § 79.

Thus was established the theory of universal grav-
itation. The general induction which immediately
follows the foregoing specific proofs is stated by
Newton as an obvious and necessary inference. He
says : " If it universally appears, by experiments and
astronomical observations, that all bodies about the
earth gravitate towards the earth, and that in pro-

176 ELEMENTS OF INDUCTIVE LOGIC

portion to the quantity of matter which they several-
ly contain ; that the moon likewise, according to the
quantity of its matter, gravitates towards the earth ;
that, on the other hand, our sea gravitates towards
the moon ; and all the planets mutually one towards
another ; and the comets in like manner towards the
sun ; we must universally allow that all bodies what-
soever are endowed with a principle of mutual grav-
itation." ^ Subsequently he says : " We have ex-
plained the phenomena of the heavens by the power
of gravity, but have not assigned the cause of this
power. Hitherto I have not been able to discover
the cause of the properties of gravity from phenom-
ena, and apart from phenomena I frame no hypoth-
eses. It is enough that gravity does really exist, and
act according to the laws which we have explained,
and abundantly serves to account for all the motions
of celestial bodies." ^

* Prindpia^ bk. iii., under Rule 3d. ^ Id., Scholium Oenerale.

XII.— NATUKAL LAW

§ 90. The ultimate essence in the generic notion
law is similarity. When a number of facts, either
beings or events, make a striking impression of simi-
larity, each is regarded as a repetition of the others.
A phenomenon is said to be repeated when the
mind of the observer receives impressions so very
similar as to be indistinguishable except as to time
or place. When several such impressions concur,
the notion of repetition is expanded into the notion
of order. This, when the order is undeviating, be-
comes the notion of strict uniformity. Law ex-
presses strict uniformity. Its most general definition
may be stated thus :

A law is a designation of a constant order
of facts determined by the constitution of
the things.'

^ The synthesis of this section is of elements obtained by an analy-
sis of the notion law. A designation simply marks out and makes
known. The things are those from which the law arises, and to which
it applies. The constitution is an assemblage of properties, which
properties, being constant causes, determine both the facts and their
constant order. The specific difference, determined^ etc., excludes
voluntary order (js. g. that discovered by statistics of crime), chance
order (§ 49), and any order discernible in primitive collocations
(§94n.).
12

178 ELEMENTS OF INDUCTIVE LOGIC

§ 91. Primarily there are two kinds of law — for-
mal law and material law.

Formal laws designate or give expression to the
forms in which the mind conceives of things. They
are strictly abstract formulas, occasioned by the or-
der of phenomena, but expressing only the conse-
quent intellectual order necessary to the understand-
ing of phenomena. Such are the primary laws of
logic, the principles of induction, the axioms of
mathematics, the fundamental principle of ethics,
and any other primary axiomatic truth of pure in-
tuition (§ 7). A formal law arising from demonstra-
tion— that is, one deduced a priori from axioms — is
a secondary formal law; as, the dicta of the syllogism,
the canons of causation, the law of a mathematical
series, and the like. Formal laws are expressive of
ultimate abstract absolute truth.

Material laws designate formal or conceptional or-
der incorporated with matter, and thereby give ex-
pression to phenomenal order. The order of phe-
nomena is always determined by the constitution of
the things themselves, which order is recognized by
the observer, and formulated as material law.

The term has in good usage such wide and varied applications that
it is difficult to formulate an accurate and adequate definition.

Montesquieu defines thus : " Laws in their most extended significa-
tion are the necessary relations arising from the nature of things."
He adds : " In this sense all beings have their laws, the Deity has His
laws, the material world has its laws, superior intelligences have their
laws, the brutes have their laws, and man has his laws." — V Esprit des
Lois, bk. L, ch. 1. This is altogether the most meritorious attempt I
have seen to construct a comprehensive definition of law.

NATURAL LAW 179

§ 92. Material law likewise is of two kinds, moral
law and natural law.

Moral law, apart from its content, has the form of
a categorical imperative: Act hy a maxim fitly uni-
versal. This materialized becomes: Trespass not y
Love thy neighbor. It is a mandate addressed to
persons, implying a possible alternative, and the re-
quired order, determined by the natural constitution
of its subjects, is sanctioned by authority, power, and
penalty. The decalogue, all civil, common, and stat-
ute law, and even the conventions of polite society,
are specialized statements of moral law.

Natural law generalizes and formulates facts of
coexistence and events of orderly succession in in-
animate things, and also in animate beings apart
from their free will. It merely states a uniformity
wliich has been found to exist in nature.

Moral law is in form imperative ; natural law is
simply indicative. The one is a uniformity enjoined,
having an alternative ; the other is a uniformity es-
tablished, having no alternative. In the one the
facts come after the law ; in the other the facts come
before the law. The one generalizes ideal facts that
ought to be ; the other, real facts that actually are.
Moral law of actions becomes known a priori by
pure intuition, and serves as a premise from which
to deduce specific rules of duty in personal conduct ;
natural laws of events become known a posteriori
by induction, and serve as premises from which to
deduce specific laws and particular facts of science,
and rules of art.

180 ELEMENTS OF INDUCTIVE LOGIC

In addition to these distinctions let us clear the
notions of two adhering misconceptions.

It is probable that the notion law is deriyed orig-
inally from the expressed will of a superior in power
and authority/ But this meaning has become specif-
ic by extending the content of the notion to include
generically various uniformities, though still retain-
ing, in perhaps all of its applications, a covert sug-
gestion of authoritative imposition. Hence, it may
be, arises the confused and inaccurate, yet very com-
mon, thinking and speaking of obedience to or vio-
lation of natural law. Persons, to whom moral law
is addressed, may obey or break it ; the alterna-
tive is possible. But neither persons nor things lit-
erally obey natural law ; for, there being no possibly
alternative, it cannot be violated, or perverted. J^y,
planet does not obey the laws of motion and gravi-
tation ; the notion of obedience is inapplicable to it.
A natural law does not convey a command, it is never
expressed in the imperative mood, but is a categor-
ical proposition indicative merely of a general fact
in general terms.'

^ The word law is cognate with lay, from the Anglo-Saxon %m, and
this from the causative leegan, to lay down. A law is that which is
laid, set, fixed ; Lat. statuere, whence statute. Austin limits it thus :
"A law, in the literal and proper sense of the word, may be defined as
a rule laid down for the guidance of an intelligent being by an in-
telligent being having power over him." — Jiirisprudence, § 2. Again
he says : " Every law or rule (taken with the largest signification
which can be given to the term properly) is a command." — Id., § 19.

* Bishop Hooker in Ecclesiastical Polity, after his famous saying of
Law, that " her seat is the bosom of God, and her voice is the har-

NATURAL LAW 181

By another very common confusion of thought,
laws of both kinds are often spoken of as though
they were themselves efficient agents. We hear of
the restraint of civil law, and of its compulsive
power. As mere metonymy this may be allowed ;
but with many who speak thus, it is not figurative, but
literal. Hence it needs to be pointed out that, while
the police and the jailor exert force and are causes,
the law which they execute does nothing beyond serv-
ing as a mandatory guide. Laws do not govern or
regulate men ; men regulate themselves, or a gov-
ernor rules them, according to law.' So, likewise,
natural laws are often confused with causes. They
relate to energy, force, cause, but are in themselves
impotent. It is true of causes, but not of laws, that
they counteract or interfere with one another, and can

mony of the world," complains that men are less subservient to the
divine order than are things. Montesquieu, in " L'Esprit des Lois," de-
claims on the stricter obedience, throughout the universe, of material
things to the laws of nature than of mankind to the divine and human
laws laid down foi" their conduct.

" The confusion of Law, in the judicial sense, with Law as a uni-
formity of nature," says Mr. Bain, "is exemplified in Butler's chapter
on the Moral Government of God [^Analogy^ etc., pt. i., ch. 3]. But-
ler calls the 'course of Nature ' a government merely on the ground
that it induces precautions to avoid pain. But these precautions have
nothing moral in them ; they may be used for criminal ends. Guy
Fawkes most faithfully obeyed [?] the laws of nature when he
placed his barrels of gunpowder so as to insure the blowing up of
Parliament, while he arranged for firing them in safety to himself."
— Logic, bk. vi., ch. 4.

^ We note, however, that enacted law inclines law-abiding subjects
to observance ; also that, as merely contemplated, it is an efficient ed-
ucator.

182 ELEMENTS OF INDUCTIVE LOGIC

be adjusted to an end. As a planet does not obey
law, so it is not governed by law, nor even according
to law as men are. Nature, amidst its apparently
unsettled vacillating diversities, is characterized by
certain established unvarying uniformities, which
natural laws merely record/

§ 93. Natural law is the product of observation.
It indicatively affirms an order of natural facts to
be universal — that is, to occur with invariable uni-
formity. Natural laws are of two kinds: primary or
ultimate, and secondary or derivative. The latter
kind is subdivided into rational and empirical." The

* Mr. Mill says : " In minds not habituated to accurate thinking,
there is often a confused notion that the general laws are the causes
of the partial ones ; that the law of general gravitation, for example,
causes the phenomenon of the fall of bodies to the earth. But to as-
sert this would be a misuse of the word cause ; terrestrial gravity is
not an effect of general gravitation, but a case of it ; that is, one kind
of the particular instances in which that general law obtains." — Logic^
p. 338. Notwithstanding this excellent statement, he uses the term
law in the sense of cause many hundred times.

The Duke of Argyll says : " Every Law of Nature is liable to coun-
teraction ; and the rule is that laws are habitually made to counteract
each other." — Reign of Law, ch. ii. (p. 100, Am. ed.). In many places
he confuses force with law; e.g., "Force ascertained according to
some measure of its operation, is one of the definitions of a scientific
Law." — 7c?., p. '71. Again : " No one Law — that is to say, no one
Force — determines anything." — /c?., p. 76.

^ The secondary or derivative laws are the axiomata media of
Bacon. The terms rational and empirical, marking the subdivision,
are not clearly distinctive, are not in proper opposition ; but good
usage sanctions this specific application of them, and we have none
better at hand.

NATURAL LAW 183

empirical are those of succession and those of coex-
istence (§ 33). We shall discuss these several kinds
in reverse order, the order of inductive generaliza-
tion.

§ 94. A uniformity of coexistence, an order of facts
observed to be simultaneous, to be associated in all
cases in wide observation without exception, is rec-
ognized as empirical law. Such is the uniform co-
existence of inertia and gravity in all bodies. These
two properties seem to be entirely independent of
each other, and yet are conjoined through all nature,
and are proportional in amount. Likewise, body and
mind coexist in all men.

In natural kinds are found many coinhering attri-
butes which are cases of uniform coexistence, and so
reducible to law; as. All animals have a nervous
superadded to a digestive system. Tlie group of
coexisting attributes marking a natural kind consti-
tutes the law of that kind as expressed in its com-
plete definition. Each being essential, if any one be
absent, we have a different kind. Thus a specific
weight of 19.3 is essential to gold ; if a metal were
found having all the other marks of gold with a
different specific weight, it would not be gold ; it
would be a new kind, with a different law.

Sometimes an accidental mark is so persistent aa
to furnish a quasi -law. Colors, for example, are
often quite constant, as that of melted or polished
gold and silver, of oak and pine leaves, of crows, and
even of men. Negroes are black, Indians are red.

184 ELEMENTS OF INDUCTIVE LOGIC

Such coexistences in many cases are properties, and
sharply characteristic, as risibility in man (§ 15).
Hence they may serve in a quasi -definition ; as, A
dog is a digitigrade quadrujpedy having fixed claws,
four toes, arid a recurved tail. But such general-
ities, however true, can rarely claim the dignity of
law.

The only method applicable to ascertain a law of
coexistence is enumeration (§ 37 sq.). Hence such
laws are attended by all the hazard and imperfection
belonging to that method, and their statements, in-
cluding definitions of kinds, often undergo modifica-
tions from wider experience/

* There is in nature a large class of coexistences, commonly spoken
of as primeval or primordial facts or original agents, which are re-
garded as ultimate, and beyond explanation or reduction to law. The
sun, as to its existence, size, gravitating force, etc., the earth, the
planets, with their various constituents of air, water, rocks, and other
distinguishable substances, simple and compound, both as to quantity
and quality, of which these various bodies are made up, are primor-
dial facts. The nebular hypothesis of Kant and Laplace seeks to go
beyond their known statm, and to explain broadly their origin. But
so long as we can give no satisfactory account of their origin, of
their distribution in space, of their relative quantities, they are pro-
visionally classed as primeval coexisting natural agents. Their dis-
tribution and relative quantities are so irregular as to seem casual
and lawless. They are mere collocations, and mere collocations
cannot be reduced to any law. Hence, what we know of them fur-
nishes no ground for an induction respecting the distribution and
quantities of similar bodies in remoter space. They are permanent
causes in nature as it is, but are themselves without assignable cause.
As the truths of pure reason are the ultimate basis of the laws of
thought, so in a sense are these permanent causes the ultimate basis
of the laws of things ; in the one case we cannot assign a reason, in
the other, a cause.

NATURAL LAW 185

§ 95. An empirical law in general is a secondary
or derivative law, the derivation of which is not yet
known. It is an ascertained uniformity attributed to
causation, and hence presumed to be resolvable into
simpler laws, but not yet resolved. It is not origi-
nal, and remains to be accounted for.

Empirical laws are inductions by the methods of
enumeration or agreement, by which methods alone
causation cannot be proved. Indeed, almost all
truths obtained by simple observation, including laws
of coexistence, are to be regarded as empirical, and
the hazard that attends them is such that scientists
hesitate to rely upon them in cases varying much
from those actually observed.

Laws of succession yet empirical are: The local
laws of tides ; Eed sunset betokens fair weather ;
Breeds are improved by crossing; Boiling tempera-
ture destroys animal life; An alloy is harder than
its components ; The number of atoms of acid neu-
tralizing an atom of base is equal to the number
of atoms of oxygen in the base. Harvey's law,
Omne vivum ex ovo, is empirical. So also is the law
of continuity, JVatura non agit per saltum, which is
illustrated in the continuity of animal and vegetable
life, and in general by the transition of matter
from one state into another, as in melting, boiling,
and their opposites. The attempt to fill apparent
gaps in nature's continuity stated in this law has
led to important discoveries, having the character of
verifications (§ 87), but the law is unproved, unex-
plained, and so empirical. True, the development

186 ELEMENTS OF INDUCTIVE LOGIC

hypothesis offers a partial explanation, which, how-
ever, is merely hypothetical (§ 85).'
' The medical sciences furnish good illustrations.
Anatomy is strictly empirical, since it is concerned
wholly with the manner of the distribution of the
various parts of the organism. Physiology, which
is concerned with the functions of these parts,
has made some progress towards rational explana-
tion, but, owing to the vast complexity of the sub-
ject, its advance is slow and hesitating. Pathology
only quite recently has given promise of passing
successfully, through hypothesis, from the empirical
to the rational stage. The old humoral hypothesis
of Galen, and the solidist hypothesis of Hoffman
and Cullen, were long rivals as explanations of dis-
ease (§*86). Both are now superseded by the germ
hypothesis, which bids fair to become established
theory (§ 88). Infectious diseases are attributed to
bacteria. The specific bacillus of tuberculosis, of

* The following Laws of the Reflection of Light are empirical :

I. The angle of reflection is equal to the angle of incidence.

II. The incident and the reflected ray are both in the same plane,
which is perpendicular to the reflecting surface.

Also Descartes' Laws of Single Refraction, as follow :

I. Whatever the obliquity of the incident ray, the ratio which the
sine of the incident angle bears to the sine of the angle of refraction
is constant for the same two media, but varies with different media.

II. The incident and the refracted ray are in the same plane which
is perpendicular to the surface separating the two media. — Ganot,
MemenU de Physique, §§ 440, 461.

These Laws of Refraction have received a fitting explanation on the
undulatory hypothesis, but it is merely hypothetical (§ 84).

NATUEAL LAW 187

cholera, of diphtheria, of typhoid fever, and others,
have been isolated, and numerous experiments
tried, with the result that no one now thinks
of humor or of disorganized tissue as the cause
of disease, but that such or such a bacillus has in-
vaded the body, and caused a specific disorder.
Therapeutics lingers in the rear. There is some
rational hygienic or constitutional treatment, but
the use of drugs is almost exclusively empirical —
their modus operandi can rarely be explained. That
quinine checks fever, that table-salt checks hemor-
rhage, are empirical facts inductively generalized.
They are doubtless derivative from some higher uni-
formities, but as yet are unexplained. Indeed, ther-
apeutics is so largely empirical that it can hardly be
deemed scientific, but is rather an art having a body
of narrow and precarious rules to guide the practi-
tioner, rules for which no aprioric reason can be as-
signed, and of which it can only be said that their ob-
servance has been remedial in similar cases. Hence
the hesitation of wise physicians, their careful, tenta-
tive, watchful procedure with each new patient.

§ 96. By the term rational law in this connection
is meant merely law that can be deductively derived
from more general laws, or, in other words, that can
be resolved into primary laws. The derived law is
thereby rationally explained.

Thus the distribution of land and water, the strat-
ification of the earth's crust, the occurrence of
heavy metals in deep mines, of corrosible metals in

188 ELEMENTS OF INDUCTIVE LOGIC

combination, of the non-corrosible, as gold and plat-
inum, in a pure state — all are cases of evident causa-
tion, and are referable to more general laws.

In the progress of knowledge it not infrequently
happens, as already intimated, that what was once
merely an empirical law is resolved into well-ascer-
tained uniformities of wider scope, and thus becomes
a rational law. The presence of snow on high moun-
tains was at one time only an empirical uniformity,
but we now resolve it into the laws of radiant heat,
and of condensation and freezing of vapor. Pre-
vious to the discovery of the pressure of the atmos-
phere, the rise of water under the action of a pump,
and the standing height of mercury in the Torricel-
lian tube, were known only as narrow empirical gen-
eralties. Now they are conjointly explained by
reference to their common cause — atmospheric press-
ure— acting in accord with Pascal's more general law
of pressure, which law, in turn, is deducible from
the still more general laws of fluidity and gravity.^

' The following is Pascal's Law of Liquid Pressure :

Pressure exerted anywhere upon a mass of liquid is transmitted
undiminished in all directions, and acts with the same force on all
equal surfaces, and in a direction at right angles to those surfaces.

Also the Laws of the Equilibrium of Floating Bodies are neat ex-
amples of rational derivative laws, as follow;

L The floating body must displace a volume of liquid whose weight
equals that of the body.

IL The centre of gravity of the floating body must be in the same
vertical line with that of the fluid displaced.

III. The equilibrium of a floating body is stable or unstable ac-
cording as the metacentre is above or below the centre of gravity. —
Ganot, Elements de Physique, §§ 89, 106.

NATURAL LAW 189

The periodical return of eclipses, as known to the
Chaldean astrologers, was an empirical law, until the
general laws of the celestial motions accounted for
it. Kepler's laws, as established by him, were mere-
ly empirical generalizations (§ 10 n.). They ceased
so to be, and became rational derivative laws when
Newton deduced them from the three laws of mo-
tion (§ 89).

Rational derivative laws are very often condi-
tioned for realization upon specific collocations of
primeval agents (§ 94 n.). The uniformity, though
invariable while the agents coexist, would cease to
be should that coexistence cease.' The orderly suc-
cession of day and night, the round of the seasons,
the ebb and flow of the sea, are dependent on the
earth's diurnal rotation, the inclination of its equator
to the ecliptic, and the relative position of earth, sun,
and moon. So long as these collocations, of which
no account can be given, are maintained, the uni-
formities result, and are rationally derivative, from
the laws of motion and of gravity. We can calculate
on finding such sequences only where we know by
direct evidence that the agents on which they depend
are present and fulfil the requisite conditions. The

^ " Derivative laws do not depend solely on the ultimate laws into
which they are resolvable ; they mostly depend on those ultimate
laws, and an ultimate fact ; namely, the mode of coexistence of some
of the component elements of the universe [§ 94, note]. The ultimate
laws of causation might be the same as at present, and yet the deriva-
tive laws completely different, if the causes coexisted in different pro-
portions, or with any difference in those of their relations by which
the effects are influenced." — Mill, Logic, p. 36Y.

190 ELEMENTS OF INDUCTIVE LOGIC

law that coal lies above red sandstone holds through-
out the earth, but cannot be applied to other planets.
The quantity and distribution of water on our globe
cannot be assigned to any other ; but the proportion
of oxygen and hydrogen in water is referable to the
ascertained universal laws of affinity or chemical com-
bination, and hence may be safely affirmed wher-
ever in the universe they unite. The coexistence
in a definite proportion of oxygen and nitrogen in
our atmosphere cannot be predicated of any other
atmosphere ; but their uniform intermixture, wher-
ever they occur, may be predicated, for the law of the
diffusion of gases is a universal natural lawy

§ 97. Something needs to be said in this connec-
tion about explanation. First, let us ask what is
meant by a mystery, a marvel, a curiosity, an unac-
countable fact, a strange event, an extraordinary phe-
nomenon. It means simply an isolated fact, one not
standing in any known order of things, not referable
to a class, or a cause, or a law, and hence exciting curi-
osity and wonder; as the zodiacal light, the aurora
borealis. Likewise a comet is not referable, perhaps,
to any narrower class than cosmical body, which refer-
ence is so far from being satisfactory that we still say
it is a curious thing. Why is its coma always turned
from the sun ? The fact is strange, wonderful, unac-
countable. Familiarity with an isolated fact will abate
emotion, still an explanation is always acceptable.*

' " It is a common illusion to regard phenomena as simple because

NATURAL LAW 191

A fact, then, either particular or general, is said to
be explained when it is assigned to a well-known
class of things, or when its cause is ascertained, or
when the law or laws of causation, of which it is an
instance, are indicated. I pick up a brilliant stone,
and am told it is a crystal of quartz ; a fire destroys a
dwelling, because a lamp was overturned ; a balloon
ascends, for the surrounding air, being heavier, push-
es it upward, in accord with the law of gravitating
fluids. These facts are thus explained, at least par-
tially. So also a law or uniformity of nature is said
to be explained when another law or laws are point-
ed out of which the law in question is a case, and
from which it could be deduced, into which it could
be resolved. An explanation very often is provi-
sionally merely hypothetical, reducible perhaps to the-
ory by subsequent proof, but commonly we have to
be content with a plausible supposition (§§ 78, 79).
Explanation, then, in a philosophical sense, is the
reference of a fact to its class, cause, or law ; or else
the resolution of an empirical uniformity into laws
of causation, real or hypothetical, from which it
logically results, or the resolving a complex law of

they are familiar. Very familiar facts seem to stand in no need of
explanation themselves, and to be the means of explaining whatever
can be assimilated to them. Thus the boiling and evaporation of a
liquid is supposed to be a very simple phenomenon requiring no
explanation, and a satisfactory medium of the explanation of rarer
phenomena. That water should dry up is, to the uninstructed mind,
a thing wholly intelligible; whereas, to the man acquainted with
physical science, the liquid state is anomalous and inexplicable." —
Bain, Logic, bk. iii., ch. 12, § 10.

192 ELEMENTS OF INDUCTIVE LOGIC

causation into simpler and more general ones from
which it is capable of being deductively inferred.'

Let it be remarked that, after all, explanation is
merely substituting one mystery for another. It
does nothing to render the general course of nature
other than mysterious ; for the highest ambition of
natural science and its loftiest reach is to attain to
primordial agents, and to such ultimate laws as are
incapable of physical explanation, and only more
mysterious because of their wider comprehension.
Natural theology with teleology, assuming the su-
pernatural, carries the explanation still further, but

^ In loose and general expression, to account for or explain any-
thing is to connect it with known things. The connection, real or hy-
pothetical, is either by similarity or by causation. We bring other
things to stand under it, and so it becomes understood by means of
them. The quasi-definition a posteriori (§ 38) in most of its forms
is merely an explanation. Says Lotze : " To explain means nothing
more than to show that a definite event is the result of its antecedents
in accordance with general rules." — Grundzuge der Fraktischen Phi-
losophies § 20.

" Scientific explanation and inductive generalization, being the same
thing, the limits of explanation are the limits of induction. The
limits to inductive generalization are the limits to the agreement or
community of facts. . . . Newton seemed unable to acquiesce in
gravity as an ultimate fact. It was inconceivable to him that mat-
ter should act upon other matter at a distance, and he therefore
desired a medium of operation, whereby gravity might be assimilated
to impact. But this assimilation has hitherto been impracticable;
if so, gravity is an ultimate fact, and its own sufficing and final ex-
planation. The acceptance of this is the proper scientific attitude of
mind. . . . We are utterly ignorant how matter and mind operate on
each other. Properly speaking, there is nothing to be known but the
fact, generalized to the utmost."— Bain, Logic^ bk. iii., ch. 12, §§ 6, 11.
See " Psychology," § 122, note.

NATURAL LAW 193

with like termination in the great mystery of mys-
teries.'

§ 98. Passing now to the class of natural laws
marked as primary or ultimate, we observe that
these are called, 'par excellence^ Laws of Nature, a
title that in usage is denied to the secondary or
derivative laws. How shall they be described so as
to distinguish them within the comprehending class
of natural laws ? First, they are free from the con-
dition, to which so many derivative laws are sub-
jected, of a special collocation of primeval agents
(§ 96). Secondly, they are the fewest and simplest

^ Dr. Whewell, in Nov. Org. Renov.., bk. iii., ch. 10, § Y, says, very
beautifully, of the Supreme Cause : " In the utterance of Science, no
cadence is heard with which the human mind can feel satisfied. Yet
we cannot but go on listening for and expecting a satisfactory close.
The notion of a cadence appears to be essential to our relish of the
music. The idea of some closing strain seems to lurk among our own
thoughts, waiting to be articulated in the notes which flow from the
knowledge of external nature. The idea of something ultimate in
our philosophical researches, something in which the mind can acqui-
esce, and which will leave us no further questions to ask, of whence
and why, and by what power^ seems as if it belonged to us, as if we
could not have it withheld from us by any imperfection or incomplete-
ness in the actual performances of science. What is the meaning of
this conviction ? What is the reality thus anticipated ? Whither does
the development of this Idea conduct us ?

"We have already seen that a difficulty of the same kind, which
arises in the contemplation of causes and effects considered as form-
ing an historical series, drives us to the assumption of a First Cause,
as an axiom to which our idea of causation in time necessarily leads.
And as we were thus guided to a First Cause in order of Succession,
the same kind of necessity directs us to a Supreme Cause in order of
Causation."
13

194 ELEMENTS OF INDUCTIVE LOGIC

general truths from which the multifarious uniform-
ities in nature may be deductively inferred, or those
widest inductions which, being granted, will account
for the existing order of nature. Accordingly, they
are reckoned as primary or ultimate — that is, original
and underived. But let us not be misled by these
expressions to understand that science claims to have
reached this high ideal. Since we are continually
discovering that uniformities, previously considered
ultimate, are derivative, resolvable into more general
laws, we cannot be sure that any of the recognized
laws of nature are strictly ultimate, though well as-
sured that there must be ultimate laws, and that
every such resolution brings us nearer to them.
Thus the laws of magnetic agency having been af-
filiated with the laws of electric action, both have
ever since been considered as special cases referable
to more general laws of electricity.

The three Laws of Motion (§ 18 n.) may be cited
as notable examples of laws of nature, their great
simplicity and wide comprehension rendering a fur-
ther reduction hardly possible. This high rank is
sustained by a special characteristic which is worthy
of remark. Whatever may have been the actual
logical process by which their discoverer evolved
them (§ 72), now that we have them they are seen to
be true ajpriori. As soon as their terms are clearly
understood, they are accepted as necessarily and uni-
versally true. They approach very nearly the char-
acter of formal laws (§ 91). Although not entirely
pure, not wholly free from empirical matter, yet

NATURAL LAW 195

they are so liigHly abstract that they deal rather
with mathematical ideas than with median ical facts.
Like the simpler theorems of geometry, they are so
directly deducible from pure axioms, combined with
the simple empirical facts of motion, change, and
force, that even a priori proof is needless, and they
are posited as the axioms of mechanics. Though
not strictly self-evident, they are evidently and ab-
solutely true, wliich means, not merely that no ex-
ception is possible, but also that no exception is
conceivable. This puts them above the plane of in-
ductive truth, whose highest reach is empirical cer-
tainty.'

The most illustrious example of a law of nature
is the Law of Universal Gravitation, the culmination
of Newton's research (§§ 79, 89). Its. statement is:
Every body of matter in the universe tends tow-
ards every other with a force that is directly as its
mass, and inversely as the square of the distance.
Consider for a moment the great number and variety
of special uniformities, both particular cases and con-
sequences, which are accounted for by this very sim-
ple and universal law of nature. The single fact of
a tendency of every particle towards every other,
varying inversely as the distance squared, explains
the fall of bodies to the ground, the revolutions of

' See §§ 7, 45. Newton's own title for these laws is Axiomata nve
Leges Motus. The laws of motion and the moral law (§ 92) are
strikingly similar in respect of this characteristic — that both may be
inductively evolved, and both are intuitively true.

196 ELEMENTS OF INDUCTIVE LOGIC

the planets and their satellites, the motion of comets,
and all the various regularities that have been ob-
served in these special phenomena, such as the ellip-
tical orbits, and the variations from exact ellipses
known as perturbations, the relation between the
solar distances of the planets and the periodic times
of their revolutions, the precession of the equinoxes,
the tidal motions, and a vast number of minor as-
tronomical and terrestrial truths.

The discovery of the universal Laws of Energy
marks an important epoch in modern science.' It ac-
complished not only a unification of many branches
of physics previously regarded as distinct, but also

* See § 17. The following are the Laws of Energy:

I. Transfer of Energy. — Energy may be transferred from one
body to another, but only by work done between them and to the ex-
tent of the work done.

II. Transformation of Energy. — Energy may be transformed (with
or without transfer) from kinetic to potential or from potential to
kinetic, or from some variety of one to a different variety of either,
but only by work and to the extent of the work done.

III. Degradation of Energy. — The quantity of energy that in any
operation takes the form of heat, is said to be dissipated. This law
is often called the law of dissipation of energy.

rV. Conservation op Energy. — In any system or collection of
bodies, the sum total of energy is not altered by the transfers and
transformations taking place between the members of the system
themselves. That sum total can be altered only by exchanges be-
tween these members and other bodies not belonging to the system.
Energy is not altered in amount by transfer or transformation. The
mutual actions of natural bodies neither create nor destroy energy.
What one body gains, some other body loses.

This statement of the Laws of Energy is taken from Outlines of
Physics (part ii., §§ 21, 23, 30, 31), by Professor F. H. Smith, LL.D.,
of the University of Virginia.

NATURAL LAW 197

has explained for the first time a multitude of spe-
cial phenomena in each branch, and by prediction
has led to lines of new research resulting in many
brilliant discoveries.

The Laws of Chemical Combination, from which
the whole science of chemistry is derived, are very
simple and very wide generalizations, which, being
regarded provisionally as ultimate, rank as laws of
nature.^

§ 99. The great object of the scientist is to obtain
by rigid induction the laws of nature, and to follow
them by rigid deduction to their consequences. A
science at first wholly inductive becomes, as soon as
a law has been proved, more or less deductive, and
as it progresses, rising to higher and wider but fewer
inductions, the deductive processes increase in num-
ber and importance, until it is no longer properly

^ The Laws of Chemical Combination are as follow :

I. Definite Proportions. — In every chemical compound the nature
and the proportions of its constituent elements are fixed, definite, and
invariable.

II. Multiple Proportions. — If two elements, A and B, unite to-
gether in more proportions than one, on comparing together quanti-
ties of the different compounds, each of which contains the same
amount of A, the quantities of B will bear a very simple relation to
each other.

III. Equivalent Proportions. — Each elementary substance, in com-
bining with other elements, or in displacing others from their combi-
nations, does so in a fixed proportion, which may be represented nu-
merically.

These laws are taken from Miller's Elenients of Chemistry^ part i.,
Chemical Physics, §§ 9, 10, 11.

198 ELEMENTS OF INDUCTIVE LOGIC

an inductive, but a deductive science. Thus hydro-
statics, acoustics, optics, and electricity, commonly
called inductive sciences, have passed under the do-
minion of mathematics, and mechanics in general
has a like history (§ 73). Celestial mechanics as
founded in the " Principia " of Newton is mainly in-
ductive, as elaborated in the " M^canique Celeste" of
Laplace is mainly deductive. By pursuing this lat-
ter process it has multiplied its matter, and reached
its present high perfection. A revolution is quiet-
ly progressing in all the natural sciences. Bacon
changed their method from deductive to inductive,
and it is now rapidly reverting from inductive to de-
ductive. The task of logic is to explicate and regu-
late these methods.*

' Bacon, in Disiributio Operis, 6th paragraph, and in Nov. Org.,
bk. i., aph. 11 sq., speaks disparagingly of the syllogism. The chief
aim of his Instauratio is to forbid the saltus, usual in previous science,
from a simple enumeration of particulars at once to the widest gener-
alities, and to require a graduated procedure. In aph. 19, he says :
*' There are and can be but two ways of investigating and discover-
ing truth. The one hurries on rapidly from the senses and particu-
lars to the most general laws ; and from them as principles and their
supposed indisputable truth derives and discovers the intermediate
laws [axiomata media]. The other constructs its laws from the
senses and particulars by ascending continuously and gradually till
it finally arrives at the most general laws, which is the true but un-
attempted way." In aph. 22, he adds : " Each of these two ways be-
gins from the senses and particulars, and ends in the greatest gener-
alities. But they are immensely different ; for the one merely touches
cursorily on particulars and experiment, whilst the other runs duly
and regularly through them ; the one, from the very outset, lays down
some abstract and useless generalities, the other gradually rises to
such as are naturally better fitted to be the object of knowledge."
Cf. aph. 104, and see the quotation in our § 40, note.

NATURAL LAW 199

§ 100. Unity, says Plato, is the end of philosophy.
It is a fair question whether the laws of nature may
not, in the advance of knowledge, be resolved into
some one all-comprehensive law, thus attaining the
philosophical ideal. In considering this, let it be
observed that all scientific investigation of natural
facts and laws is in order to obtain a philosophical
explanation of phenomena. J^ow, a phenomenon is
that which appears to an observer (§ 33). The word,
therefore, is a relative term, the name of a relation
between a natural fact and a percipient intelligence.
It follows that phenomena may be ultimately reduci-
ble to as many kinds as there are kinds of sense-per-
ception, but that they cannot be reduced to any
fewer kinds than the number of sense-perceptions
that are distinct or irreducible to one another. There-
fore, the ultimate laws of. nature are necessarily as

The limitations of human knowledge and power are indicated
in aphorisms 1-10. These the closing passage of Dist. Op. antici-
pates, saying : " Man, the minister and interpreter of nature, does
and understands as much as he has observed of the order, operation,
and mind of nature, and neither knows nor is able to do more. Neither
is it possible for any power to loosen or burst the chain of causes, nor
is nature to be overcome except by submission. Therefore these two
objects, human knowledge and power, are really the same ; and failure
in action chiefly arises from the ignorance of causes. For everything
depends on our fixing the mind's eye steadily in order to receive
their images exactly as they exist, and may God never permit us to
give out the dream of our fancy as a model of the world, but rather
in his kindness vouchsafe to us the means of writing a revelation and
true vision of the traces and stamps of the Creator on his creatures."
Then follows a Prayer which the present writer humbly makes his
own.

200 ELEMENTS OF INDUCTIVE LOGIC

many at least as the distinct kinds of perception, and
can never be reduced to one comprehensive law.

In illustration of this we note that the perception
of color is radically distinct from the perception of
sound. True, tliey are strikingly similar in several
respects, especially in their causes, both being pro-
duced by molecular vibration. But this reduction is
only apparent, for these causes, as well as their laws,
are themselves irreducibly distinct. Hence there
must always be a law connecting molecular motion
with color, and another law connecting molecular
motion with sound. Moreover, color and sound are
effects intrinsically and essentially unlike, and since
unlike effects have unlike causes (§ 23), these phe-
nomena can never be referred to causes strictly
alike, or to a common cause or law. Heat, light, and
electricity are convertible forms of energy, but
essentially distinct in their laws, because their sev-
eral phenomena are presented to distinct modes of
perception. The great generalizations of force pro-
ducing molar motion, as the laws of motion and
gravity, are all referable ultimately to muscular
sense-perception, which stands distinctly and irre-
ducibly apart from the phenomena of the other
senses. Thus it is that the ultimate laws of nature
cannot be less numerous than the ultimate powers of
perception.

INDEX

{The number refers to the page.)

Accidents, induction only of, 9.
Agent and patient, 25 n.
Agreement, canon of, llY.

— imperfections of, 122.

— yields probability, 124.

— double method of, 125.
Analogy defined, 62, 69.

— canon of, 69.

— justification of, '71.

— scientific value of, 73.
Analysis of the notion law, 177.
Analytic forms distinguished, 8.
Antecedents and consequents, 24,

— distribution of, 57 n., 121.
Approximate generalization,8 1,96.
Argyll on law of nature, 182 n.
Aristotle's view of induction, 6 n.

— four causes, 23 n.

— formula of induction, 44,

— view of analogy, 68.

— of induction vs. deduction, 142.
Axiom of change, 29.

— of uniformity, first, 31,

— of uniformity, second, 35.

— of sufficient reason, 85,
Axioms, their origin, 30,

— Mill's view criticised, 31 n.,67 n.

Bacon on induction, 6 n.

— on enumeration, 62, 67 n.

— on elimination, 104 n.

— his Organon, 142 n., 198 n.

— on crucial instances, 168 n.

— axiomata media, 182 n,, 198 n.

— on modes of research, 198 n.

— knowledge and power, 199 n,

14

Bain, definition of induction, 7 n.

— on Butler's view, 181 n.

— on familiarity, 190 n.

— on explanation, 192 n.
Butler, analogical argument, 74.

— on probability, 78.

— criticised by Bain, 181 n.

Canon of enumeration of cases, 63.

— of analogy, 69,

— of probability, 89.

— of perfect induction, 103.

— of difference, 106,

— of residue, 113,

— of agreement, 117.

— of double agreement, 126.

— of concomitant variations, 131.

— of deduction, direct, 150,
Causation, definition of, 28.

— intuitional view of, 30,
-^ empirical view of, 31 n.

— canons of, 103, 105 n.
Cause, investigation of, 19,

— its general meaning, 22.

— various kinds of, 23 n.

— definition of, 27,

— preventive, 25, 81, 109 n.

— vs. law, 181,

— the Supreme, Whewell on, 193 n.
Causes, plurality of, maxim, 37.
Certainty, strict, 76.

— empirical, 77, 83 n.
Chance, its meanings, 29, 82,

— defined ; the problem of, 84.

— Laplace's rule, 86.

— first rule of, 86.

202

INDEX

Chance, second rule of, 8*7.

— canon for distinguishing, 89.
Coexistence, phenomena of, 54.

— laws of, 183.

— of collocations, 184 n., 189.
Colligation, 16, 17.
Collocations, 184 n., 189.
Composition of causes, 147, 149.
Comte on use of hypothesis, 162 n.
Concomitant variations, 130.

— canon of, 131.

— illustrations of, 182.

— quantitative value, 135, 137.
Condition, causal, 24, 121.
Crucial instances, 168.

Darwin's hypothesis, 1 66.
Deduction distinguished, 5.

— its relation to induction, 141.

— authorities quoted on, 142 n.

— related to discovery, 144.

— direct method of, 150.

— canon of, 150.

— three stages of, 151.

— Newton's use of, 153.

— indirect method of, 160.

— three stages of, 161.

— conditions of proof, 171.

— Newton's use of, 174.
Definition of logic, 1.

— of inference, 4.

— of induction, 6, 7 n.

— of cause and of effect, 27.

— of causation, 28.

— of phenomenon, 54.

— of observation, 55.

— of analogy, 62, 69.

— of hypothesis, 162.

— of law, 177.

De8carte8,laws of refraction, 186 n.
Development hypothesis, 166.

— Mill quoted on, 166 n.

— Spencer quoted on, 167 n.
Dew, Wells's theory of, 21, 128.
Difference, canon of, 106.

— applications of, 107.

— proof of hypothesis, 173, 17^.
Discovery by deduction, 144.
Distribution of natural law, 182.
Double agreement, canon of, 126.

Double agreement, 127, 128.

Effect, definition of, 27.
Effects, plurality of, maxim, 34.

— heterogeneous, 146.

— homogeneous, 147.

Efficient cause distinguished, 23 n.
Elimination, 25, 57, 92, 117, 121.

— Bacon on, 104 n.
Empirical truth, 10.

— view of causation, 31 n.

— certainty, 77, 83 n.

— laws of coexistence, 183.

— laws of succession, 185.

— laws becoming rational, 188.
Empiricism of Mill, 31 n., 67 n.

— of medical science, 186.
Energy, conservation of, 28, 196 n.
Enumeration, divided, 62.

— of cases, canon of, 63.

— value of, 66.

— Mill and Bacon on, 67 n.

— radical defect of, 102.

— of marks, canon of, 69.

— value of, 73.
Exceptions, 14, 80, 91, 95, 168.
Experience, inference from, 10, 54.
Experimental observation, 56, 107.
Explanation, philosophical, 190.

Familiarity vs. explanation, 190 n.
Force and energy, 27.
Formal character of logic, 1.

— of law, 178.

Forms, function of, 50, 107 n.

Generalization of induction, 5.

— from experience, 10.

— beyond experience, 14.

— within experience, 15.

— approximate, 81, 96.
Gravitation, universal, 175, 195.

Hamilton on induction, 6 n.

— on syllogistic form of, 46.

— on induction vs. deduction, 143n.
Hazard, 15, 66, 73, 80, 94, 102, 139.
Herschel, on research, 105 n.

— on hazard of induction, 139 n.

— on induction vs. deduction, 142 n.

INDEX

203

Heterogeneous effects, 146.
Homogeneous effects, 147.
Hooker on obedience to law, 180 n.
Hypothesis, common use of, 155.

— Mill quoted on, 167 n.

— formal use of, 160.

— definition of, 162.

— of a vera causa, 163.

— of a vera lex, 165.

— of an ether, 164, 169, 171.

— of gases, kinetic, 165.

— of origin of species, 166.

— proof of, two steps, 171.

— of germs in disease, 173, 186.

Identification, 17, 145.
Imperfect induction, 45, 66, 102.
Induction a generalization, 5.

— definitions of, 6, 7 u.

— a synthetic process, 7.

— of accidents onlv, 9.

— exceptions, 14, 80, 91, 95, 168.

— perfect, 16, 45, 66 n., 102.

— preparation for, 20, 111, 121.

— principles of, 29.

— time not an element in, 42 n.

— an immediate inference, 43, 51.

— Aristotle's formula of, 44.

— Hamilton's syllogism, 46.

— Whately's and Mill's, 47.

— by enumeration, canon of, 63.

— by analogy, canon of, 69.

— perfect, general canon of, 103.

— quantitative, limits of, 138.

— vs. deduction, 142.

— of universal gravitation, 175.
Inductive logic formal, 1.

— sciences vs. deductive, 197.
Inference defined, 4.

— a priori and a posteriori, 12 n.

— inductive, immediate, 43, 51.
Instantia crucis, 168.
Intermixture of effects, 147, 149.
Intuitional view of causation, 30.
Intuitions, pure, distinguished, 11.

Kepler's laws, 18 n., 167, 174, 189.
Kinetic hypothesis of gases, 165.

Laplace on chance, 86.

Laplace's rule for probability, 124.
Law, definition of, 177, 178 n.

— formal and material, 178.

— moral and natural, 179, 182.

— misconceptions of, 180.

— derivative, 182, 185.

— empirical, 183, 185.

— rational, 187.
Laws of causation, 29.

— of motion, 31 n., 194.

— of light, 186 n.

— of liquid pressure, 188.

— of nature, 193.

— examples of, 194.

— ultimate number of, 199.
Leibnitz on sufiicient reason, 85 n.
Limitations of the methods, 120.

— of quantitative induction, 138.

— of knowledge and power, 199 n.

— of natural law, 199.
Logic, definition of, 1.

— formal in both laranches, 1.

— material vs. formal, 2 n.

— sole province of, 3.
Lotze on explanation, 192 n.

Mathematics, deductive, 13 n.

— applied to probabilities, 98.

— to concomitant variations, 135.

— inductions from, 138.

— in the deductive method, 152.
Metaphor vs. analogy, 68.
Method of difference, 104, 105.

— of residue, 112.

— of agreement, 104, 116.

— of double agreement, 125.

— of concomitant variations, 130.

— of deduction, 150.

Mill, material view of logic, 2 n.

— on induction vs. deduction, 5 n.

— definition of induction, 7 n.

— definition of cause, 27 n.

— empirical views of, 31 n., 67 n.

— inductive syllogism, 47 n.

— view of enumeration, 67 n.

— quoted on probability, 79, 95 n.

— canons of research, 105 n.

— on Darwm's hypothesis, 166 n.

— induction vs. deduction, 143 n.

— on law and cause, 182 n.

204

INDEX

Mill on derivative laws, 189 n.
Montesquieu, law defined, 1*78 n.

— on obedience to law, 181 n.
Moral certainty, 77, 83 n.

Natural law vs. moral law, 1*79.

— distribution of, 182.
Nature, laws of, 193.

— of motion, 31 n., 194.

— of gravitation, 195.

— of energy, 196.

— of chemical combination, 197.

— number of, ultimate, 199.
Neptune, discovery of, 20.
Newton's laws of motion, 31 n.,194.

— rules for philosophizing, 36 n.

— deductive method, 154.

— doctrine of vera causa, 163 n.

— hypothetical method, 174.

— hypothesis of light, 168.

— law of gravitation, 175, 195.

Observation, definition of, 55.

— simple, applications, 57, 107.

— experimental, 59, 109.
Order, logical vs. historical, 153 n.
Organum, as a title, 142 n.

Parcimony, law of, 36 n.
Pascal's law of pressure, 188.
Perfect induction, 16, 45, 66 n., 102.
Phenomenon, definition of, 54.

— of coexistence, 54, 183.

— of succession, 55, 185.
Philosophizing, rules for, 36 n.
Physical certainty, 77, 83 n.
Plurality of effects, 34, 123.

— of causes, 37, 122.
Prediction, power of, 170.
Preventive cause, 25, 81, 109 n.
Primeval agents, 184 n., 189.
Probability, canon of, 89.

— indefinite valuation of, 94.

— numerical valuation of, 98.

— based on statistics, 100.

— Laplace's rule for, 124.

Proof of an hypothesis, 171.
Pure logic divided, 1.

— intuitions distinguished, 1 1.

— mathematics, deductive, 13 n.

Quantitative inductions, 138.

Rational law, 182, 187.
Residue, method of, 112.

— canon of, 113.

— applications of, 114.
Rules for philosophizing, 36 n.

Sciences, quantitative, 135.

— becoming deductive, 197.
Spencer on evolution, 167 n.
Statistics, application of, 100.
Sufficient reason, axiom of, 85.
Syllogism of induction, 44.

— Hamilton's, 46.

— Whately's and Mill's, 47.

— objections to, 48.
Synthesis of induction, 7.

Theory vs. hypothesis, 165 n.

— how established, 171.

Uniformity of nature, 39.

— of coexistence, 54, 183.

— of succession, 55, 185.

Variations, concomitant, 130.

— canon of, 131.

— illustrations of, 132.

— quantitative estimates, 137.
Venn on Mill, 2 n., 27 n., 38 n.
Vera causa and lex, 163, 165.
Verification, 145, 152.

— special function of, 169.

— predictions not proof, 170.

Wells, research on dew, 21, 128.
Whately, induction defined, 6 n.

— inductive syllogism, 47.
Whewell on induction, 6 n.

— on Supreme Cause, 193 n.

FINIS

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