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Title: Logic - Deductive and Inductive
Author: Read, Carveth, 1848-1931
Language: English
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Copyright Status: Not copyrighted in the United States. If you live elsewhere check the laws of your country before downloading this ebook. See comments about copyright issues at end of book.

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First Edition, June 1898.         (Grant Richards.)
Second Edition, November 1901.    (Grant Richards.)
Third Edition, January 1906.      (A. Moring Ltd.)
Reprinted, January 1908.          (A. Moring Ltd.)
Reprinted, May 1909.              (A. Moring Ltd.)
Reprinted, July 1910.             (A. Moring Ltd.)
Reprinted, September 1911.        (A. Moring Ltd.)
Reprinted, November 1912.         (A. Moring Ltd.)
Reprinted, April 1913.            (A. Moring Ltd.)
Reprinted, May 1920.              (Simpkin.)













In this edition of my _Logic_, the text has been revised throughout,
several passages have been rewritten, and some sections added. The chief
alterations and additions occur in cc. i., v., ix., xiii., xvi., xvii.,

The work may be considered, on the whole, as attached to the school of
Mill; to whose _System of Logic_, and to Bain's _Logic_, it is deeply
indebted. Amongst the works of living writers, the _Empirical Logic_ of
Dr. Venn and the _Formal Logic_ of Dr. Keynes have given me most
assistance. To some others acknowledgments have been made as occasion

For the further study of contemporary opinion, accessible in English,
one may turn to such works as Mr. Bradley's _Principles of Logic_, Dr.
Bosanquet's _Logic; or the Morphology of Knowledge_, Prof. Hobhouse's
_Theory of Knowledge_, Jevon's _Principles of Science_, and Sigwart's
_Logic_. Ueberweg's _Logic, and History of Logical Doctrine_ is
invaluable for the history of our subject. The attitude toward Logic of
the Pragmatists or Humanists may best be studied in Dr. Schiller's
_Formal Logic_, and in Mr. Alfred Sidgwick's _Process of Argument_ and
recent _Elementary Logic_. The second part of this last work, on the
"Risks of Reasoning," gives an admirably succinct account of their
position. I agree with the Humanists that, in all argument, the
important thing to attend to is the meaning, and that the most serious
difficulties of reasoning occur in dealing with the matter reasoned
about; but I find that a pure science of relation has a necessary place
in the system of knowledge, and that the formulæ known as laws of
contradiction, syllogism and causation are useful guides in the framing
and testing of arguments and experiments concerning matters of fact.
Incisive criticism of traditionary doctrines, with some remarkable
reconstructions, may be read in Dr. Mercier's _New Logic_.

In preparing successive editions of this book, I have profited by the
comments of my friends: Mr. Thomas Whittaker, Prof. Claude Thompson, Dr.
Armitage Smith, Mr. Alfred Sidgwick, Dr. Schiller, Prof. Spearman, and
Prof. Sully, have made important suggestions; and I might have profited
more by them, if the frame of my book, or my principles, had been more

As to the present edition, useful criticisms have been received from Mr.
S.C. Dutt, of Cotton College, Assam, and from Prof. M.A. Roy, of
Midnapore; and, especially, I must heartily thank my colleague, Dr.
Wolf, for communications that have left their impress upon nearly every


_August_, 1914



PREFACE                                                                v



§1.  Definition of Logic                                               1
§2.  General character of proof                                        2
§3.  Division of the subject                                           5
§4.  Uses of Logic                                                     6
§5.  Relation of Logic to other sciences                               8
       to Mathematics (p. 8);
       to concrete Sciences (p. 10);
       to Metaphysics (p. 10);
       to regulative sciences (p. 11)
§6.  Schools of Logicians                                             11
       Relation to Psychology (p. 13)



§1.  Propositions and Sentences                                       16
§2.  Subject, Predicate and Copula                                    17
§3.  Compound Propositions                                            17
§4.  Import of Propositions                                           19
§5.  Form and Matter                                                  22
§6.  Formal and Material Logic                                        23
§7.  Symbols used in Logic                                            24



§1.  Some Account of Language necessary                               27
§2.  Logic, Grammar and Rhetoric                                      28
§3.  Words are Categorematic or Syncategorematic                      29
§4.  Terms Concrete or Abstract                                       30
§5.  Concrete Terms, Singular, General or Collective                  33



§1.  Connotation of General Names                                     37
§2.  Question of Proper Names                                         38
       other Singular Names (p. 40)
§3.  Question of Abstract Terms                                       40
§4.  Univocal and Equivocal Terms                                     41
       Connotation determined by the _suppositio_ (p. 43)
§5.  Absolute and Relative Terms                                      43
§6.  Relation of Denotation to Connotation                            46
§7.  Contradictory Terms                                              47
§8.  Positive and Negative Terms                                      50
       Infinites; Privitives; Contraries (pp. 50-51)



§1.  As to Quantity                                                   53
       Quantity of the Predicate (p. 56)
§2.  As to Quality                                                    57
       Infinite Propositions (p. 57)
§3.  A. I. E. O.                                                      58
§4.  As to Relation                                                   59
       Change of Relation (p. 60);
       Interpretation of 'either, or' (p. 63);
       Function of the hypothetical form (p. 64)
§5.  As to Modality                                                   66
§6.  Verbal and Real Propositions                                     67



§1.  Meaning of Inference                                             69
§2.  Immediate and Mediate Inference                                  70
§3.  The Laws of Thought                                              72
§4.  Identity                                                         73
§5.  Contradiction and Excluded Middle                                74
§6.  The Scope of Formal Inference                                    76



§1.  Plan of the Chapter                                              79
§2.  Subalternation                                                   79
§3.  Connotative Subalternation                                       80
§4.  Conversion                                                       82
       Reciprocality (p. 84)
§5.  Obversion                                                        85
§6.  Contrary Opposition                                              87
§7.  Contradictory Opposition                                         87
§8.  Sub-contrary Opposition                                          88
§9.  The Square of Opposition                                         89
§10. Secondary modes of Immediate Inference                           90
§11. Immediate Inferences from Conditionals                           93



§1.  Order of Terms in a proposition                                  95
§2.  Euler's Diagrams                                                 97
§3.  Propositions considered as Equations                            101
§4.  Existential Import of Propositions                              104



§1.  Nature of Mediate Inference and Syllogism                       107
§2.  General Canons of the Syllogism                                 108
       Definitions of Categorical Syllogism; Middle Term;
       Minor Term; Major Term; Minor and Major Premise (p. 109)
       Illicit Process (p. 110);
       Distribution of the Middle (p. 110);
       Negative Premises (p. 112);
       Particular Premises (p. 113)
§3.  _Dictum de omni et nullo_                                       115
§4.  Syllogism in relation to the Laws of Thought                    116
§5.  Other Kinds of Mediate Inference                                118



§1.  Illustrations of the Syllogism                                  121
§2.  Of Figures                                                      122
§3.  Of Moods                                                        123
§4.  How valid Moods are determined                                  124
§5.  Special Canons of the Four Figures                              126
§6.  Ostensive Reduction and the Mnemonic Verses                     127
§7.  Another version of the Mnemonic Verses                          132
§8.  Indirect Reduction                                              132
§9.  Uses of the several Figures                                     134
§10. Scientific Value of Reduction                                   135
§11. Euler's Diagrams for the Syllogism                              136



§1.  Popular Arguments Informal                                      138
§2.  The Enthymeme                                                   139
§3.  Monosyllogism, Polysyllogism, Prosyllogism, Episyllogism        141
§4.  The Epicheirema                                                 142
§5.  The Sorites                                                     142
§6.  The Antinomy                                                    145



§1.  The Hypothetical Syllogism                                      147
§2.  The Disjunctive Syllogism                                       152
§3.  The Dilemma                                                     154



§1.  Formal Consistency and Material Truth                           159
§2.  Real General Propositions assert more than has been
       directly observed                                             160
§3.  Hence, formally, a Syllogism's Premises seem to beg the
       Conclusion                                                    162
§4.  Materially, a Syllogism turns upon the resemblance of the
       Minor to the Middle Term;   and thus extends the
       Major Premise to new cases                                    163
§5.  Restatement of the _Dictum_ for material reasoning              165
§6.  Uses of the Syllogism                                           167
§7.  Analysis of the Uniformity of Nature, considered as the
       formal ground of all reasoning                                169
§8.  Grounds of our belief in Uniformity                             173



§1.  The most important aspect of Uniformity in relation to
       Induction is Causation                                        174
§2.  Definition of "Cause" explained: five marks of Causation        175
§3.  How strictly the conception of Cause can be applied
       depends upon the subject under investigation                  183
§4.  Scientific conception of Effect.    Plurality of Causes         185
§5.  Some condition, but not the whole cause, may long precede
       the Effect; and some co-effect, but not the whole effect,
       may long survive the Cause                                    187
§6.  Mechanical Causes and the homogeneous Intermixture of Effects;
       Chemical Causes and the heteropathic Intermixture of Effects  188
§7.  Tendency, Resultant, Counteraction, Elimination, Resolution,
       Analysis, Reciprocity                                         189



§1.  Outline of Inductive investigation                              192
§2.  Induction defined                                               196
§3.  "Perfect Induction"                                             196
§4.  Imperfect Induction methodical or immethodical                  197
§5.  Observation and Experiment, the material ground of
       Induction, compared                                           198
§6.  The principle of Causation is the formal ground of Induction    201
§7.  The Inductive Canons are derived from the principle of
       Causation, the more readily to detect it in facts observed    202



§1.  The Canon of Agreement                                          206
       Negative Instances (p. 208);
       Plurality of Causes (p. 208)
       Agreement may show connection without direct Causation (p. 209)
§2.  The Canon of Agreement in Presence and in Absence               212
       It tends to disprove a Plurality of Causes (p. 213)
§3.  The Canon of Difference                                         216
       May be applied to observations (p. 221)
§4.  The Canon of Variations                                         222
       How related to Agreement and Difference (p. 222);
       The Graphic Method (p. 227);
       Critical points (p. 230);
       Progressive effects (p. 231);
       Gradations (p. 231)
§5.  The Canon of Residues                                           232



§1.  Deductive character of Formal Induction                         236
§2.  Further complication of Deduction with Induction                238
§3.  The Direct Deductive (or Physical) Method                       240
§4.  Opportunities of Error in the Physical Method                   243
§5.  The Inverse Deductive (or Historical) Method                    246
§6.  Precautions in using the Historical Method                      251
§7.  The Comparative Method                                          255
§8.  Historical Evidence                                             261



§1.  Hypothesis defined and distinguished from Theory                266
§2.  An Hypothesis must be verifiable                                268
§3.  Proof of Hypotheses                                             270
       (1) Must an hypothetical agent be directly observable? (p. 270);
         _Vera causa_ (p. 271)
       (2) An Hypothesis must be adequate to its pretensions (p. 272);
         _Exceptio probat regulam_ (p. 274)
       (3) Every competing Hypothesis must be excluded (p. 275);
         Crucial instance (p. 277)
       (4) Hypotheses must agree with the laws of Nature (p. 279)
§4.  Hypotheses necessary in scientific investigation                280
§5.  The Method of Abstractions                                      283
       Method of Limits (p. 284);
       In what sense all knowledge is hypothetical (p. 286)



§1.  Axioms; Primary Laws; Secondary Laws, Derivative or Empirical;
       Facts                                                         288
§2.  Secondary Laws either Invariable or Approximate Generalisations 292
§3.  Secondary Laws trustworthy only in 'Adjacent Cases'             293
§4.  Secondary Laws of Succession or of Co-existence                 295
       Natural Kinds (p. 296);
       Co-existence of concrete things to be deduced from
         Causation (p. 297)
§5.  Explanation consists in tracing resemblance, especially
       of Causation                                                  299
§6.  Three modes of Explanation                                      302
       Analysis (p. 302);
       Concatenation (p. 302);
       Subsumption (p. 303)
§7.  Limits of Explanation                                           305
§8.  Analogy                                                         307



§1.  Meaning of Chance and Probability                               310
§2.  Probability as a fraction or proportion                         312
§3.  Probability depends upon experience and statistics              313
§4.  It is a kind of Induction, and pre-supposes Causation           315
§5.  Of Averages and the Law of Error                                318
§6.  Interpretation of probabilities                                 324
       Personal Equation (p. 325);
       meaning of 'Expectation' (p. 325)
§7.  Rules of the combination of Probabilities                       325
       Detection of a hidden Cause (p. 326);
       oral tradition (p. 327);
       circumstantial and analogical evidence (p. 328)



§1.  Classification, scientific, special and popular                 330
§2.  Uses of classification                                          332
§3.  Classification, Deductive and Inductive                         334
§4.  Division, or Deductive Classification:   its Rules              335
§5.  Rules for testing a Division                                    337
§6.  Inductive Classification                                        339
§7.  Difficulty of Natural Classification                            341
§8.  Darwin's influence on the theory of Classification              342
§9.  Classification of Inorganic Bodies also dependent on Causation  346



§1.  Precise thinking needs precise language                         348
§2.  Nomenclature and Terminology                                    349
§3.  Definition                                                      352
§4.  Rules for testing a Definition                                  352
§5.  Every Definition is relative to a Classification                353
§6.  Difficulties of Definition                                      356
       Proposals to substitute the Type (p. 356)
§7.  The Limits of Definition                                        357
§8.  The five Predicables                                            358
       Porphyry's Tree (p. 361)
§9.  Realism and Nominalism                                          364
§10. The Predicaments                                                366



§1.  The rigour of scientific method must be qualified               369
§2.  Still, Language comprises the Nomenclature of an imperfect
       Classification, to which every Definition is relative;        370
§3.  and an imperfect Terminology                                    374
§4.  Maxims and precautions of Definition                            375
§5.  Words of common language in scientific use                      378
§6.  How Definitions affect the cogency of arguments                 380



§1.  Fallacy defined and divided                                     385
§2.  Formal Fallacies of Deduction                                   385
§3.  Formal Fallacies of Induction                                   388
§4.  Material Fallacies classified                                   394
§5.  Fallacies of Observation                                        394
§6.  Begging the Question                                            396
§7.  Surreptitious Conclusion                                        398
§8.  Ambiguity                                                       400
§9.  Fallacies, a natural rank growth of the Human mind, not
       easy to classify, or exterminate                              403

QUESTIONS                                                            405




§ 1. Logic is the science that explains what conditions must be
fulfilled in order that a proposition may be proved, if it admits of
proof. Not, indeed, every such proposition; for as to those that declare
the equality or inequality of numbers or other magnitudes, to explain
the conditions of their proof belongs to Mathematics: they are said to
be _quantitative_. But as to all other propositions, called
_qualitative_, like most of those that we meet with in conversation, in
literature, in politics, and even in sciences so far as they are not
treated mathematically (say, Botany and Psychology); propositions that
merely tell us that something happens (as that _salt dissolves in
water_), or that something has a certain property (as that _ice is
cold_): as to these, it belongs to Logic to show how we may judge
whether they are true, or false, or doubtful. When propositions are
expressed with the universality and definiteness that belong to
scientific statements, they are called laws; and laws, so far as they
are not laws of quantity, are tested by the principles of Logic, if they
at all admit of proof.

But it is plain that the process of proving cannot go on for ever;
something must be taken for granted; and this is usually considered to
be the case (1) with particular facts that can only be perceived and
observed, and (2) with those highest laws that are called 'axioms' or
'first principles,' of which we can only say that we know of no
exceptions to them, that we cannot help believing them, and that they
are indispensable to science and to consistent thought. Logic, then, may
be briefly defined as the science of proof with respect to _qualitative_
laws and propositions, except those that are axiomatic.

§ 2. Proof may be of different degrees or stages of completeness.
Absolute proof would require that a proposition should be shown to agree
with all experience and with the systematic explanation of experience,
to be a necessary part of an all-embracing and self-consistent
philosophy or theory of the universe; but as no one hitherto has been
able to frame such a philosophy, we must at present put up with
something less than absolute proof. Logic, assuming certain principles
to be true of experience, or at least to be conditions of consistent
discourse, distinguishes the kinds of propositions that can be shown to
agree with these principles, and explains by what means the agreement
can best be exhibited. Such principles are those of Contradiction (chap.
vi.), the Syllogism (chap. ix.), Causation (chap. xiv.), and
Probabilities (chap. xx.). To bring a proposition or an argument under
them, or to show that it agrees with them, is logical proof.

The extent to which proof is requisite, again, depends upon the present
purpose: if our aim be general truth for its own sake, a systematic
investigation is necessary; but if our object be merely to remove some
occasional doubt that has occurred to ourselves or to others, it may be
enough to appeal to any evidence that is admitted or not questioned.
Thus, if a man doubts that _some acids are compounds of oxygen_, but
grants that _some compounds of oxygen are acids_, he may agree to the
former proposition when you point out that it has the same meaning as
the latter, differing from it only in the order of the words. This is
called proof by immediate inference.

Again, suppose that a man holds in his hand a piece of yellow metal,
which he asserts to be copper, and that we doubt this, perhaps
suggesting that it is really gold. Then he may propose to dip it in
vinegar; whilst we agree that, if it then turns green, it is copper and
not gold. On trying this experiment the metal does turn green; so that
we may put his argument in this way:--

    _Whatever yellow metal turns green in vinegar is copper;
    This yellow metal turns green in vinegar;
    Therefore, this yellow metal is copper._

Such an argument is called proof by mediate inference; because one
cannot see directly that the yellow metal is copper; but it is admitted
that any yellow metal is copper that turns green in vinegar, and we are
shown that this yellow metal has that property.

Now, however, it may occur to us, that the liquid in which the metal was
dipped was not vinegar, or not pure vinegar, and that the greenness was
due to the impurity. Our friend must thereupon show by some means that
the vinegar was pure; and then his argument will be that, since nothing
but the vinegar came in contact with the metal, the greenness was due to
the vinegar; or, in other words, that contact with that vinegar was the
cause of the metal turning green.

Still, on second thoughts, we may suspect that we had formerly conceded
too much; we may reflect that, although it had often been shown that
copper turned green in vinegar, whilst gold did not, yet the same might
not always happen. May it not be, we might ask, that just at this
moment, and perhaps always for the future gold turns, and will turn
green in vinegar, whilst copper does not and never will again? He will
probably reply that this is to doubt the uniformity of causation: he may
hope that we are not serious: he may point out to us that in every
action of our life we take such uniformity for granted. But he will be
obliged to admit that, whatever he may say to induce us to assent to the
principle of Nature's uniformity, his arguments will not amount to
logical proof, because every argument in some way assumes that
principle. He has come, in fact, to the limits of Logic. Just as Euclid
does not try to prove that 'two magnitudes equal to the same third are
equal to one another,' so the Logician (as such) does not attempt to
prove the uniformity of causation and the other principles of his

Even when our purpose is to ascertain some general truth, the results of
systematic inquiry may have various degrees of certainty. If Logic were
confined to strict demonstration, it would cover a narrow field. The
greater part of our conclusions can only be more or less probable. It
may, indeed, be maintained, not unreasonably, that no judgments
concerning matters of fact can be more than probable. Some say that all
scientific results should be considered as giving the average of cases,
from which deviations are to be expected. Many matters can only be
treated statistically and by the methods of Probability. Our ordinary
beliefs are adopted without any methodical examination. But it is the
aim, and it is characteristic, of a rational mind to distinguish degrees
of certainty, and to hold each judgment with the degree of confidence
that it deserves, considering the evidence for and against it. It takes
a long time, and much self-discipline, to make some progress toward
rationality; for there are many causes of belief that are not good
grounds for it--have no value as evidence. Evidence consists of (1)
observation; (2) reasoning checked by observation and by logical
principles; (3) memory--often inaccurate; (4) testimony--often
untrustworthy, but indispensable, since all we learn from books or from
other men is taken on testimony; (5) the agreement of all our results.
On the other hand, belief is caused by many influences that are not
evidence at all: such are (1) desire, which makes us believe in whatever
serves our purpose; fear and suspicion, which (paradoxically) make us
believe in whatever seems dangerous; (2) habit, which resists whatever
disturbs our prejudices; (3) vanity, which delights to think oneself
always right and consistent and disowns fallibility; (4) imitativeness,
suggestibility, fashion, which carry us along with the crowd. All these,
and nobler things, such as love and fidelity, fix our attention upon
whatever seems to support our prejudices, and prevent our attending to
any facts or arguments that threaten to overthrow them.

§ 3. Two departments of Logic are usually recognised, Deduction and
Induction; that is, to describe them briefly, proof from principles, and
proof from facts. Classification is sometimes made a third department;
sometimes its topics are distributed amongst those of the former two. In
the present work the order adopted is, Deduction in chaps. ii. to xiii.;
Induction in chaps. xiii. to xx.; and, lastly, Classification. But such
divisions do not represent fundamentally distinct and opposed aspects of
the science. For although, in discussing any question with an opponent
who makes admissions, it may be possible to combat his views with merely
deductive arguments based upon his admissions; yet in any question of
general truth, Induction and Deduction are mutually dependent and imply
one another.

This may be seen in one of the above examples. It was argued that a
certain metal must be copper, because every metal is copper that turns
green when dipped in vinegar. So far the proof appealed to a general
proposition, and was deductive. But when we ask how the general
proposition is known to be true, experiments or facts must be alleged;
and this is inductive evidence. Deduction then depends on Induction. But
if we ask, again, how any number of past experiments can prove a general
proposition, which must be good for the future as well as for the past,
the uniformity of causation is invoked; that is, appeal is made to a
principle, and that again is deductive proof. Induction then depends
upon Deduction.

We may put it in this way: Deduction depends on Induction, if general
propositions are only known to us through the facts: Induction depends
on Deduction, because one fact can never prove another, except so far as
what is true of the one is true of the other and of any other of the
same kind; and because, to exhibit this resemblance of the facts, it
must be stated in a general proposition.

§ 4. The use of Logic is often disputed: those who have not studied it,
often feel confident of their ability to do without it; those who have
studied it, are sometimes disgusted with what they consider to be its
superficial analysis of the grounds of evidence, or needless
technicality in the discussion of details. As to those who, not having
studied Logic, yet despise it, there will be time enough to discuss its
utility with them, when they know something about it; and as for those
who, having studied it, turn away in disgust, whether they are justified
every man must judge for himself, when he has attained to equal
proficiency in the subject. Meanwhile, the following considerations may
be offered in its favour:

Logic states, and partly explains and applies, certain abstract
principles which all other sciences take for granted; namely, the axioms
above mentioned--the principles of Contradiction, of the Syllogism and
of Causation. By exercising the student in the apprehension of these
truths, and in the application of them to particular propositions, it
educates the power of abstract thought. Every science is a model of
method, a discipline in close and consecutive thinking; and this merit
Logic ought to possess in a high degree.

For ages Logic has served as an introduction to Philosophy that is, to
Metaphysics and speculative Ethics. It is of old and honourable
descent: a man studies Logic in very good company. It is the warp upon
which nearly the whole web of ancient, mediæval and modern Philosophy is
woven. The history of thought is hardly intelligible without it.

As the science of proof, Logic gives an account of the _general_ nature
of evidence deductive and inductive, as applied in the physical and
social sciences and in the affairs of life. The _general_ nature of such
evidence: it would be absurd of the logician to pretend to instruct the
chemist, economist and merchant, as to the _special_ character of the
evidence requisite in their several spheres of judgment. Still, by
investigating the general conditions of proof, he sets every man upon
his guard against the insufficiency of evidence.

One application of the science of proof deserves special mention:
namely, to that department of Rhetoric which has been the most
developed, relating to persuasion by means of oratory, leader-writing,
or pamphleteering. It is usually said that Logic is useful to convince
the judgment, not to persuade the will: but one way of persuading the
will is to convince the judgment that a certain course is advantageous;
and although this is not always the readiest way, it is the most
honourable, and leads to the most enduring results. Logic is the
backbone of Rhetoric.

It has been disputed whether Logic is a science or an art; and, in fact,
it may be considered in both ways. As a statement of general truths, of
their relations to one another, and especially to the first principles,
it is a science; but it is an art when, regarding truth as an end
desired, it points out some of the means of attaining it--namely, to
proceed by a regular method, to test every judgment by the principles of
Logic, and to distrust whatever cannot be made consistent with them.
Logic does not, in the first place, teach us to reason. We learn to
reason as we learn to walk and talk, by the natural growth of our powers
with some assistance from friends and neighbours. The way to develop
one's power of reasoning is, first, to set oneself problems and try to
solve them. Secondly, since the solving of a problem depends upon one's
ability to call to mind parallel cases, one must learn as many facts as
possible, and keep on learning all one's life; for nobody ever knew
enough. Thirdly one must check all results by the principles of Logic.
It is because of this checking, verifying, corrective function of Logic
that it is sometimes called a Regulative or Normative Science. It cannot
give any one originality or fertility of invention; but it enables us to
check our inferences, revise our conclusions, and chasten the vagaries
of ambitious speculation. It quickens our sense of bad reasoning both in
others and in ourselves. A man who reasons deliberately, manages it
better after studying Logic than he could before, if he is sincere about
it and has common sense.

§ 5. The relation of Logic to other sciences:

(a) Logic is regarded by Spencer as co-ordinate with Mathematics, both
being Abstract Sciences--that is, sciences of the _relations_ in which
things stand to one another, whatever the particular things may be that
are so related; and this view seems to be, on the whole, just--subject,
however, to qualifications that will appear presently.

Mathematics treats of the relations of all sorts of things considered as
quantities, namely, as equal to, or greater or less than, one another.
Things may be quantitatively equal or unequal in _degree_, as in
comparing the temperature of bodies; or in _duration_; or in _spatial
magnitude_, as with lines, superficies, solids; or in _number_. And it
is assumed that the equality or inequality of things that cannot be
directly compared, may be proved indirectly on the assumption that
'things equal to the same thing are equal,' etc.

Logic also treats of the relations of all sorts of things, but not as to
their quantity. It considers (i) that one thing may be like or unlike
another in certain attributes, as that iron is in many ways like tin or
lead, and in many ways unlike carbon or sulphur: (ii) that attributes
co-exist or coinhere (or do not) in the same subject, as metallic
lustre, hardness, a certain atomic weight and a certain specific gravity
coinhere in iron: and (iii) that one event follows another (or is the
effect of it), as that the placing of iron in water causes it to rust.
The relations of likeness and of coinherence are the ground of
Classification; for it is by resemblance of coinhering attributes that
things form classes: coinherence is the ground of judgments concerning
Substance and Attribute, as that iron is metallic; and the relation of
succession, in the mode of Causation, is the chief subject of the
department of Induction. It is usual to group together these relations
of attributes and of order in time, and call them qualitative, in order
to contrast them with the quantitative relations which belong to
Mathematics. And it is assumed that qualitative relations of things,
when they cannot be directly perceived, may be proved indirectly by
assuming the axiom of the Syllogism (chap. ix.) and the law of Causation
(chap. xiv.).

So far, then, Logic and Mathematics appear to be co-ordinate and
distinct sciences. But we shall see hereafter that the satisfactory
treatment of that special order of events in time which constitutes
Causation, requires a combination of Logic with Mathematics; and so does
the treatment of Probability. And, again, Logic may be said to be, in a
certain sense, 'prior to' or 'above' Mathematics as usually treated. For
the Mathematics assume that one magnitude must be either equal or
unequal to another, and that it cannot be both equal and unequal to it,
and thus take for granted the principles of Contradiction and Excluded
Middle; but the statement and elucidation of these Principles are left
to Logic (chap. vi.). The Mathematics also classify and define
magnitudes, as (in Geometry) triangles, squares, cubes, spheres; but the
principles of classification and definition remain for Logic to

(b) As to the concrete Sciences, such as Astronomy, Chemistry, Zoology,
Sociology--Logic (as well as Mathematics) is implied in them all; for
all the propositions of which they consist involve causation,
co-existence, and class-likeness. Logic is therefore said to be prior to
them or above them: meaning by 'prior' not that it should be studied
earlier, for that is not a good plan; meaning by 'above' not in dignity,
for distinctions of dignity amongst liberal studies are absurd. But it
is a philosophical idiom to call the abstract 'prior to,' or 'higher
than,' the concrete (see Porphyry's Tree, chap. xxii. § 8); and Logic is
more abstract than Astronomy or Sociology. Philosophy may thank that
idiom for many a foolish notion.

(c) But, as we have seen, Logic does not investigate the truth,
trustworthiness, or validity of its own principles; nor does
Mathematics: this task belongs to Metaphysics, or Epistemology, the
criticism of knowledge and beliefs.

Logic assumes, for example, that things are what to a careful scrutiny
they seem to be; that animals, trees, mountains, planets, are bodies
with various attributes, existing in space and changing in time; and
that certain principles, such as Contradiction and Causation, are true
of things and events. But Metaphysicians have raised many plausible
objections to these assumptions. It has been urged that natural objects
do not really exist on their own account, but only in dependence on some
mind that contemplates them, and that even space and time are only our
way of perceiving things; or, again, that although things do really
exist on their own account, it is in an entirely different way from that
in which we know them. As to the principle of Contradiction--that if an
object has an attribute, it cannot at the same time and in the same way
be without it (e.g., if an animal is conscious, it is false that it is
not conscious)--it has been contended that the speciousness of this
principle is only due to the obtuseness of our minds, or even to the
poverty of language, which cannot make the fine distinctions that exist
in Nature. And as to Causation, it is sometimes doubted whether events
always have physical causes; and it is often suggested that, granting
they have physical causes, yet these are such as we can neither perceive
nor conceive; belonging not to the order of Nature as we know it, but to
the secret inwardness and reality of Nature, to the wells and reservoirs
of power, not to the spray of the fountain that glitters in our
eyes--'occult causes,' in short. Now these doubts and surmises are
metaphysical spectres which it remains for Metaphysics to lay. Logic has
no direct concern with them (although, of course, metaphysical
discussion is expected to be logical), but keeps the plain path of plain
beliefs, level with the comprehension of plain men. Metaphysics, as
examining the grounds of Logic itself, is sometimes regarded as 'the
higher Logic'; and, certainly, the study of Metaphysics is necessary to
every one who would comprehend the nature and functions of Logic, or the
place of his own mind and of Reason in the world.

(d) The relation of Logic to Psychology will be discussed in the next

(e) As a Regulative Science, pointing out the conditions of true
inference (within its own sphere), Logic is co-ordinate with (i) Ethics,
considered as assigning the conditions of right conduct, and with (ii)
Æsthetics, considered as determining the principles of criticism and
good taste.

§ 6. Three principal schools of Logicians are commonly recognised:
Nominalist, Conceptualist, and Materialist, who differ as to what it is
that Logic really treats of: the Nominalists say, 'of language'; the
Conceptualists, 'of thought'; the Materialists, 'of relations of fact.'
To illustrate these positions let us take authors who, if some of them
are now neglected, have the merit of stating their contrasted views with
a distinctness that later refinements tend to obscure.

(a) Whately, a well-known Nominalist, regarded Logic as the Science and
Art of Reasoning, but at the same time as "entirely conversant about
language"; that is to say, it is the business of Logic to discover those
modes of statement which shall ensure the cogency of an argument, no
matter what may be the subject under discussion. Thus, _All fish are
cold-blooded_, ∴ _some cold-blooded things are fish:_ this is a sound
inference by the mere manner of expression; and equally sound is the
inference, _All fish are warm-blooded_, ∴ _some warm-blooded things are
fish_. The latter proposition may be false, but it follows; and
(according to this doctrine) Logic is only concerned with the consistent
use of words: the truth or falsity of the proposition itself is a
question for Zoology. The short-coming of extreme Nominalism lies in
speaking of language as if its meaning were unimportant. But Whately did
not intend this: he was a man of great penetration and common-sense.

(b) Hamilton, our best-known Conceptualist, defined Logic as the science
of the "formal laws of thought," and "of thought as thought," that is,
without regard to the matter thought about. Just as Whately regarded
Logic as concerned merely with cogent forms of statement, so Hamilton
treated it as concerned merely with the necessary relations of thought.
This doctrine is called Conceptualism, because the simplest element of
thought is the Concept; that is, an abstract idea, such as is signified
by the word _man, planet, colour, virtue_; not a representative or
generic image, but the thought of all attributes common to any class of
things. Men, planets, colours, virtuous actions or characters, have,
severally, something in common on account of which they bear these
general names; and the thought of what they have in common, as the
ground of these names, is a Concept. To affirm or deny one concept of
another, as _Some men are virtuous_, or _No man is perfectly virtuous_,
is to form a Judgment, corresponding to the Proposition of which the
other schools of Logic discourse. Conceptualism, then, investigates the
conditions of consistent judgment.

To distinguish Logic from Psychology is most important in connection
with Conceptualism. Concepts and Judgments being mental acts, or
products of mental activity, it is often thought that Logic must be a
department of Psychology. It is recognised of course, that Psychology
deals with much more than Logic does, with sensation, pleasure and pain,
emotion, volition; but in the region of the intellect, especially in its
most deliberate and elaborate processes, namely, conception, judgment,
and reasoning, Logic and Psychology seem to occupy common ground. In
fact, however, the two sciences have little in common except a few
general terms, and even these they employ in different senses. It is
usual to point out that Psychology tries to explain the subjective
_processes_ of conception, judgment and reasoning, and to give their
natural history; but that Logic is wholly concerned with the _results_
of such processes, with concepts, judgments and reasonings, and merely
with the validity of the results, that is, with their truth or
consistency; whilst Psychology has nothing to do with their validity,
but only with their causes. Besides, the logical judgment (in Formal
Logic at least) is quite a different thing from the psychological: the
latter involves feeling and belief, whereas the former is merely a given
relation of concepts. _S is P_: that is a model logical judgment; there
can be no question of believing it; but it is logically valid if _M is
P_ and _S is M_. When, again, in Logic, one deals with belief, it
depends upon evidence; whereas, in Psychology belief is shown to depend
upon causes which may have evidentiary value or may not; for Psychology
explains quite impartially the growth of scientific insight and the
growth of prejudice.

(c) Mill, Bain, and Venn are the chief Materialist logicians; and to
guard against the error of confounding Materialism in Logic with the
ontological doctrine that nothing exists but Matter, it may suffice to
remember that in Metaphysics all these philosophers are Idealists.
Materialism in Logic consists in regarding propositions as affirming or
denying relations (_cf._ § 5) between matters-of-fact in the widest
sense; not only physical facts, but ideas, social and moral relations;
it consists, in short, in attending to the meaning of propositions. It
treats the first principles of Contradiction and Causation as true of
things so far as they are known to us, and not merely as conditions or
tendencies of thought; and it takes these principles as conditions of
right thinking, because they seem to hold good of Nature and human life.

To these differences of opinion it will be necessary to recur in the
next chapter (§ 4); but here I may observe that it is easy to exaggerate
their importance in Logic. There is really little at issue between
schools of logicians as such, and as far as their doctrines run
parallel; it is on the metaphysical grounds of their study, or as to its
scope and comprehension, that they find a battle-field. The present work
generally proceeds upon the third, or Materialist doctrine. If Deduction
and Induction are regarded as mutually dependent parts of one science,
uniting the discipline of consistent discourse with the method of
investigating laws of physical phenomena, the Materialist doctrine, that
the principles of Logic are founded on fact, seems to be the most
natural way of thinking. But if the unity of Deduction and Induction is
not disputed by the other schools, the Materialist may regard them as
allies exhibiting in their own way the same body of truths. The
Nominalist may certainly claim that his doctrine is indispensable:
consistently cogent forms of statement are necessary both to the
Conceptualist and to the Materialist; neither the relations of thought
nor those of fact can be arrested or presented without the aid of
language or some equivalent system of signs. The Conceptualist may urge
that the Nominalist's forms of statement and argument exist for the sake
of their meaning, namely, judgments and reasonings; and that the
Materialist's laws of Nature are only judgments founded upon our
conceptions of Nature; that the truth of observations and experiments
depends upon our powers of perception; that perception is inseparable
from understanding, and that a system of Induction may be constructed
upon the axiom of Causation, regarded as a principle of Reason, just as
well as by considering it as a law of Nature, and upon much the same
lines. The Materialist, admitting all this, may say that a judgment is
only the proximate meaning of a proposition, and that the ultimate
meaning, the meaning of the judgment itself, is always some
matter-of-fact; that the other schools have not hitherto been eager to
recognise the unity of Deduction and Induction or to investigate the
conditions of trustworthy experiments and observations within the limits
of human understanding; that thought is itself a sort of fact, as
complex in its structure, as profound in its relations, as subtle in its
changes as any other fact, and therefore at least as hard to know; that
to turn away from the full reality of thought in perception, and to
confine Logic to artificially limited concepts, is to abandon the effort
to push method to the utmost and to get as near truth as possible; and
that as to Causation being a principle of Reason rather than of Nature,
the distinction escapes his apprehension, since Nature seems to be that
to which our private minds turn upon questions of Causation for
correction and instruction; so that if he does not call Nature the
Universal Reason, it is because he loves severity of style.



§ 1. Since Logic discusses the proof or disproof, or (briefly) the
testing of propositions, we must begin by explaining their nature. A
proposition, then, may first be described in the language of grammar as
_a sentence indicative_; and it is usually expressed in the present

It is true that other kinds of sentences, optative, imperative,
interrogative, exclamatory, if they express or imply an assertion, are
not beyond the view of Logic; but before treating such sentences, Logic,
for greater precision, reduces them to their equivalent sentences
indicative. Thus, _I wish it were summer_ may be understood to mean,
_The coming of summer is an object of my desire_. _Thou shalt not kill_
may be interpreted as _Murderers are in danger of the judgment_.
Interrogatories, when used in argument, if their form is affirmative,
have negative force, and affirmative force if their form is negative.
Thus, _Do hypocrites love virtue?_ anticipates the answer, _No_. _Are
not traitors the vilest of mankind?_ anticipates the answer, _Yes_. So
that the logical form of these sentences is, _Hypocrites are not lovers
of virtue_; _Traitors are the vilest of mankind_. Impersonal
propositions, such as _It rains_, are easily rendered into logical forms
of equivalent meaning, thus: _Rain is falling_; or (if that be
tautology), _The clouds are raining_. Exclamations may seem capricious,
but are often part of the argument. _Shade of Chatham!_ usually means
_Chatham, being aware of our present foreign policy, is much
disgusted_. It is in fact, an appeal to authority, without the
inconvenience of stating what exactly it is that the authority declares.

§ 2. But even sentences indicative may not be expressed in the way most
convenient to logicians. _Salt dissolves in water_ is a plain enough
statement; but the logician prefers to have it thus: _Salt is soluble in
water_. For he says that a proposition is analysable into three
elements: (1) a Subject (as _Salt_) about which something is asserted or
denied; (2) a Predicate (as _soluble in water_) which is asserted or
denied of the Subject, and (3) the Copula (_is_ or _are_, or _is not_ or
_are not_), the sign of relation between the Subject and Predicate. The
Subject and Predicate are called the Terms of the proposition: and the
Copula may be called the sign of predication, using the verb 'to
predicate' indefinitely for either 'to affirm' or 'to deny.' Thus _S is
P_ means that the term _P_ is given as related in some way to the term
_S_. We may, therefore, further define a Proposition as 'a sentence in
which one term is predicated of another.'

In such a proposition as _Salt dissolves_, the copula (_is_) is
contained in the predicate, and, besides the subject, only one element
is exhibited: it is therefore said to be _secundi adjacentis_. When all
three parts are exhibited, as in _Salt is soluble_, the proposition is
said to be _tertii adjacentis_.

For the ordinary purposes of Logic, in predicating attributes of a thing
or class of things, the copula _is_, or _is not_, sufficiently
represents the relation of subject and predicate; but when it is
desirable to realise fully the nature of the relation involved, it may
be better to use a more explicit form. Instead of saying
_Salt--is--soluble_, we may say _Solubility--coinheres with--the nature
of salt_, or _The putting of salt in water--is a cause of--its
dissolving_: thus expanding the copula into a full expression of the
relation we have in view, whether coinherence or causation.

§ 3. The sentences of ordinary discourse are, indeed, for the most
part, longer and more complicated than the logical form of propositions;
it is in order to prove them, or to use them in the proof of other
propositions, that they are in Logic reduced as nearly as possible to
such simple but explicit expressions as the above (_tertii adjacentis_).
A Compound Proposition, reducible to two or more simple ones, is said to
be exponible.

The modes of compounding sentences are explained in every grammar-book.
One of the commonest forms is the copulative, such as _Salt is both
savoury and wholesome_, equivalent to two simple propositions: _Salt is
savoury; Salt is wholesome. Pure water is neither sapid nor odorous_,
equivalent to _Water is not sapid; Water is not odorous_. Or, again,
_Tobacco is injurious, but not when used in moderation_, equivalent to
_Much tobacco is injurious; a little is not_.

Another form of Exponible is the Exceptive, as _Kladderadatsch is
published daily, except on week-days_, equivalent to _Kladderadatsch is
published on Sunday; it is not published any other day_. Still another
Exponible is the Exclusive, as _Only men use fire_, equivalent to _Men
are users of fire; No other animals are_. Exceptive and exclusive
sentences are, however, equivalent forms; for we may say,
_Kladderadatsch is published only on Sunday_; and _No animals use fire,
except men_.

There are other compound sentences that are not exponible, since, though
they contain two or more verbal clauses, the construction shows that
these are inseparable. Thus, _If cats are scarce, mice are plentiful_,
contains two verbal clauses; but _if cats are scarce_ is conditional,
not indicative; and _mice are plentiful_ is subject to the condition
that _cats are scarce_. Hence the whole sentence is called a Conditional
Proposition. For the various forms of Conditional Propositions see chap.
v. § 4.

But, in fact, to find the logical force of recognised grammatical forms
is the least of a logician's difficulties in bringing the discourses of
men to a plain issue. Metaphors, epigrams, innuendoes and other figures
of speech present far greater obstacles to a lucid reduction whether for
approval or refutation. No rules can be given for finding everybody's
meaning. The poets have their own way of expressing themselves;
sophists, too, have their own way. And the point often lies in what is
unexpressed. Thus, "barbarous nations make, the civilised write
history," means that civilised nations do not make history, which none
is so brazen as openly to assert. Or, again, "Alcibiades is dead, but X
is still with us"; the whole meaning of this 'exponible' is that X would
be the lesser loss to society. Even an epithet or a suffix may imply a
proposition: _This personage_ may mean _X is a pretentious nobody_.

How shall we interpret such illusive predications except by cultivating
our literary perceptions, by reading the most significant authors until
we are at home with them? But, no doubt, to disentangle the compound
propositions, and to expand the abbreviations of literature and
conversation, is a useful logical exercise. And if it seem a laborious
task thus to reduce to its logical elements a long argument in a speech
or treatise, it should be observed that, as a rule, in a long discourse
only a few sentences are of principal importance to the reasoning, the
rest being explanatory or illustrative digression, and that a close
scrutiny of these cardinal sentences will frequently dispense us from
giving much attention to the rest.

§ 4. But now, returning to the definition of a Proposition given in § 2,
that it is 'a sentence in which one term is predicated of another,' we
must consider what is the import of such predication. For the
definition, as it stands, seems to be purely Nominalist. Is a
proposition nothing more than a certain synthesis of words; or, is it
meant to correspond with something further, a synthesis of ideas, or a
relation of facts?

Conceptualist logicians, who speak of judgments instead of
propositions, of course define the judgment in their own language.
According to Hamilton, it is "a recognition of the relation of
congruence or confliction in which two concepts stand to each other." To
lighten the sentence, I have omitted one or two qualifications
(Hamilton's _Lectures on Logic_, xiii.). "Thus," he goes on "if we
compare the thoughts _water_, _iron_, and _rusting_, we find them
congruent, and connect them into a single thought, thus: _water rusts
iron_--in that case we form a judgment." When a judgment is expressed in
words, he says, it is called a proposition.

But has a proposition no meaning beyond the judgment it expresses? Mill,
who defines it as "a portion of discourse in which a predicate is
affirmed or denied of a subject" (_Logic_, Book 1., chap. iv. § 1.),
proceeds to inquire into the import of propositions (Book 1., chap. v.),
and finds three classes of them: (a) those in which one proper name is
predicated of another; and of these Hobbes's Nominalist definition is
adequate, namely, that a proposition asserts or denies that the
predicate is a name for the same thing as the subject, as _Tully is

(b) Propositions in which the predicate means a part (or the whole) of
what the subject means, as _Horses are animals_, _Man is a rational
animal_. These are Verbal Propositions (see below: chap. v. § 6), and
their import consists in affirming or denying a coincidence between the
meanings of names, as _The meaning of 'animal' is part of the meaning of
'horse.'_ They are partial or complete definitions.

But (c) there are also Real Propositions, whose predicates do not mean
the same as their subjects, and whose import consists in affirming or
denying one of five different kinds of matter of fact: (1) That the
subject exists, or does not; as if we say _The bison exists_, _The great
auk is extinct_. (2) Co-existence, as _Man is mortal_; that is, _the
being subject to death coinheres with the qualities on account of which
we call certain objects men_. (3) Succession, as _Night follows day_.
(4) Causation (a particular kind of Succession), as _Water rusts iron_.
(5) Resemblance, as _The colour of this geranium is like that of a
soldier's coat_, or _A = B_.

On comparing this list of real predications with the list of logical
relations given above (chap. i. § 5 (a)), it will be seen that the two
differ only in this, that I have there omitted simple Existence. Nothing
simply exists, unrelated either in Nature or in knowledge. Such a
proposition as _The bison exists_ may, no doubt, be used in Logic
(subject to interpretation) for the sake of custom or for the sake of
brevity; but it means that some specimens are still to be found in N.
America, or in Zoological gardens.

Controversy as to the Import of Propositions really turns upon a
difference of opinion as to the scope of Logic and the foundations of
knowledge. Mill was dissatisfied with the "congruity" of concepts as the
basis of a judgment. Clearly, mere congruity does not justify belief. In
the proposition _Water rusts iron_, the concepts _water_, _rust_ and
_iron_ may be congruous, but does any one assert their connection on
that ground? In the proposition _Murderers are haunted by the ghosts of
their victims_, the concepts _victim_, _murderer_, _ghost_ have a high
degree of congruity; yet, unfortunately, I cannot believe it: there
seems to be no such cheap defence of innocence. Now, Mill held that
Logic is concerned with the grounds of belief, and that the scope of
Logic includes Induction as well as Deduction; whereas, according to
Hamilton, Induction is only Modified Logic, a mere appendix to the
theory of the "forms of thought as thought." Indeed, Mill endeavoured in
his _Logic_ to probe the grounds of belief deeper than usual, and
introduced a good deal of Metaphysics--either too much or not
enough--concerning the ground of axioms. But, at any rate, his great
point was that belief, and therefore (for the most part) the Real
Proposition, is concerned not merely with the relations of words, or
even of ideas, but with matters of fact; that is, both propositions and
judgments point to something further, to the relations of things which
we can examine, not merely by thinking about them (comparing them in
thought), but by observing them with the united powers of thought and
perception. This is what convinces us that _water rusts iron_: and the
difficulty of doing this is what prevents our feeling sure that
_murderers are haunted by the ghosts of their victims_. Hence, although
Mill's definition of a proposition, given above, is adequate for
propositions in general; yet that kind of proposition (the Real) with
regard to which Logic (in Mill's view) investigates the conditions of
proof, may be more explicitly and pertinently defined as 'a predication
concerning the relation of matters of fact.'

§ 5. This leads to a very important distinction to which we shall often
have to refer in subsequent pages--namely, the distinction between the
Form and the Matter of a proposition or of an argument. The distinction
between Form and Matter, as it is ordinarily employed, is easily
understood. An apple growing in the orchard and a waxen apple on the
table may have the same shape or form, but they consist of different
materials; two real apples may have the same shape, but contain distinct
ounces of apple-stuff, so that after one is eaten the other remains to
be eaten. Similarly, tables may have the same shape, though one be made
of marble, another of oak, another of iron. The form is common to
several things, the matter is peculiar to each. Metaphysicians have
carried the distinction further: apples, they say, may have not only the
same outward shape, but the same inward constitution, which, therefore,
may be called the Form of apple-stuff itself--namely, a certain
pulpiness, juiciness, sweetness, etc.; qualities common to all dessert
apples: yet their Matter is different, one being here, another
there--differing in place or time, if in nothing else. The definition of
a species is the form of every specimen of it.

To apply this distinction to the things of Logic: it is easy to see how
two propositions may have the same Form but different Matter: not using
'Form' in the sense of 'shape,' but for that which is common to many
things, in contrast with that which is peculiar to each. Thus, _All male
lions are tawny_ and _All water is liquid at 50° Fahrenheit_, are two
propositions that have the same form, though their matter is entirely
different. They both predicate something of the whole of their subjects,
though their subjects are different, and so are the things predicated of
them. Again, _All male lions have tufted tails_ and _All male lions have
manes_, are two propositions having the same form and, in their
subjects, the same matter, but different matter in their predicates. If,
however, we take two such propositions as these: _All male lions have
manes_ and _Some male lions have manes_, here the matter is the same in
both, but the form is different--in the first, predication is made
concerning _every_ male lion; in the second of only _some_ male lions;
the first is _universal_, the second is _particular_. Or, again, if we
take _Some tigers are man-eaters_ and _Some tigers are not man-eaters_,
here too the matter is the same, but the form is different; for the
first proposition is _affirmative_, whilst the second is _negative_.

§ 6. Now, according to Hamilton and Whately, pure Logic has to do only
with the Form of propositions and arguments. As to their Matter, whether
they are really true in fact, that is a question, they said, not for
Logic, but for experience, or for the special sciences. But Mill desired
so to extend logical method as to test the material truth of
propositions: he thought that he could expound a method by which
experience itself and the conclusions of the special sciences may be

To this method it may be objected, that the claim to determine Material
Truth takes for granted that the order of Nature will remain unchanged,
that (for example) water not only at present is a liquid at 50°
Fahrenheit, but will always be so; whereas (although we have no reason
to expect such a thing) the order of Nature may alter--it is at least
supposable--and in that event water may freeze at such a temperature.
Any matter of fact, again, must depend on observation, either directly,
or by inference--as when something is asserted about atoms or ether. But
observation and material inference are subject to the limitations of our
faculties; and however we may aid observation by microscopes and
micrometers, it is still observation; and however we may correct our
observations by repetition, comparison and refined mathematical methods
of making allowances, the correction of error is only an approximation
to accuracy. Outside of Formal Reasoning, suspense of judgment is your
only attitude.

But such objections imply that nothing short of absolute truth has any
value; that all our discussions and investigations in science or social
affairs are without logical criteria; that Logic must be confined to
symbols, and considered entirely as mental gymnastics. In this book
prominence will be given to the character of Logic as a formal science,
and it will also be shown that Induction itself may be treated formally;
but it will be assumed that logical forms are valuable as representing
the actual relations of natural and social phenomena.

§ 7. Symbols are often used in Logic instead of concrete terms, not only
in Symbolic Logic where the science is treated algebraically (as by Dr.
Venn in his _Symbolic Logic_), but in ordinary manuals; so that it may
be well to explain the use of them before going further.

It is a common and convenient practice to illustrate logical doctrines
by examples: to show what is meant by a Proposition we may give _salt is
soluble_, or _water rusts iron:_ the copulative exponible is exemplified
by _salt is savoury and wholesome_; and so on. But this procedure has
some disadvantages: it is often cumbrous; and it may distract the
reader's attention from the point to be explained by exciting his
interest in the special fact of the illustration. Clearly, too, so far
as Logic is formal, no particular matter of fact can adequately
illustrate any of its doctrines. Accordingly, writers on Logic employ
letters of the alphabet instead of concrete terms, (say) _X_ instead of
_salt_ or instead of _iron_, and (say) _Y_ instead of _soluble_ or
instead of _rusted by water_; and then a proposition may be represented
by _X is Y_. It is still more usual to represent a proposition by _S is
(or is not) P, S_ being the initial of Subject and _P_ of Predicate;
though this has the drawback that if we argue--_S is P_, therefore _P is
S_, the symbols in the latter proposition no longer have the same
significance, since the former subject is now the predicate.

Again, negative terms frequently occur in Logic, such as _not-water_, or
_not-iron_, and then if _water_ or _iron_ be expressed by _X_, the
corresponding negative may be expressed by _x_; or, generally, if a
capital letter stand for a positive term, the corresponding small letter
represents the negative. The same device may be adopted to express
contradictory terms: either of them being _X_, the other is _x_ (see
chap. iv., §§ 7-8); or the contradictory terms may be expressed by _x_
and _x̄_, _y_ and _ȳ_.

And as terms are often compounded, it may be convenient to express them
by a combination of letters: instead of illustrating such a case by
_boiling water_ or _water that is boiling_, we may write _XY_; or since
positive and negative terms may be compounded, instead of illustrating
this by _water that is not boiling_, we may write _Xy_.

The convenience of this is obvious; but it is more than convenient; for,
if one of the uses of Logic be to discipline the power of abstract
thought, this can be done far more effectually by symbolic than by
concrete examples; and if such discipline were the only use of Logic it
might be best to discard concrete illustrations altogether, at least in
advanced text-books, though no doubt the practice would be too severe
for elementary manuals. On the other hand, to show the practical
applicability of Logic to the arguments and proofs of actual life, or
even of the concrete sciences, merely symbolic illustration may be not
only useless but even misleading. When we speak of politics, or poetry,
or species, or the weather, the terms that must be used can rarely have
the distinctness and isolation of X and Y; so that the perfunctory use
of symbolic illustration makes argument and proof appear to be much
simpler and easier matters than they really are. Our belief in any
proposition never rests on the proposition itself, nor merely upon one
or two others, but upon the immense background of our general knowledge
and beliefs, full of circumstances and analogies, in relation to which
alone any given proposition is intelligible. Indeed, for this reason, it
is impossible to illustrate Logic sufficiently: the reader who is in
earnest about the cogency of arguments and the limitation of proofs, and
is scrupulous as to the degrees of assent that they require, must
constantly look for illustrations in his own knowledge and experience
and rely at last upon his own sagacity.



§ 1. In treating of Deductive Logic it is usual to recognise three
divisions of the subject: first, the doctrine of Terms, words, or other
signs used as subjects or predicates; secondly, the doctrine of
Propositions, analysed into terms related; and, thirdly, the doctrine of
the Syllogism in which propositions appear as the grounds of a

The terms employed are either letters of the alphabet, or the words of
common language, or the technicalities of science; and since the words
of common language are most in use, it is necessary to give some account
of common language as subserving the purposes of Logic. It has been
urged that we cannot think or reason at all without words, or some
substitute for them, such as the signs of algebra; but this is an
exaggeration. Minds greatly differ, and some think by the aid of
definite and comprehensive picturings, especially in dealing with
problems concerning objects in space, as in playing chess blindfold,
inventing a machine, planning a tour on an imagined map. Most people
draw many simple inferences by means of perceptions, or of mental
imagery. On the other hand, some men think a good deal without any
continuum of words and without any imagery, or with none that seems
relevant to the purpose. Still the more elaborate sort of thinking, the
grouping and concatenation of inferences, which we call reasoning,
cannot be carried far without language or some equivalent system of
signs. It is not merely that we need language to express our reasonings
and communicate them to others: in solitary thought we often depend on
words--'talk to ourselves,' in fact; though the words or sentences that
then pass through our minds are not always fully formed or articulated.
In Logic, moreover, we have carefully to examine the grounds (at least
the proximate grounds) of our conclusions; and plainly this cannot be
done unless the conclusions in question are explicitly stated and

Conceptualists say that Logic deals not with the process of thinking
(which belongs to Psychology) but with its results; not with conceiving
but with concepts; not with judging but with judgments. Is the concept
self-consistent or adequate? Logic asks; is the judgment capable of
proof? Now, it is only by recording our thoughts in language that it
becomes possible to distinguish between the process and the result of
thought. Without language, the act and the product of thinking would be
identical and equally evanescent. But by carrying on the process in
language and remembering or otherwise recording it, we obtain a result
which may be examined according to the principles of Logic.

§ 2. As Logic, then, must give some account of language, it seems
desirable to explain how its treatment of language differs from that of
Grammar and from that of Rhetoric.

Grammar is the study of the words of some language, their classification
and derivation, and of the rules of combining them, according to the
usage at any time recognised and followed by those who are considered
correct writers or speakers. Composition may be faultless in its
grammar, though dull and absurd.

Rhetoric is the study of language with a view to obtaining some special
effect in the communication of ideas or feelings, such as
picturesqueness in description, vivacity in narration, lucidity in
exposition, vehemence in persuasion, or literary charm. Some of these
ends are often gained in spite of faulty syntax or faulty logic; but
since the few whom bad grammar saddens or incoherent arguments divert
are not carried away, as they else might be, by an unsophisticated
orator, Grammar and Logic are necessary to the perfection of Rhetoric.
Not that Rhetoric is in bondage to those other sciences; for foreign
idioms and such figures as the ellipsis, the anacoluthon, the oxymoron,
the hyperbole, and violent inversions have their place in the
magnificent style; but authors unacquainted with Grammar and Logic are
not likely to place such figures well and wisely. Indeed, common idioms,
though both grammatically and rhetorically justifiable, both correct and
effective, often seem illogical. 'To fall asleep,' for example, is a
perfect English phrase; yet if we examine severally the words it
consists of, it may seem strange that their combination should mean
anything at all.

But Logic only studies language so far as necessary in order to state,
understand, and check the evidence and reasonings that are usually
embodied in language. And as long as meanings are clear, good Logic is
compatible with false concords and inelegance of style.

§ 3. Terms are either Simple or Composite: that is to say, they may
consist either of a single word, as 'Chaucer,' 'civilisation'; or of
more than one, as 'the father of English poetry,' or 'modern civilised
nations.' Logicians classify words according to their uses in forming
propositions; or, rather, they classify the uses of words as terms, not
the words themselves; for the same word may fall into different classes
of terms according to the way in which it is used. (Cf. Mr. Alfred
Sidgwick's _Distinction and the Criticism of Beliefs_, chap. xiv.)

Thus words are classified as Categorematic or Syncategorematic. A word
is Categorematic if used singly as a term without the support of other
words: it is Syncategorematic when joined with other words in order to
constitute the subject or predicate of a proposition. If we say _Venus
is a planet whose orbit is inside the Earth's_, the subject, 'Venus,'
is a word used categorematically as a simple term; the predicate is a
composite term whose constituent words (whether substantive, relative,
verb, or preposition) are used syncategorematically.

Prepositions, conjunctions, articles, adverbs, relative pronouns, in
their ordinary use, can only enter into terms along with other words
having a substantive, adjectival or participial force; but when they are
themselves the things spoken of and are used substantively (_suppositio
materialis_), they are categorematic. In the proposition, _'Of' was used
more indefinitely three hundred years ago than it is now_, 'of' is
categorematic. On the other hand, all substantives may be used
categorematically; and the same self-sufficiency is usually recognised
in adjectives and participles. Some, however, hold that the
categorematic use of adjectives and participles is due to an ellipsis
which the logician should fill up; that instead of _Gold is heavy_, he
should say _Gold is a heavy metal_; instead of _The sun is shining_,
_The sun is a body shining_. But in these cases the words 'metal' and
'body' are unmistakable tautology, since 'metal' is implied in gold and
'body' in sun. But, as we have seen, any of these kinds of word,
substantive, adjective, or participle, may occur syncategorematically in
connection with others to form a composite term.

§ 4. Most terms (the exceptions and doubtful cases will be discussed
hereafter) have two functions, a denotative and a connotative. A term's
denotative function is, to be the name or sign of something or some
multitude of things, which are said to be called or denoted by the term.
Its connotative function is, to suggest certain qualities and
characteristics of the things denoted, so that it cannot be used
literally as the name of any other things; which qualities and
characteristics are said to be implied or connoted by the term. Thus
'sheep' is the name of certain animals, and its connotation prevents its
being used of any others. That which a term directly indicates, then,
is its _Denotation_; that sense or customary use of it which limits the
Denotation is its _Connotation_ (ch. iv.). Hamilton and others use
'Extension' in the sense of Denotation, and 'Intension' or
'Comprehension' in the sense of Connotation. Now, terms may be
classified, first according to what they stand for or denote; that is,
according to their _Denotation_. In this respect, the use of a term is
said to be either Concrete or Abstract.

A term is Concrete when it denotes a 'thing'; that is, any person,
object, fact, event, feeling or imagination, considered as capable of
having (or consisting of) qualities and a determinate existence. Thus
'cricket ball' denotes any object having a certain size, weight, shape,
colour, etc. (which are its qualities), and being at any given time in
some place and related to other objects--in the bowler's hands, on the
grass, in a shop window. Any 'feeling of heat' has a certain intensity,
is pleasurable or painful, occurs at a certain time, and affects some
part or the whole of some animal. An imagination, indeed (say, of a
fairy), cannot be said in the same sense to have locality; but it
depends on the thinking of some man who has locality, and is definitely
related to his other thoughts and feelings.

A term is Abstract, on the other hand, when it denotes a quality (or
qualities), considered by itself and without determinate existence in
time, place, or relation to other things. 'Size,' 'shape,' 'weight,'
'colour,' 'intensity,' 'pleasurableness,' are terms used to denote such
qualities, and are then abstract in their denotation. 'Weight' is not
something with a determinate existence at a given time; it exists not
merely in some particular place, but wherever there is a heavy thing;
and, as to relation, at the same moment it combines in iron with
solidity and in mercury with liquidity. In fact, a quality is a point of
agreement in a multitude of different things; all heavy things agree in
weight, all round things in roundness, all red things in redness; and an
abstract term denotes such a point (or points) of agreement among the
things denoted by concrete terms. Abstract terms result from the
analysis of concrete things into their qualities; and conversely a
concrete term may be viewed as denoting the synthesis of qualities into
an individual thing. When several things agree in more than one quality,
there may be an abstract term denoting the union of qualities in which
they agree, and omitting their peculiarities; as 'human nature' denotes
the common qualities of men, 'civilisation' the common conditions of
civilised peoples.

Every general name, if used as a concrete term, has, or may have, a
corresponding abstract term. Sometimes the concrete term is modified to
form the abstract, as 'greedy--greediness'; sometimes a word is adapted
from another language, as 'man--humanity'; sometimes a composite term is
used, as 'mercury--the nature of mercury,' etc. The same concrete may
have several abstract correlatives, as 'man--manhood, humanity, human
nature'; 'heavy--weight, gravity, ponderosity'; but in such cases the
abstract terms are not used quite synonymously; that is, they imply
different ways of considering the concrete.

Whether a word is used as a concrete or abstract term is in most
instances plain from the word itself, the use of most words being pretty
regular one way or the other; but sometimes we must judge by the
context. 'Weight' may be used in the abstract for 'gravity,' or in the
concrete for a measure; but in the latter sense it is syncategorematic
(in the singular), needing at least the article 'a (or the) weight.'
'Government' may mean 'supreme political authority,' and is then
abstract; or, the men who happen to be ministers, and is then concrete;
but in this case, too, the article is usually prefixed. 'The life' of
any man may mean his vitality (abstract), as in "Thus following life in
creatures we dissect"; or, the series of events through which he passes
(concrete), as in 'the life of Nelson as narrated by Southey.'

It has been made a question whether the denotation of an abstract term
may itself be the subject of qualities. Apparently 'weight' may be
greater or less, 'government' good or bad, 'vitality' intense or dull.
But if every subject is modified by a quality, a quality is also
modified by making it the subject of another; and, if so, it seems then
to become a new quality. The compound terms 'great weight,' 'bad
government,' 'dull vitality,' have not the same denotation as the simple
terms 'weight, 'government,' 'vitality': they imply, and may be said to
connote, more special concrete experience, such as the effort felt in
lifting a trunk, disgust at the conduct of officials, sluggish movements
of an animal when irritated. It is to such concrete experiences that we
have always to refer in order fully to realise the meaning of abstract
terms, and therefore, of course, to understand any qualification of

§ 5. Concrete terms may be subdivided according to the number of things
they denote and the way in which they denote them. A term may denote one
thing or many: if one, it is called Singular; if many, it may do so
distributively, and then it is General; or, as taken all together, and
then it is Collective: one, then; any one of many; many in one.

Among Singular Terms, each denoting a single thing, the most obvious are
Proper Names, such as Gibraltar or George Washington, which are merely
marks of individual things or persons, and may form no part of the
common language of a country. They are thus distinguished from other
Singular Terms, which consist of common words so combined as to restrict
their denotation to some individual, such as, 'the strongest man on

Proper Terms are often said to be arbitrary signs, because their use
does not depend upon any reason that may be given for them. Gibraltar
had a meaning among the Moors when originally conferred; but no one now
knows what it was, unless he happens to have learned it; yet the name
serves its purpose as well as if it were "Rooke's Nest." Every Newton
or Newport year by year grows old, but to alter the name would cause
only confusion. If such names were given by mere caprice it would make
no difference; and they could not be more cumbrous, ugly, or absurd than
many of those that are given 'for reasons.'

The remaining kinds of Singular Terms are drawn from the common
resources of the language. Thus the pronouns 'he,' 'she,' 'it,' are
singular terms, whose present denotation is determined by the occasion
and context of discourse: so with demonstrative phrases--'the man,'
'that horse.' Descriptive names may be more complex, as 'the wisest man
of Gotham,' which is limited to some individual by the superlative
suffix; or 'the German Emperor,' which is limited by the definite
article--the general term 'German Emperor' being thereby restricted
either to the reigning monarch or to the one we happen to be discussing.
Instead of the definite, the indefinite article may be used to make
general terms singular, as 'a German Emperor was crowned at Versailles'
(_individua vaga_).

Abstract Terms are ostensively singular: 'whiteness' (e.g.) is one
quality. But their full meaning is general: 'whiteness' stands for all
white things, so far as white. Abstract terms, in fact, are only
formally singular.

General Terms are words, or combinations of words, used to denote any
one of many things that resemble one another in certain respects.
'George III.' is a Singular Term denoting one man; but 'King' is a
General Term denoting him and all other men of the same rank; whilst the
compound 'crowned head' is still more general, denoting kings and also
emperors. It is the nature of a general term, then, to be used in the
same sense of whatever it denotes; and its most characteristic form is
the Class-name, whether of objects, such as 'king,' 'sheep,' 'ghost'; or
of events, such as 'accession,' 'purchase,' 'manifestation.' Things and
events are known by their qualities and relations; and every such
aspect, being a point of resemblance to some other things, becomes a
ground of generalisation, and therefore a ground for the need and use of
general terms. Hence general terms are far the most important sort of
terms in Logic, since in them general propositions are expressed and,
moreover (with rare exceptions), all predicates are general. For,
besides these typical class-names, attributive words are general terms,
such as 'royal,' 'ruling,' 'woolly,' 'bleating,' 'impalpable,'

Infinitives may also be used as general terms, as '_To err is human_';
but for logical purposes they may have to be translated into equivalent
substantive forms, as _Foolish actions are characteristic of mankind_.
Abstract terms, too, are (as I observed) equivalent to general terms;
'folly' is abstract for 'foolish actions.' '_Honesty is the best
policy_' means _people who are honest may hope to find their account in
being so_; that is, in the effects of their honest actions, provided
they are wise in other ways, and no misfortunes attend them. The
abstract form is often much the more succinct and forcible, but for
logical treatment it needs to be interpreted in the general form.

By antonomasia proper names may become general terms, as if we say _'A
Johnson' would not have written such a book_--i.e., any man of his
genius for elaborate eloquence.

A Collective Term denotes a multitude of similar things considered as
forming one whole, as 'regiment,' 'flock,' 'nation': not distributively,
that is, not the similar things severally; to denote them we must say
'soldiers of the regiment,' 'sheep of the flock,' and so on. If in a
multitude of things there is no resemblance, except the fact of being
considered as parts of one whole, as 'the world,' or 'the town of
Nottingham' (meaning its streets and houses, open spaces, people, and
civic organisation), the term denoting them as a whole is Singular; but
'the world' or 'town of Nottingham,' meaning the inhabitants only, is

In their strictly collective use, all such expressions are equivalent to
singular terms; but many of them may also be used as general terms, as
when we speak of 'so many regiments of the line,' or discuss the
'plurality of worlds'; and in this general use they denote any of a
multitude of things of the same kind--regiments, or habitable worlds.

Names of substances, such as 'gold,' 'air,' 'water,' may be employed as
singular, collective, or general terms; though, perhaps, as singular
terms only figuratively, as when we say _Gold is king_. If we say with
Thales, '_Water is the source of all things_,' 'water' seems to be used
collectively. But substantive names are frequently used as general
terms. For example, _Gold is heavy_ means 'in comparison with other
things,' such as water. And, plainly, it does not mean that the
aggregate of gold is heavier than the aggregate of water, but only that
its specific gravity is greater; that is, bulk for bulk, any piece of
gold is heavier than water.

Finally, any class-name may be used collectively if we wish to assert
something of the things denoted by it, not distributively but
altogether, as that _Sheep are more numerous than wolves_.



§ 1. Terms are next to be classified according to their
Connotation--that is, according to what they imply as characteristic of
the things denoted. We have seen that general names are used to denote
many things in the same sense, because the things denoted resemble one
another in certain ways: it is this resemblance in certain points that
leads us to class the things together and call them by the same name;
and therefore the points of resemblance constitute the sense or meaning
of the name, or its Connotation, and limit its applicability to such
things as have these characteristic qualities. 'Sheep' for example, is
used in the same sense, to denote any of a multitude of animals that
resemble one another: their size, shape, woolly coats, cloven hoofs,
innocent ways and edibility are well known. When we apply to anything
the term 'sheep,' we imply that it has these qualities: 'sheep,'
denoting the animal, connotes its possessing these characteristics; and,
of course, it cannot, without a figure of speech or a blunder, be used
to denote anything that does not possess all these qualities. It is by a
figure of speech that the term 'sheep' is applied to some men; and to
apply it to goats would be a blunder.

Most people are very imperfectly aware of the connotation of the words
they use, and are guided in using them merely by the custom of the
language. A man who employs a word quite correctly may be sadly posed by
a request to explain or define it. Moreover, so far as we are aware of
the connotation of terms, the number and the kind of attributes we
think of, in any given case, vary with the depth of our interest, and
with the nature of our interest in the things denoted. 'Sheep' has one
meaning to a touring townsman, a much fuller one to a farmer, and yet a
different one to a zoologist. But this does not prevent them agreeing in
the use of the word, as long as the qualities they severally include in
its meaning are not incompatible.

All general names, and therefore not only class-names, like 'sheep,' but
all attributives, have some connotation. 'Woolly' denotes anything that
bears wool, and connotes the fact of bearing wool; 'innocent' denotes
anything that habitually and by its disposition does no harm (or has not
been guilty of a particular offence), and connotes a harmless character
(or freedom from particular guilt); 'edible' denotes whatever can be
eaten with good results, and connotes its suitability for mastication,
deglutition, digestion, and assimilation.

§ 2. But whether all terms must connote as well as denote something, has
been much debated. Proper names, according to what seems the better
opinion, are, in their ordinary use, not connotative. To say that they
have no meaning may seem violent: if any one is called John Doe, this
name, no doubt, means a great deal to his friends and neighbours,
reminding them of his stature and physiognomy, his air and gait, his wit
and wisdom, some queer stories, and an indefinite number of other
things. But all this significance is local or accidental; it only exists
for those who know the individual or have heard him described: whereas a
general name gives information about any thing or person it denotes to
everybody who understands the language, without any particular knowledge
of the individual.

We must distinguish, in fact, between the peculiar associations of the
proper name and the commonly recognised meaning of the general name.
This is why proper names are not in the dictionary. Such a name as
London, to be sure, or Napoleon Buonaparte, has a significance not
merely local; still, it is accidental. These names are borne by other
places and persons than those that have rendered them famous. There are
Londons in various latitudes, and, no doubt, many Napoleon Buonapartes
in Louisiana; and each name has in its several denotations an altogether
different suggestiveness. For its suggestiveness is in each application
determined by the peculiarities of the place or person denoted; it is
not given to the different places (or to the different persons) because
they have certain characteristics in common.

However, the scientific grounds of the doctrine that proper names are
non-connotative, are these: The peculiarities that distinguish an
individual person or thing are admitted to be infinite, and anything
less than a complete enumeration of these peculiarities may fail to
distinguish and identify the individual. For, short of a complete
enumeration of them, the description may be satisfied by two or more
individuals; and in that case the term denoting them, if limited by such
a description, is not a proper but a general name, since it is
applicable to two or more in the same sense. The existence of other
individuals to whom it applies may be highly improbable; but, if it be
logically possible, that is enough. On the other hand, the enumeration
of infinite peculiarities is certainly impossible. Therefore proper
names have no assignable connotation. The only escape from this
reasoning lies in falling back upon time and place, the principles of
individuation, as constituting the connotation of proper names. Two
things cannot be at the same time in the same place: hence 'the man who
was at a certain spot on the bridge of Lodi at a certain instant in a
certain year' suffices to identify Napoleon Buonaparte for that instant.
Supposing no one else to have borne the name, then, is this its
connotation? No one has ever thought so. And, at any rate, time and
place are only extrinsic determinations (suitable indeed to events like
the battle of Lodi, or to places themselves like London); whereas the
connotation of a general term, such as 'sheep,' consists of intrinsic
qualities. Hence, then, the scholastic doctrine 'that individuals have
no essence' (see chap. xxii. § 9), and Hamilton's dictum 'that every
concept is inadequate [sic] to the individual,' are justified.

General names, when used as proper names, lose their connotation, as
Euxine or Newfoundland.

Singular terms, other than Proper, have connotation; either in
themselves, like the singular pronouns 'he,' 'she,' 'it,' which are
general in their applicability, though singular in application; or,
derivatively, from the general names that combine to form them, as in
'the first Emperor of the French' or the 'Capital of the British

§ 3. Whether Abstract Terms have any connotation is another disputed
question. We have seen that they denote a quality or qualities of
something, and that is precisely what general terms connote: 'honesty'
denotes a quality of some men; 'honest' connotes the same quality,
whilst denoting the men who have it.

The denotation of abstract terms thus seems to exhaust their force or
meaning. It has been proposed, however, to regard them as connoting the
qualities they directly stand for, and not denoting anything; but surely
this is too violent. To denote something is the same as to be the name
of something (whether real or unreal), which every term must be. It is a
better proposal to regard their denotation and connotation as
coinciding; though open to the objection that 'connote' means 'to mark
along with' something else, and this plan leaves nothing else. Mill
thought that abstract terms are connotative when, besides denoting a
quality, they suggest a quality of that quality (as 'fault' implies
'hurtfulness'); but against this it may be urged that one quality cannot
bear another, since every qualification of a quality constitutes a
distinct quality in the total ('milk-whiteness' is distinct from
'whiteness,' _cf._ chap. iii. § 4). After all, if it is the most
consistent plan, why not say that abstract, like proper, terms have no

But if abstract terms must be made to connote something, should it not
be those things, indefinitely suggested, to which the qualities belong?
Thus 'whiteness' may be considered to connote either snow or vapour, or
any white thing, apart from one or other of which the quality has no
existence; whose existence therefore it implies. By this course the
denotation and connotation of abstract and of general names would be
exactly reversed. Whilst the denotation of a general name is limited by
the qualities connoted, the connotation of an abstract name includes all
the things in which its denotation is realised. But the whole difficulty
may be avoided by making it a rule to translate, for logical purposes,
all abstract into the corresponding general terms.

§ 4. If we ask how the connotation of a term is to be known, the answer
depends upon how it is used. If used scientifically, its connotation is
determined by, and is the same as, its definition; and the definition is
determined by examining the things to be denoted, as we shall see in
chap. xxii. If the same word is used as a term in different sciences, as
'property' in Law and in Logic, it will be differently defined by them,
and will have, in each use, a correspondingly different connotation. But
terms used in popular discourse should, as far as possible, have their
connotations determined by classical usage, i.e., by the sense in
which they are used by writers and speakers who are acknowledged masters
of the language, such as Dryden and Burke. In this case the classical
connotation determines the definition; so that to define terms thus used
is nothing else than to analyse their accepted meanings.

It must not, however, be supposed that in popular use the connotation
of any word is invariable. Logicians have attempted to classify
terms into Univocal (having only one meaning) and Æquivocal (or
ambiguous); and no doubt some words (like 'civil,' 'natural,' 'proud,'
'liberal,' 'humorous') are more manifestly liable to ambiguous use than
some others. But in truth all general terms are popularly and
classically used in somewhat different senses.

Figurative or tropical language chiefly consists in the transfer of
words to new senses, as by metaphor or metonymy. In the course of years,
too, words change their meanings; and before the time of Dryden our
whole vocabulary was much more fluid and adaptable than it has since
become. Such authors as Bacon, Milton, and Sir Thomas Browne often used
words derived from the Latin in some sense they originally had in Latin,
though in English they had acquired another meaning. Spenser and
Shakespeare, besides this practice, sometimes use words in a way that
can only be justified by their choosing to have it so; whilst their
contemporaries, Beaumont and Fletcher, write the perfect modern
language, as Dryden observed. Lapse of time, however, is not the chief
cause of variation in the sense of words. The matters which terms are
used to denote are often so complicated or so refined in the assemblage,
interfusion, or gradation of their qualities, that terms do not exist in
sufficient abundance and discriminativeness to denote the things and, at
the same time, to convey by connotation a determinate sense of their
agreements and differences. In discussing politics, religion, ethics,
æsthetics, this imperfection of language is continually felt; and the
only escape from it, short of coining new words, is to use such words as
we have, now in one sense, now in another somewhat different, and to
trust to the context, or to the resources of the literary art, in order
to convey the true meaning. Against this evil the having been born since
Dryden is no protection. It behoves us, then, to remember that terms are
not classifiable into Univocal and Æquivocal, but that all terms are
susceptible of being used æquivocally, and that honesty and lucidity
require us to try, as well as we can, to use each term univocally in the
same context.

The context of any proposition always proceeds upon some assumption or
understanding as to the scope of the discussion, which controls the
interpretation of every statement and of every word. This was called by
De Morgan the "universe of discourse": an older name for it, revived by
Dr. Venn, and surely a better one, is _suppositio_. If we are talking of
children, and 'play' is mentioned, the _suppositio_ limits the
suggestiveness of the word in one way; whilst if Monaco is the subject
of conversation, the same word 'play,' under the influence of a
different _suppositio_, excites altogether different ideas. Hence to
ignore the _suppositio_ is a great source of fallacies of equivocation.
'Man' is generally defined as a kind of animal; but 'animal' is often
used as opposed to and excluding man. 'Liberal' has one meaning under
the _suppositio_ of politics, another with regard to culture, and still
another as to the disposal of one's private means. Clearly, therefore,
the connotation of general terms is relative to the _suppositio_, or
"universe of discourse."

§ 5. Relative and Absolute Terms.--Some words go in couples or groups:
like 'up-down,' 'former-latter,' 'father-mother-children,'
'hunter-prey,' 'cause-effect,' etc. These are called Relative Terms,
and their nature, as explained by Mill, is that the connotations of the
members of such a pair or group are derived from the same set of facts
(the _fundamentum relationis_). There cannot be an 'up' without a
'down,' a 'father' without a 'mother' and 'child'; there cannot be a
'hunter' without something hunted, nor 'prey' without a pursuer. What
makes a man a 'hunter' is his activities in pursuit; and what turns a
chamois into 'prey' is its interest in these activities. The meaning of
both terms, therefore, is derived from the same set of facts; neither
term can be explained without explaining the other, because the relation
between them is connoted by both; and neither can with propriety be used
without reference to the other, or to some equivalent, as 'game' for

In contrast with such Relative Terms, others have been called Absolute
or Non-relative. Whilst 'hunter' and 'prey' are relative, 'man' and
'chamois' have been considered absolute, as we may use them without
thinking of any special connection between their meanings. However, if
we believe in the unity of Nature and in the relativity of knowledge
(that is, that all knowledge depends upon comparison, or a perception of
the resemblances and differences of things), it follows that nothing can
be completely understood except through its agreements or contrasts with
everything else, and that all terms derive their connotation from the
same set of facts, namely, from general experience. Thus both man and
chamois are animals; this fact is an important part of the meaning of
both terms, and to that extent they are relative terms. 'Five yards' and
'five minutes' are very different notions, yet they are profoundly
related; for their very difference helps to make both notions distinct;
and their intimate connection is shown in this, that five yards are
traversed in a certain time, and that five minutes are measured by the
motion of an index over some fraction of a yard upon the dial.

The distinction, then, between relative and non-relative terms must
rest, not upon a fundamental difference between them (since, in fact,
all words are relative), but upon the way in which words are used. We
have seen that some words, such as 'up-down,' 'cause-effect,' can only
be used relatively; and these may, for distinction, be called
Correlatives. But other words, whose meanings are only partially
interdependent, may often be used without attending to their relativity,
and may then be considered as Absolute. We cannot say 'the hunter
returned empty handed,' without implying that 'the prey escaped'; but we
may say 'the man went supperless to bed,' without implying that 'the
chamois rejoiced upon the mountain.' Such words as 'man' and 'chamois'
may, then, in their use, be, as to one another, non-relative.

To illustrate further the relativity of terms, we may mention some of
the chief classes of them.

Numerical order: 1st, 2nd, 3rd, etc.; 1st implies 2nd, and 2nd 1st;
and 3rd implies 1st and 2nd, but these do not imply 3rd; and so on.

Order in Time or Place: before-after; early-punctual-late;
right-middle-left; North-South, etc.

As to Extent, Volume, and Degree: greater-equal-less;
large-medium-small; whole and part.

Genus and Species are a peculiar case of whole and part (_cf._ chaps.
xxi.-ii.-iii.). Sometimes a term connotes all the attributes that
another does, and more besides, which, as distinguishing it, are called
differential. Thus 'man' connotes all that 'animal' does, and also (as
_differentiæ_) the erect gait, articulate speech, and other attributes.
In such a case as this, where there are well-marked classes, the term
whose connotation is included in the others' is called a Genus of that
Species. We have a Genus, triangle; and a Species, isosceles, marked off
from all other triangles by the differential quality of having two equal
sides: again--Genus, book; Species, quarto; Difference, having each
sheet folded into four leaves.

There are other cases where these expressions 'genus' and 'species'
cannot be so applied without a departure from usage, as, e.g., if we
call snow a species of the genus 'white,' for 'white' is not a
recognised class. The connotation of white (i.e., whiteness) is,
however, part of the connotation of snow, just as the qualities of
'animal' are amongst those of 'man'; and for logical purposes it is
desirable to use 'genus and species' to express that relativity of
terms which consists in the connotation of one being part of the
connotation of the other.

Two or more terms whose connotations severally include that of another
term, whilst at the same time exceeding it, are (in relation to that
other term) called Co-ordinate. Thus in relation to 'white,' snow and
silver are co-ordinate; in relation to colour, yellow and red and blue
are co-ordinate. And when all the terms thus related stand for
recognised natural classes, the co-ordinate terms are called co-ordinate
species; thus man and chamois are (in Logic) co-ordinate species of the
genus animal.

§ 6. From such examples of terms whose connotations are related as whole
and part, it is easy to see the general truth of the doctrine that as
connotation decreases, denotation increases: for 'animal,' with less
connotation than man or chamois, denotes many more objects; 'white,'
with less connotation than snow or silver, denotes many more things, It
is not, however, certain that this doctrine is always true in the
concrete: since there may be a term connoting two or more qualities, all
of which qualities are peculiar to all the things it denotes; and, if
so, by subtracting one of the qualities from its connotation, we should
not increase its denotation. If 'man,' for example, has among mammals
the two peculiar attributes of erect gait and articulate speech, then,
by omitting 'articulate speech' from the connotation of man, we could
not apply the name to any more of the existing mammalia than we can at
present. Still we might have been able to do so; there might have been
an erect inarticulate ape, and perhaps there once was one; and, if so,
to omit 'articulate' from the connotation of man would make the term
'man' denote that animal (supposing that there was no other difference
to exclude it). Hence, potentially, an increase of the connotation of
any term implies a decrease of its denotation. And, on the other hand,
we can only increase the denotation of a term, or apply it to more
objects, by decreasing its connotation; for, if the new things denoted
by the term had already possessed its whole connotation, they must
already have been denoted by it. However, we may increase the _known_
denotation without decreasing the connotation, if we can discover the
full connotation in things not formerly supposed to have it, as when
dolphins were discovered to be mammals; or if we can impose the
requisite qualities upon new individuals, as when by annexing some
millions of Africans we extend the denotation of 'British subject'
without altering its connotation.

Many of the things noticed in this chapter, especially in this section
and the preceding, will be discussed at greater length in the chapters
on Classification and Definition.

§ 7. Contradictory Relative Terms.--Every term has, or may have, another
corresponding with it in such a way that, whatever differential
qualities (§ 5) it connotes, this other connotes merely their absence;
so that one or the other is always formally predicable of any Subject,
but both these terms are never predicable of the same Subject in the
same relation: such pairs of terms are called Contradictories. Whatever
Subject we take, it is either visible or invisible, but not both; either
human or non-human, but not both.

This at least is true formally, though in practice we should think
ourselves trifled with if any one told us that 'A mountain is either
human or non-human, but not both.' It is symbolic terms, such as X and
x, that are properly said to be contradictories in relation to any
subject whatever, S or M. For, as we have seen, the ordinary use of
terms is limited by some _suppositio_, and this is true of
Contradictories. 'Human' and 'non-human' may refer to zoological
classification, or to the scope of physical, mental, or moral powers--as
if we ask whether to flourish a dumbbell of a ton weight, or to know the
future by intuition, or impeccability, be human or non-human. Similarly,
'visible' and 'invisible' refer either to the power of emitting or
reflecting light, so that the words have no hold upon a sound or a
scent, or else to power of vision and such qualifications as 'with the
naked eye' or 'with a microscope.'

Again, the above definition of Contradictories tells us that they cannot
be predicated of the same Subject "in the same relation"; that is, at
the same time or place, or under the same conditions. The lamp is
visible to me now, but will be invisible if I turn it out; one side of
it is now visible, but the other is not: therefore without this
restriction, "in the same relation," few or no terms would be

If a man is called wise, it may mean 'on the whole' or 'in a certain
action'; and clearly a man may for once be wise (or act wisely) who, on
the whole, is not-wise. So that here again, by this ambiguity, terms
that seem contradictory are predicable of the same subject, but not "in
the same relation." In order to avoid the ambiguity, however, we have
only to construct the term so as to express the relation, as 'wise on
the whole'; and this immediately generates the contradictory 'not-wise
on the whole.' Similarly, at one age a man may have black hair, at
another not-black hair; but the difficulty is practically removable by
stating the age referred to.

Still, this case easily leads us to a real difficulty in the use of
contradictory terms, a difficulty arising from the continuous change or
'flux' of natural phenomena. If things are continually changing, it may
be urged that contradictory terms are always applicable to the same
subject, at least as fast as we can utter them: for if we have just said
that a man's hair is black, since (like everything else) his hair is
changing, it must now be not-black, though (to be sure) it may still
seem black. The difficulty, such as it is, lies in this, that the human
mind and its instrument language are not equal to the subtlety of
Nature. All things flow, but the terms of human discourse assume a
certain fixity of things; everything at every moment changes, but for
the most part we can neither perceive this change nor express it in
ordinary language.

This paradox, however, may, I suppose, be easily over-stated. The change
that continually agitates Nature consists in the movements of masses or
molecules, and such movements of things are compatible with a
considerable persistence of their qualities. Not only are the molecular
changes always going on in a piece of gold compatible with its remaining
yellow, but its persistent yellowness depends on the continuance of some
of those changes. Similarly, a man's hair may remain black for some
years; though, no doubt, at a certain age its colour may begin to be
problematical, and the applicability to it of 'black' or 'not-black' may
become a matter of genuine anxiety. Whilst being on our guard, then,
against fallacies of contradiction arising from the imperfect
correspondence of fact with thought and language, we shall often have to
put up with it. Candour and humility having been satisfied by the above
acknowledgment of the subtlety of Nature, we may henceforward proceed
upon the postulate--that it is possible to use contradictory terms such
as cannot both be predicated of the same subject in the same relation,
though one of them may be; that, for example, it may be truly said of a
man for some years that his hair is black; and, if so, that during those
years to call it not-black is false or extremely misleading.

The most opposed terms of the literary vocabulary, however, such as
'wise-foolish,' 'old-young,' 'sweet-bitter,' are rarely true
contradictories: wise and foolish, indeed, cannot be predicated of the
same man in the same relation; but there are many middling men, of whom
neither can be predicated on the whole. For the comparison of
quantities, again, we have three correlative terms,
'greater--equal--less,' and none of these is the contradictory of either
of the others. In fact, the contradictory of any term is one that
denotes the sum of its co-ordinates (§ 6); and to obtain a
contradictory, the surest way is to coin one by prefixing to the given
term the particle 'not' or (sometimes) 'non': as 'wise, not-wise,'
'human, non-human,' 'greater, not-greater.'

The separate word 'not' is surer to constitute a contradictory than the
usual prefixes of negation, 'un-' or 'in-,' or even 'non'; since
compounds of these are generally warped by common use from a purely
negative meaning. Thus, 'Nonconformist' does not denote everybody who
fails to conform. 'Unwise' is not equivalent to 'not-wise,' but means
'rather foolish'; a very foolish action is not-wise, but can only be
called unwise by meiosis or irony. Still, negatives formed by 'in' or
'un' or 'non' are sometimes really contradictory of their positives; as
'visible, invisible,' 'equal, unequal.'

§ 8. The distinction between Positive and Negative terms is not of much
value in Logic, what importance would else attach to it being absorbed
by the more definite distinction of contradictories. For contradictories
are positive and negative in essence and, when least ambiguously stated,
also in form. And, on the other hand, as we have seen, when positive and
negative terms are not contradictory, they are misleading. As with
'wise-unwise,' so with many others, such as 'happy-unhappy'; which are
not contradictories; since a man may be neither happy nor unhappy, but
indifferent, or (again) so miserable that he can only be called unhappy
by a figure of speech. In fact, in the common vocabulary a formal
negative often has a limited positive sense; and this is the case with
unhappy, signifying the state of feeling in the milder shades of

When a Negative term is fully contradictory of its Positive it is said
to be Infinite; because it denotes an unascertained multitude of things,
a multitude only limited by the positive term and the _suppositio_; thus
'not-wise' denotes all except the wise, within the _suppositio_ of
'intelligent beings.' Formally (disregarding any _suppositio_), such a
negative term stands for all possible terms except its positive: x
denotes everything but X; and 'not-wise' may be taken to include stones,
triangles and hippogriffs. And even in this sense, a negative term has
some positive meaning, though a very indefinite one, not a specific
positive force like 'unwise' or 'unhappy': it denotes any and everything
that has not the attributes connoted by the corresponding positive term.

Privative Terms connote the absence of a quality that normally belongs
to the kind of thing denoted, as 'blind' or 'deaf.' We may predicate
'blind' or 'deaf' of a man, dog or cow that happens not to be able to
see or hear, because the powers of seeing and hearing generally belong
to those species; but of a stone or idol these terms can only be used
figuratively. Indeed, since the contradictory of a privative carries
with it the privative limitation, a stone is strictly 'not-blind': that
is, it is 'not-something-that-normally-having-sight-wants-it.'

Contrary Terms are those that (within a certain genus or _suppositio_)
severally connote differential qualities that are, in fact, mutually
incompatible in the same relation to the same thing, and therefore
cannot be predicated of the same subject in the same relation; and, so
far, they resemble Contradictory Terms: but they differ from
contradictory terms in this, that the differential quality connoted by
each of them is definitely positive; no Contrary Term is infinite, but
is limited to part of the _suppositio_ excluded by the others; so that,
possibly, neither of two Contraries is truly predicable of a given
subject. Thus 'blue' and 'red' are Contraries, for they cannot both be
predicated of the same thing in the same relation; but are not
Contradictories, since, in a given case, neither may be predicable: if a
flower is blue in a certain part, it cannot in the same part be red; but
it may be neither blue nor red, but yellow; though it is certainly
either blue or not-blue. All co-ordinate terms are formal Contraries;
but if, in fact, a series of co-ordinates comprises only two (as
male-female), they are empirical Contradictories; since each includes
all that area of the _suppositio_ which the other excludes.

The extremes of a series of co-ordinate terms are Opposites; as, in a
list of colours, white and black, the most strongly contrasted, are said
to be opposites, or as among moods of feeling, rapture and misery are
opposites. But this distinction is of slight logical importance.
Imperfect Positive and Negative couples, like 'happy and unhappy,' which
(as we have seen) are not contradictories, are often called Opposites.

The members of any series of Contraries are all included by any one of
them and its contradictory, as all colours come under 'red' and
'not-red,' all moods of feeling under 'happy' and 'not-happy.'



§ 1. Logicians classify Propositions according to Quantity, Quality,
Relation and Modality.

As to Quantity, propositions are either Universal or Particular; that is
to say, the predicate is affirmed or denied either of the whole subject
or of a part of it--of _All_ or of _Some S_.

    _All S is P_ (that is, _P_ is predicated of _all S_).
    _Some S is P_ (that is, _P_ is predicated of _some S_).

An Universal Proposition may have for its subject a singular term, a
collective, a general term distributed, or an abstract term.

(1) A proposition having a singular term for its subject, as _The Queen
has gone to France_, is called a Singular Proposition; and some
Logicians regard this as a third species of proposition with respect to
quantity, distinct from the Universal and Particular; but that is

(2) A collective term may be the subject, as _The Black Watch is ordered
to India_. In this case, as well as in singular propositions, a
predication is made concerning the whole subject as a whole.

(3) The subject may be a general term taken in its full denotation, as
_All apes are sagacious_; and in this case a Predication is made
concerning the whole subject distributively; that is, of each and
everything the subject stands for.

(4) Propositions whose subjects are abstract terms, though they may
seem to be formally Singular, are really as to their meaning
distributive Universals; since whatever is true of a quality is true of
whatever thing has that quality so far as that quality is concerned.
_Truth will prevail_ means that _All true propositions are accepted at
last_ (by sheer force of being true, in spite of interests, prejudices,
ignorance and indifference). To bear this in mind may make one cautious
in the use of abstract terms.

In the above paragraphs a distinction is implied between Singular and
Distributive Universals; but, technically, every term, whether subject
or predicate, when taken in its full denotation (or universally), is
said to be 'distributed,' although this word, in its ordinary sense,
would be directly applicable only to general terms. In the above
examples, then, 'Queen,' 'Black Watch,' 'apes,' and 'truth' are all
distributed terms. Indeed, a simple definition of the Universal
Proposition is 'one whose subject is distributed.'

A Particular Proposition is one that has a general term for its subject,
whilst its predicate is not affirmed or denied of everything the subject
denotes; in other words, it is one whose subject is not distributed: as
_Some lions inhabit Africa_.

In ordinary discourse it is not always explicitly stated whether
predication is universal or particular; it would be very natural to say
_Lions inhabit Africa_, leaving it, as far as the words go, uncertain
whether we mean _all_ or _some_ lions. Propositions whose quantity is
thus left indefinite are technically called 'preindesignate,' their
quantity not being stated or designated by any introductory expression;
whilst propositions whose quantity is expressed, as _All
foundling-hospitals have a high death-rate_, or _Some wine is made from
grapes_, are said to be 'predesignate.' Now, the rule is that
preindesignate propositions are, for logical purposes, to be treated as
particular; since it is an obvious precaution of the science of proof,
in any practical application, _not to go beyond the evidence_. Still,
the rule may be relaxed if the universal quantity of a preindesignate
proposition is well known or admitted, as in _Planets shine with
reflected light_--understood of the planets of our solar system at the
present time. Again, such a proposition as _Man is the paragon of
animals_ is not a preindesignate, but an abstract proposition; the
subject being elliptical for _Man according to his proper nature_; and
the translation of it into a predesignate proposition is not _All men
are paragons_; nor can _Some men_ be sufficient, since an abstract can
only be adequately rendered by a distributed term; but we must say, _All
men who approach the ideal_. Universal real propositions, true without
qualification, are very scarce; and we often substitute for them
_general_ propositions, saying perhaps--_generally, though not
universally, S is P_. Such general propositions are, in strictness,
particular; and the logical rules concerning universals cannot be
applied to them without careful scrutiny of the facts.

The marks or predesignations of Quantity commonly used in Logic are: for
Universals, _All_, _Any_, _Every_, _Whatever_ (in the negative _No_ or
_No one_, see next §); for Particulars, _Some_.

Now _Some_, technically used, does not mean _Some only,_ but _Some at
least_ (it may be one, or more, or all). If it meant '_Some only_,'
every particular proposition would be an exclusive exponible (chap. ii.
§ 3); since _Only some men are wise_ implies that _Some men are not
wise_. Besides, it may often happen in an investigation that all the
instances we have observed come under a certain rule, though we do not
yet feel justified in regarding the rule as universal; and this
situation is exactly met by the expression _Some_ (_it may be all_).

The words _Many_, _Most_, _Few_ are generally interpreted to mean
_Some_; but as _Most_ signifies that exceptions are known, and _Few_
that the exceptions are the more numerous, propositions thus
predesignate are in fact exponibles, mounting to _Some are_ and _Some
are not_. If to work with both forms be too cumbrous, so that we must
choose one, apparently _Few are_ should be treated as _Some are not_.
The scientific course to adopt with propositions predesignate by _Most_
or _Few_, is to collect statistics and determine the percentage; thus,
_Few men are wise_--say 2 per cent.

The Quantity of a proposition, then, is usually determined entirely by
the quantity of the subject, whether _all_ or _some_. Still, the
quantity of the predicate is often an important consideration; and
though in ordinary usage the predicate is seldom predesignate, Logicians
agree that in every Negative Proposition (see § 2) the predicate is
'distributed,' that is to say, is denied altogether of the subject, and
that this is involved in the form of denial. To say _Some men are not
brave_, is to declare that the quality for which men may be called brave
is not found in any of the _Some men_ referred to: and to say _No men
are proof against flattery_, cuts off the being 'proof against flattery'
entirely from the list of human attributes. On the other hand, every
Affirmative Proposition is regarded as having an undistributed
predicate; that is to say, its predicate is not affirmed exclusively of
the subject. _Some men are wise_ does not mean that 'wise' cannot be
predicated of any other beings; it is equivalent to _Some men are wise_
(_whoever else may be_). And _All elephants are sagacious_ does not
limit sagacity to elephants: regarding 'sagacious' as possibly denoting
many animals of many species that exhibit the quality, this proposition
is equivalent to '_All elephants are_ some _sagacious animals_.' The
affirmative predication of a quality does not imply exclusive possession
of it as denial implies its complete absence; and, therefore, to regard
the predicate of an affirmative proposition as distributed would be to
go beyond the evidence and to take for granted what had never been

Some Logicians, seeing that the quantity of predicates, though not
distinctly expressed, is recognised, and holding that it is the part of
Logic "to make explicit in language whatever is implicit in thought,"
have proposed to exhibit the quantity of predicates by predesignation,
thus: 'Some men are _some_ wise (beings)'; 'some men are not _any_ brave
(beings)'; etc. This is called the Quantification of the Predicate,
and leads to some modifications of Deductive Logic which will be
referred to hereafter. (See § 3; chap. vii. § 4, and chap. viii. § 3.)

§ 2. As to Quality, Propositions are either Affirmative or Negative. An
Affirmative Proposition is, formally, one whose copula is affirmative
(or, has no negative sign), as _S--is--P, All men--are--partial to
themselves_. A Negative Proposition is one whose copula is negative (or,
has a negative sign), as _S--is not--P, Some men--are not--proof against
flattery_. When, indeed, a Negative Proposition is of Universal
Quantity, it is stated thus: _No S is P, No men are proof against
flattery_; but, in this case, the detachment of the negative sign from
the copula and its association with the subject is merely an accident of
our idiom; the proposition is the same as _All men--are not--proof
against flattery_. It must be distinguished, therefore, from such an
expression as _Not every man is proof against flattery_; for here the
negative sign really restricts the subject; so that the meaning
is--_Some men at most_ (it may be _none) are proof against flattery_;
and thus the proposition is Particular, and is rendered--_Some men--are
not--proof against flattery_.

When the negative sign is associated with the predicate, so as to make
this an Infinite Term (chap. iv. § 8), the proposition is called an
Infinite Proposition, as _S is not-P_ (or _p), All men are--incapable of
resisting flattery_, or _are--not-proof against flattery_.

Infinite propositions, when the copula is affirmative, are formally,
themselves affirmative, although their force is chiefly negative; for,
as the last example shows, the difference between an infinite and a
negative proposition may depend upon a hyphen. It has been proposed,
indeed, with a view to superficial simplification, to turn all
Negatives into Infinites, and thus render all propositions Affirmative
in Quality. But although every proposition both affirms and denies
something according to the aspect in which you regard it (as _Snow is
white_ denies that it is any other colour, and _Snow is not blue_
affirms that it is some other colour), yet there is a great difference
between the definite affirmation of a genuine affirmative and the vague
affirmation of a negative or infinite; so that materially an affirmative
infinite is the same as a negative.

Generally Mill's remark is true, that affirmation and denial stand for
distinctions of fact that cannot be got rid of by manipulation of words.
Whether granite sinks in water, or not; whether the rook lives a hundred
years, or not; whether a man has a hundred dollars in his pocket, or
not; whether human bones have ever been found in Pliocene strata, or
not; such alternatives require distinct forms of expression. At the same
time, it may be granted that many facts admit of being stated with
nearly equal propriety in either Quality, as _No man is proof against
flattery_, or _All men are open to flattery_.

But whatever advantage there is in occasionally changing the Quality of
a proposition may be gained by the process of Obversion (chap. vii. §
5); whilst to use only one Quality would impair the elasticity of
logical expression. It is a postulate of Logic that the negative sign
may be transferred from the copula to the predicate, or from the
predicate to the copula, without altering the sense of a proposition;
and this is justified by the experience that not to have an attribute
and to be without it are the same thing.

§ 3. A. I. E. O.--Combining the two kinds of Quantity, Universal and
Particular, with the two kinds of Quality, Affirmative and Negative, we
get four simple types of proposition, which it is usual to symbolise by
the letters A. I. E. O., thus:

A.   Universal Affirmative   -- All S is P.
I.   Particular Affirmative  -- Some S is P.
E.   Universal Negative      -- No S is P.
O.   Particular Negative     -- Some S is not P.

As an aid to the remembering of these symbols we may observe that A. and
I. are the first two vowels in _affirmo_ and that E. and O. are the
vowels in _nego_.

It must be acknowledged that these four kinds of proposition recognised
by Formal Logic constitute a very meagre selection from the list of
propositions actually used in judgment and reasoning.

Those Logicians who explicitly quantify the predicate obtain, in all,
eight forms of proposition according to Quantity and Quality:

U.   Toto-total Affirmative      -- All X is all Y.
A.   Toto-partial Affirmative    -- All X is some Y.
Y.   Parti-total Affirmative     -- Some X is all Y.
I.   Parti-partial Affirmative   -- Some X is some Y.
E.   Toto-total Negative         -- No X is any Y.
η.   Toto-partial Negative       -- No X is some Y.
O.   Parti-total Negative        -- Some X is not any Y.
ω.   Parti-partial Negative      -- Some X is not some Y.

Here A. I. E. O. correspond with those similarly symbolised in the usual
list, merely designating in the predicates the quantity which was
formerly treated as implicit.

§ 4. As to Relation, propositions are either Categorical or Conditional.
A Categorical Proposition is one in which the predicate is directly
affirmed or denied of the subject without any limitation of time, place,
or circumstance, extraneous to the subject, as _All men in England are
secure of justice_; in which proposition, though there is a limitation
of place ('in England'), it is included in the subject. Of this kind are
nearly all the examples that have yet been given, according to the form
_S is P_.

A Conditional Proposition is so called because the predication is made
under some limitation or condition not included in the subject, as _If a
man live in England, he is secure of justice_. Here the limitation
'living in England' is put into a conditional sentence extraneous to the
subject, 'he,' representing any man.

Conditional propositions, again, are of two kinds--Hypothetical and
Disjunctive. Hypothetical propositions are those that are limited by an
explicit conditional sentence, as above, or thus: _If Joe Smith was a
prophet, his followers have been unjustly persecuted_. Or in symbols

    If A is, B is;
    If A is B, A is C;
    If A is B, C is D.

Disjunctive propositions are those in which the condition under which
predication is made is not explicit but only implied under the disguise
of an alternative proposition, as _Joe Smith was either a prophet or an
impostor_. Here there is no direct predication concerning Joe Smith, but
only a predication of one of the alternatives conditionally on the other
being denied, as, _If Joe Smith was not a prophet he was an impostor_;
or, _If he was not an impostor, he was a prophet_. Symbolically,
Disjunctives may be represented thus:

    A is either B or C,
    Either A is B or C is D.

Formally, every Conditional may be expressed as a Categorical. For our
last example shows how a Disjunctive may be reduced to two Hypotheticals
(of which one is redundant, being the contrapositive of the other; see
chap. vii. § 10). And a Hypothetical is reducible to a Categorical thus:
_If the sky is clear, the night is cold_ may be read--_The case of the
sky being clear is a case of the night being cold_; and this, though a
clumsy plan, is sometimes convenient. It would be better to say _The sky
being clear is a sign of the night being cold_, or a condition of it.
For, as Mill says, the essence of a Hypothetical is to state that one
clause of it (the indicative) may be inferred from the other (the
conditional). Similarly, we might write: _Proof of Joe Smith's not being
a prophet is a proof of his being an impostor_.

This turning of Conditionals into Categoricals is called a Change of
Relation; and the process may be reversed: _All the wise are virtuous_
may be written, _If any man is wise he is virtuous_; or, again, _Either
a man is not-wise or he is virtuous_. But the categorical form is
usually the simplest.

If, then, as substitutes for the corresponding conditionals,
categoricals are formally adequate, though sometimes inelegant, it may
be urged that Logic has nothing to do with elegance; or that, at any
rate, the chief elegance of science is economy, and that therefore, for
scientific purposes, whatever we may write further about conditionals
must be an ugly excrescence. The scientific purpose of Logic is to
assign the conditions of proof. Can we, then, in the conditional form
prove anything that cannot be proved in the categorical? Or does a
conditional require to be itself proved by any method not applicable to
the Categorical? If not, why go on with the discussion of Conditionals?
For all laws of Nature, however stated, are essentially categorical. 'If
a straight line falls on another straight line, the adjacent angles are
together equal to two right angles'; 'If a body is unsupported, it
falls'; 'If population increases, rents tend to rise': here 'if' means
'whenever' or 'all cases in which'; for to raise a doubt whether a
straight line is ever conceived to fall upon another, whether bodies are
ever unsupported, or population ever increases, is a superfluity of
scepticism; and plainly the hypothetical form has nothing to do with the
proof of such propositions, nor with inference from them.

Still, the disjunctive form is necessary in setting out the relation of
contradictory terms, and in stating a Division (chap. xxi.), whether
formal (_as A is B or not-B_) or material (as _Cats are white, or black,
or tortoiseshell, or tabby_). And in some cases the hypothetical form is
useful. One of these occurs where it is important to draw attention to
the condition, as something doubtful or especially requiring
examination. _If there is a resisting medium in space, the earth will
fall into the sun; If the Corn Laws are to be re-enacted, we had better
sell railways and buy land_: here the hypothetical form draws attention
to the questions whether there is a resisting medium in space, whether
the Corn Laws are likely to be re-enacted; but as to methods of
inference and proof, the hypothetical form has nothing to do with them.
The propositions predicate causation: _A resisting medium in space is a
condition of the earth's falling into the sun; A Corn Law is a condition
of the rise of rents, and of the fall of railway profits_.

A second case in which the hypothetical is a specially appropriate form
of statement occurs where a proposition relates to a particular matter
and to future time, as _If there be a storm to-morrow, we shall miss our
picnic_. Such cases are of very slight logical interest. It is as
exercises in formal thinking that hypotheticals are of most value;
inasmuch as many people find them more difficult than categoricals to

In discussing Conditional Propositions, the conditional sentence of a
Hypothetical, or the first alternative of a Disjunctive, is called the
Antecedent; the indicative sentence of a Hypothetical, or the second
alternative of a Disjunctive, is called the Consequent.

Hypotheticals, like Categoricals, have been classed according to
Quantity and Quality. Premising that the quantity of a Hypothetical
depends on the quantity of its Antecedent (which determines its
limitation), whilst its quality depends on the quality of its consequent
(which makes the predication), we may exhibit four forms:

    A. _If A is B, C is D;_
    I. _Sometimes when A is B, C is D;_
    E. _If A is B, C is not D;_
    O. _Sometimes when A is B, C is not D._

But I. and O. are rarely used.

As for Disjunctives, it is easy to distinguish the two quantities thus:

    A. _Either A is B, or C is D;_
    I. _Sometimes either A is B or C is D._

But I. is rarely used. The distinction of quality, however, cannot be
made: there are no true negative forms; for if we write--

    _Neither is A B, nor C D,_

there is here no alternative predication, but only an Exponible
equivalent to _No A is B, and No C is D_. And if we write--

    _Either A is not B, or C is not D,_

this is affirmative as to the alternation, and is for all methods of
treatment equivalent to A.

Logicians are divided in opinion as to the interpretation of the
conjunction 'either, or'; some holding that it means 'not both,' others
that it means 'it may be both.' Grammatical usage, upon which the
question is sometimes argued, does not seem to be established in favour
of either view. If we say _A man so precise in his walk and conversation
is either a saint or a consummate hypocrite_; or, again, _One who is
happy in a solitary life is either more or less than man_; we cannot in
such cases mean that the subject may be both. On the other hand, if it
be said that _the author of 'A Tale of a Tub' is either a misanthrope or
a dyspeptic_, the alternatives are not incompatible. Or, again, given
that _X. is a lunatic, or a lover, or a poet_, the three predicates have
much congruity.

It has been urged that in Logic, language should be made as exact and
definite as possible, and that this requires the exclusive
interpretation 'not both.' But it seems a better argument, that Logic
(1) should be able to express all meanings, and (2), as the science of
evidence, must not assume more than is given; to be on the safe side, it
must in doubtful cases assume the least, just as it generally assumes a
preindesignate term to be of particular quantity; and, therefore
'either, or' means 'one, or the other, or both.'

However, when both the alternative propositions have the same subject,
as _Either A is B, or A is C_, if the two predicates are contrary or
contradictory terms (as 'saint' and 'hypocrite,' or 'saint' and
'not-saint'), they cannot in their nature be predicable in the same way
of the same subject; and, therefore, in such a case 'either, or' means
one or the other, but not both in the same relation. Hence it seems
necessary to admit that the conjunction 'either, or' may sometimes
require one interpretation, sometimes the other; and the rule is that it
implies the further possibility 'or both,' except when both alternatives
have the same subject whilst the predicates are contrary or
contradictory terms.

If, then, the disjunctive _A is either B or C_ (_B_ and _C_ being
contraries) implies that both alternatives cannot be true, it can only
be adequately rendered in hypotheticals by the two forms--(1) _If A is
B, it is not C_, and (2)_If A is not B, it is C_. But if the disjunctive
_A is either B or C_ (_B_ and _C_ not being contraries) implies that
both may be true, it will be adequately translated into a hypothetical
by the single form, _If A is not B, it is C_. We cannot translate it
into--_If A is B, it is not C_, for, by our supposition, if '_A is B_'
is true, it does not follow that '_A is C_' must be false.

Logicians are also divided in opinion as to the function of the
hypothetical form. Some think it expresses doubt; for the consequent
depends on the antecedent, and the antecedent, introduced by 'if,' may
or may not be realised, as in _If the sky is clear, the night is cold_:
whether the sky is, or is not, clear being supposed to be uncertain. And
we have seen that some hypothetical propositions seem designed to draw
attention to such uncertainty, as--_If there is a resisting medium in
space_, etc. But other Logicians lay stress upon the connection of the
clauses as the important matter: the statement is, they say, that the
consequent may be inferred from the antecedent. Some even declare that
it is given as a necessary inference; and on this ground Sigwart rejects
particular hypotheticals, such as _Sometimes when A is B, C is D_; for
if it happens only sometimes the connexion cannot be necessary. Indeed,
it cannot even be probably inferred without further grounds. But this is
also true whenever the antecedent and consequent are concerned with
different matter. For example, _If the soul is simple, it is
indestructible_. How do you know that? Because _Every simple substance
is indestructible_. Without this further ground there can be no
inference. The fact is that conditional forms often cover assertions
that are not true complex propositions but a sort of euthymemes (chap.
xi. § 2), arguments abbreviated and rhetorically disguised. Thus: _If
patience is a virtue there are painful virtues_--an example from Dr.
Keynes. Expanding this we have--

      Patience is painful;
      Patience is a virtue:
    ∴ Some virtue is painful.

And then we see the equivocation of the inference; for though patience
be painful _to learn_, it is not painful _as a virtue_ to the patient

The hypothetical, '_If Plato was not mistaken poets are dangerous
citizens_,' may be considered as an argument against the laureateship,
and may be expanded (informally) thus: 'All Plato's opinions deserve
respect; one of them was that poets are bad citizens; therefore it
behoves us to be chary of encouraging poetry.' Or take this
disjunctive, '_Either Bacon wrote the works ascribed to Shakespeare, or
there were two men of the highest genius in the same age and country_.'
This means that it is not likely there should be two such men, that we
are sure of Bacon, and therefore ought to give him all the glory. Now,
if it is the part of Logic 'to make explicit in language all that is
implicit in thought,' or to put arguments into the form in which they
can best be examined, such propositions as the above ought to be
analysed in the way suggested, and confirmed or refuted according to
their real intention.

We may conclude that no single function can be assigned to all
hypothetical propositions: each must be treated according to its own
meaning in its own context.

§ 5. As to Modality, propositions are divided into Pure and Modal. A
Modal proposition is one in which the predicate is affirmed or denied,
not simply but _cum modo_, with a qualification. And some Logicians have
considered any adverb occurring in the predicate, or any sign of past or
future tense, enough to constitute a modal: as 'Petroleum is
_dangerously_ inflammable'; 'English _will be_ the universal language.'
But far the most important kind of modality, and the only one we need
consider, is that which is signified by some qualification of the
predicate as to the degree of certainty with which it is affirmed or
denied. Thus, 'The bite of the cobra is _probably_ mortal,' is called a
Contingent or Problematic Modal: 'Water is _certainly_ composed of
oxygen and hydrogen' is an Assertory or Certain Modal: 'Two straight
lines _cannot_ enclose a space' is a Necessary or Apodeictic Modal (the
opposite being inconceivable). Propositions not thus qualified are
called Pure.

Modal propositions have had a long and eventful history, but they have
not been found tractable by the resources of ordinary Logic, and are now
generally neglected by the authors of text-books. No doubt such
propositions are the commonest in ordinary discourse, and in some rough
way we combine them and draw inferences from them. It is understood
that a combination of assertory or of apodeictic premises may warrant an
assertory or an apodeictic conclusion; but that if we combine either of
these with a problematic premise our conclusion becomes problematic;
whilst the combination of two problematic premises gives a conclusion
less certain than either. But if we ask 'How much less certain?' there
is no answer. That the modality of a conclusion follows the less certain
of the premises combined, is inadequate for scientific guidance; so
that, as Deductive Logic can get no farther than this, it has abandoned
the discussion of Modals. To endeavour to determine the degree of
certainty attaching to a problematic judgment is not, however, beyond
the reach of Induction, by analysing circumstantial evidence, or by
collecting statistics with regard to it. Thus, instead of 'The cobra's
bite is _probably_ fatal,' we might find that it is fatal 80 times in
100. Then, if we know that of those who go to India 3 in 1000 are
bitten, we can calculate what the chances are that any one going to
India will die of a cobra's bite (chap. xx.).

§ 6. Verbal and Real Propositions.--Another important division of
propositions turns upon the relation of the predicate to the subject in
respect of their connotations. We saw, when discussing Relative Terms,
that the connotation of one term often implies that of another;
sometimes reciprocally, like 'master' and 'slave'; or by inclusion, like
species and genus; or by exclusion, like contraries and contradictories.
When terms so related appear as subject and predicate of the same
proposition, the result is often tautology--e.g., _The master has
authority over his slave; A horse is an animal; Red is not blue; British
is not foreign_. Whoever knows the meaning of 'master,' 'horse,' 'red,'
'British,' learns nothing from these propositions. Hence they are called
Verbal propositions, as only expounding the sense of words, or as if
they were propositions only by satisfying the forms of language, not by
fulfilling the function of propositions in conveying a knowledge of
facts. They are also called 'Analytic' and 'Explicative,' when they
separate and disengage the elements of the connotation of the subject.
Doubtless, such propositions may be useful to one who does not know the
language; and Definitions, which are verbal propositions whose
predicates analyse the whole connotations of their subjects, are
indispensable instruments of science (see chap. xxii.).

Of course, hypothetical propositions may also be verbal, as _If the soul
be material it is extended_; for 'extension' is connoted by 'matter';
and, therefore, the corresponding disjunctive is verbal--_Either the
soul is not material, or it is extended_. But a true divisional
disjunctive can never be verbal (chap. xxi. § 4, rule 1).

On the other hand, when there is no such direct relation between subject
and predicate that their connotations imply one another, but the
predicate connotes something that cannot be learnt from the connotation
of the subject, there is no longer tautology, but an enlargement of
meaning--e.g., _Masters are degraded by their slaves; The horse is the
noblest animal; Red is the favourite colour of the British army; If the
soul is simple, it is indestructible_. Such propositions are called
Real, Synthetic, or Ampliative, because they are propositions for which
a mere understanding of their subjects would be no substitute, since the
predicate adds a meaning of its own concerning matter of fact.

To any one who understands the language, a verbal proposition can never
be an inference or conclusion from evidence; nor can a verbal
proposition ever furnish grounds for an inference, except as to the
meaning of words. The subject of real and verbal propositions will
inevitably recur in the chapters on Definition; but tautologies are such
common blemishes in composition, and such frequent pitfalls in argument,
that attention cannot be drawn to them too early or too often.



§ 1. The word Inference is used in two different senses, which are often
confused but should be carefully distinguished. In the first sense, it
means a process of thought or reasoning by which the mind passes from
facts or statements presented, to some opinion or expectation. The data
may be very vague and slight, prompting no more than a guess or surmise;
as when we look up at the sky and form some expectation about the
weather, or from the trick of a man's face entertain some prejudice as
to his character. Or the data may be important and strongly significant,
like the footprint that frightened Crusoe into thinking of cannibals, or
as when news of war makes the city expect that Consols will fall. These
are examples of the act of inferring, or of inference as a process; and
with inference in this sense Logic has nothing to do; it belongs to
Psychology to explain how it is that our minds pass from one perception
or thought to another thought, and how we come to conjecture, conclude
and believe (_cf._ chap. i. § 6).

In the second sense, 'inference' means not this process of guessing or
opining, but the result of it; the surmise, opinion, or belief when
formed; in a word, the conclusion: and it is in this sense that
Inference is treated of in Logic. The subject-matter of Logic is an
inference, judgment or conclusion concerning facts, embodied in a
proposition, which is to be examined in relation to the evidence that
may be adduced for it, in order to determine whether, or how far, the
evidence amounts to proof. Logic is the science of Reasoning in the
sense in which 'reasoning' means giving reasons, for it shows what sort
of reasons are good. Whilst Psychology explains how the mind goes
forward from data to conclusions, Logic takes a conclusion and goes back
to the data, inquiring whether those data, together with any other
evidence (facts or principles) that can be collected, are of a nature to
warrant the conclusion. If we think that the night will be stormy, that
John Doe is of an amiable disposition, that water expands in freezing,
or that one means to national prosperity is popular education, and wish
to know whether we have evidence sufficient to justify us in holding
these opinions, Logic can tell us what form the evidence should assume
in order to be conclusive. What _form_ the evidence should assume: Logic
cannot tell us what kinds of fact are proper evidence in any of these
cases; that is a question for the man of special experience in life, or
in science, or in business. But whatever facts constitute the evidence,
they must, in order to prove the point, admit of being stated in
conformity with certain principles or conditions; and of these
principles or conditions Logic is the science. It deals, then, not with
the subjective process of inferring, but with the objective grounds that
justify or discredit the inference.

§ 2. Inferences, in the Logical sense, are divided into two great
classes, the Immediate and the Mediate, according to the character of
the evidence offered in proof of them. Strictly, to speak of inferences,
in the sense of conclusions, as immediate or mediate, is an abuse of
language, derived from times before the distinction between inference as
process and inference as result was generally felt. No doubt we ought
rather to speak of Immediate and Mediate Evidence; but it is of little
use to attempt to alter the traditional expressions of the science.

An Immediate Inference, then, is one that depends for its proof upon
only one other proposition, which has the same, or more extensive,
terms (or matter). Thus that _one means to national prosperity is
popular education_ is an immediate inference, if the evidence for it is
no more than the admission that _popular education is a means to
national prosperity:_ Similarly, it is an immediate inference that _Some
authors are vain_, if it be granted that _All authors are vain_.

An Immediate Inference may seem to be little else than a verbal
transformation; some Logicians dispute its claims to be called an
inference at all, on the ground that it is identical with the pretended
evidence. If we attend to the meaning, say they, an immediate inference
does not really express any new judgment; the fact expressed by it is
either the same as its evidence, or is even less significant. If from
_No men are gods_ we prove that _No gods are men_, this is nugatory; if
we prove from it that _Some men are not gods_, this is to emasculate the
sense, to waste valuable information, to lose the commanding sweep of
our universal proposition.

Still, in Logic, it is often found that an immediate inference expresses
our knowledge in a more convenient form than that of the evidentiary
proposition, as will appear in the chapter on Syllogisms and elsewhere.
And by transforming an universal into a particular proposition, as _No
men are gods_, therefore, _Some men are not gods_,--we get a statement
which, though weaker, is far more easily proved; since a single instance
suffices. Moreover, by drawing all possible immediate inferences from a
given proposition, we see it in all its aspects, and learn all that is
implied in it.

A Mediate Inference, on the other hand, depends for its evidence upon a
plurality of other propositions (two or more) which are connected
together on logical principles. If we argue--

      No men are gods;
      Alexander the Great is a man;
    ∴ Alexander the Great is not a god:

this is a Mediate Inference. The evidence consists of two propositions
connected by the term 'man,' which is common to both (a Middle Term),
mediating between 'gods' and 'Alexander.' Mediate Inferences comprise
Syllogisms with their developments, and Inductions; and to discuss them
further at present would be to anticipate future chapters. We must now
deal with the principles or conditions on which Immediate Inferences are
valid: commonly called the "Laws of Thought."

§ 3. The Laws of Thought are conditions of the logical statement and
criticism of all sorts of evidence; but as to Immediate Inference, they
may be regarded as the only conditions it need satisfy. They are often
expressed thus: (1) The principle of Identity--'_Whatever is, is_'; (2)
The principle of Contradiction--'_It is impossible for the same thing to
be and not be_'; (3) The principle of Excluded Middle--'_Anything must
either be or not be_.' These principles are manifestly not 'laws' of
thought in the sense in which 'law' is used in Psychology; they do not
profess to describe the actual mental processes that take place in
judgment or reasoning, as the 'laws of association of ideas' account for
memory and recollection. They are not natural laws of thought; but, in
relation to thought, can only be regarded as laws when stated as
precepts, the observance of which (consciously or not) is necessary to
clear and consistent thinking: e.g., Never assume that the same thing
can both be and not be.

However, treating Logic as the science of thought only as embodied in
propositions, in respect of which evidence is to be adduced, or which
are to be used as evidence of other propositions, the above laws or
principles must be restated as the conditions of consistent argument in
such terms as to be directly applicable to propositions. It was shown in
the chapter on the connotation of terms, that terms are assumed by
Logicians to be capable of definite meaning, and of being used
univocally in the same context; if, or in so far as, this is not the
case, we cannot understand one another's reasons nor even pursue in
solitary meditation any coherent train of argument. We saw, too, that
the meanings of terms were related to one another: some being full
correlatives; others partially inclusive one of another, as species of
genus; others mutually incompatible, as contraries; or alternatively
predicable, as contradictories. We now assume that propositions are
capable of definite meaning according to the meaning of their component
terms and of the relation between them; that the meaning, the fact
asserted or denied, is what we are really concerned to prove or
disprove; that a mere change in the words that constitute our terms, or
of construction, does not affect the truth of a proposition as long as
the meaning is not altered, or (rather) as long as no fresh meaning is
introduced; and that if the meaning of any proposition is true, any
other proposition that denies it is false. This postulate is plainly
necessary to consistency of statement and discourse; and consistency is
necessary, if our thought or speech is to correspond with the unity and
coherence of Nature and experience; and the Laws of Thought or
Conditions of Immediate Inference are an analysis of this postulate.

§ 4. The principle of Identity is usually written symbolically thus: _A
is A; not-A is not-A_. It assumes that there is something that may be
represented by a term; and it requires that, in any discussion, _every
relevant term, once used in a definite sense, shall keep that meaning
throughout_. Socrates in his father's workshop, at the battle of Delium,
and in prison, is assumed to be the same man denotable by the same name;
and similarly, 'elephant,' or 'justice,' or 'fairy,' in the same
context, is to be understood of the same thing under the same

But, further, it is assumed that of a given term another term may be
predicated again and again in the same sense under the same conditions;
that is, we may speak of the identity of meaning in a proposition as
well as in a term. To symbolise this we ought to alter the usual
formula for Identity and write it thus: _If B is A, B is A; if B is
not-A, B is not-A_. If Socrates is wise, he is wise; if fairies frequent
the moonlight, they do; if Justice is not of this world, it is not.
_Whatever affirmation or denial we make concerning any subject, we are
bound to adhere to it for the purposes of the current argument or
investigation._ Of course, if our assertion turns out to be false, we
must not adhere to it; but then we must repudiate all that we formerly
deduced from it.

Again, _whatever is true or false in one form of words is true or false
in any other_: this is undeniable, for the important thing is identity
of meaning; but in Formal Logic it is not very convenient. If Socrates
is wise, is it an identity to say 'Therefore the master of Plato is
wise'; or, further that he 'takes enlightened views of life'? If _Every
man is fallible_, is it an identical proposition that _Every man is
liable to error_? It seems pedantic to demand a separate proposition
that _Fallible is liable to error_. But, on the other hand, the
insidious substitution of one term for another speciously identical, is
a chief occasion of fallacy. How if we go on to argue: therefore, _Every
man is apt to blunder, prone to confusion of thought, inured to
self-contradiction_? Practically, the substitution of identities must be
left to candour and good-sense; and may they increase among us. Formal
Logic is, no doubt, safest with symbols; should, perhaps, content itself
with A and B; or, at least, hardly venture beyond Y and Z.

§ 5. The principle of Contradiction is usually written symbolically,
thus: _A is not not-A_. But, since this formula seems to be adapted to a
single term, whereas we want one that is applicable to propositions, it
may be better to write it thus: _B is not both A and not-A_. That is to
say: _if any term may be affirmed of a subject, the contradictory term
may, in the same relation, be denied of it_. A leaf that is green on one
side of it may be not-green on the other; but it is not both green and
not-green on the same surface, at the same time, and in the same light.
If a stick is straight, it is false that it is at the same time
not-straight: having granted that two angles are equal, we must deny
that they are unequal.

But is it necessarily false that the stick is 'crooked'; must we deny
that either angle is 'greater or less' than the other? How far is it
permissible to substitute any other term for the formal contradictory?
Clearly, the principle of Contradiction takes for granted the principle
of Identity, and is subject to the same difficulties in its practical
application. As a matter of fact and common sense, if we affirm any term
of a Subject, we are bound to deny of that Subject, in the same
relation, not only the contradictory but all synonyms for this, and also
all contraries and opposites; which, of course, are included in the
contradictory. But who shall determine what these are? Without an
authoritative Logical Dictionary to refer to, where all contradictories,
synonyms, and contraries may be found on record, Formal Logic will
hardly sanction the free play of common sense.

The principle of Excluded Middle may be written: _B is either A or
not-A_; that is, _if any term be denied of a subject, the contradictory
term may, in the same relation, be affirmed_. Of course, we may deny
that a leaf is green on one side without being bound to affirm that it
is not-green on the other. But in the same relation a leaf is either
green or not-green; at the same time, a stick is either bent or
not-bent. If we deny that A is greater than B, we must affirm that it is
not-greater than B.

Whilst, then, the principle of Contradiction (that 'of contradictory
predicates, one being affirmed, the other is denied ') might seem to
leave open a third or middle course, the denying of both
contradictories, the principle of Excluded Middle derives its name from
the excluding of this middle course, by declaring that the one or the
other must be affirmed. Hence the principle of Excluded Middle does not
hold good of mere contrary terms. If we deny that a leaf is green, we
are not bound to affirm it to be yellow; for it may be red; and then we
may deny both contraries, yellow and green. In fact, two contraries do
not between them cover the whole predicable area, but contradictories
do: the form of their expression is such that (within the _suppositio_)
each includes all that the other excludes; so that the subject (if
brought within the _suppositio_) must fall under the one or the other.
It may seem absurd to say that Mont Blanc is either wise or not-wise;
but how comes any mind so ill-organised as to introduce Mont Blanc into
this strange company? Being there, however, the principle is inexorable:
Mont Blanc is not-wise.

In fact, the principles of Contradiction and Excluded Middle are
inseparable; they are implicit in all distinct experience, and may be
regarded as indicating the two aspects of Negation. The principle of
Contradiction says: _B is not both A and not-A_, as if _not-A_ might be
nothing at all; this is abstract negation. But the principle of Excluded
Middle says: _Granting that B is not A, it is still something_--namely,
_not-A_; thus bringing us back to the concrete experience of a continuum
in which the absence of one thing implies the presence of something
else. Symbolically: to deny that B is A is to affirm that B is not A,
and this only differs by a hyphen from B is not-A.

These principles, which were necessarily to some extent anticipated in
chap. iv. § 7, the next chapter will further illustrate.

§ 6. But first we must draw attention to a maxim (also already
mentioned), which is strictly applicable to Immediate Inferences, though
(as we shall see) in other kinds of proof it may be only a formal
condition: this is the general caution _not to go beyond the evidence_.
An immediate inference ought to contain nothing that is not contained
(or formally implied) in the proposition by which it is proved. With
respect to quantity in denotation, this caution is embodied in the rule
'not to distribute any term that is not given distributed.' Thus, if
there is a predication concerning 'Some S,' or 'Some men,' as in the
forms I. and O., we cannot infer anything concerning 'All S.' or 'All
men'; and, as we have seen, if a term is given us preindesignate, we are
generally to take it as of particular quantity. Similarly, in the case
of affirmative propositions, we saw that this rule requires us to assume
that their predicates are undistributed.

As to the grounds of this maxim, not to go beyond the evidence, not to
distribute a term that is given as undistributed, it is one of the
things so plain that to try to justify is only to obscure them. Still,
we must here state explicitly what Formal Logic assumes to be contained
or implied in the evidence afforded by any proposition, such as 'All S
is P.' If we remember that in chap. iv. § 7, it was assumed that every
term may have a contradictory; and if we bear in mind the principles of
Contradiction and Excluded Middle, it will appear that such a
proposition as 'All S is P' tells us something not only about the
relations of 'S' and 'P,' but also of their relations to 'not-S' and
'not-P'; as, for example, that 'S is not not-P,' and that 'not-P is
not-S.' It will be shown in the next chapter how Logicians have
developed these implications in series of Immediate Inferences.

If it be asked whether it is true that every term, itself significant,
has a significant contradictory, and not merely a formal contradictory,
generated by force of the word 'not,' it is difficult to give any better
answer than was indicated in §§ 3-5, without venturing further into
Metaphysics. I shall merely say, therefore, that, granting that some
such term as 'Universe' or 'Being' may have no significant
contradictory, if it stand for 'whatever can be perceived or thought
of'; yet every term that stands for less than 'Universe' or 'Being' has,
of course, a contradictory which denotes the rest of the universe. And
since every argument or train of thought is carried on within a special
'universe of discourse,' or under a certain _suppositio_, we may say
that _within the given suppositio every term has a contradictory_, and
that every predication concerning a term implies some predication
concerning its contradictory. But the name of the _suppositio_ itself
has no contradictory, except with reference to a wider and inclusive

The difficulty of actual reasoning, not with symbols, but about matters
of fact, does not arise from the principles of Logic, but sometimes from
the obscurity or complexity of the facts, sometimes from the ambiguity
or clumsiness of language, sometimes from the deficiency of our own
minds in penetration, tenacity and lucidity. One must do one's best to
study the facts, and not be too easily discouraged.



§ 1. Under the general title of Immediate Inference Logicians discuss
three subjects, namely, Opposition, Conversion, and Obversion; to which
some writers add other forms, such as Whole and Part in Connotation,
Contraposition, Inversion, etc. Of Opposition, again, all recognise
four modes: Subalternation, Contradiction, Contrariety and
Sub-contrariety. The only peculiarities of the exposition upon which we
are now entering are, that it follows the lead of the three Laws of
Thought, taking first those modes of Immediate Inference in which
Identity is most important, then those which plainly involve
Contradiction and Excluded Middle; and that this method results in
separating the modes of Opposition, connecting Subalternation with
Conversion, and the other modes with Obversion. To make up for this
departure from usage, the four modes of Opposition will be brought
together again in § 9.

§ 2. Subalternation.--Opposition being the relation of propositions that
have the same matter and differ only in form (as A., E., I., O.),
propositions of the forms A. and I. are said to be Subalterns in
relation to one another, and so are E. and O.; the universal of each
quality being distinguished as 'subalternans,' and the particular as

It follows from the principle of Identity that, the matter of the
propositions being the same, if A. is true I. is true, and that if E. is
true O. is true; for A. and E. predicate something of _All S_ or _All
men_; and since I. and O. make the same predication of _Some S_ or
_Some men_, the sense of these particular propositions has already been
predicated in A. or E. If _All S is P, Some S is P_; if _No S is P, Some
S is not P_; or, if _All men are fond of laughing, Some men are_; if _No
men are exempt from ridicule, Some men are not_.

Similarly, if I. is false A. is false; if O. is false E. is false. If we
deny any predication about _Some S_, we must deny it of _All S_; since
in denying it of _Some_, we have denied it of at least part of _All_;
and whatever is false in one form of words is false in any other.

On the other hand, if I. is true, we do not know that A. is; nor if O.
is true, that E. is; for to infer from _Some_ to _All_ would be going
beyond the evidence. We shall see in discussing Induction that the great
problem of that part of Logic is, to determine the conditions under
which we may in reality transcend this rule and infer from _Some_ to
_All_; though even there it will appear that, formally, the rule is
observed. For the present it is enough that I. is an immediate inference
from A., and O. from E.; but that A. is not an immediate inference from
I., nor E. from O.

§ 3. Connotative Subalternation.--We have seen (chap. iv. § 6) that if
the connotation of one term is only part of another's its denotation is
greater and includes that other's. Hence genus and species stand in
subaltern relation, and whatever is true of the genus is true of the
species: If _All animal life is dependent on vegetation, All human life
is dependent on vegetation_. On the other hand, whatever is not true of
the species or narrower term, cannot be true of the whole genus: If it
is false that '_All human life is happy_,' it is false that '_All animal
life is happy_.'

Similar inferences may be drawn from the subaltern relation of
predicates; affirming the species we affirm the genus. To take Mill's
example, if _Socrates is a man, Socrates is a living creature_. On the
other hand, denying the genus we deny the species: if _Socrates is not
vicious, Socrates is not drunken_.

Such cases as these are recognised by Mill and Bain as immediate
inferences under the principle of Identity. But some Logicians might
treat them as imperfect syllogisms, requiring another premise to
legitimate the conclusion, thus:

      _All animal life is dependent on vegetation;
      All human life is animal life;
    ∴ All human life is dependent on vegetation._

Or again:

      _All men are living creatures;
      Socrates is a man;
    ∴ Socrates is a living creature._

The decision of this issue turns upon the question (_cf._ chap. vi. § 3)
how far a Logician is entitled to assume that the terms he uses are
understood, and that the identities involved in their meanings will be
recognised. And to this question, for the sake of consistency, one of
two answers is required; failing which, there remains the rule of thumb.
First, it may be held that no terms are understood except those that are
defined in expounding the science, such as 'genus' and 'species,'
'connotation' and 'denotation.' But very few Logicians observe this
limitation; few would hesitate to substitute 'not wise' for 'foolish.'
Yet by what right? Malvolio being foolish, to prove that he is not-wise,
we may construct the following syllogism:

      _Foolish is not-wise;
      Malvolio is foolish;
    ∴ Malvolio is not-wise._

Is this necessary? Why not?

Secondly, it may be held that all terms may be assumed as understood
unless a definition is challenged. This principle will justify the
substitution of 'not-wise' for 'foolish'; but it will also legitimate
the above cases (concerning 'human life' and 'Socrates') as immediate
inferences, with innumerable others that might be based upon the
doctrine of relative terms: for example, _The hunter missed his aim_:
therefore, _The prey escaped_. And from this principle it will further
follow that all apparent syllogisms, having one premise a verbal
proposition, are immediate inferences (_cf._ chap. ix. § 4).

Closely connected with such cases as the above are those mentioned by
Archbishop Thomson as "Immediate Inferences by added Determinants"
(_Laws of Thought_, § 87). He takes the case: '_A negro is a
fellow-creature_: therefore, _A negro in suffering is a fellow-creature
in suffering_.' This rests upon the principle that to increase the
connotations of two terms by the same attribute or determinant does not
affect the relationship of their denotations, since it must equally
diminish (if at all) the denotations of both classes, by excluding the
same individuals, if any want the given attribute. But this principle is
true only when the added attribute is not merely the same verbally, but
has the same significance in qualifying both terms. We cannot argue _A
mouse is an animal_; therefore, _A large mouse is a large animal_; for
'large' is an attribute relative to the normal magnitude of the thing

§ 4. Conversion is Immediate Inference by transposing the terms of a
given proposition without altering its quality. If the quantity is also
unaltered, the inference is called 'Simple Conversion'; but if the
quantity is changed from universal to particular, it is called
'Conversion by limitation' or '_per accidens._' The given proposition is
called the 'convertend'; that which is derived from it, the 'converse.'

Departing from the usual order of exposition, I have taken up Conversion
next to Subalternation, because it is generally thought to rest upon the
principle of Identity, and because it seems to be a good method to
exhaust the forms that come only under Identity before going on to those
that involve Contradiction and Excluded Middle. Some, indeed, dispute
the claims of Conversion to illustrate the principle of Identity; and
if the sufficient statement of that principle be 'A is A,' it may be a
question how Conversion or any other mode of inference can be referred
to it. But if we state it as above (chap. vi. § 3), that whatever is
true in one form of words is true in any other, there is no difficulty
in applying it to Conversion.

Thus, to take the simple conversion of I.,

    _Some S is P; ∴ Some P is S._
    _Some poets are business-like; ∴ Some business-like men are poets._

Here the convertend and the converse say the same thing, and this is
true if that is.

We have, then, two cases of simple conversion: of I. (as above) and of
E. For E.:

    _No S is P; ∴ No P is S._
    _No ruminants are carnivores; ∴ No carnivores are ruminants._

In converting I., the predicate (P) when taken as the new subject, being
preindesignate, is treated as particular; and in converting E., the
predicate (P), when taken as the new subject, is treated as universal,
according to the rule in chap. v. § 1.

A. is the one case of conversion by limitation:

    All S is P;
    ∴ Some P is S.

    All cats are grey in the dark;
    ∴ Some things grey in the dark are cats.

The predicate is treated as particular, when taking it for the new
subject, according to the rule not to go beyond the evidence. To infer
that _All things grey in the dark are cats_ would be palpably absurd;
yet no error of reasoning is commoner than the simple conversion of A.
The validity of conversion by limitation may be shown thus: if, _All S
is P_, then, by subalternation, _Some S is P_, and therefore, by simple
conversion, _Some P is S_.

O. cannot be truly converted. If we take the proposition: _Some S is
not P_, to convert this into _No P is S_, or _Some P is not S_, would
break the rule in chap. vi. § 6; since _S,_ undistributed in the
convertend, would be distributed in the converse. If we are told that
_Some men are not cooks_, we cannot infer that _Some cooks are not men_.
This would be to assume that '_Some men_' are identical with '_All

By quantifying the predicate, indeed, we may convert O. simply, thus:

    _Some men are not cooks_ ∴ _No cooks are some men._

And the same plan has some advantage in converting A.; for by the usual
method _per accidens_, the converse of A. being I., if we convert this
again it is still I., and therefore means less than our original
convertend. Thus:

    _All S is P ∴ Some P is S ∴ Some S is P._

Such knowledge, as that _All S_ (the whole of it) _is P_, is too
precious a thing to be squandered in pure Logic; and it may be preserved
by quantifying the predicate; for if we convert A. to Y., thus--

    _All S is P ∴ Some P is all S--_

we may reconvert Y. to A. without any loss of meaning. It is the chief
use of quantifying the predicate that, thereby, every proposition is
capable of simple conversion.

The conversion of propositions in which the relation of terms is
inadequately expressed (see chap. ii., § 2) by the ordinary copula (_is_
or _is not_) needs a special rule. To argue thus--

    _A is followed by B_ ∴ _Something followed by B is A_--

would be clumsy formalism. We usually say, and we ought to say--

    _A is followed by B_ ∴ _B follows A_ (or _is preceded by A_).

Now, any relation between two terms may be viewed from either side--_A:
B_ or _B: A_. It is in both cases the same fact; but, with the altered
point of view, it may present a different character. For example, in the
Immediate Inference--_A > B_ ∴ _B < A_--a diminishing turns into an
increasing ratio, whilst the fact predicated remains the same. Given,
then, a relation between two terms as viewed from one to the other, the
same relation viewed from the other to the one may be called the
Reciprocal. In the cases of Equality, Co-existence and Simultaneity, the
given relation and its reciprocal are not only the same fact, but they
also have the same character: in the cases of Greater and Less and
Sequence, the character alters.

We may, then, state the following rule for the conversion of
propositions in which the whole relation explicitly stated is taken as
the copula: Transpose the terms, and for the given relation substitute
its reciprocal. Thus--

    _A is the cause of B ∴ B is the effect of A._

The rule assumes that the reciprocal of a given relation is definitely
known; and so far as this is true it may be extended to more concrete

    _A is a genus of B ∴ B is a species of A
    A is the father of B ∴ B is a child of A._

But not every relational expression has only one definite reciprocal. If
we are told that _A is the brother of B_, we can only infer that _B is
either the brother or the sister of A_. A list of all reciprocal
relations is a desideratum of Logic.

§ 5. Obversion (otherwise called Permutation or Æquipollence) is
Immediate Inference by changing the quality of the given proposition and
substituting for its predicate the contradictory term. The given
proposition is called the 'obvertend,' and the inference from it the
'obverse.' Thus the obvertend being--_Some philosophers are consistent
reasoners_, the obverse will be--_Some philosophers are not inconsistent

The legitimacy of this mode of reasoning follows, in the case of
affirmative propositions, from the principle of Contradiction, that if
any term be affirmed of a subject, the contradictory term may be denied
(chap. vi. § 3). To obvert affirmative propositions, then, the rule
is--Insert the negative sign, and for the predicate substitute its
contradictory term.

    A. _All S is P ∴ No S is not-P
       All men are fallible ∴ No men are infallible._

    I. _Some S is P ∴ some S is not-P
       Some philosophers are consistent ∴ Some philosophers are not

In agreement with this mode of inference, we have the rule of modern
English grammar, that 'two negatives make an affirmative.'

Again, by the principle of Excluded Middle, if any term be denied of a
subject, its contradictory may be affirmed: to obvert negative
propositions, then, the rule is--Remove the negative sign, and for the
predicate substitute its contradictory term.

    E. _No S is P ∴ All S is not-P
       No matter is destructible ∴ All matter is indestructible._

    O. _Some S is not P ∴ Some S is not-P
       Some ideals are not attainable ∴ Some ideals are unattainable._

Thus, by obversion, each of the four propositions retains its quantity
but changes its quality: A. to E., I. to O., E. to A., O. to I. And all
the obverses are infinite propositions, the affirmative infinites having
the sense of negatives, and the negative infinites having the sense of

Again, having obtained the obverse of a given proposition, it may be
desirable to recover the obvertend; or it may at any time be requisite
to change a given infinite proposition into the corresponding direct
affirmative or negative; and in such cases the process is still
obversion. Thus, if _No S is not-P_ be given us to recover the obvertend
or to find the corresponding affirmative; the proposition being formally
negative, we apply the rule for obverting negatives: 'Remove the
negative sign, and for the predicate substitute its contradictory.' This
yields the affirmative _All S is P_. Similarly, to obtain the obvertend
of _All S is not-P_, apply the rule for obverting Affirmatives; and this
yields _No S is P_.

§ 6. Contrariety.--We have seen in chap. iv. § 8, that contrary terms
are such that no two of them are predicable in the same way of the same
subject, whilst perhaps neither may be predicable of it. Similarly,
Contrary Propositions may be defined as those of which no two are ever
both true together, whilst perhaps neither may be true; or, in other
words, both may be false. This is the relation between A. and E. when
concerned with the same matter: as A.--_All men are wise_; E.--_No men
are wise_. Such propositions cannot both be true; but they may both be
false, for some men may be wise and some not. They cannot both be true;
for, by the principle of Contradiction, if _wise_ may be affirmed of
_All men, not-wise_ must be denied; but _All men are not-wise_ is the
obverse of _No men are wise_, which therefore may also be denied.

At the same time we cannot apply to A. and E. the principle of Excluded
Middle, so as to show that one of them must be true of the same matter.
For if we deny that _All men are wise_, we do not necessarily deny the
attribute 'wise' of each and every man: to say that _Not all are wise_
may mean no more than that _Some are not_. This gives a proposition in
the form of O.; which, as we have seen, does not imply its subalternans,

If, however, two Singular Propositions, having the same matter, but
differing in quality, are to be treated as universals, and therefore as
A. and E., they are, nevertheless, contradictory and not merely
contrary; for one of them must be false and the other true.

§ 7. Contradiction is a relation between two propositions analogous to
that between contradictory terms (one of which being affirmed of a
subject the other is denied)--such, namely, that one of them is false
and the other true. This is the case with the forms A. and O., and E.
and I., in the same matter. If it be true that _All men are wise_, it is
false that _Some men are not wise_ (equivalent by obversion to _Some
men are not-wise_); or else, since the 'Some men' are included in the
'All men,' we should be predicating of the same men that they are both
'wise' and 'not-wise'; which would violate the principle of
Contradiction. Similarly, _No men are wise_, being by obversion
equivalent to _All men are not-wise_, is incompatible with _Some men are
wise_, by the same principle of Contradiction.

But, again, if it be false that _All men are wise_, it is always true
that _Some are not wise_; for though in denying that 'wise' is a
predicate of 'All men' we do not deny it of each and every man, yet we
deny it of 'Some men.' Of 'Some men,' therefore, by the principle of
Excluded Middle, 'not-wise' is to be affirmed; and _Some men are
not-wise_, is by obversion equivalent to _Some men are not wise_.
Similarly, if it be false that _No men are wise_, which by obversion is
equivalent to _All men are not-wise_, then it is true at least that
_Some men are wise_.

By extending and enforcing the doctrine of relative terms, certain other
inferences are implied in the contrary and contradictory relations of
propositions. We have seen in chap. iv. that the contradictory of a
given term includes all its contraries: 'not-blue,' for example,
includes red and yellow. Hence, since _The sky is blue_ becomes by
obversion, _The sky is not not-blue_, we may also infer _The sky is not
red_, etc. From the truth, then, of any proposition predicating a given
term, we may infer the falsity of all propositions predicating the
contrary terms in the same relation. But, on the other hand, from the
falsity of a proposition predicating a given term, we cannot infer the
truth of the predication of any particular contrary term. If it be false
that _The sky is red_, we cannot formally infer, that _The sky is blue_
(_cf._ chap. iv. § 8).

§ 8. Sub-contrariety is the relation of two propositions, concerning the
same matter that may both be true but are never both false. This is the
case with I. and O. If it be true that _Some men are wise_, it may also
be true that _Some (other) men are not wise_. This follows from the
maxim in chap. vi. § 6, not to go beyond the evidence.

For if it be true that _Some men are wise_, it may indeed be true that
_All are_ (this being the subalternans): and if _All are_, it is (by
contradiction) false that _Some are not_; but as we are only told that
_Some men are_, it is illicit to infer the falsity of _Some are not_,
which could only be justified by evidence concerning _All men_.

But if it be false that _Some men are wise_, it is true that _Some men
are not wise_; for, by contradiction, if _Some men are wise_ is false,
_No men are wise_ is true; and, therefore, by subalternation, _Some men
are not wise_ is true.

§ 9. The Square of Opposition.--By their relations of Subalternation,
Contrariety, Contradiction, and Sub-contrariety, the forms A. I. E. O.
(having the same matter) are said to stand in Opposition: and Logicians
represent these relations by a square having A. I. E. O. at its corners:

A.      Contraries      E.

S    Co             s   S
u      nt          e    u
b        ra       i     b
a          di    r      a
l            ct o       l
t            ct o       t
e          di    r      e
r        ra       i     r
n      nt          e    n
s    Co             s   s

I.   Sub-contraries     O.

As an aid to the memory, this diagram is useful; but as an attempt to
represent the logical relations of propositions, it is misleading. For,
standing at corners of the same square, A. and E., A. and I., E. and O.,
and I. and O., seem to be couples bearing the same relation to one
another; whereas we have seen that their relations are entirely
different. The following traditional summary of their relations in
respect of truth and falsity is much more to the purpose:

    (1) If A. is true,   I. is true,     E. is false,    O. is false.
    (2) If A. is false,  I. is unknown,  E. is unknown,  O. is true.
    (3) If I. is true,   A. is unknown,  E. is false,    O. is unknown.
    (4) If I. is false,  A. is false,    E. is true,     O. is true.
    (5) If E. is true,   A. is false,    I. is false,    O. is true.
    (6) If E. is false,  A. is unknown,  I. is true,     O. is unknown.
    (7) If O. is true,   A. is false,    I. is unknown,  E. is unknown.
    (8) If O. is false,  A. is true,     I. is true,     E. is false.

Where, however, as in cases 2, 3, 6, 7, alleging either the
falsity of universals or the truth of particulars, it follows that two
of the three Opposites are unknown, we may conclude further that one of
them must be true and the other false, because the two unknown are
always Contradictories.

§ 10. Secondary modes of Immediate Inference are obtained by applying
the process of Conversion or Obversion to the results already obtained
by the other process. The best known secondary form of Immediate
Inference is the Contrapositive, and this is the converse of the obverse
of a given proposition. Thus:

        DATUM.                  OBVERSE.          CONTRAPOSITIVE.

    A. _All S is P_      ∴ _No S is not-P_       ∴ _No not-P is S_
    I. _Some S is P_     ∴ _Some S is not not-P_ ∴ (none)
    E. _No S is P_       ∴ _All S is not-P_      ∴ _Some not-P is S_
    O. _Some S is not P_ ∴ _Some S is not-P_     ∴ _Some not-P is S_

There is no contrapositive of I., because the obverse of I. is in the
form of O., and we have seen that O. cannot be converted. O., however,
has a contrapositive (_Some not-P is S_); and this is sometimes given
instead of the converse, and called the 'converse by negation.'

Contraposition needs no justification by the Laws of Thought, as it is
nothing but a compounding of conversion with obversion, both of which
processes have already been justified. I give a table opposite of the
other ways of compounding these primary modes of Immediate Inference.

                     A                I               E                 O

          1 All A is B       Some A is B       No A is B        Some A is not B


Obverse   2 No A is b        Some A is not b   All A is b       Some A is b


Converse  3 Some B is A      Some B is A       No B is A


of        4 Some B is not a  Some B is not a   All B is a


positive  5 No b is A                          Some b is A      Some b is A


of        6 All b is a                         Some b is not a  Some b is not a


Obverse   7                                    Some a is B


of        8                                    Some a is not b


Obverse   9 Some a is b


of       10 Some a is not B


In this table _a_ and _b_ stand for _not-A_ and _not-B_ and had better
be read thus: for _No A is b, No A is not-B_; for _All b is a_ (col. 6),
_All not-B is not-A_; and so on.

It may not, at first, be obvious why the process of alternately
obverting and converting any proposition should ever come to an end;
though it will, no doubt, be considered a very fortunate circumstance
that it always does end. On examining the results, it will be found that
the cause of its ending is the inconvertibility of O. For E., when
obverted, becomes A.; every A, when converted, degenerates into I.;
every I., when obverted, becomes O.; O cannot be converted, and to
obvert it again is merely to restore the former proposition: so that the
whole process moves on to inevitable dissolution. I. and O. are
exhausted by three transformations, whilst A. and E. will each endure

Except Obversion, Conversion and Contraposition, it has not been usual
to bestow special names on these processes or their results. But the
form in columns 7 and 10 (_Some a is B--Some a is not B_), where the
original predicate is affirmed or denied of the contradictory of the
original subject, has been thought by Dr. Keynes to deserve a
distinctive title, and he has called it the 'Inverse.' Whilst the
Inverse is one form, however, Inversion is not one process, but is
obtained by different processes from E. and A. respectively. In this it
differs from Obversion, Conversion, and Contraposition, each of which
stands for one process.

The Inverse form has been objected to on the ground that the inference
_All A is B ∴ Some not-A is not B_, distributes _B_ (as predicate of a
negative proposition), though it was given as undistributed (as
predicate of an affirmative proposition). But Dr. Keynes defends it on
the ground that (1) it is obtained by obversions and conversions which
are all legitimate and (2) that although _All A is B_ does not
distribute _B_ in relation to _A_, it does distribute _B_ in relation to
some _not-A_ (namely, in relation to whatever _not-A_ is _not-B_). This
is one reason why, in stating the rule in chap. vi. § 6, I have
written: "an immediate inference ought to contain nothing that is not
contained, _or formally implied_, in the proposition from which it is
inferred"; and have maintained that every term formally implies its
contradictory within the _suppositio_.

§ 11. Immediate Inferences from Conditionals are those which
consist--(1) in changing a Disjunctive into a Hypothetical, or a
Hypothetical into a Disjunctive, or either into a Categorical; and (2)
in the relations of Opposition and the equivalences of Obversion,
Conversion, and secondary or compound processes, which we have already
examined in respect of Categoricals. As no new principles are involved,
it may suffice to exhibit some of the results.

We have already seen (chap. v. § 4) how Disjunctives may be read as
Hypotheticals and Hypotheticals as Categoricals. And, as to Opposition,
if we recognise four forms of Hypothetical A. I. E. O., these plainly
stand to one another in a Square of Opposition, just as Categoricals do.
Thus A. and E. (_If A is B, C is D_, and _If A is B, C is not D_) are
contraries, but not contradictories; since both may be false (_C_ may
sometimes be _D_, and sometimes not), though they cannot both be true.
And if they are both false, their subalternates are both true, being
respectively the contradictories of the universals of opposite quality,
namely, I. of E., and O. of A. But in the case of Disjunctives, we
cannot set out a satisfactory Square of Opposition; because, as we saw
(chap. v. § 4), the forms required for E. and O. are not true
Disjunctives, but Exponibles.

The Obverse, Converse, and Contrapositive, of Hypotheticals (admitting
the distinction of quality) may be exhibited thus:

      DATUM.                                        OBVERSE.

A. _If A is B, C is D_                  _If A is B, C is not d_
I. Sometimes _when A is B, C is D_      Sometimes _when A is B, C is not d_
E. _If A is B, C is not D_              _If A is B, C is d_
O. Sometimes _when A is B, C is not D_  Sometimes _when A is B, C is d_

    CONVERSE.                                   CONTRAPOSITIVE.

Sometimes _when C is D, A is B_      _If C is d, A is not B_
Sometimes _when C is D, A is B_                (none)
_If C is D, A is not B_              Sometimes _when C is d, A is B_
          (none)                     Sometimes _when C is d, A is B_

As to Disjunctives, the attempt to put them through these different
forms immediately destroys their disjunctive character. Still, given any
proposition in the form _A is either B or C_, we can state the
propositions that give the sense of obversion, conversion, etc., thus:

    DATUM.--_A is either B or C;_
    OBVERSE.--_A is not both b and c;_
    CONVERSE.--_Something, either B or C, is A;_
    CONTRAPOSITIVE.--_Nothing that is both b and c is A_.

For a Disjunctive in I., of course, there is no Contrapositive. Given a
Disjunctive in the form _Either A is B or C is D_, we may write for its
Obverse--_In no case is A b, and C at the same time d_. But no Converse
or Contrapositive of such a Disjunctive can be obtained, except by first
casting it into the hypothetical or categorical form.

The reader who wishes to pursue this subject further, will find it
elaborately treated in Dr. Keynes' _Formal Logic_, Part II.; to which
work the above chapter is indebted.



§ 1. Of the terms of a proposition which is the Subject and which the
Predicate? In most of the exemplary propositions cited by Logicians it
will be found that the subject is a substantive and the predicate an
adjective, as in _Men are mortal_. This is the relation of Substance and
Attribute which we saw (chap. i. § 5) to be the central type of
relations of coinherence; and on this model other predications may be
formed in which the subject is not a substance, but is treated as if it
were, and could therefore be the ground of attributes; as _Fame is
treacherous, The weather is changeable_. But, in literature, sentences
in which the adjective comes first are not uncommon, as _Loud was the
applause, Dark is the fate of man, Blessed are the peacemakers_, and so
on. Here, then, 'loud,' 'dark' and 'blessed' occupy the place of the
logical subject. Are they really the subject, or must we alter the order
of such sentences into _The applause was loud_, etc.? If we do, and then
proceed to convert, we get _Loud was the applause_, or (more
scrupulously) _Some loud noise was the applause_. The last form, it is
true, gives the subject a substantive word, but 'applause' has become
the predicate; and if the substantive 'noise' was not implied in the
first form, _Loud is the applause_, by what right is it now inserted?
The recognition of Conversion, in fact, requires us to admit that,
formally, in a logical proposition, the term preceding the copula is
subject and the one following is predicate. And, of course, materially
considered, the mere order of terms in a proposition can make no
difference in the method of proving it, nor in the inferences that can
be drawn from it.

Still, if the question is, how we may best cast a literary sentence into
logical form, good grounds for a definite answer may perhaps be found.
We must not try to stand upon the naturalness of expression, for _Dark
is the fate of man_ is quite as natural as _Man is mortal_. When the
purpose is not merely to state a fact, but also to express our feelings
about it, to place the grammatical predicate first may be perfectly
natural and most effective. But the grounds of a logical order of
statement must be found in its adaptation to the purposes of proof and
inference. Now general propositions are those from which most inferences
can be drawn, which, therefore, it is most important to establish, if
true; and they are also the easiest to disprove, if false; since a
single negative instance suffices to establish the contradictory. It
follows that, in re-casting a literary or colloquial sentence for
logical purposes, we should try to obtain a form in which the subject is
distributed--is either a singular term or a general term predesignate as
'All' or 'No.' Seeing, then, that most adjectives connote a single
attribute, whilst most substantives connote more than one attribute; and
that therefore the denotation of adjectives is usually wider than that
of substantives; in any proposition, one term of which is an adjective
and the other a substantive, if either can be distributed in relation to
the other, it is nearly sure to be the substantive; so that to take the
substantive term for subject is our best chance of obtaining an
universal proposition. These considerations seem to justify the practice
of Logicians in selecting their examples.

For similar reasons, if both terms of a proposition are substantive, the
one with the lesser denotation is (at least in affirmative
propositions) the more suitable subject, as _Cats are carnivores_. And
if one term is abstract, that is the more suitable subject; for, as we
have seen, an abstract term may be interpreted by a corresponding
concrete one distributed, as _Kindness is infectious_; that is, _All
kind actions suggest imitation_.

If, however, a controvertist has no other object in view than to refute
some general proposition laid down by an opponent, a particular
proposition is all that he need disentangle from any statement that
serves his purpose.

§ 2. Toward understanding clearly the relations of the terms of a
proposition, it is often found useful to employ diagrams; and the
diagrams most in use are the circles of Euler.

These circles represent the denotation of the terms. Suppose the
proposition to be _All hollow-horned animals ruminate_: then, if we
could collect all ruminants upon a prairie, and enclose them with a
circular palisade; and segregate from amongst them all the hollow-horned
beasts, and enclose them with another ring-fence inside the other; one
way of interpreting the proposition (namely, in denotation) would be
figured to us thus:

[Illustration: FIG. 1.]

An Universal Affirmative may also state a relation between two terms
whose denotation is co-extensive. A definition always does this, as _Man
is a rational animal_; and this, of course, we cannot represent by two
distinct circles, but at best by one with a thick circumference, to
suggest that two coincide, thus:

[Illustration: FIG. 2.]

The Particular Affirmative Proposition may be represented in several
ways. In the first place, bearing in mind that 'Some' means 'some at
least, it may be all,' an I. proposition may be represented by Figs. 1
and 2; for it is true that _Some horned animals ruminate_, and that
_Some men are rational_. Secondly, there is the case in which the 'Some
things' of which a predication is made are, in fact, not all; whilst the
predicate, though not given as distributed, yet might be so given if we
wished to state the whole truth; as if we say _Some men are Chinese_.
This case is also represented by Fig. 1, the outside circle representing
'Men,' and the inside one 'Chinese.' Thirdly, the predicate may
appertain to some only of the subject, but to a great many other things,
as in _Some horned beasts are domestic_; for it is true that some are
not, and that certain other kinds of animals are, domestic. This case,
therefore, must be illustrated by overlapping circles, thus:

[Illustration: FIG. 3.]

The Universal Negative is sufficiently represented by a single Fig. (4):
two circles mutually exclusive, thus:

[Illustration: FIG. 4.]

That is, _No horned beasts are carnivorous_.

Lastly, the Particular Negative may be represented by any of the Figs.
1, 3, and 4; for it is true that _Some ruminants are not hollow-horned_,
that _Some horned animals are not domestic_, and that _Some horned
beasts are not carnivorous_.

Besides their use in illustrating the denotative force of propositions,
these circles may be employed to verify the results of Obversion,
Conversion, and the secondary modes of Immediate Inference. Thus the
Obverse of A. is clear enough on glancing at Figs. 1 and 2; for if we
agree that whatever term's denotation is represented by a given circle,
the denotation of the contradictory term shall be represented by the
space outside that circle; then if it is true that _All hollow horned
animals are ruminants_, it is at the same time true that _No
hollow-horned animals are not-ruminants_; since none of the
hollow-horned are found outside the palisade that encloses the
ruminants. The Obverse of I., E. or O. may be verified in a similar

As to the Converse, a Definition is of course susceptible of Simple
Conversion, and this is shown by Fig. 2: 'Men are rational animals' and
'Rational animals are men.' But any other A. proposition is presumably
convertible only by limitation, and this is shown by Fig. 1; where _All
hollow-horned animals are ruminants_, but we can only say that _Some
ruminants are hollow-horned_.

That I. may be simply converted may be seen in Fig. 3, which represents
the least that an I. proposition can mean; and that E. may be simply
converted is manifest in Fig. 4.

As for O., we know that it cannot be converted, and this is made plain
enough by glancing at Fig. 1; for that represents the O., _Some
ruminants are not hollow-horned_, but also shows this to be compatible
with _All hollow-horned animals are ruminants_ (A.). Now in conversion
there is (by definition) no change of quality. The Converse, then, of
_Some ruminants are not hollow-horned_ must be a negative proposition,
having 'hollow-horned' for its subject, either in E. or O.; but these
would be respectively the contrary and contradictory of _All
hollow-horned animals are ruminants_; and, therefore, if this be true,
they must both be false.

But (referring still to Fig. 1) the legitimacy of contrapositing O. is
equally clear; for if _Some ruminants are not hollow-horned_, _Some
animals that are not hollow-horned are ruminants_, namely, all the
animals between the two ring-fences. Similar inferences may be
illustrated from Figs. 3 and 4. And the Contraposition of A. may be
verified by Figs. 1 and 2, and the Contraposition of E. by Fig. 4.

Lastly, the Inverse of A. is plain from Fig. 1--_Some things that are
not hollow-horned are not ruminants_, namely, things that lie outside
the outer circle and are neither 'ruminants' nor 'hollow-horned.' And
the Inverse of E may be studied in Fig. 4--_Some things that are
not-horned beasts are carnivorous_.

Notwithstanding the facility and clearness of the demonstrations thus
obtained, it may be said that a diagrammatic method, representing
denotations, is not properly logical. Fundamentally, the relation
asserted (or denied) to exist between the terms of a proposition, is a
relation between the terms as determined by their attributes or
connotation; whether we take Mill's view, that a proposition asserts
that the connotation of the subject is a mark of the connotation of the
predicate; or Dr. Venn's view, that things denoted by the subject (as
having its connotation) have (or have not) the attribute connoted by the
predicate; or, the Conceptualist view, that a judgment is a relation of
concepts (that is, of connotations). With a few exceptions artificially
framed (such as 'kings now reigning in Europe'), the denotation of a
term is never directly and exhaustively known, but consists merely in
'all things that have the connotation.' If the value of logical training
depends very much upon our habituating ourselves to construe
propositions, and to realise the force of inferences from them,
according to the connotation of their terms, we shall do well not to
turn too hastily to the circles, but rather to regard them as means of
verifying in denotation the conclusions that we have already learnt to
recognise as necessary in connotation.

§ 3. The equational treatment of propositions is closely connected with
the diagrammatic. Hamilton thought it a great merit of his plan of
quantifying the predicate, that thereby every proposition is reduced to
its true form--an equation. According to this doctrine, the proposition
_All X is all Y_ (U.) equates X and Y; the proposition _All X is some Y_
(A.) equates X with some part of Y; and similarly with the other
affirmatives (Y. and I.). And so far it is easy to follow his meaning:
the Xs are identical with some or all the Ys. But, coming to the
negatives, the equational interpretation is certainly less obvious. The
proposition _No X is Y_ (E.) cannot be said in any sense to equate X and
Y; though, if we obvert it into _All X is some not-Y_, we have (in the
same sense, of course, as in the above affirmative forms) X equated with
part at least of 'not-Y.'

But what is that sense? Clearly not the same as that in which
mathematical terms are equated, namely, in respect of some mode of
quantity. For if we may say _Some X is some Y_, these Xs that are also
Ys are not merely the same in number, or mass, or figure; they are the
same in every respect, both quantitative and qualitative, have the same
positions in time and place, are in fact identical. The proposition
2+2=4 means that any two things added to any other two are, _in respect
of number_, equal to any three things added to one other thing; and this
is true of all things that can be counted, however much they may differ
in other ways. But _All X is all Y_ means that Xs and Ys are the same
things, although they have different names when viewed in different
aspects or relations. Thus all equilateral triangles are equiangular
triangles; but in one case they are named from the equality of their
angles, and in the other from the equality of their sides. Similarly,
'British subjects' and 'subjects of King George V' are the same people,
named in one case from the person of the Crown, and in the other from
the Imperial Government. These logical equations, then, are in truth
identities of denotation; and they are fully illustrated by the
relations of circles described in the previous section.

When we are told that logical propositions are to be considered as
equations, we naturally expect to be shown some interesting developments
of method in analogy with the equations of Mathematics; but from
Hamilton's innovations no such thing results. This cannot be said,
however, of the equations of Symbolic Logic; which are the
starting-point of very remarkable processes of ratiocination. As the
subject of Symbolic Logic, as a whole, lies beyond the compass of this
work, it will be enough to give Dr. Venn's equations corresponding with
the four propositional forms of common Logic.

According to this system, universal propositions are to be regarded as
not necessarily implying the existence of their terms; and therefore,
instead of giving them a positive form, they are translated into symbols
that express what they deny. For example, the proposition _All devils
are ugly_ need not imply that any such things as 'devils' really exist;
but it certainly does imply that _Devils that are not ugly do not
exist_. Similarly, the proposition _No angels are ugly_ implies that
_Angels that are ugly do not exist_. Therefore, writing _x_ for
'devils,' _y_ for 'ugly,' and _ȳ_ for 'not-ugly,' we may express A.,
the universal affirmative, thus:

    A. _xȳ_ = 0.

That is, _x that is not y is nothing_; or, _Devils that are not-ugly do
not exist_. And, similarly, writing _x_ for 'angels' and _y_ for 'ugly,'
we may express E., the universal negative, thus:

    E. _xy_ = 0.

That is, _x that is y is nothing_; or, _Angels that are ugly do not

On the other hand, particular propositions are regarded as implying the
existence of their terms, and the corresponding equations are so framed
as to express existence. With this end in view, the symbol v is adopted
to represent 'something,' or indeterminate reality, or more than
nothing. Then, taking any particular affirmative, such as _Some
metaphysicians are obscure_, and writing _x_ for 'metaphysicians,' and
_y_ for 'obscure,' we may express it thus:

    I. _xy_ = v.

That is, _x that is y is something_; or, _Metaphysicians that are
obscure do occur in experience_ (however few they may be, or whether
they all be obscure). And, similarly, taking any particular negative,
such as _Some giants are not cruel_, and writing _x_ for 'giants' and
_y_ for 'not-cruel,' we may express it thus:

    O. _xȳ_ = v.

That is, _x that is not y is something_; or, _giants that are not-cruel
do occur_--in romances, if nowhere else.

Clearly, these equations are, like Hamilton's, concerned with
denotation. A. and E. affirm that the compound terms xȳ and xy have no
denotation; and I. and O. declare that xȳ and xy have denotation, or
stand for something. Here, however, the resemblance to Hamilton's system
ceases; for the Symbolic Logic, by operating upon more than two terms
simultaneously, by adopting the algebraic signs of operations, +,-, ×, ÷
(with a special signification), and manipulating the symbols by
quasi-algebraic processes, obtains results which the common Logic
reaches (if at all) with much greater difficulty. If, indeed, the value
of logical systems were to be judged of by the results obtainable,
formal deductive Logic would probably be superseded. And, as a mental
discipline, there is much to be said in favour of the symbolic method.
But, as an introduction to philosophy, the common Logic must hold its
ground. (Venn: _Symbolic Logic_, c. 7.)

§ 4. Does Formal Logic involve any general assumption as to the real
existence of the terms of propositions?

In the first place, Logic treats primarily of the _relations_ implied in
propositions. This follows from its being the science of proof for all
sorts of (qualitative) propositions; since all sorts of propositions
have nothing in common except the relations they express.

But, secondly, relations without terms of some sort are not to be
thought of; and, hence, even the most formal illustrations of logical
doctrines comprise such terms as S and P, X and Y, or x and y, in a
symbolic or representative character. Terms, therefore, of some sort are
assumed to exist (together with their negatives or contradictories) _for
the purposes of logical manipulation_.

Thirdly, however, that Formal Logic cannot as such directly involve the
existence of any particular concrete terms, such as 'man' or 'mountain,'
used by way of illustration, is implied in the word 'formal,' that is,
'confined to what is common or abstract'; since the only thing common to
all terms is to be related in some way to other terms. The actual
existence of any concrete thing can only be known by experience, as with
'man' or 'mountain'; or by methodically justifiable inference from
experience, as with 'atom' or 'ether.' If 'man' or 'mountain,' or
'Cuzco' be used to illustrate logical forms, they bring with them an
existential import derived from experience; but this is the import of
language, not of the logical forms. 'Centaur' and 'El Dorado' signify to
us the non-existent; but they serve as well as 'man' and 'London' to
illustrate Formal Logic.

Nevertheless, fourthly, the existence or non-existence of particular
terms may come to be implied: namely, wherever the very fact of
existence, or of some condition of existence, is an hypothesis or datum.
Thus, given the proposition _All S is P_, to be P is made a condition of
the existence of S: whence it follows that an S that is not P does not
exist (_xȳ_ = 0). On the further hypothesis that S exists, it follows
that P exists. On the hypothesis that S does not exist, the existence of
P is problematic; but, then, if P does exist we cannot convert the
proposition; since _Some P is S_ (P existing) would involve the
existence of S; which is contrary to the hypothesis.

Assuming that Universals _do not_, whilst Particulars _do_, imply the
existence of their subjects, we cannot infer the subalternate (I. or O.)
from the subalternans (A. or E.), for that is to ground the actual on
the problematic; and for the same reason we cannot convert A. _per

Assuming, again, a certain _suppositio_ or universe, to which in a given
discussion every argument shall refer, then, any propositions whose
terms lie outside that _suppositio_ are irrelevant, and for the purposes
of that discussion are sometimes called "false"; though it seems better
to call them irrelevant or meaningless, seeing that to call them false
implies that they might in the same case be true. Thus propositions
which, according to the doctrine of Opposition, appear to be
Contradictories, may then cease to be so; for of Contradictories one is
true and the other false; but, in the case supposed, both are
meaningless. If the subject of discussion be Zoology, all propositions
about centaurs or unicorns are absurd; and such specious
Contradictories as _No centaurs play the lyre--Some centaurs do play the
lyre_; or _All unicorns fight with lions--Some unicorns do not fight
with lions_, are both meaningless, because in Zoology there are no
centaurs nor unicorns; and, therefore, in this reference, the
propositions are not really contradictory. But if the subject of
discussion or _suppositio_ be Mythology or Heraldry, such propositions
as the above are to the purpose, and form legitimate pairs of

In Formal Logic, in short, we may make at discretion any assumption
whatever as to the existence, or as to any condition of the existence of
any particular term or terms; and then certain implications and
conclusions follow in consistency with that hypothesis or datum. Still,
our conclusions will themselves be only hypothetical, depending on the
truth of the datum; and, of course, until this is empirically
ascertained, we are as far as ever from empirical reality. (Venn:
_Symbolic Logic_, c. 6; Keynes: _Formal Logic_, Part II. c. 7: _cf._
Wolf: _Studies in Logic_.)



§ 1. A Mediate Inference is a proposition that depends for proof upon
two or more other propositions, so connected together by one or more
terms (which the evidentiary propositions, or each pair of them, have in
common) as to justify a certain conclusion, namely, the proposition in
question. The type or (more properly) the unit of all such modes of
proof, when of a strictly logical kind, is the Syllogism, to which we
shall see that all other modes are reducible. It may be exhibited
symbolically thus:

      M is P;
      S is M:
    ∴ S is P.

Syllogisms may be classified, as to quantity, into Universal or
Particular, according to the quantity of the conclusion; as to quality,
into Affirmative or Negative, according to the quality of the
conclusion; and, as to relation, into Categorical, Hypothetical and
Disjunctive, according as all their propositions are categorical, or one
(at least) of their evidentiary propositions is a hypothetical or a

To begin with Categorical Syllogisms, of which the following is an

      All authors are vain;
      Cicero is an author:
    ∴ Cicero is vain.

Here we may suppose that there are no direct means of knowing that
Cicero is vain; but we happen to know that all authors are vain and
that he is an author; and these two propositions, put together,
unmistakably imply that he is vain. In other words, we do not at first
know any relation between 'Cicero' and 'vanity'; but we know that these
two terms are severally related to a third term, 'author,' hence called
a Middle Term; and thus we perceive, by mediate evidence, that they are
related to one another. This sort of proof bears an obvious resemblance
(though the relations involved are not the same) to the mathematical
proof of equality between two quantities, that cannot be directly
compared, by showing the equality of each of them to some third
quantity: A = B = C ∴ A = C. Here B is a middle term.

We have to inquire, then, what conditions must be satisfied in order
that a Syllogism may be formally conclusive or valid. A specious
Syllogism that is not really valid is called a Parasyllogism.

§ 2. General Canons of the Syllogism.

(1) A Syllogism contains three, and no more, distinct propositions.

(2) A Syllogism contains three, and no more, distinct univocal terms.

These two Canons imply one another. Three propositions with less than
three terms can only be connected in some of the modes of Immediate
Inference. Three propositions with more than three terms do not show
that connection of two terms by means of a third, which is requisite for
proving a Mediate Inference. If we write--

    All authors are vain;
    Cicero is a statesman--

there are four terms and no middle term, and therefore there is no
proof. Or if we write--

      All authors are vain;
      Cicero is an author:
    ∴ Cicero is a statesman--

here the term 'statesman' occurs without any voucher; it appears in the
inference but not in the evidence, and therefore violates the maxim of
all formal proof, 'not to go beyond the evidence.' It is true that if
any one argued--

      All authors are vain;
      Cicero wrote on philosophy:
    ∴ Cicero is vain--

this could not be called a bad argument or a material fallacy; but it
would be a needless departure from the form of expression in which the
connection between the evidence and the inference is most easily seen.

Still, a mere adherence to the same form of words in the expression of
terms is not enough: we must also attend to their meaning. For if the
same word be used ambiguously (as 'author' now for 'father' and anon for
'man of letters'), it becomes as to its meaning two terms; so that we
have four in all. Then, if the ambiguous term be the Middle, no
connection is shown between the other two; if either of the others be
ambiguous, something seems to be inferred which has never been really
given in evidence.

The above two Canons are, indeed, involved in the definition of a
categorical syllogism, which may be thus stated: A Categorical Syllogism
is a form of proof or reasoning (way of giving reasons) in which one
categorical proposition is established by comparing two others that
contain together only three terms, or that have one and only one term in

The proposition established, derived, or inferred, is called the
Conclusion: the evidentiary propositions by which it is proved are
called the Premises.

The term common to the premises, by means of which the other terms are
compared, is called the Middle Term; the subject of the conclusion is
called the Minor Term; the predicate of the conclusion, the Major Term.

The premise in which the minor term occurs is called the Minor Premise;
that in which the major term occurs is called the Major Premise. And a
Syllogism is usually written thus:

Major Premise--All authors (Middle) are vain (Major);

Minor Premise--Cicero (Minor) is an author (Middle):

Conclusion--∴ Cicero (Minor) is vain (Major).

Here we have three propositions with three terms, each term occurring
twice. The minor and major terms are so called, because, when the
conclusion is an universal affirmative (which only occurs in Barbara;
see chap. x. § 6), its subject and predicate are respectively the less
and the greater in extent or denotation; and the premises are called
after the peculiar terms they contain: the expressions 'major premise'
and 'minor premise' have nothing to do with the order in which the
premises are presented; though it is usual to place the major premise

(3) No term must be distributed in the conclusion unless it is
distributed in the premises.

It is usual to give this as one of the General Canons of the Syllogism;
but we have seen (chap. vi. § 6) that it is of wider application.
Indeed, 'not to go beyond the evidence' belongs to the definition of
formal proof. A breech of this rule in a syllogism is the fallacy of
Illicit Process of the Minor, or of the Major, according to which term
has been unwarrantably distributed. The following parasyllogism
illicitly distributes both terms of the conclusion:

      All poets are pathetic;
      Some orators are not poets:
    ∴ No orators are pathetic.

(4) The Middle Term must be distributed at least once in the premises
(in order to prove a conclusion in the given terms).

For the use of mediate evidence is to show the relation of terms that
cannot be directly compared; this is only possible if the middle term
furnishes the ground of comparison; and this (in Logic) requires that
the whole denotation of the middle should be either included or excluded
by one of the other terms; since if we only know that the other terms
are related to _some_ of the middle, their respective relations may not
be with the same part of it.

It is true that in what has been called the "numerically definite
syllogism," an inference may be drawn, though our canon seems to be
violated. Thus:

        60 sheep in 100 are horned;
        60 sheep in 100 are blackfaced:
    ∴ at least 20 blackfaced sheep in 100 are horned.

But such an argument, though it may be correct Arithmetic, is not Logic
at all; and when such numerical evidence is obtainable the comparatively
indefinite arguments of Logic are needless. Another apparent exception
is the following:

      Most men are 5 feet high;
      Most men are semi-rational:
    ∴ Some semi-rational things are 5 feet high.

Here the Middle Term (men) is distributed in neither premise, yet the
indisputable conclusion is a logical proposition. The premises, however,
are really arithmetical; for 'most' means 'more than half,' or more than
50 per cent.

Still, another apparent exception is entirely logical. Suppose we are
given, the premises--_All P is M_, and _All S is M_--the middle term is
undistributed. But take the obverse of the contrapositive of both

      All m is p;
      All m is s:
    ∴ Some s is p.

Here we have a conclusion legitimately obtained; but it is not in the
terms originally given.

For Mediate Inference depending on truly logical premises, then, it is
necessary that one premise should distribute the middle term; and the
reason of this may be illustrated even by the above supposed numerical
exceptions. For in them the premises are such that, though neither of
the two premises by itself distributes the Middle, yet they always
overlap upon it. If each premise dealt with exactly half the Middle,
thus barely distributing it between them, there would be no logical
proposition inferrible. We require that the middle term, as used in one
premise, should necessarily overlap the same term as used in the other,
so as to furnish common ground for comparing the other terms. Hence I
have defined the middle term as 'that term common to both premises by
means of which the other terms are compared.'

(5) One at least of the premises must be affirmative; or, from two
negative premises nothing can be inferred (in the given terms).

The fourth Canon required that the middle term should be given
distributed, or in its whole extent, at least once, in order to afford
sure ground of comparison for the others. But that such comparison may
be effected, something more is requisite; the relation of the other
terms to the Middle must be of a certain character. One at least of them
must be, as to its extent or denotation, partially or wholly identified
with the Middle; so that to that extent it may be known to bear to the
other term, whatever relation we are told that so much of the Middle
bears to that other term. Now, identity of denotation can only be
predicated in an affirmative proposition: one premise, then, must be

If both premises are negative, we only know that both the other terms
are partly or wholly excluded from the Middle, or are not identical with
it in denotation: where they lie, then, in relation to one another we
have no means of knowing. Similarly, in the mediate comparison of
quantities, if we are told that A and C are both of them unequal to B,
we can infer nothing as to the relation of C to A. Hence the premises--

    No electors are sober;
    No electors are independent--

however suggestive, do not formally justify us in inferring any
connection between sobriety and independence. Formally to draw a
conclusion, we must have affirmative grounds, such as in this case we
may obtain by obverting both premises:

      All electors are not-sober;
      All electors are not-independent:
    ∴ Some who are not-independent are not-sober.

But this conclusion is not in the given terms.

(6) (a) If one premise be negative, the conclusion must be negative: and
(b) to prove a negative conclusion, one premise must be negative.

(a) For we have seen that one premise must be affirmative, and that thus
one term must be partly (at least) identified with the Middle. If, then,
the other premise, being negative, predicates the exclusion of the
remaining term from the Middle, this remaining term must be excluded
from the first term, so far as we know the first to be identical with
the Middle: and this exclusion will be expressed by a negative
conclusion. The analogy of the mediate comparison of quantities may here
again be noticed: if A is equal to B, and B is unequal to C, A is
unequal to C.

(b) If both premises be affirmative, the relations to the Middle of both
the other terms are more or less inclusive, and therefore furnish no
ground for an exclusive inference. This also follows from the function
of the middle term.

For the more convenient application of these canons to the testing of
syllogisms, it is usual to derive from them three Corollaries:

(i) Two particular premises yield no conclusion.

For if both premises be affirmative, _all_ their terms are
undistributed, the subjects by predesignation, the predicates by
position; and therefore the middle term must be undistributed, and there
can be no conclusion.

If one premise be negative, its predicate is distributed by position:
the other terms remaining undistributed. But, by Canon 6, the conclusion
(if any be possible) must be negative; and therefore its predicate, the
major term, will be distributed. In the premises, therefore, both the
middle and the major terms should be distributed, which is impossible:

      Some M is not P;
      Some S is M:
    ∴ Some S is not P.

Here, indeed, the major term is legitimately distributed (though the
negative premise might have been the minor); but M, the middle term, is
distributed in neither premise, and therefore there can be no

Still, an exception may be made by admitting a bi-designate conclusion:

      Some P is M;
      Some S is not M:
    ∴ Some S is not some P.

(ii) If one premise be particular, so is the conclusion.

For, again, if both premises be affirmative, they only distribute one
term, the subject of the universal premise, and this must be the middle
term. The minor term, therefore, is undistributed, and the conclusion
must be particular.

If one premise be negative, the two premises together can distribute
only two terms, the subject of the universal and the predicate of the
negative (which may be the same premise). One of these terms must be the
middle; the other (since the conclusion is negative) must be the major.
The minor term, therefore, is undistributed, and the conclusion must be

(iii) From a particular major and a negative minor premise nothing can
be inferred.

For the minor premise being negative, the major premise must be
affirmative (5th Canon); and therefore, being particular, distributes
the major term neither in its subject nor in its predicate. But since
the conclusion must be negative (6th Canon), a distributed major term is
demanded, e.g.,

      Some M is P;
      No S is M:
    ∴   ------

Here the minor and the middle terms are both distributed, but not the
major (P); and, therefore, a negative conclusion is impossible.

§ 3. First Principle or Axiom of the Syllogism.--Hitherto in this
chapter we have been analysing the conditions of valid mediate
inference. We have seen that a single step of such inference, a
Syllogism, contains, when fully expressed in language, three
propositions and three terms, and that these terms must stand to one
another in the relations required by the fourth, fifth, and sixth
Canons. We now come to a principle which conveniently sums up these
conditions; it is called the _Dictum de omni et nullo_, and may be
stated thus:

    Whatever is predicated (affirmatively or negatively) of a
    term distributed,

    With which term another term can be (partly or wholly)

    May be predicated in like manner (affirmatively or
    negatively) of the latter term (or part of it).

Thus stated (nearly as by Whately in the introduction to his _Logic_)
the _Dictum_ follows line by line the course of a Syllogism in the First
Figure (see chap. X. § 2). To return to our former example: _All authors
are vain_ is the same as--Vanity is predicated of all authors; _Cicero
is an author_ is the same as--Cicero is identified as an author;
therefore _Cicero is vain_, or--Vanity may be predicated of Cicero. The
_Dictum_ then requires: (1) three propositions; (2) three terms; (3)
that the middle term be distributed; (4) that one premise be
affirmative, since only by an affirmative proposition can one term be
identified with another; (5) that if one premise be negative the
conclusion shall be so too, since whatever is predicated of the middle
term is predicated _in like manner_ of the minor.

Thus far, then, the _Dictum_ is wholly analytic or verbal, expressing no
more than is implied in the definitions of 'Syllogism' and 'Middle
Term'; since (as we have seen) all the General Canons (except the third,
which is a still more general condition of formal proof) are derivable
from those definitions. However, the _Dictum_ makes a further statement
of a synthetic or real character, namely, that _when these conditions
are fulfilled an inference is justified_; that then the major and minor
terms are brought into comparison through the middle, and that the major
term may be predicated affirmatively or negatively of all or part of the
minor. It is this real assertion that justifies us in calling the
_Dictum_ an Axiom.

§ 4. Whether the Laws of Thought may not fully explain the Syllogism
without the need of any synthetic principle has, however, been made a
question. Take such a syllogism as the following:

      All domestic animals are useful;
      All pugs are domestic animals:
    ∴ All pugs are useful.

Here (an ingenious man might urge), having once identified pugs with
domestic animals, that they are useful follows from the Law of Identity.
If we attend to the meaning, and remember that what is true in one form
of words is true in any other form, then, all domestic animals being
useful, of course pugs are. It is merely a case of subalternation: we
may put it in this way:

      All domestic animals are useful:
    ∴ Some domestic animals (e.g., pugs) are useful.

The derivation of negative syllogisms from the Law of Contradiction (he
might add) may be shown in a similar manner.

But the force of this ingenious argument depends on the participial
clause--'having once identified pugs with domestic animals.' If this is
a distinct step of the reasoning, the above syllogism cannot be reduced
to one step, cannot be exhibited as mere subalternation, nor be brought
directly under the law of Identity. If 'pug,' 'domestic,' and 'useful'
are distinct terms; and if 'pug' and 'useful' are only known to be
connected because of their relations to 'domestic': this is something
more than the Laws of Thought provide for: it is not Immediate
Inference, but Mediate; and to justify it, scientific method requires
that its conditions be generalised. The _Dictum_, then, as we have seen,
does generalise these conditions, and declares that when such conditions
are satisfied a Mediate Inference is valid.

But, after all (to go back a little), consider again that proposition
_All pugs are domestic animals_: is it a distinct step of the reasoning;
that is to say, is it a Real Proposition? If, indeed, 'domestic' is no
part of the definition of 'pug,' the proposition is real, and is a
distinct part of the argument. But take such a case as this:

    All dogs are useful;
    All pugs are dogs.

Here we clearly have, in the minor premise, only a verbal proposition;
to be a dog is certainly part of the definition of 'pug.' But, if so,
the inference 'All pugs are useful' involves no real mediation, and the
argument is no more than this:

      All dogs are useful;
    ∴ Some dogs (e.g., pugs) are useful.

Similarly, if the major premise be verbal, thus:

    All men are rational;
    Socrates is a man--

to conclude that 'Socrates is rational' is no Mediate Inference; for so
much was implied in the minor premise, 'Socrates is a man,' and the
major premise adds nothing to this.

Hence we may conclude (as anticipated in chap. vii. § 3) that 'any
apparent syllogism, having one premise a verbal proposition, is really
an Immediate Inference'; but that, if both premises are real
propositions, the Inference is Mediate, and demands for its explanation
something more than the Laws of Thought.

The fact is that to prove the minor to be a case of the middle term may
be an exceedingly difficult operation (chap. xiii. § 7). The difficulty
is disguised by ordinary examples, used for the sake of convenience.

§ 5. Other kinds of Mediate Inference exist, yielding valid conclusions,
without being truly syllogistic. Such are mathematical inferences of
Equality, as--

    A = B = C ∴ A = C.

Here, according to the usual logical analysis, there are strictly four
terms--(1) A, (2) equal to B, (3) B, (4) equal to C.

Similarly with the argument _a fortiori_,

    A > B > C  ∴ (much more) A > C.

This also is said to contain four terms: (1) A, (2) greater than B, (3)
B, (4) greater than C. Such inferences are nevertheless intuitively
sound, may be verified by trial (within the limits of sense-perception),
and are generalised in appropriate axioms of their own, corresponding to
the _Dictum_ of the syllogism; as 'Things equal to the same thing are
equal to one another,' etc.

Now, surely, this is an erroneous application of the usual logical
analysis of propositions. Both Logic and Mathematics treat of the
_relations_ of terms; but whilst Mathematics employs the sign = for only
one kind of relation, and for that relation exclusive of the terms;
Logic employs the same signs (_is_ or _is not_) for all relations,
recognising only a difference of quality in predication, and treating
every other difference of relation as belonging to one of the terms
related. Thus Logicians read _A--is--equal to B_: as if _equal to B_
could possibly be a term co-relative with A. Whence it follows that the
argument _A = B = C ∴ A = C_ contains four terms; though everybody sees
that there are only three.

In fact (as observed in chap. ii. § 2) the sign of logical relation
(_is_ or _is not_), whilst usually adequate for class-reasoning
(coinherence) and sometimes extensible to causation (because a cause
implies a class of events), should never be stretched to include other
relations in such a way as to sacrifice intelligence to formalism. And,
besides mathematical or quantitative relations, there are others
(usually considered qualitative because indefinite) which cannot be
justly expressed by the logical copula. We ought to read propositions
expressing time-relations (and inferences drawn accordingly) thus:

      B--is before--C;
      A--is before--B:
    ∴ A--is before--C.

And in like manner _A--is simultaneous with--B; etc._ Such arguments (as
well as the mathematical) are intuitively sound and verifiable, and
might be generalised in axioms if it were worth while: but it is not,
because no method could be founded on such axioms.

The customary use of relative terms justifies some Mediate Inferences,
as, _The father of a father is a grand-father_.

Some cases, however, that at first seem obvious, are really delusive
unless further data be supplied. Thus _A co-exists with B, B with C; ∴ A
with C_--is not sound unless _B_ is an instantaneous event; for where B
is perdurable, _A_ may co-exist with it at one time and _C_ at another.

Again: _A is to the left of B, B of C; ∴ A of C_. This may pass; but it
is not a parallel argument that if _A is north of B and B west of C_,
then _A is north-west of C_: for suppose that A is a mile to the north
of B, and B a yard to the west of C, then A is practically north of C;
at least, its westward position cannot be expressed in terms of the
mariner's compass. In such a case we require to know not only the
directions but the distances of A and C from B; and then the exact
direction of A from C is an affair of mathematical calculation.

Qualitative reasoning concerning position is only applicable to things
in one dimension of space, or in time considered as having one
dimension. Under these conditions we may frame the following
generalisation concerning all Mediate Inferences: Two terms definitely
related to a third, and one of them positively, are related to one
another as the other term is related to the third (that is, positively
or negatively); provided that the relations given are of the same kind
(that is, of Time, or Coinherence, or Likeness, or Equality).

Thus, to illustrate by relations of Time--

      B is simultaneous with C;
      A is not simultaneous with B:
    ∴ A is not simultaneous with C.

Here the relations are of the same kind but of different logical
quality, and (as in the syllogism) a negative copula in the premises
leads to a negative conclusion.

An examination in detail of particular cases would show that the above
generalisation concerning all Mediate Inferences is subject to too many
qualifications to be called an Axiom; it stands to the real Axioms (the
_Dictum_, etc.) as the notion of the Uniformity of Nature does to the
definite principles of natural order (_cf._ chap. xiii. § 9).



§ 1. The type of logical, deductive, mediate, categorical Inference is a
Syllogism directly conformable with the _Dictum_: as--

      All carnivores (M) are excitable (P);
      Cats (S) are carnivores (M):
    ∴ Cats (S) are excitable (P).

In this example P is predicated of M, a term distributed; in which term,
M, S is given as included; so that P may be predicated of S.

Many arguments, however, are of a type superficially different from the
above: as--

      No wise man (P) fears death (M);
      Balbus (S) fears death (M):
    ∴ Balbus (S) is not a wise man (P).

In this example, instead of P being predicated of M, M is predicated of
P, and yet S is given as included not in P, but in M. The divergence of
such a syllogism from the _Dictum_ may, however, be easily shown to be
superficial by writing, instead of _No wise man fears death_, the
simple, converse, _No man who fears death is wise_.


      Some dogs (M) are friendly to man (P);
      All dogs (M) are carnivores (S):
    ∴ Some carnivores (S) are friendly to man (P).

Here P is predicated of M undistributed; and instead of S being included
in M, M is included in S: so that the divergence from the type of
syllogism to which the _Dictum_ directly applies is still greater than
in the former case. But if we transpose the premises, taking first

    All dogs (M) are carnivores (P),

then P is predicated of M distributed; and, simply converting the other
premise, we get--

    Some things friendly to man (S) are dogs (M):

whence it follows that--

    Some things friendly to man (S) are carnivores (P);

and this is the simple converse of the original conclusion.

Once more:

      No pigs (P) are philosophers (M);
      Some philosophers (M) are hedonists (S):
    ∴ Some hedonists (S) are not pigs (P).

In this case, instead of P being predicated of M distributed, M is
predicated of P distributed; and instead of S (or part of it) being
included in M, we are told that some M is included in S. Still there is
no real difficulty. Simply convert both the premises, and we have:

    No philosophers (M) are pigs (P);
    Some hedonists (S) are philosophers (M).

Whence the same conclusion follows; and the whole syllogism plainly
conforms directly to the _Dictum_.

Such departures as these from the normal syllogistic form are said to
constitute differences of Figure (see § 2); and the processes by which
they are shown to be unessential differences are called Reduction (see §

§ 2. Figure is determined by the position of the Middle Term in the
premises; of which position there are four possible variations. The
middle term may be subject of the major premise, and predicate of the
minor, as in the first example above; and this position, being directly
conformable to the requirements of the _Dictum_, is called the First
Figure. Or the middle term may be predicate of both premises, as in the
second of the above examples; and this is called the Second Figure. Or
the middle term may be subject of both premises, as in the third of the
above examples; and this is called the Third Figure. Or, finally, the
middle term may be predicate of the major premise, and subject of the
minor, as in the fourth example given above; and this is the Fourth

It may facilitate the recollection of this most important point if we
schematise the figures thus:

  I.             II.                III.              IV.

M---P          P---M               M---P             P---M
 \                |                 |                   /
  \               |                 |                  /
   \              |                 |                 /
S---M          S---M               M---S             M---S

The horizontal lines represent the premises, and at the angles formed
with them by the slanting or by the perpendicular lines the middle term
occurs. The schema of Figure IV. resembles Z, the last letter of the
alphabet: this helps one to remember it in contrast with Figure I.,
which is thereby also remembered. Figures II. and III. seem to stand
back to back.

§ 3. The Moods of each Figure are the modifications of it which arise
from different combinations of propositions according to quantity and
quality. In Figure I., for example, four Moods are recognised: A.A.A.,
E.A.E., A.I.I., E.I.O.

    A.  All M is P;
    A.  All S is M:
    A.  ∴ All S is P.

    E.  No M is P;
    A.  All S is M:
    E.  ∴ No S is P.

    A.  All M is P;
    I.  Some S is M:
    I.  ∴ Some S is P.

    E.  No M is P;
    I.  Some S is M:
    O.  ∴ Some S is not P.

Now, remembering that there are four Figures, and four kinds of
propositions (A. I. E. O.), each of which propositions may be major
premise, minor premise, or conclusion of a syllogism, it appears that in
each Figure there may be 64 Moods, and therefore 256 in all. On
examining these 256 Moods, however, we find that only 24 of them are
valid (i.e., of such a character that the conclusion strictly follows
from the premises), whilst 5 of these 24 are needless, because their
conclusions are 'weaker' or less extensive than the premises warrant;
that is to say, they are particular when they might be universal. Thus,
in Figure I., besides the above 4 Moods, A.A.I. and E.A.O. are valid in
the sense of being conclusive; but they are superfluous, because
included in A.A.A. and E.A.E. Omitting, then, these 5 needless Moods,
which are called 'Subalterns' because their conclusions are subaltern
(chap. vii. § 2) to those of other Moods, there remain 19 Moods that are
valid and generally recognised.

§ 4. How these 19 Moods are determined must be our next inquiry. There
are several ways more or less ingenious and interesting; but all depend
on the application, directly or indirectly, of the Six Canons, which
were shown in the last chapter to be the conditions of Mediate

(1) One way is to begin by finding what Moods of Figure I. conform to
the _Dictum_. Now, the _Dictum_ requires that, in the major premise, P
be predicated of a term distributed, from which it follows that no Mood
can be valid whose major premise is particular, as in I.A.I. or O.A.O.
Again, the _Dictum_ requires that the minor premise be affirmative
("with which term another is identified"); so that no Mood can be valid
whose minor premise is negative, as in A.E.E. or A.O.O. By such
considerations we find that in Figure I., out of 64 Moods possible, only
six are valid, namely, those above-mentioned in § 3, including the two
subalterns. The second step of this method is to test the Moods of the
Second, Third, and Fourth Figures, by trying whether they can be reduced
to one or other of the four Moods of the First (as briefly illustrated
in § 1, and to be further explained in § 6).

(2) Another way is to take the above six General or Common Canons, and
to deduce from them Special Canons for testing each Figure: an
interesting method, which, on account of its length, will be treated of
separately in the next section.

(3) Direct application of the Common Canons is, perhaps, the simplest
plan. First write out the 64 Moods that are possible without regard to
Figure, and then cross out those which violate any of the Canons or
Corollaries, thus:

AAA, [\A\A\E] (6th Can. b). AAI. [\A\A\O] (6th Can. b).
[\A\E\A] (6th Can. a) AEE, [\A\E\I] (6th Can. a) AEO,
[\A\I\A] (Cor. ii.) [\A\I\E] (6th Can. b) AII, [\A\I\O] (6th Can. b)
[\A\O\A] (6th Can. a) [\A\O\E] (Cor. ii.) [\A\O\I] (6th Can. a) AOO.

Whoever has the patience to go through the remaining 48 Moods will
discover that of the whole 64 only 11 are valid, namely:

A.A.A., A.A.I., A.E.E., A.E.O., A.I.I., A.O.O.,
E.A.E., E.A.O., E.I.O., I.A.I., O.A.O.

These 11 Moods have next to be examined in each Figure, and if valid in
every Figure there will still be 44 moods in all. We find, however, that
in the First Figure, A.E.E., A.E.O., A.O.O. involve illicit process of
the major term (3rd Can.); I.A.I., O.A.O. involve undistributed Middle
(4th Can.); and A.A.I., E.A.O. are subalterns. In the Second Figure all
the affirmative Moods, A.A.A., A.A.I., A.I.I., I.A.I., involve
undistributed Middle; O.A.O. gives illicit process of the major term;
and A.E.O., E.A.O. are subalterns. In the Third Figure, A.A.A., E.A.E.,
involve illicit process of the minor term (3rd Can.); A.E.E., A.E.O.,
A.O.O., illicit process of the major term. In the Fourth Figure, A.A.A.
and E.A.E. involve illicit process of the minor term; A.I.I., A.O.O.,
undistributed Middle; O.A.O. involves illicit process of the major term;
and A.E.O. is subaltern.

Those moods of each Figure which, when tried by these tests, are not
rejected, are valid, namely:

Fig. I.--A.A.A., E.A.E., A.I.I., E.I.O. (A.A.I., E.A.O., Subaltern);

Fig. II.--E.A.E., A.E.E., E.I.O., A.O.O. (E.A.O., A.E.O., Subaltern);

Fig. III.--A.A.I., I.A.I., A.I.I., E.A.O., O.A.O., E.I.O.;

Fig. IV.--A.A.I., A.E.E., I.A.I., E.A.O., E.I.O. (A.E.O., Subaltern).

Thus, including subaltern Moods, there are six valid in each Figure. In
Fig. III. alone there is no subaltern Mood, because in that Figure there
can be no universal conclusion.

§ 5. Special Canons of the several Figures, deduced from the Common
Canons, enable us to arrive at the same result by a somewhat different
course. They are not, perhaps, necessary to the Science, but afford a
very useful means of enabling one to thoroughly appreciate the character
of formal syllogistic reasoning. Accordingly, the proof of each rule
will be indicated, and its elaboration left to the reader. There is no
difficulty, if one bears in mind that Figure is determined by the
position of the middle term.

Fig. I., Rule (a): _The minor premise must be affirmative_.

For, if not, in negative Moods there will be illicit process of the
major term. Applying this rule to the eleven possible Moods given in §
4, as remaining after application of the Common Canons, it eliminates
A.E.E., A.E.O., A.O.O.

(b) _The major premise must be universal_.

For, if not, the minor premise being affirmative, the middle term will
be undistributed. This rule eliminates I.A.I., O.A.O.; leaving six
Moods, including two subalterns.

Fig. II. (a) _One premise must be negative._

For else neither premise will distribute the middle term. This rule
eliminates A.A.A., A.A.I., A.I.I., I.A.I.

(b) _The major premise must be universal._

For else, the conclusion being negative, there will be illicit process
of the major term. This eliminates I.A.I., O.A.O.; leaving six Moods,
including two subalterns.

Fig. III. (a) _The minor premise must be affirmative._

For else, in negative moods there will be illicit process of the major
term. This rule eliminates A.E.E., A.E.O., A.O.O.

(b) _The conclusion must be particular._

For, if not, the minor premise being affirmative, there will be illicit
process of the minor term. This eliminates A.A.A., A.E.E., E.A.E.;
leaving six Moods.

Fig. IV. (a) _When the major premise is affirmative, the minor must be

For else the middle term is undistributed. This eliminates A.I.I.,

(b) _When the minor premise is affirmative the conclusion must be

Otherwise there will be illicit process of the minor term. This
eliminates A.A.A., E.A.E.

(c) _When either premise is negative, the major must be universal._

For else, the conclusion being negative, there will be illicit process
of the major term. This eliminates O.A.O.; leaving six Moods, including
one subaltern.

§ 6. Reduction is either--(1) Ostensive or (2) Indirect. Ostensive
Reduction consists in showing that an argument given in one Mood can
also be stated in another; the process is especially used to show that
the Moods of the second, third, and fourth Figures are equivalent to one
or another Mood of the first Figure. It thus proves the validity of the
former Moods by showing that they also essentially conform to the
_Dictum_, and that all Categorical Syllogisms are only superficial
varieties of one type of proof.

To facilitate Reduction, the recognised Moods have all had names given
them; which names, again, have been strung together into mnemonic verses
of great force and pregnancy:

    Barbara, Celarent, Darii, Ferioque prioris:
    Cesare, Camestres, Festino, Baroco, secundæ:
    Tertia, Darapti, Disamis, Datisi, Felapton,
    Bocardo, Ferison, habet: Quarta insuper addit
    Bramantip, Camenes, Dimaris, Fesapo, Fresison.

In the above verses the names of the Moods of Fig. I. begin with the
first four consonants B, C, D, F, in alphabetical order; and the names
of all other Moods likewise begin with these letters, thus signifying
(except in Baroco and Bocardo) the mood of Fig. I., to which each is
equivalent, and to which it is to be reduced: as Bramantip to Barbara,
Camestres to Celarent, and so forth.

The vowels A, E, I, O, occurring in the several names, give the quantity
and quality of major premise, minor premise, and conclusion in the usual

The consonants s and p, occurring after a vowel, show that the
proposition which the vowel stands for is to be converted either (s)
simply or (p) _per accidens_; except where s or p occurs after the third
vowel of a name, the conclusion: then it refers not to the conclusion of
the given Mood (say Disamis), but to the conclusion of that Mood of the
first Figure to which the given Mood is reduced (Darii).

M (_mutare_, metathesis) means 'transpose the premises' (as of

C means 'substitute the contradictory of the conclusion for the
foregoing premise,' a process of the Indirect Reduction to be presently
explained (see Baroco, § 8).

The other consonants, r, n, t (with b and d, when not initial),
occurring here and there, have no mnemonic significance.

What now is the problem of Reduction? The difference of Figures depends
upon the position of the Middle Term. To reduce a Mood of any other
Figure to the form of the First, then, we must so manipulate its
premises that the Middle Term shall be subject of the major premise and
predicate of the minor premise.

Now in Fig. II. the Middle Term is predicate of both premises; so that
the minor premise may need no alteration, and to convert the major
premise may suffice. This is the case with Cesare, which reduces to
Celarent by simply converting the major premise; and with Festino, which
by the same process becomes Ferio. In Camestres, however, the minor
premise is negative; and, as this is impossible in Fig. I., the premises
must be transposed, and the new major premise must be simply converted:
then, since the transposition of the premises will have transposed the
terms of the conclusion (according to the usual reading of syllogisms),
the new conclusion must be simply converted in order to prove the
validity of the original conclusion. The process may be thus represented
(_s.c._ meaning 'simply convert')

  Camestres.               Celarent.

  All P is M; ----\    /---> No M is S;
                   \ c/
                   /  \
  No S is M:  ----/    \---> All P is M:

∴ No S is P.  <----------- No P is S.

The Ostensive Reduction of Baroco also needs special explanation; for as
it used to be reduced indirectly, its name gives no indication of the
ostensive process. To reduce it ostensively let us call it Faksnoko,
where k means 'obvert the foregoing premise.' By thus obverting (k) and
simply converting (s) (in sum, contrapositing) the major premise, and
obverting the minor premise, we get a syllogism in Ferio, thus:

Baroco or Faksnoko.                                  Ferio.
    All P is M;       ----------------------->  No m (not-M) is P;

    Some S is not M:  ----------------------->  Some S is m (not-M):
  ∴ Some S is not P.                            ∴ Some S is not P.

In Fig. III. the middle term is subject of both premises; so that, to
reduce its Moods to the First Figure, it may be enough to convert the
minor premise. This is the case with Darapti, Datisi, Felapton, and
Ferison. But, with Disamis, since the major premise must in the First
Figure be universal, we must transpose the premises, and then simply
convert the new minor premise; and, lastly, since the major and minor
terms have now changed places, we must simply convert the new conclusion
in order to verify the old one. Thus:

      Disamis.                      Darii.

    Some M is P; ----\    /---> All M is S;
                      /  \
    All M is S:  ----/    \---> Some P is M:

∴ Some S is P. <------------- ∴ Some P is S.

Bocardo, like Baroco, indicates by its name the indirect process. To
reduce it ostensively let its name be Doksamrosk, and proceed thus:

  Bocardo or Doksamrosk.                            Darii.

    Some M is not P; ----------\    /---------> All M is S;
                                \  /
                                 /\ _contrap_
                                /  \
    All M is S:      ----------/    \---------> Some p (not-P) is M:

                       _convert & obvert_
∴ Some S is not P. <------------------------- ∴ Some p (not-P) is S.

In Fig. IV. the position of the middle term is, in both premises, the
reverse of what it is in the First Figure; we may therefore reduce its
Moods either by transposing the premises, as with Bramantip, Camenes,
and Dimaris; or by converting both premises, the course pursued with
Fesapo and Fresison. It may suffice to illustrate by the case of

Bramantip.                                 Barbara.

All P is M;      ----------\  /------>   All M is S;
All M is S:      ----------/  \------>   All P is M:

                      convert per acc.
∴ Some S is P.   <----------------------- ∴ All P is S.

This case shows that a final significant consonant (s, p, or sk) in the
name of any Mood refers to the conclusion of the new syllogism in the
First Figure; since p in Bramantip cannot refer to that Mood's own
conclusion in I.; which, being already particular, cannot be converted
_per accidens_.

Finally, in Fig. I., Darii and Ferio differ respectively from Barbara
and Celarent only in this, that their minor premises, and consequently
their conclusions, are subaltern to the corresponding propositions of
the universal Moods; a difference which seems insufficient to give them
rank as distinct forms of demonstration. And as for Barbara and
Celarent, they are easily reducible to one another by obverting their
major premises and the new conclusions, thus:

Barbara.                                     Celarent.
All M is P;    ----------------------->    No M is p (not-P);

All S is M:    ----------------------->    All S is M:

∴ All S is P.    <-------------------    ∴ No S is p (not-P).

There is, then, only one fundamental syllogism.

§ 7. A new version of the mnemonic lines was suggested in _Mind_ No. 27,
with the object of (1) freeing them from all meaningless letters, (2)
showing by the name of each Mood the Figure to which it belongs, (3)
giving names to indicate the ostensive reduction of Baroco and Bocardo.
To obtain the first two objects, _l_ is used as the mark of Fig. I., _n_
of Fig II., _r_ of Fig. III., _t_ of Fig. IV. The verses (to be scanned
discreetly) are as follows:

Balala, Celalel, Dalii, Felioque prioris:

Cesane, Camenes, Fesinon, {        } secundæ:
                          { Banoco,}

Tertia, Darapri, Drisamis, Darisi, Ferapro,

          }, Ferisor habet: Quarta insuper addit.
Bocaro    }

Bamatip, Cametes, Dimatis, Fesapto, Fesistot.

De Morgan praised the old verses as "more full of meaning than any
others that ever were made"; and in defence of the above alteration it
may be said that they now deserve that praise still more.

§ 8. Indirect reduction is the process of proving a Mood to be valid by
showing that the supposition of its invalidity involves a contradiction.
Take Baroco, and (since the doubt as to its validity is concerned not
with the truth of the premises, but with their relation to the
conclusion) assume the premises to be true. Then, if the conclusion be
false, its contradictory is true. The conclusion being in O., its
contradictory will be in A. Substituting this A. for the minor premise
of Baroco, we have the premises of a syllogism in Barbara, which will be
found to give a conclusion in A., contradictory of the original minor
premise; thus:

     Baroco.                           Barbara.

  All P is M;      -----------------> All P is M;

  Some S is not M: <-----\    /-----> All S is P:
                          \  /
             contradictory \/
                           /\ contradictory
                          /  \
∴ Some S is not P  ------/    \------ ∴ All S is M.

But the original minor premise, _Some S is not M_, is true by
hypothesis; and therefore the conclusion of Barbara, _All S is M_, is
false. This falsity cannot, however, be due to the form of Barbara,
which we know to be valid; nor to the major premise, which, being taken
from Baroco, is true by hypothesis: it must, therefore, lie in the minor
premise of Barbara, _All S is P_; and since this is contradictory of the
conclusion of Baroco _Some S is not P_, that conclusion was true.

Similarly, with Bocardo, the Indirect Reduction proceeds by substituting
for the major premise the contradictory of the conclusion; thus again
obtaining the premises of a syllogism in Barbara, whose conclusion is
contradictory of the original major premise. Hence the initial B in
Baroco and Bocardo: it points to a syllogism in Barbara as the means of
Indirect Reduction (_Reductio ad impossibile_).

Any other Mood may be reduced indirectly: as, for example, Dimaris. If
this is supposed to be invalid and the conclusion false, substitute the
contradictory of the conclusion for the major premise, thus obtaining
the premises of Celarent:

Dimaris.                                       Celarent.

Some P is M;       <---------    -------->      No S is P;
                             \  /
All M is S:        -----------/\--------->      All M is S:
                             /  \
               contradictory/    \
∴ Some S is P. -----------      --------  ∴ No M is P}
                                                     } simply converted
                                          ∴ No P is M}

The conclusion of Celarent, simply converted, contradicts the original
major premise of Dimaris, and is therefore false. Therefore the major
premise of Celarent is false, and the conclusion of Dimaris is true. We
might, of course, construct mnemonic names for the Indirect Reduction of
all the Moods: the name of Dimaris would then be Cicari.

§ 9. The need or use of any Figure but the First has been much discussed
by Logicians. Since, in actual debate, arguments are rarely stated in
syllogistic form, and, therefore, if reduced to that form for closer
scrutiny, generally have to be treated with some freedom; why not always
throw them at once into the First Figure? That Figure has manifest
advantages: it agrees directly with the _Dictum_; it gives conclusions
in all four propositional forms, and therefore serves every purpose of
full affirmation or denial, of showing agreement or difference (total or
partial), of establishing the contradictories of universal statements;
and it is the only Figure in which the subject and predicate of the
conclusion occupy the same positions in the premises, so that the course
of argument has in its mere expression an easy and natural flow.

Still, the Second Figure also has a very natural air in some kinds of
negative arguments. The parallelism of the two premises, with the middle
term as predicate in both, brings out very forcibly the necessary
difference between the major and minor terms that is involved in their
opposite relations to the middle term. _P is not, whilst S is, M_, says
Cesare: that drives home the conviction that _S is not P_. Similarly in
Camestres: _Deer do, oxen do not, shed their horns_. What is the

The Third Figure, again, furnishes in Darapti and Felapton, the most
natural forms of stating arguments in which the middle term is singular:

      Socrates was truthful;
      Socrates was a Greek:
    ∴ Some Greek was truthful.

Reducing this to Fig I., we should get for the minor premise, _Some
Greek was Socrates_: which is certainly inelegant. Still, it might be
urged that, in relation to proof, elegance is an extraneous
consideration. And as for the other advantage claimed for Fig.
III.--that, as it yields only particular conclusions, it is useful in
establishing contradictories against universals--for that purpose none
of its Moods can be better than Darii or Ferio.

As for Fig. IV., no particular advantage has been claimed for it. It is
of comparatively late recognition (sometimes called the 'Galenian,'
after Galen, its supposed discoverer); and its scientific claim to exist
at all is disputed. It is said to be a mere inversion of Fig. I.; which
is not true in any sense in which Figs. II. and III. may not be
condemned as partial inversions of Fig. I., and as having therefore
still less claim to recognition. It is also said to invert the order of
thought; as if thought had only one order, or as if the order of thought
had anything to do with Formal Logic. Surely, if distinction of Figure
be recognised at all, the Fourth Figure is scientifically necessary,
because it is inevitably generated by an analysis of the possible
positions of the middle term.

§ 10. Is Reduction necessary, however; or have not all the Figures equal
and independent validity? In one sense not only every Figure but each
Mood has independent validity: for any one capable of abstract thinking
sees its validity by direct inspection; and this is true not only of the
abstract Moods, but very frequently of particular concrete arguments.
But science aims at unifying knowledge; and after reducing all possible
arguments that form categorical syllogisms to the nineteen Moods, it is
another step in the same direction to reduce these Moods to one form.
This is the very nature of science: and, accordingly, the efforts of
some Logicians to expound separate principles of each Figure seem to be
supererogatory. Grant that they succeed; and what can the next step be,
but either to reduce these principles to the _Dictum_, or the _Dictum_
and the rest to one of these principles? Unless this can be done there
is no science of Formal Logic. If it is done, what is gained by reducing
the principles of the other Figures to the _Dictum_, instead of the
Moods of the other Figures to those of the first Figure? It may,
perhaps, be said that to show (1) that the Moods of the second, third,
and fourth Figures flow from their own principles (though, in fact,
these principles are laboriously adapted to the Moods); and (2) that
these principles may be derived from the _Dictum_, is the more
uncompromisingly gradual and regular method: but is not Formal Logic
already sufficiently encumbered with formalities?

§ 11. Euler's diagrams are used to illustrate the syllogism, though not
very satisfactorily, thus:


[Illustration: FIG. 5.]


[Illustration: FIG. 6.]


[Illustration: FIG. 7.]

Remembering that 'Some' means 'It may be all,' it is plain that any one
of these diagrams in Fig. 7, or the one given above for Barbara, may
represent the denotative relations of P, M and S in Darii; though no
doubt the diagram we generally think of as representing Darii is No. 1
in Fig. 7.

Remembering that A may be U, and that, therefore, wherever A occurs
there may be only one circle for S and P, these syllogisms may be
represented by only two circles, and Barbara by only one.


[Illustration: FIG. 8.]

Here, again, probably, we generally think of No. 1 as the diagram
representing Ferio; but 2, or 3, or that given above for Celarent, is
compatible with the premises.

If instead of dealing with M, P, and S, a concrete example be taken of
Darii or Ferio, a knowledge of the facts of the case will show what
diagram is suitable to it. But, then, surely it must be possible to do
without the diagram. These diagrams, of course, can be used to
illustrate Moods of the other Figures.



§ 1. In ordinary discussion, whether oral or written, it is but rarely
that the forms of Logic are closely adhered to. We often leave wide gaps
in the structure of our arguments, trusting the intelligence of those
addressed to bridge them over; or we invert the regular order of
propositions, beginning with the conclusion, and mentioning the
premises, perhaps, a good while after, confident that the sagacity of
our audience will make all smooth. Sometimes a full style, like
Macaulay's, may, by means of amplification and illustration, spread the
elements of a single syllogism over several pages--a pennyworth of logic
steeped in so much eloquence. These practices give a great advantage to
sophists; who would find it very inconvenient to state explicitly in
Mood and Figure the pretentious antilogies which they foist upon the
public; and, indeed, such licences of composition often prevent honest
men from detecting errors into which they themselves have unwittingly
fallen, and which, with the best intentions, they strive to communicate
to others: but we put up with these drawbacks to avoid the inelegance
and the tedium of a long discourse in accurate syllogisms.

Many departures from the strictly logical statement of reasonings
consist in the use of vague or figurative language, or in the
substitution for one another of expressions supposed to be equivalent,
though, in fact, dangerously discrepant. Against such occasions of error
the logician can provide no safeguard, except the advice to be careful
and discriminating in what you say or hear. But as to any derangement
of the elements of an argument, or the omission of them, Logic
effectually aids the task of restoration; for it has shown what the
elements are that enter into the explicit statement of most
ratiocinations, namely, the four forms of propositions and what that
connected order of propositions is which most easily and surely exposes
the validity or invalidity of reasoning, namely, the premises and
conclusion of the Syllogism. Logic has even gone so far as to name
certain abbreviated forms of proof, which may be regarded as general
types of those that actually occur in debate, in leading articles,
pamphlets and other persuasive or polemic writings--namely, the
Enthymeme, Epicheirema and Sorites.

§ 2. The Enthymeme, according to Aristotle, is the Syllogism of probable
reasoning about practical affairs and matters of opinion, in contrast
with the Syllogism of theoretical demonstration upon necessary grounds.
But, as now commonly treated, it is an argument with one of its elements
omitted; a Categorical Syllogism, having one or other of its premises,
or else its conclusion, suppressed. If the major premise be suppressed,
it is called an Enthymeme of the First Order; if the minor premise be
wanting, it is said to be of the Second Order; if the conclusion be left
to be understood, there is an Enthymeme of the Third Order.

Let the following be a complete Syllogism:

     All free nations are enterprising;
     The Dutch are a free nation:
   ∴ The Dutch are enterprising.

Reduced to Enthymemes, this argument may be put thus:

In the First Order:

      The Dutch are a free nation:
    ∴ The Dutch are enterprising.

In the Second Order--

      All free nations are enterprising;
    ∴ The Dutch are enterprising.

In the Third Order--

    All free nations are enterprising;
    And the Dutch are a free nation.

It is certainly very common to meet with arguments whose statement may
be represented by one or other of these three forms; indeed, the
Enthymeme is the natural substitute for a full syllogism in oratory:
whence the transition from Aristotle's to the modern meaning of the
term. The most unschooled of men readily apprehend its force; and a
student of Logic can easily supply the proposition that may be wanted in
any case to complete a syllogism, and thereby test the argument's formal
validity. In any Enthymeme of the Third Order, especially, to supply the
conclusion cannot present any difficulty at all; and hence it is a
favourite vehicle of innuendo, as in Hamilton's example:

    Every liar is a coward;
    And Caius is a liar.

The frankness of this statement and its reticence, together, make it a
biting sarcasm upon Caius.

The process of finding the missing premise in an Enthymeme of either the
First or the Second Order, so as to constitute a syllogism, is sometimes
called Reduction; and for this a simple rule may be given: Take that
term of the given premise which does not occur in the conclusion (and
which must therefore be the Middle), and combine it with that term of
the conclusion which does not occur in the given premise; the
proposition thus formed is the premise which was requisite to complete
the Syllogism. If the premise thus constituted contain the predicate of
the conclusion, the Enthymeme was of the First Order; if it contain the
subject of the conclusion, the Enthymeme was of the Second Order.

That a statement in the form of a Hypothetical Proposition may really be
an Enthymeme (as observed in chap. v. § 4) can easily be shown by
recasting one of the above Enthymemes thus: _If all free nations are
enterprising, the Dutch are enterprising_. Such statements should be
treated according to their true nature.

To reduce the argument of any ordinary discourse to logical form, the
first care should be to make it clear to oneself what exactly the
conclusion is, and to state it adequately but as succinctly as possible.
Then look for the evidence. This may be of an inductive character,
consisting of instances, examples, analogies; and, if so, of course its
cogency must be evaluated by the principles of Induction, which we
shall presently investigate. But if the evidence be deductive, it will
probably consist of an Enthymeme, or of several Enthymemes one depending
on another. Each Enthymeme may be isolated and expanded into a
syllogism. And we may then inquire: (1) whether the syllogisms are
formally correct according to Barbara (or whatever the appropriate
Mood); (2) whether the premises, or the ultimate premises, are true in

§ 3. A Monosyllogism is a syllogism considered as standing alone or
without relation to other arguments. But, of course, a disputant may be
asking to prove the premises of any syllogism; in which case other
syllogisms may be advanced for that purpose. When the conclusion of one
syllogism is used to prove another, we have a chain-argument which,
stated at full length, is a Polysyllogism. In any Polysyllogism, again,
a syllogism whose conclusion is used as the premise of another, is
called in relation to that other a Prosyllogism; whilst a syllogism one
of whose premises is the conclusion of another syllogism, is in relation
to that other an Episyllogism. Two modes of abbreviating a
Polysyllogism, are usually discussed, the Epicheirema and the Sorites.

§ 4. An Epicheirema is a syllogism for one or both of whose premises a
reason is added; as--

      All men are mortal, for they are animals;
      Socrates is a man, for rational bipeds are men:
    ∴ Socrates is mortal.

The Epicheirema is called Single or Double, says Hamilton, according as
an "adscititious proposition" attaches to one or both of the premises.
The above example is of the double kind. The Single Epicheirema is said
to be of the First Order, if the adscititious proposition attach to the
major premise; if to the minor, of the Second Order. (Hamilton's
_Logic_: Lecture xix.)

An Epicheirema, then, is an abbreviated chain of reasoning, or
Polysyllogism, comprising an Episyllogism with one or two enthymematic
Prosyllogisms. The major premise in the above case, _All men are mortal,
for they are animals,_ is an Enthymeme of the First Order, suppressing
its own major premise, and may be restored thus:

      All animals are mortal;
      All men are animals:
    ∴ All men are mortal.

The minor premise, _Socrates is a man, for rational bipeds are men_, is
an Enthymeme of the Second Order, suppressing its own minor premise, and
may be restored thus:

      All rational bipeds are men;
      Socrates is a rational biped:
    ∴ Socrates is a man.

§ 5. The Sorites is a Polysyllogism in which the Conclusions, and even
some of the Premises, are suppressed until the arguments end. If the
chain of arguments were freed of its enthymematic character, the
suppressed conclusions would appear as premises of Episyllogisms.

Two varieties of Sorites are recognised, the Aristotelian (so called,
though not treated of by Aristotle), and the Goclenian (named after its
discoverer, Goclenius of Marburg, who flourished about 1600 A.D.). In
order to compare these two forms of argument, it will be convenient to
place side by side Hamilton's classical examples of them.

          Aristotelian.                     Goclenian.
      Bucephalus is a horse;          An animal is a substance;
      A horse is a quadruped;         A quadruped is an animal;
      A quadruped is an animal;       A horse is a quadruped;
      An animal is a substance:       Bucephalus is a horse:
    ∴ Bucephalus is a substance.    ∴ Bucephalus is a substance.

The reader wonders what is the difference between these two forms. In
the Aristotelian Sorites the minor term occurs in the first premise, and
the major term in the last; whilst in the Goclenian the major term
occurs in the first premise, and the minor in the last. But since the
character of premises is fixed by their terms, not by the order in which
they are written, there cannot be a better example of a distinction
without a difference. At a first glance, indeed, there may seem to be a
more important point involved; the premises of the Aristotelian Sorites
seem to proceed in the order of Fig. IV. But if that were really so the
conclusion would be, _Some Substance is Bucephalus_. That, on the
contrary, every one writes the conclusion, _Bucephalus is a substance_,
proves that the logical order of the premises is in Fig. I. Logically,
therefore, there is absolutely no difference between these two forms,
and pure reason requires either that the "Aristotelian Sorites"
disappear from the text-books, or that it be regarded as in Fig. IV.,
and its conclusion converted. It is the shining merit of Goclenius to
have restored the premises of the Sorites to the usual order of Fig. I.:
whereby he has raised to himself a monument more durable than brass, and
secured indeed the very cheapest immortality.

The common Sorites, then, being in Fig. I., its rules follow from those
of Fig. I:

(1) Only one premise can be particular; and, if any, only that in which
the minor term occurs.

For, just as in Fig I., a particular premise anywhere else involves
undistributed Middle.

(2) Only one premise can be negative; and, if any, only that in which
the major term occurs.

For if there were two negative premises, at the point where the second
entered the chain of argument there must be a syllogism with two
negative premises, which is contrary to Rule 5; whilst if one premise be
negative it must be that which contains the major term, for the same
reason as in Fig. I., namely, that the conclusion will be negative, and
that therefore only a negative major premise can prevent illicit process
of the major term.

If we expand a Sorites into its constituent syllogisms, the conclusions
successively suppressed will reappear as major premises; thus:

    (1) An animal is a substance;
        A quadruped is an animal:
      ∴ A quadruped is a substance.

    (2) A quadruped is a substance;
        A horse is a quadruped:
      ∴ A horse is a substance.

    (3) A horse is a substance:
        Bucephalus is a horse:
      ∴ Bucephalus is a substance.

This suffices to show that the Protosyllogism of a Goclenian Sorites is
an Enthymeme of the Third Order; after which the argument is a chain of
Enthymemes of the First Order, or of the First and Third combined, since
the conclusions as well as the major premises are omitted, except in the
last one.

Lest it should be thought that the Sorites is only good for arguments so
frivolous as the above, I subjoin an example collected from various
parts of Mill's _Political Economy_:--

      The cost of labour depends on the efficiency of labour;
      The rate of profits depends on the cost of labour;
      The investment of capital depends on the rate of profits;
      Wages depend on the investment of capital:
    ∴ Wages depend on the efficiency of labour.

Had it occurred to Mill to construct this Sorites, he would have
modified his doctrine of the wages-fund, and would have spared many
critics the malignant joy of refuting him.

§ 6. The Antinomy is a combination of arguments by which contradictory
attributes are proved to be predicable of the same subject. In symbols,

      All M is P;             All N is p;
      All S is M:             All S is N:
    ∴ All S is P.           ∴ All S is p.

Now, by the principle of Contradiction, S cannot be P and p (not-P):
therefore, if both of the above syllogisms are sound, S, as the subject
of contradictory attributes, is logically an impossible thing. The
contradictory conclusions are called, respectively, Thesis and

To come to particulars, we may argue: (1) that a constitution which is
at once a monarchy, an aristocracy and a democracy, must comprise the
best elements of all three forms; and must, therefore, be the best of
all forms of government: the British Constitution is, therefore, the
best of all. But (2) such a constitution must also comprise the worst
elements of monarchy, aristocracy and democracy; and, therefore, must be
the worst of all forms. Are we, then, driven to conclude that the
British Constitution, thus proved to be both the best and worst, does
not really exist at all, being logically impossible? The proofs seem
equally cogent; but perhaps neither the best nor the worst elements of
the simpler constitutions need be present in our own in sufficient force
to make it either good or bad.


    (1) Every being who is responsible for his actions is free;
        Man is responsible for his actions:
      ∴ Man is free.

    (2) Every being whose actions enter into the course of nature is not free;
        Man is such a being:
      ∴ Man is not free.

Does it, then, follow that 'Man,' as the subject of contradictory
attributes, is a nonentity? This doctrine, or something like it, has
been seriously entertained; but if to any reader it seem extravagant (as
it certainly does to me), he will no doubt find an error in the above
arguments. Perhaps the major term is ambiguous.

For other examples it is enough to refer to the _Critique of Pure
Reason_, where Kant sets out the Antinomies of Rational Cosmology. But
even if we do not agree with Kant that the human understanding, in
attempting to deal with certain subjects beyond its reach, inevitably
falls into such contradictory reasonings; yet it can hardly be doubted
that we not unfrequently hold opinions which, if logically developed,
result in Antinomies. And, accordingly, the Antinomy, if it cannot be
imputed to Reason herself, may be a very fair, and a very wholesome
_argumentum ad hominem_. It was the favourite weapon of the Pyrrhonists
against the dogmatic philosophies that flourished after the death of



§ 1. Conditional Syllogisms may be generally described as those that
contain conditional propositions. They are usually divided into two
classes, Hypothetical and Disjunctive.

A Hypothetical Syllogism is one that consists of a Hypothetical Major
Premise, a Categorical Minor Premise, and a Categorical Conclusion. Two
Moods are usually recognised the _Modus ponens_, in which the antecedent
of the hypothetical major premise is affirmed; and the _Modus tollens_
[sic], in which its consequent is denied.

(1) _Modus ponens_, or Constructive.

    If A is B, C is D;
       A is B:
     ∴ C is D.

If Aristotle's reasoning is conclusive, Plato's theory of Ideas is

      Aristotle's reasoning is conclusive:
    ∴ Plato's theory of Ideas is erroneous.

Rule of the _Modus ponens_: The antecedent of the major premise being
affirmed in the minor premise, the consequent is also affirmed in the

(2) _Modus tollens_, or Destructive.

    If A is B, C is D;
       C is not D:
     ∴ A is not B.

      If Pythagoras is to be trusted, Justice is a number;
      Justice is not a number:
    ∴ Pythagoras is not to be trusted.

Rule of the _Modus tollens_: The consequent of the major premise being
denied in the minor premise, the antecedent is denied in the conclusion.

By using negative major premises two other forms are obtainable: then,
either by affirming the antecedent or by denying the consequent, we draw
a negative conclusion.

    Thus (_Modus ponens_):                (_Modus tollens_):

    If A is B, C is not D;                 If A is B, C is not D;
       A is B:                                C is D:
     ∴ C is not D.                          ∴ A is not B.

Further, since the antecedent of the major premise, taken by itself, may
be negative, it seems possible to obtain four more forms, two in each
Mood, from the following major premises:

    (1)  If A is not B, C is D;
    (2)  If A is not B, C is not D.

But since the quality of a Hypothetical Proposition is determined by the
quality of its consequent, not at all by the quality of its antecedent,
we cannot get from these two major premises any really new Moods, that
is to say, Moods exhibiting any formal difference from the four
previously expounded.

It is obvious that, given the hypothetical major premise--

    If A is B, C is D--

we cannot, by denying the antecedent, infer a denial of the consequent.
That A is B, is a mark of C being D; but we are not told that it is the
sole and indispensable condition of it. If men read good books, they
acquire knowledge; but they may acquire knowledge by other means, as by
observation. For the same reason, we cannot by affirming the consequent
infer the affirmation of the antecedent: Caius may have acquired
knowledge; but we cannot thence conclude that he has read good books.

To see this in another light, let us recall chap. v. § 4, where it was
shown that a hypothetical proposition may be translated into a
categorical one; whence it follows that a Hypothetical Syllogism may be
translated into a Categorical Syllogism. Treating the above examples
thus, we find that the _Modus ponens_ (with affirmative major premise)
takes the form of Barbara, and the _Modus tollens_ the form of

    _Modus ponens._                  Barbara.

    If A is B, C is D;      The case of A being B is a case of C being D;
       A is B:                      This is a case of A being B:
     ∴ C is D.                    ∴ This is a case of C being D.

Now if, instead of this, we affirm the consequent, to form the new minor

    This is a case of C being D,

there will be a Syllogism in the Second Figure with two affirmative
premises, and therefore the fallacy of undistributed Middle. Again:

    _Modus tollens._                  Camestres.

    If A is B, C is D;      The case of A being B is a case of C being D:
       C is not D:               This is not a case of C being D:
     ∴ A is not B.             ∴ This is not a case of A being B.

But if, instead of this, we deny the antecedent, to form the new minor

    This is not a case of A being B,

there arises a syllogism in the First Figure with a negative minor
premise, and therefore the fallacy of illicit process of the major term.

By thus reducing the Hypothetical Syllogism to the Categorical form,
what is lost in elegance is gained in intelligibility. For, first, we
may justify ourselves in speaking of the hypothetical premise as the
major, and of the categorical premise as the minor; since in the
categorical form they contain respectively the major and minor terms.
And, secondly, we may justify ourselves in treating the Hypothetical
Syllogism as a kind of Mediate Inference, in spite of the fact that it
does not exhibit two terms compared by means of a third; since in the
Categorical form such terms distinctly appear: a new term ('This')
emerges in the position of the minor; the place of the Middle is filled
by the antecedent of the major premise in the _Modus ponens_, and by the
consequent in the _Modus tollens_.

The mediate element of the inference in a Hypothetical Syllogism
consists in asserting, or denying, the fulfilment of a given condition;
just as in a Categorical syllogism to identify the minor term with the
Middle is a condition of the major term's being predicated of it. In the
hypothetical proposition--

    If A is B, C is D--

the Antecedent, _A is B_, is the _conditio sufficiens_, or mark, of the
Consequent, _C is D_; and therefore the Consequent, _C is D_, is a
_conditio sine qua non_ of the antecedent, _A is B_; and it is by means
of affirming the former condition, or else denying the latter, that a
conclusion is rendered possible.

Indeed, we need not say that the element of mediation consists in
affirming, _or denying_, the fulfilment of a given condition: it is
enough to say 'in affirming.' For thus to explain the _Modus tollens_,
reduce it to the _Modus ponens_ (contrapositing the major premise and
obverting the minor):


      If A is B, C is D:              The case of C being not-D is
    ∴ If C is not-D, A is not B;        not a case of A being B;
      C is not-D:                     This is a case of C being
    ∴ A is not B.                             not-D:
                                      ∴ This is not a case of A
                                                being B.

The above four forms commonly treated of as Hypothetical Syllogisms, are
called by Ueberweg and Dr. Keynes 'Hypothetico-Categorical.' Ueberweg
restricts the name 'Hypothetical' simply (and Dr. Keynes the name
'Conditional') to such Syllogisms as the following, having two
Hypothetical Premises:

      If C is D, E is F;
      If A is B, C is D:
    ∴ If A is B, E is F.

If we recognise particular hypothetical propositions (see chap. v. § 4),
it is obvious that such Syllogisms may be constructed in all the Moods
and Figures of the Categorical Syllogism; and of course they may be
translated into Categoricals. We often reason in this hypothetical way.
For example:

      If the margin of cultivation be extended, rents will rise;
      If prices of produce rise, the margin of cultivation will be extended:
    ∴ If prices of produce rise, rents will rise.

But the function of the Hypothetical Syllogism (commonly so called), as
also of the Disjunctive Syllogism (to be discussed in the next section)
is to get rid of the conditional element of the premises, to pass from
suspense to certainty, and obtain a decisive categorical conclusion;
whereas these Syllogisms with two hypothetical premises leave us still
with a hypothetical conclusion. This circumstance seems to ally them
more closely with Categorical Syllogisms than with those that are
discussed in the present chapter. That they are Categoricals in disguise
may be seen by considering that the above syllogism is not materially
significant, unless in each proposition the word 'If' is equivalent to
'Whenever.' Accordingly, the name 'Hypothetical Syllogism,' is here
employed in the older usage.

§ 2. A Disjunctive Syllogism consists of a Disjunctive Major Premise, a
Categorical Minor Premise, and a Categorical Conclusion.

How many Moods are to be recognised in this kind of argument depends on
whether the alternatives of the Disjunctive Premise are regarded as
mutually exclusive or possibly coincident. In saying '_Either_ A is B,
_or_ C is D,' do we mean 'either, but not both,' or 'either, it may be
both'? (See chap. v. § 4.)

When the alternatives of the Disjunctive are not exclusive, we have only

    _Modus tollendo ponens._

    Either A is B, or C is D;
           A is not B (or C is not D):
         ∴ C is D     (or A is B).

Either wages fall, or the weaker hands are dismissed;

      Wages do not fall:
    ∴ The weaker hands are dismissed.

But we cannot argue--

      Wages fall:
    ∴ The weaker hands are not dismissed;

since in 'hard times' both events may happen together.

Rule of the _Modus tollendo ponens_: If one alternative be denied, the
other is affirmed.

When, however, the alternatives of the Disjunctive are mutually
exclusive, we have also the

    _Modus ponendo tollens._

    Either A is B, or C is D;
           A is B (or C is D):
         ∴ C is not D (or A is not B).

Either the Tories or the Whigs win the election;

      The Tories win:
    ∴ The Whigs do not win.

We may also, of course, argue as above in the _Modus tollendo ponens_--

      The Tories do not win:
    ∴ The Whigs do.

But in this example, to make the _Modus tollendo ponens_ materially
valid, it must be impossible that the election should result in a tie.
The danger of the Disjunctive Proposition is that the alternatives may
not, between them, exhaust the possible cases. Only contradictory
alternatives are sure to cover the whole ground.

Rule of the _Modus ponendo tollens:_ If one alternative be affirmed, the
other is denied.

Since a disjunctive proposition may be turned into a hypothetical
proposition (chap. v. § 4,) a Disjunctive Syllogism may be turned into a
Hypothetical Syllogism:

    _Modus tollendo ponens._                 _Modus ponens._

    Either A is B, or C is D;             If A is not B, C is D;
           A is not B:                       A is not B:
         ∴ C is D.                         ∴ C is D.

Similarly the _Modus ponendo tollens_ is equivalent to that kind of
_Modus ponens_ which may be formed with a negative major premise; for if
the alternatives of a disjunctive proposition be exclusive, the
corresponding hypothetical be affirmative or negative:

    _Modus ponendo tollens._                 _Modus ponens._

    Either A is B, or C is D;             If A is B, C is not D;
           A is B:                           A is B:
         ∴ C is not D.                     ∴ C is not D.

Hence, finally, a Disjunctive Syllogism being equivalent to a
Hypothetical, and a Hypothetical to a Categorical; a Disjunctive
Syllogism is equivalent and reducible to a Categorical. It is a form of
Mediate Inference in the same sense as the Hypothetical Syllogism is;
that is to say, the conclusion depends upon an affirmation, or denial,
of the fulfilment of a condition implied in the disjunctive major

§ 3. The Dilemma is perhaps the most popularly interesting of all forms
of proof. It is a favourite weapon of orators and wits; and "impaled
upon the horns of a dilemma" is a painful situation in which every one
delights to see his adversary. It seems to have been described by
Rhetoricians before finding its way into works on Logic; and Logicians,
to judge from their diverse ways of defining it, have found some
difficulty in making up their minds as to its exact character.

There is a famous Dilemma employed by Demosthenes, from which the
general nature of the argument may be gathered:

    If Æschines joined in the public rejoicings, he is
    inconsistent; if he did not, he is unpatriotic;

     But either he joined, or he did not join:

     Therefore he is either inconsistent or unpatriotic.

That is, reduced to symbols:

    If A is B, C is D; and if E is F, G is H:
      But either A is B, or E is F;
    ∴ Either C is D or G is H (_Complex Constructive_).

This is a compound Conditional Syllogism, which may be analysed as

    Either A is B or E is F.

    Suppose that E is not F:        Suppose that A is not B:
    Then A is B.                    Then E is F.
    But if A is B, C is D;          But if E is F, G is H;
        (A is B):                        (E is F):
      ∴ C is D.                     ∴ G is H.

    ∴ Either C is D or G is H.

A Dilemma, then, is a compound Conditional Syllogism, having for its
Major Premise two Hypothetical Propositions, and for its Minor Premise a
Disjunctive Proposition, whose alternative terms either affirm the
Antecedents or deny the Consequents of the two Hypothetical Propositions
forming the Major Premise.

The hypothetical propositions in the major premise, may have all four
terms distinct (as in the above example); and then the conclusion is a
disjunctive proposition, and the Dilemma is said to be Complex. Or the
two hypothetical propositions may have a common antecedent or a common
consequent; and then the conclusion is a categorical proposition, and
the Dilemma is said to be Simple.

Again, the alternatives of the disjunctive minor premise may be
affirmative or negative: if affirmative, the Dilemma is called
Constructive; and if negative, Destructive.

Using, then, only affirmative hypothetical propositions in the major
premise, there are four Moods:

1. The Simple Constructive--

    If A is B, C is D; and if E is F, C is D:
      But either A is B, or E is F:
    ∴ C is D.

    If the Tories win the election, the Government will avoid
    innovation; and if the Whigs win, the House of Lords will
    prevent them innovating:

     But either the Tories or the Whigs will win:

     ∴ There will be no innovation.

2. The Complex Constructive--

    If A is B, C is D; and if E is F, G is H:
      But either A is B, or E is F:
    ∴ Either C is D or G is H.

    If appearance is all that exists, reality is a delusion; and
    if there is a substance beyond consciousness, knowledge of
    reality is impossible:

     But either appearance is all, or there is a substance beyond

     ∴ Either reality is a delusion, or a knowledge of it is

3. Simple Destructive--

    If A is B, C is D; and if A is B, E is F:
      But either C is not D, or E is not F:
    ∴ A is not B.

    If table-rappers are to be trusted, the departed are spirits;
    and they also exert mechanical energy:

     But either the departed are not spirits, or they do not
    exert mechanical energy:

     ∴ Table-rappers are not to be trusted.

4. Complex Destructive--

    If A is B, C is D; and if E is F, G is H:
      But either C is not D, or G is not H:
    ∴ Either A is not B, or E is not F.

    If poetic justice is observed, virtue is rewarded; and if the
    mirror is held up to Nature, the villain triumphs:

     But either virtue is not rewarded, or the villain does not

     ∴ Either poetic justice is not observed, or the mirror is
    not held up to Nature.

Such are the four Moods of the Dilemma that emerge if we only use
affirmative hypotheticals for the major premise; but, certainly, it is
often quite as natural to employ two negative hypotheticals (indeed,
one might be affirmative and the other negative; but waive that); and
then four more moods emerge, all having negative conclusions. It is
needless to intimidate the reader by drawing up these four moods in
battle array: they always admit of reduction to the foregoing moods by
obverting the hypotheticals. Still, by the same process we may greatly
decrease the number of moods of the Categorical Syllogism; and just as
some Syllogisms are most simply expressed in Celarent or Cesare, so some
Dilemmas are most simply stated with negative major premises--e.g.,
The example of a Simple Constructive Dilemma above given would run more
naturally thus: _If the Tories win, the Government will not innovate;
and if the Whigs, the Lords will not let them_: and similarly
Demosthenes' Dilemma--_If Æschines joined, he is not consistent; and if
he did not, he is not patriotic_. Moreover, the propriety of recognising
Dilemmas with negative major premises, follows from the above analysis
of the Dilemma into a combination of Conditional Syllogisms, even if (as
in § 1 of this chapter) we take account of only four Moods of the
Hypothetical Syllogism.

In the rhetorical use of the Dilemma, it may be observed that the
disjunction in the minor premise ought to be obvious, or (at any rate)
easily acceptable to the audience. Thus, _Either the Tories or the Whigs
will win; Either Æschines joined in the rejoicings, or he did not_; such
propositions are not likely to be disputed. But if the orator must stop
to prove his minor premise, the smacking effect of this figure (if the
expression be allowed) will be lost. Hence the minor premises of other
examples given above are only fit for a select audience. That _Either
ghosts are not spirits, or they do not exert mechanical energy_,
supposes a knowledge of the principle, generally taught by physical
philosophers, that only matter is the vehicle of energy; and that
_Either appearance is all, or there is substance beyond consciousness_,
is a doctrine which only metaphysical philosophers could be expected to
understand, and upon which they could not be expected to agree. However,
the chief danger is that a plausible disjunction may not be really such
as to exclude any middle ground: _Either the Tories or the Whigs win_,
is bad, if a tie be possible; though in the above argument this is
negligible, seeing that a tie cannot directly cause innovations. _Either
Æschines joined in the rejoicings, or he did not_, does not allow for a
decent conformity with the public movement where resistance would be
vain; yet such conformity as need not be inconsistent with subsequent
condemnation of the proceedings, nor incompatible with patriotic reserve
founded on a belief that the rejoicings are premature and ominous.

Another rhetorical consideration is, that the alternatives of the
disjunctive conclusion of a Complex Dilemma should both point the same
way, should be equally distasteful or paradoxical. 'Either inconsistent
or unpatriotic': horrid words to a politician! 'Either no reality or no
possible knowledge of it': very disappointing to an anxious inquirer!
Thus the disjunctive conclusion is as bad for an opponent as the
categorical one in a Simple Dilemma.

Logicians further speak of the Trilemma, with three Hypotheticals and a
corresponding triple Disjunction; and of a Polylemma, with any further
number of perplexities. But anyone who has a taste for logical forms may
have it amply gratified in numerous text-books.



§ 1. Having now discussed Terms, Propositions, Immediate and Mediate
Inferences, and investigated the conditions of formal truth or
consistency, we have next to consider the conditions of material truth:
whether (or how far) it is possible to arrive at propositions that
accurately represent the course of nature or of human life. Hitherto we
have dealt with no sort of proof that gives any such assurance. A valid
syllogism guarantees the truth of its conclusion, provided the premises
be true: but what of the premises? The relation between the premises of
a valid syllogism and its conclusion is the same as the relation between
the antecedent and consequent of a hypothetical proposition. If A is B,
C is D: grant that A is B, and it follows that C is D; and, similarly,
grant the premises of a syllogism, and the conclusion follows. Again,
grant that C is not D, and it follows that A is not B; and, similarly,
if the conclusion of a valid syllogism be false, it follows that one, or
other, or both of the premises must be false. But, once more, grant that
C is D, and it does not follow that A is B; so neither, if the
conclusion of a syllogism be true, does it follow that the premises are.
For example:--

      Sociology is an exact science;
      Mathematics is a branch of Sociology:
    ∴ Mathematics is an exact science.

Here the conclusion is true although the premises are absurd. Or

      Mathematics is an exact science;
      Sociology is a branch of Mathematics:
    ∴ Sociology is an exact science.

Here the major premise is true, but the minor is false, and the
conclusion is false. In both cases, however, whether the conclusion be
true or false, it equally follows from the premises, if there is any
cogency in Barbara. The explanation of this is, that Barbara has only
formal cogency; and that whether the conclusion of that, or any other
valid mood, shall be true according to fact and experience, depends upon
how the form is filled up. How to establish the premises, then, is a
most important problem; and it still remains to be solved.

§ 2. We may begin by recalling the distinction between the denotation
and connotation of a general term: the denotation comprising the things
or events which the term is a name for; the connotation comprising the
common qualities on account of which these things are called by the same
name. Obviously, there are very few general terms whose denotation is
exhaustively known; since the denotation of a general term comprises all
the things that have its connotation, or that ever have had, or that
ever will have it, whether they exist here, or in Australia, or in the
Moon, or in the utmost stars. No one has examined all men, all mammoths,
all crystals, all falling bodies, all cases of fever, all revolutions,
all stars--nor even all planets, since from time to time new ones are
discerned. We have names for animals that existed long before there were
men to observe them, and of which we know only a few bones, the remains
of multitudinous species; and for others that may continue to exist when
men have disappeared from the earth.

If, indeed, we definitely limit the time, or place, or quantity of
matter to be explored, we may sometimes learn, within the given limits,
all that there is to know: as all the bones of a particular animal, or
the list of English monarchs hitherto, or the names of all the members
of the House of Commons at the present time. Such cases, however, do not
invalidate the above logical truth that few general terms are
exhaustively known in their denotation; for the very fact of assigning
limits of time and place impairs the generality of a term. The bones of
a certain animal may be all examined, but not the bones of all animals,
nor even of one species. The English monarchs that have reigned hitherto
may be known, but there may be many still to reign.

The general terms, then, with which Logic is chiefly concerned, the
names of Causes and Kinds, such as gravitation, diseases, social events,
minerals, plants and animals, stand for some facts that are, or have
been, known, and for a great many other similar ones that have not been,
and never will be, known. The use of a general term depends not upon our
direct knowledge of everything comprised in its denotation, but upon our
readiness to apply it to anything that has its connotation, whether we
have seen the thing or not, and even though we never can perceive it; as
when a man talks freely of the ichthyosaurus, or of the central heat of
planets, or of atoms and ether.

Hence Universal Propositions, which consist of general terms, deceive
us, if we suppose that their predicates are directly known to be related
to all the facts denoted by their subjects. In exceptional cases, in
which the denotation of a subject is intentionally limited, such
exhaustive direct knowledge may be possible; as that "all the bones of a
certain animal consist of phosphate of lime," or that every member of
the present Parliament wears a silk hat. But what predication is
possible concerning the hats of all members of Parliament from the
beginning? Ordinarily, then, whilst the relation of predicate to subject
has been observed in some cases, in much the greater number of cases
our belief about it depends upon something besides observation, or may
be said (in a certain sense) to be taken on trust.

'All rabbits are herbivorous': why do we believe that? We may have seen
a few wild rabbits feeding: or have kept tame ones, and tried
experiments with their diet; or have read of their habits in a book of
Natural History; or have studied the anatomy and physiology of the
digestive system in many sorts of animals: but with whatever care we add
testimony and scientific method to our own observation, it still remains
true that the rabbits observed by ourselves and others are few in
comparison with those that live, have lived and will live. Similarly of
any other universal proposition; that it 'goes beyond the evidence' of
direct observation plainly follows from the fact that the general terms,
of which such propositions consist, are never exhaustively known in
their denotation. What right have we then to state Universal
Propositions? That is the problem of Inductive Logic.

§ 3. Universal Propositions, of course, cannot always be proved by
syllogisms; because to prove a universal proposition by a syllogism, its
premises must be universal propositions; and, then, these must be proved
by others. This process may sometimes go a little way, thus: _All men
are mortal_, because _All animals are_; and _All animals are mortal_,
because _All composite bodies are subject to dissolution._ Were there no
limit to such sorites, proof would always involve a _regressus ad
infinitum_, for which life is too short; but, in fact, prosyllogisms
soon fail us.

Clearly, the form of the Syllogism must itself be misleading if the
universal proposition is so: if we think that premises prove the
conclusion because they themselves have been established by detailed
observation, we are mistaken. The consideration of any example will show
this. Suppose any one to argue:

      All ruminants are herbivorous;
      Camels are ruminants:
    ∴ Camels are herbivorous.

Have we, then, examined all ruminants? If so, we must have examined all
camels, and cannot need a syllogism to prove their herbivorous nature:
instead of the major premise proving the conclusion, the proof of the
conclusion must then be part of the proof of the major premise. But if
we have not examined all ruminants, having omitted most giraffes, most
deer, most oxen, etc., how do we know that the unexamined (say, some
camels) are not exceptional? Camels are vicious enough to be
carnivorous; and indeed it is said that Bactrian camels will eat flesh
rather than starve, though of course their habit is herbivorous.

Or, again, it is sometimes urged that--

      All empires decay:
    ∴ Britain will decay.

This is manifestly a prediction: at present Britain flourishes, and
shows no signs of decay. Yet a knowledge of its decay seems necessary,
to justify any one in asserting the given premise. If it is a question
whether Britain will decay, to attempt (while several empires still
flourish) to settle the matter by asserting that _all_ empires decay,
seems to be 'a begging of the question.' But although this latter case
is a manifest prediction, it does not really differ from the former one;
for the proof that camels are herbivorous has no limits in time. If
valid, it shows not only that they are, but also that they will be,

Hence, to resort to a dilemma, it may be urged: If _all_ the facts of
the major premise of any syllogism have been examined, the syllogism is
needless; and if _some_ of them have not been examined, it is a _petitio
principii_. But either all have been examined, or some have not.
Therefore; the syllogism is either useless or fallacious.

§ 4. A way of escape from this dilemma is provided by distinguishing
between the formal and material aspects of the syllogism considered as a
means of proof. It begs the question formally, but not materially; that
is to say, if it be a question whether camels are herbivorous, and to
decide it we are told that '_all_ ruminants are,' laying stress upon the
'all,' as if all had been examined, though in fact camels have not been,
then the question as to camels is begged. The form of a universal
proposition is then offered as evidence, when in fact the evidence has
not been universally ascertained. But if in urging that 'all ruminants
are herbivorous' no more is meant than that so many other ruminants of
different species are known to be herbivorous, and that the ruminant
stomach is so well adapted to a coarse vegetable diet, that the same
habit may be expected in other ruminants, such as camels, the argument
then rests upon material evidence without unfairly implying the case in
question. Now the nature of the material evidence is plainly this, that
the resemblance of camels to deer, oxen, etc., in chewing the cud,
justifies us in believing that they have a further resemblance in
feeding on herbs; in other words, we assume that _resemblance is a
ground of inference_.

Another way of putting this difficulty which we have just been
discussing, with regard to syllogistic evidence, is to urge that by the
Laws of Syllogism a conclusion must never go beyond the premises, and
that therefore no progress in knowledge can ever be established, except
by direct observation. Now, taking the syllogism formally, this is true:
if the conclusion go beyond the premises, there must be either four
terms, or illicit process of the major or minor term. But, taking it
materially, the conclusion may cover facts which were not in view when
the major premise was laid down; facts of which we predicate something
not as the result of direct observation, but because they resemble in a
certain way those facts which had been shown to carry the predicate when
the major premise was formed.

'What sort of resemblance is a sufficient ground of inference?' is,
therefore, the important question alike in material Deduction and in
Induction; and in endeavouring to answer it we shall find that the
surest ground of inference is resemblance of causation. For example, it
is due to causation that ruminants are herbivorous. Their instincts make
them crop the herb, and their stomachs enable them easily to digest it;
and in these characters camels are like the other ruminants.

§ 5. In ch. ix, § 3, the _Dictum de omni et nullo_ was stated: 'Whatever
may be predicated of a term distributed may be predicated of anything
that can be identified with that term.' Nothing was there said (as
nothing was needed) of the relations that might be implied in the
predication. But now that it comes to the ultimate validity of
predication, we must be clear as to what these relations are; and it
will also be convenient to speak no longer of terms, as in Formal Logic,
but of the things denoted. What relations, then, can be determined
between concrete facts or phenomena (physical or mental) with the
greatest certainty of general truth; and what axioms are there that
sanction mediate inferences concerning those relations?

In his _Logic_ (B. II. c. 2, § 3) Mill gives as the axiom of syllogistic
reasoning, instead of the _Dictum_: "A thing which co-exists with
another thing, which other co-exists with a third thing, also co-exists
with that third thing." Thus the peculiar properties of Socrates
co-exist with the attributes of man, which co-exist with mortality:
therefore, Socrates is mortal. But, again, he says that the ground of
the syllogism is Induction; that man is mortal is an induction. And,
further, the ground of Induction is causation; the law of causation is
the ultimate major premise of every sound induction. Now causation is
the principle of the succession of phenomena: how, then, can the
syllogism rest on an axiom concerning co-existence? On reflection, too,
it must appear that 'Man is mortal' predicates causation: the human
constitution issues in death.

The explanation of this inconsistency may perhaps be found in the
history of Mill's work. Books I. and II. were written in 1831; but being
unable at that time to explain Induction, he did not write Book III.
until 1837-8. Then, no doubt, he revised the earlier Books, but not
enough to bring his theory of the syllogism into complete agreement with
the theory of Induction; so that the axiom of co-existence was allowed
to stand.

Mill also introduced the doctrine of Natural Kinds as a ground of
Induction supplementary, at least provisionally, to causation; and to
reasoning about Kinds, or Substance and Attribute, his axiom of
co-existence is really adapted. Kinds are groups of things that agree
amongst themselves and differ from all others in a multitude of
qualities: these qualities co-exist, or co-inhere, with a high degree of
constancy; so that where some are found others may be inferred. Their
co-inherence is not to be considered an ultimate fact; for, "since
everything which occurs is determined by laws of causation and
collocations of the original causes, it follows that the co-existences
observable amongst effects cannot themselves be the subject of any
similar set of laws distinct from laws of causation" (B. III. c. 5, §
9). According to the theory of evolution (worked out since Mill wrote),
Kinds--that is, species of plants, animals and minerals--with their
qualities are all due to causation. Still, as we can rarely, or never,
trace the causes with any fullness or precision, a great deal of our
reasoning, as, e.g., about men and camels, does in fact trust to the
relative permanence of natural Kinds as defined by co-inhering

To see this more clearly, we should consider that causation and natural
Kinds are not at present separable; propositions about causation in
concrete phenomena (as distinct from abstract 'forces') always involve
the assumption of Kinds. For example--'Water rusts iron,' or the oxygen
of water combines with iron immersed in it to form rust: this statement
of causation assumes that water, oxygen, iron, and iron-rust are known
Kinds. On the other hand, the constitution of every concrete thing, and
manifestly of every organised body, is always undergoing change, that
is, causation, upon which fact its properties depend.

How, then, can we frame principles of mediate reasoning, about such
things? So far as we consider them as Kinds, it is enough to say:
_Whatever can be identified as a specimen of a known substance or Kind
has the properties of that Kind_. So far as we consider them as in the
relation of causation, we may say: _Whatever relation of events can be
identified with the relation of cause and effect is constant_. And these
principles may be generalised thus: _Whatever is constantly related to a
phenomenon (cause or Kind), determined by certain characters, is related
in the same way to any phenomenon, that has the same characters_. Taking
this as axiom of the syllogism materially treated, we see that
herbivorousness, being constantly related to ruminants, is constantly
related to camels; mortality to man and, therefore, to Socrates; rusting
to the immersion of iron in water generally and, therefore, to this
piece of iron. _Nota notæ, nota rei ipsius_ is another statement of the
same principle; still another is Mill's axiom, "Whatever has a mark has
what it is a mark of." A mark is anything (A) that is never found
without something else (B)--a phenomenon constantly related to another
phenomenon--so that wherever A is found, B may be expected: human nature
is a mark of mortality.

§ 6. The Syllogism has sometimes been discarded by those who have only
seen that, as formally stated, it is either useless or fallacious: but
those who also perceive its material grounds retain and defend it. In
fact, great advantages are gained by stating an argument as a formal
syllogism. For, in the first place, we can then examine separately the
three conditions on which the validity of the argument depends:

(1) Are the Premises so connected that, _if they are true_, the
Conclusion follows? This depends upon the formal principles of chap. x.

(2) Is the Minor Premise true? This question can only arise when the
minor premise is a real proposition; and then it may be very difficult
to answer. Water rusts iron; but is the metal we are now dealing with a
fair specimen of iron? Few people, comparatively, know how to determine
whether diamonds, or even gold or silver coins, are genuine. That
_Camels are ruminants_ is now a verbal proposition to a Zoologist, but
not to the rest of us; and to the Zoologist the ascertaining of the
relation in which camels stand to such ruminants as oxen and deer, was
not a matter of analysing words but of dissecting specimens. What a long
controversy as to whether the human race constitutes a Family of the
Primates! That 'the British Empire is an empire' affords no matter for
doubt or inquiry; but how difficult to judge whether the British Empire
resembles Assyria, Egypt, Rome, Spain in those characters and
circumstances that caused their downfall!

(3) Is the Major Premise true? Are all ruminants herbivorous? If there
be any exceptions to the rule, camels are likely enough to be among the
exceptions. And here the need of Inductive Logic is most conspicuous:
how can we prove our premises when they are universal propositions?
Universal propositions, however, are also involved in proving the minor
premise: to prove a thing to be iron, we must know the constant
reactions of iron.

A second advantage of the syllogism is, that it makes us fully aware of
what an inference implies. An inference must have some grounds, or else
it is a mere prejudice; but whatever the grounds, if sufficient in a
particular case, they must be sufficient for all similar cases, they
must admit of being generalised; and to generalise the grounds of the
inference, is nothing else than to state the major premise. If the
evidence is sufficient to justify the argument that camels are
herbivorous _because_ they are ruminants, it must also justify the major
premise, _All ruminants are herbivorous_; for else the inference cannot
really depend merely upon the fact of ruminating. To state our evidence
syllogistically, then, must be possible, if the evidence is mediate and
of a logical kind; and to state it in this formal way, as depending on
the truth of a general principle (the major premise), increases our
sense of responsibility for the inference that is thus seen to imply so
much; and if any negative instances lie within our knowledge, we are the
more likely to remember them. The use of syllogisms therefore tends to
strengthen our reasonings.

A third advantage is, that to formulate an accurate generalisation may
be useful to others: it is indeed part of the systematic procedure of
science. The memoranda of our major premises, or reasons for believing
anything, may be referred to by others, and either confirmed or refuted.
When such a memorandum is used for further inferences, these inferences
are said, in the language of Formal Logic, to be drawn _from_ it, as if
the conclusion were contained in our knowledge of the major premise;
but, considering the limited extent of the material evidence, it is
better to say that the inference is drawn _according to_ the memorandum
or major premise, since the grounds of the major premise and of the
conclusion are in fact the same (Mill: _Logic_, B. II. c. 3). Inductive
proofs may be stated in Syllogisms, and inductive inferences are drawn
_according to_ the Law of Causation.

§ 7. To assume that resemblance is a ground of inference, and that
substance and attribute, or cause and effect, are phenomena constantly
related, implies belief in the Uniformity of Nature. The Uniformity of
Nature cannot be defined, and is therefore liable to be misunderstood.
In many ways Nature seems not to be uniform: there is great variety in
the sizes, shapes, colours and all other properties of things: bodies
falling in the open air--pebbles, slates, feathers--descend in different
lines and at different rates; the wind and weather are proverbially
uncertain; the course of trade or of politics, is full of surprises. Yet
common maxims, even when absurd, testify to a popular belief that the
relations of things are constant: the doctrine of St. Swithin and the
rhyme beginning 'Evening red and morning grey,' show that the weather is
held to be not wholly unpredictable; as to human affairs, it is
said that 'a green Yule makes a fat churchyard,' that 'trade follows the
flag,' and that 'history repeats itself'; and Superstition knows that
witches cannot enter a stable-door if a horse-shoe is nailed over it,
and that the devil cannot cross a threshold inscribed with a perfect
pentagram. But the surest proof of a belief in the uniformity of nature
is given by the conduct of men and animals; by that adherence to habit,
custom and tradition, to which in quiet times they chiefly owe their
safety, but which would daily disappoint and destroy them, if it were
not generally true that things may be found where they have been left
and that in similar circumstances there are similar events.

Now this general belief, seldom distinctly conceived, for the most part
quite unconscious (as a principle), merely implied in what men do, is
also the foundation of all the Sciences; which are entirely occupied in
seeking the Laws (that is, the Uniformities) of Nature. As the
uniformity of nature cannot be defined, it cannot be proved; the most
convincing evidence in its favour is the steady progress made by Science
whilst trusting in it. Nevertheless, what is important is not the
comprehensive but indeterminate notion of Uniformity so much as a number
of First Principles, which may be distinguished in it as follows:

(1) The Principles of Contradiction and Excluded Middle (ch. vi. § 3)
declare that in a given relation to a given phenomenon any two or more
other phenomena are incompatible (_B is not A and a_); whilst the given
phenomenon either stands related to another phenomenon or not (_B is
either A or a_). It is not only a matter of Logic but of fact that, if a
leaf is green, it is not under the same conditions red or blue, and that
if it is not green it is some other colour.

(2) Certain Axioms of Mediate Evidence: as, in Mathematics, 'that
magnitudes equal to the same magnitude are equal to one another'; and,
in Logic, the _Dictum_ or its material equivalent.

(3) That all Times and all Spaces are commensurable; although in certain
relations of space (as π) the unit of measurement must be infinitely
small.--If Time really trotted with one man and galloped with another,
as it seems to; if space really swelled in places, as De Quincey dreamed
that it did; life could not be regulated, experience could not be
compared and science would be impossible. The Mathematical Axioms would
then never be applicable to space or time, or to the objects or
processes that fill them.

(4) The Persistence of Matter and Energy: the physical principle that,
in all changes of the universe, the quantities of Matter and Energy
(actual and potential, so-called) remain the same.--For example, as to
matter, although dew is found on the grass at morning without any
apparent cause, and although a candle seems to burn away to a scrap of
blackened wick, yet every one knows that the dew has been condensed from
vapour in the air, and that the candle has only turned into gas and
smoke. As to energy, although a stone thrown up to the housetop and
resting there has lost actual energy, it has gained such a position that
the slightest touch may bring it to the earth again in the same time as
it took to travel upwards; so on the house-top it is said to have
potential energy. When a boiler works an engine, every time the piston
is thrust forward (mechanical energy), an equivalent in heat (molecular
energy) is lost. But for the elucidation of these principles, readers
must refer to treatises of Chemistry and Physics.

(5) Causation, a special form of the foregoing principles of the
persistence of matter and energy, we shall discuss in the next chapter.
It is not to be conceived of as anything occult or noumenal, but merely
as a special mode of the uniformity of Nature or experience.

(6) Certain Uniformities of Co-existence; but for want of a general
principle of Co-existence, corresponding to Causation (the principle of
Succession), we can only classify these uniformities as follows:

(a) The Geometrical; as that, in a four-sided figure, if the opposite
angles are equal, the opposite sides are equal and parallel.--Countless
similar uniformities of co-existence are disclosed by Geometry. The
co-existent facts do not cause one another, nor are they jointly caused
by something else; they are mutually involved: such is the nature of

(b) Universal co-inherences among the properties of concrete
things.--The chief example is the co-inherence of gravity with inertia
in all material bodies. There is, I believe, no other entirely
satisfactory case; but some good approximations to such uniformity are
known to physical science.

(c) Co-existence due to Causation; such as the positions of objects in
space at any time.--The houses of a town are where they are, because
they were put there; and they remain in their place as long as no other
causes arise strong enough to remove or destroy them. Similarly, the
relative positions of rocks in geological strata, and of trees in a
forest, are due to causes.

(d) The co-inherence of properties in Natural Kinds; which we call the
constitution, defining characters, or specific nature of such
things.--Oxygen, platinum, sulphur and the other elements; water, common
salt, alcohol and other compounds; the various species of plants and
animals: all these are known to us as different groups of co-inherent
properties. It may be conjectured that these groupings of properties are
also due to causation, and sometimes the causes can be traced: but very
often the causes are still unknown; and, until resolved into their
causes, they must be taken as necessary data in the investigation of
nature. Laws of the co-inherence of the properties of Kinds do not, like
laws of causation, admit of methodical proof upon their own principles,
but only by constancy in experience and statistical probability (c. xix,
§ 4).

(e) There are also a few cases in which properties co-exist in an
unaccountable way, without being co-extensive with any one species,
genus, or order: as most metals are whitish, and scarlet flowers are
wanting in fragrance. (On this § 7, see Venn's _Empirical Logic_, c. 4.)

§ 8. Inasmuch as Axioms of Uniformity are ultimate truths, they cannot
be deduced; and inasmuch as they are universal, no proof by experience
can ever be adequate. The grounds of our belief in them seem to be

(1) Every inference takes for granted an order of Nature corresponding
with it; and every attempt to explain the origin of anything assumes
that it is the transformation of something else: so that uniformity of
order and conservation of matter and energy are necessary
presuppositions of reasoning.

(2) On the rise of philosophic reflection, these tacit presuppositions
are first taken as dogmas, and later as postulates of scientific
generalisation, and of the architectonic unification of science. Here
they are indispensable.

(3) The presuppositions or postulates are, in some measure, verifiable
in practical life and in scientific demonstration, and the better
verifiable as our methods become more exact.

(4) There is a cause of this belief that cannot be said to contain any
evidence for it, namely, the desire to find in Nature a foundation for
confidence in our own power to foresee and to control events.



§ 1. For the theory of Induction, the specially important aspect of the
Uniformity of Nature is Causation.

For (1) the Principles of Contradiction and Excluded Middle are implied
in all logical operations, and need no further explication.

(2) That one thing is a mark of another or constantly related to it,
must be established by Induction; and the surest of all marks is a
Cause. So that the application of the axiom of the Syllogism in
particular cases requires, when most valid, a previous appeal to

(3) The uniformity of Space and of Time is involved in Causation, so far
as we conceive Causation as essentially matter in motion--for motion is
only known as a traversing of space in time; and so far as forces vary
in any way according to the distance between bodies; so that if space
and time were not uniform, causation would be irregular. Not that time
and space are agents, but they are conditions of every agent's

(4) The persistence of Matter and Energy, being nothing else than
Causation in the general movement of the world, is applied under the
name of that principle in explaining any particular limited phenomenon,
such as a soap-bubble, or a thunderstorm, or the tide.

(5) As to co-existences, the Geometrical do not belong to Logic: those
involved in the existence of plants, animals, and inorganic bodies,
must, as far as possible, be traced to causes; and so, of course, must
the relative positions of objects in space at any time: and what
Co-existences remain do not admit of methodical inductive treatment;
they will be briefly discussed in chap. xix.

Causation, then, is that mode or aspect of the Uniformity of Nature
which especially concerns us in Induction; and we must make it as
definite as possible. It is nothing occult, but merely a convenient name
for phenomena in a particular relation to other phenomena, called their
effect. Similarly, if the word 'force' is sometimes used for convenience
in analysing causation, it means nothing more than something in time and
space, itself moving, or tending to move, or hindering or accelerating
other things. If any one does not find these words convenient for the
purpose, he can use others.

§ 2. A Cause, according to Mill, is "the invariable unconditional
antecedent" of a given phenomenon. To enlarge upon this:

(1) A Cause is _relative to a given phenomenon_, called the Effect.
Logic has no method for investigating the cause of the universe as a
whole, but only of a part or epoch of it: we select from the infinite
continuum of Nature any portion that is neither too large nor too small
for a trained mind to comprehend. The magnitude of the phenomenon may be
a matter of convenience. If the cause of disease in general be too wide
a problem, can fevers be dealt with; or, if that be too much, is typhus
within the reach of inquiry? In short, how much can we deal with

(2) The given phenomenon is always _an event_; that is to say, not a new
thing (nothing is wholly new), but a change in something, or in the
relative position of things. We may ask the cause of the phases of the
moon, of the freezing of water, of the kindling of a match, of a deposit
of chalk, of the differentiation of species. To inquire the cause of
France being a republic, or Russia an autocracy, implies that these
countries were once otherwise governed, or had no government: to inquire
the cause of the earth being shaped like an orange, implies that the
matter of the earth had once another shape.

(3) The Cause is _antecedent_ to the Effect, which accordingly is often
called its _consequent_. This is often misunderstood and sometimes
disputed. It has been said that the meaning of 'cause' implies an
'effect,' so that until an effect occurs there can be no cause. But this
is a blunder; for whilst the word 'cause' implies 'effect,' it also
implies the relative futurity of the effect; and effect implies the
relative priority of the cause. The connotation of the words, therefore,
agrees well enough with Mill's doctrine. In fact, the danger is that any
pair of contrasted words may suggest too strongly that the phenomena
denoted are separate in Nature; whereas every natural process is
continuous. If water, dripping from the roof wears away a stone, it fell
on the roof as rain; the rain came from a condensing cloud; the cloud
was driven by the wind from the sea, whence it exhaled; and so on. There
is no known beginning to this, and no break in it. We may take any one
of these changes, call it an effect, and ask for its cause; or call it a
cause, and ask for its effect. There is not in Nature one set of things
called causes and another called effects; but every change is both cause
(or a condition) of the future and effect of the past; and whether we
consider an event as the one or the other, depends upon the direction of
our curiosity or interest.

Still, taking the event as effect, its cause is the antecedent process;
or, taking it as a cause, its effect is the consequent process. This
follows from the conception of causation as essentially motion; for that
_motion takes time_ is (from the way our perceptive powers grow) an
ultimate intuition. But, for the same reason, there is no interval of
time between cause and effect; since all the time is filled up with

Nor must it be supposed that the whole cause is antecedent to the
effect as a whole: for we often take the phenomenon on such a scale that
minutes, days, years, ages, may elapse before we consider the cause as
exhausted (e.g., an earthquake, a battle, an expansion of credit,
natural selection operating on a given variety); and all that time the
effect has been accumulating. But we may further consider such a cause
as made up of moments or minute factors, and the effect as made up of
corresponding moments; and then the cause, taken in its moments, is
antecedent throughout to the effect, taken in its corresponding moments.

(4) The Cause is the _invariable_ antecedent of the effect; that is to
say, whenever a given cause occurs it always has the same effect: in
this, in fact, consists the Uniformity of Causation. Accordingly, not
every antecedent of an event is its Cause: to assume that it is so, is
the familiar fallacy of arguing '_post hoc ergo propter hoc_.' Every
event has an infinite number of antecedents that have no ascertainable
connection with it: if a picture falls from the wall in this room, there
may have occurred, just previously, an earthquake in New Zealand, an
explosion in a Japanese arsenal, a religious riot in India, a political
assassination in Russia and a vote of censure in the House of Commons,
besides millions of other less noticeable events, between none of which
and the falling of the picture can any direct causation be detected;
though, no doubt, they are all necessary occurrences in the general
world-process, and remotely connected. The cause, however, was that a
door slammed violently in the room above and shook the wall, and that
the picture was heavy and the cord old and rotten. Even if two events
invariably occur one after the other, as day follows night, or as the
report follows the flash of a gun, they may not be cause and effect,
though it is highly probable that they are closely connected by
causation; and in each of these two examples the events are co-effects
of a common cause, and may be regarded as elements of its total effect.
Still, whilst it is not true that every antecedent, or that every
invariable antecedent, of an event is its cause, the cause is conceived
of as some change in certain conditions, or some state and process of
things, such that should it exactly recur the same event would
invariably follow. If we consider the antecedent state and process of
things very widely or very minutely, it never does exactly recur; nor
does the consequent. But the purpose of induction is to get as near the
truth as possible within the limits set by our faculties of observation
and calculation. Complex causal instances that are most unlikely to
recur as a whole, may be analysed into the laws of their constituent

(5) The Cause is the Unconditional Antecedent. A cause is never simple,
but may be analysed into several conditions; and 'Condition' means any
necessary factor of a Cause: any thing or agent that exerts, absorbs,
transforms, or deflects energy; or any relation of time or space in
which agents stand to one another. A positive condition is one that
cannot be omitted without frustrating the effect; a negative condition
is one that cannot be introduced without frustrating the effect. In the
falling of the picture, e.g., the positive conditions were the picture
(as being heavy), the slamming of the door, and the weakness of the
cord: a negative condition was that the picture should have no support
but the cord. When Mill, then, defines the Cause of any event as its
"unconditional" antecedent, he means that it is that group of conditions
(state and process of things) which, without any further condition, is
followed by the event in question: it is the least antecedent that
suffices, positive conditions being present and negative absent.

Whatever item of the antecedent can be left out, then, without affecting
the event, is no part of the cause. Earthquakes have happened in New
Zealand and votes of censure in the House of Commons without a picture's
falling in this room: they were not unconditional antecedents;
something else was needed to bring down a picture. Unconditionality also
distinguishes a true cause from an invariable antecedent that is only a
co-effect: for when day follows night something else happens; the Earth
rotates upon her axis: a flash of gunpowder is not an unconditional
antecedent of a report; the powder must be ignited in a closed chamber.

By common experience, and more precisely by experiment, it is found
possible to select from among the antecedents of an event a certain
number upon which, so far as can be perceived, it is dependent, and to
neglect the rest: to purge the cause of all irrelevant antecedents is
the great art of inductive method. Remote or minute conditions may
indeed modify the event in ways so refined as to escape our notice.
Subject to the limitations of our human faculties, however, we are able
in many cases to secure an unconditional antecedent upon which a certain
event invariably follows. Everybody takes this for granted: if the gas
will not burn, or a gun will not go off, we wonder 'what can be wrong
with it,' that is, what positive condition is wanting, or what negative
one is present. No one now supposes that gunnery depends upon those
"remotest of all causes," the stars, or upon the sun being in
Sagittarius rather than in Aquarius, or that one shoots straightest with
a silver bullet, or after saying the alphabet backwards.

(6) That the Cause of any event is an Immediate Antecedent follows from
its being an unconditional one. For if there are three events, A B C,
causally connected, it is plain that A is not the unconditional
antecedent of C, but requires the further condition of first giving rise
to B. But that is not all; for the B that gives rise to C is never
merely the effect of A; it involves something further. Take such a
simple case as the motion of the earth round the sun (neglecting all
other conditions, the other planets, etc.); and let the earth's
motion at three successive moments be A B C: A is not the whole cause of
B in velocity and direction; we must add relation to the sun, say x. But
then, again, the cause of C will not be merely Bx, for the relation to
the sun will have altered; so that we must represent it as Bx'. The
series, therefore, is Ax Bx' C. What is called a "remote cause" is,
therefore, doubly conditional; first, because it supposes an intervening
cause; and secondly, because it only in part determines the conditions
that constitute this intervening cause.

The immediacy of a cause being implied in its unconditionalness, is an
important clue to it; but as far as the detection of causes depends upon
sense-perception, our powers (however aided by instruments) are unequal
to the subtlety of Nature. Between the event and what seems to us the
immediate antecedent many things (molecular or etherial changes) may
happen in Chemistry or Physics. The progress of science would be
impossible were not observation supplemented by hypothesis and
calculation. And where phenomena are treated upon a large scale, as in
the biological and social sciences, immediacy, as a mark of causation,
must be liberally interpreted. So far, then, as to the qualitative
character of Causation.

(7) But to complete our account of it, we must briefly consider its
quantitative character. As to the Matter contained, and as to the Energy
embodied, Cause and Effect are conceived to be _equal_. As to matter,
indeed, they may be more properly called identical; since the effect is
nothing but the cause redistributed. When oxygen combines with hydrogen
to form water, or with mercury to form red precipitate, the weight of
the compound is exactly equal to the weight of the elements combined in
it; when a shell explodes and knocks down a wall, the materials of the
shell and wall are scattered about. As to energy, we see that in the
heavenly bodies, which meet with no sensible impediment, it remains the
same from age to age: with things 'below the moon' we have to allow for
the more or less rapid conversion of the visible motion of a mass into
other forms of energy, such as sound and heat. But the right
understanding of this point involves physical considerations of some
difficulty, as to which the reader must refer to appropriate books, such
as Balfour Stewart's on _The Conservation of Energy_.

The comprehension of the quantitative aspect of causation is greatly
aided by Bain's analysis of any cause into a 'Moving or an Inciting
Power' and a 'Collocation' of circumstances. When a demagogue by making
a speech stirs up a mob to a riot, the speech is the moving or inciting
power; the mob already in a state of smouldering passion, and a street
convenient to be wrecked, are the collocation. When a small quantity of
strychnine kills a man, the strychnine is the inciting power; the nature
of his nervo-muscular system, apt to be thrown into spasms by that drug,
and all the organs of his body dependent on that system, are the
collocation. Now any one who thinks only of the speech, or the drug, in
these cases, may express astonishment at the disproportion of cause and

    "What great events from trivial causes spring!"

But, remembering that the whole cause of the riot included the excited
mob, every one sees that its muscular power is enough to wreck a street;
and remembering that breathing depends upon the normal action of the
intercostal muscles, it is plain that if this action is stopped by
strychnine, a man must die. Again, a slight rise of temperature may be a
sufficient inciting power to occasion extensive chemical changes in a
collocation of elements otherwise stable; a spark is enough to explode a
powder magazine. Hence, when sufficient energy to account for any effect
cannot be found in the inciting power, or manifestly active condition,
we must look for it in the collocation which is often supposed to be

And that reminds us of another common misapprehension, namely, that in
Nature some things are passive and others active: the distinction
between 'agent' and 'patient.' This is a merely relative distinction: in
Nature all things are active. To the eye some things seem at rest and
others in motion; but we know that nothing is really at rest, that
everything palpitates with molecular change, and whirls with the planet
through space. Everything that is acted upon reacts according to its own
nature: the quietest-looking object (say, a moss-covered stone), if we
try to push or lift it, pushes or pulls us back, assuring us that
'action and reaction are equal and opposite.' 'Inertia' does not mean
want of vigour, but may be metaphorically described as the inexpugnable
resolve of everything to have its own way.

The equality of cause and effect defines and interprets the
unconditionality of causation. The cause, we have seen, is that group of
conditions which, without any further condition, is followed by a given
event. But how is such a group to be conceived? Unquantified, it admits
only of a general description: quantified, it must mean a group of
conditions equal to the effect in mass and energy, the essence of the
physical world. Apparently, a necessary conception of the human mind:
for if a cause seem greater than its effect, we ask what has become of
the surplus matter and energy; or if an effect seem greater than its
cause, we ask whence the surplus matter and energy has arisen. So
convinced of this truth is every experimenter, that if his results
present any deviation from it, he always assumes that it is he who has
made some mistake or oversight, never that there is indeterminism or
discontinuity in Nature.

The transformation of matter and energy, then, is the essence of
causation: because it is continuous, causation is immediate; and because
in the same circumstances the transformation always follows the same
course, a cause has invariably the same effect. If a fire be lit
morning after morning in the same grate, with coal, wood, and paper of
the same quality and similarly arranged, there will be each day the same
flaming of paper, crackling of wood and glowing of coal, followed in
about the same time by the same reduction of the whole mass partly to
ashes and partly to gases and smoke that have gone up the chimney. The
flaming, crackling and glowing are, physically, modes of energy; and the
change of materials into gas and ashes is a chemical and physical
redistribution: and, if some one be present, he will be aware of all
this; and then, besides the physical changes, there will be sensations
of light, sound and heat; and these again will be always the same in the
same circumstances.

The Cause of any event, then, when exactly ascertainable, has five
marks: it is (quantitatively) _equal_ to the effect, and (qualitatively)
_the immediate, unconditional, invariable antecedent of the effect_.

§ 3. This scientific conception of causation has been developed and
rendered definite by the investigations of those physical sciences that
can avail themselves of exact experiments and mathematical calculation;
and it is there, in Chemistry and Physics, that it is most at home. The
conception can indeed be carried into the Biological and Social
Sciences, even in its quantitative form, by making the proper
allowances. For the limbs of animals are levers, and act upon mechanical
principles; and digestion and the aeration of the blood by breathing are
partly chemical processes. There is a quantitative relation between the
food a man eats and the amount of work he can do. The numbers of any
species of plant or animal depend upon the food supply. The value of a
country's imports is equal to the value of its exports and of the
services it renders to foreigners. But, generally, the less experiment
and exact calculation are practicable in any branch of inquiry, the less
rigorously can the conception of causation be applied there, the more
will its application depend upon the qualitative marks, and the more
need there will be to use it judiciously. In every inquiry the greatest
possible precision must be aimed at; but it is unreasonable to expect in
any case more precise proof than the subject admits of in the existing
state of culture.

Wherever mental action is involved, there is a special difficulty in
applying the physical notion of causation. For if a Cause be conceived
of as matter in motion, a thought, or feeling, or volition can be
neither cause nor effect. And since mental action is involved in all
social affairs, and in the life of all men and animals, it may seem
impossible to interpret social or vital changes according to laws of
causation. Still, animals and men are moving bodies; and it is
recognised that their thoughts and feelings are so connected with their
movements and with the movements of other things acting upon them, that
we can judge of one case by another; although the connection is by no
means well understood, and the best words (such as all can agree to use)
have not yet been found to express even what we know about it. Hence, a
regular connection being granted, I have not hesitated, to use
biological and social events and the laws of them, to illustrate
causation and induction; because, though less exact than chemical or
mechanical examples, they are to most people more familiar and

In practical affairs, it is felt that everything depends upon causation;
how to play the fiddle, or sail a yacht, or get one's living, or defeat
the enemy. The price of pig-iron six months hence, the prospects of the
harvest, the issue in a Coroner's Court, Home Rule and Socialism, are
all questions of causation. But, in such cases, the conception of a
cause is rarely applied in its full scientific acceptation, as the
unconditional antecedent, or 'all the conditions' (neither more nor
less) upon which the event depends. This is not because men of affairs
are bad logicians, or incapable of scientific comprehension; for very
often the reverse is conspicuously true; but because practical affairs
call for promptitude and a decisive seizing upon what is predominantly
important. How learn to play the fiddle? "Go to a good teacher." (Then,
beginning young enough, with natural aptitude and great diligence, all
may be well.) How defeat the enemy? "Be two to one at the critical
juncture." (Then, if the men are brave, disciplined, well armed and well
fed, there is a good chance of victory.) Will the price of iron improve?
"Yes: for the market is oversold": (that is, many have sold iron who
have none to deliver, and must at some time buy it back; and that will
put up the price--if the stock is not too great, if the demand does not
fall off, and if those who have bought what they cannot pay for are not
in the meanwhile obliged to sell.) These prompt and decisive judgments
(with the parenthetic considerations unexpressed) as to what is the
Cause, or predominantly important condition, of any event, are not as
good as a scientific estimate of all the conditions, when this can be
obtained; but, when time is short, the insight of trained sagacity may
be much better than an imperfect theoretical treatment of such problems.

§ 4. To regard the Effect of certain antecedents in a narrow selective
way, is another common mistake. In the full scientific conception of an
Effect it is the sum of the unconditional consequences of a given state
and process of things: the consequences immediately flowing from that
situation without further conditions. Always to take account of all the
consequences of any cause would no doubt be impracticable; still the
practical, as well as the scientific interest, often requires that we
should enlarge our views of them; and there is no commoner error in
private effort or in legislation than to aim at some obvious good,
whilst overlooking other consequences of our action, the evil of which
may far outweigh that good. An important consequence of eating is to
satisfy hunger, and this is the ordinary motive to eat; but it is a
poor account of the physiological consequences. An important consequence
of firing a gun is the propulsion of the bullet or shell; but there are
many other consequences in the whole effect, and one of them is the
heating of the barrel, which, accumulating with rapid firing, may at
last put the gun out of action. The tides have consequences to shipping
and in the wear and tear of the coast that draw every one's attention;
but we are told that they also retard the rotation of the earth, and at
last may cause it to present always the same face to the sun, and,
therefore, to be uninhabitable. Such concurrent consequences of any
cause may be called its Co-effects: the Effect being the sum of them.

The neglect to take account of the whole effect (that is, of all the
co-effects) in any case of causation is perhaps the reason why many
philosophers have maintained the doctrine of a "Plurality of Causes":
meaning not that more than one condition is operative in the antecedent
of every event (which is true), but that the same event may be due at
different times to different antecedents, that in fact there may be
_vicarious_ causes. If, however, we take any effect as a whole, this
does not seem to be true. A fire may certainly be lit in many ways: with
a match or a flint and steel, or by rubbing sticks together, or by a
flash of lightning: have we not here a plurality of causes? Not if we
take account of the whole effect; for then we shall find it modified in
each case according to the difference of the cause. In one case there
will be a burnt match, in another a warm flint, in the last a changed
state of electrical tension. And similar differences are found in cases
of death under different conditions, as stabbing, hanging, cholera; or
of shipwreck from explosion, scuttling, tempest. Hence a Coroner's Court
expects to find, by examining a corpse, the precise cause of death. In
short, if we knew the facts minutely enough, it would be found that
there is only one Cause (sum of conditions) for each Effect (sum of
co-effects), and that the order of events is as uniform backwards as

Still, as we are far from knowing events minutely, it is necessary in
practical affairs, and even in the more complex and unmanageable
scientific investigations, especially those that deal with human life,
to acknowledge a possible plurality of causes for any effect. Indeed,
forgetfulness of this leads to many rash generalisations; as that
'revolutions always begin in hunger'; or that 'myths are a disease of
language.' Then there is great waste of ingenuity in reconciling such
propositions with the recalcitrant facts. A scientific method recognises
that there may be other causes of effects thus vaguely conceived, and
then proceeds to distinguish in each class of effects the peculiarities
due to different causes.

§ 5. The understanding of the complex nature of Causes and Effects helps
us to overcome some other difficulties that perplex the use of these
words. We have seen that the true cause is an _immediate_ antecedent;
but if the cause is confounded with _one_ of its constituent conditions,
it may seem to have long preceded the event which is regarded as its
effect. Thus, if one man's death is ascribed to another's desire of
revenge, this desire may have been entertained for years before the
assassination occurred: similarly, if a shipwreck is ascribed to a
sunken reef, the rock was waiting for ages before the ship sailed that
way. But, of course, neither the desire of revenge nor the sunken rock
was 'the sum of the conditions' on which the one or the other event
depended: as soon as this is complete the effect appears.

We have also seen the true effect of any state and process of things is
the immediate consequence; but if the effect be confounded with _one_ of
its constituent factors, it may seem to long outlive the cessation of
the cause. Thus, in nearly every process of human industry and art, one
factor of the effect--a road, a house, a tool, a picture--may, and
generally does, remain long after the work has ceased: but such a
result is not the whole effect of the operations that produce it. The
other factors may be, and some always are, evanescent. In most of such
works some heat is produced by hammering or friction, and the labourers
are fatigued; but these consequences soon pass off. Hence the effect as
a whole only momentarily survives the cause. Consider a pendulum which,
having been once set agoing, swings to and fro in an arc, under the
joint control of the shaft, gravitation and its own inertia: at every
moment its speed and direction change; and each change may be considered
as an effect, of which the antecedent change was one condition. In such
a case as this, which, though a very simple, is a perfectly fair example
of all causation, the duration of either cause or effect is quite
insensible: so that, as Dr. Venn says, an Effect, rigorously conceived,
is only "the initial tendency" of its Cause.

§ 6. Mill contrasted two forms under which causation appears to us: that
is to say, the conditions constituting a cause may be modified, or
'intermixed' in the effect, in two ways, which are typified respectively
by Mechanical and Chemical action. In mechanical causation, which is
found in Astronomy and all branches of Physics, the effects are all
reducible to modes of energy, and are therefore commensurable with their
causes. They are either directly commensurable, as in the cases treated
of in the consideration of the mechanical powers; or, if different forms
of energy enter into cause and effect, such as mechanical energy,
electrical energy, heat, these different forms are severally reducible
to units, between which equivalents have been established. Hence Mill
calls this the "homogeneous intermixture of effects," because the
antecedents and consequents are fundamentally of the same kind.

In chemical causation, on the other hand, cause and effect (at least, as
they present themselves to us) differ in almost every way: in the act of
combination the properties of elements (except weight) disappear, and
are superseded by others in the compound. If, for example, mercury (a
heavy, silvery liquid) be heated in contact with oxygen (a colourless
gas), oxide of mercury is formed (red precipitate, which is a powder).
This compound presents very different phenomena from those of its
elements; and hence Mill called this class of cases "the heteropathic
intermixture of effects." Still, in chemical action, the effect is not
(in Nature) heterogeneous with the cause: for the weight of a compound
is equal to the sum of the weights of the elements that are merged in
it; and an equivalence has been ascertained between the energy of
chemical combination and the heat, light, etc., produced in the act of

The heteropathic intermixture of effects is also found in organic
processes (which, indeed, are partly chemical): as when a man eats bread
and milk, and by digestion and assimilation converts them into nerve,
muscle and bone. Such phenomena may make us wonder that people should
ever have believed that 'effects resemble their causes,' or that 'like
produces like.' A dim recognition of the equivalence of cause and effect
in respect of matter and motion may have aided the belief; and the
resemblance of offspring to parents may have helped: but it is probably
a residuum of magical rites; in which to whistle may be regarded as a
means of raising the wind, because the wind whistles; and rain-wizards
may make a victim shed tears that the clouds also may weep.

§ 7. Another consideration arises out of the complex character of causes
and effects. When a cause consists of two or more conditions or forces,
we may consider what effect any one of them would have if it operated
alone, that is to say, its _Tendency_. This is best illustrated by the
Parallelogram of Forces: if two forces acting upon a point, but not in
the same direction, be represented by straight lines drawn in the
direction of the forces, and in length proportional to their
magnitudes, these lines, meeting in an angle, represent severally the
tendencies of the forces; whilst if the parallelogram be completed on
these lines, the diagonal drawn from the point in which they meet
represents their _Resultant_ or effect.

Again, considering the tendency of any force if it operated alone, we
may say that, when combined with another force (not in the same
direction) in any resultant, its tendency is _counteracted_: either
partially, when the direction of the resultant is different; or wholly
when, the other force being equal and opposite, the resultant is
equilibrium. If the two forces be in the same direction, they are merely
added together. Counteraction is only one mode of combination; in no
case is any force destroyed.

Sometimes the separate tendencies of combined forces can only be
theoretically distinguished: as when the motion of a projectile is
analysed into a tendency to travel in the straight line of its
discharge, and a tendency to fall straight to the ground. But sometimes
a tendency can be isolated: as when,--after dropping a feather in some
place sheltered from the wind, and watching it drift to and fro, as the
air, offering unequal resistances to its uneven surface, counteracts its
weight with varying success, until it slowly settles upon the
ground,--we take it up and drop it again in a vacuum, when it falls like
lead. Here we have the tendency of a certain cause (namely, the relation
between the feather and the earth) free from counteraction: and this is
called the _Elimination_ of the counteracting circumstances. In this
case indeed there is physical elimination; whereas, in the case of a
projectile, when we say that its actual motion is resolvable (neglecting
the resistance of the air) into two tendencies, one in the line of
discharge, the other earthwards, there is only theoretical elimination
of either tendency, considered as counteracting the other; and this is
more specifically called the _Resolution_ or Analysis of the total
effect into its component conditions. Now, Elimination and Resolution
may be said to be the essential process of Induction in the widest sense
of the term, as including the combination of Induction with Deduction.

The several conditions constituting any cause, then, by aiding or
counteracting one another's tendencies, jointly determine the total
effect. Hence, viewed in relation one to another, they may be said to
stand in _Reciprocity_ or mutual influence. This relation at any moment
is itself one of co-existence, though it is conceived with reference to
a possible effect. As Kant says, all substances, as perceived in space
at the same time, are in reciprocal activity. And what is true of the
world of things at any moment (as connected, say, by gravity), is true
of any selected group of circumstances which we regard as the particular
cause of any event to come. The use of the concept of reciprocity, then,
lies in the analysis of a cause: we must not think of reciprocity as
obtaining in the succession of cause and effect, as if the effect could
turn back upon its cause; for as the effect arises its cause disappears,
and is irrecoverable by Nature or Magic. There are many cases of
rhythmic change and of moving equilibria, in which one movement or
process produces another, and this produces something closely resembling
the former, and so on in long series; as with the swing of a pendulum or
the orbit of a planet: but these are series of cause and effect, not of



§ 1. It is necessary to describe briefly the process of investigating
laws of causation, not with the notion of teaching any one the Art of
Discovery, which each man pursues for himself according to his natural
gifts and his experience in the methods of his own science, but merely
to cast some light upon the contents of the next few chapters. Logic is
here treated as a process of proof; proof supposes that some general
proposition or hypothesis has been suggested as requiring proof; and the
search for such propositions may spring from scientific curiosity or
from practical interests.

We may, as Bain observes (_Logic_: B. iii. ch. 5), desire to detect a
process of causation either (1) amidst circumstances that have no
influence upon the process but only obscure it; as when, being pleased
with a certain scent in a garden, we wish to know from what flower it
rises; or, being attracted by the sound of some instrument in an
orchestra, we desire to know which it is: or (2) amidst circumstances
that alter the effect from what it would have been by the sole operation
of some cause; as when the air deflects a falling feather; or in some
more complex case, such as a rise or fall of prices that may extend over
many years.

To begin with, we must form definite ideas as to what the phenomenon is
that we are about to investigate; and in a case of any complexity this
is best done by writing a detailed description of it: e.g., to
investigate the cause of a recent fall of prices, we must describe
exactly the course of the phenomenon, dating the period over which it
extends, recording the successive fluctuations of prices, with their
maxima and minima, and noting the classes of goods or securities that
were more or less affected, etc.

Then the first step of elimination (as Bain further observes) is "to
analyse the situation mentally," in the light of analogies suggested by
our experience or previous knowledge. Dew, for example, is moisture
formed upon the surface of bodies from no apparent source. But two
possible sources are easily suggested by common experience: is it
deposited from the air, like the moisture upon a mirror when we breathe
upon it; or does it exude from the bodies themselves, like gum or
turpentine? Or, again, as to a fall of prices, a little experience in
business, or knowledge of Economics, readily suggests two possible
explanations: either cheaper production in making goods or carrying
them; or a scarcity of that in which the purchasing power of the chief
commercial nations is directly expressed, namely, gold.

Having thus analysed the situation and considered the possibility of
one, two, three, or more possible causes, we fix upon one of them for
further investigation; that is to say, we frame an hypothesis that this
is the cause. When an effect is given to find its cause, an inquirer
nearly always begins his investigations by thus framing an hypothesis as
to the cause.

The next step is to try to _verify_ this Hypothesis. This we may
sometimes do by _varying the circumstances_ of the phenomenon, according
to the Canons of direct Inductive Proof to be discussed in the next
chapter; that is to say, by _observing_ or _experimenting_ in such a way
as to get rid of or eliminate the obscuring or disturbing conditions.
Thus, to find out which flower in a garden gives a certain scent, it is
usually enough to rely on observation, going up to the likely flowers
one after the other and smelling them: at close quarters, the greater
relative intensity of the scent is sufficiently decisive. Or we may
resort to a sort of experiment, plucking a likely flower, as to which we
frame the hypothesis (this is the cause), and carrying it to some place
where the air is free from conflicting odours. Should observation or
experiment disprove our first hypothesis we try a second; and so on
until we succeed, or exhaust the known possibilities.

But if the phenomenon is so complex and extensive as a continuous fall
of prices, direct observation or experiment is a useless or impossible
method; and we must then resort to Deduction; that is, to indirect
Induction. If, for example, we take the hypothesis that the fall is due
to a scarcity of gold, we must show that there is a scarcity; what
effect such a scarcity may be expected to have upon prices from the
acknowledged laws of prices, and from the analogy of other cases of an
expanded or restricted currency; that this expectation agrees with the
statistics of recent commerce: and finally, that the alternative
hypothesis that the fall is due to cheaper production is not true;
either because there has not been a sufficient cheapening of general
production; or because, if there has been, the results to be rationally
expected from it are not such as to agree with the statistics of recent
commerce. (Ch. xviii.)

But now suppose that, a phenomenon having been suggested for
explanation, we are unable at the time to think of any cause--to frame
any hypothesis about it; we must then wait for the phenomenon to occur
again, and, once more observing its course and accompaniments and trying
to recall its antecedents, do our best to conceive an hypothesis, and
proceed as before. Thus, in the first great epidemic of influenza, some
doctors traced it to a deluge in China, others to a volcanic eruption
near Java; some thought it a mild form of Asiatic plague, and others
caught a specific microbe. As the disease often recurred, there were
fresh opportunities of framing hypotheses; and the microbe was

Again, the investigation may take a different form: given a supposed
Cause to find its Effect; e.g., a new chemical element, to find what
compounds it forms with other elements; or, the spots on the sun--have
they any influence upon our weather?

Here, if the given cause be under control, as a new element may be, it
is possible to try experiments with it according to the Canons of
Inductive Proof. The inquirer may form some hypothesis or expectation as
to the effects, to guide his observation of them, but will be careful
not to hold his expectation so confidently as to falsify his observation
of what actually happens.

But if the cause be, like the sun-spots, not under control, the inquirer
will watch on all sides what events follow their appearance and
development; he must watch for consequences of the new cause he is
studying in many different circumstances, that his observations may
satisfy the canons of proof. But he will also resort for guidance to
deduction; arguing from the nature of the cause, if anything is known of
its nature, what consequences may be expected, and comparing the results
of this deduction with any consequent which he suspects to be connected
with the cause. And if the results of deduction and observation agree,
he will still consider whether the facts observed may not be due to some
other cause.

A cause, however, may be under control and yet be too dangerous to
experiment with; such as the effects of a poison--though, if too
dangerous to experiment with upon man, it may be tried upon animals; or
such as a proposed change of the constitution by legislation; or even
some minor Act of Parliament, for altering the Poor Law, or regulating
the hours of labour. Here the first step must be deductive. We must ask
what consequences are to be expected from the nature of the change
(comparing it with similar changes), and from the laws of the special
circumstances in which it is to operate? And sometimes we may partially
verify our deduction by trying experiments upon a small scale or in a
mild form. There are conflicting deductions as to the probable effect of
giving Home Rule to Ireland; and experiments have been made in more or
less similar cases, as in the Colonies and in some foreign countries. As
to the proposal to make eight hours the legal limit of a day's labour in
all trades, we have all tried to forecast the consequences of this; and
by way of verification we might begin with nine hours; or we might
induce some other country to try the experiment first. Still, no
verification by experiments on a small scale, or in a mild form, or in
somewhat similar yet different circumstances, can be considered
logically conclusive. What proofs are conclusive we shall see in the
following chapters.

§ 2. To begin with the conditions of direct Induction.--An Induction is
an universal real proposition, based on observation, in reliance on the
uniformity of Nature: when well ascertained, it is called a Law. Thus,
that all life depends on the presence of oxygen is (1) an universal
proposition; (2) a real one, since the 'presence of oxygen' is not
connoted by 'life'; (3) it is based on observation; (4) it relies on the
uniformity of Nature, since all cases of life have not been examined.

Such a proposition is here called 'an induction,' when it is inductively
proved; that is, proved by facts, not merely deduced from more general
premises (except the premise of Nature's uniformity): and by the
'process of induction' is meant the method of inductive proof. The
phrase 'process of induction' is often used in another sense, namely for
the inference or judgment by which such propositions are arrived at. But
it is better to call this 'the process of hypothesis,' and to regard it
as a preliminary to the process of induction (that is, proof), as
furnishing the hypothesis which, if it can stand the proper tests,
becomes an induction or law.

§ 3. Inductive proofs are usually classed as Perfect and Imperfect.
They are said to be perfect when all the instances within the scope of
the given proposition have been severally examined, and the proposition
has been found true in each case. But we have seen (chap. xiii. § 2)
that the instances included in universal propositions concerning Causes
and Kinds cannot be exhaustively examined: we do not know all planets,
all heat, all liquids, all life, etc.; and we never can, since a man's
life is never long enough. It is only where the conditions of time,
place, etc., are arbitrarily limited that examination can be exhaustive.
Perfect induction might show (say) that every member of the present
House of Commons has two Christian names. Such an argument is sometimes
exhibited as a Syllogism in Darapti with a Minor premise in U., which
legitimates a Conclusion in A., thus:

      A.B. to Z have two Christian names;
      A.B. to Z are all the present M.P.'s:
    ∴ All the present M.P.'s have two Christian names.

But in such an investigation there is no need of logical method to find
the major premise; it is mere counting: and to carry out the syllogism
is a hollow formality. Accordingly, our definition of Induction excludes
the kind unfortunately called Perfect, by including in the notion of
Induction a reliance on the uniformity of Nature; for this would be
superfluous if every instance in question had been severally examined.
Imperfect Induction, then, is what we have to deal with: the method of
showing the credibility of an universal real proposition by an
examination of _some_ of the instances it includes, generally a small
fraction of them.

§ 4. Imperfect Induction is either Methodical or Immethodical. Now,
Method is procedure upon a principle; and if the method is to be precise
and conclusive, the principle must be clear and definite.

There is a Geometrical Method, because the axioms of Geometry are clear
and definite, and by their means, with the aid of definitions, laws are
deduced of the equality of lines and angles and other relations of
position and magnitude in space. The process of proof is purely
Deductive (the axioms and definitions being granted). Diagrams are used
not as facts for observation, but merely to fix our attention in
following the general argument; so that it matters little how badly they
are drawn, as long as their divergence from the conditions of the
proposition to be proved is not distracting. Even the appeal to
"superposition" to prove the equality of magnitudes (as in Euclid I. 4),
is not an appeal to observation, but to our judgment of what is implied
in the foregoing conditions. Hence no inference is required from the
special case to all similar ones; for they are all proved at once.

There is also, as we have seen, a method of Deductive Logic resting on
the Principles of Consistency and the _Dictum de omni et nullo_. And we
shall find that there is a method of Inductive Logic, resting on the
principle of Causation.

But there are a good many general propositions, more or less trustworthy
within a certain range of conditions, which cannot be methodically
proved for want of a precise principle by which they may be tested; and
they, therefore, depend upon Immethodical Induction, that is, upon the
examination of as many instances as can be found, relying for the rest
upon the undefinable principle of the Uniformity of Nature, since we are
not able to connect them with any of its definite modes enumerated in
chap. xiii. § 7. To this subject we shall return in chap. xix., after
treating of Methodical Induction, or the means of determining that a
relation of events is of the nature of cause and effect, because the
relation can be shown to have the marks of causation, or some of them.

§ 5. Observations and Experiments are the _material_ grounds of
Induction. An experiment is an observation made under prepared, and
therefore known, conditions; and, when obtainable, it is much to be
preferred. Simple observation shows that the burning of the fire
depends, for one thing, on the supply of air; but it cannot show us that
it depends on oxygen. To prove this we must make experiments as by
obtaining pure oxygen and pure nitrogen (which, mixed in the proportion
of one to four, form the air) in separate vessels, and then plunging a
burning taper into the oxygen--when it will blaze fiercely; and again
plunging it into the nitrogen--when it will be extinguished. This shows
that the greater part of the air does nothing to keep the fire alight,
except by diminishing its intensity and so making it last longer.
Experiments are more perfect the more carefully they are prepared, and
the more completely the conditions are known under which the given
phenomenon is to be observed. Therefore, they become possible only when
some knowledge has already been gained by observation; for else the
preparation which they require could not be made.

Observation, then, was the first material ground of Induction, and in
some sciences it remains the chief ground. The heavenly bodies, the
winds and tides, the strata of the earth, and the movements of history,
are beyond our power to experiment with. Experiments upon the living
body or mind are indeed resorted to when practicable, even in the case
of man, as now in all departments of Psychology; but, if of a grave
nature, they are usually thought unjustifiable. And in political affairs
experiments are hindered by the reflection, that those whose interests
are affected must bear the consequences and may resent them. Hence, it
is in physical and chemical inquiries and in the physiology of plants
and animals (under certain conditions) that direct experiment is most
constantly practised.

Where direct experiment is possible, however, it has many advantages
over unaided observation. If one experiment does not enable us to
observe the phenomenon satisfactorily, we may try again and again;
whereas the mere observer, who wishes to study the bright spots on Mars,
or a commercial crisis, must wait for a favourable opportunity. Again,
in making experiments we can vary the conditions of the phenomenon, so
as to observe its different behaviour in each case; whereas he who
depends solely on observation must trust the bounty of nature to supply
him with a suitable diversity of instances. It is a particular advantage
of experiment that a phenomenon may sometimes be 'isolated,' that is,
removed from the influence of all agents except that whose operation we
desire to observe, or except those whose operation is already known:
whereas a simple observer, who has no control over the conditions of the
subject he studies, can never be quite sure that its movements or
changes are not due to causes that have never been conspicuous enough to
draw his attention. Finally, experiment enables us to observe coolly and
circumspectly and to be precise as to what happens, the time of its
occurrence, the order of successive events, their duration, intensity
and extent.

But whether we proceed by observation or experiment, the utmost
attainable exactness of measurements and calculation is requisite; and
these presuppose some Unit, in multiples or divisions of which the
result may be expressed. This unit cannot be an abstract number as in
Arithmetic, but must be one something--an hour, or a yard, or a
pound--according to the nature of the phenomenon to be measured. But
what is an hour, or a yard or a pound? There must in each case be some
constant Standard of reference to give assurance that the unit may
always have the same value. "The English pound is defined by a certain
lump of platinum preserved at Westminster." The unit may be identical
with the standard or some division or multiple of it; and, in measuring
the same kind of phenomena, different units may be used for different
purposes as long as each bears a constant relation to the standard.
Thus, taking the rotation of the earth as the standard of Time, the
convenient unit for long periods is a year (which is a multiple); for
shorter periods, a day (which is identical); for shorter still, an hour
(which is a division), or a second, or a thousandth of a second. (See
Jevons' _Principles of Science_, ch. 14.)

§ 6. The principle of Causation is the _formal_ ground of Induction; and
the Inductive Canons derived from it are means of testing the formal
sufficiency of observations to justify the statement of a Law. If we can
observe the process of cause and effect in nature we may generalise our
observation into a law, because that process is invariable. First, then,
can we observe the course of cause and effect? Our power to do so is
limited by the refinement of our senses aided by instruments, such as
lenses, thermometers, balances, etc. If the causal process is
essentially molecular change, as in the maintenance of combustion by
oxygen, we cannot directly observe it; if the process is partly cerebral
or mental, as in social movements which depend on feeling and opinion,
it can but remotely be inferred; even if the process is a collision of
moving masses (billiard-balls), we cannot really observe what happens,
the elastic yielding, and recoil and the internal changes that result;
though no doubt photography will throw some light upon this, as it has
done upon the galloping of horses and the impact of projectiles. Direct
observation is limited to the effect which any change in a phenomenon
(or its index) produces upon our senses; and what we believe to be the
causal process is a matter of inference and calculation. The meagre and
abstract outlines of Inductive Logic are apt to foster the notion, that
the evidence on which Science rests is simple; but it is amazingly
intricate and cumulative.

Secondly, so far as we can observe the process of nature, how shall we
judge whether a true causal instance, a relation of cause and effect,
is before us? By looking for the five marks of Causation. Thus, in the
experiment above described, showing that oxygen supports combustion, we
find--(1) that the taper which only glowed before being plunged into the
oxygen, bursts into flame when there--Sequence; (2) that this begins to
happen at once without perceptible interval--Immediacy; (3) that no
other agent or disturbing circumstance was present (the preparation of
the experiment having excluded any such thing)--Unconditionalness; (4)
the experiment may be repeated as often as we like with the same
result--Invariableness. Invariableness, indeed, I do not regard as
formally necessary to be shown, supposing the other marks to be clear;
for it can only be proved within our experience; and the very object of
Induction is to find grounds of belief beyond actual experience.
However, for material assurance, to guard against his own liability to
error, the inquirer will of course repeat his experiments.

The above four are the qualitative marks of Causation: the fifth and
quantitative mark is the Equality of Cause and Effect; and this, in the
above example, the Chemist determines by showing that, instead of the
oxygen and wax that have disappeared during combustion, an equivalent
weight of carbon dioxide, water, etc., has been formed.

Here, then, we have all the marks of causation; but in the ordinary
judgments of life, in history, politics, criticism, business, we must
not expect such clear and direct proofs; in subsequent chapters it will
appear how different kinds of evidence are combined in different
departments of investigation.

§ 7. The Inductive Canons, to be explained in the next chapter, describe
the character of observations and experiments that justify us in drawing
conclusions about causation; and, as we have mentioned, they are derived
from the principle of Causation itself. According to that principle,
cause and effect are invariably, immediately and unconditionally
antecedent and consequent, and are equal as to the matter and energy

Invariability can only be observed, in any of the methods of induction,
by collecting more and more instances, or repeating experiments. Of
course it can never be exhaustively observed.

Immediacy, too, in direct Induction, is a matter for observation the
most exact that is possible.

Succession, or the relation itself of antecedent and consequent, must
either be directly observed (or some index of it); or else ascertained
by showing that energy gained by one phenomenon has been lost by
another, for this implies succession.

But to determine the unconditionality of causation, or the
indispensability of some condition, is the great object of the methods,
and for that purpose the meaning of unconditionality may be further
explicated by the following rules for the determination of a Cause.


_I.--For Positive Instances._

To prove a supposed Cause: (a) Any agent whose introduction among
certain conditions (without further change) is followed by a given
phenomenon; or, (b) whose removal is followed by the cessation (or
modification) of that phenomenon, is (so far) the cause or an
indispensable condition of it.

To find the Effect: (c) Any event that follows a given phenomenon, when
there is no further change; or, (d) that does not occur when the
conditions of a former occurrence are exactly the same, except for the
absence of that phenomenon, is the effect of it (or is dependent on

_II.--For Negative Instances._

To exclude a supposed Cause: (a) Any agent that can be introduced among
certain conditions without being followed by a given phenomenon (or that
is found without that phenomenon); or (b) that can be removed when that
phenomenon is present without impairing it (or that is absent when that
phenomenon is present), is not the cause, or does not complete the
cause, of that phenomenon in those circumstances.

To exclude a supposed Effect: (c) Any event that occurs without the
introduction (or presence) of a given phenomenon; or (d) that does not
occur when that phenomenon is introduced (or is present), is not the
effect of that phenomenon.

       *       *       *       *       *

Subject to the conditions thus stated, the rules may be briefly put as

I. (a) That which (without further change) is followed by a given event
is its cause.

II. (a) That which is not so followed is not the cause.

I. (b) That which cannot be left out without impairing a phenomenon is a
condition of it.

II. (b) That which can be left out is not a condition of it.


The Equality of Cause and Effect may be further explained by these

III. (a) When a cause (or effect) increases or decreases, so does its
effect (or cause).

III. (b) If two phenomena, having the other marks of cause and effect,
seem unequal, the less contains an unexplored factor.

III. (c) If an antecedent and consequent do not increase or decrease
correspondingly, they are not cause and effect, so far as they vary.

It will next be shown that these propositions are variously combined in
Mill's five Canons of Induction: Agreement, the Joint Method,
Difference, Variations, Residues. The first three are sometimes called
Qualitative Methods, and the two last Quantitative; and although this
grouping is not quite accurate, seeing that Difference is often used
quantitatively, yet it draws attention to an important distinction
between a mere description of conditions and determination by exact

To avoid certain misunderstandings, some slight alterations have been
made in the wording of the Canons. It may seem questionable whether the
Canons add anything to the above propositions: I think they do. They are
not discussed in the ensuing chapter merely out of reverence for Mill,
or regard for a nascent tradition; but because, as describing the
character of observations and experiments that justify us in drawing
conclusions about causation, they are guides to the analysis of
observations and to the preparation of experiments. To many eminent
investigators the Canons (as such) have been unknown; but they prepared
their work effectively so far only as they had definite ideas to the
same purport. A definite conception of the conditions of proof is the
necessary antecedent of whatever preparations may be made for proving



§ 1. Let me begin by borrowing an example from Bain (_Logic_: B. III. c.
6). The North-East wind is generally detested in this country: as long
as it blows few people feel at their best. Occasional well-known causes
of a wind being injurious are violence, excessive heat or cold,
excessive dryness or moisture, electrical condition, the being laden
with dust or exhalations. Let the hypothesis be that the last is the
cause of the North-East wind's unwholesome quality; since we know it is
a ground current setting from the pole toward the equator and bent
westward by the rotation of the earth; so that, reaching us over
thousands of miles of land, it may well be fraught with dust, effluvia,
and microbes. Now, examining many cases of North-East wind, we find that
this is the only circumstance in which all the instances agree: for it
is sometimes cold, sometimes hot; generally dry, but sometimes wet;
sometimes light, sometimes violent, and of all electrical conditions.
Each of the other circumstances, then, can be omitted without the N.E.
wind ceasing to be noxious; but one circumstance is never absent,
namely, that it is a ground current. That circumstance, therefore, is
probably the cause of its injuriousness. This case illustrates:--


_If two or more instances of a phenomenon under investigation have only
one other circumstance (antecedent or consequent) in common, that
circumstance is probably the cause (or an indispensable condition) or
the effect of the phenomenon, or is connected with it by causation._

This rule of proof (so far as it is used to establish direct causation)
depends, first, upon observation of an invariable connection between the
given phenomenon and one other circumstance; and, secondly, upon I. (a)
and II. (b) among the propositions obtained from the unconditionality of
causation at the close of the last chapter.

To prove that A is causally related to _p_, suppose two instances of the
occurrence of A, an antecedent, and _p_, a consequent, with concomitant
facts or events--and let us represent them thus:

    Antecedents:     A B C          A D E
    Consequents:    _p q r_        _p s t_;

and suppose further that, in this case, the immediate succession of
events can be observed. Then A is probably the cause, or an
indispensable condition, of _p_. For, as far as our instances go, A is
the invariable antecedent of _p_; and _p_ is the invariable consequent
of A. But the two instances of A or _p_ agree in no other circumstance.
Therefore A is (or completes) the unconditional antecedent of _p_. For B
and C are not indispensable conditions of _p_, being absent in the
second instance (Rule II. (b)); nor are D and E, being absent in the
first instance. Moreover, _q_ and _r_ are not effects of A, being absent
in the second instance (Rule II. (d)); nor are _s_ and _t_, being absent
in the first instance.

It should be observed that the cogency of the proof depends entirely
upon its tending to show the unconditionality of the sequence A-_p_, or
the indispensability of A as a condition of _p_. That _p_ follows A,
even immediately, is nothing by itself: if a man sits down to study and,
on the instant, a hand-organ begins under his window, he must not infer
malice in the musician: thousands of things follow one another every
moment without traceable connection; and this we call 'accidental.' Even
invariable sequence is not enough to prove direct causation; for, in
our experience does not night invariable follow day? The proof requires
that the instances be such as to show not merely what events _are_ in
invariable sequence, but also what _are not_. From among the occasional
antecedents of _p_ (or consequents of A) we have to eliminate the
accidental ones. And this is done by finding or making 'negative
instances' in respect of each of them. Thus the instance

     A D E
    _p s t_

is a negative instance of B and C considered as supposable causes of _p_
(and of _q_ and _r_ as supposable effects of A); for it shows that they
are absent when _p_ (or A) is present.

To insist upon the cogency of 'negative instances' was Bacon's great
contribution to Inductive Logic. If we neglect them, and merely collect
examples of the sequence A-_p_, this is 'simple enumeration'; and
although simple enumeration, when the instances of agreement are
numerous enough, may give rise to a strong belief in the connection of
phenomena, yet it can never be a methodical or logical proof of
causation, since it does not indicate the unconditionalness of the
sequence. For simple enumeration of the sequence A-_p_ leaves open the
possibility that, besides A, there is always some other antecedent of
_p_, say X; and then X may be the cause of _p_. To disprove it, we must
find, or make, a negative instance of X--where _p_ occurs, but X is

So far as we recognise the possibility of a plurality of causes, this
method of Agreement cannot be quite satisfactory. For then, in such
instances as the above, although D is absent in the first, and B in the
second, it does not follow that they are not the causes of _p_; for they
may be alternative causes: B may have produced _p_ in the first
instance, and D in the second; A being in both cases an accidental
circumstance in relation to _p_. To remedy this shortcoming by the
method of Agreement itself, the only course is to find more instances of
_p_. We may never find a negative instance of A; and, if not, the
probability that A is the cause of _p_ increases with the number of
instances. But if there be no antecedent that we cannot sometimes
exclude, yet the collection of instances will probably give at last all
the causes of _p_; and by finding the proportion of instances in which
A, B, or X precedes _p_, we may estimate the probability of any one of
them being the cause of _p_ in any given case of its occurrence.

But this is not enough. Since there cannot really be vicarious causes,
we must define the effect (_p_) more strictly, and examine the cases to
find whether there may not be varieties of _p_, with each of which one
of the apparent causes is correlated: A with _p_^{1} B with _p_^{11}, X
with _p_^{111}. Or, again, it may be that none of the recognised
antecedents is effective: as we here depend solely on observation, the
true conditions may be so recondite and disguised by other phenomena as
to have escaped our scrutiny. This may happen even when we suppose that
the chief condition has been isolated: the drinking of foul water was
long believed to cause dysentery, because it was a frequent antecedent;
whilst observation had overlooked the bacillus, which was the
indispensable condition.

Again, though we have assumed that, in the instances supposed above,
immediate sequence is observable, yet in many cases it may not be so, if
we rely only on the canon of Agreement; if instances cannot be obtained
by experiment, and we have to depend on observation. The phenomena may
then be so mixed together that A and _p_ seem to be merely concomitant;
so that, though connection of some sort may be rendered highly probable,
we may not be able to say which is cause and which is effect. We must
then try (as Bain says) to trace the expenditure of energy: if _p_ gains
when A loses, the course of events if from A to _p_.

Moreover, where succession cannot be traced, the method of Agreement may
point to a connection between two or more facts (perhaps as co-effects
of a remote cause) where direct causation seems to be out of the
question: e.g., that Negroes, though of different tribes, different
localities, customs, etc., are prognathous, woolly-haired and

The Method of Agreement, then, cannot by itself prove causation. Its
chief use (as Mill says) is to suggest hypotheses as to the cause; which
must then be used (if possible) experimentally to try if it produces the
given effect. A bacillus, for example, being always found with a certain
disease, is probably the chief condition of it: give it to a guinea-pig,
and observe whether the disease appears in that animal.

Men often use arguments which, if they knew it, might be shown to
conform more or less to this canon; for they collect many instances to
show that two events are connected; but usually neglect to bring out the
negative side of the proof; so that their arguments only amount to
simple enumeration. Thus Ascham in his _Toxophilus_, insisting on the
national importance of archery, argues that victory has always depended
on superiority in shooting; and, to prove it, he shows how the Parthians
checked the Romans, Sesostris conquered a great part of the known world,
Tiberius overcame Arminius, the Turks established their empire, and the
English defeated the French (with many like examples)--all by superior
archery. But having cited these cases to his purpose, he is content;
whereas he might have greatly strengthened his proof by showing how one
or the other instance excludes other possible causes of success. Thus:
the cause was not discipline, for the Romans were better disciplined
than the Parthians; nor yet the boasted superiority of a northern
habitat, for Sesostris issued from the south; nor better manhood, for
here the Germans probably had the advantage of the Romans; nor superior
civilisation, for the Turks were less civilised than most of those they
conquered; nor numbers, nor even a good cause, for the French were more
numerous than the English, and were shamefully attacked by Henry V. on
their own soil. Many an argument from simple enumeration may thus be
turned into an induction of greater plausibility according to the Canon
of Agreement.

Still, in the above case, the effect (victory) is so vaguely conceived,
that a plurality of causes must be allowed for: although, e.g.,
discipline did not enable the Romans to conquer the Parthians, it may
have been their chief advantage over the Germans; and it was certainly
important to the English under Henry V. in their war with the French.

Here is another argument, somewhat similar to the above, put forward by
H. Spencer with a full consciousness of its logical character. States
that make war their chief object, he says, assume a certain type of
organisation, involving the growth of the warrior class and the
treatment of labourers as existing solely to sustain the warriors; the
complete subordination of individuals to the will of the despotic
soldier-king, their property, liberty and life being at the service of
the State; the regimentation of society not only for military but also
for civil purposes; the suppression of all private associations, etc.
This is the case in Dahomey and in Russia, and it was so at Sparta, in
Egypt, and in the empire of the Yncas. But the similarity of
organisation in these States cannot have been due to race, for they are
all of different races; nor to size, for some are small, some large; nor
to climate or other circumstances of habitat, for here again they differ
widely: the one thing they have in common is the military purpose; and
this, therefore, must be the cause of their similar organisation.
(_Political Institutions._)

By this method, then, to prove that one thing is causally connected with
another, say A with _p_, we show, first, that in all instances of _p_, A
is present; and, secondly, that any other supposable cause of _p_ may be
absent without disturbing _p_. We next come to a method the use of which
greatly strengthens the foregoing, by showing that where _p_ is absent
A is also absent, and (if possible) that A is the only supposable cause
that is always absent along with _p_.


_If_ (1) _two or more instances in which a phenomenon occurs have only
one other circumstance (antecedent or consequent) in common, while_ (2)
_two or more instances in which it does not occur (though in important
points they resemble the former set of instances) have nothing else in
common save the absence of that circumstance--the circumstance in which
alone the two sets of instances differ throughout (being present in the
first set and absent in the second) is probably the effect, or the
cause, or an indispensable condition of the phenomenon._

The first clause of this Canon is the same as that of the method of
Agreement, and its significance depends upon the same propositions
concerning causation. The second clause, relating to instances in which
the phenomenon is absent, depends for its probative force upon Prop. II.
(a), and I. (b): its function is to exclude certain circumstances (whose
nature or manner of occurrence gives them some claim to consideration)
from the list of possible causes (or effects) of the phenomenon
investigated. It might have been better to state this second clause
separately as the Canon of the Method of Exclusions.

To prove that A is causally related to _p_, let the two sets of
instances be represented as follows:

Instances of Presence.     Instances of Absence.
        A B C                       C H F
       _p q r_                     _r x v_
        A D E                       B D K
       _p s t_                     _q y s_
        A F G                       E G M
       _p u v_                     _t f u_

Then A is probably the cause or a condition of _p_, or _p_ is dependent
upon A: first, by the Canon of Agreement in Presence, as represented by
the first set of instances; and, secondly, by Agreement in Absence in
the second set of instances. For there we see that C, H, F, B, D, K, E,
G, M occur without the phenomenon _p_, and therefore (by Prop. II. (a))
are not its cause, or not the whole cause, unless they have been
counteracted (which is a point for further investigation). We also see
that _r, v, q, s, t, u_ occur without A, and therefore are not the
effects of A. And, further, if the negative instances represent all
possible cases, we see that (according to Prop. I. (b)) A is the cause
of _p_, because it cannot be omitted without the cessation of _p_. The
inference that A and _p_ are cause and effect, suggested by their being
present throughout the first set of instances, is therefore strengthened
by their being both absent throughout the second set.

So far as this Double Method, like the Single Method of Agreement,
relies on observation, sequence may not be perceptible in the instances
observed, and then, direct causation cannot be proved by it, but only
the probability of causal connection; and, again, the real cause, though
present, may be so obscure as to evade observation. It has, however, one
peculiar advantage, namely, that if the second list of instances (in
which the phenomenon and its supposed antecedent are both absent) can be
made exhaustive, it precludes any hypothesis of a plurality of causes;
since all possible antecedents will have been included in this list
without producing the phenomenon. Thus, in the above symbolic example,
taking the first set of instances, the supposition is left open that B,
C, D, E, F, G may, at one time or another, have been a condition of _p_;
but, in the second list, these antecedents all occur, here or there,
without producing _p_, and therefore (unless counteracted somehow)
cannot be a condition of _p_. A, then, stands out as the one thing that
is present whenever _p_ is present, and absent whenever _p_ is absent.

Stated in this abstract way, the Double Method may seem very elaborate
and difficult; yet, in fact, its use may be very simple. Tyndall, to
prove that dispersed light in the air is due to motes, showed by a
number of cases (1) that any gas containing motes is luminous; (2) that
air in which the motes had been destroyed by heat, and any gas so
prepared as to exclude motes, are not luminous. All the instances are of
gases, and the result is: motes--luminosity; no motes--no luminosity.
Darwin, to show that cross-fertilisation is favourable to flowers,
placed a net about 100 flower-heads, and left 100 others of the same
varieties exposed to the bees: the former bore no seed, the latter
nearly 3,000. We must assume that, in Darwin's judgment, the net did not
screen the flowers from light and heat sufficiently to affect the

There are instructive applications of this Double Method in Wallace's
_Darwinism_. In chap. viii., on _Colour in Animals_, he observes, that
the usefulness of their coloration to animals is shown by the fact that,
"as a rule, colour and marking are constant in each species of wild
animal, while, in almost every domesticated animal, there arises great
variability. We see this in our horses and cattle, our dogs and cats,
our pigeons and poultry. Now the essential difference between the
conditions of life of domesticated and wild animals is, that the former
are protected by man, while the latter have to protect themselves." Wild
animals protect themselves by acquiring qualities adapted to their mode
of life; and coloration is a very important one, its chief, though not
its only use, being concealment. Hence a useful coloration having been
established in any species, individuals that occasionally may vary from
it, will generally, perish; whilst, among domestic animals, variation of
colour or marking is subject to no check except the taste of owners. We
have, then, two lists of instances; first, innumerable species of wild
animals in which the coloration is constant and which depend upon their
own qualities for existence; secondly, several species of domestic
animals in which the coloration is _not_ constant, and which do _not_
depend upon their own qualities for existence. In the former list two
circumstances are present together (under all sorts of conditions); in
the latter they are absent together. The argument may be further
strengthened by adding a third list, parallel to the first, comprising
domestic animals in which coloration is approximately constant, but
where (as we know) it is made a condition of existence by owners, who
only breed from those specimens that come up to a certain standard of

Wallace goes on to discuss the colouring of arctic animals. In the
arctic regions, he says, some animals are wholly white all the year
round, such as the polar bear, the American polar hare, the snowy owl
and the Greenland falcon: these live amidst almost perpetual snow.
Others, that live where the snow melts in summer, only turn white in
winter, such as the arctic hare, the arctic fox, the ermine and the
ptarmigan. In all these cases the white colouring is useful, concealing
the herbivores from their enemies, and also the carnivores in
approaching their prey; this usefulness, therefore, is a condition of
the white colouring. Two other explanations have, however, been
suggested: first, that the prevalent white of the arctic regions
directly colours the animals, either by some photographic or chemical
action on the skin, or by a reflex action through vision (as in the
chameleon); secondly, that a white skin checks radiation and keeps the
animals warm. But there are some exceptions to the rule of white
colouring in arctic animals which refute these hypotheses, and confirm
the author's. The sable remains brown throughout the winter; but it
frequents trees, with whose bark its colour assimilates. The musk-sheep
is brown and conspicuous; but it is gregarious, and its safety depends
upon its ability to recognise its kind and keep with the herd. The raven
is always black; but it fears no enemy and feeds on carrion, and
therefore does not need concealment for either defence or attack. The
colour of the sable, then, though not white, serves for concealment; the
colour of the musk-sheep serves a purpose more important than
concealment; the raven needs no concealment. There are thus two sets of
instances:--in one set the animals are white (a) all the year, (b) in
winter; and white conceals them (a) all the year, (b) in winter; in the
other set, the animals are _not_ white, and to them either whiteness
would _not_ give concealment, or concealment would _not_ be
advantageous. And this second list refutes the rival hypotheses: for the
sable, the musk-sheep and the raven are as much exposed to the glare of
the snow, and to the cold, as the other animals are.


_If an instance in which a phenomenon occurs, and an instance in which
it does not occur, have every other circumstance in common save one,
that one (whether consequent or antecedent) occurring only in the
former; the circumstance in which alone the two instances differ is the
effect, or the cause, or an indispensable condition of the phenomenon._

This follows from Props. I (a) and (b), in chapter xv. § 7. To prove
that A is a condition of _p_, let two instances, such as the Canon
requires, be represented thus:

     A B C                          B C
    _p q r_                        _q r_

Then A is the cause or a condition of _p_. For, in the first instance, A
being introduced (without further change), _p_ arises (Prop. I. (a));
and, in the second instance, A having been removed (without other
change), _p_ disappears (Prop. I. (b)). Similarly we may prove, by the
same instances, that _p_ is the effect of A.

The order of the phenomena and the immediacy of their connection is a
matter for observation, aided by whatever instruments and methods of
inspection and measurement may be available.

As to the invariability of the connection, it may of course be tested by
collecting more instances or making more experiments; but it has been
maintained, that a single perfect experiment according to this method is
sufficient to prove causation, and therefore implies invariability
(since causation is uniform), though no other instances should ever be
obtainable; because it establishes once for all the unconditionality of
the connection

     A B C
    _p q r_.

Now, formally this is true; but in any actual investigation how shall we
decide what is a satisfactory or perfect experiment? Such an experiment
requires that in the negative instance

     B C
    _q r_,

BC shall be the least assemblage of conditions necessary to co-operate
with A in producing _p_; and that it is so cannot be ascertained without
either general prior knowledge of the nature of the case or special
experiments for the purpose. So that invariability will not really be
inferred from a single experiment; besides that every prudent inquirer
repeats his experiments, if only to guard against his own liability to

The supposed plurality of causes does not affect the method of
Difference. In the above symbolic case, A is clearly _one_ cause (or
condition) of _p_, whatever other causes may be possible; whereas with
the Single Method of Agreement, it remained doubtful (admitting a
plurality of causes) whether A, in spite of being always present with
_p_, was ever a cause or condition of it.

This method of Difference without our being distinctly aware of it, is
oftener than any other the basis of ordinary judgments. That the sun
gives light and heat, that food nourishes and fire burns, that a stone
breaks a window or kills a bird, that the turning of a tap permits or
checks the flow of water or of gas, and thousands of other propositions
are known to be true by rough but often emphatic applications of this
method in common experience.

The method of Difference may be applied either (1) by observation, on
finding two instances (distinct assemblages of conditions) differing
only in one phenomenon together with its antecedent or consequent; or
(2) by experiment, and then, either (a) by preparing two instances that
may be compared side by side, or (b) by taking certain conditions, and
then introducing (or subtracting) some agent, supposed to be the cause,
to see what happens: in the latter case the "two instances" are the same
assemblage of conditions considered before and, again, after, the
introduction of the agent. As an example of (a) there is an experiment
to show that radium gives off heat: take two glass tubes, in one put
some chloride of radium, in both thermometers, and close them with
cotton-wool. Soon the thermometer in the tube along with radium reads
54° F. higher than the other one. The tube without the radium, whose
temperature remains unaltered, is called the "control" experiment. Most
experiments are of the type (b); and since the Canon, which describes
two co-existing instances, does not readily apply to this type, an
alternative version may be offered: _Any agent whose introduction into
known circumstances (without further change) is immediately followed by
a definite phenomenon is a condition of the occurrence of that

The words _into known circumstances_ are necessary to emphasise what is
required by this Method, namely, that the two instances differ in only
one thing; for this cannot be ascertained unless all the other
conditions are known; and this further implies that they have been
prepared. It is, therefore, not true (as Sigwart asserts) that this
method determines only one condition of a phenomenon, and that it is
then necessary to inquire into the other conditions. If they were not
known they must be investigated; but then the experiment would not have
been made upon this method. Practically, experiments have to be made in
all degrees of imperfection, and the less perfect they are, that is, the
less the circumstances are known beforehand, the more remains to be
done. A common imperfection is delay, or the occurrence of a latent
period between the introduction of an agent and the manifestation of its
effects; it cannot then be the unconditional cause; though it may be an
indispensable remote condition of whatever change occurs. If, feeling
out of sorts, you take a drug and some time afterwards feel better, it
is not clear on this ground alone that the drug was the cause of
recovery, for other curative processes may have been active
meanwhile--food, or sleep, or exercise.

Any book of Physics or of Chemistry will furnish scores of examples of
the method of Difference: such as Galileo's experiment to show that air
has weight, by first weighing a vessel filled with ordinary air, and
then filling it with condensed air and weighing it again; when the
increased weight can only be due to the greater quantity of air
contained. The melting-point of solids is determined by heating them
until they do melt (as silver at 1000° C., gold at 1250°, platinum at
2000°); for the only difference between bodies at the time of melting
and just before is the addition of so much heat. Similarly with the
boiling point of liquids. That the transmission of sound depends upon
the continuity of an elastic ponderable medium, is proved by letting a
clock strike in a vacuum (under a glass from which the air has been
withdrawn by an air pump), and standing upon a non-elastic pedestal:
when the clock be seen to strike, but makes only such a faint sound as
may be due to the imperfections of the vacuum and the pedestal.

The experiments by which the chemical analysis or synthesis of various
forms of matter is demonstrated are simple or compound applications of
this method of Difference, together with the quantitative mark of
causation (that cause and effect are equal); since the bodies resulting
from an analysis are equal in weight to the body analysed, and the body
resulting from a synthesis is equal in weight to the bodies synthesised.
That an electric current resolves water into oxygen and hydrogen may be
proved by inserting the poles of a galvanic battery in a vessel of
water; when this one change is followed by another, the rise of bubbles
from each pole and the very gradual decrease of the water. If the
bubbles are caught in receivers placed over them, it can be shown that
the joint weight of the two bodies of gas thus formed is equal to the
weight of the water that has disappeared; and that the gases are
respectively oxygen and hydrogen may then be shown by proving that they
have the properties of those gases according to further experiments by
the method of Difference; as (e.g.) that one of them is oxygen because
it supports combustion, etc.

When water was first decomposed by the electric current, there appeared
not only oxygen and hydrogen, but also an acid and an alkali. These
products were afterwards traced to impurities of the water and of the
operator's hands. Mill observes that in any experiment the effect, or
part of it, may be due, not to the supposed agent, but to the means
employed in introducing it. We should know not only the other conditions
of an experiment, but that the agent or change introduced is nothing
else than what it is supposed to be.

In the more complex sciences the method of Difference is less easily
applicable, because of the greater difficulty of being sure that only
one circumstance at a time has altered; still, it is frequently used.
Thus, if by dividing a certain nerve certain muscles are paralysed, it
is shown that normally that nerve controls those muscles. That the sense
of smell in flies and cockroaches is connected with the antennae has
been shown by cutting them off: whereupon the insects can no longer find
carrion. In his work on _Earthworms_, Darwin shows that, though
sensitive to mechanical tremors, they are deaf (or, at least, not
sensitive to sonorous vibrations transmitted through the air), by the
following experiment. He placed a pot containing a worm that had come to
the surface, as usual at night, upon a table, whilst close by a piano
was violently played; but the worm took no notice of the noise. He then
placed the pot upon the piano, whilst it was being played, when the
worm, probably feeling mechanical vibrations, hastily slid back into its

When, instead of altering one circumstance in an instance (which we have
done our best not otherwise to disturb) and then watching what follows,
we try to find two ready-made instances of a phenomenon, which only
differ in one other circumstance, it is, of course, still more difficult
to be sure that there is only one other circumstance in which they
differ. It may be worth while, however, to look for such instances.
Thus, that the temperature of ocean currents influences the climate of
the shores they wash, seems to be shown by the fact that the average
temperature of Newfoundland is lower than that of the Norwegian coast
some 15° farther north. Both regions have great continents at their
back; and as the mountains of Norway are higher and capped with
perennial snow, we might expect a colder climate there: but the shore of
Norway is visited by the Gulf Stream, whilst the shore of Newfoundland
is traversed by a cold current from Greenland. Again, when in 1841 the
railway from Rouen to Paris was being built, gangs of English and gangs
of French workmen were employed upon it, and the English got through
about one-third more work per man than the French. It was suspected that
this difference was due to one other difference, namely, that the
English fed better, preferring beef to thin soup. Now, logically, it
might have been objected that the evidence was unsatisfactory, seeing
that the men differed in other things besides diet--in 'race' (say),
which explains so much and so easily. But the Frenchmen, having been
induced to try the same diet as the English, were, in a few days, able
to do as much work: so that the "two instances" were better than they
looked. It often happens that evidence, though logically questionable,
is good when used by experts, whose familiarity with the subject makes
it good.


_Whatever phenomenon varies in any manner whenever another phenomenon
(consequent or antecedent) varies in some particular manner [no other
change having concurred] is either the cause or effect of that
phenomenon [or is connected with it through some fact of causation]._

This is not an entirely fresh method, but may be regarded as a special
case either of Agreement or of Difference, to prove the cause or effect,
not of a phenomenon as a whole, but of some increment of it (positive or
negative). There are certain forces, such as gravitation, heat,
friction, that can never be eliminated altogether, and therefore can
only be studied in their degrees. To such phenomena the method of
Difference cannot be applied, because there are no negative instances.
But we may obtain negative instances of a given quantity of such a
phenomenon (say, heat), and may apply the method of Difference to that
quantity. Thus, if the heat of a body increases 10 degrees, from 60 to
70, the former temperature of 60 was a negative instance in respect of
those 10 degrees; and if only one other circumstance (say, friction)
has altered at the same time, that circumstance (if an antecedent) is
the cause. Accordingly, if in the above Canon we insert, after
'particular manner,' "[no other change having concurred,]" it is a
statement of the method of Difference as applicable to the increment of
a phenomenon, instead of to the phenomenon as a whole; and we may then
omit the last clause--"[or is connected, etc.]." For these words are
inserted to provide for the case of co-effects of a common cause (such
as the flash and report of a gun); but if no other change (such as the
discharge of a gun) has concurred with the variations of two phenomena,
there cannot have been a common cause, and they are therefore cause and

If, on the other hand, we omit the clause "[no other change having
concurred,]" the Canon is a statement of the method of Agreement as
applicable to the increment of a phenomenon instead of to the phenomenon
as a whole; and it is then subject to the imperfections of that method:
that is to say, it leaves open the possibilities, that an inquirer may
overlook a plurality of causes; or may mistake a connection of two
phenomena, which (like the flash and report of a gun) are co-effects of
a common cause, for a direct relation of cause and effect.

It may occur to the reader that we ought also to distinguish Qualitative
and Quantitative Variations as two orders of phenomena to which the
present method is applicable. But, in fact, Qualitative Variations may
be adequately dealt with by the foregoing methods of Agreement, Double
Agreement, and Difference; because a change of quality or property
entirely gets rid of the former phase of that quality, or substitutes
one for another; as when the ptarmigan changes from brown to white in
winter, or as when a stag grows and sheds its antlers with the course of
the seasons. The peculiar use of the method of Variations, however, is
to formulate the conditions of proof in respect of those causes or
effects which cannot be entirely got rid of, but can be obtained only in
greater or less amount; and such phenomena are or course, quantitative.

Even when there are two parallel series of phenomena the one
quantitative and the other qualitative--like the rate of air-vibration
and the pitch of sound, or the rate of ether-vibration and the
colour-series of the spectrum--the method of Variations is not
applicable. For (1) two such series cannot be said to vary together,
since the qualitative variations are heterogeneous: 512: 576 is a
definite ratio; but the corresponding notes, C, D, in the treble clef,
present only a difference. Hence (2) the correspondence of each note
with each number is a distinct fact. Each octave even is a distinct
fact; there is a difference between C 64 and C 128 that could never have
been anticipated without the appropriate experience. There is,
therefore, no such law of these parallel series as there is for
temperature and change of volume (say) in mercury. Similar remarks apply
to the physical and sensitive light-series.

We may illustrate the two cases of the method thus (putting a dash
against any letter, A' or _p_', to signify an increase or decrease of
the phenomenon the letter stands for): Agreement in Variations (other
changes being admissible)--

     A B C           A' D E            A'' F G
    _p q r_         _p' s t_          _p'' u v_

Here the accompanying phenomena (_B C q r, D E s t, F G u v_) change
from time to time, and the one thing in which the instances agree
throughout is that any increase of A (A' or A'') is followed or
accompanied by an increase of _p (p' or p'')_: whence it is argued that
A is the cause of _p_, according to Prop. III. (a) (ch. xv. § 7). Still,
it is supposable that, in the second instance, D or E may be the cause
of the increment of _p_; and that, in the third instance, F or G may be
its cause: though the probability of such vicarious causation decreases
rapidly with the increase of instances in which A and _p_ vary together.
And, since an actual investigation of this type must rely on
observation, it is further possible that some undiscovered cause, X, is
the real determinant of both A and _p_ and of their concomitant

Professor Ferri, in his _Criminal Sociology_, observes: "I have shown
that in France there is a manifest correspondence of increase and
decrease between the number of homicides, assaults and malicious
wounding, and the more or less abundant vintage, especially in the years
of extraordinary variations, whether of failure of the vintage (1853-5,
1859, 1867, 1873, 1878-80), attended by a remarkable diminution of crime
(assaults and wounding), or of abundant vintages (1850, 1856-8, 1862-3,
1865, 1868, 1874-5), attended by an increase of crime" (p. 117, Eng.
trans.). And earlier he had remarked that such crimes also "in their
oscillations from month to month display a characteristic increase
during the vintage periods, from June to December, notwithstanding the
constant diminution of other offences" (p. 77). This is necessarily an
appeal to the canon of Concomitant Variations, because France is never
without her annual vintage, nor yet without her annual statistics of
crime. Still, it is an argument whose cogency is only that of Agreement,
showing that probably the abuse of the vintage is a cause of crimes of
violence, but leaving open the supposition, that some other circumstance
or circumstances, arising or varying from year to year, may determine
the increase or decrease of crime; or that there is some unconsidered
agent which affects both the vintage and crimes of violence. French
sunshine, it might be urged, whilst it matures the generous grape, also
excites a morbid fermentation in the human mind.

Difference in Variations may be symbolically represented thus (no other
change having concurred):

     A B      A' B      A'' B
    _p q_,   _p' q_,   _p'' q_.

Here the accompanying phenomena are always the same B/q; and the only
point in which the successive instances differ is in the increments of A
(A', A'') followed by corresponding increments of _p_ (_p', p''_): hence
the increment of A is the cause of the increment of _p_.

For examples of the application of this method, the reader should refer
to some work of exact science. He will find in Deschanel's _Natural
Philosophy_, c. 32, an account of some experiments by which the
connection between heat and mechanical work has been established. It is
there shown that "whenever work is performed by the agency of heat" [as
in driving an engine], "an amount of heat disappears equivalent to the
work performed; and whenever mechanical work is spent in generating
heat" [as in rubbing two sticks together], "the heat generated is
equivalent to the work thus spent." And an experiment of Joule's is
described, which consisted in fixing a rod with paddles in a vessel of
water, and making it revolve and agitate the water by means of a string
wound round the rod, passed over a pulley and attached to a weight that
was allowed to fall. The descent of the weight was measured by a
graduated rule, and the rise of the water's temperature by a
thermometer. "It was found that the heat communicated to the water by
the agitation amounted to one pound-degree Fahrenheit for every 772
foot-pounds of work" expended by the falling weight. As no other
material change seems to take place during such an experiment, it shows
that the progressive expenditure of mechanical energy is the cause of
the progressive heating of the water.

The thermometer itself illustrates this method. It has been found that
the application of heat to mercury expands it according to a law; and
hence the volume of the mercury, measured by a graduated index, is used
to indicate the temperature of the air, water, animal body, etc., in
which the thermometer is immersed, or with which it is brought into
contact. In such cases, if no other change has taken place, the heat of
the air, water, or body is the cause of the rise of the mercury in its
tube. If some other substance (say spirit) be substituted for mercury in
constructing a thermometer, it serves the same purpose, provided the
index be graduated according to the law of the expansion of that
substance by heat, as experimentally determined.

Instances of phenomena that do not vary together indicate the exclusion
of a supposed cause (by Prop. III (c)). The stature of the human race
has been supposed to depend on temperature; but there is no
correspondence. The "not varying together," however, must not be
confused with "varying inversely," which when regular indicates a true
concomitance. It is often a matter of convenience whether we regard
concomitant phenomena as varying directly or inversely. It is usual to
say--'the greater the friction the less the speed'; but it is really
more intelligible to say--'the greater the friction the more rapidly
molar is converted into molecular motion.'

The Graphic Method exhibits Concomitant Variations to the eye, and is
extensively used in physical and statistical inquiries. Along a
horizontal line (the abscissa) is measured one of the conditions (or
agents) with which the inquiry is concerned, called the Variable; and
along perpendiculars (ordinates) is measured some phenomenon to be
compared with it, called the Variant.

Thus, the expansion of a liquid by heat may be represented by measuring
degrees of temperature along the horizontal, and the expansion of a
column of the liquids in units of length along the perpendicular.

[Illustration: FIG. 9.]

In the next diagram (Fig. 10), reduced from one given by Mr. C.H. Denyer
in an article on the Price of Tea (_Economic Journal_, No. 9), the
condition measured horizontally is Time; and, vertically, three variants
are measured simultaneously, so that their relations to one another from
time to time may be seen at a glance. From this it is evident that, as
the duty on tea falls, the price of tea falls, whilst the consumption of
tea rises; and, in spite of some irregularity of correspondence in the
courses of the three phenomena, their general causal connection can
hardly be mistaken. However, the causal connection may also be inferred
by general reasoning; the statistical Induction can be confirmed by a
Deduction; thus illustrating the combined method of proof to be
discussed in the next chapter. Without such confirmation the proof by
Concomitant Variations would not be complete; because, from the
complexity of the circumstances, social statistics can only yield
evidence according to the method of Agreement in Variations. For,
besides the agents that are measured, there may always be some other
important influence at work. During the last fifty years, for example,
crime has decreased whilst education has increased: true, but at the
same time wages have risen and many other things have happened.

[Illustration: FIG. 10.]

It will be noticed that in the diagram the three lines, especially those
of Price and Consumption (which may be considered _natural_ resultants,
in contrast with the arbitrary fixation of a Tax), do not depart widely
from regular curves; and accordingly, assuming the causes at work to
vary continuously during the intervals between points of measurement,
curves may be substituted. In fact, a curve often represents the course
of a phenomenon more truthfully than can be done by a line that zigzags
along the exact measurements; because it is less influenced by temporary
and extraordinary causes that may obscure the operation of those that
are being investigated. On the other hand, the abrupt deviations of a
punctilious zigzag may have their own logical value, as will appear in
the next section.

In working with the Method of Variations one must allow for the
occurrence in a series of 'critical points,' at which sudden and
sometimes heterogeneous changes may take place. Every substance exists
at different temperatures in three states, gaseous, liquid, solid; and
when the change takes place, from one state to another, the series of
variations is broken. Water, e.g., follows the general law that
cooling is accompanied by decrease of volume between 212° and 39° F.:
but above 212°, undergoes a sudden expansion in becoming a gas; and
below 39° begins to expand, until at 32° the expansion is considerable
on its becoming solid. This illustrates a common experience that
concomitant variations are most regular in the 'median range,' and are
apt to become irregular at the extremities of the series, where new
conditions begin to operate.

The Canon of Variations, again, deals not with sudden irruptions of a
cause, force or agent, but with some increase or decrease of an agent
already present, and a corresponding increase or decrease of some other
phenomenon--say an increase of tax and a rise of price. But there are
cases in which the energy of a cause is not immediately discharged and
dissipated. Whilst a tax of 6_d._ per lb. on tea raises the price per
lb. by about 6_d._, however long it lasts, the continuous application of
friction to a body may gradually raise its temperature to the point of
combustion; because heat is received faster than it is radiated, and
therefore accumulates. Such cases are treated by Mill under the title of
'progressive effects' (_Logic_: B. III., c. 15): he gives as an example
of it the acceleration of falling bodies. The storage of effects is a
fact of the utmost importance in all departments of nature, and is
especially interesting in Biology and Sociology, where it is met with as
heredity, experience, tradition. Evolution of species of plants and
animals would (so far as we know) be impossible, if the changes (however
caused) that adapt some individuals better than others to the conditions
of life were not inherited by, and accumulated in, their posterity. The
eyes in the peacock's tail are supposed to have reached their present
perfection gradually, through various stages that may be illustrated by
the ocelli in the wings of the Argus pheasant and other genera of
_Phasianidæ_. Similarly the progress of societies would be impossible
without tradition, whereby the improvements made in any generation may
be passed on to the next, and the experience of mankind may be gradually
accumulated in various forms of culture. The earliest remains of culture
are flint implements and weapons; in which we can trace the effect of
tradition in the lives of our remote forefathers, as they slowly through
thousands of years learnt to improve the chipping of flints, until the
first rudely shaped lumps gave place to works of unmistakable design,
and these to the beautiful weapons contemporary with the Bronze Age.

The Method of Gradations, the arranging of any phenomena to be studied
in series, according to the degree in which some character is exhibited,
is, perhaps, the most definite device in the Art of Discovery. (Bain:
_Induction_, c. 6, and App. II.) If the causes are unknown it is likely
to suggest hypotheses: and if the causes are partly known, variation in
the character of the series is likely to indicate a corresponding
variation of the conditions.


_Subduct from any phenomenon such part as previous inductions have shown
to be the effect of certain antecedents, and the residue of the
phenomenon is the effect of the remaining antecedents_.

The phenomenon is here assumed to be an effect: a similar Canon may be
framed for residuary causes.

This also is not a fresh method, but a special case of the method of
Difference. For if we suppose the phenomenon to be _p q r_, and the
antecedent to be A B C, and that we already know B and C to have (either
severally or together) the consequents _q r_, in which their efficacy is
exhausted; we may regard

 B C
_q r_

as an instance of the absence of _p_ obtained deductively from the whole

 A B C
_p q r_

by our knowledge of the laws of B and C; so that

 A B C
_p q r_

is an instance of the presence of _p_, differing otherwise from

 B C
_q r_

in nothing except that A is also present. By the Canon of Difference,
therefore A is the cause of _p_. Or, again, when phenomena thus treated
are strictly quantitative, the method may be based on Prop. III. (b),
ch. xv. § 7.

Of course, if A can be obtained apart from B C and directly experimented
with so as to produce _p_, so much the better; and this may often be
done; but the special value of the method of Residues appears, when some
complex phenomenon has been for the most part accounted for by known
causes, whilst there remains some excess, or shortcoming, or deviation
from the result which those causes alone would lead us to expect, and
this residuary fact has to be explained in relation to the whole. Here
the negative instance is constituted by deduction, showing what would
happen but for the interference of some unknown cause which is to be
investigated; and this prominence of the deductive process has led some
writers to class the method as deductive. But we have seen that all the
Canons involve deduction; and, considering how much in every experiment
is assumed as already known (what circumstances are 'material,' and when
conditions may be called 'the same'), the wonder is that no one has
insisted upon regarding every method as concerned with residues. In
fact, as scientific explanation progresses, the phenomena that may be
considered as residuary become more numerous and the importance of this
method increases.

Examples: The recorded dates of ancient eclipses having been found to
differ from those assigned by calculation, it appears that the average
length of a day has in the meanwhile increased. This is a residuary
phenomenon not accounted for by the causes formerly recognised as
determining the rotation of the earth on its axis; and it may be
explained by the consideration that the friction of the tides reduces
the rate of the earth's rotation, and thereby lengthens the day.
Astronomy abounds in examples of the method of Residues, of which the
discovery of Neptune is the most famous.

Capillarity seems to be a striking exception to the principle that water
(or any liquid) 'finds its level,' that being the condition of
equilibrium; yet capillarity proves to be only a refined case of
equilibrium when account is taken of the forces of adhesion exerted by
different kinds of bodies in contact.

"Many of the new elements of Chemistry," says Herschel, "have been
detected in the investigation of residual phenomena." Thus, Lord
Rayleigh and Sir W. Ramsay found that nitrogen from the atmosphere was
slightly heavier than nitrogen got from chemical sources; and, seeking
the cause of this difference, discovered argon.

The Economist shows that when a country imports goods the chief means of
paying for them is to export other goods. If this were all, imports and
exports would be of equal value: yet the United Kingdom imports about
£400,000,000 annually, and exports about £300,000,000. Here, then, is a
residuary phenomenon of £100,000,000 to be accounted for. But foreign
countries owe us about £50,000,000 for the use of shipping, and
£70,000,000 as interest on the capital we have lent them, and
£15,000,000 in commissions upon business transacted for them. These sums
added together amount to £135,000,000; and that is £35,000,000 too much.
Thus another residuary phenomenon emerges; for whilst foreigners seem to
owe us £435,000,000 they only send us £400,000,000 of imports. These
£35,000,000 are accounted for by the annual investment of our capital
abroad, in return for which no immediate payment is due; and, these
being omitted, exports and imports balance. Since this was written the
figures of our foreign trade have greatly risen; but the character of
the explanation remains the same.

When, in pursuing the method of Variations, the phenomena compared do
not always correspond in their fluctuations, the irregular movements of
that phenomenon which we regard as the effect may often be explained by
treating them as residuary phenomena, and then seeking for exceptional
causes, whose temporary interference has obscured the influence of the
general cause. Thus, returning to the diagram of the Price of Tea in §
4, it is clear that generally the price falls as the duty falls; but in
Mr. Denyer's more minutely wrought diagram, from which this is reduced,
it may be seen that in 1840 the price of tea rose from 3_s._ 9_d._ to
4_s._ 9_d._ without any increase of duty. This, however, is readily
explained by the Chinese War of that year, which checked the supply.
Again, from 1869 to 1889 the duty was constant, whilst the price of tea
fell as much as 8_d._ per lb.; but this residuary phenomenon is
explained by the prodigiously increased production of tea during that
period in India and Ceylon.

The above examples of the method of Residues are all quantitative; but
the method is often employed where exact estimates are unobtainable.
Thus Darwin, having found certain modifications of animals in form,
coloration and habits, that were not clearly derivable from their
struggle for existence in relation to other species or to external
conditions, suggested that they were due to Sexual Selection.

The 'vestiges' and 'survivals' so common in Biology and Sociology are
residuary phenomena. It is a general inference from the doctrine of
Natural Selection that every organ of a plant, animal, or society is in
some way useful to it. There occur, however, organs that have at present
no assignable utility, are at least wasteful, and sometimes even
injurious. And the explanation is that formerly they were useful; but
that, their uses having lapsed, they are now retained by the force of
heredity or tradition. Either they are not injurious enough to be
eliminated by natural selection; or they are correlated with other
organs, whose utility outweighs their disutility.



§ 1. We have now reviewed Mill's five Canons of Inductive Proof. At
bottom, as he observes, there are only two, namely, Agreement and
Difference: since the Double Method, Variations and Residues are only
special forms of the other two. Indeed, in their function of _proof_,
they are all reducible to one, namely, Difference; for the cogency of
the method of Agreement (as distinguished from a simple enumeration of
instances agreeing in the coincidence of a supposed cause and its
effect), depends upon the omission, in one instance after another, of
all other circumstances; which omission is a point of difference.

The Canons are an analysis of the conditions of proving directly (where
possible), by means of observation or experiment, any proposition that
predicates causation. But if we say 'by means of observation or
experiment,' it is not to be understood that these are the only means
and that nothing else is involved; for it has been shown that the Law of
Causation is itself an indispensable foundation of the evidence. In fact
Inductive Logic may be considered as having a purely formal character.
It consists (1) in a statement of the Law of Cause and Effect; (2) in
certain immediate inferences from this Law, expanded into the Canons;
(3) in the syllogistic application of the Canons to special predications
of causation by means of minor premises, showing that certain instances
satisfy the Canons.

At the risk of some pedantry, we may exhibit the process as follows
(_cf._ Prof. Ray's _Logic_: Appendix D):

Whatever relation of events has certain marks is a case of causation;

    The relation A: _p_ has some or all of these marks (as shown
    by observation and by the conformity of instances to such or
    such a Canon):

Therefore, the relation A: _p_ is a case of causation. Now, the
parenthesis, "as shown by the conformity, etc.," is an adscititious
member of an Epicheirema, which may be stated, as a Prosyllogism, thus:

    If an instance, etc. (Canon of Difference);

    The instances A B C  B C are of the kind required:
                 _p q r' q r_

    Therefore, A, present where _p_ occurs and absent where it
    does not occur, is an indispensable antecedent of _p_.

Such is the bare Logic of Induction: so that, strictly speaking,
observation or experiment is no part of the logic, but a means of
applying the logic to actual, that is, not merely symbolical,
propositions. The Formal Logic of Induction is essentially deductive;
and it has been much questioned whether any transition from the formal
to the material conditions of proof is possible. As long as we are
content to illustrate the Canons with symbols, such as A and _p_, all
goes well; but can we in any actual investigation show that the relevant
facts or 'instances' correspond with those symbols?

In the first place, as Dr. Venn shows, natural phenomena want the
distinctness and capability of isolation that belong to symbols.
Secondly, the observing whether instances conform to a Canon, must
always be subject at last to the limits of our faculties. How can we
ascertain exact equality, immediate sequence? The Canon of Difference,
in its experimental application, is usually considered the most cogent
sort of proof: yet when can the two sequent instances, before and after
the introduction of a certain agent, be said to differ in nothing else?
Are not earth and stars always changing position; is not every molecule
in the room and apparatus always oscillating? It is true that our senses
are now aided by elaborate instruments; but the construction of these
depends on scientific theories, which again depend on experiments.

It is right to touch upon this well-known sceptical topic; but to insist
much upon it is not a sign of good sense. The works of Herschel,
Whewell, and Jevons should be consulted for the various methods of
correcting observations, by repeating them, averaging them, verifying
one experimental process by another, always refining the methods of
exact measurement, multiplying the opportunities of error (that if any
exist it may at last show itself), and by other devices of what may be
called Material Logic or Methodology. But only direct experience and
personal manipulation of scientific processes, can give a just sense of
their effectiveness; and to stand by, suggesting academic doubts, is
easier and more amusing.

§ 2. Still, it is not so much in laws based upon direct observation or
experiment, that the material validity of scientific reasoning appears,
as in the cumulative evidence that arises from the co-ordination of laws
within each science, and the growing harmony and coherence of all
sciences. This requires a more elaborate combination of deduction with
observation and experiment. During the last three hundred years many
departments of science have been reduced under principles of the
greatest generality, such as the Conservation of Energy, the Law of
Gravitation, the Undulatory theory of Light, the Law of combining
Equivalents, and the Theory of Natural Selection; connecting and
explaining the less general laws, which, again, are said to connect and
explain the facts. Meanwhile, those sciences that were the first to make
progress have helped to develop others which, like Biology and
Sociology, present greater difficulties; and it becomes more and more
apparent that the distinctions drawn among sciences are entirely for the
convenience of study, and that all sciences tend to merge in one
universal Science of Nature. Now, this process of the 'unification of
knowledge' is almost another name for deduction; but at the same time it
depends for its reality and solidity upon a constant reference to
observation and experiment. Only a very inadequate notion of it can be
given in the ensuing chapters.

We saw in chap. xiv. § 6, that when two or more agents or forces combine
to produce a phenomenon, their effects are intermixed in it, and this in
one of two ways according to their nature. In chemical action and in
vegetable and animal life, the causal agents concerned are blended in
their results in such a way that most of the qualities which they
exhibited severally are lost, whilst new qualities appear instead. Thus
chlorine (a greenish-yellow gas) and sodium (a metal) unite to form
common salt NaCl; which is quite unlike either of them: a man eats
bread, and it becomes muscle, nerve and bone. In such cases we cannot
trace the qualities of the causal agents in the qualities of the
effects; given such causes, we can prove experimentally, according to
the canons of induction, that they have such effects; but we may not be
able in any new case to calculate what the effects will be.

On the other hand, in Astronomy and Physics, the causes treated of are
mechanical; at least, it is the aim of Physics to attain to a mechanical
conception of phenomena; so that, in every new combination of forces,
the intermixed effect, or resultant, may be calculated beforehand;
provided that the forces concerned admit of being quantitatively
estimated, and that the conditions of their combination are not so
complex as to baffle the powers of mathematicians. In such cases, when
direct observation or experiment is insufficient to resolve an effect
into the laws of its conditions, the general method is to calculate
what may be expected from a combination of its conditions, as either
known or hypothetically assumed, and to compare this anticipation with
the actual phenomenon.

§ 3. This is what Mill calls the Direct Deductive Method; or, the
Physical Method, because it is so much relied on in treating of Light,
Heat, Sound, etc.; it is also the method of Astronomy and much used in
Economics: Deduction leads the way, and its results are tested
inductively by experiments or observations. Given any complex mechanical
phenomenon, the inquirer considers--(1) what laws already ascertained
seem likely to apply to it (in default of known laws, hypotheses are
substituted: _cf._ chap. xviii.); he then--(2) computes the effect that
will follow from these laws in circumstances similar to the case before
him; and (3) he verifies his conclusion by comparing it with the actual

A simple example of this method is the explanation of the rise of water
in the 'common pump.' We know three laws applicable to this case: (a)
that the atmosphere weighs upon the water outside the pump with a
pressure of 15 lb. to the square inch; (b) that a liquid (and therefore
the water) transmits pressure equally in all directions (upwards as well
as downwards and sideways); and (c) that pressure upon a body in any
direction, if not counteracted by an opposite pressure, produces motion.
Hence, when the rise of the piston of the pump removes the pressure upon
the water within the cylinder, tending to produce a vacuum there, this
water is pushed up by the pressure of the air upon the water outside the
cylinder, and follows the rising piston, until the column of water
inside the cylinder exerts a pressure equal to that of the atmosphere
upon an equal area. So much for the computation; does it correspond with
the fact? It is found that at the sea level water can be pumped to the
height of 33 ft; and that such a column of water has a pressure of 15
lb. to the square inch. We may show further that, at the sea level,
spirits of wine may be pumped higher according to its less specific
gravity; and that if we attempt to pump water at successive altitudes
above the sea level, we can only raise it to less and less heights,
corresponding with the lessened atmospheric pressure at those altitudes,
where the column of air producing the pressure is shorter. Finally, if
we try to work a pump, having first produced a vacuum over the water
outside the cylinder, we shall find that the water inside will not rise
at all; the piston can be raised, but the water does not follow it. The
verification thus shows that the computed effect corresponds with the
phenomenon to be explained; that the result does not depend upon the
nature of water only, but is true (allowing for differences of specific
gravity) of other liquids; that if the pressure of the outside air is
diminished, the height of pumping is so too (canon of Variations); and
that if that pressure is entirely removed, pumping becomes impossible
(canon of Difference).

Any text-book of Astronomy or Physics furnishes numerous illustrations
of the deductive method. Take, for example, the first chapter of
Deschanel's _Optics_, where are given three methods of determining the
velocity of Light. This was first deduced from observation of Jupiter's
satellites. The one nearest the planet passes behind it, or into its
shadow, and is eclipsed, at intervals of about 42-1/2 hours. But it can
be shown that, when Jupiter and the Earth are nearest together on the
same side of the Sun, an eclipse of this satellite is visible from the
earth 16 min. 26.6 sec. earlier than when Jupiter and the earth are
furthest apart on opposite sides of the Sun: 16 min. 26.6 sec, then, is
the time in which light traverses the diameter of the Earth's orbit.
Therefore, supposing the Earth's distance from the Sun to be 92 millions
of miles, light travels about 186,000 miles a second. Another deduction,
agreeing with this, starts from the fact of aberration, or the
displacement of the apparent from the actual position of the stars in
the direction of the earth's motion. Aberration depends partly on the
velocity of light, partly on the velocity of the Earth; and the latter
being known, the former can be computed. Now, these two deductive
arguments, verifying each other, have also been verified experimentally.
Foucault's experiment to measure the velocity of light is too elaborate
to be described here: a full account of it will be found in the treatise
above cited, § 687.

When the phenomena to be explained are of such a character, so vast in
extent, power or duration, that it is impossible, in the actual
circumstances of the case, to frame experiments in order to verify a
deductive explanation, it may still be possible to reproduce a similar
phenomenon upon a smaller scale. Thus Monge's explanation of mirage by
the great heat of the desert sand, which makes the lowest stratum of air
less dense than those above it, so that rays of light from distant
objects are refracted in descending, until they are actually turned
upwards again to the eye of the beholders, giving him inverted images of
the objects as if they were reflected in water, is manifestly incapable
of being verified by experiment in the natural conditions of the
phenomenon. But by heating the bottom of "a sheet-iron box, with its
ends cut away," the rarefied air at the bottom of the box may sometimes
be made to yield reflections; and this shows at least that the supposed
cause is a possible one (Deschanel, _Optics,_ § 726). Similarly as to
the vastest of all phenomena, the evolution of the stellar system, and
of the solar system as part of it, from an immense cloudlike volume of
matter: H. Spencer, in his Essay on _The Nebular Hypothesis_, says,
amidst a great array of deductive arguments from mechanical principles,
that "this _a priori_ reasoning harmonises with the results of
experiment. Dr. Plateau has shown that when a mass of fluid is, as far
as may be, protected from the action of external forces, it will, if
made to rotate with adequate velocity, form detached rings; and that
these rings will break up into spheroids, which turn on their axes in
the same direction with the central mass." The theory of the evolution
of species of plants and animals by Natural Selection, again, though, of
course, it cannot be verified by direct experiment (since experiment
implies artificial arrangement), and the process is too slow for
observation, is, nevertheless, to some extent confirmed by the practice
of gardeners and breeders of animals: since, by taking advantage of
accidental variations of form and colour in the plants or animals under
their care, and relying on the inheritability of these variations they
obtain extensive modifications of the original stocks, and adapt them to
the various purposes for which flowers and cereals, poultry, dogs and
cattle are domesticated. This shows, at least, that living forms are
plastic, and extensively modifiable in a comparatively short time.

§ 4. Suppose, however, that, in verifying a deductive argument, the
effect as computed from the laws of the causes assigned, does not
correspond with the facts observed: there must then be an error
somewhere. If the fact has been accurately observed, the error must lie
either in the process of deduction and computation, or else in the
premises. As to the process of deduction, it may be very simple and
easily revised, as in the above explanation of the common pump; or it
may be very involved and comprise long trains of mathematical
calculation. If, however, on re-examining the computations, we find them
correct, it remains to look for some mistake in the premises.

(1) We may not have accurately ascertained the laws, or the modes of
operation, or the amounts of the forces present. Thus, the rate at which
bodies fall was formerly believed to vary in proportion to their
relative weights; and any estimate based upon this belief cannot agree
with the facts. Again, the corpuscular theory of light, namely, that
the physical cause of light is a stream of fine particles projected in
straight lines from the luminous object, though it seemed adequate to
the explanation of many optical phenomena, could not be made to agree
with the facts of interference and double refraction.

(2) The circumstances in which the agents are combined may not have been
correctly conceived. When Newton began to inquire whether the attraction
of the earth determined the orbit of the moon, he was at first
disappointed. "According to Newton's calculations, made at this time,"
says Whewell, "the moon, by her motion in her orbit, was deflected from
the tangent every minute through a space of thirteen feet. But by
noticing the space which bodies would fall in one minute at the earth's
surface, and supposing this to be diminished in the ratio of the inverse
square, it appeared that gravity would, at the moon's orbit, draw a body
through more than fifteen feet." In view of this discrepancy he gave up
the inquiry for sixteen years, until in 1682, having obtained better
data, he successfully renewed it. "He had been mistaken in the magnitude
of the earth, and consequently in the distance of the moon, which is
determined by measurements of which the earth's radius is the base." It
was not, therefore, a mistake as to the law or as to the nature of the
forces concerned (namely, the law of the inverse square and the identity
of celestial with terrestrial gravity), but as to the circumstances in
which the agents (earth and moon) were combined, that prevented his
calculations being verified. (_Hist. Ind. Sc._: VII. ii. 3.)

(3) One or more of the agents affecting the result may have been
overlooked and omitted from the estimate. Thus, an attempt to explain
the tides by taking account only of the earth and the moon, will not
entirely agree with the facts, since the sun also influences the tides.
This illustration, however, shows that when the conclusion of a
deductive explanation does not entirely agree with the facts, it is not
always to be inferred that the reasoning is, properly speaking, wrong;
it may be right as far as it goes, and merely inadequate. Hence (a) in
such cases an opportunity occurs of applying the Method of Residues, by
discovering the agent that must be allowed for in order to complete the
explanation. And (b) the investigation of a phenomenon is often
designedly begun upon an imperfect basis for the sake of simplicity; the
result being regarded as a first approximation, to be afterwards
corrected by including, one by one, the remaining agents or
circumstances affecting the phenomenon, until the theory is complete;
that is, until its agreement with the facts is satisfactory.

(4) We may have included among the data of our reasonings agents or
circumstances that do not exist or do not affect the phenomenon in
question. In the early days of science purely fanciful powers were much
relied upon: such as the solid spheres that carried the planets and
stars; the influence of the planets upon human destiny; the tendency of
everything to seek "its own place," so that fire rises to heaven, and
solids fall to the earth; the "plastic virtue" of the soil, which was
once thought to have produced fossils. When, however, such conceptions
hindered the progress of explanation, it was not so much by vitiating
the deductive method as by putting men off from exact inquiries. More to
our present purpose were the supposed cataclysms, or extraordinary
convulsions of the earth, a belief in which long hindered the progress
of Geology. Again, in Biology, Psychology, and Sociology many
explanations have depended upon the doctrine that any improvement of
structure or faculty acquired by an individual may be inherited by his
descendants: as that, if an animal learns to climb trees, his offspring
have a greater aptitude for that mode of life; that if a man tries to be
good, his children find it easier to be virtuous; that if the
inhabitants of a district carry on cloth-work, it becomes easier for
each successive generation to acquire dexterity in that art. But now the
inheritability of powers acquired by the individual through his own
efforts, is disputed; and, if the denial be made good, all such
explanations as the above must be revised.

If, then, the premises of a deductive argument be vitiated in any of
these four ways, its conclusion will fail to agree with the results of
observation and experiment, unless, of course, one kind of error happen
to be cancelled by another that is 'equal and opposite.' We now come to
a variation of the method of combining Induction with Deduction, so
important as to require separate treatment.

§ 5. The Inverse or Historical Method has of late years become
remarkably fruitful. When the forces determining a phenomenon are too
numerous, or too indefinite, to be combined in a direct deduction, we
may begin by collecting an empirical law of the phenomenon (as that 'the
democracies of City-States are arbitrary and fickle'), and then
endeavour to show by deductions from "the nature of the case," that is,
from a consideration of the circumstances and forces known to be
operative (of which, in the above instance, the most important is
sympathetic contagion), that such a law was to be expected. Deduction is
thus called in to verify a previous induction; whereas in the 'Physical
Method' a deduction was verified by comparing it with an induction or an
experiment; hence the method now to be discussed has been named the
Inverse Deductive Method.

But although it is true that, in such inquiries as we are now dealing
with, induction generally takes the lead; yet I cannot think that the
mere order in which the two logical processes occur is the essential
distinction between the two ways of combining them. For, in the first
place, in investigations of any complexity both induction and deduction
recur again and again in whatever order may be most convenient; and, in
the second place, the so-called 'inverse order' is sometimes resorted to
in Astronomy and Physics. For example, Kepler's Laws were first
collected empirically from observations of the planetary motions, and
afterwards deduced by Newton from the Law of Gravitation; this, then,
was the Inverse Method; but the result is something very different from
any that can be obtained by the Historical Method. The essential
difference between the Physical and Historical Methods is that, in the
former, whether Direct or Inverse, the deductive process, when complete,
amounts to exact demonstration; whereas, in the latter, the deductions
may consist of qualitative reasonings, and the results are indefinite.
They establish--(1) a merely probable connection between the phenomena
according to an empirical law (say, between City-democracy and fickle
politics); (2) connect this with other historical or social
generalisations, by showing that they all alike flow from the same
causes, namely, from the nature of races of men under certain social and
geographical conditions; and (3) explain why such empirical laws may
fail, according to the differences that prevail among races of men and
among the conditions under which they live. Thus, seeing how rapidly
excitement is propagated by the chatter, grimacing, and gesticulation of
townsmen, it is probable enough that the democracy of a City-state
should be fickle (and arbitrary, because irresponsible). A similar
phenomenon of panic, sympathetic hope and despair, is exhibited by every
stock-exchange, and is not peculiar to political life. And when
political opinion is not manufactured solely in the reverberating
furnace of a city, fickleness ceases to characterise democracy; and, in
fact, is not found in Switzerland, or the United States, nor in France
so far as politics, depend upon the peasantry.

This is called the Historical Method, then, because it is especially
useful in explaining the movements of history, and in verifying the
generalisations of political and social science. We must not, however,
suppose that its use is confined to such studies. Only a ridiculous
pedantry would allot to each subject its own method and forbid the use
of any other; as if it were not our capital object to establish truth by
any means. Wherever the forces determining a phenomenon are too numerous
or too indefinite to be combined in a deductive demonstration, there the
Historical Method is likely to be useful; and this seems often to be the
case in Geology and Biology, as well as in the Science of History, or
Sociology, and its various subsidiary studies.

Consider upon what causes historical events depend: the customs,
character, and opinions of all the people concerned; the organisation of
their government, and the character of their religious institutions; the
development of industry among them, of the military art, of fine art,
literature and science; their relations, commercial, political and
social, with other nations; the physical conditions of climate and
geographical position amidst which they live. Hardly an event of
importance occurs in any nation that is not, directly or indirectly,
influenced by every one of these circumstances, and that does not react
upon them. Now, from the nature of the Canons of direct Induction, a
satisfactory employment of them in such a complex and tangled situation
as history presents, is rarely possible; for they all require the actual
or virtual isolation of the phenomenon under investigation. They also
require the greatest attainable immediacy of connection between cause
and effect; whereas the causes of social events may accumulate during
hundreds of years. In collecting empirical laws from history, therefore,
only very rough inductions can be hoped for, and we may have to be
content with simple enumeration. Hence the importance of supporting such
laws by deduction from the nature of the case, however faint a
probability of the asserted connection is thereby raised; and this even
if each law is valued merely for its own sake. Still more, if anything
worth the name of Historical Science is to be constructed, must a mere
collection of such empiricisms fail to content us; and the only way to
give them a scientific character is to show deductively their common
dependence upon various combinations of the same causes. Yet even those
who profess to employ the Historical Method often omit the deductive
half of it; and of course 'practical politicians' boast of their entire
contentment with what they call 'the facts.'

Sometimes, however, politicians, venturing upon deductive reasoning,
have fallen into the opposite error of omitting to test their results by
any comparison with the facts: arguing from certain 'Rights of Man,' or
'Interests of Classes,' or 'Laws of Supply and Demand,' that this or
that event will happen, or ought to happen, without troubling themselves
to observe whether it does happen or ever has happened. This method of
Deduction without any empirical verification, is called by Mill the
Geometrical; and, plainly, it can be trustworthy only where there is no
actual conflict of forces to be considered. In pure mathematical
reasoning about space, time, and number, provided the premises and the
reasoning be correct, verification by a comparison with the facts may be
needless, because there is no possibility of counteraction. But when we
deal with actual causes, no computation of their effects can be relied
upon without comparing our conclusions with the facts: not even in
Astronomy and Physics, least of all in Politics.

Burke, then, has well said that "without the guide and light of sound,
well-understood principles all our reasoning in politics, as in
everything else, would be only a confused jumble of particular facts and
details without the means of drawing any sort of theoretical or
practical conclusion"; but that, on the other hand, the statesman, who
does not take account of circumstances, infinite and infinitely
combined, "is not erroneous, but stark mad--he is metaphysically mad"
(_On the Petition of the Unitarians_). There is, or ought to be, no
logical difference between the evidence required by a statesman and that
appealed to by a philosopher; and since, as we have seen, the
combination of principles with circumstances cannot, in solving problems
of social science, be made with the demonstrative precision that belongs
to astronomical and physical investigations, there remains the
Historical Method as above described.

Examples of the empirical laws from which this method begins abound in
histories, newspapers, and political discussions, and are of all shades
of truth or half-truth: as that 'History consists in the biographies of
great men'; in other words, that the movements of society are due to
exceptional personal powers, not to general causes; That at certain
epochs great men occur in groups; That every Fine Art passes through
periods of development, culmination and decline; That Democracies tend
to change into Despotisms; That the possession of power, whether by
classes or despots, corrupts the possessor: That 'the governments most
distinguished for sustained vigour and abilities have generally been
aristocracies'; That 'revolutions always begin in hunger'; That
civilisation is inimical to individuality; That the civilisation of the
country proceeds from the town; That 'the movement of progressive
societies has hitherto been a movement from _Status_ to _Contract_
(i.e., from a condition in which the individual's rights and duties
depend on his caste, or position in his family as slave, child, or
patriarch, to a condition in which his rights and duties are largely
determined by the voluntary agreements he enters into)'; and this last
is treated by H. Spencer as one aspect of the law first stated by Comte,
that the progress of societies is from the military to the industrial

The deductive process we may illustrate by Spencer's explanation of the
co-existence in the military state of those specific characters, the
inductive proof of which furnished an illustration of the method of
Agreement (ch. xvi. § 1). The type of the military State involves the
growth of the warrior class, and the treatment of labourers as existing
solely to support the warriors; the complete subordination of all
individuals to the will of the despotic soldier-king, their property,
liberty and life being at the service of the State; the regimentation of
society, not only for military, but also for civil purposes; the
suppression of all private associations, etc. Now all these
characteristics arise from their utility for the purpose of war, a
utility amounting to necessity if war is the State's chief purpose. For
every purpose is best served when the whole available force co-operates
toward it: other things equal, the bigger the army the better; and to
increase it, men must be taken from industry, until only just enough
remain to feed and equip the soldiers. As this arrangement is not to
everybody's taste, there must be despotic control; and this control is
most effective through regimentation by grades of command. Private
associations, of course, cannot live openly in such a State, because
they may have wills of their own and are convenient for conspiracy. Thus
the induction of characteristics is verified by a deduction of them from
the nature of the case.

§ 6. The greater indefiniteness of the Historical compared with the
Physical Method, both in its inductions and in its deductions, makes it
even more difficult to work with. It wants much sagacity and more
impartiality; for the demon of Party is too much with us. Our first care
should be to make the empirical law as nearly true as possible,
collecting as many as we can of the facts which the law is supposed to
generalise, and examining them according to the canons of Induction,
with due allowance for the imperfect applicability of those canons to
such complex, unwieldy, and indefinite instances. In the examples of
such laws given above, it is clear that in some cases no pains have been
taken to examine the facts. What is the inductive evidence that
Democracies change into Despotisms; that revolutions always begin in
hunger; or that civilisation is inimical to individuality? Even Mill's
often quoted saying, "that the governments remarkable in history for
sustained vigour and ability have generally been aristocracies," is
oddly over-stated. For if you turn to the passage (_Rep. Gov._ chap.
vi.), the next sentence tells you that such governments have always been
aristocracies of public functionaries; and the next sentence but one
restricts, apparently, the list of such remarkable governments to
two--Rome and Venice. Whence, then, comes the word "generally" into
Mill's law?

As to deducing our empirical law from a consideration of the nature of
the case, it is obvious that we ought--(a) to take account of all the
important conditions; (b) to allow weight to them severally in
proportion to their importance; and (c) not to include in our estimates
any condition which we cannot show to be probably present and operative.
Thus the Great-Man-Theory of history must surely be admitted to assign a
real condition of national success. The great man organises, directs,
inspires: is that nothing? On the other hand, to recognise no other
condition of national success is the manifest frenzy of a mind in the
mythopœic age. We must allow the great man his due weight, and then
inquire into the general conditions that (a) bring him to birth in one
nation rather than another, and (b) give him his opportunity.

Mill's explanation of the success of the aristocratic governments of
Rome and Venice is, that they were, in fact, bureaucracies; that is to
say, their members were trained in the science and art of
administration and command. Here, again, we have, no doubt, a real
condition; but is it the only one? The popular mind, which little
relishes the scaling down of Mill's original law to those two remote
cases, is persuaded that an aristocracy is the depository of hereditary
virtue, especially with reference to government, and would at once
ascribe to this circumstance the greater part of the success of any
aristocratic constitution. Now, if the effects of training are
inherited, they must, in an hereditary aristocracy, increase the energy
of the cause assigned by Mill; but, if not, such heredity is a condition
"not present or not operative." Still, if families are ennobled for
their extraordinary natural powers of administration or command (as
sometimes happens), it is agreed on all hands that innate qualities are
inheritable; at least, if care be taken to intermarry with families
similarly distinguished, and if by natural or artificial selection all
the failures among the offspring be eliminated. The Spartans had some
crude notion of both these precautions; and if such measures had been
widely adopted, we might deduce from the doctrine of heredity a
probability in favour of Mill's original proposition, and thereby verify
it in its generality, if it could be collected from the facts.

The Historical Method may be further illustrated by the course adopted
in that branch of Social Science which has been found susceptible of the
most extensive independent development, namely, Economics. First, by way
of contrast, I should say that the abstract, or theoretical treatment of
Economics follows the Physical Method; because, as Mill explains,
although the phenomena of industry are no doubt influenced, like other
social affairs, by all the other circumstances of Society, government,
religion, war, art, etc.; yet, where industry is most developed, as in
England and the United States, certain special conditions affecting it
are so much the most important that, for the purpose at least of a
first outline of the science, they may conveniently be considered as the
only ones. These conditions are: (1) the general disposition of men to
obtain wealth with as little trouble as possible, and (2) to spend it so
as to obtain the greatest satisfaction of their various desires; (3) the
facts that determine population; and (4) the tendency of extractive
industry, when pushed beyond a certain limit without any improvement in
the industrial arts, to yield "diminishing returns." From these premises
it is easy to infer the general laws of prices, of wages and interest
(which are the prices of labour and of the use of capital), and of rent;
and it remains to verify these laws by comparing them with the facts in
each case; and (if they fail to agree with the facts) to amend them,
according to the Method of Residues, by taking account of those
influential conditions which were omitted from the first draft of the

Whilst, however, this is usually the procedure of those inquirers who
have done most to give Economics its scientific character, to insist
that no other plan shall be adopted would be sheer pedantry; and Dr.
Keynes has shown, in his _Scope and Method of Political Economy_, that
Mill has himself sometimes solved economic problems by the Historical
Method. With an analysis of his treatment of Peasant Proprietorship
(_Political Economy_, B. II., cc. 7 and 8) we may close this section.
Mill first shows inductively, by collecting evidence from Switzerland,
Germany, Norway, Belgium, and France (countries differing in race,
government, climate and situation), that peasant proprietors are
superhumanly industrious; intelligent cultivators, and generally
intelligent men; prudent, temperate, and independent, and that they
exercise self-control in avoiding improvident marriages. This group of
empirical generalisations as to the character of peasant proprietors he
then deduces from the nature of the case: their industry, he says, is a
natural consequence of the fact that, however much they produce, it is
all their own; they cultivate intelligently, because for generations
they have given their whole mind to it; they are generally intelligent
men, because the variety of work involved in small farming, requiring
foresight and calculation, necessarily promotes intelligence; they are
prudent, because they have something to save, and by saving can improve
their station and perhaps buy more land; they are temperate, because
intemperance is incompatible with industry and prudence; they are
independent, because secure of the necessaries of life, and from having
property to fall back upon; and they avoid improvidence in marriage,
because the extent and fertility of their fields is always plainly
before them, and therefore how many children they can maintain is easily
calculated. The worst of them is that they work too hard and deny
themselves too much: but, over the greater part of the world, other
peasantry work too hard; though they can scarcely be said to deny
themselves too much; since all their labour for others brings them no
surplus to squander upon self-indulgence.

§ 7. The foregoing account of the Historical Method is based upon Mill's
discussions in B. VI. of his _Logic_, especially cc. 6 to 11. Mill
ascribes to Comte the first clear statement of the method; and it is
highly scientific, and important in generalising the connections of
historical events. But perhaps the expression, 'Historical Method,' is
more frequently applied to the Comparative Method, as used in
investigating the history of institutions or the true sense of legends.

(1) Suppose we are trying to explain the institution of capital
punishment as it now exists in England. (1) We must try to trace the
history of it back to the earliest times; for _social custom and
tradition is one line of causation_. At present the punishment of death
is legally incident only to murder and high treason. But early in the
last century malefactors were hung for forgery, sheep-stealing, arson
and a long list of other offences down to pocket-picking: earlier still
the list included witchcraft and heresy. At present hanging is the only
mode of putting a malefactor to death; but formerly the ways of putting
to death included also burning, boiling, pressing, beheading, and mixed
modes. Before the Restoration, however, the offences punishable with
death were far fewer than they afterwards became; and until the twelfth
century, the penalty of death might be avoided by paying compensation,
the wer-geld.

(2) Every change in the history of an institution must be explained by
pointing to _the special causes_ in operation during the time when the
change was in progress. Thus the restriction of the death penalty, in
the nineteenth century, to so few offences was due partly to the growth
of humane feelings, partly to the belief that the infliction, or threat,
of the extreme penalty had failed to enforce the law and had demoralised
the administration of Justice. The continual extension of the death
penalty throughout the eighteenth century may be attributed to a belief
that it was the most effectual means of deterring evil-doers when the
means of detecting and apprehending criminals were feeble and
ill-organised. The various old brutal ways of execution were adopted
sometimes to strike terror, sometimes for vengeance, sometimes from
horror of the crime, or even from 'conscientious scruples';--which last
were the excuse for preferring the burning of heretics to any sort of

(3) The causes of any change in the history of an institution in any
country may not be directly discoverable: they must then be investigated
by the Comparative Method. Again, the recorded history of a nation, and
of all its institutions, followed backwards, comes at last to an end:
then the antecedent history must also be supplied by the Comparative
Method; whose special use is to indicate the existence of facts for
which there is no direct evidence.

This method rests upon the principle that where the causes are alike the
effects will be alike, and that similar effects are traceable to similar
causes. Every department of study--Astronomy, Chemistry, Zoology,
Sociology--is determined by the fact that the phenomena it investigates
have certain common characteristics; and we are apt to infer that any
process observed in some of these phenomena, if depending on those
common characteristics, will be found in others. For example, the
decomposition, or radio-activity, of certain elements prepares one to
believe that all elements may exhibit it. Where the properties of an
object are known to be closely interdependent, as in the organisation of
plants, animals and societies, we are especially justified in inferring
from one case to another. The whole animal Kingdom has certain common
characters--the metabolic process, dependence upon oxygen, upon
vegetable food (ultimately), heredity, etc., and, upon this ground, any
process (say, the differentiation of species by Natural Selection) that
has been established for some kinds of animal is readily extended to
others. If instead of the whole animal Kingdom we take some district of
it--Class, Order, Family--our confidence in such inferences increases;
because the common characters are more numerous and the conditions of
life are more alike; or, in other words, the common causes are more
numerous that initiate and control the development of nearly allied
animals. For such reasons a few fragmentary remains of an extinct animal
enable the palæontologist to reconstruct with some probability an
outline of its appearance, organisation, food, habitat and habits.

Applied to History, the Comparative Method rests upon an assumption
(which the known facts of (say) 6,000 years amply justify) that human
nature, after attaining a recognisable type as _homo sapiens_, is
approximately uniform in all countries and in all ages, though more
especially where states of culture are similar. Men living in society
are actuated by similar motives and reasons in similar ways; they are
all dependent upon the supply of food and therefore on the sun and the
seasons and the weather and upon means of making fire, and so on.
Accordingly, they entertain similar beliefs, and develop similar
institutions through similar series of changes. Hence, if in one nation
some institution has been altered for reasons that we cannot directly
discover, whereas we know the reasons why a similar change was adopted
elsewhere, we may conjecture with more or less probability, after making
allowance for differences in other circumstances, that the motives or
causes in the former case were similar to those in the latter, or in any
cases that are better known. Or, again, if in one nation we cannot trace
an institution beyond a certain point, but can show that elsewhere a
similar institution has had such or such an antecedent history, we may
venture to reconstruct with more or less probability the earlier history
of that institution in the nation we are studying.

Amongst the English and Saxon tribes that settled in Britain, death was
the penalty for murder, and the criminal was delivered to the
next-of-kin of his victim for execution; he might, however, compound for
his crime by paying a certain compensation. Studying the history of
other tribes in various parts of the world, we are able, with much
probability, to reconstruct the antecedents of this death-penalty in our
own prehistoric ages, and to trace it to the blood-feud; that is, to a
tribal condition in which the next-of-kin of a murdered man was socially
and religiously bound to avenge him by slaying the murderer or one of
his kindred. This duty of revenge is sometimes (and perhaps was at first
everywhere) regarded as necessary to appease the ghost of the victim;
sometimes as necessary to compensate the surviving members of his
family. In the latter case, it is open to them to accept compensation in
money or cattle, etc. Whether the kin will be ready to accept
compensation must depend upon the value they set upon wealth in
comparison with revenge; but for the sake of order and tribal strength,
it is the interest of the tribe, or its elders, or chieftain, to
encourage or even to enforce such acceptance. It is also their interest
to take the questions--whether a crime has been committed, by whom, and
what compensation is due--out of the hands of the injured party, and to
submit them to some sort of court or judicial authority. At first,
following ancient custom as much as possible, the act of requital, or
the choice of accepting compensation, is left to the next-of-kin; but
with the growth of central power these things are entrusted to ministers
of the Government. Then revenge has undergone its full transformation
into punishment. Very likely the wrong itself will come to be treated as
having been done not to the kindred of the murdered man, but to the
State or the King, as in fact a "breach of the King's peace." This
happened in our own history.

(4) The Comparative Method assumes that human nature is approximately
the same in different countries and ages; but, of course,
'approximately' is an important word. Although there is often a striking
and significant resemblance between the beliefs and institutions of
widely separated peoples, we expect to draw the most instructive
parallels between those who are nearly related by descent, or
neighbourhood, or culture. To shed light upon our own manners, we turn
first to other Teutons, then to Slavonians and Kelts, or other Aryans,
and so on; and we prefer evidence from Europe to examples from Africa.

(5) As to national culture, that it exhibits certain 'stages' of
development is popularly recognised in the distinction drawn between
savages, barbarians and civilised folk. But the idea remains rather
vague; and there is not space here to define it. I refer, therefore, to
the classifications of stages of culture given by A. Sutherland,
(_Origin and Growth of Moral Instinct_, Vol. I, p. 103), and L.T.
Hobhouse (_Morals in Evolution_, c. 2). That in any 'state of Society,'
its factors--religion, government, science, etc.--are mutually
dependent, was a leading doctrine with Comte, adopted by Mill. There
must be some truth in it; but in some cases we do not understand social
influences sufficiently well to trace the connection of factors; and
whilst preferring to look for historical parallels between nations of
similar culture, we find many cases in which barbarous or savage customs
linger in a civilised country.

(6) It was another favourite doctrine with Comte, also adopted by
Mill--that the general state of culture is chiefly determined by the
prevailing intellectual condition of a people, especially by the
accepted ground of explanation--whether the will of supernatural beings,
or occult powers, or physical antecedents: the "law of three stages,"
Fetichism, Metaphysics, Positivism. And this also is, at least, so far
true, that it is useless to try to interpret the manners and
institutions of any nation until we know its predominant beliefs. Magic
and animism are beliefs everywhere held by mankind in early stages of
culture, and they influence every action of life. But that is not all:
these beliefs retain their hold upon great multitudes of civilised men
and affect the thoughts of the most enlightened. Whilst the saying 'that
human nature is the same in all ages' seems to make no allowance for the
fact that, in some nations, a considerable number of individuals has
attained to powers of deliberation, self-control, and exact reasoning,
far above the barbarous level, it is yet so far true that, even in
civilised countries, masses of people, were it not for the example and
instruction of those individuals, would fall back upon magic and animism
and the manners that go with those beliefs. The different degrees of
enlightenment enjoyed by different classes of the population often
enable the less educated to preserve a barbarous custom amidst many
civilised characteristics of the national life.

§ 8. Historical reasoning must start from, or be verified by,
observations. If we are writing the history of ourselves: if of another
time or country, we can observe some of the present conditions of the
country, its inhabitants, language, manners, institutions, which are
effects of the past and must be traceable to it; we may also be able to
observe ancient buildings or their ruins, funerary remains, coins,
dating from the very times we are to treat of. Our own observations, of
course, are by no means free from error.

But even in treating of our own age and country, most of our information
must be derived from the testimony of others, who may have made mistakes
of observation and further mistakes in reporting their observations, or
may have intentionally falsified them. Testimony is of two kinds: Oral;
and Written, inscribed or printed. In investigating the events of a
remote age, nearly all our direct evidence must be some sort of

(1) Oral testimony depends upon the character of the witness; and the
best witness is not perfectly trustworthy; for he may not have observed
accurately, or he may not have reported correctly; especially if some
time elapsed between the event and his account of it; for no man's
memory is perfect. Since witnesses vary widely in capacity and
integrity, we must ask concerning any one of them--was he a good judge
of what he saw, and of what was really important in the event? Had he
good opportunities of knowing the circumstances? Had he any interest in
the event--personal, or partisan, or patriotic? Such interests would
colour his report; and so would the love of telling a dramatic story, if
that was a weakness of his. Nay, a love of truth might lead him to
modify the report of what he remembered if--as he remembered it--the
matter seemed not quite credible. We must also bear in mind that, for
want of training, precision in speaking the truth is not understood or
appreciated by many honest people even now, still less in unscientific

Oral tradition is formed by passing a report from one to another,
generation by generation; and it is generally true that such a tradition
loses credit at every step, because every narrator has some weakness.
However, the value of tradition depends upon the motives people have to
report correctly, and on the form of the communication, and on whether
monuments survive in connection with the story. Amongst the things best
remembered are religious and magic formulæ, heroic poems, lists of
ancestors, popular legends about deeply impressive events, such as
migrations, conquests, famines, plagues. We are apt now to underrate the
value of tradition, because the use of writing has made tradition less
important, and therefore less pains are taken to preserve it. In the
middle of last century, it was usual (and then quite justifiable) to
depreciate oral tradition as nearly worthless; but the spread of
archæological and anthropological research, and the growth of the
Comparative Method, have given new significance to legends and
traditions which, merely by themselves, could not deserve the slightest

(2) As to written evidence, contemporary inscriptions--such as are found
on rocks and stones and bricks in various parts of the world, and most
abundantly in Egypt and Western Asia--are of the highest value, because
least liable to fraudulent abuse; but must be considered with reference
to the motives of those who set them forth. Manuscripts and books give
rise to many difficulties. We have to consider whether they were
originally written by some one contemporary with the events recorded: if
so they have the same value as immediate oral testimony, provided they
have not been tampered with since. But if not contemporary records, they
may have been derived from other records that were contemporary, or only
from oral tradition. In the latter case they are vitiated by the
weakness of oral tradition. In the former case, we have to ask what was
the trustworthiness of the original records, and how far do the extant
writings fairly represent those records?

Our answers to these questions will partly depend upon what we know or
can discover of the authors of the MSS. or books. Who was the author? If
a work bears some man's name, did he really write it? The evidence
bearing upon this question is usually divided into internal, external
and mixed; but perhaps no evidence is purely internal, if we define it
as that which is derived entirely from the work itself. Under the name
of internal evidence it is usual to put the language, the style,
consistency of ideas; but if we had no grounds of judgment but the book
itself, we could not possibly say whether the style was the author's:
this requires us to know his other works. Nor could we say whether the
language was that of his age, unless we knew other literature of the
same age; nor even that different passages seem to be written in the
manner of different ages, but for our knowledge of change in other
literatures. There must in every case be some external reference. Thus
we judge that a work is not by the alleged author, nor contemporary with
him, if words are used that only became current at a later date, or are
used in a sense that they only later acquired, or if later writers are
imitated, or if events are mentioned that happened later
('anachronism'). Books are sometimes forged outright, that is, are
written by one man and deliberately fathered upon another; but sometimes
books come to be ascribed to a well-known name, which were written by
some one else without fraudulent intent, dramatically or as a
rhetorical exercise.

As to external evidence, if from other sources we have some knowledge of
the facts described in a given book, and if it presents no serious
discrepancies with those facts, this is some confirmation of a claim to
contemporaneity. But the chief source of external evidence is other
literature, where we may find the book in question referred to or
quoted. Such other literature may be by another author, as when
Aristotle refers to a dialogue of Plato's, or Shakespeare quotes
Marlowe; or may be other work of the author himself, as when Aristotle
in the _Ethics_ refers to his own _Physics_, or Chaucer in _The
Canterbury Tales_ mentions as his own _The Legend of Good Women_, and in
_The Legend_ gives a list of other works of his. This kind of argument
assumes that the authorship of the work we start from is undisputed;
which is practically the case with the _Ethics_ and _The Canterbury

But, now, granting that a work is by a good author, or contemporary with
the events recorded, or healthily related to others that were
contemporary, it remains to consider whether it has been well preserved
and is likely to retain its original sense. It is, therefore, desirable
to know the history of a book or MS., and through whose hands it has
passed. Have there been opportunities of tampering with it; and have
there been motives to do so? In reprinting books, but still more in
copying MSS., there are opportunities of omitting or interpolating
passages, or of otherwise altering the sense. In fact, slight changes
are almost sure to be made even without meaning to make them, especially
in copying MSS., through the carelessness or ignorance of transcribers.
Hence the oldest MS. is reckoned the best.

If a work contains stories that are physically impossible, it shows a
defect of judgment in the author, and decreases our confidence in his
other statements; but it does not follow that these others are to be
rejected. We must try to compare them with other evidence. Even
incredible stories are significant: they show what people were capable
of believing, and, therefore, under what conditions they reasoned and
acted. One cause of the incredibility of popular stories is the fusion
of legend with myth. A legend is a traditionary story about something
that really happened: it may have been greatly distorted by stupidity,
or exaggeration, or dramatisation, or rationalisation, but may still
retain a good deal of the original fact. A myth, however, has not
necessarily any basis of fact: it may be a sort of primitive philosophy,
an hypothesis freely invented to explain some fact in nature, such as
eclipses, or to explain some social custom whose origin is forgotten,
such as the sacrificing of a ram.

All historical conclusions, then, depend on a sum of convergent and
conflicting probabilities in the nature of circumstantial evidence. The
best testimony is only highly probable, and it is always incomplete. To
complete the picture of any past age there is no resource but the
Comparative Method. We use this method without being aware of it,
whenever we make the records of the last generation intelligible to
ourselves by our own experience. Without it nothing would be
intelligible: an ancient coin or weapon would have no meaning, were we
not acquainted with the origins and uses of other coins and weapons.
Generally, the further we go back in history, the more the evidence
needs interpretation and reconstruction, and the more prominent becomes
the appeal to the Comparative Method. Our aim is to construct a history
of the world, and of the planet as part of the world, and of mankind as
part of the life of the planet, in such a way that every event shall be
consistent with, and even required by, the rest according to the
principle of Causation.



§ 1. An Hypothesis, sometimes employed instead of a known law, as a
premise in the deductive investigation of nature, is defined by Mill as
"any supposition which we make (either without actual evidence, or on
evidence avowedly insufficient) in order to endeavour to deduce from it
conclusions in accordance with facts which are known to be real; under
the idea that if the conclusions to which the hypothesis leads are known
truths, the hypothesis itself either must be, or at least is likely to
be, true." The deduction of known truths from an hypothesis is its
Verification; and when this has been accomplished in a good many cases,
and there are no manifest failures, the hypothesis is often called a
Theory; though this term is also used for the whole system of laws of a
certain class of phenomena, as when Astronomy is called the 'theory of
the heavens.' Between hypothesis and theory in the former sense no
distinct line can be drawn; for the complete proof of any speculation
may take a long time, and meanwhile the gradually accumulating evidence
produces in different minds very different degrees of satisfaction; so
that the sanguine begin to talk of 'the theory,' whilst the circumspect
continue to call it 'the hypothesis.'

An Hypothesis may be made concerning (1) an Agent, such as the ether; or
(2) a Collocation, such as the plan of our solar system--whether
geocentric or heliocentric; or (3) a Law of an agent's operation, as
that light is transmitted by a wave motion of such lengths or of such
rates of vibration.

The received explanation of light involves both an agent, the ether, as
an all-pervading elastic fluid, and also the law of its operation, as
transmitting light in waves of definite form and length, with definite
velocity. The agreement between the calculated results of this complex
hypothesis and the observed phenomena of light is the chief part of the
verification; which has now been so successfully accomplished that we
generally hear of the 'Undulatory Theory.' Sometimes a new agent only is
proposed; as the planet Neptune was at first assumed to exist in order
to account for perturbations in the movements of Uranus, influencing it
according to the already established law of gravitation. Sometimes the
agents are known, and only the law of their operation is hypothetical,
as was at first the case with the law of gravitation itself. For the
agents, namely, Earth, falling bodies on the Earth, Moon, Sun, and
planets were manifest; and the hypothesis was that their motions might
be due to their attracting one another with a force inversely
proportional to the squares of the distances between them. In the
Ptolemaic Astronomy, again, there was an hypothesis as to the
collocation of the heavenly bodies (namely, that our Earth was the
centre of the universe, and that Moon, Sun, planets and stars revolved
around her): in the early form of the system there was also an
hypothesis concerning agents upon which this arrangement depended
(namely, the crystalline spheres in which the heavenly bodies were
fixed, though these were afterwards declared to be imaginary); and an
hypothesis concerning the law of operation (namely, that circular motion
is the most perfect and eternal, and therefore proper to celestial

Hypotheses are by no means confined to the physical sciences: we all
make them freely in private life. In searching for anything, we guess
where it may be before going to look for it: the search for the North
Pole was likewise guided by hypotheses how best to get there. In
estimating the characters or explaining the conduct of acquaintances or
of public men, we frame hypotheses as to their dispositions and
principles. 'That we should not impute motives' is a peculiarly absurd
maxim, as there is no other way of understanding human life. To impute
bad motives, indeed, when good are just as probable, is to be wanting in
the scientific spirit, which views every subject in 'a dry light.' Nor
can we help 'judging others by ourselves'; for self-knowledge is the
only possible starting-point when we set out to interpret the lives of
others. But to understand the manifold combinations of which the
elements of character are susceptible, and how these are determined by
the breeding of race or family under various conditions, and again by
the circumstances of each man's life, demands an extraordinary union of
sympathetic imagination with scientific habits of thought. Such should
be the equipment of the historian, who pursues the same method of
hypothesis when he attempts to explain (say) the state of parties upon
the Exclusion Bill, or the policy of Louis XI. Problems such as the
former of these are the easier; because, amidst the compromises of a
party, personal peculiarities obliterate one another, and expose a
simpler scheme of human nature with fewer fig-leaves. Much more
hazardous hypotheses are necessary in interpreting the customs of
savages, and the feelings of all sorts of animals. Literary criticisms,
again, abound with hypotheses: e.g., as to the composition of the
Homeric poems, the order of the Platonic dialogues, the authorship of
the Cædmonic poems, or the Ossianic, or of the letters of Junius. Thus
the method of our everyday thoughts is identical with that of our most
refined speculations; and in every case we have to find whether the
hypothesis accounts for the facts.

§ 2. It follows from the definition of an hypothesis that none is of
any use that does not admit of verification (proof or disproof), by
comparing the results that may be deduced from it with facts or laws. If
so framed as to elude every attempt to test it by facts, it can never be
proved by them nor add anything to our understanding of them.

Suppose that a conjurer asserts that his table is controlled by the
spirit of your deceased relative, and makes it rap out an account of
some adventure that could not easily have been within a stranger's
knowledge. So far good. Then, trying again, the table raps out some
blunder about your family which the deceased relative could not have
committed; but the conjurer explains that 'a lying spirit' sometimes
possesses the table. This amendment of the hypothesis makes it equally
compatible with success and with failure. To pass from small things to
great, not dissimilar was the case of the Ptolemaic Astronomy: by
successive modifications, its hypothesis was made to correspond with
accumulating observations of the celestial motions so ingeniously that,
until the telescope was invented, it may be said to have been
unverifiable. Consider, again, the sociological hypothesis, that civil
order was at first founded on a Contract which remains binding upon all
mankind: this is reconcilable with the most opposite institutions. For
we have no record of such an event: and if the institutions of one State
(say the British) include ceremonies, such as the coronation oath and
oath of allegiance, which may be remnants of an original contract, they
may nevertheless be of comparatively recent origin; whereas if the
institutions of another State (say the Russian) contain nothing that
admits of similar interpretation, yet traces of the contract once
existing may long since have been obliterated. Moreover, the actual
contents of the contract not having been preserved, every adherent of
this hypothesis supplies them at his own discretion, 'according to the
dictates of Reason'; and so one derives from it the duty of passive
obedience, and another with equal cogency establishes the right of

To be verifiable, then, an hypothesis must be definite; if somewhat
vague in its first conception (which is reasonably to be expected), it
must be made definite in order to be put to the proof. But, except this
condition of verifiability, and definiteness for the sake of
verifiability, without which a proposition does not deserve the name of
an hypothesis, it seems inadvisable to lay down rules for a 'legitimate'
hypothesis. The epithet is misleading. It suggests that the Logician
makes rules for scientific inquirers; whereas his business is to
discover the principles which they, in fact, employ in what are
acknowledged to be their most successful investigations. If he did make
rules for them, and they treated him seriously, they might be
discouraged in the exercise of that liberty of hypothesising which is
the condition of all originality; whilst if they paid no attention to
him, he must suffer some loss of dignity. Again, to say that a
'legitimate hypothesis' must explain all the facts, at least in the
department for which it is invented, is decidedly discouraging. No doubt
it may be expected to do this in the long run when (if ever) it is
completely established; but this may take a long time: is it meanwhile
illegitimate? Or can this adjective be applied to Newton's corpuscular
theory of light, even though it has failed to explain all the facts?

§ 3. Given a verifiable hypothesis, however, what constitutes proof or

(1) _If a new agent be proposed, it is desirable that we should be able
directly to observe it, or at least to obtain some evidence of its
existence of a different kind from the very facts which it has been
invented to explain._ Thus, in the discovery of Neptune, after the
existence of such a planet outside the orbit of Uranus had been
conjectured (to account for the movements of the latter), the place in
the heavens which such a body should occupy at a certain time was
calculated, and there by means of the telescope it was actually seen.

Agents, however, are assumed and reasoned upon very successfully which,
by their nature, never can be objects of perception: such are the atoms
of Chemistry and the ether of Optics. But the severer methodologists
regard them with suspicion: Mill was never completely convinced about
the ether; the defining of which has been found very difficult. He was
willing, however, to make the most of the evidence that has been adduced
as indicating a certain property of it distinct from those by which it
transmits radiation, namely, mechanical inertia, whereby it has been
supposed to retard the career of the heavenly bodies, as shown
especially by the history of Encke's comet. This comet returned sooner
than it should, as calculated from the usual data; the difference was
ascribed to the influence of a resisting medium in reducing the extent
of its orbit; and such a medium may be the ether. If this conjecture
(now of less credit) should gain acceptance, the ether might be regarded
as a _vera causa_ (that is, a condition whose existence may be proved
independently of the phenomena it was intended to explain), in spite of
its being excluded by its nature from the sphere of direct perception.
However, science is not a way of perceiving things, but essentially a
way of thinking about them. It starts, indeed, from perception and
returns to it, and its thinking is controlled by the analogies of
perception. Atoms and ether are thought about as if they could be seen
or felt, not as noumena; and if still successful in connecting and
explaining perceptions, and free from contradiction, they will stand as
hypotheses on that ground.

On the other hand, a great many agents, once assumed in order to explain
phenomena, have since been explained away. Of course, a _fact_ can never
be 'explained away': the phrase is properly applicable to the fate of
erroneous hypotheses, when, not only are they disproved, but others are
established in their places. Of the Aristotelian spheres, which were
supposed to support and translate sun, moon and planets, no trace has
ever been found: they would have been very much in the way of the
comets. Phlogiston, again, an agent much in favour with the earlier
Chemists, was found, Whewell tells us, when their theories were tested
by exact weighing, to be not merely non-existent but a minus quantity;
that is to say, it required the assumption of its absolute lightness "so
that it diminished the weight of the compounds into which it entered."
These agents, then, the spheres and phlogiston, have been explained
away, and instead of them we have the laws of motion and oxygen.

(2) _Whether the hypothetical agent be perceptible or not, it cannot be
established as a cause, nor can a supposed law of such an agent be
accepted as sufficient to the given inquiry, unless it is adequate to
account for the effects which it is called upon to explain, at least so
far as it pretends to explain them._ The general truth of this is
sufficiently obvious, since to explain the facts is the purpose of an
hypothesis; and we have seen that Newton gave up his hypothesis that the
moon was a falling body, as long as he was unable to show that the
amount of its deflection from a tangent (or fall) in a given time, was
exactly what it should be, if the Moon was controlled by the same force
as falling bodies on the Earth.

It is important to observe the limitations to this canon. In the first
place, it says that, unless adequate to explain the facts in question,
an hypothesis cannot be '_established_'; but, for all that, such an
hypothesis may be a very promising one, not to be hastily rejected,
since it may take a very long time fully to verify an hypothesis. Some
facts may not be obtainable that are necessary to show the connection of
others: as, for example, the hypothesis that all species of animals
have arisen from earlier ones by some process of gradual change, can be
only imperfectly verified by collecting the fossil remains of extinct
species, because immense depths and expanses of fossiliferous strata
have been destroyed. Or, again, the general state of culture may be such
as to prevent men from tracing the consequences of an hypothesis; for
which reason, apparently, the doctrine that the Sun is the centre of our
planetary system remained a discredited hypothesis for 2000 years. This
should instruct us not to regard an hypothesis as necessarily erroneous
or illegitimate merely because we cannot yet see how it works out: but
neither can we in such a case regard it as established, unless we take
somebody's word for it.

Secondly, the canon says that an hypothesis is not established, unless
it accounts for the phenomena _so far as it professes to_. But it
implies a complete misunderstanding to assail a doctrine for not
explaining what lies beyond its scope. Thus, it is no objection to a
theory of the origin of species, that it does not explain the origin of
life: it does not profess to. For the same reason, it is no objection to
the theory of Natural Selection, that it does not account for the
variations which selection presupposes. But such objections might be
perfectly fair against a general doctrine of Evolution.

An interesting case in Wallace's _Darwinism_ (chap. x.) will illustrate
the importance of attending to the exact conditions of an hypothesis. He
says that in those groups of "birds that need protection from enemies,"
"when the male is brightly coloured and the female sits exposed on the
nest, she is always less brilliant and generally of quite sober and
protective hues"; and his hypothesis is, that these sober hues have been
acquired or preserved by Natural Selection, because it is important to
the family that the sitting bird should be inconspicuous. Now to this it
might be objected that in some birds both sexes are brilliant or
conspicuous; but the answer is that the female of such species _does not
sit exposed on the nest_; for the nests are either domed over, or made
in a hole; so that the sitting bird does not need protective colouring.
If it be objected, again, that some sober-coloured birds build domed
nests, it may be replied that the proposition 'All conspicuously
coloured birds are concealed in the nest,' is not to be converted simply
into 'All birds that sit concealed in the nest are conspicuously
coloured.' In the cases alleged the domed nests are a protection against
the weather, and the sober colouring is a general protection to the
bird, which inhabits an open country. It may be urged, however, that
jays, crows, and magpies are conspicuous birds, and yet build open
nests: but these are aggressive birds, _not needing protection from
enemies_. Finally, there are cases, it must be confessed, in which the
female is more brilliant than the male, and which yet have open nests.
Yes: but _then the male sits upon the eggs_, and the female is stronger
and more pugnacious!

Thus every objection is shown to imply some inattention to the
conditions of the hypothesis; and in each case it may be said, _exceptio
probat regulam_--the exception _tests_ the rule. (Of course, the usual
translation "proves the rule," in the restricted modern sense of
"prove," is absurd.) That is to say, it appears on examination: (1) that
the alleged exception is not really one, and (2) that it stands in such
relation to the rule as to confirm it. For to all the above objections
it is replied that, granting the phenomenon in question (special
protective colouring for the female) to be absent, the alleged cause
(need of protection) is also absent; so that the proof is, by means of
the objections, extended, from being one by the method of Agreement,
into one by the Double Method.

Thirdly, an hypothesis originally intended to account for the whole of a
phenomenon and failing to do so, though it cannot be established in that
sense, may nevertheless contain an essential part of the explanation.
The Neptunian Hypothesis in Geology, was an attempt to explain the
formation of the Earth's outer crust, as having been deposited from an
universal ocean of mud. In the progress of the science other causes,
seismic, fluvial and atmospheric, have been found necessary in order to
complete the theory of the history of the Earth's crust; but it remains
true that the stratified rocks, and some that have lost their stratified
character, were originally deposited under water. Inadequacy, therefore,
is not a reason for entirely rejecting an hypothesis or treating it as

(3) Granting that the hypothetical cause is real and adequate, the
investigation is not complete. Agreement with the facts is a very
persuasive circumstance, the more so the more extensive the agreement,
especially if no exceptions are known. Still, if this is all that can be
said in favour of an hypothesis, it amounts to proof at most by the
method of Agreement; it does not exclude the possibility of vicarious
causes; and if the hypothesis proposes a new agent that cannot be
directly observed, an equally plausible hypothesis about another
imagined agent may perhaps be invented.

According to Whewell, it is a strong mark of the truth of an hypothesis
when it agrees with distinct inductions concerning different classes of
facts, and he calls this the 'Consilience of Inductions,' because they
jump together in the unity of the hypothesis. It is particularly
convincing when this consilience takes place easily and naturally
without necessitating the mending and tinkering of the hypothesis; and
he cites the Theory of Gravitation and the Undulatory Theory of Light as
the most conspicuous examples of such ever-victorious hypotheses. Thus,
gravitation explains the fall of bodies on the Earth, and the orbits of
the planets and their satellites; it applies to the tides, the comets,
the double stars, and gives consistency to the Nebular Hypothesis,
whence flow important geological inferences; and all this without any
need of amendment. Nevertheless, Mill, with his rigorous sense of duty,
points out, that an induction is merely a proposition concerning many
facts, and that a consilience of inductions is merely a multiplication
of the facts explained; and that, therefore, if the proof is merely
Agreement in each case, there can be no more in the totality; the
possibility of vicarious causes is not precluded; and the hypothesis
may, after all, describe an accidental circumstance.

Whewell also laid great stress upon prediction as a mark of a true
hypothesis. Thus, Astronomers predict eclipses, occultations, transits,
long beforehand with the greatest precision; and the prediction of the
place of Neptune by sheer force of deduction is one of the most
astonishing things in the history of science. Yet Mill persisted in
showing that a predicted fact is only another fact, and that it is
really not very extraordinary that an hypothesis, that happens to agree
with many known facts, should also agree with some still undiscovered.
Certainly, there seems to be some illusion in the common belief in the
probative force of prediction. Prediction surprises us, puts us off our
guard, and renders persuasion easy; in this it resembles the force of an
epigram in rhetoric. But cases can be produced in which erroneous
hypotheses have led to prediction; and Whewell himself produces them.
Thus, he says that the Ptolemaic theory was confirmed by its predicting
eclipses and other celestial phenomena, and by leading to the
construction of Tables in which the places of the heavenly bodies were
given at every moment of time. Similarly, both Newton's theory of light
and the chemical doctrine of phlogiston led to predictions which came

What sound method demands in the proof of an hypothesis, then, is _not
merely that it be shown to agree with the facts, but that every other
hypothesis be excluded._ This, to be sure, may be beyond our power;
there may in some cases be no such negative proof except the exhaustion
of human ingenuity in the course of time. The present theory of colour
has in its favour the failure of Newton's corpuscular hypothesis and of
Goethe's anti-mathematical hypothesis; but the field of conjecture
remains open. On the other hand, Newton's proof that the solar system is
controlled by a central force, was supported by the demonstration that a
force having any other direction could not have results agreeing with
Kepler's second law of the planetary motions, namely, that, as a planet
moves in its orbit, the areas described by a line drawn from the sun to
the planet are proportional to the times occupied in the planet's
motion. When a planet is nearest to the sun, the area described by such
a line is least for any given distance traversed by the planet; and then
the planet moves fastest: when the planet is furthest from the sun, the
area described by such a line is greatest for an equal distance
traversed; and then the planet moves slowest. This law may be deduced
from the hypothesis of a central force, but not from any other; the
proof, therefore, as Mill says, satisfies the method of Difference.

Apparently, to such completeness of demonstration certain conditions are
necessary: the possibilities must lie between alternatives, such as A or
not-A, or amongst some definite list of cases that may be exhausted,
such as equal, greater or less. He whose hypothesis cannot be brought to
such a definite issue, must try to refute whatever other hypotheses are
offered, and naturally he will attack first the strongest rivals. With
this object in view he looks about for a "crucial instance," that is, an
observation or experiment that stands like a cross (sign-post) at the
parting of the ways to guide us into the right way, or, in plain words,
an instance that can be explained by one hypothesis but not by another.
Thus the phases of Venus, similar to those of the Moon, but concurring
with great changes of apparent size, presented, when discovered by
Galileo, a crucial instance in favour of the Copernican hypothesis, as
against the Ptolemaic, so far at least as to prove that Venus revolved
around the Sun inside the orbit of the Earth. Foucault's experiment
determining the velocity of Light (cited in the last chapter) was at
first intended as an _experimentum crucis_ to decide between the
corpuscular and undulatory theories; and answered this purpose, by
showing that the velocity of a beam passed through water was less than
it should be by the former, but in agreement with the latter doctrine
(Deschanel: § 813).

Perhaps experiments of this decisive character are commonest in
Chemistry: chemical tests, says Herschel, "are almost universally
crucial experiments." The following is abridged from Playfair (_Encycl.
Met., Diss._ III.): The Chemists of the eighteenth century observed that
metals were rendered heavier by calcination; and there were two ways of
accounting for this: either something had been added in the process,
though what, they could not imagine; or, something had been driven off
that was in its nature light, namely, phlogiston. To decide between
these hypotheses, Lavoisier hermetically sealed some tin in a glass
retort, and weighed the whole. He then heated it; and, when the tin was
calcined, weighed the whole again, and found it the same as before. No
substance, therefore, either light or heavy, had escaped. Further, when
the retort was cooled and opened, the air rushed in, showing that some
of the air formerly within had disappeared or lost its elasticity. On
weighing the whole again, its weight was now found to have increased by
ten grains; so that ten grains of air had entered when it was opened.
The calcined tin was then weighed separately, and proved to be exactly
ten grains heavier than when it was placed in the retort; showing that
the ten grains of air that had disappeared had combined with the metal
during calcination. This experiment, then, decided against phlogiston,
and led to an analysis of common air confirming Priestley's discovery of

(4) _An hypothesis must agree with the rest of the laws of Nature; and,
if not itself of the highest generality, must be derivable from primary
laws_ (chap. xix. § 1). Gravitation and the diffusion of heat, light and
sound from a centre, all follow the 'law of the inverse square,' and
agree with the relation of the radius of a sphere to its surface. Any
one who should think that he had discovered a new central force would
naturally begin to investigate it on the hypothesis that it conformed to
the same law as gravitation or light. A Chemist again, who should
believe himself to have discovered a new element, would expect it to
fill one of the vacant places in the Periodic Table. Conformity, in such
cases, is strong confirmation, and disagreement is an occasion of

A narrower hypothesis, as 'that the toad's ugliness is protective',
would be supported by the general theory of protective colouring and
figure, and by the still more general theory of Natural Selection, if
facts could be adduced to show that the toad's appearance does really
deter its enemies. Such an hypothesis resembles an Empirical Law in its
need of derivation (chap. xix. §§ 1, 2). If underivable from, or
irreconcilable with, known laws, it is a mere conjecture or prejudice.
The absolute leviation of phlogiston, in contrast with the gravitation
of all other forms of matter, discredited that supposed agent. That
Macpherson should have found the Ossianic poems extant in the Gaelic
memory, was contrary to the nature of oral tradition; except where
tradition is organised, as it was for ages among the Brahmins. The
suggestion that xanthochroid Aryans were "bleached" by exposure during
the glacial period, does not agree with Wallace's doctrine concerning
the coloration of Arctic animals. That our forefathers being predatory,
like bears, white variations amongst them were then selected by the
advantage of concealment, is a more plausible hypothesis.

Although, then, the consilience of Inductions or Hypotheses is not a
sufficient proof of their truth, it is still a condition of it;
nonconsilience is a suspicious circumstance, and resilience (so to
speak), or mutual repugnance, is fatal to one or all.

§ 4. We have now seen that a scientific hypothesis, to deserve the name,
must be verifiable and therefore definite; and that to establish itself
as a true theory, it must present some symptom of reality, and be
adequate and exclusive and in harmony with the system of experience.
Thus guarded, hypotheses seem harmless enough; but some people have a
strong prejudice against them, as against a tribe of savages without
government, or laws, or any decent regard for vested interests. It is
well known, too, that Bacon and Newton disparaged them. But Bacon, in
his examples of an investigation according to his own method, is
obliged, after a preliminary classification of facts, to resort to an
hypothesis, calling it _permissio intellectus_, _interpretatio inchoata_
or _vindemiatio prima_. And Newton when he said _hypotheses non fingo_,
meant that he did not deal in fictions, or lay stress upon supposed
forces (such as 'attraction'), that add nothing to the law of the facts.
Hypotheses are essential aids to discovery: speaking generally,
deliberate investigation depends wholly upon the use of them.

It is true that we may sometimes observe a train of events that chances
to pass before us, when either we are idle or engaged with some other
inquiry, and so obtain a new glimpse of the course of nature; or we may
try experiments haphazard, and watch the results. But, even in these
cases, before our new notions can be considered knowledge, they must be
definitely framed in hypotheses and reobserved or experimented upon,
with whatever calculations or precautions may be necessary to ensure
accuracy or isolation. As a rule, when inquiring deliberately into the
cause of an event, whether in nature or in history, we first reflect
upon the circumstances of the case and compare it with similar ones
previously investigated, and so are guided by a preconception more or
less definite of 'what to look for,' what the cause is likely to be,
that is, by an hypothesis. Then, if our preconception is justified, or
something which we observe leads to a new hypothesis, either we look for
other instances to satisfy the canons of Agreement; or (if the matter
admits of experiment) we endeavour, under known conditions according to
the canon of Difference, to reproduce the event by means of that which
our hypothesis assigns as the cause; or we draw remote inferences from
our hypothesis, and try to test these by the Inductive Canons.

If we argue from an hypothesis and express ourselves formally, it will
usually appear as the major premise; but this is not always the case. In
extending ascertained laws to fresh cases, the minor premise may be an
hypothesis, as in testing the chemical constitution of any doubtful
substance, such as a piece of ore. Some solution or preparation, A, is
generally made which (it is known) will, on the introduction of a
certain agent, B, give a reaction, C, if the preparation contains a
given substance, X. The major premise is the law of reaction--

    Whenever A is X, if treated with B it is C.

The minor premise is an hypothesis that the preparation contains X. An
experiment then treats A with B. If C result, a probability is raised in
favour of the hypothesis that A is X; or a certainty, if we know that C
results on that condition only.

So important are hypotheses to science, that Whewell insists that they
have often been extremely valuable even though erroneous. Of the
Ptolemaic system he says, "We can hardly imagine that Astronomy could,
in its outset, have made so great a progress under any other form." It
served to connect men's thoughts on the subject and to sustain their
interest in working it out; by successive corrections "to save
appearances," it attained at last to a descriptive sort of truth, which
was of great practical utility; it also occasioned the invention of
technical terms, and, in general digested the whole body of observations
and prepared them for assimilation by a better hypothesis in the fulness
of time. Whewell even defends the maxim that "Nature abhors a vacuum,"
as having formerly served to connect many facts that differ widely in
their first aspect. "And in reality is it not true," he asks, "that
nature _does_ abhor a vacuum, and does all she can to avoid it?" Let no
forlorn cause despair of a champion! Yet no one has accused Whewell of
Quixotry; and the sense of his position is that the human mind is a
rather feeble affair, that can hardly begin to think except with

The progress of science may be plausibly attributed to a process of
Natural Selection; hypotheses are produced in abundance and variety, and
those unfit to bear verification are destroyed, until only the fittest
survive. Wallace, a practical naturalist, if there ever was one, as well
as an eminent theorist, takes the same view as Whewell of such
inadequate conjectures. Of 'Lemuria,' an hypothetical continent in the
Indian Ocean, once supposed to be traceable in the islands of
Madagascar, Seychelles, and Mauritius, its surviving fragments, and
named from the Lemurs, its characteristic denizens, he says (_Island
Life_, chap. xix.) that it was "essentially a provisional hypothesis,
very useful in calling attention to a remarkable series of problems in
geographical distribution [of plants and animals], but not affording the
true solution of those problems." We see, then, that 'provisional
hypotheses,' or working hypotheses,' though erroneous, may be very
useful or (as Whewell says) necessary.

Hence, to be prolific of hypotheses is the first attribute of scientific
genius; the first, because without it no progress whatever can be made.
And some men seem to have a marked felicity, a sort of instinctive
judgment even in their guesses, as if their heads were made according to
Nature. But others among the greatest, like Kepler, guess often and are
often wrong before they hit upon the truth, and themselves, like Nature,
destroy many vain shoots and seedlings of science for one that they find
fit to live. If this is how the mind works in scientific inquiry (as it
certainly is, with most men, in poetry, in fine art, and in the scheming
of business), it is useless to complain. We should rather recognise a
place for fools' hypotheses, as Darwin did for "fools' experiments." But
to complete the scientific character, there must be great patience,
accuracy, and impartiality in examining and testing these conjectures,
as well as great ingenuity in devising experiments to that end. The want
of these qualities leads to crude work and public failure and brings
hypotheses into derision. Not partially and hastily to believe in one's
own guesses, nor petulantly or timidly to reject them, but to consider
the matter, to suspend judgment, is the moral lesson of science:
difficult, distasteful, and rarely mastered.

§ 5. The word 'hypothesis' is often used also for the scientific device
of treating an Abstraction as, for the purposes of argument, equivalent
to the concrete facts. Thus, in Geometry, a line is treated as having no
breadth; in Mechanics, a bar may be supposed absolutely rigid, or a
machine to work without friction; in Economics, man is sometimes
regarded as actuated solely by love of gain and dislike of exertion. The
results reached by such reasoning may be made applicable to the concrete
facts, if allowance be made for the omitted circumstances or properties,
in the several cases of lines, bars, and men; but otherwise all
conclusions from abstract terms are limited by their definitions.
Abstract reasoning, then (that is, reasoning limited by definitions),
is often said to imply 'the hypothesis' that things exist as their names
are defined, having no properties but those enumerated in their
definitions. This seems, however, a needless and confusing extension of
the term; for an hypothesis proposes an agent, collocation, or law
hitherto unknown; whereas abstract reasoning proposes to exclude from
consideration a good deal that is well known. There seems no reason why
the latter device should not be plainly called an Abstraction.

Such abstractions are necessary to science; for no object is
comprehensible by us in all its properties at once. But if we forget the
limitations of our abstract data, we are liable to make strange blunders
by mistaking the character of the results: treating the results as
simply true of actual things, instead of as true of actual things only
so far as they are represented by the abstractions. In addressing
abstract reasoning, therefore, to those who are unfamiliar with
scientific methods, pains should be taken to make it clear what the
abstractions are, what are the consequent limitations upon the argument
and its conclusions, and what corrections and allowances are necessary
in order to turn the conclusions into an adequate account of the
concrete facts. The greater the number, variety, and subtlety of the
properties possessed by any object (such as human nature), the greater
are the qualifications required in the conclusions of abstract
reasoning, before they can hold true of such an object in practical

Closely allied to this method of Abstraction is the Mathematical Method
of Limits. In his _History of Scientific Ideas_ (B. II. c. 12), Whewell
says: "The _Idea of a Limit_ supplies a new mode of establishing
mathematical truths. Thus with regard to the length of any portion of a
curve, a problem which we have just mentioned; a curve is not made up of
straight lines, and therefore we cannot by means of any of the
doctrines of elementary geometry measure the length of any curve. But we
may make up a figure nearly resembling any curve by putting together
many short straight lines, just as a polygonal building of very many
sides may nearly resemble a circular room. And in order to approach
nearer and nearer to a curve, we may make the sides more and more small,
more and more numerous. We may then possibly find some mode of
measurement, some relation of these small lines to other lines, which is
not disturbed by the multiplication of the sides, however far it be
carried. And thus we may do what is equivalent to measuring the curve
itself; for by multiplying the sides we may approach more and more
closely to the curve till no appreciable difference remains. The curve
line is the _Limit_ of the polygon; and in this process we proceed on
the _Axiom_ that 'What is true up to the Limit is true at the Limit.'"

What Whewell calls the Axiom here, others might call an Hypothesis; but
perhaps it is properly a Postulate. And it is just the obverse of the
Postulate implied in the Method of Abstractions, namely, that 'What is
true of the Abstraction is true of concrete cases the more nearly they
approach the Abstraction.' What is true of the 'Economic Man' is truer
of a broker than of a farmer, of a farmer than of a labourer, of a
labourer than of the artist of romance. Hence the Abstraction may be
called a Limit or limiting case, in the sense that it stands to concrete
individuals, as a curve does to the figures made up "by putting together
many short straight lines." Correspondingly, the Proper Name may be
called the Limit of the class-name; since its attributes are infinite,
whereas any name whose attributes are less than infinite stands for a
possible class. In short, for logical purposes, a Limit may be defined
as any extreme case to which actual examples may approach without ever
reaching it. And in this sense 'Method of Limits' might be used as a
term including the Method of Abstractions; though it would be better to
speak of them generically as 'Methods of Approximation.'

We may also notice the Assumptions (as they may be called) that are
sometimes employed to facilitate an investigation, because some definite
ground must be taken and nothing better can be thought of: as in
estimating national wealth, that furniture is half the value of the

It is easy to conceive of an objector urging that such devices as the
above are merely ways of avoiding the actual problems, and that they
display more cunning than skill. But science, like good sense, puts up
with the best that can be had; and, like prudence, does not reject the
half-loaf. The position, that a conceivable case that can be dealt with
may, under certain conditions, be substituted for one that is
unworkable, is a touchstone of intelligence. To stand out for ideals
that are known to be impossible, is only an excuse for doing nothing at

In another sense, again, the whole of science is sometimes said to be
hypothetical, because it takes for granted the Uniformity of Nature; for
this, in its various aspects, can only be directly ascertained by us as
far as our experience extends; whereas the whole value of the principle
of Uniformity consists in its furnishing a formula for the extension of
our other beliefs beyond our actual experience. Transcendentalists,
indeed, call it a form of Reason, just because it is presupposed in all
knowledge; and they and the Empiricists agree that to adduce material
evidence for it, in its full extent, is impossible. If, then, material
evidence is demanded by any one, he cannot regard the conclusions of
Mathematics and Physical Science as depending on what is itself
unproved; he must, with Mill, regard these conclusions as drawn "not
from but according to" the axioms of Equality and Causation. That is to
say, if the axioms are true, the conclusions are; the material evidence
for both the axioms and the conclusions being the same, namely,
uncontradicted experience. Now when we say, 'If Nature is uniform,
science is true,' the hypothetical character of science appears in the
form of the statement. Nevertheless, it seems undesirable to call our
confidence in Nature's uniformity an 'hypothesis': it is incongruous to
use the same term for our tentative conjectures and for our most
indispensable beliefs. 'The Universal Postulate' is a better term for
the principle which, in some form or other, every generalisation takes
for granted.

We are now sometimes told that, instead of the determinism and
continuity of phenomena hitherto assumed by science, we should recognise
indeterminism and discontinuity. But it will be time enough to fall in
with this doctrine when its advocates produce a new Logic of Induction,
and explain the use of the method of Difference and of control
experiments according to the new postulates.



§ 1. Laws are classified, according to their degrees of generality, as
higher and lower, though the grades may not be decisively

First, there are Axioms or Principles, that is real, universal,
self-evident propositions. They are--(1) real propositions; not, like
'The whole is greater than any of its parts,' merely definitions, or
implied in definitions. (2) They are regarded as universally true of
phenomena, as far as the form of their expression extends; that is, for
example, Axioms concerning quantity are true of everything that is
considered in its quantitative aspect, though not (of course) in its
qualitative aspect. (3) They are self-evident; that is, each rests upon
its own evidence (whatever that may be); they cannot be derived from one
another, nor from any more general law. Some, indeed, are more general
than others: the Logical Principle of Contradiction, 'if A is B, it is
not not-B', is true of qualities as well as of quantities; whereas the
Axioms of Mathematics apply only to quantities. The Mathematical Axioms,
again, apply to time, space, mental phenomena, and matter and energy;
whereas the Law of Causation is only true of concrete events in the
redistribution of matter and energy: such, at least, is the strict limit
of Causation, if we identify it with the Conservation of Energy;
although our imperfect knowledge of life and mind often drives us to
speak of feelings, ideas, volitions, as causes. Still, the Law of
Causation cannot be derived from the Mathematical Axioms, nor these from
the Logical. The kind of evidence upon which Axioms rest, or whether any
evidence can be given for them, is (as before observed) a question for
Metaphysics, not for Logic. Axioms are the upward limit of Logic, which,
like all the special sciences, necessarily takes them for granted, as
the starting point of all deduction and the goal of all generalisation.

Next to Axioms, come Primary Laws of Nature: these are of less
generality than the Axioms, and are subject to the conditions of
methodical proof; being universally true only of certain forces or
properties of matter, or of nature under certain conditions; so that
proof of them by logical or mathematical reasoning is expected, because
they depend upon the Axioms for their formal evidence. Such are the law
of gravitation, in Astronomy; the law of definite proportions, in
Chemistry; the law of heredity, in Biology; and in Psychology, the law
of relativity.

Then, there are Secondary Laws, of still less generality, resulting from
a combination of conditions or forces in given circumstances, and
therefore conceivably derivable from the laws of those conditions or
forces, if we can discover them and compute their united effects.
Accordingly, Secondary Laws are either--(1) Derivative, having been
analysed into, and deduced from, Primary Laws; or (2) Empirical, those
that have not yet been deduced (though from their comparatively special
and complex character, it seems probable they may be, given sufficient
time and ingenuity), and that meanwhile rest upon some unsatisfactory
sort of induction by Agreement or Simple Enumeration.

Whether laws proved only by the canon of Difference are to be considered
Empirical, is perhaps a question: their proof derives them from the
principle of Causation; but, being of narrow scope, some more special
account of them seems requisite in relation to the Primary Laws before
we can call them Derivative in the technical sense.

Many Secondary Laws, again, are partially or imperfectly Derivative; we
can give general reasons for them, without being able to determine
theoretically the precise relations of the phenomena they describe.
Meteorologists can explain the general conditions of all sorts of
weather, but have made little progress toward predicting the actual
course of it (at least, for our island): Geologists know the general
causes of mountain ranges, but not why they rise just where we find
them: Economists explain the general course of a commercial crisis, but
not why the great crises recurred at intervals of about ten years.

Derivative Laws make up the body of the exact sciences, having been
assimilated and organised; whilst Empirical Laws are the undigested
materials of science. The theorems of Euclid are good examples of
derivative laws in Mathematics; in Astronomy, Kepler's laws and the laws
of the tides; in Physics, the laws of shadows, of perspective, of
harmony; in Biology, the law of protective coloration; in Economics, the
laws of prices, wages, interest, and rent.

Empirical Laws are such as Bode's law of the planetary distances; the
laws of the expansion of different bodies by heat, and formulæ
expressing the electrical conductivity of each substance as a function
of the temperature. Strictly speaking, I suppose, all the laws of
chemical combination are empirical: the law of definite proportions is
verifiable in all cases that have been examined, except for variations
that may be ascribed to errors of experiment. Much the same is true in
Biology; most of the secondary laws are empirical, except so far as
structures or functions may be regarded as specialised cases in Physics
or Chemistry and deducible from these sciences. The theory of Natural
Selection, however, has been the means of rendering many laws, that were
once wholly empirical, at least partially derivative; namely, the laws
of the geographical distribution of plants and animals, and of their
adaptation in organisation, form and colour, habits and instincts, to
their various conditions of life. The laws that remain empirical in
Biology are of all degrees of generality from that of the tendency to
variation in size and in every other character shown by every species
(though as to the reason of this there are promising hypotheses), down
to such curious cases as that the colour of roses and carnations never
varies into blue, that scarlet flowers are never sweet-scented, that
bullfinches fed on hemp-seed turn black, that the young of white, yellow
and dun pigeons are born almost naked (whilst others have plenty of
down); and so on. The derivation of empirical laws is the greater part
of the explanation of Nature (§§ 5, 6).

A 'Fact,' in the common use of the word, is a particular observation: it
is the material of science in its rawest state. As perceived by a mind,
it is, of course, never absolutely particular: for we cannot perceive
anything without classing it, more or less definitely, with things
already known to us; nor describe it without using connotative terms
which imply a classification of the things denoted. Still, we may
consider an observation as particular, in comparison with a law that
includes it with numerous others in one general proposition. To turn an
observation into an experiment, or (where experiment is impracticable)
to repeat it with all possible precautions and exactness, and to
describe it as to the duration, quantity, quality and order of
occurrence of its phenomena, is the first stage of scientific
manufacture. Then comes the formulation of an empirical law; and lastly,
if possible, deduction or derivation, either from higher laws previously
ascertained, or from an hypothesis. However, as a word is used in
various senses, we often speak of laws as 'facts': we say the law of
gravitation is a fact, meaning that it is real, or verifiable by
observations or experiments.

§ 2. Secondary Laws may also be classified according to their constancy
into--(1) the Invariable (as far as experience reaches), and (2)
Approximate Generalisations in the form--Most X's are Y. Of the
invariable we have given examples above. The following are approximate
generalisations: Most comets go round the Sun from East to West; Most
metals are solid at ordinary temperatures; Most marsupials are
Australasian; Most arctic animals are white in winter; Most cases of
plague are fatal; Most men think first of their own interests. Some of
these laws are empirical, as that 'Most metals are solid at ordinary
temperatures': at present no reason can be given for this; nor do we
know why most cases of plague are fatal. Others, however, are at least
partially derivative, as that 'Most arctic animals are white'; for this
seems to be due to the advantage of concealment in the snow; whether, as
with the bear, the better to surprise its prey, or, with the hare, to
escape the notice of its enemies.

But the scientific treatment of such a proposition requires that we
should also explain the exceptions: if 'Most are,' this implies that
'Some are not'; why not, then? Now, if we can give reasons for all the
exceptions, the approximate generalisation may be converted into an
universal one, thus: 'All arctic animals are white, unless (like the
raven) they need no concealment either to prey or to escape; or unless
mutual recognition is more important to them than concealment (as with
the musk-sheep)'. The same end of universal statement may be gained by
including the conditions on which the phenomenon depends, thus: 'All
arctic animals to whom concealment is of the utmost utility are white.'

When statistics are obtainable, it is proper to convert an approximate
generalisation into a proportional statement of the fact, thus: instead
of 'Most attacks of plague are fatal', we might find that in a certain
country 70 per cent. were so. Then, if we found that in another country
the percentage of deaths was 60, in another 40, we might discover, in
the different conditions of these countries, a clue to the high rate of
mortality from this disease. Even if the proportion of cases in which
two facts are connected does not amount to 'Most,' yet, if any definite
percentage is obtainable, the proposition has a higher scientific value
than a vague 'Some': as if we know that 2 per cent. of the deaths in
England are due to suicide, this may be compared with the rates of
suicide in other countries; from which perhaps inferences may be drawn
as to the causes of suicide.

In one department of life, namely, Politics, there is a special
advantage in true approximate generalisations amounting to 'Most cases.'
The citizens of any State are so various in character, enlightenment,
and conditions of life, that we can expect to find few propositions
universally true of them: so that propositions true of the majority must
be trusted as the bases of legislation. If most men are deterred from
crime by fear of punishment; if most men will idle if they can obtain
support without industry; if most jurymen will refuse to convict of a
crime for which the prescribed penalties seem to them too severe; these
are most useful truths, though there should be numerous exceptions to
them all.

§ 3. Secondary Laws can only be trusted in 'Adjacent Cases'; that is,
where the circumstances are similar to those in which the laws are known
to be true.

A Derivative Law will be true wherever the forces concerned exist in the
combinations upon which the law depends, if there are no counteracting
conditions. That water can be pumped to about 33 feet at the sea-level,
is a derivative law on this planet: is it true in Mars? That depends on
whether there are in Mars bodies of a liquid similar to our water;
whether there is an atmosphere there, and how great its pressure is;
which will vary with its height and density. If there is no atmosphere
there can be no pumping; or if there is an atmosphere of less pressure
than ours, water such as ours can only be pumped to a less height than
33 feet. Again, we know that there are arctic regions in Mars; if there
are also arctic animals, are they white? That may depend upon whether
there are any beasts of prey. If not, concealment seems to be of no use.

An Empirical Law, being one whose conditions we do not know, the extent
of its prevalence is still less ascertainable. Where it has not been
actually observed to be true, we cannot trust it unless the
circumstances, on the whole, resemble so closely those amongst which it
has been observed, that the unknown causes, whatever they may be, are
likely to prevail there. And, even then, we cannot have much confidence
in it; for there may be unknown circumstances which entirely frustrate
the effect. The first naturalist who travelled (say) from Singapore
eastward by Sumatra and Java, or Borneo, and found the mammalia there
similar to those of Asia, may naturally have expected the same thing in
Celebes and Papua; but, if so, he was entirely disappointed; for in
Papua the mammalia are marsupials like those of Australia. Thus his
empirical law, 'The mammalia of the Eastern Archipelago are Asiatic,'
would have failed for no apparent reason. According to Mr. Wallace,
there is a reason for it, though such as could only be discovered by
extensive researches; namely, that the sea is deep between Borneo and
Celebes, so that they must have been separated for many ages; whereas it
is shallow from Borneo westward to Asia, and also southward from Celebes
to Australia; so that these regions, respectively, may have been
recently united: and the true law is that similar mammalia belong to
those tracts which at comparatively recent dates have formed parts of
the same continents (unless they are the remains of a former much wider

A considerable lapse of time may make an empirical law no longer
trustworthy; for the forces from whose combination it resulted may have
ceased to operate, or to operate in the same combination; and since we
do not know what those forces were, even the knowledge that great
changes have taken place in the meantime cannot enable us, after an
interval, to judge whether or not the law still holds true. New stars
shine in the sky and go out; species of plants and animals become
extinct; diseases die out and fresh ones afflict mankind: all these
things doubtless have their causes, but if we do not know what they are,
we have no measure of the effects, and cannot tell when or where they
will happen.

Laws of Concomitant Variations may hold good only within certain limits.
That bodies contract as the temperature falls, is not true of water
below 39° F. In Psychology, Weber's Law is only true within the median
range of sensation-intensities, not for very faint, nor for very strong,
stimuli. In such cases the failure of the laws may depend upon something
imperfectly understood in the collocation: as to water, on its molecular
constitution; as to sensation, upon the structure of the nervous system.

§ 4. Secondary Laws, again, are either of Succession or of Co-existence.

Those of Succession are either--(1) of direct causation, as that 'Water
quenches fire,' or (more strictly) that 'Evaporation reduces
temperature'; or (2) of the effect of a remote cause, as 'Bad harvests
tend to raise the price of bread'; or (3) of the joint effects of the
same cause, as that 'Night follows day' (from the revolution of the
earth), or the course of the seasons (from the inclination of the
earth's axis).

Laws of Co-existence are of several classes. (1) One has the generality
of a primary law, though it is proved only by Agreement, namely, 'All
gravitating bodies are inert'. Others, though less general than this,
are of very extensive range, as that 'All gases that are not decomposed
by rise of temperature have the same rate of expansion'; and, in Botany
that 'All monocotyledonous plants are endogenous'. These laws of
Co-existence are concerned with fundamental properties of bodies.

(2) Next come laws of the Co-existence of those properties which are
comprised in the definitions of Natural Kinds. Mill distinguished
between (α) classes of things that agree among themselves and differ
from others only in one or a few attributes (such as 'red things,'
'musical notes', 'carnivorous animals', 'soldiers'), and (β) classes of
things that agree among themselves and differ from others in a multitude
of characters: and the latter he calls Natural Kinds. These comprise the
chemical elements and their pure compounds (such as water, alcohol,
rock-salt), and the species of plants and animals. Clearly, each of
these is constituted by the co-existence or co-inherence of a multitude
of properties, some of which are selected as the basis of their
definitions. Thus, Gold is a metal of high specific gravity, atomic
weight 197.2, high melting point, low chemical affinities, great
ductility, yellow colour, etc.: a Horse has 'a vertebral column,
mammæ, a placental embryo, four legs, a single well-developed toe in
each foot provided with a hoof, a bushy tail, and callosities on the
inner sides of both the fore and the hind legs' (Huxley).

Since Darwinism has obtained general acceptance, some Logicians have
doubted the propriety of calling the organic species 'Kinds,' on the
ground that they are not, as to definiteness and permanence, on a par
with the chemical elements or such compounds as water and rock-salt;
that they vary extensively, and that it is only by the loss of former
generations of animals that we are able to distinguish species at all.
But to this it may be replied that species are often approximately
constant for immense periods of time, and may be called permanent in
comparison with human generations; and that, although the leading
principles of Logic are perhaps eternal truths, yet upon a detail such
as this, the science may condescend to recognise a distinction if it is
good for (say) only 100,000 years. That if former generations of plants
and animals were not lost, all distinctions of species would disappear,
may be true; but they are lost--for the most part beyond hope of
recovery; and accordingly the distinction of species is still
recognised; although there are cases, chiefly at the lower stages of
organisation, in which so many varieties occur as to make adjacent
species almost or quite indistinguishable. So far as species are
recognised, then, they present a complex co-inherence of qualities,
which is, in one aspect, a logical problem; and, in another, a logical
datum; and, coming more naturally under the head of Natural Kinds than
any other, they must be mentioned in this place.

(3) There are, again, certain coincidences of qualities not essential to
any kind, and sometimes prevailing amongst many different kinds: such as
'Insects of nauseous taste have vivid (warning) colours'; 'White
tom-cats with blue eyes are deaf'; 'White spots and patches, when they
appear in domestic animals, are most frequent on the left side.'

(4) Finally, there may be constancy of relative position, as of sides
and angles in Geometry; and also among concrete things (at least for
long periods of time), as of the planetary orbits, the apparent
positions of fixed stars in the sky, the distribution of land and water
on the globe, opposite seasons in opposite hemispheres.

All these cases of Co-existence (except the geometrical) present the
problem of deriving them from Causation; for there is no general Law of
Co-existence from which they can be derived; and, indeed, if we conceive
of the external world as a perpetual redistribution of matter and
energy, it follows that the whole state of Nature at any instant, and
therefore every co-existence included in it, is due to causation issuing
from some earlier distribution of matter and energy. Hence, indeed, it
is not likely that the problems of co-existence as a whole will ever be
solved, since the original distribution of matter is, of course,
unknown. Still, starting with any given state of Nature, we may hope to
explain some of the co-existences in any subsequent state. We do not,
indeed, know why heavy bodies are always inert, nor why the chemical
elements are what they are; but it is known that "the properties of the
elements are functions of their atomic weight," which (though, at
present, only an empirical law) may be a clue to some deeper
explanation. As to plants and animals, we know the conditions of their
generation, and can trace a connection between most of their
characteristics and the conditions of their life: as that the teeth and
stomach of animals vary with their food, and that their colour generally
varies with their habitat.

Geometrical Co-existence, when it is not a matter of definition (as 'a
square is a rectangle with four equal sides'), is deduced from the
definitions and axioms: as when it is shown that in triangles the
greater side is opposite the greater angle. The deductions of theorems
or secondary laws, in Geometry is a type of what is desirable in the
Physical Sciences: the demonstration, namely, that all the connections
of phenomena, whether successive or co-existent, are consequences of the
redistribution of matter and energy according to the principle of

Coincidences of Co-existence (Group (3)) may sometimes be deduced and
sometimes not. That 'nauseous insects have vivid coloration' comes under
the general law of 'protective coloration'; as they are easily
recognised and therefore avoided by insectivorous birds and other
animals. But why white tom-cats with blue-eyes should be deaf, is (I
believe) unknown. When co-existences cannot be derived from causation,
they can only be proved by collecting examples and trusting vaguely to
the Uniformity of Nature. If no exceptions are found, we have an
empirical law of considerable probability within the range of our
exploration. If exceptions occur, we have at most an approximate
generalisation, as that 'Most metals are whitish,' or 'Most domestic
cats are tabbies' (but this probably is the ancestral colouring). We may
then resort to statistics for greater definiteness, and find that in
Hampshire (say) 90 _per cent._ of the domestic cats are tabby.

§ 5. Scientific Explanation consists in discovering, deducing, and
assimilating the laws of phenomena; it is the analysis of that
Heracleitan 'flux' which so many philosophers have regarded as
intractable to human inquiry. In the ordinary use of the word,
'explanation' means the satisfying a man's understanding; and what may
serve this purpose depends partly upon the natural soundness of his
understanding, and partly on his education; but it is always at last an
appeal to the primary functions of cognition, discrimination and

Generally, what we are accustomed to seems to need no explanation,
unless our curiosity is particularly directed to it. That boys climb
trees and throw stones, and that men go fox-hunting, may easily pass for
matters of course. If any one is so exacting as to ask the reason, there
is a ready answer in the 'need of exercise.' But this will not explain
the peculiar zest of those exercises, which is something quite different
from our feelings whilst swinging dumb-bells or tramping the highway.
Others, more sophisticated, tell us that the civilised individual
retains in his nature the instincts of his remote ancestors, and that
these assert themselves at stages of his growth corresponding with
ancestral periods of culture or savagery: so that if we delight to climb
trees, throw stones, and hunt, it is because our forefathers once lived
in trees, had no missiles but stones, and depended for a livelihood
upon killing something. To some of us, again, this seems an
explanation; to others it merely gives annoyance, as a superfluous
hypothesis, the fruit of a wanton imagination and too much leisure.

However, what we are not accustomed to immediately excites curiosity. If
it were exceptional to climb trees, throw stones, ride after foxes,
whoever did such things would be viewed with suspicion. An eclipse, a
shooting star, a solitary boulder on the heath, a strange animal, or a
Chinaman in the street, calls for explanation; and among some nations,
eclipses have been explained by supposing a dragon to devour the sun or
moon; solitary boulders, as the missiles of a giant; and so on. Such
explanations, plainly, are attempts to regard rare phenomena as similar
to others that are better known; a snake having been seen to swallow a
rabbit, a bigger one may swallow the sun: a giant is supposed to bear
much the same relation to a boulder as a boy does to half a brick. When
any very common thing seems to need no explanation, it is because the
several instances of its occurrence are a sufficient basis of
assimilation to satisfy most of us. Still, if a reason for such a thing
be demanded, the commonest answer has the same implication, namely, that
assimilation or classification is a sufficient reason for it. Thus, if
climbing trees is referred to the need of exercise, it is assimilated to
running, rowing, etc.; if the customs of a savage tribe are referred
to the command of its gods, they are assimilated to those things that
are done at the command of chieftains.

Explanation, then, is a kind of classification; it is the finding of
resemblance between the phenomenon in question and other phenomena. In
Mathematics, the explanation of a theorem is the same as its proof, and
consists in showing that it repeats, under different conditions, the
definitions and axioms already assumed and the theorems already
demonstrated. In Logic, the major premise of every syllogism is an
explanation of the conclusion; for the minor premise asserts that the
conclusion is an example of the major premise.

In Concrete Sciences, to discover the cause of a phenomenon, or to
derive an empirical law from laws of causation, is to explain it;
because a cause is an invariable antecedent, and therefore reminds us
of, or enables us to conceive, an indefinite number of cases similar to
the present one wherever the cause exists. It classifies the present
case with other instances of causation, or brings it under the universal
law; and, as we have seen that the discovery of the laws of nature is
essentially the discovery of causes, the discovery and derivation of
laws is scientific explanation.

The discovery of quantitative laws is especially satisfactory, because
it not only explains why an event happens at all, but why it happens
just in this direction, degree, or amount; and not only is the given
relation of cause and effect definitely assimilated to other causal
instances, but the effect is identified with the cause as the same
matter and energy redistributed; wherefore, whether the conservation of
matter and energy be universally true or not, it must still be an
universal postulate of scientific explanation.

The mere discovery of an empirical law of co-existence, as that 'white
tom-cats with blue eyes are deaf', is indeed something better than an
isolated fact: every general proposition relieves the mind of a load of
facts; and, for many people, to be able to say--'It is always so'--may
be enough; but for scientific explanation we require to know the reason
of it, that is, the cause. Still, if asked to explain an axiom, we can
only say, 'It is always so:' though it is some relief to point out
particular instances of its realisation, or to exhibit the similarity of
its form to that of other axioms--as of the _Dictum_ to the axiom of

§ 6. There are three modes of scientific Explanation; First, the
analysis of a phenomenon into the laws of its causes and the concurrence
of those causes.

The pumping of water implies (1) pressure of the air, (2) distribution
of pressure in a liquid, (3) that motion takes the direction of least
resistance. Similarly, that thunder follows forked lightning, and that
the report of a gun follows the flash, are resolvable into (1) the
discharge of electricity, or the explosion of gunpowder; (2) distance of
the observer from the event; (3) that light travels faster than sound.
The planetary orbits are analysable into the tendency of planets to fall
into the sun, and their tendency to travel in a straight line. When this
conception is helped out by swinging a ball round by a string, and then
letting it go, to show what would happen to the earth if gravitation
ceased, we see how the recognition of resemblance lies at the bottom of

Secondly, the discovery of steps of causation between a cause and its
remote effects; the interpolation and concatenation of causes.

The maxim 'No cats no clover' is explained by assigning the intermediate
steps in the following series; that the fructification of red clover
depends on the visits of humble-bees, who distribute the pollen in
seeking honey; that if field-mice are numerous they destroy the
humble-bees' nests; and that (owls and weasels being exterminated by
gamekeepers) the destruction of field-mice depends upon the supply of
cats; which, therefore, are a remote condition of the clover crop.
Again, the communication of thought by speech is an example of something
so common that it seems to need no explanation; yet to explain it is a
long story. A thought in one man's mind is the remote cause of a similar
thought in another's: here we have (1) a thought associated with mental
words; (2) a connection between these thoughts and some tracts of the
brain; (3) a connection between these tracts of the brain and the
muscles of the larynx, the tongue and the lips; (4) movements of the
chest, larynx and mouth, propelling and modifying waves of air; (5) the
impinging of these air-waves upon another man's ear, and by a complex
mechanism exciting the aural nerve; (6) the transfer of this excitation
to certain tracts of his brain; (7) a connection there with sounds of
words and their associated thoughts. If one of these links fail, there
is no communication.

Thirdly, the subsumption of several laws under one more general

The tendency of bodies to fall to the earth and the tendency of the
earth itself (with the other planets) to fall into the sun, are subsumed
under the general law that 'All matter gravitates.' The same law
subsumes the movements of the tide. By means of the notion of specific
gravity, it includes 'levitation,' or the actual rising of some bodies,
as of corks in water, of balloons, or flames in the air: the fact being
that these things do not tend to rise, but to fall like everything else;
only as the water or air weighs more in proportion to its volume than
corks or balloons, the latter are pushed up.

This process of subsumption bears the same relation to secondary laws,
that these do to particular facts. The generalisation of many particular
facts (that is, a statement of that in which they agree) is a law; and
the generalisation of these laws (that is, again, a statement of that in
which they agree) is a higher law; and this process, upwards or
downwards, is characteristic of scientific progress. The perfecting of
any science consists in comprehending more and more of the facts within
its province, and in showing that they all exemplify a smaller and
smaller number of principles, which express their most profound

These three modes of explanation (analysis, interpolation, subsumption)
all consist in generalising or assimilating the phenomena. The pressure
of the air, of a liquid, and motion in the direction of least
resistance, are all commoner facts than pumping; that light travels
faster than sound is a commoner fact than a thunderstorm or gun-firing.
Each of the laws--'Cats kill mice,' 'Mice destroy humble-bees' nests,'
'Humble-bees fructify red clover'--is wider and expresses the
resemblance of more numerous cases than the law that 'Clover depends on
cats'; because each of them is less subject to further conditions.
Similarly, every step in the communication of thought by language is
less conditional, and therefore more general, than the completion of the

In all the above cases, again, each law into which the phenomenon
(whether pumping or conversation) is resolved, suggests a host of
parallel cases: as the modifying of air-waves by the larynx and lips
suggests the various devices by which the strings and orifices of
musical instruments modify the character of notes.

Subsumption consists entirely in proving the existence of an essential
similarity between things where it was formerly not observed: as that
the gyrations of the moon, the fall of apples, and the flotation of
bubbles are all examples of gravitation: or that the purifying of the
blood by breathing, the burning of a candle, and the rusting of iron are
all cases of oxidation: or that the colouring of the underside of a
red-admiral's wings, the spots of the giraffe, the shape and attitude of
a stick-caterpillar, the immobility of a bird on its nest, and countless
other cases, though superficially so different, agree in this, that they
conceal and thereby protect the organism.

Not any sort of likeness, however, suffices for scientific explanation:
the only satisfactory explanation of concrete things or events, is to
discover their likeness to others in respect of Causation. Hence
attempts to help the understanding by familiar comparisons are often
worse than useless. Any of the above examples will show that the first
result of explanation is not to make a phenomenon seem familiar, but to
put (as the saying is) 'quite a new face upon it.' When, indeed, we have
thought it over in all its newly discovered relations, we feel more at
home with it than ever; and this is one source of our satisfaction in
explaining things; and hence to substitute immediate familiarisation for
radical explanation, is the easily besetting sin of human understanding:
the most plausible of fallacies, the most attractive, the most difficult
to avoid even when we are on our guard against it.

§ 7. The explanation of Nature (if it be admitted to consist in
generalisation, or the discovery of resemblance amidst differences) can
never be completed. For--(1) there are (as Mill says) facts, namely,
fundamental states or processes of consciousness, which are distinct; in
other words, they do not resemble one another, and therefore cannot be
generalised or subsumed under one explanation. Colour, heat, smell,
sound, touch, pleasure and pain, are so different that there is one
group of conditions to be sought for each; and the laws of these
conditions cannot be subsumed under a more general one without leaving
out the very facts to be explained. A general condition of sensation,
such as the stimulating of the sensory organs of a living animal, gives
no account of the _special_ characters of colour, smell, etc.; which
are, however, the phenomena in question; and each of them has its own
law. Nay, each distinct sensation-quality, or degree, must have its own
law; for in each ultimate difference there is something that cannot be
assimilated. Such differences amount, according to experimental
Psychologists, to more than 50,000. Moreover, a neural process can never
explain a conscious process in the way of cause and effect; for there is
no equivalence between them, and one can never absorb the other.

(2) When physical science is treated objectively (that is, with as
little reference as possible to the fact that all phenomena are only
known in relation to the human mind), colour, heat, smell, sound
(considered as sensations) are neglected, and attention is fixed upon
certain of their conditions: extension, figure, resistance, weight,
motion, with their derivatives, density, elasticity, etc. These are
called the Primary Qualities of Matter; and it is assumed that they
belong to matter by itself, whether we look on or not: whilst colour,
heat, sound, etc., are called Secondary Qualities, as depending
entirely upon the reaction of some conscious animal. By physical science
the world is considered in the abstract, as a perpetual redistribution
of matter and energy, and the distracting multiplicity of sensations
seems to be got rid of.

But, not to dwell upon the difficulty of reducing the activities of life
and chemistry to mechanical principles--even if this were done, complete
explanation could not be attained. For--(a) as explanation is the
discovery of causes, we no sooner succeed in assigning the causes of the
present state of the world than we have to inquire into the causes of
those causes, and again the still earlier causes, and so on to infinity.
But, this being impossible, we must be content, wherever we stop, to
contemplate the uncaused, that is, the unexplained; and then all that
follows is only relatively explained.

Besides this difficulty, however, there is another that prevents the
perfecting of any theory of the abstract material world, namely (b),
that it involves more than one first principle. For we have seen that
the Uniformity of Nature is not really a principle, but a merely nominal
generalisation, since it cannot be definitely stated; and, therefore,
the principles of Contradiction, Mediate Equality, and Causation remain
incapable of subsumption; nor can any one of them be reduced to another:
so that they remain unexplained.

(3) Another limit to explanation lies in the infinite character of every
particular fact; so that we may know the laws of many of its properties
and yet come far short of understanding it as a whole. A lump of
sandstone in the road: we may know a good deal about its specific
gravity, temperature, chemical composition, geological conditions; but
if we inquire the causes of the particular modifications it exhibits of
these properties, and further why it is just so big, containing so many
molecules, neither more nor less, disposed in just such relations to one
another as to give it this particular figure, why it lies exactly there
rather than a yard off, and so forth, we shall get no explanation of all
this. The causes determining each particular phenomenon are infinite,
and can never be computed; and, therefore, it can never be fully

§ 8. Analogy is used in two senses: (1) for the resemblance of relations
between terms that have little or no resemblance--as _The wind drives
the clouds as a shepherd drives his sheep_--where wind and shepherd,
clouds and sheep are totally unlike. Such analogies are a favourite
figure in poetry and rhetoric, but cannot prove anything. For valid
reasoning there must be parallel cases, according to substance and
attribute, or cause and effect, or proportion: e.g. _As cattle and deer
are to herbivorousness, so are camels; As bodies near the earth fall
toward it, so does the moon; As 2 is to 3 so is 4 to 6._

(2) Analogy is discussed in Logic as a kind of probable proof based upon
imperfect similarity (as the best that can be discovered) between the
_data_ of comparison and the subject of our inference. Like Deduction
and Induction, it assumes that things which are alike in some respects
are also alike in others; but it differs from them in not appealing to a
definite general law assigning the essential points of resemblance upon
which the argument relies. In Deductive proof, this is done by the major
premise of every syllogism: if the major says that 'All fat men are
humorists,' and we can establish the minor, 'X is a fat man,' we have
secured the essential resemblance that carries the conclusion. In
induction, the Law of Causation and its representatives, the Canons,
serve the same purpose, specifying the essential marks of a cause. But,
in Analogy, the resemblance relied on cannot be stated categorically.

If we argue that Mars is inhabited because it resembles the datum, our
Earth, (1) in being a planet, (2) neither too hot nor too cold for life,
(3) having an atmosphere, (4) land and water, etc., we are not
prepared to say that 'All planets having these characteristics are
inhabited.' It is, therefore, not a deduction; and since we do not know
the original causes of life on the Earth, we certainly cannot show by
induction that adequate causes exist in Mars. We rely, then, upon some
such vague notion of Uniformity as that 'Things alike in some points are
alike in others'; which, plainly, is either false or nugatory. But if
the linear markings upon the surface of Mars indicate a system of
canals, the inference that he has intelligent inhabitants is no longer
analogical, since canals can have no other cause.

The cogency of any proof depends upon the _character_ and _definiteness_
of the likeness which one phenomenon bears to another; but Analogy
trusts to the general _quantity_ of likeness between them, in ignorance
of what may be the really important likeness. If, having tried with a
stone, an apple, a bullet, etc., we find that they all break an
ordinary window, and thence infer that a cricket ball will do so, we do
not reason by analogy, but make instinctively a deductive extension of
an induction, merely omitting the explicit generalisation, 'All missiles
of a certain weight, size and solidity break windows.' But if, knowing
nothing of snakes except that the viper is venomous, a child runs away
from a grass-snake, he argues by analogy; and, though his conduct is
prudentially justifiable, his inference is wrong: for there is no law
that 'All snakes are venomous,' but only that those are venomous that
have a certain structure of fang; a point which he did not stay to

The discovery of an analogy, then, may suggest hypotheses; it states a
problem--to find the causes of the analogy; and thus it may lead to
scientific proof; but merely analogical argument is only probable in
various degrees. (1) The greater the number and importance of the points
of agreement, the more probable is the inference. (2) The greater the
number and importance of the points of difference, the less probable is
the inference. (3) The greater the number of unknown properties in the
subject of our argument, the less the value of any inference from those
that we do know. Of course the number of unknown properties can itself
be estimated only by analogy. In the case of Mars, they are probably
very numerous; and, apart from the evidence of canals, the prevalent
assumption that there are intelligent beings in that planet, seems to
rest less upon probability than on a curiously imaginative extension of
the gregarious sentiment, the chilly discomfort of mankind at the
thought of being alone in the universe, and a hope that there may be
conversable and 'clubable' souls nearer than the Dog-star.



§ 1. Chance was once believed to be a distinct power in the world,
disturbing the regularity of Nature; though, according to Aristotle, it
was only operative in occurrences below the sphere of the moon. As,
however, it is now admitted that every event in the world is due to some
cause, if we can only trace the connection, whilst nevertheless the
notion of Chance is still useful when rightly conceived, we have to find
some other ground for it than that of a spontaneous capricious force
inherent in things. For such a conception can have no place in any
logical interpretation of Nature: it can never be inferred from a
principle, seeing that every principle expresses an uniformity; nor,
again, if the existence of a capricious power be granted, can any
inference be drawn from it. Impossible alike as premise and as
conclusion, for Reason it is nothing at all.

Every event is a result of causes: but the multitude of forces and the
variety of collocations being immeasurably great, the overwhelming
majority of events occurring about the same time are only related by
Causation so remotely that the connection cannot be followed. Whilst my
pen moves along the paper, a cab rattles down the street, bells in the
neighbouring steeple chime the quarter, a girl in the next house is
practising her scales, and throughout the world innumerable events are
happening which may never happen together again; so that should one of
them recur, we have no reason to expect any of the others. This is
Chance, or chance coincidence. The word Coincidence is vulgarly used
only for the inexplicable concurrence of _interesting_ events--"quite a

On the other hand, many things are now happening together or coinciding,
that will do so, for assignable reasons, again and again; thousands of
men are leaving the City, who leave at the same hour five days a week.
But this is not chance; it is causal coincidence due to the custom of
business in this country, as determined by our latitude and longitude
and other circumstances. No doubt the above chance
coincidences--writing, cab-rattling, chimes, scales, etc.--are
causally connected at some point of past time. They were predetermined
by the condition of the world ten minutes ago; and that was due to
earlier conditions, one behind the other, even to the formation of the
planet. But whatever connection there may have been, we have no such
knowledge of it as to be able to deduce the coincidence, or calculate
its recurrence. Hence Chance is defined by Mill to be: Coincidence
giving no ground to infer uniformity.

Still, some chance coincidences do recur according to laws of their own:
I say _some_, but it may be all. If the world is finite, the possible
combinations of its elements are exhaustible; and, in time, whatever
conditions of the world have concurred will concur again, and in the
same relation to former conditions. This writing, that cab, those
chimes, those scales will coincide again; the Argonautic expedition, and
the Trojan war, and all our other troubles will be renewed. But let us
consider some more manageable instance, such as the throwing of dice.
Every one who has played much with dice knows that double sixes are
sometimes thrown, and sometimes double aces. Such coincidences do not
happen once and only once; they occur again and again, and a great
number of trials will show that, though their recurrence has not the
regularity of cause and effect, it yet has a law of its own, namely--a
tendency to average regularity. In 10,000 throws there will be some
number of double sixes; and the greater the number of throws the more
closely will the average recurrence of double sixes, or double aces,
approximate to one in thirty-six. Such a law of average recurrence is
the basis of Probability. Chance being the fact of coincidence without
assignable cause, Probability is expectation based on the average
frequency of its happening.

§ 2. Probability is an ambiguous term. Usually, when we say that an
event is 'probable,' we mean that it is more likely than not to happen.
But, scientifically, an event is probable if our expectation of its
occurrence is less than certainty, as long as the event is not
impossible. Probability, thus conceived, is represented by a fraction.
Taking 1 to stand for certainty, and 0 for impossibility, probability
may be 999/1000, or 1/1000, or (generally) 1/_m_. The denominator
represents the number of times that an event happens, and the numerator
the number of times that it coincides with another event. In throwing a
die, the probability of ace turning up is expressed by putting the
number of throws for the denominator and the number of times that ace is
thrown for the numerator; and we may assume that the more trials we make
the nearer will the resulting fraction approximate to 1/6.

Instead of speaking of the 'throwing of the die' and its 'turning up
ace' as two events, the former is called 'the event' and the latter 'the
way of its happening.' And these expressions may easily be extended to
cover relations of distinct events; as when two men shoot at a mark and
we desire to represent the probability of both hitting the bull's eye
together, each shot may count as an event (denominator) and the
coincidence of 'bull's-eyes' as the way of its happening (numerator).

It is also common to speak of probability as a proportion. If the
fraction expressing the probability of ace being cast is 1/6, the
proportion of cases in which it happens is 1 to 5; or (as it is,
perhaps, still more commonly put) 'the chances are 5 to 1 against it.'

§ 3. As to the grounds of probability opinions differ. According to one
view the ground is subjective: probability depends, it is said, upon the
quantity of our Belief in the happening of a certain event, or in its
happening in a particular way. According to the other view the ground is
objective, and, in fact, is nothing else than experience, which is most
trustworthy when carefully expressed in statistics.

To the subjective view it may be objected, (a) that belief cannot by
itself be satisfactorily measured. No one will maintain that belief,
merely as a state of mind, always has a definite numerical value of
which one is conscious, as 1/100 or 1/10. Let anybody mix a number of
letters in a bag, knowing nothing of them except that one of them is X,
and then draw them one by one, endeavouring each time to estimate the
value of his belief that the next will be X; can he say that his belief
in the drawing of X next time regularly increases as the number of
letters left decreases?

If not, we see that (b) belief does not uniformly correspond with the
state of the facts. If in such a trial as proposed above, we really wish
to draw X, as when looking for something in a number of boxes, how
common it is, after a few failures, to feel quite hopeless and to say:
"Oh, of course it will be in the last." For belief is subject to hope
and fear, temperament, passion, and prejudice, and not merely to
rational considerations. And it is useless to appeal to 'the Wise Man,'
the purely rational judge of probability, unless he is producible. Or,
if it be said that belief is a short cut to the evaluation of
experience, because it is the resultant of all past experience, we may
reply that this is not true. For one striking experience, or two or
three recent ones, will immensely outweigh a great number of faint or
remote experiences. Moreover, the experience of two men may be
practically equal, whilst their beliefs upon any question greatly
differ. Any two Englishmen have about the same experience, personal and
ancestral, of the weather; yet their beliefs in the saw that 'if it rain
on St. Swithin's Day it will rain for forty days after,' may differ as
confident expectation and sheer scepticism. Upon which of these beliefs
shall we ground the probability of forty days' rain?

But (c) at any rate, if Probability is to be connected with Inductive
Logic, it must rest upon the same ground, namely--observation.
Induction, in any particular case, is not content with beliefs or
opinions, but aims at testing, verifying or correcting them by appealing
to the facts; and Probability has the same object and the same basis.

In some cases, indeed, the conditions of an event are supposed to be
mathematically predetermined, as in tossing a penny, throwing dice,
dealing cards. In throwing a die, the ways of happening are six; in
tossing a penny only two, head and tail: and we usually assume that the
odds with a die are fairly 5 to 1 against ace, whilst with a penny 'the
betting is even' on head or tail. Still, this assumption rests upon
another, that the die is perfectly fair, or that the head and tail of a
penny are exactly alike; and this is not true. With an ordinary die or
penny, a very great number of trials would, no doubt, give an average
approximating to 1/6 or 1/2; yet might always leave a certain excess one
way or the other, which would also become more definite as the trials
went on; thus showing that the die or penny did not satisfy the
mathematical hypothesis. Buffon is said to have tossed a coin 4040
times, obtaining 1992 heads and 2048 tails; a pupil of De Morgan tossed
4092 times, obtaining 2048 heads and 2044 tails.

There are other important cases in which probability is estimated and
numerically expressed, although statistical evidence directly bearing
upon the point in question cannot be obtained; as in betting upon a
race; or in the prices of stocks and shares, which are supposed to
represent the probability of their paying, or continuing to pay, a
certain rate of interest. But the judgment of experts in such matters is
certainly based upon experience; and great pains are taken to make the
evidence as definite as possible by comparing records of speed, or by
financial estimates; though something must still be allowed for reports
of the condition of horses, or of the prospects of war, harvests, etc.

However, where statistical evidence is obtainable, no one dreams of
estimating probability by the quantity of his belief. Insurance offices,
dealing with fire, shipwreck, death, accident, etc., prepare elaborate
statistics of these events, and regulate their rates accordingly. Apart
from statistics, at what rate ought the lives of men aged 40 to be
insured, in order to leave a profit of 5 per cent. upon £1000 payable at
each man's death? Is 'quantity of belief' a sufficient basis for doing
this sum?

§ 4. The ground of probability is experience, then, and, whenever
possible, statistics; which are a kind of induction. It has indeed been
urged that induction is itself based upon probability; that the
subtlety, complexity and secrecy of nature are such, that we are never
quite sure that we fully know even what we have observed; and that, as
for laws, the conditions of the universe at large may at any moment be
completely changed; so that all imperfect inductions, including the law
of causation itself, are only probable. But, clearly, this doctrine
turns upon another ambiguity in the word 'probable.' It may be used in
the sense of 'less than absolutely certain'; and such doubtless is the
condition of all human knowledge, in comparison with the comprehensive
intuition of arch-angels: or it may mean 'less than certain according
to _our_ standard of certainty,' that is, in comparison with the law of
causation and its derivatives.

We may suppose some one to object that "by this relative standard even
empirical laws cannot be called 'only probable' as long as we 'know no
exception to them'; for that is all that can be said for the boasted law
of causation; and that, accordingly, we can frame no fraction to
represent their probability. That 'all swans are white' was at one time,
from this point of view, not probable but certain; though we now know it
to be false. It would have been an indecorum to call it only probable as
long as no other-coloured swan had been discovered; not merely because
the quantity of belief amounted to certainty, but because the number of
events (seeing a swan) and the number of their happenings in a certain
way (being white) were equal, and therefore the evidence amounted to 1
or certainty." But, in fact, such an empirical law is only probable; and
the estimate of its probability must be based on the number of times
that similar laws have been found liable to exceptions. Albinism is of
frequent occurrence; and it is common to find closely allied varieties
of animals differing in colour. Had the evidence been duly weighed, it
could never have seemed more than probable that 'all swans are white.'
But what law, approaching the comprehensiveness of the law of causation,
presents any exceptions?

Supposing evidence to be ultimately nothing but accumulated experience,
the amount of it in favour of causation is incomparably greater than the
most that has ever been advanced to show the probability of any other
kind of event; and every relation of events which is shown to have the
marks of causation obtains the support of that incomparably greater body
of evidence. Hence the only way in which causation can be called
probable, for us, is by considering it as the upward limit (1) to which
the series of probabilities tends; as impossibility is the downward
limit (0). Induction, 'humanly speaking,' does not rest on probability;
but the probability of concrete events (not of mere mathematical
abstractions like the falling of absolutely true dice) rests on
induction and, therefore, on causation. The inductive evidence
underlying an estimate of probability may be of three kinds: (a) direct
statistics of the events in question; as when we find that, at the age
of 20, the average expectation of life is 39-40 years. This is an
empirical law, and, if we do not know the causes of any event, we must
be content with an empirical law. But (b) if we do know the causes of an
event, and the causes which may prevent its happening, and can estimate
the comparative frequency of their occurring, we may deduce the
probability that the effect (that is, the event in question) will occur.
Or (c) we may combine these two methods, verifying each by means of the
other. Now either the method (b) or (_a fortiori_) the method (c) (both
depending on causation) is more trustworthy than the method (a) by

But, further, a merely empirical statistical law will only be true as
long as the causes influencing the event remain the same. A die may be
found to turn ace once in six throws, on the average, in close
accordance with mathematical theory; but if we load it on that facet the
results will be very different. So it is with the expectation of life,
or fire, or shipwreck. The increased virulence of some epidemic such as
influenza, an outbreak of anarchic incendiarism, a moral epidemic of
over-loading ships, may deceive the hopes of insurance offices. Hence we
see, again, that probability depends upon causation, not causation upon

That uncertainty of an event which arises not from ignorance of the law
of its cause, but from our not knowing whether the cause itself does or
does not occur at any particular time, is Contingency.

§ 5. The nature of an average supposes deviations from it. Deviations
from an average, or "errors," are assumed to conform to the law (1) that
the greater errors are less frequent than the smaller, so that most
events approximate to the average; and (2) that errors have no "bias,"
but are equally frequent and equally great in both directions from the
mean, so that they are scattered symmetrically. Hence their distribution
may be expressed by some such figure as the following:

[Illustration: FIG. 11.]

Here _o_ is the average event, and _oy_ represents the number of average
events. Along _ox_, in either direction, deviations are measured. At _p_
the amount of error or deviation is _op_; and the number of such
deviations is represented by the line or ordinate _pa_. At _s_ the
deviation is _os_; and the number of such deviations is expressed by
_sb_. As the deviations grow greater, the number of them grows less. On
the other side of _o_, toward _-x_, at distances, _op'_, _os'_ (equal to
_op_, _os_) the lines _p'a'_, _s'b'_ represent the numbers of those
errors (equal to _pa_, _sb_).

If _o_ is the average height of the adult men of a nation, (say) 5 ft. 6
in., _s'_ and _s_ may stand for 5 ft. and 6 ft.; men of 4 ft. 6 in. lie
further toward _-x_, and men of 6 ft. 6 in. further toward _x_. There
are limits to the stature of human beings (or to any kind of animal or
plant) in both directions, because of the physical conditions of
generation and birth. With such events the curve _b'yb_ meets the
abscissa at some point in each direction; though where this occurs can
only be known by continually measuring dwarfs and giants. But in
throwing dice or tossing coins, whilst the average occurrence of ace is
once in six throws, and the average occurrence of 'tail' is once in two
tosses, there is no necessary limit to the sequences of ace or of 'tail'
that may occur in an infinite number of trials. To provide for such
cases the curve is drawn as if it never touched the abscissa.

That some such figure as that given above describes a frequent
characteristic of an average with the deviations from it, may be shown
in two ways: (1) By arranging the statistical results of any homogeneous
class of measurements; when it is often found that they do, in fact,
approximately conform to the figure; that very many events are near the
average; that errors are symmetrically distributed on either side, and
that the greater errors are the rarer. (2) By mathematical demonstration
based upon the supposition that each of the events in question is
influenced, more or less, by a number of unknown conditions common to
them all, and that these conditions are independent of one another. For
then, in rare cases, all the conditions will operate favourably in one
way, and the men will be tall; or in the opposite way, and the men will
be short; in more numerous cases, many of the conditions will operate in
one direction, and will be partially cancelled by a few opposing them;
whilst in still more cases opposed conditions will approximately balance
one another and produce the average event or something near it. The
results will then conform to the above figure.

From the above assumption it follows that the symmetrical curve
describes only a 'homogeneous class' of measurements; that is, a class
no portion of which is much influenced by conditions peculiar to itself.
If the class is not homogeneous, because some portion of it is subject
to _peculiar_ conditions, the curve will show a hump on one side or the
other. Suppose we are tabulating the ages at which Englishmen die who
have reached the age of 20, we may find that the greatest number die at
39 (19 years being the average expectation of life at 20) and that as
far as that age the curve upwards is regular, and that beyond the age of
39 it begins to descend regularly, but that on approaching 45 it bulges
out some way before resuming its regular descent--thus:

[Illustration: FIG. 12.]

Such a hump in the curve might be due to the presence of a considerable
body of teetotalers, whose longevity was increased by the peculiar
condition of abstaining from alcohol, and whose average age was 45, 6
years more than the average for common men.

Again, if the group we are measuring be subject to selection (such as
British soldiers, for which profession all volunteers below a certain
height--say, 5 ft. 5 in.--are rejected), the curve will fall steeply on
one side, thus:

[Illustration: FIG. 13.]

If, above a certain height, volunteers are also rejected, the curve will
fall abruptly on both sides. The average is supposed to be 5 ft. 8 in.

The distribution of events is described by 'some such curve' as that
given in Fig. 11; but different groups of events may present figures or
surfaces in which the slopes of the curves are very different, namely,
more or less steep; and if the curve is very steep, the figure runs into
a peak; whereas, if the curve is gradual, the figure is comparatively
flat. In the latter case, where the figure is flat, fewer events will
closely cluster about the average, and the deviations will be greater.

Suppose that we know nothing of a given event except that it belongs to
a certain class or series, what can we venture to infer of it from our
knowledge of the series? Let the event be the cephalic index of an
Englishman. The cephalic index is the breadth of a skull × 100 and
divided by the length of it; e.g. if a skull is 8 in. long and 6 in.
broad, (6×100)/8=75. We know that the average English skull has an index
of 78. The skull of the given individual, therefore, is more likely to
have that index than any other. Still, many skulls deviate from the
average, and we should like to know what is the probable error in this
case. The probable error is the measurement that divides the deviations
from the average in either direction into halves, so that there are as
many events having a greater deviation as there are events having a less
deviation. If, in Fig. 11 above, we have arranged the measurements of
the cephalic index of English adult males, and if at _o_ (the average or
mean) the index is 78, and if the line _pa_ divides the right side of
the fig. into halves, then _op_ is the probable error. If the
measurement at _p_ is 80, the probable error is 2. Similarly, on the
left hand, the probable error is _op'_, and the measurement at _p'_ is
76. We may infer, then, that the skull of the man before us is more
likely to have an index of 78 than any other; if any other, it is
equally likely to lie between 80 and 76, or to lie outside them; but as
the numbers rise above 80 to the right, or fall below 76 to the left, it
rapidly becomes less and less likely that they describe this skull.

In such cases as heights of men or skull measurements, where great
numbers of specimens exist, the average will be actually presented by
many of them; but if we take a small group, such as the measurements of
a college class, it may happen that the average height (say, 5 ft. 8
in.) is not the actual height of any one man. Even then there will
generally be a closer cluster of the actual heights about that number
than about any other. Still, with very few cases before us, it may be
better to take the median than the average. The median is that event on
either side of which there are equal numbers of deviations. One
advantage of this procedure is that it may save time and trouble. To
find approximately the average height of a class, arrange the men in
order of height, take the middle one and measure him. A further
advantage of this method is that it excludes the influence of
extraordinary deviations. Suppose we have seven cephalic indices, from
skeletons found in the same barrow, 75-1/2, 76, 78, 78, 79, 80-1/2, 86.
The average is 79; but this number is swollen unduly by the last
measurement; and the median, 78, is more fairly representative of the
series; that is to say, with a greater number of skulls the average
would probably have been nearer 78.

To make a single measurement of a phenomenon does not give one much
confidence. Another measurement is made; and then, if there is no
opportunity for more, one takes the mean or average of the two. But why?
For the result may certainly be worse than the first measurement.
Suppose that the events I am measuring are in fact fairly described by
Fig. II, although (at the outset) I know nothing about them; and that my
first measurement gives _p_, and my second _s_; the average of them is
worse than _p_. Still, being yet ignorant of the distribution of these
events, I do rightly in taking the average. For, as it happens, 3/4 of
the events lie to the left of _p_; so that if the first trial gives _p_,
then the average of _p_ and any subsequent trial that fell nearer than
(say) _s'_ on the opposite side, would be better than _p_; and since
deviations greater than _s'_ are rare, the chances are nearly 3 to 1
that the taking of an average will improve the observation. Only if the
first trial give _o_, or fall within a little more than 1/2 _p_ on
either side of _o_, will the chances be against any improvement by
trying again and taking an average. Since, therefore, we cannot know the
position of our first trial in relation to _o_, it is always prudent to
try again and take the average; and the more trials we can make and
average, the better is the result. The average of a number of
observations is called a "Reduced Observation."

We may have reason to believe that some of our measurements are better
than others because they have been taken by a better trained observer,
or by the same observer in a more deliberate way, or with better
instruments, and so forth. If so, such observations should be
'weighted,' or given more importance in our calculations; and a simple
way of doing this is to count them twice or oftener in taking the

§ 6. These considerations have an important bearing upon the
interpretation of probabilities. The average probability for any
_general class_ or series of events cannot be confidently applied to any
_one instance_ or to any _special class_ of instances, since this one,
or this special class, may exhibit a striking error or deviation; it
may, in fact, be subject to special causes. Within the class whose
average is first taken, and which is described by general characters as
'a man,' or 'a die,' or 'a rifle shot,' there may be classes marked by
special characters and determined by special influences. Statistics
giving the average for 'mankind' may not be true of 'civilised men,' or
of any still smaller class such as 'Frenchmen.' Hence life-insurance
offices rely not merely on statistics of life and death in general, but
collect special evidence in respect of different ages and sexes, and
make further allowance for teetotalism, inherited disease, etc.
Similarly with individual cases: the average expectation for a class,
whether general or special, is only applicable to any particular case if
that case is adequately described by the class characters. In England,
for example, the average expectation of life for males at 20 years of
age is 39.40; but at 60 it is still 13.14, and at 73 it is 7.07; at 100
it's 1.61. Of men 20 years old those who live more or less than 39.40
years are deviations or errors; but there are a great many of them. To
insure the life of a single man at 20, in the expectation of his dying
at 60, would be a mere bet, if we had no special knowledge of him; the
safety of an insurance office lies in having so many clients that
opposite deviations cancel one another: the more clients the safer the
business. It is quite possible that a hundred men aged 20 should be
insured in one week and all of them die before 25; this would be
ruinous, if others did not live to be 80 or 90.

Not only in such a practical affair as insurance, but in matters purely
scientific, the minute and subtle peculiarities of individuals have
important consequences. Each man has a certain cast of mind, character,
physique, giving a distinctive turn to all his actions even when he
tries to be normal. In every employment this determines his Personal
Equation, or average deviation from the normal. The term Personal
Equation is used chiefly in connection with scientific observation, as
in Astronomy. Each observer is liable to be a little wrong, and this
error has to be allowed for and his observations corrected accordingly.

The use of the term 'expectation,' and of examples drawn from insurance
and gambling, may convey the notion that probability relates entirely to
future events; but if based on laws and causes, it can have no reference
to point of time. As long as conditions are the same, events will be the
same, whether we consider uniformities or averages. We may therefore
draw probable inferences concerning the past as well as the future,
subject to the same hypothesis, that the causes affecting the events in
question were the same and similarly combined. On the other hand, if we
know that conditions bearing on the subject of investigation, have
changed since statistics were collected, or were different at some time
previous to the collection of evidence, every probable inference based
on those statistics must be corrected by allowing for the altered
conditions, whether we desire to reason forwards or backwards in time.

§ 7. The rules for the combination of probabilities are as follows:

(1) If two events or causes do not concur, the probability of one or the
other occurring is the sum of the separate probabilities. A die cannot
turn up both ace and six; but the probability in favour of each is 1/6:
therefore, the probability in favour of one or the other is 1/3. Death
can hardly occur from both burning and drowning: if 1 in 1000 is burned
and 2 in 1000 are drowned, the probability of being burned or drowned is

(2) If two events are independent, having neither connection nor
repugnance, the probability of their concurring is found by multiplying
together the separate probabilities of each occurring. If in walking
down a certain street I meet A once in four times, and B once in three
times, I ought (by mere chance) to meet both once in twelve times: for
in twelve occasions I meet B four times; but once in four I meet A.

This is a very important rule in scientific investigation, since it
enables us to detect the presence of causation. For if the coincidence
of two events is more or less frequent than it would be if they were
entirely independent, there is either connection or repugnance between
them. If, e.g., in walking down the street I meet both A and B oftener
than once in twelve times, they may be engaged in similar business,
calling them from their offices at about the same hour. If I meet them
both less often than once in twelve times, they may belong to the same
office, where one acts as a substitute for the other. Similarly, if in a
multitude of throws a die turns six oftener than once in six times, it
is not a fair one: that is, there is a cause favouring the turning of
six. If of 20,000 people 500 see apparitions and 100 have friends
murdered, the chance of any man having both experiences is 1/8000; but
if each lives on the average 300,000 hours, the chance of both events
occurring in the same hour is 1/2400000000. If the two events occur in
the same hour oftener than this, there is more than a chance

The more minute a cause of connection or repugnance between events, the
longer the series of trials or instances necessary to bring out its
influence: the less a die is loaded, the more casts must be made before
it can be shown that a certain side tends to recur oftener than once in

(3) The rule for calculating the probability of a dependent event is the
same as the above; for the concurrence of two independent events is
itself dependent upon each of them occurring. My meeting with both A and
B in the street is dependent on my walking there and on my meeting one
of them. Similarly, if A is sometimes a cause of B (though liable to be
frustrated), and B sometimes of C (C and B having no causes independent
of B and A respectively), the occurrence of C is dependent on that of B,
and that again on the occurrence of A. Hence we may state the rule: If
two events are dependent each on another, so that if one occur the
second may (or may not), and if the second a third; whilst the third
never occurs without the second, nor the second without the first; the
probability that if the first occur the third will, is found by
multiplying together the fractions expressing the probability that the
first is a mark of the second and the second of the third.

Upon this principle the value of hearsay evidence or tradition
deteriorates, and generally the cogency of any argument based upon the
combination of approximate generalisations dependent on one another or
"self-infirmative." If there are two witnesses, A and B, of whom A saw
an event, whilst B only heard A relate it (and is therefore dependent on
A), what credit is due to B's recital? Suppose the probability of each
man's being correct as to what he says he saw, or heard, is 3/4: then
(3/4 × 3/4 = 9/16) the probability that B's story is true is a little
more than 1/2. For if in 16 attestations A is wrong 4 times, B can only
be right in 3/4 of the remainder, or 9 times in 16. Again, if we have
the Approximate Generalisations, 'Most attempts to reduce wages are met
by strikes,' and 'Most strikes are successful,' and learn, on
statistical inquiry, that in every hundred attempts to reduce wages
there are 80 strikes, and that 70 p.c. of the strikes are successful,
then 56 p.c. of attempts to reduce wages are unsuccessful.

Of course this method of calculation cannot be quantitatively applied if
no statistics are obtainable, as in the testimony of witnesses; and even
if an average numerical value could be attached to the evidence of a
certain class of witnesses, it would be absurd to apply it to the
evidence of any particular member of the class without taking account of
his education, interest in the case, prejudice, or general capacity.
Still, the numerical illustration of the rapid deterioration of hearsay
evidence, when less than quite veracious, puts us on our guard against
rumour. To retail rumour may be as bad as to invent an original lie.

(4) If an event may coincide with two or more other independent events,
the probability that they will together be a sign of it, is found by
multiplying together the fractions representing the improbability that
each is a sign of it, and subtracting the product from unity.

This is the rule for estimating the cogency of circumstantial evidence
and analogical evidence; or, generally, for combining approximate
generalisations "self-corroboratively." If, for example, each of two
independent circumstances, A and B, indicates a probability of 6 to 1 in
favour of a certain event; taking 1 to represent certainty, 1-6/7 is the
improbability of the event, notwithstanding each circumstance. Then 1/7
× 1/7 = 1/49, the improbability of both proving it. Therefore the
probability of the event is 48 to 1. The matter may be plainer if put
thus: A's indication is right 6 times in 7, or 42 in 49; in the
remaining 7 times in 49, B's indication will be right 6 times.
Therefore, together they will be right 48 times in 49. If each of two
witnesses is truthful 6 times in 7, one or the other will be truthful 48
times in 49. But they will not be believed unless they agree; and in
the 42 cases of A being right, B will contradict him 6 times; so that
they only concur in being right 36 times. In the remaining 7 times in
which A is wrong, B will contradict him 6 times, and once they will both
be wrong. It does not follow that when both are wrong they will concur;
for they may tell very different stories and still contradict one

If in an analogical argument there were 8 points of comparison, 5 for
and 3 against a certain inference, and the probability raised by each
point could be quantified, the total value of the evidence might be
estimated by doing similar sums for and against, and subtracting the
unfavourable from the favourable total.

When approximate generalisations that have not been precisely quantified
combine their evidence, the cogency of the argument increases in the
same way, though it cannot be made so definite. If it be true that most
poets are irritable, and also that most invalids are irritable, a still
greater proportion will be irritable of those who are both invalids and

On the whole, from the discussion of probabilities there emerge four
principal cautions as to their use: Not to make a pedantic parade of
numerical probability, where the numbers have not been ascertained; Not
to trust to our feeling of what is likely, if statistics can be
obtained; Not to apply an average probability to special classes or
individuals without inquiring whether they correspond to the average
type; and Not to trust to the empirical probability of events, if their
causes can be discovered and made the basis of reasoning which the
empirical probability may be used to verify.

The reader who wishes to pursue this subject further should read a work
to which the foregoing chapter is greatly indebted, Dr. Venn's _Logic of



§ 1. Classification, in its widest sense, is a mental grouping of facts
or phenomena according to their resemblances and differences, so as best
to serve some purpose. A "mental grouping": for although in museums we
often see the things themselves arranged in classes, yet such an
arrangement only contains specimens representing a classification. The
classification itself may extend to innumerable objects most of which
have never been seen at all. Extinct animals, for example, are
classified from what we know of their fossils; and some of the fossils
may be seen arranged in a museum; but the animals themselves have
disappeared for many ages.

Again, things are classed according to their resemblances and
differences: that is to say, those that most closely resemble one
another are classed together on that ground; and those that differ from
one another in important ways, are distributed into other classes. The
more the things differ, the wider apart are their classes both in
thought and in the arrangements of a museum. If their differences are
very great, as with animals, vegetables and minerals, the classing of
them falls to different departments of thought or science, and is often
represented in different museums, zoological, botanical, mineralogical.

We must not, however, suppose that there is only one way of classifying
things. The same objects may be classed in various ways according to
the purpose in view. For gardening, we are usually content to classify
plants into trees, shrubs, flowers, grasses and weeds; the ordinary
crops of English agriculture are distinguished, in settling their
rotation, into white and green; the botanist divides the higher plants
into gymnosperms and angiosperms, and the latter into monocotyledons and
dicotyledons. The principle of resemblance and difference is recognised
in all these cases; but what resemblances or differences are important
depends upon the purpose to be served.

Purposes are either (α) special or practical, as in gardening or
hunting, or (β) general or scientific, as in Botany or Zoology. The
scientific purpose is merely knowledge; it may indeed subserve all
particular or practical ends, but has no other end than knowledge
directly in view. And whilst, even for knowledge, different
classifications may be suitable for different lines of inquiry, in
Botany and Zoology the Morphological Classification is that which gives
the most general and comprehensive knowledge (see Huxley, _On the
Classification of Animals_, ch. 1). Most of what a logician says about
classification is applicable to the practical kind; but the scientific
(often called 'Natural Classification'), as the most thorough and
comprehensive, is what he keeps most constantly before him.

Scientific classification comes late in human history, and at first
works over earlier classifications which have been made by the growth of
intelligence, of language, and of the practical arts. Even in the
distinctions recognised by animals, may be traced the grounds of
classification: a cat does not confound a dog with one of its own
species, nor water with milk, nor cabbage with fish. But it is in the
development of language that the progress of instinctive classification
may best be seen. The use of general names implies the recognition of
classes of things corresponding to them, which form their denotation,
and whose resembling qualities, so far as recognised, form their
connotation; and such names are of many degrees of generality. The use
of abstract names shows that the objects classed have also been
analysed, and that their resembling qualities have been recognised
amidst diverse groups of qualities.

Of the classes marked by popular language it is worth while to
distinguish two sorts (_cf._ chap. xix. § 4): Kinds, and those having
but few points of agreement.

But the popular classifications, made by language and the primitive
arts, are very imperfect. They omit innumerable things which have not
been found useful or noxious, or have been inconspicuous, or have not
happened to occur in the region inhabited by those who speak a
particular language; and even things recognised and named may have been
very superficially examined, and therefore wrongly classed, as when a
whale or porpoise is called a fish, or a slowworm is confounded with
snakes. A scientific classification, on the other hand, aims at the
utmost comprehensiveness, ransacking the whole world from the depths of
the earth to the remotest star for new objects, and scrutinising
everything with the aid of crucible and dissecting knife, microscope and
spectroscope, to find the qualities and constitution of everything, in
order that it may be classed among those things with which it has most
in common and distinguished from those other things from which it
differs. A scientific classification continually grows more
comprehensive, more discriminative, more definitely and systematically
coherent. Hence the uses of classification may be easily perceived.

§ 2. The first use of classification is the better understanding of the
facts of Nature (or of any sphere of practice); for understanding
consists in perceiving and comprehending the likeness and difference of
things, in assimilating and distinguishing them; and, in carrying out
this process systematically, new correlations of properties are
continually disclosed. Thus classification is closely analogous to
explanation. Explanation has been shown (chap. xix. § 5) to consist in
the discovery of the laws or causes of changes in Nature; and laws and
causes imply similarity, or like changes under like conditions: in the
same way classification consists in the discovery of resemblances in the
things that undergo change. We may say (subject to subsequent
qualifications) that Explanation deals with Nature in its dynamic,
Classification in its static aspect. In both cases we have a feeling of
relief. When the cause of any event is pointed out, or an object is
assigned its place in a system of classes, the gaping wonder, or
confusion, or perplexity, occasioned by an unintelligible thing, or by a
multitude of such things, is dissipated. Some people are more than
others susceptible of this pleasure and fastidious about its purity.

A second use of classification is to aid the memory. It strengthens
memory, because one of the conditions of our recollecting things is,
that they resemble what we last thought of; so that to be accustomed to
study and think of things in classes must greatly facilitate
recollection. But, besides this, a classification enables us easily to
run over all the contrasted and related things that we want to think of.
Explanation and classification both tend to rationalise the memory, and
to organise the mind in correspondence with Nature.

Every one knows how a poor mind is always repeating itself, going by
rote through the same train of words, ideas, actions; and that such a
mind is neither interesting nor practical. It is not practical, because
the circumstances of life are rarely exactly repeated, so that for a
present purpose it is rarely enough to remember only one former case; we
need several, that by comparing (perhaps automatically) their
resemblances and differences with the one before us, we may select a
course of action, or a principle, or a parallel, suited to our
immediate needs. Greater fertility and flexibility of thought seem
naturally to result from the practice of explanation and classification.
But it must be honestly added, that the result depends upon the spirit
in which such study is carried on; for if we are too fond of finality,
too eager to believe that we have already attained a greater precision
and comprehension than are in fact attainable, nothing can be more
petrific than 'science,' and our last state may be worse than the first.
Of this, students of Logic have often furnished examples.

§ 3. Classification may be either Deductive or Inductive; that is to
say, in the formation of classes, as in the proof of propositions, we
may, on the whole, proceed from the more to the less, or from the less
to the more general; not that these two processes are entirely

If we begin with some large class, such as 'Animal,' and subdivide it
deductively into Vertebrate and Invertebrate, yet the principle of
division (namely, central structure) has first been reached by a
comparison of examples and by generalisation; if, on the other hand,
beginning with individuals, we group them inductively into classes, and
these again into wider ones (as dogs, rats, horses, whales and monkeys
into mammalia) we are guided both in special cases by hypotheses as to
the best grounds of resemblance, and throughout by the general principle
of classification--to associate things that are alike and to separate
things that are unlike. This principle holds implicitly a place in
classification similar to that of causation in explanation; both are
principles of intelligence. Here, then, as in proof, induction is
implied in deduction, and deduction in induction. Still, the two modes
of procedure may be usefully distinguished: in deduction, we proceed
from the idea of a whole to its parts, from general to special; in
induction, from special (or particular) to general, from parts to the
idea of a whole.

§ 4. The process of Deductive Classification, or Formal Division, may be
represented thus:

          |                 |
         A B               A b
          |                 |
      ----------        ----------
      |        |        |        |
    A B C    A B c    A b C    A b c

Given any class (A) to be divided:

1. Select one important character, attribute, or quality (B), not common
to all the individuals comprehended in the class, as the basis of
division (_fundamentum divisionis_).

2. Proceed by Dichotomy; that is, cut the given class into two, one
having the selected attribute (say, B), the other not having it (b).
This, like all formal processes, assumes the principles of Contradiction
and Excluded Middle, that 'No A is both B and not-B,' and that 'Every A
is either B or not-B' (chap. vi. § 3); and if these principles are not
true, or not applicable, the method fails.

When a class is thus subdivided, it may be called, in relation to its
subclasses, a Genus; and in relation to it, the subclasses may be called
Species: thus--genus A, species AB and Ab, etc.

3. Proceed gradually in the order of the importance of characters; that
is, having divided the given class, subdivide on the same principle the
two classes thence arising; and so again and again, step by step, until
all the characters are exhausted: _Divisio ne fiat per saltum_.

Suppose we were to attempt an exhaustive classification of things by
this method, we must begin with 'All Things,' and divide them (say) into
phenomenal and not-phenomenal, and then subdivide phenomena, and so on,

                                     All Things
                                 |                |
                             Phenomenal    Not-phenomenal
                        |                  |
                     Extended          Unextended
                        |      (e.g., Pleasure and Pain)
                 |              |
             Resistant    Not-resistant
             (Matter)        (Space)
        |                 |
   Gravitating    Not-gravitating
   |         |
Simple    Compound

Having subdivided 'Simple' by all possible characters, we must then go
back and similarly subdivide Not-phenomenal, Unextended, Not-resistant,
Not-gravitating, and Compound. Now, if we knew all possible characters,
and the order of their importance, we might prepare _a priori_ a
classification of all possible things; at least, of all things that come
under the principles of Contradiction and Excluded Middle. Many of our
compartments might contain nothing actual; there may, for example, be
nothing that is not phenomenal to some mind, or nothing that is extended
and not-resistant (no vacuum), and so forth. This would imply a breach
of the rule, that the dividing quality be not common to the whole class;
but, in fact, doubts have been, and are, seriously entertained whether
these compartments are filled or not. If they are not, we have concepts
representing nothing, which have been generated by the mere force of
grammatical negation, or by the habit of thinking according to the
principle of Excluded Middle; and, on the strength of these empty
concepts, we have been misled into dividing by an attribute, which
(being universal) cannot be a _fundamentum divisionis_. But though in
such a classification places might be empty, there would be a place for
everything; for whatever did not come into some positive class (such as
Gravitating) must fall under one of the negative classes (the 'Nots')
that run down the right-hand side of the Table and of its subdivisions.

This is the ideal of classification. Unfortunately we have to learn what
characters or attributes are possible, by experience and comparison; we
are far from knowing them all: and we do not know the order of their
importance; nor are we even clear what 'important' means in this
context, whether 'widely prevalent,' or 'ancient,' or 'causally
influential,' or 'indicative of others.' Hence, in classifying actual
things, we must follow the inductive method of beginning with
particulars, and sorting them according to their likeness and difference
as discovered by investigation. The exceptional cases, in which
deduction is really useful, occur where certain limits to the number and
combination of qualities happen to be known, as they may be in human
institutions, or where there are mathematical conditions. Thus, we might
be able to classify orders of Architecture, or the classical metres and
stanzas of English poetry; though, in fact, these things are too free,
subtle and complex for deductive treatment: for do not the Arts grow
like trees? The only sure cases are mathematical; as we may show that
there are possible only three kinds of plane triangles, four conic
sections, five regular solids.

§ 5. The rules for _testing_ a Division are as follows:

1. Each Sub-class, or Species, should comprise less than the Class, or
Genus, to be divided. This provides that the division shall be a real
one, and not based upon an attribute common to the whole class; that,
therefore, the first rule for making a division shall have been adhered
to. But, as in § 4, we are here met by a logical difficulty. Suppose
that the class to be divided is A, and that we attempt to divide upon
the attribute B, into AB and Ab; is this a true division, if we do not
know any A that is not B? As far as our knowledge extends, we have not
divided A at all. On the other hand, our knowledge of concrete things is
never exhaustive; so that, although we know of no A that is not B, it
may yet exist, and we have seen that it is a logical caution not to
assume what we do not know. In a deductive classification, at least, it
seems better to regard every attribute as a possible ground of division.
Hence, in the above division of 'All Things,'--'Not-phenomenal,'
'Extended-Not-resistant,' 'Resistant-Not-gravitating,' appear as
negative classes (that is, classes based on the negation of an
attribute), although their real existence may be doubtful. But, if this
be justifiable, we must either rewrite the first test of a division
thus: 'Each sub-class should _possibly_ comprise less than the class to
be divided'; or else we must confine the test to (a) thoroughly
empirical divisions, as in dividing Colour into Red and Not-red, where
we know that both sub-classes are real; and (b) divisions under
demonstrable conditions--as in dividing the three kinds of triangles by
the quality equilateral, we know that it is only applicable to
acute-angled triangles, and do not attempt to divide the right-angled or
obtuse-angled by it.

2. The Sub-classes taken together should be equal to the Class to be
divided: the sum of the Species constitutes the Genus. This provides
that the division shall be exhaustive; which dichotomy always secures,
according to the principle of Excluded Middle; because whatever is not
in the positive class, must be in the negative: Red and Not-red include
all colours.

3. The Sub-classes must be opposed or mutually exclusive: Species must
not overlap. This again is secured by dichotomy, according to the
principle of Contradiction, provided the division be made upon one
attribute at a time. But, if we attempt to divide simultaneously upon
two attributes, as 'Musicians' upon 'nationality' and 'method,' we get
what is called a Cross-division, thus 'German Musicians.' 'Not-German,'
'Classical,' 'Not-Classical;' for these classes may overlap, the same
men sometimes appearing in two groups--Bach in 'German' and 'Classical,'
Pergolesi in 'Not-German' and 'Classical.' If, however, we divide
Musicians upon these attributes successively, cross division will be
avoided, thus:

         |                      |
     Classical            Not-classical
         |                      |
   ------------            ------------
   |          |            |          |
German    Not-German    German    Not-German

Here no Musician will be found in two classes, _unless_ he has written
works in two styles, _or unless_ there are works whose style is
undecided. This "unless--or unless" may suggest caution in using
dichotomy as a short cut to the classification of realities.

4. No Sub-class must include anything that is not comprised in the class
to be divided: the Genus comprises all the Species. We must not divide
Dogs into fox-terriers and dog-fish.

§ 6. The process of Inductive Classification may be represented thus:

Given any multitude of individuals to be classified:

(1) Place together in groups (or in thought) those things that have in
common the most, the most widely diffused and the most important

(2) Connect those groups which have, as groups, the greater resemblance,
and separate those that have the greater difference.

(3) Demarcate, as forming higher or more general classes, those groups
of groups that have important characters in common; and, if possible, on
the same principle, form these higher classes into classes higher still:
that is to say, graduate the classification upwards.

Whilst in Division the terms 'Genus' and 'Species' are entirely relative
to one another and have no fixed positions in a gradation of classes, it
has been usual, in Inductive Classification, to confine the term
'Species' to classes regarded as lowest in the scale, to give the term
'Genera' to classes on the step above, and at each higher step to find
some new term such as 'Tribe,' 'Order,' 'Sub-kingdom,' 'Kingdom'; as may
be seen by turning to any book on Botany or Zoology. If, having fixed
our Species, we find them subdivisible, it is usual to call the
Sub-species 'Varieties.'

Suppose an attempt to classify by this method the objects in a
sitting-room. We see at a glance carpets, mats, curtains, grates,
fire-irons, coal-scuttles, chairs, sofas, tables, books, pictures,
musical instruments, etc. These may be called 'Species.' Carpets and
mats go together; so do chairs and sofas; so do grates, fire-irons, and
coal-scuttles and so on. These greater groups, or higher classes, are
'Genera.' Putting together carpets, mats and curtains as
'warmth-fabrics'; chairs, sofas and tables as 'supports'; books,
pictures and musical instruments as 'means of culture'; these groups we
may call Orders. Sum up the whole as, from the housewife's point of
view, 'furniture.' If we then subdivide some of the species, as books
into poetry, novels, travels, etc., these Sub-species may be
considered 'Varieties.'

A Classification thus made, may be tested by the same rules as those
given for testing a Division; but if it does not stand the test, we must
not infer that the classification is a bad one. If the best possible, it
is good, though formally imperfect: whatever faults are found must then
be charged upon the 'matter,' which is traditionally perverse and
intractable. If, for example, there is a hammock in the room, it must be
classed not with the curtains as a warmth-fabric, but with the sofas as
a support; and books and pictures may be classed as, in a peculiar
sense, means of culture, though all the objects in the room may have
been modified and assorted with a view to gratifying and developing good

§ 7. The difficulty of classifying natural objects is very great. It is
not enough to consider their external appearance: exhaustive knowledge
of their internal structure is necessary, and of the functions of every
part of their structure. This is a matter of immense research, and has
occupied many of the greatest minds for very many years. The following
is a tabular outline of the classification of the

                      Animal Kingdom
                   |                 |
SUB-KINGDOM,  Vertebrates    Invertebrates (10 Sub-kingdom 5)
    or             |
  Phylum:          |-------------------------------
                   |           |                  |
                   |      Sauropsida        Ichthyopsida
                   |           |                  |
                   |       ---------         -----------
                   |       |       |         |         |
CLASS:         Mammalia  Birds  Reptiles  Amphibia  Fishes
                |                       |
SUB-CLASS:  Placental              Implacental
                |                       |
                |              -------------------
                |              |                 |
DIVISION:  Monodelphia     Didelphia       Ornithodelphia
            |          |       |         |        |
ORDER:  Quadrumana Rodentia Carnivora Ungulata Cætacea, etc.
              |                |               |
SECTION:  Pinnigrada      Plantigrada     Digitigrada
              |                |               |
        (Seals, etc.)  (Bears, etc.)           |
             |          |         |        |      |
GENUS:   Mustelidæ  Viverridæ  Hyaænidæ  Canidæ  Felidæ
             |            |                       |
    (Weasels, etc.)   (Civets, etc.)              |
              |      |        |       |      |      |
SPECIES:    Lion   Tiger   Leopard   Puma   Lynx   Cat, etc.
             |         |         |
VARIETY:  African    Syrian    Cave-lion (extinct)

As there is not space enough to tabulate such a classification in full,
I have developed at each step the most interesting groups: Vertebrates,
Mammalia, Monodelphia Carnivora, Digitigrada, Felidæ, Lion. Most of the
other groups in each grade are also subdivisible, though some of them
contain far fewer sub-classes than others.

To see the true character of this classification, we must consider that
it is based chiefly upon knowledge of existing animals. Some extinct
animals, known by their fossils, find places in it; for others new
places have been made. But it represents, on the whole, a cross-section,
or cross-sections of Nature as developing in time; and, in order to give
a just view of the relations of animals, it must be seen in the light of
other considerations. The older systems of classification, and the rules
for making them, seem to have assumed that an actual system of classes,
or of what Mill calls 'Kinds,' exists in nature, and that the relations
of Kinds in this system are determined by quantity of resemblance in
co-inherent qualities, as the ground of their affinity.

§ 8. Darwin's doctrine of the origin of species affects the conception
of natural classification in several ways, (1) If all living things are
blood-relations, modified in the course of ages according to their
various conditions of life, 'affinity' must mean 'nearness of common
descent'; and it seems irrational to propose a classification upon any
other basis. We have to consider the Animal (or the Vegetable) Kingdom
as a family tree, exhibiting a long line of ancestors, and (descended
from them) all sorts of cousins, first, second, third, etc., perhaps
once, twice, or oftener 'removed.' Animals in the relation of first
cousins must be classed as nearer than second cousins, and so on.

But, if we accept this principle, and are able to trace relationship, it
may not lead to the same results as would be reached by simply relying
upon the present 'quantity of resemblance,' unless we understand this
in a very particular way. For the most obvious features of an animal may
have been recently acquired; which often happens with those characters
that adapt an animal to its habits of life, as the wings of a bat, or
the fish-like shape of a dolphin; or as in cases of 'mimicry.' Some
butterflies, snakes, etc., have grown to resemble closely, in a
superficial way, other butterflies and snakes, from which a stricter
investigation widely separates them; and this superficial resemblance is
probably a recent acquisition, for the sake of protection; the imitated
butterflies being nauseous, and the imitated snakes poisonous. On the
other hand, ancient and important traits of structure may, in some
species, have dwindled into inconspicuous survivals or be still found
only in the embryo; so that only great knowledge and sagacity can
identify them; yet upon ancient traits, though hidden, classification
depends. The seal seems nearer allied to the porpoise than to the tiger,
the shrew nearer to the mouse than to the hedgehog; and the Tasmanian
wolf looks more like a true wolf, the Tasmanian devil more like a
badger, than like a kangaroo: yet the seal is nearer akin to the tiger,
the shrew to the hedgehog, and the Tasmanian flesh-eaters are marsupial,
like the kangaroo. To overcome this difficulty we must understand the
resemblance upon which classification is based to include resemblance of
Causation, that is, the fact itself of descent from common ancestors.
For organic beings, all other rules of classification are subordinate to
one: trace the genealogy of every form.

By this rule we get a definite meaning for the phrase 'important or
fundamental attribute' as determining organic classes; namely, most
ancient, or 'best serving to indicate community of origin.' Grades of
classification will be determined by such fundamental characters, and
may correspond approximately to the more general types (now extinct)
from which existing animals have descended.

(2) By the hypothesis of development the fixity of species is
discredited. The lowest grade of a classification is made up not of
well-defined types unchanging from age to age, but of temporary species,
often connected by uncertain and indistinct varieties: some of which
may, in turn, if the conditions of their existence alter, undergo such
changes as to produce new species. Hence the notion that Kinds exist in
organic nature must be greatly modified. During a given period of a few
thousand years, Kinds may be recognised, because, under such conditions
as now prevail in the world, that period of time is insufficient to
bring about great changes. But, if it be true that lions, tigers, and
leopards have had a common ancestor, from whose type they have gradually
diverged, it is plain that their present distinctness results only from
the death of intermediate specimens and the destruction of intermediate
varieties. Were it possible to restore, by the evidence of fossils, all
the ranks of the great processions that have descended from the common
ancestor, there would nowhere occur a greater difference than between
offspring and parents; and the appearance of Kinds existing in nature,
which is so striking in a museum or zoological garden, would entirely

A classification, then, as formerly observed, represents a cross-section
of nature as developing in time: could we begin at the beginning and
follow this development down the course of time, we should find no
classes, but an ever-moving, changing, spreading, branching continuum.
It may be represented thus: Suppose an animal (or plant) A, extending
over a certain geographical area, subject to different influences and
conditions of climate, food, hill and plain, wood and prairie, enemies
and rivals, and undergoing modifications here and there in adaptation to
the varying conditions of life: then varieties appear. These varieties,
diverging more and more, become distinct species (AB, AC, AD, AX). Some
of these species, the more widely diffused, again produce varieties;
which, in turn become species (ABE, ABF, ADG, ADH). From these, again,
ABE, ABFI, ABFJ, AC, ADHK, ADHL, ADHM, the extant species, descend.

                   / | \ \__
                  /  |  \   \_____
         ________/   |   \__      \
        /            |      \      \
       AB            AC     AD     AX
       /\            |      /\
      /  \           |     /  \
     /    \          |    /    \
   ABE    ABF        |  ADG    ADH
   /      / \        |         /|\
  /      /   \       |        / | \
 /      /     \      |       /  |  \

If in this age a classifier appears, he finds seven living species,
which can be grouped into four genera (ABE, ABF, AC, ADH), and these
again into three Families (AB, AC, AD), all forming one Order. But the
animals which were their ancestors are all extinct. If the fossils of
any of them--say AB, ADG and AX--can be found, he has three more
species, one more genus (ADG), and one more family (AX). For AC, which
has persisted unchanged, and AX, which has become extinct, are both of
them Families, each represented by only one species. It seems necessary
to treat such ancient types as species on a level with extant forms; but
the naturalist draws our attention to their archaic characteristics, and
tries to explain their places in the order of evolution and their

But now suppose that he could find a fossil specimen of every generation
(hundreds of thousands of generations), from ABFI, etc., up to A;
then, as each generation would only differ from the preceding as
offspring from parents, he would be unable at any point to distinguish a
species; at most, he would observe a slightly marked variety. ABFI and
ABFJ would grow more and more alike, until they became indistinguishable
in ABF; ABF and ABE would merge into AB; AB, AC, AD and AX would merge
into A. Hence, the appearance of species is due to our taking
cross-sections of time, or comparing forms that belong to periods remote
from one another (like AX, ADG, and ADHK, or AD, ADH and ADHK), and this
appearance of species depends upon the destruction of ancestral
intermediate forms.

(3) The hypothesis of development modifies the logical character of
classification: it no longer consists in a direct induction of
co-inherent characters, but is largely a deduction of these from the
characters of earlier forms, together with the conditions of variation;
in other words, the definition of a species must, with the progress of
science, cease to be a mere empirical law of co-inherence and become a
derivative law of Causation. But this was already implied in the
position that causation is the fundamental principle of the explanation
of concrete things; and accordingly, the derivative character of species
or kinds extends beyond organic nature.

§ 9. The classification of inorganic bodies also depends on causation.
There is the physical classification into Solids, Liquids, and Gases.
But these states of matter are dependent on temperature; at different
temperatures, the same body may exist in all three states. They cannot
therefore be defined as solid, liquid, or gaseous absolutely, but only
within certain degrees of temperature, and therefore as dependent upon
causation. Similarly, the geological classification of rocks, according
to relative antiquity (primary, secondary, tertiary, with their
subdivisions), and mode of formation (igneous and aqueous), rests upon
causation; and so does the chemical classification of compound bodies
according to the elements that enter into them in definite proportions.
Hence, only the classification of the elements themselves (amongst
concrete things), at present, depends largely upon empirical
Coinherence. If the elements remain irresolvable into anything simpler,
the definitions of the co-inherent characters that distinguish them must
be reckoned amongst the ultimate Uniformities of Nature. But if a
definite theory of their origin both generally and severally, whether
out of ether-vortices, or groups of electric corpuscles, or whatnot,
shall ever gain acceptance, similarity of genesis or causation will
naturally be the leading consideration in classifying the chemical
elements. To find common principles of causation, therefore, constitutes
the verification of every Natural Classification. The ultimate
explanation of nature is always causation; the Law of Causation is the
backbone of the system of Experience.



§ 1. Precision of thought needs precision of language for the recording
of such thought and for communicating it to others. We can often
remember with great vividness persons, things, landscapes, changes and
actions of persons or things, without the aid of language (though words
are often mixed with such trains of imagery), and by this means may form
judgments and inferences in particular cases; but for general notions,
judgments and inferences, not merely about this or that man, or thing,
but about all men or all kinds of things, we need something besides the
few images we can form of them from observation. Even if we possess
generic images, say, of 'horse' or 'cat' (that is, images formed, like
composite photographs, by a coalescence of the images of all the horses
or cats we have seen, so that their common properties stand out and
their differences frustrate and cancel one another), these are useless
for precise thought; for the generic image will not correspond with the
general appearance of horse or cat, unless we have had proportional
experience of all varieties and have been impartially interested in all;
and, besides, what we want for general thought is not a generic image of
the appearance of things, though it were much more definite and fairly
representative than such images ever are, but a general representation
of their important characters; which may be connected with internal
organs, such as none but an anatomist ever sees. We require a symbol
connected with the general character of a thing, or quality, or process,
as scientifically determined, whose representative truth may be trusted
in ordinary cases, or may be verified whenever doubt arises. Such
symbols are for most purposes provided by language; Mathematics and
Chemistry have their own symbols.

§ 2. First there should be "a name for every important meaning": (a) A
Nomenclature, or system of the names of all classes of objects, adapted
to the use of each science. Thus, in Geology there are names for classes
of rocks and strata, in Chemistry for the elements and their compounds,
in Zoology and Botany for the varieties and species of animals and
plants, their genera, families and orders.

To have such names, however, is not the whole aim in forming a
scientific language; it is desirable that they should be systematically
significant, and even elegant. Names, like other instruments, ought to
be efficient, and the efficiency of names consists in conveying the most
meaning with the least effort. In Botany and Zoology this result is
obtained by giving to each species a composite name which includes that
of the genus to which it belongs. The species of Felidæ given in chap.
xvii. § 7, are called _Felis leo_ (lion), _Felis tigris_ (tiger), _Felis
leopardus_ (leopard), _Felis concolor_ (puma), _Felis lyncus_ (European
lynx), _Felis catus_ (wild cat). In Chemistry, the nomenclature is
extremely efficient. Names of the simpler compounds are formed by
combining the names of the elements that enter into them; as Hydrogen
Chloride, Hydrogen Sulphide, Carbon Dioxide; and these can be given
still more briefly and efficiently in symbols, as HCl, H_{2}S, CO_{2}.
The symbolic letters are usually initials of the names of the elements:
as C = Carbon, S = Sulphur; sometimes of the Latin name, when the common
name is English, as Fe = Iron. Each letter represents a fixed quantity
of the element for which it stands, viz., the atomic weight. The
number written below a symbol on the right-hand side shows how many
atoms of the element denoted enter into a molecule of the compound.

(b) A Terminology is next required, in order to describe and define the
things that constitute the classes designated by the nomenclature, and
to describe and explain their actions.

(i) A name for every integral part of an object, as head, limb,
vertebra, heart, nerve, tendon; stalk, leaf, corolla, stamen, pistil;
plinth, frieze, etc. (ii) A name for every metaphysical part or
abstract quality of an object, and for its degrees and modes; as
extension, figure, solidity, weight; rough, smooth, elastic, friable;
the various colours, red, blue, yellow, in all their shades and
combinations and so with sounds, smells, tastes, temperatures. The terms
of Geometry are employed to describe the modes of figure, as angular,
curved, square, elliptical; and the terms of Arithmetic to express the
degrees of weight, elasticity, temperature, pitch of sound. When other
means fail, qualities are suggested by the names of things which exhibit
them in a salient way; figures by such terms as amphitheatre, bowl-like,
pear-shaped, egg-shaped; colours by lias-blue, sky-blue, gentian-blue,
peacock-blue; and similarly with sounds, smells and tastes. It is also
important to express by short terms complex qualities, as harmony,
fragrance, organisation, sex, symmetry, stratification.

(iii) In the explanation of Nature we further require suitable names for
processes and activities: as deduction, conversion, verification,
addition, integration, causation, tendency, momentum, gravitation,
aberration, refraction, conduction, affinity, combination, germination,
respiration, attention, association, development.

There may sometimes be a difficulty in distinguishing the terms which
stand for qualities from those that express activities, since all
qualities imply activities: weight, for example, implies gravitation;
and the quality heat is also a kind of motion. The distinction aimed at
lies between a quality as perceived by means of an effect upon our
senses (as weight is resistance to our effort in lifting; heat, a
sensation when we approach fire), and that property of a body which is
conceived to account for its energy (as gravitation that brings a body
to the ground, or physical heat that expands an iron bar or works an
engine). The former class of words, expressing qualities, are chiefly
used in description: the latter class, expressing activities, are
chiefly needed in explanation. They correspond respectively, like
classification and explanation, with the static and dynamic aspects of

The terms of ordinary language fall into the same classes as those of
science: they stand for things, classes of things, parts, or qualities,
or activities of things; but they are far less precise in their
signification. As long as popular thought is vague its language must be
vague; nor is it desirable too strictly to correct the language whilst
the thought is incorrigible. Much of the effect of poetry and eloquence
depends upon the elasticity and indirect suggestiveness of common terms.
Even in reasoning upon some subjects, it is a mistake to aim at an
unattainable precision. It is better to be vaguely right than exactly
wrong. In the criticism of manners, of fine art, or of literature, in
politics, religion and moral philosophy, what we are anxious to say is
often far from clear to ourselves; and it is better to indicate our
meaning approximately, or as we feel about it, than to convey a false
meaning, or to lose the warmth and colour that are the life of such
reflections. It is hard to decide whether more harm has been done by
sophists who take a base advantage of the vagueness of common terms, or
by honest paralogists (if I may use the word) who begin by deceiving
themselves with a plausible definiteness of expression, and go on to
propagate their delusions amongst followers eager for systematic insight
but ignorant of the limits of its possibility.

§ 3. A Definition is necessary (if possible) for every scientific name.
To define a name is to give a precise statement of its meaning or
connotation. The name to be defined is the subject of a proposition,
whose predicate is a list of the fundamental qualities common to the
things or processes which the subject denotes, and on account of
possessing which qualities this name is given to them.

Thus, a curve is a line of which no part is straight. The momentum of a
moving body is the product of its mass and its velocity (these being
expressed in numbers of certain units). Nitrogen is a transparent
colourless gas, atomic weight 14, specific gravity .9713, not readily
combining, etc. A lion is a monodelphian mammal, predatory, walking on
its toes, of nocturnal habits, with a short rounded head and muzzle;
dental formula: Incisors (3-3)/(3-3), canines (1-1)/(1/1), præmolars
(3-3)/(2-2), molars (1-1)/(1-1) = 30; four toes on the hind and five on
the fore foot, retractile claws, prickly tongue, light and muscular in
build, about 9-1/2 feet from muzzle to tip of tail, tawny in colour, the
males maned, with a tufted tail. If anything answers to this
description, it is called a lion; if not, not: for this is the meaning
of the name.

For ordinary purposes, it may suffice to give an Incomplete Definition;
that is, a list of qualities not exhaustive, but containing enough to
identify the things denoted by the given name; as if we say that a lion
is 'a large tawny beast of prey with a tufted tail.' Such purposes may
also be served by a Description; which is technically, a proposition
mentioning properties sufficient to distinguish the things denoted, but
not the properties that enter into the definition; as if nitrogen be
indicated as the gas that constitutes 4/5 of the atmosphere.

§ 4. The rules for testing a Definition are: I.--As to its Contents--

(1) It must state the whole connotation of the name to be defined.

(2) It must not include any quality derivative from the connotation.
Such a quality is called a Proprium. A breach of this rule can do no
positive harm, but it is a departure from scientific economy. There is
no need to state in the definition what can be derived from it; and
whatever can be derived by causation, or by mathematical demonstration,
should be exhibited in that manner.

(3) It must not mention any circumstance that is not a part of the
connotation, even though it be universally found in the things denoted.
Such a circumstance, if not derivable from the connotation, is called an
Accident. That, for example, the lion at present only inhabits the Old
World, is an accident: if a species otherwise like a lion were found in
Brazil, it would not be refused the name of lion on the score of
locality. Whilst, however, the rules of Logic have forbidden the
inclusion of proprium or accident in a definition, in fact the
definitions of Natural History often mention such attributes when
characteristic. Indeed, definitions of superordinate classes--Families
and Orders--not infrequently give qualities as generally found in the
subordinate classes, and at the same time mention exceptional cases in
which they do not occur.

II.--As to its Expression--

(4) A Definition must not include the very term to be defined, nor any
cognate. In defining 'lion' we must not repeat 'lion,' nor use
'leonine'; it would elucidate nothing.

(5) It must not be put in vague language.

(6) It must not be in a negative form, if a positive form be obtainable.
We must not be content to say that a lion is 'no vegetarian,' or 'no
lover of daylight.' To define a curve as a line 'always changing its
direction' may be better than as 'in no part straight.'

§ 5. The process of determining a Definition is inseparable from
classification. We saw that classification consists in distributing
things into groups according to their likenesses and differences,
regarding as a class those individuals which have most qualities in
common. In doing so we must, of course, recognise the common qualities
or points of likeness; and to enumerate these is to define the name of
the class. If we discover the qualities upon which a class is based by
direct observation and induction, by the same method we discover the
definition of its name.

We saw also that classification is not merely the determination of
isolated groups of things, but a systematic arrangement of such groups
in relation to one another. Hence, again, Definitions are not
independent, but relative to one another; and, of course, in the same
way as classes are relative. That is to say, as a class is placed in
subordination to higher or more comprehensive groups, so the definition
of its name is subordinate to that of their names; and as a class stands
in contrast with co-ordinate classes (those that are in the same degree
of subordination to the same higher groups), so the definition of its
name is in contrast or co-ordination with the definitions of their
names. Lion is subordinate to _Felis_, to Digitigrade, to Carnivore and
so on up to Animal; and, beyond the Animal Kingdom, to Phenomenon; it is
co-ordinate with tiger, puma, etc.; and more remotely it is
co-ordinate with dog, jackal, wolf, which come under _Canis_--a genus
co-ordinate with _Felis_. The definition of lion, therefore, is
subordinate to that of _Felis_, and to all above it up to Phenomenon;
and is co-ordinate with that of tiger, and with all species in the same
grade. This is the ground of the old method of definition _per genus et

The genus being the next class above any species, the _differentia_ or
Difference consists of the qualities which mark that species in addition
to those that mark the genus, and which therefore distinguish it from
all other species of the same genus. In the above definition of lion,
for example, all the properties down to "light and muscular in build"
are generic, that is, are possessed by the whole genus, _Felis_; and the
remaining four (size, colour, tufted tail, and mane in the male) are the
Difference or specific properties, because in those points the lion
contrasts with the other species of that genus. Differences may be
exhibited thus:

     _Lion._                                 _Tiger._
SIZE: about 9-1/2 feet from nose   | About 10 feet.
  to tip of tail.                  |
COLOUR: tawny.                     | Warm tawny, striped with black.
TAIL: tufted.                      | Tapering.
MANE: present in the male.         | Both sexes maneless.

There are other differences in the shape of the skull. In defining lion,
then, it would have been enough to mention the genus and the properties
making up the Difference; because the properties of the genus may be
found by turning to the definition of the genus; and, on the principle
of economy, whatever it is enough to do it is right to do. To define 'by
genus and difference' is a point of elegance, when the genus is known;
but the only way of knowing it is to compare the individuals comprised
in it and in co-ordinate genera, according to the methods of scientific
classification. It may be added that, as the genus represents ancestral
derivation, the predication of genus in a definition indicates the
remote causes of the phenomena denoted by the name defined. And this way
of defining corresponds with the method of double naming by genus and
species: _Felis leo_, _Felis tigris_, etc.; _Vanessa Atalanta_,
_Vanessa Io_, etc.

The so-called Genetic Definition, chiefly used in Mathematics, is a rule
for constructing that which a name denotes, in such a way as to ensure
its possessing the tributes connoted by the name. Thus, for a circle:
Take any point and, at any constant distance from it, trace a line
returning into itself. In Chemistry a genetic definition of any compound
might be given in the form of directions for the requisite synthesis of

§ 6. The chief difficulty in the definition of scientific names consists
in determining exactly the nature of the things denoted by them, as in
classifying plants and animals. If organic species are free growths,
continually changing, however gradually, according as circumstances give
some advantage to one form over others, we may expect to find such
species branching into varieties, which differ considerably from one
another in some respects, though not enough to constitute distinct
species. This is the case; and, consequently, there arises some
uncertainty in collecting from all the varieties those attributes which
are common to the species as a whole; and, therefore, of course,
uncertainty in defining the species. The same difficulty may occur in
defining a genus, on account of the extent to which some of its species
differ from others, whilst having enough of the common character to
deter the classifier from forming a distinct genus on their account. On
the other hand the occurrence of numerous intermediate varieties may
make it difficult to distinguish genera or species at all. Even the
Kingdoms of plants and animals are hard to discriminate at the lowest
levels of organisation. Now, where there is a difficulty of
classification there must be a corresponding difficulty of definition.

It has been proposed in such cases to substitute a Type for a
Definition; to select some variety of a species, or species of a genus,
as exhibiting its character in an eminent degree, and to regard other
groups as belonging to the same species or genus, according as they
agree more with this type than with other types representing other
species or genera. But the selection of one group as typical implies a
recognition of its attributes as prevailing generally (though not
universally) throughout the species or genus; and to recognise these
attributes and yet refuse to enumerate them in a definition, seems to be
no great gain. To enumerate the attributes of the type as an Approximate
Definition of the species or genus, true of _most_ of the groups
constituting the species or genus, answers the same purpose, is more
explicit, and can mislead no one who really attends to the exposition.
An approximate definition is, indeed, less misleading than the
indication of a type; for the latter method seems to imply that the
group which is now typical has a greater permanence or reality than its
co-ordinate groups; whereas, for aught we know, one of the outside
varieties or species may even now be superseding and extinguishing it.
But the statement of a definition as approximate, is an honest
confession that both the definition and the classification are (like a
provisional hypothesis) merely the best account we can give of the
matter according to our present knowledge.

§ 7. The limits of Definition are twofold: (a) A name whose meaning
cannot be analysed cannot be defined. This limitation meets us only in
dealing with the names of the metaphysical parts or simple qualities of
objects under the second requisite of a Terminology. Resistance and
weight, colour and its modes, many names of sounds, tastes, smells, heat
and cold--in fact, whatever stands for an unanalysable perception,
cannot be made intelligible to any one who has not had experience of the
facts denoted; they cannot be defined, but only exemplified. A sort of
genetic definition may perhaps be attempted, as if we say that colour is
the special sensation of the cones of the retina, or that blue is the
sensation produced by a ray of light vibrating about 650,000,000,000,000
times a second; but such expressions can give no notion of our meaning
to a blind man, or to any one who has never seen a blue object. Nor can
we explain what heat is like, or the smell of tobacco, to those who have
never experienced them; nor the sound of C 128 to one who knows nothing
of the musical scale.

If we distinguish the property of an object from the sensation it
excites in us, we may define any simple property as 'the power of
producing the sensation'; the colour of a flower as the power of
exciting the sensation of colour in us. Still, this gives no information
to the blind nor to the colour-blind. Abstract names may be defined by
defining the corresponding concrete: the definition of 'human nature' is
the same as of 'man.' But if the corresponding concrete be a simple
sensation (as blue), this being indefinable, the abstract (blueness) is
also indefinable.

(b) The second limit of Definition is the impossibility of exhausting
infinity, which would be necessary in order to convey the meaning of the
name of any individual thing or person. For, as we saw in chap. iv., if
in attempting to define a proper name we stop short of infinity, our
list of qualities or properties may possibly be found in two
individuals, and then it becomes the definition of a class-name or
general name, however small the actual class. Hence we can only give a
Description of that which a proper name denotes, enumerating enough of
its properties to distinguish it from everything else as far as our
knowledge goes.

§ 8. The five Predicables (Species, Genus, Difference, Proprium,
Accident) may best be discussed in connection with Classification and
Definition; and in giving an account of Classification, most of what has
to be said about them has been anticipated. Their name, indeed, connects
them with the doctrine of Propositions; for Predicables are terms that
may be predicated, classified according to their connotative relation to
the subject of a proposition (that is, according to the relation in
which their connotation stands to the connotation of the subject):
nevertheless, the significance of the relations of such predicates to a
subject is derivative from the general doctrine of classification.

For example, in the proposition 'X is Y,' Y must be one of the five
sorts of predicables in relation to X; but of what sort, depends upon
what X (the subject) is, or means. The subject of the proposition must
be either a definition, or a general connotative name, or a singular

If X be a definition, Y must be a species; for nothing but a general
name can be predicated of a definition: and, strictly speaking, it is
only in relation to a definition (as subject) that species can be a
predicable; when it is called _Species predicabilis_ (1).

If X be a connotative name, it is itself a species (_Species
subjicibilis_); and the place of the subject of a proposition is the
usual one for species. The predicate, Y, may then be related to the
species in three different ways. First, it may be a definition, exactly
equivalent to the species;--in fact, nothing else than the species in an
explicit form, the analysis of its connotation. Secondly, the predicate
may be, or connote, some _part only_ of the definition or connotation of
the species; and then it is either genus (2), or difference (3).
Thirdly, the predicate may connote _no part_ of the definition, and then
it is either derivable from it, being a proprium (4), or not derivable
from it, being an accident (5). These points of doctrine will be
expanded and illustrated in subsequent pages.

If X be a singular name, deriving connotation from its constituent terms
(chap. iv. § 2), as 'The present Emperor of China,' it may be treated as
a _Species subjicibilis_. Then that he is 'an absolute monarch,'
predicates a genus; because that is a genus of 'Emperor,' a part of the
singular name that gives it connotation. That he wears a yellow robe is
a proprium, derivable from the ceremonial of his court. That he is
thirty years of age is an accident.

But if X be a proper name, having no connotation, Y must always be an
accident; since there can then be no definition of X, and therefore
neither species, genus, difference, nor proprium. Hence, that 'John Doe
is a man' is an accidental proposition: 'man' is not here a _Species
predicabilis_; for the name might have been given to a dog or a
mountain. That is what enables the proposition to convey information: it
would be useless if the proper name implied 'humanity.'

'Species' is most frequently used (as in Zoology) for the _class
denoted_ by a general name; but in Logic it is better to treat it as a
general name used connotatively for the attributes possessed in common
by the things denoted, and on account of which they are regarded as a
class: it is sometimes called the Essence (§ 9). In this connotative
sense, a species is implicitly what the definition is explicitly; and
therefore the two are always simply convertible. Thus, 'A plane
triangle' (species) is 'a figure enclosed by three straight lines'
(definition): clearly we may equally say, 'A figure enclosed by three
straight lines is a plane triangle.' It is a simple identity.

A genus is also commonly viewed denotatively, as a class containing
smaller classes, its species; but in Logic it is, again, better to treat
it connotatively, as a name whose definition is part of the definition
of a given species.

A difference is the remainder of the definition of any species after
subtracting a given genus. Hence, the genus and difference together make
up the species; whence the method of definition _per genus et
differentiam_ (_ante_, § 5).

Whilst in Botany and Zoology the species is fixed at the lowest step of
the classification (varieties not being reckoned as classes), and the
genus is also fixed on the step next above it, in Logic these
predicables are treated as movable up and down the ladder: any lower
class being species in relation to any higher; which higher class,
wherever taken, thus becomes a genus. Lion may logically be regarded as
a species of digitigrade, or mammal, or animal; and then each of these
is a genus as to lion: or, again, digitigrade may be regarded as a
species of mammal, or mammal as a species of animal. The highest class,
however, is never a species; wherefore it is called a _Summum Genus_:
and the lowest class is never a genus; wherefore it is called an _Infima
Species_. Between these two any step may be either species or genus,
according to the relation in which it is viewed to other classes, and is
then called Subaltern. The _summum genus_, again, may be viewed in
relation to a _given_ universe or _suppositio_ (that is, any limited
area of existence now the object of attention), or to the _whole_
universe. If we take the animal kingdom as our _suppositio_, Animal is
the _summum genus_; but if we take the whole universe, 'All things' is
the _summum genus_.

"Porphyry's tree" is used to illustrate this doctrine. It begins with a
_summum genus_, 'Substance,' and descends by adding differences, step by
step, to the _infima species_, 'Man.' It also illustrates Division by

                      /       \
                        /   \
                 ANIMATE || INANIMATE
                    LIVING BODY
                    /         \
                SENSIBLE || INSENSIBLE
                       /    \
                RATIONAL || IRRATIONAL
                      // || \\
                     //  ||  \\
                    //   ||   \\
                   //    ||    \\
                  //     ||     \\
                 //      ||      \\
         _Socrates_   _Plato_   _Aristotle_

Beginning with 'Substance,' as _summum genus_, and adding the
difference 'Corporeal,' we frame the species 'Body.' Taking 'Body' as
the genus and adding the difference 'Animate,' we frame the species
'Living Body;' and so on till 'Man' is reached; which, being _infima
species_, is only subdivisible into individuals. But the division of Man
into individuals involves a change of principle; it is a division of the
denotation, not an increase of the connotation as in the earlier steps.
Only one side of each dichotomy is followed out in the 'tree': if the
other side had been taken, Incorporeal Substance would be 'Spirit';
which might be similarly subdivided.

Genus and species, then, have a double relation. In denotation the genus
includes the species; in connotation the species includes the genus.
Hence the doctrine that by increasing the connotation of a name we
decrease its denotation: if, for example, to the definition of 'lion' we
add 'inhabiting Africa,' Asiatic lions are no longer denoted by it. On
the other hand, if we use a name to denote objects that it did not
formerly apply to, some of the connotation must be dropped: if, for
example, the name 'lion' be used to include 'pumas,' the tufted tail and
mane can no longer be part of the meaning of the word; since pumas have
not these properties.

This doctrine is logically or formally true, but it may not always be
true in fact. It is logically true; because wherever we add to the
connotation of a name, it is possible that some things to which it
formerly applied are now excluded from its denotation, though we may not
know of any such things. Still, as a matter of fact, an object may be
discovered to have a property previously unknown, and this property may
be fundamental and co-extensive with the denotation of its name, or even
more widely prevalent. The discovery that the whale is a mammal did not
limit the class 'whale'; nor did the discovery that lions, dogs, wolves,
etc., walk upon their toes, affect the application of any of these

Similarly, the extension of a name to things not previously denoted by
it, may not in fact alter its definition; for the extension may be made
on the very ground that the things now first denoted by it have been
found to have the properties enumerated in its definition, as when the
name 'mammal' was applied to whales, dolphins, etc. If, however,
'mammal' had formerly been understood to apply only to land animals, so
that its definition included (at least, popularly) the quality of
'living on the land,' this part of the connotation was of course lost
when the denotation came to include certain aquatic animals.

A proprium is an attribute derived from the definition: being either (a)
implied in it, or deducible from it, as 'having its three angles equal
to two right angles' may be proved from the definition of a triangle; or
(b) causally dependent on it, as being 'dangerous to flocks' results
from the nature of a wolf, and as 'moving in an ellipse' results from
the nature of a planet in its relation to the sun.

An accident is a property accompanying the defining attributes without
being deducible from them. The word suggests that such a property is
merely 'accidental,' or there 'by chance'; but it only means that we do
not understand the connection.

Proprium and Accident bear the same relation to one another as
Derivative and Empirical Laws: the predication of a proprium is a
derivative law, and the predication of an accident is an empirical law.
Both accidents and empirical laws present problems, the solution of
which consists in reducing them, respectively, to propria and derivative
laws. Thus the colour of animals was once regarded as an accident for
which no reason could be given; but now the colour of animals is
regarded as an effect of their nature and habits, the chief determinants
of it being the advantage of concealment; whilst in other cases, as
among brightly coloured insects and snakes, the determinant may be the
advantage of advertising their own noxiousness. If such reasoning is
sound, colour is a proprium (and if so, it cannot _logically_ be
included in a definition; but it is better to be judicious than formal).

If the colour of animals is a proprium, we must recognise a distinction
between Inseparable and Separable Propria, according as they do, or do
not, always accompany the essence: for mankind is regarded as one
species; but each colour, white, black or yellow, is separable from it
under different climatic conditions; whilst tigers are everywhere
coloured and striped in much the same way; so that we may consider their
colouring as inseparable, in spite of exceptional specimens black or
white or clouded.

The same distinction may be drawn between accidents. 'Inhabiting Asia'
is an Inseparable Accident of tiger, but a Separable Accident of lion.
Even the occasional characteristics and occupations of individuals are
sometimes called separable accidents of the species; as, of man, being
colour-blind, carpentering, or running.

A proprium in the original signification of the term ἴδιον was peculiar
to a species, never found with any other, and was therefore convertible
with the subject; but this restriction is no longer insisted on.

§ 9. Any predication of a genus, difference or definition, is a verbal,
analytic, or essential proposition: and any predication of a proprium or
accident, is a real, synthetic, or accidental proposition (chap. v. §
6). A proposition is called verbal or analytic when the predicate is a
part, or the whole, of the meaning of the subject; and the subject being
species, a genus or difference is part, and a definition is the whole,
of its meaning or connotation. Hence such a proposition has also been
called explicative. Again, a proposition is called real or synthetic
when the predicate is no part of the meaning of the subject; and, the
subject being species, a proprium or accident is no part of its meaning
or connotation. Hence such a proposition has been called ampliative.

As to Essential and Accidental, these terms are derived from the
doctrine of Realism. Realists maintain that the essence of a thing, or
that which makes a thing to be what (or of what kind) it is, also makes
everything else of the same kind to be what it is. The essence, they
say, is not proper to each thing or separately inherent in it, but is an
'Universal' common to all things of that kind. Some hold that the
universal nature of things of any kind is an Idea existing (apart from
the things) in the intelligible world, invisible to mortal eye and only
accessible to thought; whence the Idea is called a noumenon: that only
the Idea is truly real, and that the things (say, trees, bedsteads and
cities) which appear to us in sense-perception, and which therefore are
called phenomena, only exist by participating in, or imitating, the Idea
of each kind of them. The standard of this school bears the legend
_Universalia ante rem_.

But others think that the Universal does not exist apart from particular
things, but is their present essence; gives them actuality as individual
substances; "informs" them, or is their formal cause, and thus makes
them to be what they are of their kind according to the definition: the
universal lion is in all lions, and is not merely similar, but identical
in all; for thus the Universal Reason thinks and energises in Nature.
This school inscribes upon its banners, _Universalia in re_.

To define anything, then, is to discover its essence, whether
transcendent or immanent; and to predicate the definition, or any part
of it (genus or difference), is to enounce an essential proposition. But
a proprium, being no part of a definition, though it always goes along
with it, does not show what a thing is; nor of course does an accident;
so that to predicate either of these is to enounce an accidental

Another school of Metaphysicians denies the existence of Universal Ideas
or Forms; the real things, according to them, are individuals; which,
so far as any of them resemble one another, are regarded as forming
classes; and the only Universal is the class-name, which is applied
universally in the same sense. Hence, they are called Nominalists. The
sense in which any name is applied, they say, is derived from a
comparison of the individuals, and by abstraction of the properties they
have in common; and thus the definition is formed. _Universalia post
rem_ is their motto. Some Nominalists, however, hold that, though
Universals do not exist in nature, they do in our minds, as Abstract
Ideas or Concepts; and that to define a term is to analyse the concept
it stands for; whence, these philosophers are called Conceptualists.

Such questions belong to Metaphysics rather than to Logic; and the
foregoing is a commonplace account of a subject upon every point of
which there is much difference of opinion.

§ 10. The doctrine of the Predicaments, or Categories, is so interwoven
with the history of speculation and especially of Logic that, though its
vitality is exhausted, it can hardly be passed over unmentioned. The
predicaments of Aristotle are the heads of a classification of terms as
possible predicates of a particular thing or individual. Hamilton
(_Logic_: Lect. xi.) has given a classification of them; which, if it
cannot be found in Aristotle, is an aid to the memory, and may be thrown
into a table thus:

    Substance                     οὐσία        (1)
               {Quantity          ποσόν        (2)
    Attribute  {Quality           ποιόν        (3)
               {Relation          πρόσ τι       (4)

                       { Where    ποῡ         (5)
                       { When     πότε         (6)

                       { Action   ποιεῑν        (7)
    Modes of Relation  { Passion  πάσχειν       (8)

                       { Posture  κεῑσθαι        (9)
                       { Habit    ἔχειν         (10)

Taking a particular thing or individual, as 'Socrates,' this is
Substance in the proper sense of the word, and can never be a predicate,
but is the subject of all predicates. We may assert of him (1) Substance
in the secondary sense (species or genus) that he is a man or an animal;
(2) Quantity, of such a height or weight; (3) Quality, fair or dark; (4)
Relation, shorter or taller than Xanthippe; (5) Where, at Athens; (6)
When, two thousand and odd years ago; (7) Action, that he questions or
pleads; (8) Passion, that he is answered or condemned; (9) Posture, that
he sits or stands; (10) Habit, that he is clothed or armed.

Thus illustrated (_Categoriæ_: c. 4), the predicaments seem to be a list
of topics, generally useful for the analysis and description of an
individual, but wanting in the scientific qualities of rational
arrangement, derivation and limitation. Why are there just these heads,
and just so many? It has been suggested that they were determined by
grammatical forms: for Substance is expressed by a substantive;
Quantity, Quality and Relation are adjectival; Where and When,
adverbial; and the remaining four are verbal. It is true that the parts
of speech were not systematically discriminated until some years after
Aristotle's time; but, as they existed, they may have unconsciously
influenced his selection and arrangement of the predicaments. Where a
principle is so obscure one feels glad of any clue to it (_cf._ Grote's
_Aristotle_, c. 3, and Zeller's _Aristotle_, c. 6). But whatever the
origin and original meaning of the predicaments, they were for a long
time regarded as a classification of things; and it is in this sense
that Mill criticises them (_Logic_: Bk. I. c. 3).

If, however, the predicaments are heads of a classification of terms
predicable, we may expect to find some connection with the predicables;
and, in fact, secondary Substances are species and genus; whilst the
remaining nine forms are generally accidents. But, again, we may expect
some agreement between them and the fundamental forms of predication
(_ante_, chap. i. § 5, and chap. ii § 4): Substance, whether as the
foundation of attributes, or as genus and species, implies the
predication of co-inherence, which is one mode of _Co-existence_.
Quantity is predicated as equality (or inequality) a mode of _Likeness_;
and the other mode of _Likeness_ is involved in the predication of
Quality. Relation, indeed, is the abstract of all predication, and ought
not to appear in a list along with special forms of itself. 'Where' is
position, or _Co-existence_ in space; and 'When' is position in time, or
_Succession_. Action and Passion are the most interesting aspect of
_Causation_. Posture and Habit are complex modes of _Co-existence_, but
too specialised to have any philosophic value. Now, I do not pretend
that this is what Aristotle meant and was trying to say: but if
Likeness, Co-existence, Succession and Causation are fundamental forms
of predication, a good mind analysing the fact of predication is likely
to happen upon them in one set of words or another.

By Kant the word 'Category' has been appropriated to the highest forms
of judgment, such as Unity, Reality, Substance, and Cause, under which
the understanding reduces phenomena to order and thereby constitutes
Nature. This change of meaning has not been made without a certain
continuity of thought; for forms of judgment are modes of predication.
But besides altering the lists of categories and greatly improving it,
Kant has brought forward under an old title a doctrine so original and
suggestive that it has extensively influenced the subsequent history of
Philosophy. At the same time, and probably as a result of the vogue of
the Kantian philosophy, the word 'category' has been vulgarised as a
synonym for 'class,' just as 'predicament' long ago passed from
Scholastic Logic into common use as a synonym for 'plight.' A minister
is said to be 'in a predicament,' or to fall under the 'category of



§ 1. Ordinary words may need definition, if in the course of exposition
or argument their meaning is liable to be mistaken. But as definition
cannot give one the sense of a popular word for all occasions of its
use, it is an operation of great delicacy. Fixity of meaning in the use
of single words is contrary to the genius of the common vocabulary;
since each word, whilst having a certain predominant character, must be
used with many shades of significance, in order to express the different
thoughts and feelings of multitudes of men in endlessly diversified
situations; and its force, whenever it is used, is qualified by the
other words with which it is connected in a sentence, by its place in
the construction of the sentence, by the emphasis, or by the pitch of
its pronunciation compared with the other words.

Clearly, the requisite of a scientific language, 'that every word shall
have one meaning well defined,' is too exacting for popular language;
because the other chief requisite of scientific language cannot be
complied with, 'that there be no important meaning without a name.'
'Important meanings,' or what seem such, are too numerous to be thus
provided for; and new ones are constantly arising, as each of us pursues
his business or his pleasure, his meditations or the excursions of his
fancy. It is impossible to have a separate term for each meaning; and,
therefore, the terms we have must admit of variable application.

An attempt to introduce new words is generally disgusting. Few men have
mastered the uses of half the words already to be found in our classics.
Much more would be lost than gained by doubling the dictionary. It is
true that, at certain stages in the growth of a people, a need may be
widely felt for the adoption of new words: such, in our own case, was
the period of the Tudors and early Stuarts. Many fresh words, chiefly
from the Latin, then appeared in books, were often received with
reprobation and derision, sometimes disappeared again, sometimes
established their footing in the language: see _The Art of English
Poetry_ (ascribed to Puttenham), Book III. chap. 4, and Ben Jonson's
_Poetaster_, Act. V. sc. I. Good judges did not know whether a word was
really called for: even Shakespeare thought 'remuneration' and
'accommodate' ridiculous. But such national exigencies rarely arise; and
in our own time great authors distinguish themselves by the plastic
power with which they make common words convey uncommon meanings.

Fluid, however, as popular language is and ought to be, it may be
necessary for the sake of clear exposition, or to steady the course of
an argument, to avoid either sophistry or unintentional confusion, that
words should be defined and discriminated; and we must discuss the means
of doing so.

§ 2. Scientific method is applicable, with some qualifications, to the
definition of ordinary words. Classification is involved in any problem
of definition: at least, if our object is to find a meaning that shall
be generally acceptable and intelligible. No doubt two disputants may,
for their own satisfaction, adopt any arbitrary definition of a word
important in their controversy; or, any one may define a word as he
pleases, at the risk of being misunderstood, provided he has no
fraudulent intention. But in exposition or argument addressed to the
public, where words are used in some of their ordinary senses, it should
be recognised that the meaning of each one involves that of many
others. For language has grown with the human mind, as representing its
knowledge of the world: this knowledge consists of the resemblances and
differences of things and of the activities of things, that is, of
classes and causes; and as there is such order in the world, so there
must be in language: language, therefore, embodies an irregular
classification of things with their attributes and relations according
to our knowledge and beliefs. The best attempt (known to me) to carry
out this view is contained in Roget's _Thesaurus_, which is a
classification of English words according to their meanings: founded, as
the author tells us, on the models of Zoology and Botany, it has some of
the requisites of a Logical Dictionary.

Popular language, indeed, having grown up with a predominantly practical
purpose, represents a very imperfect classification philosophically
considered. Things, or aspects, or processes of things, that have
excited little interest, have often gone unnamed: so that scientific
discoverers are obliged, for scientific purposes, to invent thousands of
new names. Strong interests, on the other hand, give such a colour to
language, that, where they enter, it is difficult to find any
indifferent expressions. _Consistency_ being much prized, though often
the part of a blockhead, _inconsistency_ implies not merely the absence
of the supposed virtue, but a positive vice: _Beauty_ being attractive
and _ugliness_ the reverse, if we invent a word for that which is
neither, 'plainness,' it at once becomes tinged with the ugly. We seem
to love beauty and morality so much as to be almost incapable of
signifying their absence without expressing aversion.

Again, the erroneous theories of mankind have often found their way into
popular speech, and their terms have remained there long after the
rejection of the beliefs they embodied: as--lunatic, augury, divination,
spell, exorcism: though, to be sure, such words may often be turned to
good account, besides the interest of preserving their original sense.
Language is a record as well as an index of ideas.

Language, then, being essentially classificatory, any attempt to
ascertain the meaning of a word, far from neglecting its relations to
others, should be directed toward elucidating them.

Every word belongs to a group, and this group to some other larger
group. A group is sometimes formed by derivation, at least so far as
different meanings are marked merely by inflections, as _short_,
_shorter_, _shorten_, _shortly_; but, for the most part, is a conflux of
words from many different sources. _Repose_, _depose_, _suppose_,
_impose_, _propose_, are not nearly connected in meaning; but are
severally allied in sense much more closely with words philologically
remote. Thus _repose_ is allied with _rest_, _sleep_, _tranquillity_;
_disturbance_, _unrest_, _tumult_; whilst _depose_ is, in one sense,
allied with _overthrow_, _dismiss_, _dethrone_; _restore_, _confirm_,
_establish_; and, in another sense, with _declare_, _attest_, _swear_,
_prove_, etc. Groups of words, in fact, depend on their meanings, just
as the connection of scientific names follows the resemblance in
character of the things denoted.

Words, accordingly, stand related to one another, for the most part,
though very irregularly, as genus, species, and co-ordinate species.
Taking _repose_ as a genus, we have as species of it, though not exactly
co-ordinate with one another, _tranquillity_ with a mental differentia
(repose of mind), _rest_, whether of mind or body, _sleep_, with the
differentia of unconsciousness (privative). Synonyms are species, or
varieties, wherever any difference can be detected in them; and to
discriminate them we must first find the generic meaning; for which
there may, or may not, be a single word. Thus, _equality_, _sameness_,
_likeness_, _similarity_, _resemblance_, _identity_, are synonyms; but,
if we attend to the ways in which they are actually used, perhaps none
of them can claim to be a genus in relation to the rest. If so, we must
resort to a compound term for the genus, such as 'absence of some sort
of difference.' Then _equality_ is absence of difference in quantity;
_sameness_ is often absence of difference in quality, though the usage
is not strict: _likeness_, _similarity_, and _resemblance_, in their
actual use, perhaps, cannot be discriminated; unless _likeness_ be the
more concrete, _similarity_ the more abstract; but they may all be used
compatibly with the recognition of more or less difference in the things
compared, and even imply this. _Identity_ is the absence of difference
of origin, a continuity of existence, with so much sameness from moment
to moment as is compatible with changes in the course of nature; so that
egg, caterpillar, chrysalis, butterfly may be identical for the run of
an individual life, in spite of differences quantitative and
qualitative, as truly as a shilling that all the time lies in a drawer.

Co-ordinate Species, when positive, have the least contrariety; but
there are also opposites, namely, negatives, contradictories and fuller
contraries. These may be regarded as either co-ordinate genera or the
species of co-ordinate genera. Thus, _repose_ being a genus,
_not-repose_ is by dichotomy a co-ordinate genus and is a negative and
contradictory; then _activity_ (implying an end in view), _motion_
(limited to matter), _disturbance_ (implying changes from a state of
calm), _tumult_, etc., are co-ordinate species of _not-repose_, and
are therefore co-ordinate opposites, or contraries, of the species of

As for correlative words, like _master and slave_, _husband and wife_,
etc., it may seem far-fetched to compare them with the sexes of the
same species of plants or animals; but there is this resemblance between
the two cases, that sexual names are correlative, as 'lioness,' and that
one sex of a species, like a correlative name, cannot be defined without
implying the other; for if a distinctive attribute of one sex be
mentioned (as the lion's mane), it is implied that the other wants it,
and apart from this implication the species is not defined: just as the
definition of 'master' implies a 'slave' to obey.

Common words, less precise than the terms of a scientific nomenclature,
differ from them also in this, that the same word may occur in different
genera. Thus, _sleep_ is a species of _repose_ as above; but it is also
a species of _unconsciousness_, with co-ordinate species _swoon_,
_hypnotic state_, etc. In fact, every word stands under as many
distinct genera, at least, as there are simple or indefinable qualities
to be enumerated in its definition.

§ 3. Partially similar to a scientific nomenclature, ordinary language
has likewise a terminology for describing things according to their
qualities and structure. Such is the function of all the names of
colours, sounds, tastes, contrasts of temperature, of hardness, of
pleasantness; in short, of all descriptive adjectives, and all names for
the parts and processes of things. Any word connoting a quality may be
used to describe many very different things, as long as they agree in
that quality.

But the quality connoted by a word, and treated as always the same
quality, is often only analogically the same. We speak of a _great_
storm, a _great_ man, a _great_ book; but _great_ is in each case not
only relative, implying small, and leaving open the possibility that
what we call great is still smaller than something else of its kind, but
it is also predicated with reference to some quality or qualities, which
may be very different in the several cases of its application. If the
book is prized for wisdom, or for imagination, its greatness lies in
that quality; if the man is distinguished for influence, or for courage,
his greatness is of that nature; if the storm is remarkable for
violence, or for duration, its greatness depends on that fact. The word
_great_, therefore, is not used for these things in the same sense, but
only analogically and elliptically. Similarly with good, pure, free,
strong, rich, and so on. 'Rest' has not the same meaning in respect of
a stone and of an animal, nor 'strong' in respect of thought and muscle,
nor 'sweet' in respect of sugar and music. But here we come to the
border between literal and figurative use; every one sees that
figurative epithets are analogical; but by custom any figurative use may
become literal.

Again, many general names of widely different meaning, are brought
together in describing any concrete object, as an animal, or a
landscape, or in defining any specific term. This is the sense of the
doctrine, that any concrete thing is a conflux of generalities or
universals: it may at least be considered in this way; though it seems
more natural to say, that an object presents these different aspects to
a spectator, who, fully to comprehend it, must classify it in every

§ 4. The process of seeking a definition may be guided by the following

(1) Find the usage of good modern authors; that is (as they rarely
define a word explicitly), consider what in various relations they use
it to denote; from which uses its connotation may be collected.

(2) But if this process yield no satisfactory result, make a list of the
things denoted, and of those denoted by the co-ordinate and opposite
words; and observe the qualities in which the things denoted agree, and
in which they differ from those denoted by the contraries and opposites.
If 'civilisation' is to be defined, make lists of civilised peoples, of
semi-civilised, of barbarous, and of savage: now, what things are common
to civilised peoples and wanting in the others respectively? This is an
exercise worth attempting. If poetry is to be defined, survey some
typical examples of what good critics recognise as poetry, and compare
them with examples of bad 'poetry,' literary prose, oratory, and
science. Having determined the characteristics of each kind, arrange
them opposite one another in parallel columns. Whoever tries to define
by this method a few important, frequently occurring words, will find
his thoughts the clearer for it, and will collect by the way much
information which may be more valuable than the definition itself,
should he ever find one.

(3) If the genus of a word to be defined is already known, the process
may be shortened. Suppose the genus of poetry to be _belles lettres_
(that is, 'appealing to good taste'), this suffices to mark it off from
science; but since literary prose and oratory are also _belles lettres_,
we must still seek the differentia of poetry by a comparison of it with
these co-ordinate species. A compound word often exhibits genus and
difference upon its face: as 're-turn,' 'inter-penetrate,'
'tuning-fork,' 'cricket-bat'; but the two last would hardly be
understood without inspection or further description. And however a
definition be discovered, it is well to state it _per genus et

(4) In defining any term we should avoid encroaching upon the meaning of
any of the co-ordinate terms; for else their usefulness is lessened: as
by making 'law' include 'custom,' or 'wealth' include 'labour' or

(5) If two or more terms happen to be exactly synonymous, it may be
possible (and, if so, it is a service to the language) to divert one of
them to any neighbouring meaning that has no determinate expression.
Thus, Wordsworth and Coleridge took great pains to distinguish between
Imagination and Fancy, which had become in common usage practically
equivalent; and they sought to limit 'imagination' to an order of poetic
effect, which (they said) had prevailed during the Elizabethan age, but
had been almost lost during the Gallo-classic, and which it was their
mission to restore. Co-ordinate terms often tend to coalesce and become
synonymous, or one almost supersedes the other, to the consequent
impoverishment of our speech. At present _proposition_ (that something
is the fact) has almost driven out _proposal_ (that it is desirable to
co-operate in some action). Even good writers and speakers, by their own
practice, encourage this confusion: they submit to Parliament certain
'propositions' (proposals for legislation), or even make 'a proposition
of marriage.' Definition should counteract such a tendency.

(6) We must avoid the temptation to extend the denotation of a word so
far as to diminish or destroy its connotation; or to increase its
connotation so much as to render it no longer applicable to things which
it formerly denoted: we should neither unduly generalise, nor unduly
specialise, a term. Is it desirable to define _education_ so as to
include the 'lessons of experience'; or is it better to restrict it as
implying a personal educator? If any word implies blame or praise, we
are apt to extend it to everything we hate or approve. But _coward_
cannot be so defined as to include all bullies, nor _noble_ so as to
include every honest man, without some loss in distinctness of thought.

The same impulses make us specialise words; for, if two words express
approval, we wish to apply both to whatever we admire and to refuse both
to whatever displeases us. Thus, a man may resolve to call no one great
who is not good: greatness, according to him, connotes goodness: whence
it follows that (say) Napoleon I. was not great. Another man is
disgusted with greatness: according to him, good and great are mutually
exclusive classes, sheep and goats, as in Gray's wretched clench:
"Beneath the good how far, yet far above the great." In feet, however
'good' and 'great' are descriptive terms, sometimes applicable to the
same object, sometimes to different: but 'great' is the wider term and
applicable to goodness itself and also to badness; whereas by making
'great' connote goodness it becomes the narrower term. And as we have
seen (§ 3), such epithets may be applicable to objects on account of
different qualities: _good_ is not predicated on the same ground of a
man and of a horse.

(7) In defining any word, it is desirable to bear in mind its
derivation, and to preserve the connection of meaning with its origin;
unless there are preponderant reasons for diverting it, grounded on our
need of the word to express a certain sense, and the greater difficulty
of finding any other word for the same purpose. It is better to lean to
the classical than to the vulgar sense of 'indifferent,' 'impertinent,'
'aggravating,' 'phenomenal.'

(8) Rigorous definition should not be attempted where the subject does
not admit of it. Some kinds of things are so complex in their qualities,
and each quality may manifest itself in so many degrees without ever
admitting of exact measurement, that we have no means of marking them
off precisely from other things nearly allied, similarly complex and
similarly variable. If so we cannot precisely define their names.
Imagination and fancy are of this nature, civilisation and barbarism,
poetry and other kinds of literary expression. As to poetry, some think
it only exists in metre, but hardly maintain that the metre must be
strictly regular: if not, how much irregularity of rhythm is admissible?
Others regard a certain mood of impassioned imagination as the essence
of poetry; but they have never told us how great intensity of this mood
is requisite. We also hear that poetry is of such a nature that the
enjoyment of it is an end in itself; but as it is not maintained that
poetry must be wholly impersuasive or uninstructive, there seems to be
no means of deciding what amount or prominence of persuasion or
instruction would transfer the work to the region of oratory or science.
Such cases make the method of defining by the aid of a type really
useful: the difficulty can hardly be got over without pointing to
typical examples of each meaning, and admitting that there may be many
divergences and unclassifiable instances on the border between allied

§ 5. As science began from common knowledge, the terms of the common
vocabulary have often been adopted into the sciences, and many are still
found there: such as weight, mass, work, attraction, repulsion,
diffusion, reflection, absorption, base, salt, and so forth. In the more
exact sciences, the vague popular associations with such words are
hardly an inconvenience: since those addicted to such studies do not
expect to master them without undergoing special discipline; and, having
precisely defined the terms, they acquire the habit of thinking with
them according to their assigned signification in those investigations
to which they are appropriate. It is in the Social Sciences, especially
Economics and Ethics, that the use of popular terminology is at once
unavoidable and prejudicial. For the subject-matters, industry and the
conduct of life, are every man's business; and, accordingly, have always
been discussed with a consciousness of their direct practical bearing
upon public and private interests, and therefore in the common language,
in order that everybody may as far as possible benefit by whatever light
can be thrown upon them. The general practice of Economists and
Moralists, however, shows that, in their judgment, the good derived from
writing in the common vocabulary outweighs the evil: though it is
sometimes manifest that they themselves have been misled by
extra-scientific meanings. To reduce the evil as much as possible, the
following precautions seem reasonable:

(1) To try to find and adopt the central meaning of the word (say rent
or money) in its current or traditionary applications: so as to lessen
in the greater number of cases the jar of conflicting associations. But
if the central popular meaning does not correspond with the scientific
conception to be expressed, it may be better to invent a new term.

(2) To define the term with sufficient accuracy to secure its clear and
consistent use for scientific purposes.

(3) When a popular term has to be used in a sense that departs from the
ordinary one in such a way as to incur the danger of misunderstanding,
to qualify it by some adjunct or "interpretation-clause."

The first of these rules is not always adhered to; and, in the progress
of a science, as subtler and more abstract relations are discovered
amongst the facts, the meaning of a term may have to be modified and
shifted further and further from its popular use. The term 'rent,' for
example, is used by economists, in such a sense that they have to begin
the discussion of the facts it denotes, by explaining that it does not
imply any actual payment by one man to another. Here, for most readers,
the meaning they are accustomed to, seems already to have entirely
disappeared. Difficulties may, however, be largely overcome by
qualifying the term in its various relations, as produce-rents,
ground-rents, customary rents, and so forth, (Cf. Dr. Keynes' _Scope
and Method of Political Economy_, chap. 5.)

§ 6. Definitions affect the cogency of arguments in many ways, whether
we use popular or scientific language. If the definitions of our terms
are vague, or are badly abstracted from the facts denoted, all arguments
involving these terms are inconclusive. There can be no confidence in
reasoning with such terms; since, if vague, there is nothing to protect
us from ambiguity; or, if their meaning has been badly abstracted, we
may be led into absurdity--as if 'impudence' should be defined in such a
way as to confound it with honesty.

Again, it is by definitions that we can best distinguish between Verbal
and Real Propositions. Whether a term predicated is implied in the
definition of the subject, or adds something to its meaning, deserves
our constant attention. We often persuade ourselves that statements are
profound and important, when, in fact, they are mere verbal
propositions. "It is just to give every man his due"; "the greater good
ought to be preferred to the less"; such dicta sound well--indeed, too
well! For 'a man's due' means nothing else than what it is just to give
him; and 'the greater good' may mean the one that ought to be preferred:
these, therefore, are Truisms. The investigation of a definition may be
a very valuable service to thought; but, once found, there is no merit
in repeating it. To put forward verbal or analytic propositions, or
truisms, as information (except, of course, in explaining terms to the
uninstructed), shows that we are not thinking what we say; for else we
must become aware of our own emptiness. Every step forward in knowledge
is expressed in a real or synthetic proposition; and it is only by means
of such propositions that information can be given (except as to the
meaning of words) or that an argument or train of reasoning can make any

Opposed to a truism is a Contradiction in Terms; that is, the denying of
a subject something which it connotes (or which belongs to its
definition), or the affirming of it something whose absence it connotes
(or which is excluded by its definition). A verbal proposition is
necessarily true, because it is tautologous; a contradiction in terms is
necessarily false, because it is inconsistent. Yet, as a rhetorical
artifice, or figure, it may be effective: that 'the slave is not bound
to obey his master' may be a way of saying that there ought to be no
slaves; that 'property is theft,' is an uncompromising assertion of the
communistic ideal. Similarly a truism may have rhetorical value: that 'a
Negro is a man' has often been a timely reminder, or even that "a man's
a man." It is only when we fall into such contradiction or tautology by
lapse of thought, by not fully understanding our own words, that it
becomes absurd.

Real Propositions comprise the predication of Propria and Accidentia.
Accidentia, implying a sort of empirical law, can only be established by
direct induction. But propria are deduced from (or rather by means of)
the definition with the help of real propositions, and this is what is
called 'arguing from a Definition.' Thus, if increasing capacity for
co-operation be a specific character of civilisation, 'great wealth' may
be considered as a proprium of civilised as compared with barbarous
nations. For co-operation is made most effectual by the division of
labour, and that this is the chief condition of producing wealth is a
real proposition. Such arguments from definitions concerning concrete
facts and causation require verification by comparing the conclusion
with the facts. The verification of this example is easy, if we do not
let ourselves be misled in estimating the wealth of barbarians by the
ostentatious "pearl and gold" of kings and nobles, where 99 per cent. of
the people live in penury and servitude. The wealth of civilisation is
not only great but diffused, and in its diffusion its greatness must be

To argue from a definition may be a process of several degrees of
complexity. The simplest case is the establishing of a proprium as the
direct consequence of some connoted attribute, as in the above example.
If the definition has been correctly abstracted from the particulars,
the particulars have the attributes summarised in the definition; and,
therefore, they have whatever can be shown to follow from those
attributes. But it frequently happens that the argument rests partly on
the qualities connoted by the class name and partly on many other facts.

In Geometry, the proof of a theorem depends not only upon the definition
of the figure or figures directly concerned, but also upon one or more
axioms, and upon propria or constructions already established. Thus, in
Euclid's fifth Proposition, the proof that the angles at the base of an
isosceles triangle are equal, depends not only on the equality of the
opposite sides, but upon this together with the construction that shows
how from the greater of two lines a part may be cut off equal to the
less, the proof that triangles that can be conceived to coincide are
equal, and the axiom that if equals be taken from equals the remainders
are equal. Similarly, in Biology, if colouring favourable to concealment
is a proprium of carnivorous animals, it is not deducible merely from
their predatory character or any other attribute entering into the
definition of any species of them, but from their predatory character
together with the causes summarised in the phrase 'Natural Selection';
that is, competition for a livelihood, and the destruction of those that
labour under any disadvantages, of which conspicuous colouring would be
one. The particular coloration of any given species, again, can only be
deduced by further considering its habitat (desert, jungle or
snowfield): a circumstance lying wholly outside the definition of the

The validity of an argument based partly or wholly on a definition
depends, in the first place, on the existence of things corresponding
with the definition--that is, having the properties connoted by the name
defined. If there are no such things as isosceles triangles, Euclid's
fifth Proposition is only formally true, like a theorem concerning the
fourth dimension of space: merely consistent with his other assumptions.
But if there be any triangles only approximately isosceles, the proof
applies to them, making allowance for their concrete imperfection: the
nearer their sides approach straightness and equality the more nearly
equal will the opposite angles be.

Again, as to the things corresponding with terms defined, according to
Dr. Venn, their 'existence' may be understood in several senses: (1)
merely for the reason, like the pure genera and species of Porphyry's
tree; the sole condition of whose being is logical consistency: or (2)
for the imagination, like the giants and magicians of romance, the
heroes of tragedy and the fairies of popular superstition; whose
properties may be discussed, and verified by appeal to the right
documents and authorities (poems and ballads): or (3) for perception,
like plants, animals, stones and stars. Only the third class exist in
the proper sense of the word. But under a convention or hypothesis of
existence, we may argue from the definition of a fairy, or a demigod, or
a dragon, and deduce various consequences without absurdity, if we are
content with poetic consistency and the authority of myths and romances
as the test of truth.

In the region of concrete objects, whose properties are causes, and
neither merely fictions nor determinations of space (as in Geometry), we
meet with another condition of the validity of any argument depending on
a definition: there must not only be objects corresponding to the
definition, but there must be no other causes counteracting those
qualities on whose agency our argument relies. Thus, though we may infer
from the quality of co-operation connoted by civilisation, that a
civilised country will be a wealthy one, this may not be found true of
such a country recently devastated by war or other calamity. Nor can
co-operation always triumph over disadvantageous circumstances.
Scandinavia is so poor in the gifts of nature favourable to industry,
that it is not wealthy in spite of civilisation: still, it is far
wealthier than it would be in the hands of a barbarous people. In short,
when arguing from a definition, we can only infer the _tendency_ of any
causal characteristics included in it; the unqualified realisation of
such a tendency must depend upon the absence of counteracting causes. As
soon as we leave the region of pure conceptions and make any attempt to
bring our speculations home to the actual phenomena of nature or of
human life, the verification of every inference becomes an unremitting



§ 1. A Fallacy is any failure to fulfil the conditions of proof. If we
neglect or mistake the conditions of proof unintentionally, whether in
our private meditations or in addressing others, it is a Paralogism: but
if we endeavour to pass off upon others evidence or argument which we
know or suspect to be unsound, it is a Sophism.

Fallacies, whether paralogisms or sophisms, may be divided into two
classes: (a) the Formal, or those that can be shown to conflict with one
or more of the truths of Logic, whether Deductive or Inductive; as if we
attempt to prove an universal affirmative in the Third Figure; or to
argue that, as the average expectation of life for males at the age of
20 is 19-1/2 years, therefore Alcibiades, being 20 years of age, will
die when he is 39-1/2; (b) the Material, or those that cannot be clearly
exhibited as transgressions of any logical principle, but are due to
superficial inquiry or confused reasoning; as in adopting premises on
insufficient authority, or without examining the facts; or in mistaking
the point to be proved.

§ 2. Formal Fallacies of Deduction and Induction are, all of them,
breaches of the rule 'not to go beyond the evidence.' As a detailed
account of them would be little else than a repetition of the foregoing
chapters, it may suffice to recall some of the places at which it is
easiest to go astray.

(1) It is not uncommon to mistake the Contrary for the Contradictory,
as--A is not taller than B, ∴ he is shorter.

(2) To convert _A._ or _O._ simply, as--

    All Money is Wealth ∴ All Wealth is Money;

or--Some Wealth is not Money ∴ Some Money is not Wealth.

In both these cases, Wealth, though undistributed in the convertend, is
distributed in the converse.

(3) To attempt to syllogise with two premises containing four terms, as

      The Papuans are savages;
      The Javanese are neighbours of the Papuans:
    ∴ The Javanese are savages.

Such an argument is excluded by the definition of a Syllogism, and
presents no formal evidence whatever. We should naturally assume that
any man who advanced it merely meant to raise some probability that
'neighbourhood is a sign of community of ideas and customs.' But, if so,
he should have been more explicit. There would, of course, be the same
failure of connection, if a fourth term were introduced into the
conclusion, instead of into the premises.

(4) To distribute in the conclusion a term that was undistributed in the
premises (an error essentially the same as (2) above), i.e., Illicit
process of the major or minor term, as--

      Every rational agent is accountable;
      Brutes are not rational agents:
    ∴ Brutes are not accountable.

In this example (from Whately), an illegitimate mood of Fig. I., the
major term, 'accountable,' has suffered the illicit process; since, in
the premise, it is predicate of an affirmative proposition and,
therefore, undistributed; but, in the conclusion, it is predicate of a
negative proposition and, therefore, distributed. The fact that nearly
everybody would accept the conclusion as true, might lead one to
overlook the formal inconclusiveness of the proof.


      All men are two-handed;
      All two-handed animals are cooking animals:
    ∴ All cooking animals are men.

Here we have Bramantip concluding in A.; and there is, formally, an
illicit process of the minor; though the conclusion is true; and the
evidence, such as it is, is materially adequate. ('Two-handed,' being a
peculiar differentia, is nugatory as a middle term, and may be cut out
of both premises; whilst 'cooking' is a proprium peculiar to the species
Man; so that these terms might be related in U., _All men are all
cookers_; whence, by conversion, _All cookers are men_.)

(5) To omit to distribute the middle term in one or the other premise,

      All verbal propositions are self-evident;
      All axioms are self-evident:
    ∴ All axioms are verbal propositions.

This is an illegitimate mood in Fig. II.; in which, to give any
conclusion, one premise must be negative. It may serve as a formal
illustration of Undistributed Middle; though, as both premises are
verbal propositions, it is (materially) not syllogistic at all, but an
error of classification; a confounding of co-ordinate species by assuming
their identity because they have the generic attribute in common.

(6) To simply convert an hypothetical proposition, as--

      If trade is free, it prospers;
    ∴ If trade prospers, it is free.

This is similar to the simple conversion of the categorical A.; since it
takes for granted that the antecedent is co-extensive with the
consequent, or (in other words) that the freedom of trade is the sole
condition of, or (at least) inseparable from, its prosperity.

The same assumption is made if, in an hypothetical syllogism, we try to
ground an inference on the affirmation of the consequent or denial of
the antecedent, as--

    If trade is free it prospers:
      It does prosper;
    ∴ It is free.
      It is not free;
    ∴ It does not prosper.

Neither of these arguments is formally good; nor, of course, is either
of them materially valid, if it be possible for trade to prosper in
spite of protective tariffs.

An important example of this fallacy is the prevalent notion, that if
the conclusion of an argument is true the premises must be trustworthy;
or, that if the premises are false the conclusion must be erroneous.
For, plainly, that--

If the premises are true, the conclusion is true, is a hypothetical
proposition; and we argue justly--

        The premises are true;
      ∴ The conclusion is true;
    or, The conclusion is false;
      ∴ The premises are false (or one of them is).

This is valid for every argument that is formally correct; but that we
cannot trust the premises on the strength of the conclusion, nor reject
the conclusion because the premises are absurd, the following example
will show:

      All who square the circle are great mathematicians;
      Newton squared the circle:
    ∴ Newton was a great mathematician.

The conclusion is true; but the premises are intolerable.

How the taking of Contraries for Contradictories may vitiate Disjunctive
Syllogisms and Dilemmas has been sufficiently explained in the twelfth

§ 3. Formal Fallacies of Induction consist in supposing or inferring
Causation without attempting to prove it, or in pretending to prove it
without satisfying the Canons of observation and experiment: as--

(1) To assign the Cause of anything that is not a concrete event: as,
e.g., why two circles can touch only in one point. We should give the
'reason'; for this expression includes, besides evidence of causation,
the principles of formal deduction, logical and mathematical.

(2) To argue, as if on inductive grounds, concerning the cause of the
Universe as a whole. This may be called the fallacy of transcendent
inference: since the Canons are only applicable to instances of events
that can be compared; they cannot deal with that which is in its nature

(3) To mistake co-existent phenomena for cause and effect: as when a
man, wearing an amulet and escaping shipwreck, regards the amulet as the
cause of his escape. To prove his point, he must either get again into
exactly the same circumstances without his amulet, and be
drowned--according to the method of Difference; or, shirking the only
satisfactory test, and putting up with mere Agreement, he must show, (a)
that all who are shipwrecked and escape wear amulets, and (b) that their
cases agree in nothing else; and (c), by the Joint Method, that all who
are shipwrecked without amulets are drowned. And even if his evidence,
according to Agreement, seemed satisfactory at all these points, it
would still be fallacious to trust to it as proof of direct causation;
since we have seen that unaided observation is never sufficient for
this: it is only by experiment in prepared circumstances that we can
confidently trace sequence and the transfer of energy.

There is the reverse error of mistaking causal connection for
independent co-existence: as if any one regards it as merely a curious
coincidence that great rivers generally flow past great towns. In this
case, however, the evidence of connection does not depend merely upon
direct Induction.

(4) _Post hoc, ergo propter hoc_: to accept the mere sequence of
phenomena, even though often repeated, as proving that the phenomena are
cause and effect, or connected by causation. This is a very natural
error: for although, the antecedents of a phenomenon being numerous,
most of them cannot be its cause, yet it is among them that the cause
must be sought. Indeed, if there is neither time nor opportunity for
analysis, it may seem better to accept any antecedent as a cause (or, at
least, as a sign) of an important event than to go without any guide.
And, accordingly, the vast and complicated learning of omens, augury,
horoscopy and prophetic dreams, relies upon this maxim; for whatever the
origin of such superstitions, a single coincidence in their favour
triumphantly confirms them. It is the besetting delusion of everybody
who has wishes or prejudices; that is, of all of us at some time or
other; for then we are ready to believe without evidence. The fallacy
consists in judging off-hand, without any attempt, either by logic or by
common sense, to eliminate the irrelevant antecedents; which may include
all the most striking and specious.

(5) To regard the Co-Effects (whether simultaneous or successive) of a
common cause as standing in the direct relation of cause and effect.
Probably no one supposes that the falling of the mercury in his
thermometer causes the neighbouring lake to freeze. True, it is the
antecedent, and (within a narrow range of experience) may be the
invariable antecedent of the formation of ice; but, besides that the two
events are so unequal, every one is aware that there is another
antecedent, the fall of temperature, which causes both. To justify
inductively our belief in causation, the instances compared must agree,
or differ, in one circumstance only (besides the effect). The flowing
tide is an antecedent of the ebbing tide; it is invariably so, and is
equal to it; but it is not the cause of it: other circumstances are
present; and the moon is the chief condition of both flow and ebb. In
several instances, States that have grown outrageously luxurious have
declined in power: that luxury caused their downfall may seem obvious,
and capable of furnishing a moral lesson to the young. Hence other
important circumstances are overlooked, such as the institution of
slavery, the corruption and rapacity of officials and tax-gatherers, an
army too powerful for discipline; any or all of which may be present,
and sufficient to explain both the luxury and the ruin.

(6) To mistake one condition of a phenomenon for the whole cause. To
speak of an indispensable condition of any phenomenon as the cause of
it, may be a mere conventional abbreviation; and in this way such a mode
of expression is common not only in popular but also in scientific
discussion. Thus we say that a temperature of 33° F. is a cause of the
melting of ice; although that ice melts at 33° F., must further depend
upon something in the nature of water; for every solid has its own
melting-point. As long, then, as we remember that 'cause,' used in this
sense, is only a convenient abbreviation, no harm is done; but, if we
forget it, fallacy may result: as when a man says that the cause of a
financial crisis was the raising of the rate of discount, neglecting the
other conditions of the market; whereas, in some circumstances, a rise
of the Bank-rate may increase public confidence and prevent a crisis.

We have seen that the direct use of the Canons of Agreement and
Difference may only enable us to say that a certain antecedent is a
cause or an indispensable condition of the phenomenon under
investigation. If, therefore, it is important to find the whole cause,
we must either experiment directly upon the other conditions, or resort
to the Method of Residues and deductive reasoning; nor must we be
content, without showing (where such precision is possible) that the
alleged cause and the given phenomenon are equal.

(7) To mistake a single consequence of a given cause for the whole
effect, is a corresponding error; and none so common. Nearly all the
mistakes of private conduct and of legislation are due to it: To cure
temporary lassitude by a stimulant, and so derange the liver; to
establish a new industry by protective duties, and thereby impoverish
the rest of the country; to gag the press, and so drive the discontented
into conspiracy; to build an alms-house, and thereby attract paupers
into the parish, raise the rates, and discourage industry.

(8) To demand greater exactness in the estimate of causes or effects
than a given subject admits of. In the more complex sciences, Biology,
Psychology, Sociology, it is often impossible to be confident that all
the conditions of a given phenomenon have been assigned, or that all its
consequences have been traced. The causes of the origin of species and
of the great French Revolution have been carefully investigated, and
still we may doubt whether they have all been discovered, or whether
their comparative importance has been rightly determined; but it would
be very unreasonable to treat those things as miraculous and
unintelligible. We read in the _Ethics_, that a properly cultivated mind
knows what degree of precision is to be expected in each science. The
greatest possible precision is always to be sought; but what is possible
depends partly on the nature of the study and partly upon the state of
scientific preparation.

(9) To treat an agent or condition remote in time as an unconditional
cause: for every moment of time gives an opportunity for new
combinations of forces and, therefore, for modifications of the effect.
Thus, although we often say that Napoleon's Russian expedition was the
cause of his downfall, yet the effect was subject to numerous further
conditions. Had the natives not burnt Moscow, had the winter been
exceptionally mild, had the Prussians and Austrians not risen against
him, the event might have been very different. It is rash to trace the
liberties of modern Europe to the battle of Marathon. Indeed, our powers
of perception are so unequal to the subtlety of nature, that even in
experimental science there is time for molecular changes to occur
between what we treat as a cause and the effect as we perceive it; and,
in such cases, the strictly unconditional cause has not been discovered.

(10) To neglect the negative conditions to which a cause is subject.
When we say that water boils at 212° F., we mean "provided the pressure
be the same as that of the atmosphere at about the sea-level"; for under
a greater pressure water will not boil at that temperature, whilst under
less pressure it boils at a lower temperature. In the usual statement of
a law of causation, 'disturbing,' 'frustrating,' 'counteracting'
circumstances (that is, negative conditions) are supposed to be absent;
so that the strict statement of such a law, whether for a remote cause,
or for an immediate cause (when only positive conditions are included),
is that the agent or assemblage of conditions, _tends_ to produce such
an effect, other conditions being favourable, or in the absence of
contrary forces.

(11) It is needless to repeat what has already been said of other
fallacies that beset inductive proof; such as the neglect of a possible
plurality of causes where the effect has been vaguely conceived; the
extension of empirical laws beyond adjacent cases; the chief errors to
which the estimate of analogies and probabilities, or the application of
the principles of classification are liable; and the reliance upon
direct Induction where the aid of Deduction may be obtained, or upon
observation where experiment may be employed. As to formal fallacies
that may be avoided by adhering to the rules of logical method, this may

§ 4. There remain many ways in which arguments fall short of a tolerable
standard of proof, though they cannot be exhibited as definite breaches
of logical principles. Logicians, therefore, might be excused from
discussing them; but out of the abundance of their pity for human
infirmity they usually describe and label the chief classes of these
'extra-logical fallacies,' and exhibit a few examples.

We may adopt Whately's remark, that a fallacy lies either (1) in the
premises, or (2) in the conclusion, or (3) in the attempt to connect a
conclusion with the premises.

(1) Now the premises of a sound argument must either be valid
deductions, or valid inductions, or particular observations, or axioms.
In an unsound argument, then, whose premises are supported by either
deduction or induction, the evidence may be reduced to logical rules;
and its failure is therefore a 'logical fallacy' such as we have already
discussed. It follows that an extra-logical fallacy of the premises must
lie in what cannot be reduced to rules of evidence, that is, in bad
observations (§ 5), or sham axioms (§ 6).

(2) As to the conclusion, this can only be fallacious if some other
conclusion has been substituted for that which was to have been proved
(§ 7).

(3) Fallacies in the connection between premises and conclusion, if all
the propositions are distinctly and explicitly stated, become manifest
upon applying the rules of Logic. Fallacies, therefore, which are not
thus manifest, and so are extra-logical, must depend upon some sort of
slurring, confusion, or ambiguity of thought or speech (§ 8).

§ 5. Amongst Fallacies of Observation, Mill distinguishes (1) those of
Non-observation, where either instances of the presence or absence of
the phenomenon under investigation, or else some of the circumstances
constituting it or attending upon it, though important to the induction,
are overlooked. These errors are implied in the Formal Fallacies of
Induction already treated of in § 3 (paragraphs (3) to (7)).

Mill's class (2) comprises fallacies of Malobservation. Malobservation
may be due to obtuseness or slowness of perception; and it is one
advantage of the physical sciences as means of education, that the
training involved in studying them tends to cure these defects--at
least, within their own range.

But the occasion of error upon which Mill most insists, is our proneness
to substitute a hasty inference for a just representation of the fact
before us; as when a yachtsman, eager for marvels, sees a line of
porpoises and takes them for the sea-serpent. Every one knows what it is
to mistake a stranger for a friend, a leaf for a sparrow, one word for
another. The wonder is that we are not oftener wrong; considering how
small a part present sensation has in perception, and how much of every
object observed is supplied by a sort of automatic judgment. You see
something brown, which your perceptive mechanism classes with the
appearance of a cow at such a distance; and instantly all the other
properties of a cow are supplied from the resources of former
experience: but on getting nearer, it turns out to be a log of wood. It
is some protection against such errors to know that we are subject to
them; and the Logician fulfils his duty in warning us accordingly. But
the matter belongs essentially to Psychology; and whoever wishes to
pursue it will find a thorough explanation in Prof. Sully's volume on

Another error is the accumulation of useless, irrelevant observations,
from which no proof of the point at issue can be derived. It has been
said that an important part of an inductive inquirer's equipment
consists in knowing what to observe. The study of any science educates
this faculty by showing us what observations have been effective in
similar cases; but something depends upon genius. Observation is
generally guided by hypotheses: he makes the right observations who can
frame the right hypotheses; whilst another overlooks things, or sees
them all awry, because he is confused and perverted by wishes,
prejudices or other false preconceptions; and still another gropes about
blindly, noting this and docketing that to no purpose, because he has no
hypothesis, or one so vague and ill-conceived that it sheds no light
upon his path.

§ 6. The second kind of extra-logical Fallacy lying in the premises,
consists in offering as evidence some assertion entirely baseless or
nugatory, but expressed in such a way as to seem like a general truth
capable of subsuming the proposition in dispute: it is generally known
as _petitio principii_, or begging the question. The question may be
begged in three ways:

(1) There are what Mill calls Fallacies _a priori_, mere assertions,
pretending to be self-evident, and often sincerely accepted as such by
the author and some infatuated disciples, but in which the cool
spectator sees either no sense at all, or palpable falsity. These sham
axioms are numerous; and probably every one is familiar with the
following examples: That circular motion is the most perfect; That every
body strives toward its natural place; That like cures like; That every
bane has its antidote; That what is true of our conceptions is true of
Nature; That pleasure is nothing but relief from pain; That the good,
the beautiful and the true are the same thing; That, in trade, whatever
is somewhere gained is somewhere lost; That only in agriculture does
nature assist man; That a man may do what he will with his own; That
some men are naturally born to rule and others to obey. Some of these
doctrines are specious enough; whilst, as to others, how they could ever
have been entertained arouses a wonder that can only be allayed by a
lengthy historical and psychological disquisition.

(2) Verbal propositions offered as proof of some matter of fact. These
have, indeed, one attribute of axioms; they are self-evident to any one
who knows the language; but as they only dissect the meaning of words,
nothing but the meaning of words can be inferred from them. If anything
further is arrived at, it must be by the help of real propositions. How
common is such an argument as this: 'Lying is wrong, because it is
vicious'--the implied major premise being that 'what is vicious is
wrong.' All three propositions are verbal, and we merely learn from them
that lying is _called_ vicious and wrong; and to make that knowledge
deterrent, it must be supplemented by a further premise, that 'whatever
is called wrong ought to be avoided.' This is a real proposition; but it
is much more difficult to prove it than 'that lying ought to be
avoided.' Still, such arguments, though bad Logic, often have a
rhetorical force: to call lying not only wrong but vicious, may be
dissuasive by accumulating associations of shame and ignominy.

Definitions, being the most important of verbal propositions (since they
imply the possibility of as many other verbal propositions as there are
defining attributes and combinations of them), need to be watched with
especial care. If two disputants define the same word in different ways,
with each of the different attributes included in their several
definitions they may bring in a fresh set of real propositions as to the
agency or normal connection of that attribute. Hence their conclusions
about the things denoted by the word defined, diverge in all directions
and to any extent. And it is generally felt that a man who is allowed to
define his terms as he pleases, may prove anything to those who, through
ignorance or inadvertence, grant that the things that those terms stand
for have the attributes that figure in his definitions.

(3) _Circulus in demonstrando_, the pretence of giving a reason for an
assertion, whilst in fact only repeating the assertion itself--generally
in other words. In such cases the original proposition is, perhaps,
really regarded as self-evident, but by force of habit a man says
'because'; and then, after vainly fumbling in his empty pocket for the
coin of reason, the habit of symbolic thinking in words only, without
reference to the facts, comes to his rescue, and he ends with a
paraphrase of the same assertion. Thus a man may try to prove the
necessity of Causation: 'Every event must have a cause; because an event
is a change of phenomena, and this implies a transformation of something
pre-existing; which can only have been possible, if there were forces in
operation capable of transforming it.' Or, again: 'We ought not to go to
war, because it is wrong to shed blood.' But, plainly, if war did not
imply bloodshed, the unlawfulness of this could be nothing against war.
The more serious any matter is, the more important it becomes either to
reason thoroughly about it, or to content ourselves with wholesome
assertions. How many 'arguments' are superfluous!

§ 7. The Fallacy of surreptitious conclusion (_ignoratio elenchi_), the
mistaking or obscuring of the proposition really at issue, whilst
proving something else instead. This may be done by substituting a
particular proposition for an universal, or an universal for a
particular. Thus, he who attacks the practice of giving in charity must
not be content to show that it has, in this or that case, degraded the
recipient; who may have been exceptionally weak. Or, again, to dissuade
another from giving alms in a particular case, it is not enough to show
that the general tendency of almsgiving is injurious; for, by taking
pains in a particular case, the general tendency may often be

Sometimes an argument establishing a wholly irrelevant conclusion is
substituted for an _argumentum ad rem_. Macaulay complains of those
apologists for Charles I. who try to defend him as a king, by urging
that he was a good judge of paintings and indulgent to his wife.

To this class of Fallacies belongs the _argumentum ad hominem_, which
consists in showing not that a certain proposition is true, but that
Critias ought to accept it in consistency with his other opinions. Thus:
'In every parish the cost of education ought to be paid out of the
rates: you, at least, have said that there can be no sound economy,
unless local expenses are defrayed from local funds.' But whether this
is a fallacy depends, as Whately observes, upon whether it is urged as
actually proving the point at issue, or merely as convicting the
opponent of inconsistency. In the latter case, the argument is quite
fair: whatever such a conclusion may be worth.

Similarly with the _argumentum ad populum_: 'this measure is favourable
to such or such a class; let them vote for it.' An appeal to private
greed, however base, is not fallacious, as long as the interest of the
class is not _fraudulently_ substituted for the good of the nation. And
much the same may be said for the _argumentum ad verecundiam_. When a
question of morals is debated as a question of honour among thieves,
there is no fallacy, if the moral issue is frankly repudiated. The
argument from authority is often brought under this head: 'such is the
opinion of Aristotle.' Although this does not establish the truth of any
proposition, it may be fairly urged as a reason for not hastily adopting
a contrary conclusion: that is, if the subject under discussion be one
as to which Aristotle (or whoever the authority may be) had materials
for forming a judgment.

A negative use of this fallacy is very common. Some general doctrine,
such as Positivism, Transcendentalism, Utilitarianism, or Darwinism, is
held in common by a group of men; who, however, all judge independently,
and therefore are likely to differ in details. An opponent exhibits
their differences of opinion, and thereupon pretends to have refuted the
theory they agree in supporting. This is an _argumentum ad scholam_, and
pushes too far the demand for consistency. In fact it recoils upon the
sophist; for there is no sense in quoting men against one another,
unless both (or all) are acknowledged to speak with the authority of
learning and judgment, and therefore the general doctrine which they
hold in common is the more confirmed.

This is an example of the paralogism of 'proving too much'; when a
disputant is so eager to refute an opponent as to lay down, or imply,
principles from which an easy inference destroys his own position. To
appeal to a principle of greater sweep than the occasion requires may
easily open the way to this pitfall: as if a man should urge that 'all
men are liars,' as the premise of an argument designed to show that
another's assertion is less credible than his own.

A common form of _ignoratio elenchi_ is that which Whately called the
'fallacy of objections': namely, to lay stress upon all the
considerations against any doctrine or proposal, without any attempt to
weigh them against the considerations in its favour; amongst which
should be reckoned all the considerations that tell against the
alternative doctrines or proposals. Incontestable demonstration can
rarely be expected even in science, outside of the Mathematics; and in
practical affairs, as Butler says, 'probability is the very guide of
life'; so that every conclusion depends upon the balance of evidence,
and to allow weight to only a part of it is an evasion of the right

§ 8. Fallacies in the connection of premises and conclusion, that cannot
be detected by reducing the arguments to syllogistic form, must depend
upon some juggling with language to disguise their incoherence. They may
be generally described as Fallacies of Ambiguity, whether they turn upon
the use of the same word in different senses, or upon ellipsis. Thus it
may be argued that all works written in a classical language are
classical, and that, therefore, the history of Philosophy by Diogenes
Lærtius, being written in Greek, is a classic. Such ambiguities are
sometimes serious enough; sometimes are little better than jokes. For
jokes, as Whately observes, are often fallacies; and considered as a
propædeutic to the art of sophistry, punning deserves the ignominy that
has overtaken it.

Fallacies of ellipsis usually go by learned names, as; (1) _a dicto
secundum quid ad dictum simpliciter_. It has been argued that since,
according to Ricardo, the value of goods depends solely upon the
quantity of labour necessary to produce them, the labourers who are
employed upon (say) cotton cloth ought to receive as wages the whole
price derived from its sale, leaving nothing for interest upon capital.
Ricardo, however, explained that by 'the quantity of labour necessary to
produce goods' he meant not only what is immediately applied to them,
but also the labour bestowed upon the implements and buildings with
which the immediate labour is assisted. Now these buildings and
implements are capital, the labour which produced them was paid for, and
it was far enough from Ricardo's mind to suppose that the capital which
assists present labour upon (say) cotton cloth has no claim to
remuneration out of the price of it. In this argument, then, the word
labour in the premise is used _secundum quid_, that is, with the
suppressed qualification of including past as well as present labour;
but in the conclusion labour is used _simpliciter_ to mean present
labour only.

(2) _A dicto secundum quid ad dictum secundum alterum quid_. It may be
urged that, since the tax on tea is uniform, therefore all consumers
contribute equally to the revenue for their enjoyment of it. But written
out fairly this argument runs thus: Since tea is taxed uniformly _4d.
per lb._, all consumers pay equally for their enjoyment of it _whatever
quantity they use_. These qualifications introduced, nobody can be

(3) _A dicto simpliciter ad dictum secundum quid_, also called _fallacia
accidentis_. Thus: To take interest upon a loan is perfectly just,
therefore, I do right to exact it from my own father in distress. The
popular answer to this sort of blunder is that 'circumstances alter
cases.' We commit this error in supposing that what is true of the
average is likely to be true of each case; as if one should say: 'The
offices are ready to insure my house [with thousands of others] against
fire at a rate per annum which will leave them heavy losers unless it
lasts a hundred years; so, as we are told not to take long views of
life, I shall not insure.'

The Fallacy of Division and Composition consists in suggesting, or
assuming, that what is true of things severally denoted by a term is
true of them taken together. That every man is mortal is generally
admitted, but we cannot infer that, therefore, the human race will
become extinct. That the remote prospects of the race are tragic may be
plausibly argued, but not from that premise.

Changing the Premises is a fallacy usually placed in this division;
although, instead of disguising different meanings under similar words,
it generally consists in using words or phrases ostensibly differing, as
if they were equivalent: those addressed being expected to renounce
their right to reduce the argument to strict forms of proof, as needless
pedantry in dealing with an author so palpably straightforward. If an
orator says--'Napoleon conquered Europe; in other words, he murdered
five millions of his fellow creatures'--and is allowed to go on, he may
infer from the latter of these propositions many things which the former
of them would hardly have covered. This is a sort of hyperbole, and
there is a corresponding meiosis, as: 'Mill _admits_ that the Syllogism
is useful'; when, in fact, that is Mill's _contention_. It may be
supposed that, if a man be fool enough to be imposed upon by such
transparent colours, it serves him right; but this harsh judgment will
not be urged by any one who knows and considers the weaker brethren.

§ 9. The above classification of Fallacies is a rearrangement of the
plans adopted by Whately and Mill. But Fallacies resemble other
spontaneous natural growths in not submitting to precise and definite
classification. The same blunders, looked at from different points of
view, may seem to belong to different groups. Thus, the example given
above to illustrate _fallacia accidentis_, 'that, since it is just to
take interest, it is right to exact it from one's own father,' may also
be regarded as _petitio principii_, if we consider the unconditional
statement of the premise--'to take interest upon a loan is perfectly
just'; for, surely, this is only conditionally true. Or, again, the
first example given of simple ambiguity--'that whatever is written in a
classical language is classical, etc.,' may, if we attend merely to
the major premise, be treated as a bad generalisation, an undue
extension of an inference, founded upon a simple enumeration of the
first few Greek and Latin works that one happened to remember.

It must also be acknowledged that genuine wild fallacies, roaming the
jungle of controversy, are not so easily detected or evaded as specimens
seem to be when exhibited in a Logician's collection; where one surveys
them without fear, like a child at a menagerie. To assume the succinct
mode of statement that is most convenient for refutation, is not the
natural habit of these things. But to give reality to his account of
fallacies an author needs a large space, that he may quote no
inconsiderable part of literature ancient and modern.

As to the means of avoiding fallacies, a general increase of sincerity
and candour amongst mankind may be freely recommended. With more honesty
there would be fewer bad arguments; but there is such a thing as
well-meaning incapacity that gets unaffectedly fogged in converting A.,
and regards the refractoriness of O., as more than flesh and blood can
endure. Mere indulgence in figurative language, again, is a besetting
snare. "One of the fathers, in great severity called poesy _vinum
dæmonum_," says Bacon: himself too fanciful for a philosopher. Surely,
to use a simile for the discovery of truth is like studying beauty in
the bowl of a spoon.

The study of the natural sciences trains and confirms the mind in a
habit of good reasoning, which is the surest preservative against
paralogism, as long as the terms in use are, like those of science, well
defined; and where they are ill defined, so that it is necessary to
guard against ambiguity, a thorough training in politics or metaphysics
may be useful. Logic seems to me to serve, in some measure, both these
purposes. The conduct of business, or experience, a sufficient time
being granted, is indeed the best teacher, but also the most austere and
expensive. In the seventeenth century some of the greatest philosophers
wrote _de intellectus emendatione_; and if their successors have given
over this very practical inquiry, the cause of its abandonment is not
success and satiety but despair. Perhaps the right mind is not to be
made by instruction, but can only be bred: a slow, haphazard process;
and meanwhile the rogue of a sophist may count on a steady supply of
dupes to amuse the tedium of many an age.



_The following questions are chiefly taken from public examination
papers: Civil Service_ [S], _Oxford_ [O], _Cambridge_ [C], _London_ [L].


1. What is a Term? Explain and illustrate the chief divisions of Terms.
What is meant by the Connotation of a Term? Illustrate. [S]

2. "The connotation and denotation of terms vary inversely." Examine
this assertion, explaining carefully the limits within which it is true,
if at all. [S]

3. Exemplify the false reasoning arising from the confusion of Contrary
and Contradictory Terms. [S]

4. Discuss the claims of the doctrine of Terms to be included in a
Logical System. Distinguish between a General and an Abstract Term. [S]

5. Explain and illustrate what is meant by the Denotation and
Connotation of a Term. What terms have both, and what have one only? [S]

6. Distinguish between Abstract and Concrete Names. To which of these
classes belong (a) adjectives, (b) names of states of consciousness? Are
any abstract names connotative? [S]

7. Distinguish between (a) Proper and Singular Terms, (b) Negative and
Privative, (c) Absolute and Relative. Illustrate.

8. What connection is there between the Connotation and the Relativity
of Names?

9. Examine the logical relations between the following pairs of terms:
(a) happy and happiness; (b) happy and unhappy; (c) 'the juryman' and
'the jury'; (d) parent and offspring.

Explain the technical words used in your answer. [C]

10. Distinguish between _name_; _part of speech_; _term_: and illustrate
by reference to the following--use, useful, usefully. [C]

11. Describe the nature of _Collective_ terms; examine in particular any
difficulties in distinguishing between these and general or abstract
terms. [C]

12. Distinguish between _positive_, _negative_, and _privative_ names.
Of what kind are the following, and why--parallel, alien, idle, unhappy?
What ambiguity is there in the use of such a term as "not-white"? [C]


13. What is meant by (1) the Conversion, and (2) the Contra-position of
a proposition? Apply these processes, as far as admissible, to the

    (a) All invertebrates have cold blood.

    (b) Some cold-blooded animals are not invertebrates.

    (c) No wingless birds are songsters.

    (d) Some winged birds are not songsters.

What can you infer from (a) and (b) jointly, and what from (c) and (d)
jointly? [S]

14. "The author actually supposes that, because Professor Fawcett denies
that all wealth is money, he denies that all money is wealth." Analyse
the differences of opinion implied in the above passage. [S]

15. Take any universal affirmative proposition; convert it by obversion
(contraposition); attach the negative particle to the predicate, and
again convert. Interpret the result exactly, and say whether it is or is
not equivalent to the original proposition. [S]

16. What information about the term "solid body" can we derive from the
proposition, "No bodies which are not solids are crystals"? [S]

17. Discuss the proposal to treat all propositions as affirmative.

18. Convert the proposition "A is probably B." What information does the
proposition give us concerning B? [S]

19. Show in how many ways you can deny the following assertions: All
cathedral towns are all cities; Canterbury is the Metropolitan see. [S]

20. Explain the nature of a _hypothetical_ (or conditional) proposition.
What do you consider the radical difference between it and a
categorical? [S]

21. What is the function of the _copula_? In what different manners has
it been treated? [S]

22. Convert "A killed C unjustly"; "All Knowledge is probably useful";
"The exception proves the rule"; "Birds of a feather flock together."

23. What is modality? How are modals treated by (a) formal logic and (b)
by the theory of induction? [S]

24. What is the subject of an impersonal proposition? Give reasons for
your answer. [S]

25. Is the categorical proposition sufficiently described as referring a
thing or things to a class? [S]

26. Enumerate the cases in which the truth or falsity of one proposition
may be formally inferred from the truth or falsity of another.
Illustrate these cases, and give to each its technical name. [S]

27. Illustrate the relation of Immediate Inferences to the Laws of

28. Explain what is meant by (a) Symbolic Logic; (b) the Logic of
Relatives. Describe some method of representing propositions by means
of diagrams; and indicate how far any particular theory of the import of
propositions is involved in such representation. [S]

29. Explain the exact nature of the relation between two _Contradictory_
propositions; and define Conversion by Contraposition, determining what
kind of propositions admit of such conversion.

Give the contradictory and the contrapositive of each of the following

    (a) All equilateral triangles are equiangular;

    (b) No vertebrate animal has jaws opening sideways;

    (c) Wherever A and B are both present, either C or D is also
    present. [S]

30. Define Obversion and Inversion, and apply these processes also to
the above three propositions.

31. Propositions can be understood either in extension or in intension.
Explain this, and discuss the relative value of the two interpretations.

32. Distinguish between real and verbal propositions; and explain the
importance of the distinction.

33. Illustrate the process called 'change of Relation.'


34. What is a Syllogism? Find, without reference to the mnemonic verses,
in what different ways it is possible to prove syllogistically the
conclusion _No S is P_; and show the equivalence between these different
ways. [S]

35. From what points of view can the syllogism be regarded

    (1) as being, (2) as not being, a _petitio principii_? [S]

36. What are the figures of syllogism? For what kind of arguments are
they severally adapted? [S]

37. What is meant by Mood and Figure? How can the validity of a Mood be
tested? Should there be four Figures or three? [S]

38. Construct syllogisms in Camenes, Datisi and Baroco, and reduce them
to the corresponding moods of the first figure.

39. Explain the meaning of "ostensive" and "indirect" Reduction. Show
that any Mood of the second Figure may be reduced in either way.

40. Show that A cannot be proved except in the First Figure. Express the
following reasoning in as many syllogistic figures as you can: Some
theorists cannot be trusted, for they are unwise. [S]

41. Discuss the possibility of reducing the argument _a fortiori_ to the
syllogistic form. [S]

42. Can a false conclusion be reached through true premises, or a true
conclusion through false premises? Give reasons for your answer. [S]

43. Can we under any circumstances infer a relation between X and Z from
the premises--

    Some Y's are X's
    Some Y's are Z's?    [S]

44. Take an apparent syllogism subject to the fallacy of negative
premises, and inquire whether you can correct the reasoning by
converting one or both of the premises into the affirmative form. [S]

45. Enumerate the faults to which a syllogism is liable, giving
instances of each. [S]

46. State any Enthymeme, and expand it into (1) a Syllogism, (2) an
Epicheirema, (3) a Sorites; and give in each case the technical name of
the Mood or Order that results.

47. State any Disjunctive Syllogism, and change it (1) into a
Hypothetical, (2) into a Categorical; and discuss the loss or gain, in
cogency or significance involved in this process.

48. Can the Syllogism be treated as merely a consequence of the "Laws of
Thought"? If not, why not; and what else does it imply?

49. Prove that with three given propositions (of the forms A., E., I.,
O.) it is never possible to construct more than one valid syllogism. [C]

50. Distinguish between a Constructive and a Destructive Hypothetical
Syllogism; and show how one may be reduced to the other. [C]


51. What constitutes a Valid Induction? Distinguish it from a legitimate
hypothesis. [S]

52. Is it possible to form true universal propositions about facts if we
have not actually observed all the individuals designated by the subject
of the proposition? If so, how? [S]

53. "Perfect induction is demonstrative and syllogistic; imperfect
induction is neither." Explain the difference between perfect and
imperfect induction, and examine the truth of this assertion. [S]

54. Why is it that one should not regard night as the cause, nor even as
a universal condition of day? Explain "cause" and "condition." [S]

55. What do you understand by an experiment? Can you say how many
experiments are required to establish (1) a fact, (2) a law of nature?

56. How would you define _antecedent_, _cause_, _effect_, _consequent_?

57. England is the richest country in the world, and has a gold
currency. Russia and India, in proportion to population, are poor
countries and have little or no gold currency. How far are such kind of
facts logically sufficient to prove that a gold currency is the cause of
a nation's wealth? [S]

58. A man having been shot through the heart immediately falls dead.
Investigate the logical value of such a fact as proving that all men
shot through the heart will fall dead. [S]

59. Explain the process of induction called the Method of Difference,
and give some new instances of its application. How is it related to the
Method of Concomitant Variations? What is the Major Premise implied in
all these methods? [S]

59A. Examine the position, that the Canons of Experiment are useless,
because the work of preparing the experiments must have been done before
the canons can be applied.

60. Explain the logical cogency of experiments in the search for
physical causes. [S]

61. If the effects of A B C D are fully expressed by a b c d, and those
of B C D by b c d, what inductive inference can be drawn and on what
principle? State the canon according to which it is drawn. [S]

62. Compare the advantage of observation and experiment as means of
gaining data for Reasoning. [S]

63. Compare the cogency of different Inductive Methods, showing the kind
of evidence each requires, and the principle on which it is based. [S]

64. Compare the Canons of Agreement and Difference (1) as to the
difficulty of finding or preparing actual Instances for them, and (2) as
to their conclusiveness.

65. Describe what is meant by residual phenomena, and estimate their
value in inductive science. [S]

66. What is the argument from Analogy? How does it differ from (a)
Induction, (b) metaphorical argument? [S]

67. What are the various senses in which the word Analogy has been used?
Distinguish, giving instances, between good and bad analogies. [S]

68. How do you distinguish between what Mill calls the Geometrical,
Physical, and Historical Methods?

68A. The Comparative Method is appealed to where direct evidence is
wanting. Explain this.

69. What is meant by a doctrine being unverifiable? If a conclusion
reached by deduction does not agree with the facts, where must we look
for error?

70. There are certain cases in which failure of verification is fatal to
a theory, and other cases in which it is of comparatively little
cogency. How would you distinguish between these classes of cases? [S]

71. Taking the "evolution," or any other proposed hypothesis, how should
one proceed (a) to show whether it satisfies the conditions of a
legitimate hypothesis sufficiently to entitle it to investigation, and
(b) to test it with a view to its acceptance or rejection as a truth of
science? [S]

72. What do you mean by saying that "a phenomenon has been
satisfactorily explained"?

73. Explain and illustrate the Historical Method of Sociological
inquiry. [S]

74. What is the relation of the theory of Probability to Logic? [S]

75. Explain and discuss the doctrine that Induction is based upon the
Theory of Probability. [S]

75A. What are the logical grounds of the Law of Error?

76. Explain the nature and use of Classification, the means to, and
tests of, its successful performance. [S]

77. What is Definition and what is its use? Mention various difficulties
that occur in the process, and show how they are to be met. [S]

78. Propose rules for a good Division and a good Definition, and
exemplify the breach of them. [S]

79. Examine the validity of the idea of Real Kinds. [O]

80. What kind of words are indefinable, and why? When do we define by
negation and by example? [S]

81. Distinguish between the province and aims of classification and
(logical) division. Illustrate. [S]

82. What is an _infima species_ or _species specialissima_? Compare the
use of the terms _genus_ and _species_ in Logic with that which is
common in speaking of animals or plants. [S]

83. How far does the formation of Definitions and Classifications
constitute the end of Science? [S]

84. Examine the methodological relations between Definition,
Classification and Nomenclature. [S]

85. Give instances of "Differentia," "Property," "Inseparable Accident";
and examine, with reference to your instances, how far it is possible to
distinguish them. [S]


86. "People can reason without the help of Logic." Why is this not a
sufficient objection to the study? In your answer show distinctly why
Logic should be studied. [S]

87. What is the meaning of the assertion that Logic is concerned with
the form, and not with the matter, of thought? [S]

88. "Neither by deductive nor inductive reasoning can we add a tittle to
our implicit knowledge." (Jevons.) Explain and criticise. [S]

89. What is the logical foundation of the indirect method or _reductio
ad absurdum_? Is it applicable to non-mathematical subjects? [S]

90. On what grounds do we believe in the reality of an historical event?

91. "Facts are familiar theories." Explain and discuss this. [O]

92. Wherein lies the difficulty of proving a negative? [O]

93. Can any limits be assigned to the possible unification of the
sciences? [O]

94. Are the results of inductive inference necessarily certain? [O]

95. The method of deductive science is hypothetical. Explain and
discuss. [O]

96. "The uniformity of Nature can never be more than a working
hypothesis." Explain and criticise.

97. "Without speculation there is no good and original observation."
Why? [O]

98. Can the provinces of induction and deduction be kept separate? [O]

99. How far is the relation of logical dependence identical with that of
causation? [O]

99A. Discuss the position that the forms of Logic are meaningless apart
from their application.

100. State in syllogistic form (mood and figure) the following

    (a) As polygamy is in many countries legal, we may infer the
    variability of the moral standard.

    (b) If gold is wealth, to export it diminishes the national

    (c) If all good people are happy, unhappiness is an
    indication of vice.

    (d) One may be sure of the benefits of inuring young
    children to cold, from the strength exhibited by all men and
    women thus treated in infancy.

    (e) Where there is no law, there is no injustice.

    (f) "Dissimulation is but a faint kind of policy or wisdom;
    for it asketh a strong wit and a strong heart to know when to
    tell the truth, and to do it; therefore it is the weaker sort
    of politicians that are the greatest dissemblers." (Bacon.)

    (g) Money being a barren product, it is contrary to nature
    to make it reproduce itself. Usury, therefore, is unnatural,
    and, being unnatural, is unjustifiable.

    (h) The study of mathematics is essential to a complete
    course of education, because it induces a habit of close and
    regular reasoning. [S]

101. Explain and illustrate the following terms:--Subalternans, _Vera
Causa_, Plurality of Causes, Law of Nature, Empirical Law, _Summum
Genus_, Predicament, _Arbor Porphyriana_, Axiom, Universe of discourse
(_suppositio_), Antinomy, Dilemma, Realism, Dichotomy, etc.

102. Is there any distinction and, if so, what, between a complete
Description and an Explanation? [C]

103. On what principles have fallacies been classified? To what extent
do you think a satisfactory classification of Fallacies possible? [C]

104. Examine how far conceptions of Persistence and of Invariable
Concomitance of Properties are involved in the methodological
application of the conception of Cause.

104A. Inquire whether the two following propositions can be reconciled
with one another: (a) The same conjunction of antecedents is invariably
followed by the same consequent; (b) We never find the same concurrence
of phenomena a second time. [C]

105. Using the term Logic in a wide sense, so as to include Methodology,
inquire how far a Logic of Observation is possible, and show in what it
will consist. [C]

106. What is Proof?

Explain and discuss the following dicta:--(a) _Qui nimium probat, nihil
probat_: (b) A bad proof is worse than no proof; (c) The exception
proves the rule; (d) Negatives cannot be proved. [C]

107. Examine how far the rules of immediate and syllogistic inference
are modified by differences of interpretation of the categorical
proposition in respect of the existence of the subject. [S]

108. "An effect is but the sum of all the partial causes, the
concurrence of which constitutes its existence." "The cause of an event
is its invariable and unconditional antecedent." Explain and compare
these two theories of causation. Does either alone exhaust the
scientific conception of cause? [S]

109. Under what logical conditions are statistical inferences
authorised, and what is the nature of their conclusions? [S]

110. Distinguish between Psychology, Metaphysics, and Logic; and discuss
briefly their mutual relations. [S]

111. All processes of inference in which the ultimate premises are
particular cases are equally induction.

Induction is an inverse deduction.

Explain and contrast these two theories of the relation of induction to
deduction. [S]

112. What are the Fallacies specially incident to Induction?--or to the
application of the theory of Probabilities? [S]

113. What is meant by the _personal error_ (or _personal equation_) in
observation? Discuss its importance in different branches of knowledge.

114. Define and illustrate:--Paralogism, _ignoratio elenchi_, _fallacia
accidentis_, _argumentum ad verecundiam_, illicit process, undistributed
middle, etc.

115. State the three fundamental laws of thought, explain their meaning,
and consider how far they are independent of each other? [L]

116. Enumerate the "Heads of Predicables" and define their meaning.
Discuss their logical importance. [L]

117. Upon what grounds has it been asserted that the conclusion of a
syllogism is drawn, not from, but according to, the major premise? Are
they valid? [L]

118. "Experiment is always preferable to observation." Why is this?
Explain from the example of any science how observation and experiment
supplement each other. [L]

119. What is a hypothesis? Distinguish between a working hypothesis and
an established hypothesis, so as to bring out the conditions on which
the latter depends. [L]

120. Explain how good scientific nomenclature and terminology are
connected with the purposes of good classification. [L]

_Printed in Great Britain by Hazell, Watson & Viney, Ld., London and
Aylesbury for Simpkin, Marshall, Hamilton, Kent & Co. Ltd._

[Transcriber's Note:

The following corrections were made to the original text:

Page 40: "inedequate" changed to "inadequate"
Page 42: "classily" changed to "classify"
Page 90: "alledging" changed to "alleging"
Page 128: missing comma added: "Camenes, Dimaris"
Page 141: "evalued" changed to "evaluated"
Page 147: "tellens" changed to "tollens": "and the Modus tollens"
Page 170: "impredictable" changed to "unpredictable"
Page 210: missing word 'a' added: "Sesostris conquered a great"
Page 307: "either" changed to "neither"
Page 315: "inductions" changed to "induction"
Page 401: "quality" changed to "quantity"
Page 401: "propœdeutic" changed to "propædeutic"]

*** End of this Doctrine Publishing Corporation Digital Book "Logic - Deductive and Inductive" ***

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