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Title: Logic, Inductive and Deductive
Author: Minto, William, 1845-1893
Language: English
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UNIVERSITY MANUALS
EDITED BY PROFESSOR KNIGHT
LOGIC
INDUCTIVE AND DEDUCTIVE
PUBLISHED MAY, 1893
_Reprinted December, 1893_
" _November, 1894_
" _January, 1899_
" _August, 1904_
" _June, 1909_
" _September, 1912_
" _July, 1913_
" _January, 1915_
LOGIC
INDUCTIVE AND DEDUCTIVE
BY WILLIAM MINTO, M.A.
HON. LL.D. ST. ANDREWS
LATE PROFESSOR OF LOGIC IN THE UNIVERSITY OF ABERDEEN
LONDON
JOHN MURRAY, ALBEMARLE STREET, W.
1915
PREFACE.
In this little treatise two things are attempted that at first might
appear incompatible. One of them is to put the study of logical
formulæ on a historical basis. Strangely enough, the scientific
evolution of logical forms, is a bit of history that still awaits the
zeal and genius of some great scholar. I have neither ambition nor
qualification for such a _magnum opus_, and my life is already more
than half spent; but the gap in evolutionary research is so obvious
that doubtless some younger man is now at work in the field unknown to
me. All that I can hope to do is to act as a humble pioneer according
to my imperfect lights. Even the little I have done represents work
begun more than twenty years ago, and continuously pursued for the
last twelve years during a considerable portion of my time.
The other aim, which might at first appear inconsistent with this,
is to increase the power of Logic as a practical discipline. The main
purpose of this practical science, or scientific art, is conceived to
be the organisation of reason against error, and error in its various
kinds is made the basis of the division of the subject. To carry out
this practical aim along with the historical one is not hopeless,
because throughout its long history Logic has been a practical
science; and, as I have tried to show at some length in introductory
chapters, has concerned itself at different periods with the risks of
error peculiar to each.
To enumerate the various books, ancient and modern, to which I have
been indebted, would be a vain parade. Where I have consciously
adopted any distinctive recent contribution to the long line of
tradition, I have made particular acknowledgment. My greatest
obligation is to my old professor, Alexander Bain, to whom I owe my
first interest in the subject, and more details than I can possibly
separate from the general body of my knowledge.
W. M.
ABERDEEN, _January, 1893_.
Since these sentences were written, the author of this book has
died; and Professor Minto's _Logic_ is his last contribution to the
literature of his country. It embodies a large part of his teaching in
the philosophical class-room of his University, and doubtless reflects
the spirit of the whole of it.
Scottish Philosophy has lost in him one of its typical
representatives, and the University of the North one of its most
stimulating teachers. There have been few more distinguished men than
William Minto in the professoriate of Aberdeen; and the memory of what
he was, of his wide and varied learning, his brilliant conversation,
his urbanity, and his rare power of sympathy with men with whose
opinions he did not agree, will remain a possession to many who mourn
his loss.
It will be something if this little book keeps his memory alive, both
amongst the students who owed so much to him, and in the large circle
of friends who used to feel the charm of his personality.
WILLIAM KNIGHT.
_GENERAL PLAN OF THE SERIES._
_This Series is primarily designed to aid the University Extension
Movement throughout Great Britain and America, and to supply the need
so widely felt by students, of Text-books for study and reference, in
connexion with the authorised Courses of Lectures._
_The Manuals differ from those already in existence in that they are
not intended for School use, or for Examination purposes; and that
their aim is to educate, rather than to inform. The statement of
details is meant to illustrate the working of general laws, and the
development of principles; while the historical evolution of the
subject dealt with is kept in view, along with its philosophical
significance._
_The remarkable success which has attended University Extension in
Britain has been partly due to the combination of scientific treatment
with popularity, and to the union of simplicity with thoroughness.
This movement, however, can only reach those resident in the larger
centres of population, while all over the country there are thoughtful
persons who desire the same kind of teaching. It is for them also that
this Series is designed. Its aim is to supply the general reader with
the same kind of teaching as is given in the Lectures, and to reflect
the spirit which has characterised the movement, viz., the combination
of principles with facts, and of methods with results._
_The Manuals are also intended to be contributions to the Literature
of the Subjects with which they respectively deal, quite apart from
University Extension; and some of them will be found to meet a general
rather than a special want._
_They will be issued simultaneously in England and America. Volumes
dealing with separate sections of Literature, Science, Philosophy,
History, and Art have been assigned to representative literary men,
to University Professors, or to Extension Lecturers connected with
Oxford, Cambridge, London, and the Universities of Scotland and
Ireland._
_A list of the works in this Series will be found at the end of the
volume._
CONTENTS.
INTRODUCTION.
I.
PAGE
The Origin and Scope of Logic, 1
II.
Logic as a Preventive of Error or Fallacy--The Inner
Sophist, 17
III.
The Axioms of Dialectic and of Syllogism, 29
BOOK I.
THE LOGIC OF CONSISTENCY--SYLLOGISM AND
DEFINITION.
PART I.
THE ELEMENTS OF PROPOSITIONS.
CHAPTER I.
General Names and Allied Distinctions, 43
CHAPTER II.
The Syllogistic Analysis of Proposition, into Terms. (1)
The Bare Analytic Forms. (2) The Practice of Syllogistic
Analysis. (3) Some Technical Difficulties, 62
PART II.
DEFINITION.
CHAPTER I.
(1) Imperfect Understanding of Words. (2) Verification of
the Meaning--Dialectic. (3) Fixation of the Meaning--Division
or Classification, Definition, Naming, 82
CHAPTER II.
The Five Predicables--Verbal and Real Predication, 105
CHAPTER III.
Aristotle's Categories, 112
CHAPTER IV.
The Controversy about Universals--Difficulties concerning
the Relation of General Names to Thought and to Reality, 120
PART III.
THE INTERPRETATION OF PROPOSITIONS.
CHAPTER I.
Theories of Predication--Theories of Judgment, 131
CHAPTER II.
The "Opposition" of Propositions--The Interpretation of
"No," 139
CHAPTER III.
The Implication of Propositions--Immediate Formal Inference
--Eduction, 146
CHAPTER IV.
The Counter-Implication of Propositions, 156
PART IV.
THE INTERDEPENDENCE OF PROPOSITIONS.
CHAPTER I.
The Syllogism, 167
CHAPTER II.
The Figures and Moods of the Syllogism. (1) The First
Figure. (2) The Minor Figures and their Reduction to
the First. (3) Sorites, 173
CHAPTER III.
The Demonstration of the Syllogistic Moods--The Canons
of the Syllogism, 185
CHAPTER IV.
The Analysis of Arguments into Syllogistic Forms, 196
CHAPTER V.
Enthymemes, 205
CHAPTER VI.
The Utility of the Syllogism, 209
CHAPTER VII.
Conditional Arguments--Hypothetical Syllogism, Disjunctive
Syllogism and Dilemma, 215
CHAPTER VIII.
Fallacies in Deductive Argument--_Petitio Principii_ and
_Ignoratio Elenchi_, 226
CHAPTER IX.
Formal or Aristotelian Induction--Inductive Argument--The
Inductive Syllogism, 235
BOOK II.
INDUCTIVE LOGIC, OR THE LOGIC OF SCIENCE.
Introduction, 243
CHAPTER I.
The Data of Experience as Grounds of Inference or Rational
Belief, 273
CHAPTER II.
Ascertainment of Simple Facts in their Order--Personal
Observation--Hearsay Evidence--Method of Testing
Traditional Evidence, 285
CHAPTER III.
Ascertainment of Facts of Causation. (1) _Post Hoc Ergo
Propter Hoc._ (2) Meaning of Cause--Methods of Observation
--Mill's Experimental Methods, 295
CHAPTER IV.
Method of Observation--Single Difference. (1) The Principle
of Single Difference. (2) Application of the Principle, 308
CHAPTER V.
Methods of Observation--Elimination--Single Agreement.
(1) The Principle of Elimination. (2) The Principle of
Single Agreement. (3) Mill's "Joint Method of Agreement
and Difference," 318
CHAPTER VI.
Methods of Observation--Minor Methods. (1) Concomitant
Variations. (2) Single Residue, 329
CHAPTER VII.
The Method of Explanation. (1) The Four Stages of Orderly
Procedure. (2) Obstacles to Explanation--Plurality of Causes
and Intermixture of Effects. (3) The Proof of a Hypothesis, 334
CHAPTER VIII.
Supplementary Methods of Investigation. (1) The Maintenance
of Averages--Supplement to the Method of Difference.
(2) The Presumption from Extra-Casual Coincidence, 351
CHAPTER IX.
Probable Inference to Particulars--The Measurement of
Probability, 362
CHAPTER X.
Inference from Analogy, 367
INTRODUCTION.
I.--THE ORIGIN AND SCOPE OF LOGIC.
The question has sometimes been asked, Where should we begin in Logic?
Particularly within the present century has this difficulty been felt,
when the study of Logic has been revived and made intricate by the
different purposes of its cultivators.
Where did the founder of Logic begin? Where did Aristotle begin? This
seems to be the simplest way of settling where we should begin, for
the system shaped by Aristotle is still the trunk of the tree, though
there have been so many offshoots from the old stump and so many
parasitic plants have wound themselves round it that Logic is now
almost as tangled a growth as the Yews of Borrowdale--
An intertwisted mass of fibres serpentine
Upcoiling and inveterately convolved.
It used to be said that Logic had remained for two thousand years
precisely as Aristotle left it. It was an example of a science or
art perfected at one stroke by the genius of its first inventor.
The bewildered student must often wish that this were so: it is only
superficially true. Much of Aristotle's nomenclature and his central
formulæ have been retained, but they have been very variously
supplemented and interpreted to very different purposes--often to no
purpose at all.
The Cambridge mathematician's boast about his new theorem--"The best
of it all is that it can never by any possibility be made of the
slightest use to anybody for anything"--might be made with truth about
many of the later developments of Logic. We may say the same, indeed,
about the later developments of any subject that has been a playground
for generation after generation of acute intellects, happy in
their own disinterested exercise. Educational subjects--subjects
appropriated for the general schooling of young minds--are
particularly apt to be developed out of the lines of their original
intention. So many influences conspire to pervert the original aim.
The convenience of the teacher, the convenience of the learner,
the love of novelty, the love of symmetry, the love of subtlety;
easy-going indolence on the one hand and intellectual restlessness
on the other--all these motives act from within on traditional
matter without regard to any external purpose whatever. Thus in
Logic difficulties have been glossed over and simplified for the dull
understanding, while acute minds have revelled in variations and new
and ingenious manipulations of the old formulæ, and in multiplication
and more exact and symmetrical definition of the old distinctions.
To trace the evolution of the forms and theories of Logic under these
various influences during its periods of active development is a task
more easily conceived than executed, and one far above the ambition of
an introductory treatise. But it is well that even he who writes for
beginners should recognise that the forms now commonly used have been
evolved out of a simpler tradition. Without entering into the details
of the process, it is possible to indicate its main stages, and thus
furnish a clue out of the modern labyrinthine confusion of purposes.
How did the Aristotelian Logic originate? Its central feature is the
syllogistic forms. In what circumstances did Aristotle invent these?
For what purpose? What use did he contemplate for them? In rightly
understanding this, we shall understand the original scope or province
of Logic, and thus be in a position to understand more clearly how it
has been modified, contracted, expanded, and supplemented.
Logic has always made high claims as the _scientia scientiarum_,
the science of sciences. The builders of this Tower of Babel are
threatened in these latter days with confusion of tongues. We may
escape this danger if we can recover the designs of the founder, and
of the master-builders who succeeded him.
Aristotle's Logic has been so long before the world in abstract
isolation that we can hardly believe that its form was in any way
determined by local accident. A horror as of sacrilege is excited by
the bare suggestion that the author of this grand and venerable work,
one of the most august monuments of transcendent intellect, was in his
day and generation only a pre-eminent tutor or schoolmaster, and
that his logical writings were designed for the accomplishment of his
pupils in a special art in which every intellectually ambitious young
Athenian of the period aspired to excel. Yet such is the plain fact,
baldly stated. Aristotle's Logic in its primary aim was as practical
as a treatise on Navigation, or "Cavendish on Whist". The latter is
the more exact of the two comparisons. It was in effect in its various
parts a series of handbooks for a temporarily fashionable intellectual
game, a peculiar mode of disputation or dialectic,[1] the game of
Question and Answer, the game so fully illustrated in the Dialogues of
Plato, the game identified with the name of Socrates.
We may lay stress, if we like, on the intellectuality of the game, and
the high topics on which it was exercised. It was a game that could
flourish only among a peculiarly intellectual people; a people less
acute would find little sport in it. The Athenians still take a
singular delight in disputation. You cannot visit Athens without being
struck by it. You may still see groups formed round two protagonists
in the cafés or the squares, or among the ruins of the Acropolis, in
a way to remind you of Socrates and his friends. They do not argue as
Gil Blas and his Hibernians did with heat and temper, ending in blows.
They argue for the pure love of arguing, the audience sitting
or standing by to see fair play with the keenest enjoyment of
intellectual thrust and parry. No other people could argue like the
Greeks without coming to blows. It is one of their characteristics
now, and so it was in old times two thousand years ago. And about
a century before Aristotle reached manhood, they had invented this
peculiarly difficult and trying species of disputative pastime, in
which we find the genesis of Aristotle's logical treatises.
To get a proper idea of this debate by Question and Answer, which we
may call Socratic disputation after its most renowned master, one
must read some of the dialogues of Plato. I will indicate merely
the skeleton of the game, to show how happily it lent itself to
Aristotle's analysis of arguments and propositions.
A thesis or proposition is put up for debate, _e.g._, that knowledge
is nothing else than sensible perception,[2] that it is a greater
evil to do wrong than to suffer wrong,[3] that the love of gain is not
reprehensible.[4] There are two disputants, but they do not speak on
the question by turns, so many minutes being allowed to each as in
a modern encounter of wits. One of the two, who may be called the
Questioner, is limited to asking questions, the other, the Respondent,
is limited to answering. Further, the Respondent can answer only "Yes"
or "No," with perhaps a little explanation: on his side the Questioner
must ask only questions that admit of the simple answer "Yes" or "No".
The Questioner's business is to extract from the Respondent admissions
involving the opposite of what he has undertaken to maintain. The
Questioner tries in short to make him contradict himself. Only a very
stupid Respondent would do this at once: the Questioner plies him
with general principles, analogies, plain cases; leads him on from
admission to admission, and then putting the admissions together
convicts him out of his own mouth of inconsistency.[5]
Now mark precisely where Aristotle struck in with his invention of the
Syllogism, the invention on which he prided himself as specially his
own, and the forms of which have clung to Logic ever since, even
in the usage of those who deride Aristotle's Moods and Figures as
antiquated superstitions. Suppose yourself the Questioner, where did
he profess to help you with his mechanism? In effect, as the word
Syllogism indicates, it was when you had obtained a number of
admissions, and wished to reason them together, to demonstrate how
they bore upon the thesis in dispute, how they hung together, how they
necessarily involved what you were contending for. And the essence of
his mechanism was the reduction of the admitted propositions to common
terms, and to certain types or forms which are manifestly equivalent
or inter-dependent. Aristotle advised his pupils also in the
tactics of the game, but his grand invention was the form or type
of admissions that you should strive to obtain, and the effective
manipulation of them when you had got them.
An example will show the nature of this help, and what it was worth.
To bring the thing nearer home, let us, instead of an example from
Plato, whose topics often seem artificial to us now, take a thesis
from last century, a paradox still arguable, Mandeville's famous--some
would say infamous--paradox that Private Vices are Public Benefits.
Undertake to maintain this, and you will have no difficulty in getting
a respondent prepared to maintain the negative. The plain men, such as
Socrates cross-questioned, would have declared at once that a vice is
a vice, and can never do any good to anybody. Your Respondent denies
your proposition simply: he upholds that private vices never are
public benefits, and defies you to extract from him any admission
inconsistent with this. Your task then is to lure him somehow into
admitting that in some cases what is vicious in the individual may
be of service to the State. This is enough: you are not concerned to
establish that this holds of all private vices. A single instance
to the contrary is enough to break down his universal negative. You
cannot, of course, expect him to make the necessary admission in
direct terms: you must go round about. You know, perhaps, that he
has confidence in Bishop Butler as a moralist. You try him with the
saying: "To aim at public and private good are so far from being
inconsistent that they mutually promote each other". Does he admit
this?
Perhaps he wants some little explanation or exemplification to enable
him to grasp your meaning. This was within the rules of the game. You
put cases to him, asking for his "Yes" or "No" to each. Suppose a man
goes into Parliament, not out of any zeal for the public good, but in
pure vainglory, or to serve his private ends, is it possible for him
to render the State good service? Or suppose a milk-seller takes
great pains to keep his milk pure, not because he cares for the public
health, but because it pays, is this a benefit to the public?
Let these questions be answered in the affirmative, putting you in
possession of the admission that some actions undertaken for private
ends are of public advantage, what must you extract besides to make
good your position as against the Respondent? To see clearly at this
stage what now is required, though you have to reach it circuitously,
masking your approach under difference of language, would clearly be
an advantage. This was the advantage that Aristotle's method offered
to supply. A disputant familiar with his analysis would foresee at
once that if he could get the Respondent to admit that all actions
undertaken for private ends are vicious, the victory was his, while
nothing short of this would serve.
Here my reader may interject that he could have seen this without any
help from Aristotle, and that anybody may see it without knowing
that what he has to do is, in Aristotelian language, to construct a
syllogism in Bokardo. I pass this over. I am not concerned at this
point to defend the utility of Aristotle's method. All that I want
is to illustrate the kind of use that it was intended for. Perhaps if
Aristotle had not habituated men's minds to his analysis, we should
none of us have been able to discern coherence and detect incoherence
as quickly and clearly as we do now.
But to return to our example. As Aristotle's pupil, you would have
seen at the stage we are speaking of that the establishment of your
thesis must turn upon the definition of virtue and vice. You must
proceed, therefore, to cross-examine your Respondent about this. You
are not allowed to ask him what he means by virtue, or what he means
by vice. In accordance with the rules of the dialectic, it is your
business to propound definitions, and demand his Yes or No to them.
You ask him, say, whether he agrees with Shaftesbury's definition of a
virtuous action as an action undertaken purely for the good of others.
If he assents, it follows that an action undertaken with any suspicion
of a self-interested motive cannot be numbered among the virtues.
If he agrees, further, that every action must be either vicious
or virtuous, you have admissions sufficient to prove your original
thesis. All that you have now to do to make your triumph manifest, is
to display the admissions you have obtained in common terms.
Some actions done with a self-interested motive are public
benefits. All actions done with a self-interested motive are
private vices.
From these premisses it follows irresistibly that
Some private vices are public benefits.
This illustration may serve to show the kind of disputation for which
Aristotle's logic was designed, and thus to make clear its primary
uses and its limitations.
To realise its uses, and judge whether there is anything analogous
to them in modern needs, conceive the chief things that it behoved
Questioner and Respondent in this game to know. All that a proposition
necessarily implies; all that two propositions put together imply; on
what conditions and to what extent one admission is inconsistent with
another; when one admission necessarily involves another; when two
necessarily involve a third. And to these ends it was obviously
necessary to have an exact understanding of the terms used, so as to
avoid the snares of ambiguous language.
That a Syllogistic or Logic of Consistency should emerge out of
Yes-and-No Dialectic was natural. Things in this world come when they
are wanted: inventions are made on the spur of necessity. It was above
all necessary in this kind of debate to avoid contradicting yourself:
to maintain your consistency. A clever interrogator spread out
proposition after proposition before you and invited your assent,
choosing forms of words likely to catch your prejudices and lure
you into self-contradiction. An organon, instrument, or discipline
calculated to protect you as Respondent and guide you as Questioner
by making clear what an admission led to, was urgently called for,
and when the game had been in high fashion for more than a century
Aristotle's genius devised what was wanted, meeting at the same time,
no doubt, collateral needs that had arisen from the application of
Dialectic to various kinds of subject-matter.
The thoroughness of Aristotle's system was doubtless due partly to
the searching character of the dialectic in which it had its birth.
No other mode of disputation makes such demands upon the disputant's
intellectual agility and precision, or is so well adapted to lay bare
the skeleton of an argument.
The uses of Aristotle's logical treatises remained when the fashion
that had called them forth had passed.[6] Clear and consistent
thinking, a mastery of the perplexities and ambiguities of language,
power to detect identity of meaning under difference of expression, a
ready apprehension of all that a proposition implies, all that may
be educed or deduced from it--whatever helps to these ends must be of
perpetual use. "To purge the understanding of those errors which lie
in the confusion and perplexities of an inconsequent thinking," is
a modern description of the main scope of Logic.[7] It is a good
description of the branch of Logic that keeps closest to the
Aristotelian tradition.
The limitations as well as the uses of Aristotle's logic may be traced
to the circumstances of its origin. Both parties to the disputation,
Questioner and Respondent alike, were mainly concerned with the
inter-dependence of the propositions put forward. Once the Respondent
had given his assent to a question, he was bound in consistency to all
that it implied. He must take all the consequences of his admission.
It might be true or it might be false as a matter of fact: all the
same he was bound by it: its truth or falsehood was immaterial to his
position as a disputant. On the other hand, the Questioner could not
go beyond the admissions of the Respondent. It has often been alleged
as a defect in the Syllogism that the conclusion does not go beyond
the premisses, and ingenious attempts have been made to show that
it is really an advance upon the premisses. But having regard to the
primary use of the syllogism, this was no defect, but a necessary
character of the relation. The Questioner could not in fairness assume
more than had been granted by implication. His advance could only be
an argumentative advance: if his conclusion contained a grain more
than was contained in the premisses, it was a sophistical trick, and
the Respondent could draw back and withhold his assent. He was bound
in consistency to stand by his admissions; he was not bound to go a
fraction of an inch beyond them.
We thus see how vain it is to look to the Aristotelian tradition
for an organon of truth or a criterion of falsehood. Directly and
primarily, at least, it was not so; the circumstances of its origin
gave it a different bent. Indirectly and secondarily, no doubt, it
served this purpose, inasmuch as truth was the aim of all serious
thinkers who sought to clear their minds and the minds of others by
Dialectic. But in actual debate truth was represented merely by the
common-sense of the audience. A dialectician who gained a triumph
by outraging this, however cleverly he might outwit his antagonist,
succeeded only in amusing his audience, and dialecticians of the
graver sort aimed at more serious uses and a more respectful homage,
and did their best to discountenance merely eristic disputation.
Further, it would be a mistake to conclude because Aristotle's Logic,
as an instrument of Dialectic, concerned itself with the syllogism
of propositions rather than their truth, that it was merely an art of
quibbling. On the contrary, it was essentially the art of preventing
and exposing quibbling. It had its origin in quibbling, no doubt,
inasmuch as what we should call verbal quibbling was of the essence
of Yes-and-No Dialectic, and the main secret of its charm for an
intellectual and disputatious people; but it came into being as a
safeguard against quibbling, not a serviceable adjunct.
The mediæval developments of Logic retained and even exaggerated
the syllogistic character of the original treatises. Interrogative
dialectic had disappeared in the Middle Ages whether as a diversion or
as a discipline: but errors of inconsistency still remained the errors
against which principally educated men needed a safeguard. Men had
to keep their utterances in harmony with the dogmas of the Church. A
clear hold of the exact implications of a proposition, whether singly
or in combination with other propositions, was still an important
practical need. The Inductive Syllogism was not required, and its
treatment dwindled to insignificance in mediæval text-books, but the
Deductive Syllogism and the formal apparatus for the definition of
terms held the field.
It was when observation of Nature and its laws became a paramount
pursuit that the defects of Syllogistic Logic began to be felt. Errors
against which this Logic offered no protection then called for a
safeguard--especially the errors to which men are liable in the
investigation of cause and effect. "Bring your thoughts into harmony
one with another," was the demand of Aristotle's age. "Bring your
thoughts into harmony with authority," was the demand of the Middle
Ages. "Bring them into harmony with fact," was the requirement most
keenly felt in more recent times. It is in response to this demand
that what is commonly but not very happily known as Inductive Logic
has been formulated.
In obedience to custom, I shall follow the now ordinary division of
Logic into Deductive and Inductive. The titles are misleading in many
ways, but they are fixed by a weight of usage which it would be vain
to try to unsettle. Both come charging down the stream of time
each with its cohort of doctrines behind it, borne forward with
irresistible momentum.
The best way of preventing confusion now is to retain the established
titles, recognise that the doctrines behind each have a radically
different aim or end, and supply the interpretation of this end from
history. What they have in common may be described as the prevention
of error, the organisation of reason against error. I have shown
that owing to the bent impressed upon it by the circumstances of its
origin, the errors chiefly safeguarded by the Aristotelian logic
were the errors of inconsistency. The other branch of Logic, commonly
called Induction, was really a separate evolution, having its origin
in a different practical need. The history of this I will trace
separately after we have seen our way through the Aristotelian
tradition and its accretions. The Experimental Methods are no less
manifestly the germ, the evolutionary centre or starting-point, of
the new Logic than the Syllogism is of the old, and the main errors
safeguarded are errors of fact and inference from fact.
At this stage it will be enough to indicate briefly the broad
relations between Deductive Logic and Inductive Logic.
Inductive Logic, as we now understand it--the Logic of Observation and
Explanation--was first formulated and articulated to a System of Logic
by J. S. Mill. It was he that added this wing to the old building.
But the need of it was clearly expressed as early as the thirteenth
century. Roger Bacon, the Franciscan friar (1214-1292), and not his
more illustrious namesake Francis, Lord Verulam, was the real founder
of Inductive Logic. It is remarkable that the same century saw
Syllogistic Logic advanced to its most complete development in the
system of Petrus Hispanus, a Portuguese scholar who under the title of
John XXI. filled the Papal Chair for eight months in 1276-7.
A casual remark of Roger Bacon's in the course of his advocacy of
Experimental Science in the _Opus Majus_ draws a clear line between
the two branches of Logic. "There are," he says, "two ways of knowing,
by Argument and by Experience. Argument concludes a question, but
it does not make us feel certain, unless the truth be also found in
experience."
On this basis the old Logic may be clearly distinguished from the new,
taking as the general aim of Logic the protection of the mind against
the errors to which it is liable in the acquisition of knowledge.
All knowledge, broadly speaking, comes either from Authority, _i.e._,
by argument from accepted premisses, or from Experience. If it comes
from Authority it comes through the medium of words: if it comes from
Experience it comes through the senses. In taking in knowledge through
words we are liable to certain errors; and in taking in knowledge
through the senses we are liable to certain errors. To protect against
the one is the main end of "Deductive" Logic: to protect against the
other is the main end of "Inductive" Logic. As a matter of fact the
pith of treatises on Deduction and Induction is directed to those ends
respectively, the old meanings of Deduction and Induction as formal
processes (to be explained afterwards) being virtually ignored.
There is thus no antagonism whatever between the two branches of
Logic. They are directed to different ends. The one is supplementary
to the other. The one cannot supersede the other.
Aristotelian Logic can never become superfluous as long as men are apt
to be led astray by words. Its ultimate business is to safeguard in
the interpretation of the tradition of language. The mere syllogistic,
the bare forms of equivalent or consistent expression, have a very
limited utility, as we shall see. But by cogent sequence syllogism
leads to proposition, and proposition to term, and term to a close
study of the relations between words and thoughts and things.
[Footnote 1: We know for certain--and it is one of the
evidences of the importance attached to this trivial-looking
pastime--that two of the great teacher's logical treatises,
the Topics and the Sophistical Refutations, were written
especially for the guidance of Questioners and Respondents.
The one instructs the disputant how to qualify himself
methodically for discussion before an ordinary audience, when
the admissions extracted from the respondent are matters of
common belief, the questioner's skill being directed to make
it appear that the respondent's position is inconsistent
with these. The other is a systematic exposure of sophistical
tricks, mostly verbal quibbles, whereby a delusive appearance
of victory in debate may be obtained. But in the concluding
chapter of the _Elenchi_, where Aristotle claims not only
that his method is superior to the empirical methods of
rival teachers, but that it is entirely original, it is the
Syllogism upon which he lays stress as his peculiar and
chief invention. The Syllogism, the pure forms of which are
expounded in his Prior Analytics, is really the centre of
Aristotle's logical system, whether the propositions to
which it is applied are matters of scientific truth as in the
Posterior Analytics, or matters of common opinion as in
the Topics. The treatise on Interpretation, _i.e._, the
interpretation of the Respondent's "Yes" and "No," is
preliminary to the Syllogism, the reasoning of the admissions
together. Even in the half-grammatical half-logical treatise
on the Categories, the author always keeps an eye on the
Syllogistic analysis.]
[Footnote 2: Theætetus, 151 E.]
[Footnote 3: Gorgias, 473 D.]
[Footnote 4: Hipparchus, 225 A.]
[Footnote 5: In its leading and primary use, this was a mode
of debate, a duel of wits, in which two men engaged before
an audience. But the same form could be used, and was used,
notably by Socrates, not in an eristic spirit but as a means
of awakening people to the consequences of certain admissions
or first principles, and thus making vague knowledge explicit
and clear. The mind being detained on proposition after
proposition as assent was given to it, dialectic was a
valuable instrument of instruction and exposition. But
whatever the purpose of the exercise, controversial triumph,
or solid grounding in the first principles--"the evolution
of in-dwelling conceptions"--the central interest lay in the
syllogising or reasoning together of the separately assumed or
admitted propositions.]
[Footnote 6: Like every other fashion, Yes-and-No Dialectic
had its period, its rise and fall. The invention of it is
ascribed to Zeno the Eleatic, the answering and questioning
Zeno, who flourished about the middle of the fifth century
B.C. Socrates (469-399) was in his prime at the beginning of
the great Peloponnesian War when Pericles died in 429. In that
year Plato was born, and lived to 347, "the olive groves of
Academe" being established centre of his teaching from
about 386 onwards. Aristotle (384-322), who was the tutor of
Alexander the Great, established his school at the Lyceum
when Alexander became king in 336 and set out on his career
of conquest. That Yes-and-No Dialectic was then a prominent
exercise, his logical treatises everywhere bear witness. The
subsequent history of the game is obscure. It is probable that
Aristotle's thorough exposition of its legitimate arts and
illegitimate tricks helped to destroy its interest as an
amusement.]
[Footnote 7: Hamilton's _Lectures_, iii. p. 37.]
II.--LOGIC AS A PREVENTIVE OF ERROR OR FALLACY.--THE INNER SOPHIST.
Why describe Logic as a system of defence against error? Why say that
its main end and aim is the organisation of reason against confusion
and falsehood? Why not rather say, as is now usual, that its end is
the attainment of truth? Does this not come to the same thing?
Substantially, the meaning is the same, but the latter expression
is more misleading. To speak of Logic as a body of rules for the
investigation of truth has misled people into supposing that Logic
claims to be an art of Discovery, that it claims to lay down rules
by simply observing which investigators may infallibly arrive at new
truths. Now, this does not hold even of the Logic of Induction, still
less of the older Logic, the precise relation of which to truth will
become apparent as we proceed. It is only by keeping men from going
astray and by disabusing them when they think they have reached their
destination that Logic helps men on the road to truth. Truth often
lies hid in the centre of a maze, and logical rules only help the
searcher onwards by giving him warning when he is on the wrong track
and must try another. It is the searcher's own impulse that carries
him forward: Logic does not so much beckon him on to the right path
as beckon him back from the wrong. In laying down the conditions of
correct interpretation, of valid argument, of trustworthy evidence,
of satisfactory explanation, Logic shows the inquirer how to test and
purge his conclusions, not how to reach them.
To discuss, as is sometimes done, whether Fallacies lie within the
proper sphere of Logic, is to obscure the real connexion between
Fallacies and Logic. It is the existence of Fallacies that calls Logic
into existence; as a practical science Logic is needed as a protection
against Fallacies. Such historically is its origin. We may, if we
like, lay down an arbitrary rule that a treatise on Logic should be
content to expound the correct forms of interpretation and reasoning
and should not concern itself with the wrong. If we take this view we
are bound to pronounce Fallacies extra-logical. But to do so is
simply to cripple the usefulness of Logic as a practical science.
The manipulation of the bare logical forms, without reference to
fallacious departures from them, is no better than a nursery exercise.
Every correct form in Logic is laid down as a safeguard against some
erroneous form to which men are prone, whether in the interpretation
of argument or the interpretation of experience, and the statement and
illustration of the typical forms of wrong procedure should accompany
_pari passu_ the exposition of the right procedure.
In accordance with this principle, I shall deal with special
fallacies, special snares or pitfalls--misapprehension of words,
misinterpretation of propositions, misunderstanding of arguments,
misconstruction of facts, evidences, or signs--each in connexion
with its appropriate safeguard. This seems to me the most profitable
method. But at this stage, it may be worth while, by way of
emphasising the need for Logic as a science of rational belief, to
take a survey of the most general tendencies to irrational belief,
the chief kinds of illusion or bias that are rooted in the human
constitution. We shall then better appreciate the magnitude of the
task that Logic attempts in seeking to protect reason against its own
fallibility and the pressure of the various forces that would usurp
its place.
It is a common notion that we need Logic to protect us against the
arts of the Sophist, the dishonest juggler with words and specious
facts. But in truth the Inner Sophist, whose instruments are our own
inborn propensities to error, is a much more dangerous enemy. For once
that we are the victims of designing Sophists, we are nine times the
victims of our own irrational impulses and prejudices. Men generally
deceive themselves before they deceive others.
Francis Bacon drew attention to these inner perverting influences,
these universal sources of erroneous belief, in his _De Augmentis_
and again in his _Novum Organum_, under the designation of _Idola_
([Greek: eidôla]), deceptive appearances of truth, illusions. His
classification of _Idola--Idola Tribus_, illusions common to all men,
illusions of the race; _Idola Specus_, personal illusions, illusions
peculiar to the "den" in which each man lives; _Idola Fori_, illusions
of conversation, vulgar prejudices embodied in words; _Idola Theatri_,
illusions of illustrious doctrine, illusions imposed by the dazzling
authority of great names--is defective as a classification inasmuch as
the first class includes all the others, but like all his writings it
is full of sagacious remarks and happy examples. Not for the sake of
novelty, but because it is well that matters so important should be
presented from more than one point of view, I shall follow a division
adapted from the more scientific, if less picturesque, arrangement
of Professor Bain, in his chapter on the Fallacious Tendencies of the
Human Mind.[1]
The illusions to which we are all subject may best be classified
according to their origin in the depths of our nature. Let us try to
realise how illusory beliefs arise.
What is a belief? One of the uses of Logic is to set us thinking about
such simple terms. An exhaustive analysis and definition of belief is
one of the most difficult of psychological problems. We cannot enter
upon that: let us be content with a few simple characters of belief.
First, then, belief is a state of mind. Second, this state of mind is
outward-pointing: it has a reference beyond itself, a reference to the
order of things outside us. In believing, we hold that the world as it
is, has been, or will be, corresponds to our conceptions of it.
Third, belief is the guide of action: it is in accordance with what we
believe that we direct our activities. If we want to know what a man
really believes, we look at his action. This at least is the clue to
what he believes at the moment. "I cannot," a great orator once said,
"read the minds of men." This was received with ironical cheers. "No,"
he retorted, "but I can construe their acts." Promoters of companies
are expected to invest their own money as a guarantee of good faith.
If a man says he believes the world is coming to an end in a year,
and takes a lease of a house for fifteen years, we conclude that his
belief is not of the highest degree of strength.
The close connexion of belief with our activities, enables us to
understand how illusions, false conceptions of reality, arise. The
illusions of Feeling and the illusions of Custom are well understood,
but other sources of illusion, which may be designated Impatient
Impulse and Happy Exercise, are less generally recognised. An example
or two will show what is meant. We cannot understand the strength of
these perverting influences till we realise them in our own case. We
detect them quickly enough in others. Seeing that in common speech the
word illusion implies a degree of error amounting almost to insanity,
and the illusions we speak of are such as no man is ever quite free
from, it is perhaps less startling to use the word _bias_.
_The Bias of Impatient Impulse._
As a being formed for action, not only does healthy man take
a pleasure in action, physical and mental, for its own sake,
irrespective of consequences, but he is so charged with energy that
he cannot be comfortable unless it finds a free vent. In proportion
to the amount and excitability of his energy, restraint, obstruction,
delay is irksome, and soon becomes a positive and intolerable pain.
Any bar or impediment that gives us pause is hateful even to think of:
the mere prospect annoys and worries.
Hence it arises that belief, a feeling of being prepared for action,
a conviction that the way is clear before us for the free exercise of
our activities, is a very powerful and exhilarating feeling, as much
a necessity of happy existence as action itself. We see this when we
consider how depressing and uncomfortable a condition is the opposite
state to belief, namely, doubt, perplexity, hesitation, uncertainty as
to our course. And realising this, we see how strong a bias we have in
this fact of our nature, this imperious inward necessity for action;
how it urges us to act without regard to consequences, and to jump at
beliefs without inquiry. For, unless inquiry itself is our business, a
self-sufficient occupation, it means delay and obstruction.
This ultimate fact of our nature, this natural inbred constitutional
impatience, explains more than half of the wrong beliefs that we form
and persist in. We must have a belief of some kind: we cannot be happy
till we get it, and we take up with the first that seems to show the
way clear. It may be right or it may be wrong: it is not, of course,
necessarily always wrong: but that, so far as we are concerned, is a
matter of accident. The pressing need for a belief of some sort, upon
which our energies may proceed in anticipation at least, will not
allow us to stop and inquire. Any course that offers a relief from
doubt and hesitation, any conviction that lets the will go free, is
eagerly embraced.
It may be thought that this can apply only to beliefs concerning
the consequences of our own personal actions, affairs in which we
individually play a part. It is from them, no doubt, that our nature
takes this set: but the habit once formed is extended to all sorts
of matters in which we have no personal interest. Tell an ordinary
Englishman, it has been wittily said, that it is a question whether
the planets are inhabited, and he feels bound at once to have a
confident opinion on the point. The strength of the conviction bears
no proportion to the amount of reason spent in reaching it, unless it
may be said that as a general rule the less a belief is reasoned the
more confidently it is held.
"A grocer," writes Mr. Bagehot in an acute essay on "The Emotion of
Conviction,"[2] "has a full creed as to foreign policy, a young lady a
complete theory of the Sacraments, as to which neither has any doubt.
A girl in a country parsonage will be sure that Paris never can be
taken, or that Bismarck is a wretch." An attitude of philosophic
doubt, of suspended judgment, is repugnant to the natural man. Belief
is an independent joy to him.
This bias works in all men. While there is life, there is pressure
from within on belief, tending to push reason aside. The force of
the pressure, of course, varies with individual temperament, age, and
other circumstances. The young are more credulous than the old, as
having greater energy: they are apt, as Bacon puts it, to be "carried
away by the sanguine element in their temperament". Shakespeare's
Laertes is a study of the impulsive temperament, boldly contrasted
with Hamlet, who has more discourse of reason. When Laertes hears that
his father has been killed, he hurries home, collects a body of armed
sympathisers, bursts into the presence of the king, and threatens with
his vengeance--the wrong man. He never pauses to make inquiry: like
Hotspur he is "a wasp-stung and impatient fool"; he must wreak his
revenge on somebody, and at once. Hamlet's father also has been
murdered, but his reason must be satisfied before he proceeds to his
revenge, and when doubtful proof is offered, he waits for proof more
relative.
Bacon's _Idola Tribus_ and Dr. Bain's illustrations of incontinent
energy, are mostly examples of unreasoning intellectual activity,
hurried generalisations, unsound and superficial analogies, rash
hypotheses. Bacon quotes the case of the sceptic in the temple of
Poseidon, who, when shown the offerings of those who had made vows
in danger and been delivered, and asked whether he did not now
acknowledge the power of the god, replied: "But where are they who
made vows and yet perished?" This man answered rightly, says Bacon.
In dreams, omens, retributions, and such like, we are apt to remember
when they come true and to forget the cases when they fail. If we
have seen but one man of a nation, we are apt to conclude that all his
countrymen are like him; we cannot suspend our judgment till we have
seen more. Confident belief, as Dr. Bain remarks, is the primitive
attitude of the human mind: it is only by slow degrees that this is
corrected by experience. The old adage, "Experience teaches fools,"
has a meaning of its own, but in one sense it is the reverse of the
truth. The mark of a fool is that he is not taught by experience, and
we are all more or less intractable pupils, till our energies begin to
fail.
_The Bias of Happy Exercise._
If an occupation is pleasant in itself, if it fully satisfies our
inner craving for action, we are liable to be blinded thereby to its
consequences. Happy exercise is the fool's Paradise. The fallacy lies
not in being content with what provides a field for the full activity
of our powers: to be content in such a case may be the height of
wisdom: but the fallacy lies in claiming for our occupation results,
benefits, utilities that do not really attend upon it. Thus we see
subjects of study, originally taken up for some purpose, practical,
artistic, or religious, pursued into elaborate detail far beyond their
original purpose, and the highest value, intellectual, spiritual,
moral, claimed for them by their votaries, when in truth they merely
serve to consume so much vacant energy, and may be a sheer waste of
time that ought to be otherwise employed.
But as I am in danger of myself furnishing an illustration of this
bias--it is nowhere more prevalent than in philosophy--I will pass to
our next head.
_The Bias of the Feelings._
This source of illusion is much more generally understood. The
blinding and perverting influence of passion on reason has been a
favourite theme with moralists ever since man began to moralise, and
is acknowledged in many a popular proverb. "Love is blind;" "The wish
is father to the thought;" "Some people's geese are all swans;" and so
forth.
We need not dwell upon the illustration of it. Fear and Sloth magnify
dangers and difficulties; Affection can see no imperfection in its
object: in the eyes of Jealousy a rival is a wretch. From the nature
of the case we are much more apt to see examples in others than in
ourselves. If the strength of this bias were properly understood by
everybody, the mistake would not so often be committed of suspecting
bad faith, conscious hypocrisy, when people are found practising
the grossest inconsistencies, and shutting their eyes apparently in
deliberate wilfulness to facts held under their very noses. Men are
inclined to ascribe this human weakness to women. Reasoning from
feeling is said to be feminine logic. But it is a human weakness.
To take one very powerful feeling, the feeling of self-love or
self-interest--this operates in much more subtle ways than most people
imagine, in ways so subtle that the self-deceiver, however honest,
would fail to be conscious of the influence if it were pointed out to
him. When the slothful man saith, There is a lion in the path, we can
all detect the bias to his belief, and so we can when the slothful
student says that he will work hard to-morrow, or next week, or next
month; or when the disappointed man shows an exaggerated sense of
the advantages of a successful rival or of his own disadvantages. But
self-interest works to bias belief in much less palpable ways than
those. It is this bias that accounts for the difficulty that men of
antagonistic interests have in seeing the arguments or believing
in the honesty of their opponents. You shall find conferences
held between capitalists and workmen in which the two sides, both
represented by men incapable of consciously dishonest action, fail
altogether to see the force of each other's arguments, and are
mutually astonished each at the other's blindness.
_The Bias of Custom._
That custom, habits of thought and practice, affect belief, is also
generally acknowledged, though the strength and wide reach of the bias
is seldom realised. Very simple cases of unreasoning prejudice were
adduced by Locke, who was the first to suggest a general explanation
of them in the "Association of Ideas" (_Human Understanding_, bk. ii.
ch. xxxiii.). There is, for instance, the fear that overcomes many
people when alone in the dark. In vain reason tells them that there is
no real danger; they have a certain tremor of apprehension that they
cannot get rid of, because darkness is inseparably connected in
their minds with images of horror. Similarly we contract unreasonable
dislikes to places where painful things have happened to us. Equally
unreasoning, if not unreasonable, is our attachment to customary
doctrines or practices, and our invincible antipathy to those who do
not observe them.
Words are very common vehicles for the currency of this kind of
prejudice, good or bad meanings being attached to them by custom. The
power of words in this way is recognised in the proverb: "Give a dog
a bad name, and then hang him". These verbal prejudices are Bacon's
_Idola Fori_, illusions of conversation. Each of us is brought up in a
certain sect or party, and accustomed to respect or dishonour
certain sectarian or party names, Whig, Tory, Radical, Socialist,
Evolutionist, Broad, Low, or High Church. We may meet a man without
knowing under what label he walks and be charmed with his company:
meet him again when his name is known, and all is changed.
Such errors are called Fallacies of Association to point to the
psychological explanation. This is that by force of association
certain ideas are brought into the mind, and that once they are there,
we cannot help giving them objective reality. For example, a doctor
comes to examine a patient, and finds certain symptoms. He has lately
seen or heard of many cases of influenza, we shall say; influenza is
running in his head. The idea once suggested has all the advantage of
possession.
But why is it that a man cannot get rid of an idea? Why does it force
itself upon him as a belief? Association, custom, explains how it got
there, but not why it persists in staying.
To explain this we must call in our first fallacious principle, the
Impatience of Doubt or Delay, the imperative inward need for a belief
of some sort.
And this leads to another remark, that though for convenience of
exposition, we separate these various influences, they are not
separated in practice. They may and often do act all together, the
Inner Sophist concentrating his forces.
Finally, it may be asked whether, seeing that illusions are the
offspring of such highly respectable qualities as excess of energy,
excess of feeling, excess of docility, it is a good thing for man to
be disillusioned. The rose-colour that lies over the world for youth
is projected from the abundant energy and feeling within: disillusion
comes with failing energies, when hope is "unwilling to be fed". Is
it good then to be disillusioned? The foregoing exposition would
be egregiously wrong if the majority of mankind did not resent the
intrusion of Reason and its organising lieutenant Logic. But really
there is no danger that this intrusion succeeds to the extent of
paralysing action and destroying feeling, and uprooting custom.
The utmost that Logic can do is to modify the excess of these good
qualities by setting forth the conditions of rational belief. The
student who masters those conditions will soon see the practical
wisdom of applying his knowledge only in cases where the grounds of
rational belief are within his reach. To apply it to the consequences
of every action would be to yield to that bias of incontinent activity
which is, perhaps, our most fruitful source of error.
[Footnote 1: Bain's _Logic_, bk. vi. chap. iii. Bacon intended
his _Idola_ to bear the same relation to his _Novum Organum_
that Aristotle's Fallacies or Sophistical Tricks bore to the
old Organum. But in truth, as I have already indicated, what
Bacon classifies is our inbred tendencies to form _idola_ or
false images, and it is these same tendencies that make
us liable to the fallacies named by Aristotle. Some of
Aristotle's, as we shall see, are fallacies of Induction.]
[Footnote 2: Bagehot's _Literary Studies_, ii. 427.]
III.--THE AXIOMS OF DIALECTIC AND OF SYLLOGISM.
There are certain principles known as the Laws of Thought or the
Maxims of Consistency. They are variously expressed, variously
demonstrated, and variously interpreted, but in one form or another
they are often said to be the foundation of all Logic. It is even said
that all the doctrines of Deductive or Syllogistic Logic may be educed
from them. Let us take the most abstract expression of them, and see
how they originated. Three laws are commonly given, named respectively
the Law of Identity, the Law of Contradiction and the Law of Excluded
Middle.
1. _The Law of Identity._ A is A. Socrates is Socrates. Guilt is
guilt.
2. _The Law of Contradiction._ A is not not-A. Socrates is not other
than Socrates. Guilt is not other than guilt. Or A is not at once _b_
and not-_b_. Socrates is not at once good and not-good. Guilt is not
at once punishable and not-punishable.
3. _The Law of Excluded Middle._ Everything is either A or not-A;
or, A is either _b_ or not-_b_. A given thing is either Socrates
or not-Socrates, either guilty or not-guilty. It must be one or the
other: no middle is possible.
Why lay down principles so obvious, in some interpretations, and so
manifestly sophistical in others? The bare forms of modern Logic have
been reached by a process of attenuation from a passage in Aristotle's
_Metaphysics_[1] (iii. 3, 4, 1005_b_ - 1008). He is there laying down
the first principle of demonstration, which he takes to be that "it is
impossible that the same predicate can both belong, and not belong,
to the same subject, at the same time, and in the same sense".[2]
That Socrates knows grammar, and does not know grammar--these two
propositions cannot both be true at the same time, and in the same
sense. Two contraries cannot exist together in the same subject. The
double answer Yes and No cannot be given to one and the same question
understood in the same sense.
But why did Aristotle consider it necessary to lay down a principle
so obvious? Simply because among the subtle dialecticians who
preceded him the principle had been challenged. The Platonic dialogue
Euthydemus shows the farcical lengths to which such quibbling was
carried. The two brothers vanquish all opponents, but it is by
claiming that the answer No does not preclude the answer Yes. "Is not
the honourable honourable, and the base base?" asks Socrates. "That is
as I please," replies Dionysodorus. Socrates concludes that there
is no arguing with such men: they repudiate the first principles of
dialectic.
There were, however, more respectable practitioners who canvassed on
more plausible grounds any form into which ultimate doctrines about
contraries and contradictions, truth and falsehood, could be put, and
therefore Aristotle considered it necessary to put forth and defend
at elaborate length a statement of a first principle of demonstration.
"Contradictions cannot both be true of the same subject at the same
time and in the same sense." This is the original form of the Law of
Contradiction.
The words "of the same subject," "at the same time," and "in the
same sense," are carefully chosen to guard against possible quibbles.
"_Socrates knows grammar._" By Socrates we must mean the same
individual man. And even of the same man the assertion may be true at
one time and not at another. There was a time when Socrates did not
know grammar, though he knows it now. And the assertion may be true
in one sense and not in another. It may be true that Socrates knows
grammar, yet not that he knows everything that is to be known about
grammar, or that he knows as much as Aristarchus.
Aristotle acknowledges that this first principle cannot itself be
demonstrated, that is, deduced from any other. If it is denied, you
can only reduce the denier to an absurdity. And in showing how to
proceed in so doing, he says you must begin by coming to an agreement
about the words used, that they signify the same for one and the other
disputant.[3] No dialectic is possible without this understanding.
This first principle of Dialectic is the original of the Law of
Identity. While any question as to the truth or falsehood of a
question is pending, from the beginning to the end of any logical
process, the words must continue to be accepted in the same sense.
Words must have an identical reference to things.
Incidentally in discussing the Axiom of Contradiction ([Greek:
axiôma tês antiphaseôs]),[4] Aristotle lays down what is now known
as the Law of Excluded Middle. Of two contradictories one or other
must be true: we must either affirm or deny any one thing of any
other: no mean or middle is possible.
In their origin, then, these so-called Laws of Thought were simply the
first principles of Dialectic and Demonstration. Consecutive argument,
coherent ratiocination, is impossible unless they are taken for
granted.
If we divorce or abstract them from their original application, and
consider them merely as laws of thinking or of being, any abstract
expression, or illustration, or designation of them may easily be
pushed into antagonism with other plain truths or first principles
equally rudimentary. Without entering into the perplexing and
voluminous discussion to which these laws have been subjected by
logicians within the last hundred years, a little casuistry is
necessary to enable the student to understand within what limits they
hold good.
_Socrates is Socrates._ The name Socrates is a name for something to
which you and I refer when we use the name. Unless we have the same
reference, we cannot hold any argument about the thing, or make any
communication one to another about it.
But if we take _Socrates is Socrates_ to mean that, "An object of
thought or thing is identical with itself," "An object of thought or
thing cannot be other than itself," and call this a law of thought, we
are met at once by a difficulty. Thought, properly speaking, does
not begin till we pass beyond the identity of an object with itself.
Thought begins only when we recognise the likeness between one object
and others. To keep within the self-identity of the object is to
suspend thought. "Socrates was a native of Attica," "Socrates was
a wise man," "Socrates was put to death as a troubler of the
commonweal"--whenever we begin to think or say anything about
Socrates, to ascribe any attributes to him, we pass out of his
self-identity into his relations of likeness with other men, into what
he has in common with other men.
Hegelians express this plain truth with paradoxical point when they
say: "Of any definite existence or thought, therefore, it may be said
with quite as much truth that it _is not_, as that it _is_, its own
bare self".[5] Or, "A thing must other itself in order to be itself".
Controversialists treat this as a subversion of the laws of Identity
and Contradiction. But it is only Hegel's fun--his paradoxical way of
putting the plain truth that any object has more in common with other
objects than it has peculiar to itself. Till we enter into those
aspects of agreement with other objects, we cannot truly be said to
think at all. If we say merely that a thing is itself, we may as well
say nothing about it. To lay down this is not to subvert the Law of
Identity, but to keep it from being pushed to the extreme of appearing
to deny the Law of Likeness, which is the foundation of all the
characters, attributes, or qualities of things in our thoughts.
That self-same objects are like other self-same objects, is an
assumption distinct from the Law of Identity, and any interpretation
of it that excludes this assumption is to be repudiated. But does not
the law of Identity as well as the law of the likeness of mutually
exclusive identities presuppose that there are objects self-same,
like others, and different from others? Certainly: this is one of the
presuppositions of Logic.[6] We assume that the world of which we talk
and reason is separated into such objects in our thoughts. We assume
that such words as _Socrates_ represent individual objects with a
self-same being or substance; that such words as _wisdom_, _humour_,
_ugliness_, _running_, _sitting_, _here_, _there_, represent
attributes, qualities, characters or predicates of individuals; that
such words as _man_ represent groups or classes of individuals.
Some logicians in expressing the Law of Identity have their eye
specially upon the objects signified by general names or abstract
names, _man_, _education_.[7] "A concept is identical with the sum
of its characters," or, "Classes are identical with the sum of
the individuals composing them". The assumptions thus expressed in
technical language which will hereafter be explained are undoubtedly
assumptions that Logic makes: but since they are statements of the
internal constitution of some of the identities that words represent,
to call them the Law of Identity is to depart confusingly from
traditional usage.[8]
That throughout any logical process a word must signify the same
object, is one proposition: that the object signified by a general
name is identical with the sum of the individuals to each of whom it
is applicable, or with the sum of the characters that they bear in
common, is another proposition. Logic assumes both: Aristotle assumed
both: but it is the first that is historically the original of all
expressions of the Law of Identity in modern text-books.
Yet another expression of a Law of Identity which is really distinct
from and an addition to Aristotle's original. _Socrates was an
Athenian, a philosopher, an ugly man, an acute dialectician, etc._
Let it be granted that the word Socrates bears the same signification
throughout all these and any number more of predicates, we may still
ask: "But what is it that Socrates signifies?" The title Law of
Identity is sometimes given[9] to a theory on this point. _Socrates is
Socrates._ "An individual is the identity running through the totality
of its attributes." Is this not, it may be asked, to confuse thought
and being, to resolve Socrates into a string of words? No: real
existence is one of the admissible predicates of Socrates: one of
the attributes under which we conceive him. But whether we accept
or reject this "Law of Identity," it is an addition to Aristotle's
dialectical "law of identity"; it is a theory of the metaphysical
nature of the identity signified by a Singular name. And the same may
be said of yet another theory of Identity, that, "An individual is
identical with the totality of its predicates," or (another way
of putting the same theory), "An individual is a conflux of
generalities".
To turn next to the Laws of Contradiction and Excluded Middle. These
also may be subjected to Casuistry, making clearer what they assert by
showing what they do not deny.
They do not deny that things change, and that successive states of the
same thing may pass into one another by imperceptible degrees. A thing
may be neither here nor there: it may be on the passage from here to
there: and, while it is in motion, we may say, with equal truth, that
it is neither here nor there, or that it is both here and there. Youth
passes gradually into age, day into night: a given man or a given
moment may be on the borderland between the two.
Logic does not deny the existence of indeterminate margins: it merely
lays down that for purposes of clear communication and coherent
reasoning the line must be drawn somewhere between _b_, and not-_b_.
A difference, however, must be recognised between logical negation
and the negations of common thought and common speech. The latter are
definite to a degree with which the mere Logic of Consistency does
not concern itself. To realise this is to understand more clearly the
limitations of Formal Logic.
In common speech, to deny a quality of anything is by implication to
attribute to it some other quality of the same kind. Let any man tell
me that "the streets of such and such a town are not paved with wood,"
I at once conclude that they are paved with some other material. It
is the legitimate effect of his negative proposition to convey this
impression to my mind. If, proceeding on this, I go on to ask: "Then
they are paved with granite or asphalt, or this or that?" and he turns
round and says: "I did not say they were paved at all," I should be
justified in accusing him of a quibble. In ordinary speech, to deny
one kind of pavement is to assert pavement of some kind. Similarly, to
deny that So-and-so is not in the Twenty-first Regiment, is to
imply that he is in another regiment, that he is in the army in some
regiment. To retort upon this inference: "He is not in the army at
all," is a quibble: as much so as it would be to retort: "There is no
such person in existence".
Now Logic does not take account of this implication, and nothing has
contributed more to bring upon it the reproach of quibbling. In Logic,
to deny a quality is simply to declare a repugnance between it and the
subject; negation is mere sublation, taking away, and implies nothing
more. Not-_b_ is entirely indefinite: it may cover anything except
_b_.
Is Logic then really useless, or even misleading, inasmuch as it
ignores the definite implication of negatives in ordinary thought
and speech? In ignoring this implication, does Logic oppose this
implication as erroneous? Does Logic shelter the quibbler who
trades upon it? By no means: to jump to this conclusion were a
misunderstanding. The fact only is that nothing beyond the logical
Law of Contradiction needs to be assumed for any of the processes
of Formal Logic. Aristotle required to assume nothing more for his
syllogistic formulæ, and Logic has not yet included in its scope any
process that requires any further assumption. "If not-_b_ represent
everything except _b_, everything outside _b_, then that A is _b_, and
that A is not-_b_, cannot both be true, and one or other of them must
be true."
Whether the scope of Logic ought to be extended is another question.
It seems to me that the scope of Logic may legitimately be extended so
as to take account both of the positive implication of negatives and
the negative implication of positives. I therefore deal with this
subject in a separate chapter following on the ordinary doctrines of
Immediate Inference, where I try to explain the simple Law of Thought
involved. When I say that the extension is legitimate, I mean that it
may be made without departing from the traditional view of Logic as
a practical science, conversant with the nature of thought and its
expression only in so far as it can provide practical guidance against
erroneous interpretations and inferences. The extension that I propose
is in effect an attempt to bring within the fold of Practical Logic
some of the results of the dialectic of Hegel and his followers,
such as Mr. Bradley and Mr. Bosanquet, Professor Caird and Professor
Wallace.[10]
The logical processes formulated by Aristotle are merely stages in the
movement of thought towards attaining definite conceptions of reality.
To treat their conclusions as positions in which thought may dwell and
rest, is an error, against which Logic itself as a practical science
may fairly be called upon to guard. It may even be conceded that the
Aristotelian processes are artificial stages, courses that thought
does not take naturally, but into which it has to be forced for a
purpose. To concede this is not to concede that the Aristotelian logic
is useless, as long as we have reason on our side in holding that
thought is benefited and strengthened against certain errors by
passing through those artificial stages.
[Footnote 1: The first statement of the Law of Identity in the
form _Ens est ens_ is ascribed by Hamilton (_Lectures_, iii.
91) to Antonius Andreas, a fourteenth century commentator
on the _Metaphysics_. But Andreas is merely expounding what
Aristotle sets forth in iii. 4, 1006 _a, b_. _Ens est ens_
does not mean in Andreas what A is A means in Hamilton.]
[Footnote 2: Greek: to gar auto hama huparchein te kai mê
huparchein adynaton tô autô kai kata to auto, . . . ahutê dê
pasôn esti bebaiotatê tôn archôn. iii. 3, 1005_b_, 19-23.]
[Footnote 3: Hamilton credits Andreas with maintaining,
"against Aristotle," that "the principle of Identity, and not
the principle of Contradiction, is the one absolutely first".
Which comes first, is a scholastic question on which ingenuity
may be exercised. But in fact Aristotle put the principle
of Identity first in the above plain sense, and Andreas only
expounded more formally what Aristotle had said.]
[Footnote 4: [Greek: Metaxu antiphaseôs endechetai einai outhen,
all' anankê ê phanai ê apophanai en kath henos hotioun.]
_Metaph._ iii. 7, 1011_b_, 23-4.]
[Footnote 5: Prof. Caird's _Hegel_, p. 138.]
[Footnote 6: See Venn, _Empirical Logic_, 1-8.]
[Footnote 7: _E.g._, Hamilton, lect. v.; Veitch's _Institutes
of Logic_, chaps, xii., xiii.]
[Footnote 8: The confusion probably arises in this way. First,
these "laws" are formulated as laws of thought that Logic
assumes. Second, a notion arises that these laws are the
only postulates of Logic: that all logical doctrines can be
"evolved" from them. Third, when it is felt that more than the
identical reference of words or the identity of a thing
with itself must be assumed in Logic, the Law of Identity is
extended to cover this further assumption.]
[Footnote 9: _E.g._, Bosanquet's _Logic_, ii. 207.]
[Footnote 10: Bradley, _Principles of Logic_; Bosanquet,
_Logic or The Morphology of Knowledge_; Caird, _Hegel_ (in
Blackwood's Philosophical Classics); Wallace, _The Logic of
Hegel_.]
BOOK I.
THE LOGIC OF CONSISTENCY. SYLLOGISM AND DEFINITION.
PART I.
THE ELEMENTS OF PROPOSITIONS.
CHAPTER I.
GENERAL NAMES AND ALLIED DISTINCTIONS.
To discipline us against the errors we are liable to in receiving
knowledge through the medium of words--such is one of the objects of
Logic, the main object of what may be called the Logic of Consistency.
Strictly speaking, we may receive knowledge about things through signs
or single words, as a nod, a wink, a cry, a call, a command. But an
assertory sentence, proposition, or predication, is the unit with
which Logic concerns itself--a sentence in which a subject is
named and something is said or predicated about it. Let a man once
understand the errors incident to this regular mode of communication,
and he may safely be left to protect himself against the errors
incident to more rudimentary modes.
A proposition, whether long or short, is a unit, but it is an
analysable unit. And the key to syllogistic analysis is the General
Name. Every proposition, every sentence in which we convey knowledge
to another, contains a general name or its equivalent. That is to say,
every proposition may be resolved into a form in which the predicate
is a general name. A knowledge of the function of this element of
speech is the basis of all logical discipline. Therefore, though we
must always remember that the proposition is the real unit of speech,
and the general name only an analytic element, we take the general
name and its allied distinctions in thought and reality first.
How propositions are analysed for syllogistic purposes will be shown
by-and-by, but we must first explain various technical terms that
logicians have devised to define the features of this cardinal
element. The technical terms CLASS, CONCEPT, NOTION, ATTRIBUTE,
EXTENSION or DENOTATION, INTENSION or CONNOTATION, GENUS, SPECIES,
DIFFERENTIA, SINGULAR NAME, COLLECTIVE NAME, ABSTRACT NAME, all centre
round it.
A general name is a name applicable to a number of different things on
the ground of some likeness among them, as _man_, _ratepayer_, _man of
courage_, _man who fought at Waterloo_.
From the examples it will be seen that a general name logically is
not necessarily a single word. Any word or combination of words that
serves a certain function is technically a general name. The different
ways of making in common speech the equivalent of a general name
logically are for the grammarian to consider.
In the definition of a general name attention is called to two
distinct considerations, the individual objects to each of which
the name is applicable, and the points of resemblance among them, in
virtue of which they have a common name. For those distinctions there
are technical terms.
CLASS is the technical term for the objects, different yet agreeing,
to each of which a general name may be applied.
The points of resemblance are called the common ATTRIBUTES of the
class.
A class may be constituted on one attribute or on several.
_Ratepayer_, _woman ratepayer_, _unmarried woman ratepayer_;
_soldier_, _British soldier_, _British soldier on foreign service_.
But every individual to which the general name can be applied must
possess the common attribute or attributes.
These common attributes are also called the NOTION of the class,
inasmuch as it is these that the mind notes or should note when the
general name is applied. CONCEPT is a synonym perhaps in more common
use than notion; the rationale of this term (derived from _con_ and
_capere_, to take or grasp together) being that it is by means of
the points of resemblance that the individuals are grasped or held
together by the mind. These common points are the one in the many, the
same amidst the different, the identity signified by the common name.
The name of an attribute as thought of by itself without reference to
any individual or class possessing it, is called an ABSTRACT name. By
contradistinction, the name of an individual or a class is CONCRETE.
Technical terms are wanted also to express the relation of the
individuals and the attributes to the general name. The individuals
jointly are spoken of as the DENOTATION, or EXTENSION or SCOPE of
the name; the common attributes as its CONNOTATION, INTENSION,
COMPREHENSION, or GROUND. The whole denotation, etc., is the class;
the whole connotation, etc., is the concept.[1] The limits of a
"class" in Logic are fixed by the common attributes. Any individual
object that possesses these is a member. The statement of them is the
DEFINITION.
To predicate a general name of any object, as, "This is a cat," "This
is a very sad affair," is to refer that object to a class, which is
equivalent to saying that it has certain features of resemblance with
other objects, that it reminds us of them by its likeness to them.
Thus to say that the predicate of every proposition is a general name,
expressed or implied, is the same as to say that every predication may
be taken as a reference to a class.
Ordinarily our notion or concept of the common features signified by
general names is vague and hazy. The business of Logic is to make them
clear. It is to this end that the individual objects of the class are
summoned before the mind. In ordinary thinking there is no definite
array or muster of objects: when we think of "dog" or "cat,"
"accident," "book," "beggar," "ratepayer," we do not stop to call
before the mind a host of representatives of the class, nor do we take
precise account of their common attributes. The concept of "house" is
what all houses have in common. To make this explicit would be no
easy matter, and yet we are constantly referring objects to the
class "house". We shall see presently that if we wish to make the
connotation or concept clear we must run over the denotation or class,
that is to say, the objects to which the general name is applied in
common usage. Try, for example, to conceive clearly what is meant
by house, tree, dog, walking-stick. You think of individual objects,
so-called, and of what they have in common.
A class may be constituted on one property or on many. There are
several points common to all houses, enclosing walls, a roof, a means
of exit and entrance. For the full concept of the natural kinds,
_men_, _dogs_, _mice_, etc., we should have to go to the natural
historian.
DEGREES OF GENERALITY. One class is said to be of higher generality
than another when it includes that other and more. Thus animal
includes man, dog, horse, etc.; man includes Aryan, Semite, etc.;
Aryan includes Hindoo, Teuton, Celt, etc.
The technical names for higher and lower classes are GENUS and
SPECIES. These terms are not fixed as in Natural History to certain
grades, but are purely relative one to another, and movable up and
down a scale of generality. A class may be a species relatively to one
class, which is above it, and a genus relatively to one below it. Thus
Aryan is a species of the genus man, Teuton a species of the genus
Aryan.
In the graded divisions of Natural History genus and species are fixed
names for certain grades. Thus: Vertebrates form a "division"; the
next subdivision, _e.g._, Mammals, Birds, Reptiles, etc., is called a
"class"; the next, _e.g_., Rodents, Carnivora, Ruminants, an "order";
the next, _e.g._, Rats, Squirrels, Beavers, a "genus"; the next,
_e.g._, Brown rats, Mice, a "species".
Vertebrates (division).
|
Mammals, Birds, Reptiles, etc. (class).
|
Rodents, Ruminants, Carnivors, etc. (order).
|
Rats, Squirrels, Beavers, etc. (genus).
|
Brown rats, Mice, etc. (species).
If we subdivide a large class into smaller classes, and, again,
subdivide these subdivisions, we come at last to single objects.
Men
|
------------
Europeans, Asiatics, etc.
|
-------------
Englishmen, Frenchmen, etc.
|
---------------------
John Doe, Richard Roe, etc.
A table of higher and lower classes arranged in order has been known
from of old as a _tree_ of division or classification. The following
is Porphyry's "tree":--
Being
/ \
Corporeal Incorporeal
\ /
(Body)
/ \
Animate Inanimate
\ /
(Living Being)
/ \
Sensible Insensible
\ /
(Animal)
/ \
Rational Irrational
\ /
(Man)
|
---------------------------------------
Socrates, Plato, and other individuals.
The single objects are called INDIVIDUALS, because the division cannot
be carried farther. The highest class is technically the SUMMUM GENUS,
or _Genus generalissimum_; the next highest class to any species is
the PROXIMUM GENUS; the lowest group before you descend to individuals
is the INFIMA SPECIES, or _Species specialissima_.
The attribute or attributes whereby a species is distinguished
from other species of the same genus, is called its DIFFERENTIA or
DIFFERENTIÆ. The various species of houses are differentiated by their
several uses, dwelling-house, town-house, ware-house, public-house.
Poetry is a species of Fine Art, its differentia being the use of
metrical language as its instrument.
A lower class, indicated by the name of its higher class qualified by
adjectives or adjective phrases expressing its differential property
or properties, is said to be described PER GENUS ET DIFFERENTIAM.
Examples: "Black-bird," "note-book," "clever man," "man of Kent,"
"eminent British painter of marine subjects". By giving a combination
of attributes common to him with nobody else, we may narrow down the
application of a name to an individual: "The Commander-in-Chief of the
British forces at the battle of Waterloo".
Other attributes of classes as divided and defined, have received
technical names.
An attribute common to all the individuals of a class, found in that
class only, and following from the essential or defining attributes,
though not included among them, is called a PROPRIUM.
An attribute that belongs to some, but not to all, or that belongs to
all, but is not a necessary consequence of the essential attributes,
is called an ACCIDENT.
The clearest examples of Propria are found in mathematical figures.
Thus, the defining property of an equilateral triangle is the equality
of the sides: the equality of the angles is a proprium. That the
three angles of a triangle are together equal to two right angles is
a proprium, true of all triangles, and deducible from the essential
properties of a triangle.
Outside Mathematics, it is not easy to find _propria_ that satisfy
the three conditions of the definition. It is a useful exercise of
the wits to try for such. Educability--an example of the proprium
in mediæval text-books--is common to men, and results from man's
essential constitution; but it is not peculiar; other animals are
educable. That man cooks his food is probably a genuine proprium.
That horses run wild in Thibet: that gold is found in California: that
clergymen wear white ties, are examples of Accidents. Learning is an
accident in man, though educability is a proprium.
What is known technically as an INSEPARABLE ACCIDENT, such as the
black colour of the crow or the Ethiopian, is not easy to distinguish
from the Proprium. It is distinguished only by the third character,
deducibility from the essence.[2]
Accidents that are both common and peculiar are often useful for
distinguishing members of a class. Distinctive dresses or badges, such
as the gown of a student, the hood of a D.D., are accidents, but mark
the class of the individual wearer. So with the colours of flowers.
_Genus_, _Species_, _Differentia_, _Proprium_, and _Accidens_ have
been known since the time of Porphyry as the FIVE PREDICABLES. They
are really only terms used in dividing and defining. We shall return
to them and endeavour to show that they have no significance except
with reference to fixed schemes, scientific or popular, of Division or
Classification.
Given such a fixed scheme, very nice questions may be raised as to
whether a particular attribute is a defining attribute, or a proprium,
or an accident, or an inseparable accident. Such questions afford
great scope for the exercise of the analytic intellect.
We shall deal more particularly with degrees of generality when
we come to Definition. This much has been necessary to explain an
unimportant but much discussed point in Logic, what is known as the
inverse variation of Connotation and Denotation.
Connotation and Denotation are often said to vary inversely in
quantity. The larger the connotation the smaller the denotation, and
_vice versâ_. With certain qualifications the statement is correct
enough, but it is a rough compendious way of expressing the facts and
it needs qualification.
The main fact to be expressed is that the more general a name is, the
thinner is its meaning. The wider the scope, the shallower the ground.
As you rise in the scale of generality, your classes are wider but the
number of common attributes is less. Inversely, the name of a species
has a smaller denotation than the name of its genus, but a richer
connotation. _Fruit-tree_ applies to fewer objects than _tree_,
but the objects denoted have more in common: so with _apple_ and
_fruit-tree_, _Ribston Pippin_ and _apple_.
Again, as a rule, if you increase the connotation you contract the
area within which the name is applicable. Take any group of things
having certain attributes in common, say, _men of ability_: add
_courage_, _beauty_, _height of six feet_, _chest measurement of 40
inches_, and with each addition fewer individuals are to be found
possessing all the common attributes.
This is obvious enough, and yet the expression inverse variation is
open to objection. For the denotation may be increased in a sense
without affecting the connotation. The birth of an animal may be said
to increase the denotation: every year thousands of new houses are
built: there are swarms of flies in a hot summer and few in a cold.
But all the time the connotation of _animal_, _house_, or _fly_
remains the same: the word does not change its meaning.
It is obviously wrong to say that they vary in inverse proportion.
Double or treble the number of attributes, and you do not necessarily
reduce the denotation by one-half or one-third.
It is, in short, the meaning or connotation that is the main thing.
This determines the application of a word. As a rule if you increase
meaning, you restrict scope. Let your idea, notion, or concept of
_culture_ be a knowledge of Mathematics, Latin and Greek: your _men of
culture_ will be more numerous than if you require from each of them
these qualifications _plus_ a modern language, an acquaintance with
the Fine Arts, urbanity of manners, etc.
It is just possible to increase the connotation without decreasing the
denotation, to thicken or deepen the concept without diminishing
the class. This is possible only when two properties are exactly
co-extensive, as equilaterality and equiangularity in triangles.
SINGULAR and PROPER NAMES. A Proper or Singular name is a name used to
designate an individual. Its function, as distinguished from that of
the general name, is to be used purely for the purpose of distinctive
reference.
A man is not called Tom or Dick because he is like in certain respects
to other Toms or other Dicks. The Toms or the Dicks do not form a
logical class. The names are given purely for purposes of distinction,
to single out an individual subject. The Arabic equivalent for a
Proper name, _alam_, "a mark," "a sign-post," is a recognition of
this.
In the expressions "a Napoleon," "a Hotspur," "a Harry," the names
are not singular names logically, but general names logically, used to
signify the possession of certain attributes.
A man may be nicknamed on a ground, but if the name sticks and is
often used, the original meaning is forgotten. If it suggests
the individual in any one of his qualities, any point in which he
resembles other individuals, it is no longer a Proper or Singular name
logically, that is, in logical function. That function is fulfilled
when it has called to mind the individual intended.
To ask, as is sometimes done, whether Proper names are connotative or
denotative, is merely a confusion of language. The distinction between
connotation and denotation, extension and intension, applies only to
general names. Unless a name is general, it has neither extension nor
intension:[3] a Proper or Singular name is essentially the opposite of
a general name and has neither the one nor the other.
A nice distinction may be drawn between Proper and Singular names,
though they are strict synonyms for the same logical function. It is
not essential to the discharge of that function that the name should
be strictly appropriated to one object. There are many Toms and many
Dicks. It is enough that the word indicates the individual without
confusion in the particular circumstances.
This function may be discharged by words and combinations of words
that are not Proper in the grammatical sense. "This man," "the cover
of this book," "the Prime Minister of England," "the seer of Chelsea,"
may be Singular names as much as Honolulu or Lord Tennyson.
In common speech Singular names are often manufactured _ad hoc_
by taking a general name and narrowing it down by successive
qualifications till it applies only to one individual, as "The leading
subject of the Sovereign of England at the present time". If it so
happens that an individual has some attribute or combination peculiar
to himself, he may be suggested by the mention of that attribute
or combination:--"the inventor of the steam-engine," "the author of
Hudibras".
Have such names a connotation? The student may exercise his wits
on the question. It is a nice one, an excellent subject of debate.
Briefly, if we keep rigid hold of the meaning of connotation, this
Singular name has none. The combination is a singular name only
when it is the subject of a predication or an attribution, as in
the sentences, "The position of the leading subject of etc., is a
difficult one," or "The leading subject of etc., wears an eyeglass".
In such a sentence as "So-and-so is the leading subject of etc.," the
combined name has a connotation, but then it is a general and not a
singular name.
COLLECTIVE NAMES, as distinguished from General Names. A collective
name is a name for a number of similar units taken as a whole--a name
for a totality of similar units, as army, regiment, mob, mankind,
patrimony, personal estate.
A group or collection designated by a collective name is so far like
a class that the individual objects have something in common: they
are not heterogeneous but homogeneous. A mob is a collection of human
beings; a regiment of soldiers; a library of books.
The distinction lies in this, that whatever is said of a collective
name is said about the collection as a whole, and does not apply to
each individual; whatever is said of a general name applies to each
individual. Further, the collective name can be predicated only of
the whole group, as a whole; the general name is predicable of each,
distributively. "Mankind has been in existence for thousands of
years;" "The mob passed through the streets." In such expressions as
"An honest man's the noblest work of God," the subject is functionally
a collective name.
A collective name may be used as a general name when it is extended on
the ground of what is common to all such totalities as it designates.
"An excited mob is dangerous;" "An army without discipline is
useless." The collective name is then "connotative" of the common
characters of the collection.
MATERIAL OR SUBSTANTIAL NAMES. The question has been raised whether
names of material, gold, water, snow, coal, are general or collective
singular. In the case of pieces or bits of a material, it is true that
any predicate made concerning the material, such as "Sugar is sweet,"
or "Water quenches thirst," applies to any and every portion. But
the separate portions are not individuals in the whole signified by
a material name as individuals are in a class. Further, the name of
material cannot be predicated of a portion as a class name can be of
an individual. We cannot say, "This is a sugar". When we say, "This is
a piece of sugar," sugar is a collective name for the whole material.
There are probably words on the borderland between general names and
collective names. In such expressions as "This is a _coal_," "The
bonnie _water_ o' Urie," the material name is used as a general
name. The real distinction is between the distributive use and the
collective use of a name; as a matter of grammatical usage, the same
word may be used either way, but logically in any actual proposition
it must be either one or the other.
ABSTRACT NAMES are names for the common attributes or concepts on
which classes are constituted. A concrete name is a name directly
applicable to an individual in all his attributes, that is, as he
exists in the concrete. It may be written on a ticket and pinned to
him. When we have occasion to speak of the point or points in which a
number of individuals resemble one another, we use what is called an
abstract name. "Generous man," "clever man," "timid man," are concrete
names; "generosity," "cleverness," "timidity," are abstract names.
It is disputed whether abstract names are connotative. The question
is a confused one: it is like asking whether the name of a town is
municipal. An abstract name is the name of a connotation as a separate
object of thought or reference, conceived or spoken of in abstraction
from individual accidents. Strictly speaking it is notative rather
than _con_notative: it cannot be said to have a connotation because
it is itself the symbol of what is called the connotation of a general
name.[4]
The distinction between abstract names and concrete names is
virtually a grammatical distinction, that is, a distinction in mode
of predication. We may use concrete names or abstract names at our
pleasure to express the same meaning. To say that "John is a timid
man" is the same thing as saying that "Timidity is one of the
properties or characteristics or attributes of John". "Pride and
cruelty generally go together;" "Proud men are generally cruel men."
General names are predicable of individuals because they possess
certain attributes: to predicate the possession of those attributes is
the same thing as to predicate the general name.
Abstract forms of predication are employed in common speech quite as
frequently as concrete, and are, as we shall see, a great source of
ambiguity and confusion.
[Footnote 1: It has been somewhat too hastily assumed on the
authority of Mansel (Note to Aldrich, pp. 16, 17) that Mill
inverted the scholastic tradition in his use of the word
_Connotative_. Mansel puts his statement doubtfully, and
admits that there was some licence in the use of the word
Connotative, but holds that in Scholastic Logic an adjective
was said to "signify _primarily_ the attribute, and to
_connote_ or _signify secondarily_ ([Greek: prossêmainein])
the subject of inhesion". The truth is that Mansel's view was
a theory of usage not a statement of actual usage, and he had
good reason for putting it doubtfully.
As a matter of fact, the history of the distinction follows
the simple type of increasing precision and complexity, and
Mill was in strict accord with standard tradition. By the
Nominalist commentators on the _Summulæ_ of Petrus Hispanus
certain names, adjectives grammatically, are called
_Connotativa_ as opposed to _Absoluta_, simply because they
have a double function. White is connotative as signifying
both a subject, such as Socrates, of whom "whiteness" is an
attribute, and an attribute "whiteness": the names "Socrates"
and "whiteness" are Absolute, as having but a single
signification. Occam himself speaks of the subject as the
primary signification, and the attribute as the secondary,
because the answer to "What is white?" is "Something informed
with whiteness," and the subject is in the nominative case
while the attribute is in an oblique case (_Logic_, part I.
chap. x.). Later on we find that Tataretus (_Expositio in
Summulas_, A.D. 1501), while mentioning (Tract. Sept. _De
Appellationibus_) that it is a matter of dispute among
Doctores whether a connotative name _connotat_ the subject or
the attribute, is perfectly explicit in his own definition,
"Terminus connotativus est qui præter illud pro quo supponit
connotat aliquid adjacere vel non adjacere rei pro qua
supponit" (Tract. Sept. _De Suppositionibus_). And this
remained the standard usage as long as the distinction
remained in logical text-books. We find it very clearly
expressed by Clichtoveus, a Nominalist, quoted as an authority
by Guthutius in his _Gymnasium Speculativum_, Paris, 1607 (_De
Terminorum Cognitione_, pp. 78-9). "Terminus absolutus est,
qui solum illud pro quo in propositione supponit, significat.
Connotativus autem, qui ultra idipsum, aliud importat." Thus
_man_ and _animal_ are absolute terms, which simply stand
for (supponunt pro) the things they signify. _White_ is a
connotative name, because "it stands for (supponit pro) a
subject in which it is an accident: and beyond this, still
signifies an accident, which is in that subject, and is
expressed by an abstract name". Only Clichtoveus drops the
verb _connotat_, perhaps as a disputable term, and says simply
_ultra importat_.
So in the Port Royal Logic (1662), from which possibly Mill
took the distinction: "Les noms qui signifient les choses
comme modifiées, marquant premièrement et directement la
chose, quoique plus confusément, et indirectement le mode,
quoique plus distinctement, sont appelés _adjectifs_ ou
_connotatifs_; comme rond, dur, juste, prudent" (part i. chap
ii.).
What Mill did was not to invert Scholastic usage but to revive
the distinction, and extend the word connotative to general
names on the ground that they also imported the possession
of attributes. The word has been as fruitful of meticulous
discussion as it was in the Renaissance of Logic, though
the ground has changed. The point of Mill's innovation was,
premising that general names are not absolute but are applied
in virtue of a meaning, to put emphasis on this meaning as
the cardinal consideration. What he called the connotation had
dropped out of sight as not being required in the Syllogistic
Forms. This was as it were the point at which he put in his
horn to toss the prevalent conception of Logic as Syllogistic.
The real drift of Mill's innovation has been obscured by
the fact that it was introduced among the preliminaries
of Syllogism, whereas its real usefulness and significance
belongs not to Syllogism in the strict sense but to
Definition. He added to the confusion by trying to devise
forms of Syllogism based on connotation, and by discussing
the Axiom of the Syllogism from this point of view. For
syllogistic purposes, as we shall see, Aristotle's forms are
perfect, and his conception of the proposition in extension
the only correct conception. Whether the centre of gravity in
Consistency Logic should not be shifted back from Syllogism to
Definition, the latter being the true centre of consistency,
is another question. The tendency of Mill's polemic was to
make this change. And possibly the secret of the support it
has recently received from Mr. Bradley and Mr. Bosanquet is
that they, following Hegel, are moving in the same direction.
In effect, Mill's doctrine of Connotation helped to fix
a conception of the general name first dimly suggested by
Aristotle when he recognised that names of genera and species
signify Quality, in showing what sort a thing is. Occam
carried this a step farther towards clear light by including
among Connotative Terms such general names as "monk," name of
classes that at once suggest a definite attribute. The third
step was made by Mill in extending the term Connotation to
such words as "man," "horse," the _Infimæ Species_ of the
Schoolmen, the Species of modern science.
Whether connotation was the best term to use for this purpose,
rather than extension, may be questioned: but at least it was
in the line of tradition through Occam.]
[Footnote 2: The history of the definition of the _Proprium_
is an example of the tendency of distinctions to become more
minute and at the same time more purposeless. Aristotle's
[Greek: idion] was an attribute, such as the laugh of the man
or the bark of the dog, common to all of a class and peculiar
to the class (_quod convenit omni soli et semper_) yet not
comprised in the definition of the class. Porphyry recognised
three varieties of [Greek: idia] besides this, four in all,
as follows:--(1) an attribute peculiar to a species but not
possessed by all, as knowledge of medicine or geometry; (2)
possessed by a whole species but not peculiar to it, as being
a biped in man; (3) peculiar to a species, and possessed
by all at a certain time, as turning grey in old age; (4)
Aristotle's "proprium," peculiar and possessed by all, as
risibility. The idea of the Proprium as deducible from or
consequent on the essence would seem to have originated in
the desire to find something common to all Poryphyry's four
varieties.]
[Footnote 3: It is a plausible contention that in the case
of the Singular name the extension is at a minimum and the
intension at a maximum, the extension being one individual,
and the intension the totality of his attributes. But this is
an inexact and confused use of words. A name does not _extend_
beyond the individual except when it is used to signify one
or more of his prominent qualities, that is, is used with the
function of a general name. The _ex_tension of a Singular name
is zero: it has no extension. On the other hand, it suggests,
in its function as a Singular name, no properties or
qualities; it suggests only a subject; _i.e._, it has no
intension. The ambiguity of the term Denotation helps the
confusion in the case of Singular names. "Denote" in common
speech means to indicate, to distinguish. But when in Logic
we say that a general name denotes individuals, we have no
thought of indicating or distinguishing: we mean only that
it is applicable to any one, without respect of individuals,
either in predication or epithetic description.]
[Footnote 4: Strictly speaking, as I have tried to indicate
all along, the words Connotation and Denotation, or Extension
and Intension, apply only to general names. Outside general
names, they have no significance. An adjective with its noun
is a general name, of which the adjective gives part of the
Connotation. If we apply the word connotation to signify
merely the suggestion of an attribute in whatever grammatical
connexion, then an abstract name is undoubtedly as much
connotative as an adjective. The word _Sweetness_ has the same
meaning as _Sweet_: it indicates or signifies, conveys to the
mind of the reader the same attribute: the only difference is
that it does not at the same time indicate a subject in which
the attribute is found, as _sweet apple_. The meaning is not
_con_noted.]
CHAPTER II.
THE SYLLOGISTIC ANALYSIS OF PROPOSITIONS INTO TERMS.
I.--THE BARE ANALYTIC FORMS.
The word "term" is loosely used as a mere synonym for a name: strictly
speaking, a term ([Greek: horos], a boundary) is one of the parts of
a proposition as analysed into Subject and Predicate. In Logic, a term
is a technical word in an analysis made for a special purpose, that
purpose being to test the mutual consistency of propositions.
For this purpose, the propositions of common speech may be viewed as
consisting of two TERMS, a linkword called the copula (positive or
negative) expressing a relation between them, and certain symbols of
quantity used to express that relation more precisely.
Let us indicate the Subject term by S, and the Predicate term by P.
All propositions may be analysed into one or other of four forms:--
All S is P,
No S is P,
Some S is P,
Some S is not P.
All S is P is called the UNIVERSAL AFFIRMATIVE, and is indicated by
the symbol A (the first vowel of Affirmo).
No S is P is called the UNIVERSAL NEGATIVE, symbol E (the first vowel
of Nego).
Some S is P is called the PARTICULAR AFFIRMATIVE, symbol I (the second
vowel of _aff_Irmo).
Some S is not P is called the PARTICULAR NEGATIVE, symbol O (the
second vowel of _neg_O).
The distinction between Universal and Particular is called a
distinction in QUANTITY; between Affirmative and Negative, a
distinction in QUALITY. A and E, I and O, are of the same quantity,
but of different quality: A and I, E and O, same in quality, different
in quantity.
In this symbolism, no provision is made for expressing degrees of
particular quantity. _Some_ stands for any number short of all: it may
be one, few, most, or all but one. The debates in which Aristotle's
pupils were interested turned mainly on the proof or disproof of
general propositions; if only a proposition could be shown to be not
universal, it did not matter how far or how little short it came. In
the Logic of Probability, the degree becomes of importance.
Distinguish, in this Analysis, to avoid subsequent confusion, between
the Subject and the Subject Term, the Predicate and the Predicate
Term. The Subject is the Subject Term quantified: in A and E,[1] "All
S"; in I and O, "Some S". The Predicate is the Predicate Term with the
Copula, positive or negative: in A and I, "is P"; in E and O, "is not
P".
It is important also, in the interest of exactness, to note that S and
P, with one exception, represent general names. They are symbols
for classes. P is so always: S also except when the Subject is an
individual object. In the machinery of the Syllogism, predications
about a Singular term are treated as Universal Affirmatives. "Socrates
is a wise man" is of the form All S is P.
S and P being general names, the signification of the symbol "is" is
not the same as the "is" of common speech, whether the substantive
verb or the verb of incomplete predication. In the syllogistic form,
"is" means _is contained in_, "is not," _is not contained in_.
The relations between the terms in the four forms are represented by
simple diagrams known as Euler's circles.
[Illustration:
1 concentric circles of P and S - S in centre A
2 S and P in the same circle A
3 S and P each in a circle, overlapping circle. I & O
4 S in one circle and P in another circle. E
5 concentric circles of S and P - P in centre I?
]
Diagram 5 is a purely artificial form, having no representative in
common speech. In the affirmations of common speech, P is always a
term of greater extent than S.
No. 2 represents the special case where S and P are coextensive, as in
All equiangular triangles are equilateral.
S and P being general names, they are said to be DISTRIBUTED when the
proposition applies to them in their whole extent, that is, when the
assertion covers every individual in the class.
In E, the Universal Negative, both terms are distributed: "No S is P"
wholly excludes the two classes one from the other, imports that not
one individual of either is in the other.
In A, S is distributed, but not P. S is wholly in P, but nothing is
said about the extent of P beyond S.
In O, S is undistributed, P is distributed. A part of S is declared to
be wholly excluded from P.
In I, neither S nor P is distributed.
It will be seen that the Predicate term of a Negative proposition is
always distributed, of an Affirmative, always undistributed.
A little indistinctness in the signification of P crept into mediæval
text-books, and has tended to confuse modern disputation about the
import of Predication. Unless P is a class name, the ordinary doctrine
of distribution is nonsense; and Euler's diagrams are meaningless. Yet
many writers who adopt both follow mediæval usage in treating P as the
equivalent of an adjective, and consequently "is" as identical with
the verb of incomplete predication in common speech.
It should be recognised that these syllogistic forms are purely
artificial, invented for a purpose, namely, the simplification of
syllogising. Aristotle indicated the precise usage on which his
syllogism is based (_Prior Analytics_, i. 1 and 4). His form[2] for
All S is P, is S is wholly in P; for No S is P, S is wholly not in P.
His copula is not "is," but "is in," and it is a pity that this usage
was not kept. "All S is in P" would have saved much confusion. But,
doubtless for the sake of simplicity, the besetting sin of tutorial
handbooks, All S is P crept in instead, illustrated by such examples
as "All men are mortal".
Thus the "is" of the syllogistic form became confused with the "is"
of common speech, and the syllogistic view of predication as
being equivalent to inclusion in, or exclusion from a class, was
misunderstood. The true Aristotelian doctrine is not that predication
consists in referring subjects to classes, but only that for certain
logical purposes it may be so regarded. The syllogistic forms are
artificial forms. They were not originally intended to represent the
actual processes of thought expressed in common speech. To argue
that when I say "All crows are black," I do not form a class of
black things, and contemplate crows within it as one circle is within
another, is to contradict no intelligent logical doctrine.
The root of the confusion lies in quoting sentences from common speech
as examples of the logical forms, forgetting that those forms are
purely artificial. "Omnis homo est mortalis," "All men are mortal," is
not an example formally of All S is P. P is a symbol for a substantive
word or combination of words, and mortal is an adjective. Strictly
speaking, there is no formal equivalent in common speech, that is,
in the forms of ordinary use--no strict grammatical formal
equivalent--for the syllogistic propositional symbols. We can make
an equivalent, but it is not a form that men would use in ordinary
intercourse. "All man is in mortal being" would be a strict
equivalent, but it is not English grammar.
Instead of disputing confusedly whether All S is P should be
interpreted in extension or in comprehension, it would be better to
recognise the original and traditional use of the symbols S and P
as class names, and employ other symbols for the expression in
comprehension or connotation. Thus, let _s_ and _p_ stand for the
connotation. Then the equivalent for All S is P would be All S has
_p_, or _p_ always accompanies _s_, or _p_ belongs to all S.
It may be said that if predication is treated in this way, Logic is
simplified to the extent of childishness. And indeed, the manipulation
of the bare forms with the help of diagrams and mnemonics is a very
humble exercise. The real discipline of Syllogistic Logic lies in the
reduction of common speech to these forms.
This exercise is valuable because it promotes clear ideas about the
use of general names in predication, their ground in thought and
reality, and the liabilities to error that lurk in this fundamental
instrument of speech.
[Footnote 1: For perfect symmetry, the form of E should be
All S is not P. "No S is P" is adopted for E to avoid conflict
with a form of common speech, in which All S is not P conveys
the meaning of the Particular Negative. "All advices are not
safe" does not mean that safeness is denied of all advices,
but that safeness cannot be affirmed of all, _i.e._, Not all
advices are safe, _i.e._, some are not.]
[Footnote 2: His most precise form, I should say, for in "P is
predicated of every S" he virtually follows common speech.]
II.--THE PRACTICE OF SYLLOGISTIC ANALYSIS.
The basis of the analysis is the use of general names in predication.
To say that in predication a subject is referred to a class, is only
another way of saying that in every categorical sentence the predicate
is a general name express or implied: that it is by means of general
names that we convey our thoughts about things to others.
"Milton is a great poet." "Quoth Hudibras, _I smell a rat_." _Great
poet_ is a general name: it means certain qualities, and applies to
anybody possessing them. _Quoth_ implies a general name, a name for
persons _speaking_, connoting or meaning a certain act and applicable
to anybody in the performance of it. _Quoth_ expresses also past time:
thus it implies another general name, a name for persons _in past
time_, connoting a quality which differentiates a species in the genus
persons speaking, and making the predicate term "persons speaking in
past time". Thus the proposition _Quoth Hudibras_, analysed into the
syllogistic form S is in P, becomes S (Hudibras) is in P (persons
speaking in past time). The Predicate term P is a class constituted on
those properties. _Smell a rat_ also implies a general name, meaning
an act or state predicable of many individuals.
Even if we add the grammatical object of _Quoth_ to the analysis, the
Predicate term is still a general name. Hudibras is only one of the
persons speaking in past time who have spoken of themselves as being
in a certain mood of suspicion.[1]
The learner may well ask what is the use of twisting plain speech into
these uncouth forms. The use is certainly not obvious. The analysis
may be directly useful, as Aristotle claimed for it, when we wish
to ascertain exactly whether one proposition contradicts another, or
forms with another or others a valid link in an argument. This is
to admit that it is only in perplexing cases that the analysis is of
direct use. The indirect use is to familiarise us with what the
forms of common speech imply, and thus strengthen the intellect for
interpreting the condensed and elliptical expression in which common
speech abounds.
There are certain technical names applied to the components of
many-worded general names, CATEGOREMATIC and SYNCATEGOREMATIC, SUBJECT
and ATTRIBUTIVE. The distinctions are really grammatical rather than
logical, and of little practical value.
A word that can stand by itself as a term is said to be Categorematic.
_Man_, _poet_, or any other common noun.
A word that can only form part of a term is Syncategorematic. Under
this definition come all adjectives and adverbs.
The student's ingenuity may be exercised in applying the
distinction to the various parts of speech. A verb may be said to be
_Hypercategorematic_, implying, as it does, not only a term, but also
a copula.
A nice point is whether the Adjective is categorematic or
syncategorematic. The question depends on the definition of "term"
in Logic. In common speech an adjective may stand by itself as a
predicate, and so might be said to be Categorematic. "This heart is
merry." But if a term is a class, or the name of a class, it is not
Categorematic in the above definition. It can only help to specify a
class when attached to the name of a higher genus.
Mr. Fowler's words SUBJECT and ATTRIBUTIVE express practically the
same distinction, except that Attributive is of narrower extent than
syncategorematic. An Attributive is a word that connotes an attribute
or property, as _hot_, _valorous_, and is always grammatically an
adjective.
The EXPRESSION OF QUANTITY, that is, of Universality or
non-universality, is all-important in syllogistic formulæ. In them
universality is expressed by _All_ or _None_. In ordinary speech
universality is expressed in various forms, concrete and abstract,
plain and figurative, without the use of "all" or "none".
Uneasy lies the head that wears a crown.
He can't be wrong whose life is in the right.
What cat's averse to fish?
Can the leopard change his spots?
The longest road has an end.
Suspicion ever haunts the guilty mind.
Irresolution is always a sign of weakness.
Treason never prospers.
A proposition in which the quantity is not expressed is called by
Aristotle INDEFINITE ([Greek: adioristos]). For "indefinite"[2]
Hamilton suggests PREINDESIGNATE, undesignated, that is, before being
received from common speech for the syllogistic mill. A proposition is
PREDESIGNATE when the quantity is definitely indicated. All the above
propositions are "Predesignate" universals, and reducible to the form
All S is P, or No S is P.
The following propositions are no less definitely particular,
reducible to the form I or O. In them as in the preceding quantity
is formally expressed, though the forms used are not the artificial
syllogistic forms:--
Afflictions are often salutary.
Not every advice is a safe one.
All that glitters is not gold.
Rivers generally[3] run into the sea.
Often, however, it is really uncertain from the form of common speech
whether it is intended to express a universal or a particular. The
quantity is not formally expressed. This is especially the case
with proverbs and loose floating sayings of a general tendency. For
example:--
Haste makes waste.
Knowledge is power.
Light come, light go.
Left-handed men are awkward antagonists.
Veteran soldiers are the steadiest in fight.
Such sayings are in actual speech for the most part delivered as
universals.[4] It is a useful exercise of the Socratic kind to decide
whether they are really so. This can only be determined by a survey of
facts. The best method of conducting such a survey is probably (1)
to pick out the concrete subject, "hasty actions," "men possessed of
knowledge," "things lightly acquired"; (2) to fix the attribute or
attributes predicated; (3) to run over the individuals of the subject
class and settle whether the attribute is as a matter of fact meant to
be predicated of each and every one.
This is the operation of INDUCTION. If one individual can be found of
whom the attribute is not meant to be predicated, the proposition is
not intended as Universal.
Mark the difference between settling what is intended and settling
what is true. The conditions of truth and the errors incident to the
attempt to determine it, are the business of the Logic of Rational
Belief, commonly entitled Inductive Logic. The kind of "induction"
here contemplated has for its aim merely to determine the quantity of
a proposition in common acceptation, which can be done by considering
in what cases the proposition would generally be alleged. This
corresponds nearly as we shall see to Aristotelian Induction, the
acceptance of a universal statement when no instance to the contrary
is alleged.
It is to be observed that for this operation we do not practically use
the syllogistic form All S is P. We do not raise the question Is All
S, P? That is, we do not constitute in thought a class P: the class in
our minds is S, and what we ask is whether an attribute predicated of
this class is truly predicated of every individual of it.
Suppose we indicate by _p_ the attribute, knot of attributes, or
concept on which the class P is constituted, then All S is P is
equivalent to "All S has _p_": and Has All S _p_? is the form of a
question that we have in our minds when we make an inductive survey on
the above method. I point this out to emphasise the fact that there is
no prerogative in the form All S is P except for syllogistic purposes.
This inductive survey may be made a useful COLLATERAL DISCIPLINE. The
bare forms of Syllogistic are a useless item of knowledge unless they
are applied to concrete thought. And determining the quantity of a
common aphorism or saw, the limits within which it is meant to hold
good, is a valuable discipline in exactness of understanding. In
trying to penetrate to the inner intention of a loose general maxim,
we discover that what it is really intended to assert is a general
connexion of attributes, and a survey of concrete cases leads to
a more exact apprehension of those attributes. Thus in considering
whether _Knowledge is power_ is meant to be asserted of all knowledge,
we encounter along with such examples as the sailor's knowledge that
wetting a rope shortens it, which enabled some masons to raise a stone
to its desired position, or the knowledge of French roads possessed by
the German invaders,--along with such examples as these we encounter
cases where a knowledge of difficulties without a knowledge of the
means of overcoming them is paralysing to action. Samuel Daniel
says:--
Where timid knowledge stands considering
Audacious ignorance has done the deed.
Studying numerous cases where "Knowledge is power" is alleged or
denied, we find that what is meant is that a knowledge of the right
means of doing anything is power--in short, that the predicate is not
made of all knowledge, but only of a species of knowledge.
Take, again, _Custom blunts sensibility_. Putting this in the
concrete, and inquiring what predicate is made about "men accustomed
to anything" (S), we have no difficulty in finding examples where such
men are said to become indifferent to it. We find such illustrations
as Lovelace's famous "Paradox":--
Through foul we follow fair
For had the world one face
And earth been bright as air
We had known neither place.
Indians smell not their nest
The Swiss and Finn taste best
The spices of the East.
So men accustomed to riches are not acutely sensible of their
advantages: dwellers in noisy streets cease to be distracted by the
din: the watchmaker ceases to hear the multitudinous ticking in his
shop: the neighbours of chemical works are not annoyed by the smells
like the casual passenger. But we find also that wine-tasters
acquire by practice an unusual delicacy of sense; that the eyes once
accustomed to a dim light begin to distinguish objects that were at
first indistinguishable; and so on. What meanings of "custom" and of
"sensibility" will reconcile these apparently conflicting examples?
What are the exact attributes signified by the names? We should
probably find that by sensibility is meant emotional sensibility as
distinguished from intellectual discrimination, and that by custom is
meant familiarity with impressions whose variations are not attended
to, or subjection to one unvarying impression.
To verify the meaning of abstract proverbs in this way is to travel
over the road by which the Greek dialecticians were led to feel the
importance of definition. Of this more will be said presently. If
it is contended that such excursions are beyond the bounds of Formal
Logic, the answer is that the exercise is a useful one and that it
starts naturally and conveniently from the formulæ of Logic. It is the
practice and discipline that historically preceded the Aristotelian
Logic, and in the absence of which the Aristotelian formulæ would have
a narrowing and cramping effect.
CAN ALL PROPOSITIONS BE REDUCED TO THE SYLLOGISTIC FORM? Probably: but
this is a purely scientific inquiry, collateral to Practical Logic.
The concern of Practical Logic is chiefly with forms of proposition
that favour inaccuracy or inexactness of thought. When there is no
room for ambiguity or other error, there is no virtue in artificial
syllogistic form. The attempt so to reduce any and every proposition
may lead, however, to the study of what Mr. Bosanquet happily calls
the "Morphology" of Judgment, Judgment being the technical name for
the mental act that accompanies the utterance of a proposition. Even
in such sentences as "How hot it is!" or "It rains," the rudiment of
subject and predicate may be detected. When a man says "How hot it
is," he conveys the meaning, though there is no definitely formed
subject in his mind, that the outer world at the moment of his
speaking has a certain quality or attribute. So with "It rains". The
study of such examples in their context, however, reveals the fact
that the same form of Common speech may cover different subjects and
predicates in different connexions. Thus in the argument:--
"Whatever is, is best.
It rains!"--
the Subject is _Rain_ and the Predicate _is now_, "is at the present
time," "is in the class of present events".
[Footnote 1: Remember that when we speak of a general name,
we do not necessarily mean a single word. A general name,
logically viewed, is simply the name of a _genus_, kind, or
class: and whether this is single-worded or many-worded
is, strictly speaking, a grammatical question. "Man,"
"man-of-ability," "man-of-ability-and-courage,"
"man-of-ability-and-courage-and-gigantic-stature,"
"man-who-fought-at-Marathon"--these are all general names
in their logical function. No matter how the constitutive
properties of the class are indicated, by one word or in
combination, that word or combination is a general name. In
actual speech we can seldom indicate by a single word the
meaning predicated.]
[Footnote 2: The objection taken to the word "indefinite,"
that the quantity of particular propositions is indefinite,
_some_ meaning any quantity less than all, is an example of
the misplaced and frivolous subtlety that has done so much
to disorder the tradition of Logic. By "indefinite" is
simply meant not definitely expressed as either Universal
or Particular, Total or Partial. The same objection might be
taken to any word used to express the distinction: the degree
of quantity in Some S is not "designate" any more than it is
"definite" or "dioristic".]
[Footnote 3: _Generally._ In this word we have an instance of
the frequent conflict between the words of common speech and
logical terminology. How it arises shall be explained in next
chapter. A General proposition is a synonym for a Universal
proposition (if the forms A and E are so termed): but
"generally" in common speech means "for the most part," and is
represented by the symbol of particular quantity, _Some_.]
[Footnote 4: With some logicians it is a mechanical rule in
reducing to syllogistic form to treat as I or O all sentences
in which there is no formal expression of quantity. This is to
err on the safe side, but common speakers are not so guarded,
and it is to be presumed rather that they have a universal
application in their minds when they do not expressly
qualify.]
III.--SOME TECHNICAL DIFFICULTIES.
_The formula for_ EXCLUSIVE PROPOSITIONS. "None but the brave deserve
the fair": "No admittance except on business": "Only Protestants can
sit on the throne of England".
These propositions exemplify different ways in common speech of naming
a subject _exclusively_, the predication being made of all outside a
certain term. "None that are not brave, etc.;" "none that are not on
business, etc.;" "none that are not Protestants, etc.". No not-S is
P. It is only about all outside the given term that the universal
assertion is made: we say nothing universally about the individuals
within the term: we do not say that all Protestants are eligible, nor
that all persons on business are admitted, nor that every one of the
brave deserves the fair. All that we say is that the possession of the
attribute named is an indispensable condition: a person may possess
the attribute, and yet on other grounds may not be entitled to the
predicate.
The justification for taking special note of this form in Logic is
that we are apt by inadvertence to make an inclusive inference from
it. Let it be said that None but those who work hard can reasonably
expect to pass, and we are apt to take this as meaning that all who
work hard may reasonably expect to pass. But what is denied of every
Not-S is not necessarily affirmed of every S.
_The expression of_ TENSE or TIME _in the Syllogistic Forms_. Seeing
that the Copula in S is P or S is in P does not express time, but only
a certain relation between S and P, the question arises Where are we
to put time in the analytic formula? "Wheat is dear;" "All had fled;"
time is expressed in these propositions, and our formula should
render the whole content of what is given. Are we to include it in
the Predicate term or in the Subject term? If it must not be left out
altogether, and we cannot put it with the copula, we have a choice
between the two terms.
It is a purely scholastic question. The common technical treatment is
to view the tense as part of the predicate. "All had fled," All S is
P, _i.e._, the whole subject is included in a class constituted on the
attributes of flight at a given time. It may be that the Predicate is
solely a predicate of time. "The Board met yesterday at noon." S is P,
_i.e._, the meeting of the Board is one of the events characterised by
having happened at a certain time, agreeing with other events in that
respect.
But in some cases the time is more properly regarded as part of the
subject. _E.g._, "Wheat is dear". S does not here stand for wheat
collectively, but for the wheat now in the market, the wheat of the
present time: it is concerning this that the attribute of dearness is
predicated; it is this that is in the class of dear things.
_The expression of_ MODALITY _in the Syllogistic Forms_. Propositions
in which the predicate is qualified by an expression of necessity,
contingency, possibility or impossibility [_i.e._, in English by
_must_, _may_, _can_, or _cannot_], were called in Mediæval Logic
_Modal_ Propositions. "Two and two _must_ make four." "Grubs _may_
become butterflies." "Z _can_ paint." "Y _cannot_ fly."
There are two recognised ways of reducing such propositions to the
form S is P. One is to distinguish between the _Dictum_ and the
_Mode_, the proposition and the qualification of its certainty, and
to treat the _Dictum_ as the Subject and the _Mode_ as the Predicate.
Thus: "That two and two make four is necessary"; "That Y can fly is
impossible".
The other way is to treat the Mode as part of the predicate.
The propriety of this is not obvious in the case of Necessary
propositions, but it is unobjectionable in the case of the other three
modes. Thus: "Grubs are things that have the potentiality of becoming
butterflies"; "Z has the faculty of painting"; "Y has not the faculty
of flying".
The chief risk of error is in determining the quantity of the subject
about which the Contingent or Possible predicate is made. When it is
said that "Victories may be gained by accident," is the predicate made
concerning All victories or Some only? Here we are apt to confuse the
meaning of the contingent assertion with the matter of fact on which
in common belief it rests. It is true only that some victories have
been gained by accident, and it is on this ground that we assert in
the absence of certain knowledge concerning any victory that it
may have been so gained. The latter is the effect of the contingent
assertion: it is made about any victory in the absence of certain
knowledge, that is to say, formally about all.
The history of Modals in Logic is a good illustration of intricate
confusion arising from disregard of a clear traditional definition.
The treatment of them by Aristotle was simple, and had direct
reference to tricks of disputation practised in his time. He specified
four "modes," the four that descended to mediæval logic, and he
concerned himself chiefly with the import of contradicting these
modals. What is the true contradictory of such propositions as, "It
is possible to be" ([Greek: dynaton einai]), "It admits of being"
([Greek: endechetai einai]), "It must be" ([Greek: anankaion einai]),
"It is impossible to be" ([Greek: adynaton einai])? What is implied in
saying "No" to such propositions put interrogatively? "Is it possible
for Socrates to fly?" "No." Does this mean that it is not possible for
Socrates to fly, or that it is possible for Socrates not to fly?
A disputant who had trapped a respondent into admitting that it is
possible for Socrates not to fly, might have pushed the concession
farther in some such way as this: "Is it possible for Socrates not to
walk?" "Certainly." "Is it possible for him to walk?" "Yes." "When you
say that it is possible for a man to do anything do you not believe
that it is possible for him to do it?" "Yes." "But you have admitted
that it is possible for Socrates not to fly?"
It was in view of such perplexities as these that Aristotle set forth
the true contradictories of his four Modals. We may laugh at such
quibbles now and wonder that a grave logician should have thought them
worth guarding against. But historically this is the origin of the
Modals of Formal Logic, and to divert the names of them to signify
other distinctions than those between modes of qualifying the
certainty of a statement is to introduce confusion.
Thus we find "Alexander was a great general," given as an example of
a Contingent Modal, on the ground that though as a matter of fact
Alexander was so he might have been otherwise. It was not _necessary_
that Alexander should be a great general: therefore the proposition
is _contingent_. Now the distinction between Necessary truth and
Contingent truth may be important philosophically: but it is merely
confusing to call the character of propositions as one or the other by
the name of Modality. The original Modality is a mode of expression:
to apply the name to this character is to shift its meaning.
A more simple and obviously unwarrantable departure from tradition
is to extend the name Modality to any grammatical qualification of
a single verb in common speech. On this understanding "Alexander
conquered Darius" is given by Hamilton as a _Pure_ proposition, and
"Alexander conquered Darius honourably" as a _Modal_. This is a merely
grammatical distinction, a distinction in the mode of composing the
predicate term in common speech. In logical tradition Modality is a
mode of qualifying the certainty of an affirmation. "The conquest
of Darius by Alexander was honourable," or "Alexander in conquering
Darius was an honourable conqueror," is the syllogistic form of the
proposition: it is simply assertory, not qualified in any "mode".
There is a similar misunderstanding in Mr. Shedden's treatment of
"generally" as constituting a Modal in such sentences, as "Rivers
_generally_ flow into the sea". He argues that as _generally_ is not
part either of the Subject term or of the Predicate term, it must
belong to the Copula, and is therefore a _modal_ qualification of the
whole assertion. He overlooked the fact that the word "generally" is
an expression of Quantity: it determines the quantity of the Subject
term.
Finally it is sometimes held (_e.g._, by Mr. Venn) that the question
of Modality belongs properly to Scientific or Inductive Logic, and is
out of place in Formal Logic. This is so far accurate that it is
for Inductive Logic to expound the conditions of various degrees of
certainty. The consideration of Modality is pertinent to Formal Logic
only in so far as concerns special perplexities in the expression of
it. The treatment of it by Logicians has been rendered intricate by
torturing the old tradition to suit different conceptions of the end
and aim of Logic.
PART II.
DEFINITION.
CHAPTER I.
IMPERFECT UNDERSTANDING OF WORDS AND THE REMEDIES
THEREFOR.--DIALECTIC.--DEFINITION.
We cannot inquire far into the meaning of proverbs or traditional
sayings without discovering that the common understanding of general
and abstract names is loose and uncertain. Common speech is a
quicksand.
Consider how we acquire our vocabulary, how we pick up the words that
we use from our neighbours and from books, and why this is so soon
becomes apparent. Theoretically we know the full meaning of a name
when we know all the attributes that it connotes, and we are not
justified in extending it except to objects that possess all the
attributes. This is the logical ideal, but between the _ought to be_
of Logic and the _is_ of practical life, there is a vast difference.
How seldom do we conceive words in their full meaning! And who is to
instruct us in the full meaning? It is not as in the exact sciences,
where we start with a knowledge of the full meaning. In Geometry, for
example, we learn the definitions of the words used, _point_, _line_,
_parallel_, etc., before we proceed to use them. But in common speech,
we learn words first in their application to individual cases. Nobody
ever defined _good_ to us, or _fair_, or _kind_, or _highly educated_.
We hear the words applied to individual objects: we utter them in the
same connexion: we extend them to other objects that strike us as like
without knowing the precise points of likeness that the convention
of common speech includes. The more exact meaning we learn by
gradual induction from individual cases. _Ugly_, _beautiful_, _good_,
_bad_--we learn the words first as applicable to things and persons:
gradually there arises a more or less definite sense of what the
objects so designated have in common. The individual's extension of
the name proceeds upon what in the objects has most impressed him
when he caught the word: this may differ in different individuals; the
usage of neighbours corrects individual eccentricities. The child
in arms shouts _Da_ at the passing stranger who reminds him of his
father: for him at first it is a general name applicable to every man:
by degrees he learns that for him it is a singular name.
The mode in which words are learnt and extended may be studied most
simply in the nursery. A child, say, has learnt to say _mambro_ when
it sees its nurse. The nurse works a hand-turned sewing machine,
and sings to it as she works. In the street the child sees an
organ-grinder singing as he turns his handle: it calls _mambro_:
the nurse catches the meaning and the child is overjoyed. The
organ-grinder has a monkey: the child has an india-rubber monkey toy:
it calls this also _mambro_. The name is extended to a monkey in
a picture-book. It has a toy musical box with a handle: this also
becomes _mambro_, the word being extended along another line of
resemblance. A stroller with a French fiddle comes within the
denotation of the word: a towel-rail is also called _mambro_ from some
fancied resemblance to the fiddle. A very swarthy hunch-back _mambro_
frightens the child: this leads to the transference of the word to a
terrific coalman with a bag of coals on his back. In a short time
the word has become a name for a great variety of objects that have
nothing whatever common to all of them, though each is strikingly like
in some point to a predecessor in the series. When the application
becomes too heterogeneous, the word ceases to be of use as a sign and
is gradually abandoned, the most impressive meaning being the last
to go. In a child's vocabulary where the word _mambro_ had a run of
nearly two years, its last use was as an adjective signifying ugly or
horrible.
The history of such a word in a child's language is a type of what
goes on in the language of men. In the larger history we see similar
extensions under similar motives, checked and controlled in the same
way by surrounding usage.
It is obvious that to avoid error and confusion, the meaning or
connotation of names, the concepts, should somehow be fixed: names
cannot otherwise have an identical reference in human intercourse. We
may call this ideal fixed concept the LOGICAL CONCEPT: or we may call
it the SCIENTIFIC CONCEPT, inasmuch as one of the main objects of the
sciences is to attain such ideals in different departments of study.
But in actual speech we have also the PERSONAL CONCEPT, which varies
more or less with the individual user, and the POPULAR or VERNACULAR
CONCEPT, which, though roughly fixed, varies from social sect to
social sect and from generation to generation.
The variations in Popular Concepts may be traced in linguistic
history. Words change with things and with the aspects of things,
as these change in public interest and importance. As long as the
attributes that govern the application of words are simple, sensible
attributes, little confusion need arise: the variations are matters of
curious research for the philologist, but are logically insignificant.
Murray's Dictionary, or such books as Trench's _English Past and
Present_, supply endless examples, as many, indeed as there are
words in the language. _Clerk_ has almost as many connotations as
our typical _mambro_: clerk in holy orders, church clerk, town clerk,
clerk of assize, grocer's clerk. In Early English, the word meant "man
in a religious order, cleric, clergyman"; ability to read, write, and
keep accounts being a prominent attribute of the class, the word was
extended on this simple ground till it has ceased altogether to cover
its original field except as a formal designation. But no confusion is
caused by the variation, because the property connoted is simple.[1]
So with any common noun: street, carriage, ship, house, merchant,
lawyer, professor. We might be puzzled to give an exact definition of
such words, to say precisely what they connote in common usage; but
the risk of error in the use of them is small.
When we come to words of which the logical concept is a complex
relation, an obscure or intangible attribute, the defects of the
popular conception and its tendencies to change and confusion, are
of the greatest practical importance. Take such words as _Monarchy_,
_tyranny_, _civil freedom_, _freedom of contract_, _landlord_,
_gentleman_, _prig_, _culture_, _education_, _temperance_,
_generosity_. Not merely should we find it difficult to give an
analytic definition of such words: we might be unable to do so, and
yet flatter ourselves that we had a clear understanding of their
meaning. But let two men begin to discuss any proposition in which any
such word is involved, and it will often be found that they take the
word in different senses. If the relation expressed is complex, they
have different sides or lines of it in their minds; if the meaning
is an obscure quality, they are guided in their application of it by
different outward signs.
Monarchy, in its original meaning, is applied to a form of government
in which the will of one man is supreme, to make laws or break them,
to appoint or dismiss officers of state and justice, to determine
peace or war, without control of statute or custom. But supreme power
is never thus uncontrolled in reality; and the word has been extended
to cover governments in which the power of the titular head is
controlled in many different modes and degrees. The existence of a
head, with the title of King or Emperor, is the simplest and most
salient fact: and wherever this exists, the popular concept of a
monarchy is realised. The President of the United States has more real
power than the Sovereign of Great Britain; but the one government
is called a Republic and the other a Monarchy. People discuss the
advantages and disadvantages of monarchy without first deciding
whether they take the word in its etymological sense of unlimited
power, or its popular sense of titular kingship, or its logical sense
of power definitely limited in certain ways. And often in debate,
monarchy is really a singular term for the government of Great
Britain.
_Culture_, _religious_, _generous_, are names for inward states or
qualities: with most individuals some simple outward sign directs
the application of the word--it may be manner, or bearing, or routine
observances, or even nothing more significant than the cut of the
clothes or of the hair. Small things undoubtedly are significant, and
we must judge by small things when we have nothing else to go by: but
instead of trying to get definite conceptions for our moral epithets,
and suspending judgment till we know that the use of the epithet is
justified, the trifling superficial sign becomes for us practically
the whole meaning of the word. We feel that we must have a judgment of
some sort at once: only simple signs are suited to our impatience.
It was with reference to this state of things that Hegel formulated
his paradox that the true abstract thinker is the plain man who laughs
at philosophy as what he calls abstract and unpractical. He holds
decided opinions for or against this or the other abstraction,
_freedom_, _tyranny_, _revolution_, _reform_, _socialism_, but what
these words mean and within what limits the things signified are
desirable or undesirable, he is in too great a hurry to pause and
consider.
The disadvantages of this kind of "abstract" thinking are obvious.
The accumulated wisdom of mankind is stored in language. Until we have
cleared our conceptions, and penetrated to the full meaning of words,
that wisdom is a sealed book to us. Wise maxims are interpreted by
us hastily in accordance with our own narrow conceptions. All the
vocables of a language may be more or less familiar to us, and yet we
may not have learnt it as an instrument of thought. Outside the very
limited range of names for what we see and use in the daily routine of
life, food and clothes and the common occupations of men, words
have little meaning for us, and are the vehicles merely of thin
preconceptions and raw prejudices.
The remedy for "abstract" thinking is more thinking, and in pursuing
this two aims may be specified for the sake of clearness, though they
are closely allied, and progress towards both may often be made by
one and the same operation. (1) We want to reach a clear and full
conception of the meaning of names as used now or at a given time.
Let us call this the _Verification of the Meaning_. (2) We want to fix
such conceptions, and if necessary readjust their boundaries. This is
the province of _Definition_, which cannot be effectually performed
without _Scientific Classification_ or _Division_.
I.--VERIFICATION OF THE MEANING--DIALECTIC.
This can only be done by assembling the objects to which the words are
applied, and considering what they have in common. To ascertain the
actual connotation we must run over the actual denotation. And since
in such an operation two or more minds are better than one, discussion
or dialectic is both more fruitful and more stimulating than solitary
reflection or reading.
The first to practise this process on a memorable scale, and with
a distinct method and purpose, was Socrates. To insist upon the
necessity of clear conceptions, and to assist by his dialectic
procedure in forming them, was his contribution to philosophy.
His plan was to take a common name, profess ignorance of its meaning,
and ask his interlocutor whether he would apply it in such and such
an instance, producing one after another. According to Xenophon's
_Memorabilia_ he habitually chose the commonest names, _good_,
_unjust_, _fitting_, and so forth, and tried to set men thinking
about them, and helped them by his questions to form an intelligent
conception of the meaning.
For example, what is the meaning of injustice? Would you say that
the man who cheats or deceives is unjust? Suppose a man deceives his
enemies, is there any injustice in that? Can the definition be that
a man who deceives his friends is unjust? But there are cases where
friends are deceived for their own good: are these cases of injustice?
A general may inspirit his soldiers by a falsehood. A man may cajole
a weapon out of his friend's hand when he sees him about to commit
suicide. A father may deceive his son into taking medicine. Would you
call these men unjust? By some such process of interrogation we
are brought to the definition that a man is unjust who deceives his
friends to their hurt.
Observe that in much of his dialectic the aim of Socrates was merely
to bring out the meaning lying vague and latent, as it were, in
the common mind. His object was simply what we have called the
verification of the meaning. And a dialectic that confines itself to
the consideration of what is ordinarily meant as distinct from what
ought to be meant may often serve a useful purpose. Disputes about
words are not always as idle as is sometimes supposed. Mr. H. Sidgwick
truly remarks (_à propos_ of the terms of Political Economy) that
there is often more profit in seeking a definition than in finding it.
Conceptions are not merely cleared but deepened by the process. Mr.
Sidgwick's remarks are so happy that I must take leave to quote them:
they apply not merely to the verification of ordinary meaning but also
to the study of special uses by authorities, and the reasons for those
special uses.
"The truth is--as most readers of Plato know, only it is a
truth difficult to retain and apply--that what we gain by
discussing a definition is often but slightly represented in
the superior fitness of the formula that we ultimately adopt;
it consists chiefly in the greater clearness and fulness in
which the characteristics of the matter to which the formula
refers have been brought before the mind in the process of
seeking for it. While we are apparently aiming at definitions
of terms, our attention should be really fixed on distinctions
and relations of fact. These latter are what we are concerned
to know, contemplate, and as far as possible arrange and
systematise; and in subjects where we cannot present them to
the mind in ordinary fulness by the exercise of the organs of
sense, there is no way of surveying them so convenient as
that of reflecting on our use of common terms.... In comparing
different definitions our aim should be far less to decide
which we ought to adopt, than to apprehend and duly consider
the grounds on which each has commended itself to reflective
minds. We shall generally find that each writer has noted some
relation, some resemblance or difference, which others have
overlooked; and we shall gain in completeness, and often
in precision, of view by following him in his observations,
whether or not we follow him in his conclusions."[2]
Mr. Sidgwick's own discussions of _Wealth_, _Value_, and _Money_ are
models. A clue is often found to the meaning in examining startlingly
discrepant statements connected with the same leading word. Thus
we find some authorities declaring that "style" cannot be taught or
learnt, while others declare that it can. But on trying to ascertain
what they mean by "style," we find that those who say it cannot be
taught mean either a certain marked individual character or manner of
writing--as in Buffon's saying, _Le style c'est l'homme même_--or a
certain felicity and dignity of expression, while those who say style
can be taught mean lucid method in the structure of sentences or in
the arrangement of a discourse. Again in discussions on the rank of
poets, we find different conceptions of what constitutes greatness in
poetry lying at the root of the inclusion of this or the other poet
among great poets. We find one poet excluded from the first rank of
greatness because his poetry was not serious; another because his
poetry was not widely popular; another because he wrote comparatively
little; another because he wrote only songs or odes and never
attempted drama or epic. These various opinions point to different
conceptions of what constitutes greatness in poets, different
connotations of "great poet". Comparing different opinions concerning
"education" we may be led to ask whether it means more than
instruction in the details of certain subjects, whether it does not
also import the formation of a disposition to learn or an interest in
learning or instruction in a certain method of learning.
Historically, dialectic turning on the use of words preceded the
attempt to formulate principles of Definition, and attempts at precise
definition led to Division and Classification, that is to systematic
arrangement of the objects to be defined. Attempt to define any such
word as "education," and you gradually become sensible of the needs in
respect of method that forced themselves upon mankind in the history
of thought. You soon become aware that you cannot define it by itself
alone; that you are beset by a swarm of more or less synonymous words,
_instruction_, _discipline_, _culture_, _training_, and so on; that
these various words represent distinctions and relations among things
more or less allied; and that, if each must be fixed to a definite
meaning, this must be done with reference to one another and to the
whole department of things that they cover.
The first memorable attempts at scientific arrangement were
Aristotle's treatises on Ethics and Politics, which had been the
subjects of active dialectic for at least a century before. That
these the most difficult of all departments to subject to scientific
treatment should have been the first chosen was due simply to their
preponderating interest: "The proper study of mankind is man". The
systems of what are known as the Natural Sciences are of modern
origin: the first, that of Botany, dates from Cesalpinus in the
sixteenth century. But the principles on which Aristotle proceeded in
dividing and defining, principles which have gradually themselves been
more precisely formulated, are principles applicable to all systematic
arrangements for purposes of orderly study. I give them in the precise
formulæ which they have gradually assumed in the tradition of Logic.
The principles of Division are often given in Formal Logic, and the
principles of Classification in Inductive Logic, but there is no
valid reason for the separation. The classification of objects in the
Natural Sciences, of animals, plants, and stones, with a view to
the thorough study of them in form, structure, and function, is
more complex than classifications for more limited purposes, and the
tendency is to restrict the word classification to these elaborate
systems. But really they are only a series of divisions and
subdivisions, and the same principles apply to each of the subordinate
divisions as well as to the division of the whole department of study.
II.--PRINCIPLES OF DIVISION OR CLASSIFICATION AND DEFINITION.
Confusion in the boundaries of names arises from confused ideas
regarding the resemblances and differences of things. As a protective
against this confusion, things must be clearly distinguished in their
points of likeness and difference, and this leads to their arrangement
in systems, that is, to division and classification. A name is not
secure against variation until it has a distinct place in such a
system as a symbol for clearly distinguished attributes. Nor must we
forget, further, that systems have their day, that the best system
attainable is only temporary, and may have to be recast to correspond
with changes of things and of man's way of looking at them.
The leading principles of DIVISION may be stated as follows:--
I. Every division is made on the ground of differences in
some attribute common to all the members of the whole to be
divided.
This is merely a way of stating what a logical division is. It is
a division of a generic whole or _genus_, an indefinite number of
objects thought of together as possessing some common character or
attribute. All have this attribute, which is technically called the
_fundamentum divisionis_, or generic attribute. But the whole is
divisible into smaller groups (_species_), each of which possesses the
common character with a difference (_differentia_). Thus, mankind may
be divided into White men, Black men, Yellow men, on ground of the
differences in the colour of their skins: all have skins of some
colour: this is the _fundamentum divisionis_: but each subdivision or
species has a different colour: this is the _differentia_. Rectilineal
figures are divided into triangles, quadrangles, pentagons, etc., on
the ground of differences in the number of angles.
Unless there is a _fund. div._, _i.e._, unless the differences are
differences in a common character, the division is not a logical
division. To divide men into Europeans, opticians, tailors, blondes,
brunettes, and dyspeptics is not to make a logical division. This is
seen more clearly in connexion with the second condition of a perfect
division.
II. In a perfect division, the subdivisions or species are
mutually exclusive.
Every object possessing the common character should be in one or other
of the groups, and none should be in more than one.
Confusion between classes, or overlapping, may arise from two
causes. It may be due (1) to faulty division, to want of unity in
the _fundamentum divisionis_; (2) to the indistinct character of the
objects to be defined.
(1) Unless the division is based upon a single ground, unless each
species is based upon some mode of the generic character, confusion is
almost certain to arise. Suppose the field to be divided, the objects
to be classified, are three-sided rectilineal plane figures, each
group must be based upon some modification of the three sides. Divide
them into equilateral, isosceles, and scalene according as the three
sides are all of equal length, or two of equal length, or each of
different length, and you have a perfect division. Similarly you can
divide them perfectly according to the character of the angles into
acute-angled, right-angled and obtuse-angled. But if you do not keep
to a single basis, as in dividing them into equilateral, isosceles,
scalene, and right-angled, you have a cross-division. The same
triangle might be both right-angled and isosceles.
(2) Overlapping, however, may be unavoidable in practice owing to
the nature of the objects. There may be objects in which the dividing
characters are not distinctly marked, objects that possess the
differentiæ of more than one group in a greater or less degree. Things
are not always marked off from one another by hard and fast lines.
They shade into one another by imperceptible gradations. A clear
separation of them may be impossible. In that case you must allow a
certain indeterminate margin between your classes, and sometimes it
may be necessary to put an object into more than one class.
To insist that there is no essential difference unless a clear
demarcation can be made is a fallacy. A sophistical trick called the
_Sorites_ or Heap from the classical example of it was based upon this
difficulty of drawing sharp lines of definition. Assuming that it is
possible to say how many stones constitute a heap, you begin by asking
whether three stones form a heap. If your respondent says No, you
ask whether four stones form a heap, then five, and so on and he is
puzzled to say when the addition of a single stone makes that a heap
which was not a heap before. Or you may begin by asking whether twenty
stones form a heap, then nineteen, then eighteen, and so on, the
difficulty being to say when what was a heap ceases to be so.
Where the objects classified are mixed states or affections, the
products of interacting factors, or differently interlaced or
interfused growths from common roots, as in the case of virtues, or
emotions, or literary qualities, sharp demarcations are impossible.
To distinguish between wit and humour, or humour and pathos, or pathos
and sublimity is difficult because the same composition may partake
of more than one character. The specific characters cannot be made
rigidly exclusive one of another.
Even in the natural sciences, where the individuals are concrete
objects of perception, it may be difficult to decide in which of
two opposed groups an object should be included. Sydney Smith has
commemorated the perplexities of Naturalists over the newly
discovered animals and plants of Botany Bay, in especial with the
_Ornithorynchus_,--"a quadruped as big as a large cat, with the
eyes, colour, and skin of a mole, and the bill and web-feet of a
duck--puzzling Dr. Shaw, and rendering the latter half of his life
miserable, from his utter inability to determine whether it was a bird
or a beast".
III. The classes in any scheme of division should be of
co-ordinate rank.
The classes may be mutually exclusive, and yet the division imperfect,
owing to their not being of equal rank. Thus in the ordinary division
of the Parts of Speech, parts, that is, of a sentence, Prepositions
and Conjunctions are not co-ordinate in respect of function, which is
the basis of the division, with Nouns, Adjectives, Verbs, and Adverbs.
The preposition is a part of a phrase which serves the same function
as an adjective, _e.g._, _royal_ army, army _of the king_; it is thus
functionally part of a part, or a particle. So with the conjunction:
it also is a part of a part, _i.e._, part of a clause serving the
function of adjective or adverb.
IV. The basis of division (_fundamentum divisionis_) should be
an attribute admitting of important differences.
The importance of the attribute chosen as basis may vary with the
purpose of the division. An attribute that is of no importance in one
division, may be important enough to be the basis of another division.
Thus in a division of houses according to their architectural
attributes, the number of windows or the rent is of little importance;
but if houses are taxed or rated according to the number of windows
or the rent, these attributes become important enough to be a basis
of division for purposes of taxation or rating. They then admit of
important differences.
That the importance is relative to the purpose of the division should
be borne in mind because there is a tendency to regard attributes that
are of importance in any familiar or pre-eminent division as if they
had an absolute importance. In short, disregard of this relativity is
a fallacy to be guarded against.
In the sciences, the purpose being the attainment and preservation
of knowledge, the objects of study are divided so as to serve that
purpose. Groups must be formed so as to bring together the objects
that have most in common. The question, Who are to be placed together?
in any arrangement for purposes of study, receives the same answer
for individuals and for classes that have to be grouped into higher
classes, namely, Those that have most in common. This is what Dr. Bain
happily calls "the golden rule" of scientific classification: "Of the
various groupings of resembling things, preference is given to such as
have the greatest number of attributes in common". I slightly
modify Dr. Bain's statement: he says "the most numerous and the most
important attributes in common". But for scientific purposes number of
attributes constitutes importance, as is well recognised by Dr. Fowler
when he says that the test of importance in an attribute proposed as a
basis of classification is the number of other attributes of which it
is an index or invariable accompaniment. Thus in Zoology the
squirrel, the rat, and the beaver are classed together as Rodents, the
difference between their teeth and the teeth of other Mammalia being
the basis of division, because the difference in teeth is accompanied
by differences in many other properties. So the hedge-hog, the
shrew-mouse, and the mole, though very unlike in outward appearance
and habits, are classed together as Insectivora, the difference in
what they feed on being accompanied by a number of other differences.
_The Principles of_ DEFINITION. The word "definition" as used in Logic
shows the usual tendency of words to wander from a strict meaning and
become ambiguous. Throughout most of its uses it retains this much of
a common signification, the fixing or determining of the boundaries
of a class[3] by making clear its constituent attributes. Now in this
making clear two processes may be distinguished, a material process
and a verbal process. We have (1) the clearing up of the common
attributes by a careful examination of the objects included in the
class: and we have (2) the statement of these common attributes in
language. The rules of definition given by Dr. Bain, who devotes a
separate Book in his Logic to the subject of Definition, concern the
first of these processes: the rules more commonly given concern mainly
the second.
One eminent merit in Dr. Bain's treatment is that it recognises the
close connexion between Definition and Classification. His cardinal
rules are reduced to two.
I. _Assemble for comparison representative individuals of the
class._
II. _Assemble for comparison representative individuals of the
contrasted class or classes._
Seeing that the contrasted classes are contrasted on some basis of
division, this is in effect to recognise that you cannot clearly
define any class except in a scheme of classification. You must have
a wide _genus_ with its _fundamentum divisionis_, and, within this,
_species_ distinguished by their several _differentiæ_.
Next, as to the verbal process, rules are commonly laid down mostly
of a trifling and obvious character. That "a definition should state
neither more nor less than the common attributes of the class," or
than the attributes signified by the class-name, is sometimes given
as a rule of definition. This is really an explanation of what a
definition is, a definition of a definition. And as far as mere
statement goes it is not strictly accurate, for when the attributes
of a genus are known it is not necessary to give all the attributes of
the species, which include the generic attributes as well, but it is
sufficient to give the generic name and the differentia. Thus
Poetry may be defined as "a Fine Art having metrical language as its
instrument". This is technically known as definition _per genus
et differentiam_. This mode of statement is a recognition of the
connexion between Definition and Division.
The rule that "a definition should not be a synonymous repetition of
the name of the class to be defined," is too obvious to require formal
statement. To describe a Viceroy as a man who exercises viceregal
functions, may have point as an epigram in the case of a _faineant_
viceroy, but it is not a definition.
So with the rule that "a definition should not be couched in ambiguous
unfamiliar, or figurative language". To call the camel "the ship of
the desert" is a suggestive and luminous description of a property,
but it is not a definition. So with the noble description of Faith as
"the substance of things hoped for, the evidence of things not seen".
But if one wonders why so obvious a "rule" should be laid down, the
answer is that it has its historical origin in the caprices of
two classes of offenders, mystical philosophers and pompous
lexicographers.[4]
That "the definition should be simply convertible with the term for
the class defined," so that we may say, for example, either: "Wine is
the juice of the grape," or, "The juice of the grape is wine," is an
obvious corollary from the nature of definition, but should hardly be
dignified with the name of a "rule".
_The Principles of_ NAMING. Rules have been formulated for the choice
of names in scientific definition and classification, but it may be
doubted whether such choice can be reduced to precise rule. It is
enough to draw attention to certain considerations obvious enough on
reflection.
We may take for granted that there should be distinct names for every
defining attribute (a _Terminology_) and for every group or class (a
_Nomenclature_). What about the selection of the names? Suppose an
investigator is struck with likenesses and differences that seem to
him important enough to be the basis of a new division, how should he
be guided in his choice of names for the new groups that he proposes?
Should he coin new names, or should he take old names and try to fit
them with new definitions?
The balance of advantages is probably in favour of Dr. Whewell's
dictum that "in framing scientific terms, the appropriation of old
words is preferable to the invention of new ones". Only care must be
taken to keep as close as possible to the current meaning of the
old word, and not to run counter to strong associations. This is
an obvious precept with a view to avoiding confusion. Suppose, for
example, that in dividing Governments you take the distribution of
political power as your basis of division and come to the conclusion
that the most important differences are whether this power is vested
in a few or in the majority of the community. You want names to
fix this broad division. You decide instead of coining the new word
_Pollarchy_ to express the opposite of _Oligarchy_ to use the old
words _Republic_ and _Oligarchy_. You would find, as Sir George
Cornewall Lewis found, that however carefully you defined the word
Republic, a division under which the British Government had to be
ranked among Republics would not be generally understood and accepted.
Using the word in the sense explained above, Mr. Bagehot maintained
that the constitution of Great Britain was more Republican than that
of the United States, but his meaning was not taken except by a few.
The difficulty in choosing between new words and old words to express
new meanings is hardly felt in the exact sciences. It is at least at
a minimum. The innovator may encounter violent prejudice, but, arguing
with experts, he can at least make sure of being understood, if his
new division is based upon real and important differences. But in
other subjects the difficulty of transmitting truth or of expressing
it in language suited for precise transmission, is almost greater than
the difficulty of arriving at truth. Between new names and old
names redefined, the possessor of fresh knowledge, assuming it to be
perfectly verified, is in a quandary. The objects with which he deals
are already named in accordance with loose divisions resting on strong
prejudices. The names in current use are absolutely incapable of
conveying his meaning. He must redefine them if he is to use them. But
in that case he runs the risk of being misunderstood from people being
too impatient to master his redefinition. His right to redefine may
even be challenged without any reference to the facts to be expressed:
he may simply be accused of circulating false linguistic coin, of
debasing the verbal currency. The other alternative open to him is
to coin new words. In that case he runs the risk of not being read at
all. His contribution to verified knowledge is passed by as pedantic
and unintelligible. There is no simple rule of safety: between Scylla
and Charybdis the mariner must steer as best he may. Practically the
advantage lies with old words redefined, because thereby discussion is
provoked and discussion clears the air.
Whether it is best to attempt a formal definition or to use words in a
private, peculiar, or esoteric sense, and leave this to be gathered by
the reader from the general tenor of your utterances, is a question of
policy outside the limits of Logic. It is for the logician to expound
the method of Definition and the conditions of its application: how
far there are subjects that do not admit of its application profitably
must be decided on other grounds. But it is probably true that no
man who declines to be bound by a formal definition of his terms is
capable of carrying them in a clear unambiguous sense through a heated
controversy.
[Footnote 1: Except, perhaps, in new offices to which the
name is extended, such as _Clerk_ of School Board. The name,
bearing its most simple and common meaning, may cause popular
misapprehension of the nature of the duties. Any uncertainty
in meaning may be dangerous in practice: elections have been
affected by the ambiguity of this word.]
[Footnote 2: Sidgwick's _Political Economy_, pp. 52-3. Ed.
1883.]
[Footnote 3: Some logicians, however, speak of defining a
thing, and illustrate this as if by a thing they meant a
concrete individual, the realistic treatment of Universals
lending itself to such expressions. But though the authority
of Aristotle might be claimed for this, it is better to
confine the name in strictness to the main process of defining
a class. Since, however, the method is the same whether it is
an individual or a class that we want to make distinct, there
is no harm in the extension of the word definition to both
varieties. See Davidson's _Logic of Definition_, chap. ii.]
[Footnote 4: See Davidson's _Logic of Definition_, chap. iii.]
CHAPTER II.
THE FIVE PREDICABLES.--VERBAL AND REAL PREDICATION.
We give a separate chapter to this topic out of respect for the space
that it occupies in the history of Logic. But except as an exercise in
subtle distinction for its own sake, all that falls to be said about
the Predicables might be given as a simple appendix to the chapter on
Definition.
Primarily, the Five Predicables or Heads of Predicables--Genus,
Species, Differentia, Proprium, and Accidens--are not predicables at
all, but merely a list or enumeration of terms used in dividing and
defining on the basis of attributes. They have no meaning except in
connexion with a fixed scheme of division. Given such a scheme, and
we can distinguish in it the whole to be divided (the _genus_), the
subordinate divisions (the _species_), the attribute or combination of
attributes on which each species is constituted (the _differentia_),
and other attributes, which belong to some or all of the individuals
but are not reckoned for purposes of division and definition
(_Propria_ and _Accidentia_). The list is not itself a logical
division: it is heterogeneous, not homogeneous; the two first are
names of classes, the three last of attributes. But corresponding to
it we might make a homogeneous division of attributes, as follows:--
Attributes
___________|____________
| |
Defining Non-defining
_____|______ ____|__________
| | | |
Generic Specific Proprium Accidens
(Differentia)
The origin of the title Predicables as applied to these five terms is
curious, and may be worth noting as an instance of the difficulty of
keeping names precise, and of the confusion arising from forgetting
the purpose of a name. Porphyry in his [Greek: eisagôgê] or
Introduction explains the five words ([Greek: phônai]) simply as terms
that it is useful for various purposes to know, expressly mentioning
definition and division. But he casually remarks that Singular names,
"This man," "Socrates," can be predicated only of one individual,
whereas _Genera_, _Species_, _Differentiæ_, etc., are predicables
of many. That is to say he describes them as Predicables simply by
contradistinction from Singular names. A name, however, was wanted for
the five terms taken all together, and since they are not a logical
division, but merely a list of terms used in dividing and defining,
there was no apt general designation for them such as would occur
spontaneously. Thus it became the custom to refer to them as
the Predicables, a means of reference to them collectively being
desiderated, while the meaning of this descriptive title was
forgotten.
To call the five divisional elements or _Divisoria_ Predicables is to
present them as a division of Predicate Terms on the basis of their
relation to the Subject Term: to suggest that every Predicate Term
must be either a Genus or a Species, or a Differentia, or a Proprium,
or an Accidens of the Subject Term. They are sometimes criticised
as such, and it is rightly pointed out that the Predicate is never a
species of or with reference to the Subject. But, in truth, the five
so-called Predicables were never meant as a division of predicates
in relation to the subject: it is only the title that makes this
misleading suggestion.
To complete the confusion it so happens that Aristotle used three of
the Five terms in what was virtually a division of Predicates inasmuch
as it was a division of Problems or Questions. In expounding the
methods of Dialectic in the Topica he divided Problems into four
classes according to the relation of the Predicate to the Subject. The
Predicate must either be simply convertible with the subject or not.
If simply convertible, the two must be coextensive, and the
Predicate must be either a Proprium or the Definition. If not simply
convertible, the Predicate must either be part of the Definition or
not. If part of the Definition it must be either a Generic Property
or a Differentia (both of which in this connexion Aristotle includes
under Genus): if not part of the Definition, it is an Accident.
Aristotle thus arrives at a fourfold division of Problems or
Predicates:--[Greek: genos] (_Genus_, including _Differentia_,
[Greek: diaphora]); [Greek: horos] (Definition); [Greek: to idion]
(_Proprium_); and [Greek: to symbebêkos] (_Accidens_). The object of
it was to provide a basis for his systematic exposition; each of the
four kinds admitted of differences in dialectic method. For us it is
a matter of simple curiosity and ingenuity. It serves as a monument
of how much Greek dialectic turned on Definition, and it corresponds
exactly to the division of attributes into Defining and Non-defining
given above. It is sometimes said that Aristotle showed a more
scientific mind than Porphyry in making the Predicables four instead
of five. This is true if Porphyry's list had been meant as a division
of attributes: but it was not so meant.
The distinction between VERBAL or ANALYTIC and REAL or SYNTHETIC
Predication corresponds to the distinction between Defining and
Non-defining attributes, and also has no significance except with
reference to some scheme of Division, scientific and precise or loose
and popular.
When a proposition predicates of a subject something contained in the
full notion, concept, or definition of the subject term, it is called
Verbal, Analytic, or Explicative: _verbal_, inasmuch as it merely
explains the meaning of a name; _explicative_ for the same reason;
_analytic_, inasmuch as it unties the bundle of attributes held
together in the concept and pays out one, or all one by one.
When the attributes of the Predicate are not contained in the concept
of the Subject, the proposition is called _Real_, _Synthetic_, or
_Ampliative_, for parallel reasons.
Thus: "A triangle is a three-sided rectilinear figure" is Verbal or
Analytic; "Triangles have three angles together equal to two right
angles," or "Triangles are studied in schools," is Real or Synthetic.
According to this distinction, predications of the whole Definition
or of a Generic attribute or of a Specific attribute are Verbal;
predications of Accident are Real. A nice point is whether Propria are
Verbal or Real. They can hardly be classed with Verbal, inasmuch as
one may know the full meaning of the name without knowing them: but
it might be argued that they are Analytic, inasmuch as they are
implicitly contained in the defining attributes as being deducible
from them.
Observe, however, that the whole distinction is really valid only
in relation to some fixed or accepted scheme of classification or
division. Otherwise, what is Verbal or Analytic to one man may be
Real or Synthetic to another. It might even be argued that every
proposition is Analytic to the man who utters it and Synthetic to the
man who receives it. We must make some analysis of a whole of thought
before paying it out in words: and in the process of apprehending the
meaning of what we hear or read we must add the other members of the
sentence on to the subject. Whether or not this is super-subtle, it
clearly holds good that what is Verbal (in the sense defined) to the
learned man of science may be Real to the learner. That the horse has
six incisors in each jaw or that the domestic dog has a curly tail,
is a Verbal Proposition to the Natural Historian, a mere exposition
of defining marks; but the plain man has a notion of horse or dog into
which this defining attribute does not enter, and to him accordingly
the proposition is Real.
But what of propositions that the plain man would at once recognise as
Verbal? Charles Lamb, for example, remarks that the statement that "a
good name shows the estimation in which a man is held in the world" is
a verbal proposition. Where is the fixed scheme of division there?
The answer is that by a fixed scheme of division we do not necessarily
mean a scheme that is rigidly, definitely and precisely fixed. To make
such schemes is the business of Science. But the ordinary vocabulary
of common intercourse as a matter of fact proceeds upon schemes
of division, though the names used in common speech are not always
scientifically accurate, not always the best that could be devised for
the easy acquisition and sure transmission of thorough knowledge. The
plain man's vocabulary, though often twisted aside by such causes
as we have specified, is roughly moulded on the most marked
distinguishing attributes of things. This was practically recognised
by Aristotle when he made one of his modes of definition consist in
something like what we have called verifying the meaning of a name,
ascertaining the attributes that it signifies in common speech or in
the speech of sensible men. This is to ascertain the essence, [Greek:
ousia], or _Substantia_, of things, the most salient attributes that
strike the common eye either at once or after the closer inspection
that comes of long companionship, and form the basis of the ordinary
vocabulary. "Properly speaking," Mansel says,[1] "All Definition is an
inquiry into _Attributes_. Our complex notions of Substances can only
be resolved into various Attributes, with the addition of an unknown
_substratum_: a something to which we are compelled to regard
these attributes as belonging. _Man_, for example, is analysed
into Animality, Rationality, and the something which exhibits these
phenomena. Pursue the analysis and the result is the same. We have a
something corporeal, animated, sensible, rational. An unknown constant
must always be added to complete the integration." This "unknown
constant" was what Locke called the _Real_ Essence, as distinguished
from the _Nominal_ Essence, or complex of attributes. It is upon this
nominal essence, upon divisions of things according to attributes,
that common speech rests, and if it involves many cross-divisions,
this is because the divisions have been made for limited and
conflicting purposes.
[Footnote 1: Aldrich's Compendium, Appendix, Note C. The
reader may be referred to Mansel's Notes A and C for valuable
historical notices of the Predicables and Definition.]
CHAPTER III.
ARISTOTLE'S CATEGORIES.
In deference to tradition a place must be found in every logical
treatise for Aristotle's Categories. No writing of the same length
has exercised a tithe of its influence on human thought. It governed
scholastic thought and expression for many centuries, being from its
shortness and consequent easiness of transcription one of the
few books in every educated man's library. It still regulates the
subdivisions of Parts of Speech in our grammars. Its universality of
acceptance is shown in the fact that the words _category_ ([Greek:
katêgoria]) and _predicament_, its Latin translation, have passed into
common speech.
The Categories have been much criticised and often condemned as a
division, but, strange to say, few have inquired what they originally
professed to be a division of, or what was the original author's
basis of division. Whether the basis is itself important, is another
question: but to call the division imperfect, without reference to the
author's intention, is merely confusing, and serves only to illustrate
the fact that the same objects may be differently divided on different
principles of division. Ramus was right in saying that the Categories
had no logical significance, inasmuch as they could not be made a
basis for departments of logical method; and Kant and Mill in saying
that they had no philosophical significance, inasmuch as they are
founded upon no theory of Knowing and Being: but this is to condemn
them for not being what they were never intended to be.
The sentence in which Aristotle states the objects to be divided, and
his division of them is so brief and bold that bearing in mind the
subsequent history of the Categories, one first comes upon it with a
certain surprise. He says simply:--
"Of things expressed without syntax (_i.e._, single words), each
signifies either substance, or quantity, or quality, or relation,
or place, or time, or disposition (_i.e._, attitude or internal
arrangement), or appurtenance, or action (doing), or suffering (being
done to)."[1]
The objects, then, that Aristotle proposed to classify were single
words (the _themata simplicia_ of the Schoolmen). He explains that by
"out of syntax" ([Greek: aneu symplokês]) he means without reference
to truth or falsehood: there can be no declaration of truth or
falsehood without a sentence, a combination, or syntax: "man runs" is
either true or false, "man" by itself, "runs" by itself, is neither.
His division, therefore, was a division of single words according to
their differences of signification, and without reference to the truth
or falsehood of their predication.[2]
Signification was thus the basis of division. But according to what
differences? The Categories themselves are so abstract that this
question might be discussed on their bare titles interminably. But
often when abstract terms are doubtful, an author's intention may be
gathered from his examples. And when Aristotle's examples are ranged
in a table, certain principles of subdivision leap to the eyes.
Thus:--
Substance Man } COMMON { Substance
([Greek: ousia]) ([Greek: anthrôpos]) } NOUN {
(_Substantia_)
--------------------------------------------------------------
Quantity Five-feet-five } {
([Greek: poson]) ([Greek: tripêchu]) } {
(_Quantitas_) } {
Quality Scholarly } { Permanent
([Greek: poion]) ([Greek: grammatikon])} ADJECTIVE { Attribute
(_Qualitas_) } {
Relation Bigger } {
([Greek: pros ti]) ([Greek: meizon]) } {
(_Relatio_) } {
--------------------------------------------------------------
Place In-the-Lyceum } {
([Greek: pou]) ([Greek: en Lykeiô]) } {
(_Ubi_) } ADVERB { Temporary
Time Yesterday } { Attribute
([Greek: pote]) ([Greek: chthes]) } {
(_Quando_) } {
--------------------------------------------------------------
Disposition Reclines } {
([Greek: keisthai]) ([Greek: anakeitai]) } {
(_Positio_) } {
Appurtenance Has-shoes-on } {
([Greek: echein]) ([Greek: hypodedetai])} {
(_Habitus_) } VERB {
Action Cuts } { Temporary
([Greek: poiein]) ([Greek: temnei]) } { Attribute
(_Actio_) } {
Passion Is cut } {
([Greek: paschein]) ([Greek: temnetai]) } {
(_Passio_) } {
In looking at the examples, our first impression is that Aristotle has
fallen into a confusion. He professes to classify words out of syntax,
yet he gives words with the marks of syntax on them. Thus his division
is accidentally grammatical, a division of parts of speech, parts of
a sentence, into Nouns, Adjectives, Adverbs, and Verbs. And his
subdivisions of these parts are still followed in our grammars. But
really it is not the grammatical function that he attends to, but
the signification: and looking further at the examples, we see what
differences of signification he had in his mind. It is differences
relative to a concrete individual, differences in the words applied to
him according as they signify the substance of him or his attributes,
permanent or temporary.
Take any concrete thing, Socrates, this book, this table. It must
be some kind of a thing, a man, a book. It must have some size
or quantity, six feet high, three inches broad. It must have some
quality, white, learned, hard. It must have relations with other
things, half this, double that, the son of a father. It must be
somewhere, at some time, in some attitude, with some "havings,"
appendages, appurtenances, or belongings, doing something, or having
something done to it. Can you conceive any name (simple or composite)
applicable to any object of perception, whose signification does not
fall into one or other of these classes? If you cannot, the categories
are justified as an exhaustive division of significations. They are
a complete list of the most general resemblances among individual
things, in other words, of the _summa genera_, the _genera
generalissima_ of predicates concerning this, that or the other
concrete individual. No individual thing is _sui generis_: everything
is like other things: the categories are the most general likenesses.
The categories are exhaustive, but do they fulfil another requisite of
a good division--are they mutually exclusive? Aristotle himself
raised this question, and some of his answers to difficulties are
instructive. Particularly his discussion of the distinction between
Second Substances or Essences and Qualities. Here he approximates to
the modern doctrine of the distinction between Substance and Attribute
as set forth in our quotation from Mansel at p. 110. Aristotle's
Second Essences ([Greek: deuterai ousiai]) are common nouns or general
names, Species and Genera, _man_, _horse_, _animal_, as distinguished
from Singular names, _this man_, _this horse_, which he calls First
Substances ([Greek: prôtai ousiai]), essences _par excellence_, to
which real existence in the highest sense is attributed. Common nouns
are put in the First Category because they are predicated in answer to
the question, What is this? But he raises the difficulty whether they
may not rather be regarded as being in the Third Category, that of
Quality ([Greek: to poion]). When we say, "This is a man," do we not
declare what sort of a thing he is? do we not declare his Quality? If
Aristotle had gone farther along this line, he would have arrived
at the modern point of view that a man is a man in virtue of his
possessing certain attributes, that general names are applied in
virtue of their connotation. This would have been to make the line
of distinction between the First Category and the Third pass between
First Essence and Second, ranking the Second Essences with Qualities.
But Aristotle did not get out of the difficulty in this way. He solved
it by falling back on the differences in common speech. "Man" does not
signify the quality simply, as "whiteness" does. "Whiteness" signifies
nothing but the quality. That is to say, there is no separate name in
common speech for the common attributes of man. His further obscure
remark that general names "define quality round essence" [Greek: peri
ousian], inasmuch as they signify what sort a certain essence is, and
that genera make this definition more widely than species, bore fruit
in the mediæval discussions between Realists and Nominalists by which
the signification of general names was cleared up.
Another difficulty about the mutual exclusiveness of the Categories
was started by Aristotle in connexion with the Fourth Category,
Relation ([Greek: pros ti] _Ad aliquid_, _To something_). Mill remarks
that "that could not be a very comprehensive view of the nature of
Relation which would exclude action, passivity, and local situation
from that Category," and many commentators, from Simplicius down to
Hamilton, have remarked that all the last six Categories might be
included under Relation. This is so far correct that the word Relation
is one of the vaguest and most extensive of words; but the criticism
ignores the strictness with which Aristotle confined himself in
his Categories to the forms of common speech. It is clear from
his examples that in his Fourth Category he was thinking only of
"relation" as definitely expressed in common speech. In his meaning,
any word is a relative which is joined with another in a sentence by
means of a preposition or a case-inflection. Thus "disposition" is a
relative: it is the disposition _of_ something. This kind of relation
is perfect when the related terms reciprocate grammatically; thus
"master," "servant," since we can say either "the master of the
servant," or "the servant of the master". In mediæval logic the term
_Relata_ was confined to these perfect cases, but the Category had
a wider scope with Aristotle. And he expressly raised the question
whether a word might not have as much right to be put in another
Category as in this. Indeed, he went further than his critics in his
suggestions of what Relation might be made to include. Thus: "big"
signifies Quality; yet a thing is big with reference to something
else, and is so far a Relative. Knowledge must be knowledge of
something, and is a relative: why then should we put "knowing"
(_i.e._, learned) in the Category of Quality. "Hope" is a relative,
as being the hope _of_ a man and the hope of something. Yet we say,
"I have hope," and there hope would be in the category of Having,
Appurtenance. For the solution of all such difficulties, Aristotle
falls back upon the forms of common speech, and decides the place of
words in his categories according to them. This was hardly consistent
with his proposal to deal with separate words out of syntax, if by
this was meant anything more than dealing with them without reference
to truth or falsehood. He did not and could not succeed in dealing
with separate words otherwise than as parts of sentences, owing their
signification to their position as parts of a transient plexus of
thought. In so far as words have their being in common speech, and
it is their being in this sense that Aristotle considers in the
Categories, it is a transient being. What being they represent besides
is, in the words of Porphyry, a very deep affair, and one that needs
other and greater investigation.
[Footnote 1: [Greek: tôn kata mêdemian symplokên legomenôn
hekaston êtoi ousian sêmainei, ê poson, ê poion, ê pros ti,
ê pou, ê pote, ê keisthai, ê echein, ê poiein, ê paschein.]
(Categ. ii. 5.)]
[Footnote 2: To describe the Categories as a grammatical
division, as Mansel does in his instructive Appendix C to
Aldrich, is a little misleading without a qualification.
They are non-logical inasmuch as they have no bearing on any
logical purpose. But they are grammatical only in so far as
they are concerned with words. They are not grammatical in
the sense of being concerned with the function of words
in predication. The unit of grammar in this sense is the
sentence, a combination of words in syntax; and it is
expressly with words out of syntax that Aristotle deals, with
single words not in relation to the other parts of a sentence,
but in relation to the things signified. In any strict
definition of the provinces of Grammar and Logic, the
Categories are neither grammatical nor logical: the
grammarians have appropriated them for the subdivision of
certain parts of the sentence, but with no more right than the
logicians. They really form a treatise by themselves, which
is in the main ontological, a discussion of substances and
attributes as underlying the forms of common speech. In saying
this I use the word substance in the modern sense: but it
must be remembered that Aristotle's [Greek: ousia], translated
substantia, covered the word as well as the thing signified,
and that his Categories are primarily classes of words. The
union between names and things would seem to have been closer
in the Greek mind than we can now realise. To get at it we
must note that every separate word [Greek: to legomenon]
is conceived as having a being or thing [Greek: to on]
corresponding to it, so that beings or things [Greek: ta
onta] are coextensive with single words: a being or thing is
whatever receives a separate name. This is clear and simple
enough, but perplexity begins when we try to distinguish
between this nameable being and concrete being, which last is
Aristotle's category of [Greek: ousia], the being signified by
a Proper or a Common as distinguished from an Abstract Noun.
As we shall see, it is relatively to the highest sense of this
last kind of being, namely, the being signified by a Proper
name, that he considers the other kinds of being.]
CHAPTER IV.
THE CONTROVERSY ABOUT UNIVERSALS.--DIFFICULTIES CONCERNING THE
RELATION OF GENERAL NAMES TO THOUGHT AND TO REALITY.
In the opening sentences of his Isagoge, before giving his simple
explanation of the Five Predicables, Porphyry mentions certain
questions concerning Genera and Species, which he passes over as being
too difficult for the beginner. "Concerning genera and species,"
he says, "the question whether they subsist (_i.e._, have real
substance), or whether they lie in the mere thoughts only, or whether,
granting them to subsist, they are corporeal or incorporeal, or
whether they subsist apart, or in sensible things and cohering round
them--this I shall pass over, such a question being a very deep affair
and one that needs other and greater investigation."
This passage, written about the end of the third century, A.D., is a
kind of isthmus between Greek Philosophy and Mediæval: it summarises
questions which had been turned over on every side and most
intricately discussed by Plato and Aristotle and their successors,
and the bald summary became a starting-point for equally intricate
discussions among the Schoolmen, among whom every conceivable variety
of doctrine found champions. The dispute became known as the dispute
about Universals, and three ultra-typical forms of doctrine
were developed, known respectively as Realism, Nominalism, and
Conceptualism. Undoubtedly the dispute, with all its waste of
ingenuity, had a clearing effect, and we may fairly try now what
Porphyry shrank from, to gather some simple results for the better
understanding of general names and their relations to thoughts and to
things. The rival schools had each some aspect of the general name in
view, which their exaggeration served to render more distinct.
What does a general name signify? For logical purposes it is
sufficient to answer--the points of resemblance as grasped in
the mind, fixed by a name applicable to each of the resembling
individuals. This is the signification of the general name
_logically_, its connotation or concept, the identical element of
objective reference in all uses of a general name.
But other questions may be asked that cannot be so simply answered.
What is this concept in thought? What is there in our minds
corresponding to the general name when we utter it? How is its
signification conceived? What is the signification _psychologically_?
We may ask, further, What is there in nature that the general name
signifies? What is its relation to reality? What corresponds to it in
the real world? Has the unity that it represents among individuals
no existence except in the mind? Calling this unity, this one in
the many, the Universal (_Universale_, [Greek: to pan]), what is the
Universal _ontologically_?
It was this ontological question that was so hotly and bewilderingly
debated among the Schoolmen. Before giving the ultra-typical answers
to it, it may be well to note how this question was mixed up with
still other questions of Theology and Cosmogony. Recognising that
there is a unity signified by the general name, we may go on to
inquire into the ground of the unity. Why are things essentially like
one another? How is the unity maintained? How is it continued? Where
does the common pattern come from? The question of the nature of
the Universal thus links itself with metaphysical theories of the
construction of the world, or even with the Darwinian theory of the
origin of species.
Passing by these remoter questions, we may give the answers of
the three extreme schools to the ontological question, What is a
Universal?
The answer of the Ultra-Realists, broadly put, was that a Universal is
a substance having an independent existence in nature.
Of the Ultra-Nominalists, that the Universal is a name and nothing
else, _vox et præterea nihil_; that this name is the only unity among
the individuals of a species, all that they have in common.
Of the Ultra-Conceptualists, that the individuals have more in common
than the name, that they have the name plus the meaning,
_vox_ + _significatio_, but that the Universals, the genera and
species, exist only in the mind.
Now these extreme doctrines, as literally interpreted by opponents,
are so easily refuted and so manifestly untenable, that it may be
doubted whether they were ever held by any thinker, and therefore I
call them Ultra-Realism, Ultra-Nominalism, and Ultra-Conceptualism.
They are mere exaggerations or caricatures, set up by opponents
because they can be easily knocked down.
To the Ultra-Realists, it is sufficient to say that if there existed
anywhere a substance having all the common attributes of a species
and only these, having none of the attributes peculiar to any of the
individuals of that species, corresponding to the general name as an
individual corresponds to a Proper or Singular name, it would not be
the Universal, the unity pervading the individuals, but only another
individual.
To the Ultra-Nominalists, it is sufficient to say that the individuals
must have more in common than the name, because the name is not
applied arbitrarily, but on some ground. The individuals must have in
common that on account of which they receive the common name: to call
them by the same name is not to make them of the same species.
To the Ultra-Conceptualists, it is sufficient to say that when we
employ a general name, as when we say "Socrates is a man," we do
not refer to any passing thought or state of mind, but to certain
attributes independent of what is passing in our minds. We cannot make
a thing of this or that species by merely thinking of it as such.
The ultra-forms of these doctrines are thus easily shown to
be inadequate, yet each of the three, Realism, Nominalism, and
Conceptualism, represents a phase of the whole truth.
Thus, take Realism. Although it is not true that there is anything
in reality corresponding to the general name such as there is
corresponding to the singular name, the general name merely signifying
attributes of what the singular name signifies, it does not follow, as
the opponents of Ultra-Realism hastily assume, that there is nothing
in the real world corresponding to the general name. Three senses may
be particularised in which Realism is justified.
(1) The points of resemblance from which the concept is formed are as
real as the individuals themselves. It is true in a sense that it
is our thought that gives unity to the individuals of a class, that
gathers the many into one, and so far the Conceptualists are right.
Still we should not gather them into one if they did not resemble one
another: that is the reason why we think of them together: and the
respects in which they resemble one another are as much independent of
us and our thinking as the individuals themselves, as much beyond the
power of our thought to change. We must go behind the activity of the
mind in unifying to the reason for the unification: and the ground of
unity is found in what really exists. We do not confer the unity:
we do not make all men or all dogs alike: we find them so. The curly
tails in a thousand domestic dogs, which serve to distinguish them
from wolves and foxes, are as real as the thousand individual domestic
dogs. In this sense the Aristotelian doctrine, _Universalia in re_,
expresses a plain truth.
(2) The Platonic doctrine, formulated by the Schoolmen as _Universalia
ante rem_, has also a plain validity. Individuals come and go, but
the type, the Universal, is more abiding. Men are born and die: man
remains throughout. The snows of last year have vanished, but snow is
still a reality to be faced. Wisdom does not perish with the wise
men of any generation. In this plain sense, at least, it is true that
Universals exist before Individuals, have a greater permanence, or, if
we like to say so, a higher, as it is a more enduring, reality.
(3) Further, the "idea," concept, or universal, though it cannot be
separated from the individual, and whether or not we ascribe to it
the separate suprasensual existence of the archetypal forms of Plato's
poetical fancy, is a very potent factor in the real world. Ideals of
conduct, of manners, of art, of policy, have a traditional life: they
do not pass away with the individuals in whom they have existed,
in whom they are temporarily materialised: they survive as potent
influences from age to age. The "idea" of Chaucer's Man of Law,
who always "seemed busier than he was," is still with us. Mediæval
conceptions of chivalry still govern conduct. The Universal enters
into the Individual, takes possession of him, makes of him its
temporary manifestation.
Nevertheless, the Nominalists are right in insisting on the importance
of names. What we call the real world is a common object of perception
and knowledge to you and me: we cannot arrive at a knowledge of it
without some means of communication with one another: our means of
communication is language. It may be doubted whether even thinking
could go far without symbols with the help of which conceptions may
be made definite. A concept cannot be explained without reference to
a symbol. There is even a sense in which the Ultra-Nominalist doctrine
that the individuals in a class have nothing in common but the name
is tenable. Denotability by the same name is the only respect in which
those individuals are absolutely identical: in this sense the name
alone is common to them, though it is applied in virtue of their
resemblance to one another.
Finally, the Conceptualists are right in insisting on the mind's
activity in connexion with general names. Genera and species are not
mere arbitrary subjective collections: the union is determined by the
characters of the things collected. Still it is with the concept in
each man's mind that the name is connected: it is by the activity of
thought in recognising likenesses and forming concepts that we are
able to master the diversity of our impressions, to introduce unity
into the manifold of sense, to reduce our various recollections to
order and coherence.
So much for the Ontological question. Now for the PSYCHOLOGICAL. What
is in the mind when we employ a general name? What is the Universal
psychologically? How is it conceived?
What breeds confusion in these subtle inquiries is the want of fixed
unambiguous names for the things to be distinguished. It is only by
means of such names that we can hold on to the distinctions, and
keep from puzzling ourselves. Now there are three things to be
distinguished in this inquiry, which we may call the Concept, the
Conception, and the Conceptual or Generic Image. Let us call them by
these names, and proceed to explain them.
By the Concept, I understand the meaning of the general name, what the
general name signifies: by the Conception, the mental act or state
of him who conceives this meaning. The concept of "triangle," _i.e._,
what you and I mean by the word, is not my act of mind or your act of
mind when we think or speak of a triangle. The Conception, which is
this act, is an event or incident in our mental history, a psychical
act or state, a distinct occurrence, a particular fact in time as much
as the battle of Waterloo. The concept is the objective reference of
the name, which is the same, or at least is understood to be the
same, every time we use it. I make a figure on paper with ink or on a
blackboard with chalk, and recognise or conceive it as a triangle: you
also conceive it as such: we do the same to-morrow: we did the same
yesterday: each act of conception is a different event, but the
concept is the same throughout.
Now the psychological question about the Universal is, What is this
conception? We cannot define it positively further than by saying that
it consists in realising the meaning of a general name: the act being
unique, we can only make it intelligible by producing an example of
it. But we may define it negatively by distinguishing it from the
conceptual image. Whenever we conceive anything, "man," "horse," there
is generally present to our minds an image of a man or horse, with
accidents of size, colour, position or other categories. But this
conceptual image is not the concept, and the mental act of forming it
is not conception.
This distinction between mental picturing or imaging and the
conception of common attributes is variously expressed. The
correlative terms _Intuitive_ and _Symbolical_ Thinking,
_Presentative_ and _Representative_ Knowledge have been employed.[1]
But whatever terms we use, the distinction itself is vital, and the
want of it leads to confusion.
Thus the fact that we cannot form a conceptual image composed
solely of common attributes has been used to support the argument of
Ultra-Nominalism, that the individuals classed under a common name
have nothing in common but the name. What the word "dog" signifies,
_i.e._, the "concept" of dog, is neither big nor little, neither black
nor tan, neither here nor there, neither Newfoundland, nor Retriever,
nor Terrier, nor Greyhound, nor Pug, nor Bulldog. The concept consists
only of the attributes common to all dogs apart from any that are
peculiar to any variety or any individual. Now we cannot form any such
conceptual image. Our conceptual image is always of some definite
size and shape. Therefore, it is argued, we cannot conceive what a dog
means, and dogs have nothing in common but the name. This, however,
does not follow. The concept is not the conceptual image, and
forming the image is not conception. We may even, as in the case of
a chiliagon, or thousand-sided figure, conceive the meaning without
being able to form any definite image.
How then, do we ordinarily proceed in conceiving, if we cannot picture
the common attributes alone and apart from particulars? We attend, or
strive to attend, only to those aspects of an image which it has in
common with the individual things denoted. And if we want to make our
conception definite, we pass in review an indefinite number of the
individuals, case after case.
A minor psychological question concerns the nature of the conceptual
image. Is it a copy of some particular impression, or a confused blur
or blend of many? Possibly neither: possibly it is something like
one of Mr. Galton's composite photographs, photographs produced by
exposing the same surface to the impressions of a number of different
photographs in succession. If the individuals are nearly alike,
the result is an image that is not an exact copy of any one of the
components and yet is perfectly distinct. Possibly the image that
comes into our mind's eye when we hear such a word as "horse" or "man"
is of this character, the result of the impressions of a number of
similar things, but not identical with any one. As, however, different
persons have different conceptual images of the same concept, so we
may have different conceptual images at different times. It is only
the concept that remains the same.
But how, it may be asked, can the concept remain the same? If the
universal or concept psychologically is an intellectual act, repeated
every time we conceive, what guarantee have we for the permanence
of the concept? Does this theory not do away with all possibility of
defining and fixing concepts?
This brings us back to the doctrine already laid down about the truth
of Realism. The theory of the concept is not exhausted when it is
viewed only psychologically, as a psychic act. If we would understand
it fully, we must consider the act in its relations to the real
experience of ourselves and others. To fix this act, we give it a
separate name, calling it the conception: and then we must go behind
the activity of the mind to the objects on which it is exercised.
The element of fixity is found in them. And here also the truth of
Nominalism comes in. By means of words we enter into communication
with other minds. It is thus that we discover what is real, and what
is merely personal to ourselves.
[Footnote 1: The only objection to these terms is that they
have slipped from their moorings in philosophical usage. Thus
instead of Leibnitz's use of Intuitive and Symbolical, which
corresponds to the above distinction between Imaging and
Conception, Mr. Jevons employs the terms to express a
distinction among conceptions proper. We can understand what
a chiliagon means, but we cannot form an image of it in our
minds, except in a very confused and imperfect way; whereas we
can form a distinct image of a triangle. Mr. Jevons would call
the conception of the triangle _Intuitive_, of the chiliagon
_Symbolical_.
Again, while Mansel uses the words Presentative and
Representative to express our distinction, a more common
usage is to call actual Perception Presentative Knowledge, and
ideation or recollection in idea Representative.]
PART III.
THE INTERPRETATION OF PROPOSITIONS.--OPPOSITION AND IMMEDIATE
INFERENCE.
CHAPTER I.
THEORIES OF PREDICATION.--THEORIES OF JUDGMENT.
We may now return to the Syllogistic Forms, and the consideration of
the compatibility or incompatibility, implication, and interdependence
of propositions.
It was to make this consideration clear and simple that what we have
called the Syllogistic Form of propositions was devised. When are
propositions incompatible? When do they imply one another? When do
two imply a third? We have seen in the Introduction how such questions
were forced upon Aristotle by the disputative habits of his time.
It was to facilitate the answer that he analysed propositions into
Subject and Predicate, and viewed the Predicate as a reference to a
class: in other words, analysed the Predicate further into a Copula
and a Class Term.
But before showing how he exhibited the interconnexion of propositions
on this plan, we may turn aside to consider various so-called Theories
of Predication or of Judgment. Strictly speaking, they are not
altogether relevant to Logic, that is to say, as a practical science:
they are partly logical, partly psychological theories: some of
them have no bearing whatever on practice, but are matters of pure
scientific curiosity: but historically they are connected with the
logical treatment of propositions as having been developed out of
this.
The least confusing way of presenting these theories is to state
them and examine them both logically and psychologically. The logical
question is, Has the view any advantage for logical purposes? Does it
help to prevent error, to clear up confusion? Does it lead to firmer
conceptions of the truth? The psychological question is, Is this a
correct theory of how men actually think when they make propositions?
It is a question of _what is_ in the one case, and of _what ought to
be for a certain purpose_ in the other.
Whether we speak of Proposition or of Judgment does not materially
affect our answer. A Judgment is the mental act accompanying a
Proposition, or that may be expressed in a proposition and cannot be
expressed otherwise: we can give no other intelligible definition or
description of a judgment. So a proposition can only be defined as the
expression of a judgment: unless there is a judgment underneath them,
a form of words is not a proposition.
Let us take, then, the different theories in turn. We shall find that
they are not really antagonistic, but only different: that each is
substantially right from its own point of view: and that they seem to
contradict one another only when the point of view is misunderstood.
I. _That the Predicate term may be regarded as a class in or
from which the Subject is included or excluded._ Known as the
Class-Inclusion, Class-Reference, or Denotative view.
This way of analysing propositions is possible, as we have seen,
because every statement implies a general name, and the extension
or denotation of a general name is a class defined by the common
attribute or attributes. It is useful for syllogistic purposes:
certain relations among propositions can be most simply exhibited in
this way.
But if this is called a Theory of Predication or Judgment, and taken
psychologically as a theory of what is in men's minds whenever they
utter a significant Sentence, it is manifestly wrong. When discussed
as such, it is very properly rejected. When a man says "P struck Q,"
he has not necessarily a class of "strikers of Q" definitely in his
mind. What he has in his mind is the logical equivalent of this,
but it is not this directly. Similarly, Mr. Bradley would be quite
justified in speaking of Two Terms and a Copula as a superstition, if
it were meant that these analytic elements are present to the mind of
an ordinary speaker.
II. _That every Proposition may be regarded as affirming or denying
an attribute of a subject._ Known sometimes as the Connotative or
the Denotative-Connotative view. This also follows from the implicit
presence of a general name in every sentence. But it should not be
taken as meaning that the man who says: "Tom came here yesterday,"
or "James generally sits there," has a clearly analysed Subject and
Attribute in his mind. Otherwise it is as far wrong as the other view.
III. _That every proposition may be regarded as an equation between
two terms._ Known as the Equational View.
This is obviously not true for common speech or ordinary thought.
But it is a possible way of regarding the analytic components of
a proposition, legitimate enough if it serves any purpose. It is a
modification of the Class-Reference analysis, obtained by what is
known as Quantification of the Predicate. In "All S is in P," P
is undistributed, and has no symbol of Quantity. But since the
proposition imports that All S is a part of P, _i.e._, Some P, we may,
if we choose, prefix the symbol of Quantity, and then the proposition
may be read "All S = Some P". And so with the other forms.
Is there any advantage in this? Yes: it enables us to subject the
formulæ to algebraic manipulation. But any logical advantage--any
help to thinking? None whatever. The elaborate syllogistic systems of
Boole, De Morgan, and Jevons are not of the slightest use in helping
men to reason correctly. The value ascribed to them is merely an
illustration of the Bias of Happy Exercise. They are beautifully
ingenious, but they leave every recorded instance of learned
Scholastic trifling miles behind.
IV. _That every proposition is the expression of a comparison between
concepts._ Sometimes called the Conceptualist View.
"To judge," Hamilton says, "is to recognise the relation of congruence
or confliction in which two concepts, two individual things, or a
concept and an individual compared together stand to each other."
This way of regarding propositions is permissible or not according to
our interpretation of the words "congruence" and "confliction," and
the word "concept". If by concept we mean a conceived attribute of
a thing, and if by saying that two concepts are congruent or
conflicting, we mean that they may or may not cohere in the same
thing, and by saying that a concept is congruent or conflicting with
an individual that it may or may not belong to that individual, then
the theory is a corollary from Aristotle's analysis. Seeing that we
must pass through that analysis to reach it, it is obviously not
a theory of ordinary thought, but of the thought of a logician
performing that analysis.
The precise point of Hamilton's theory was that the logician does not
concern himself with the question whether two concepts are or are
not as a matter of fact found in the same subject, but only with the
question whether they are of such a character that they may be found,
or cannot be found, in the same subject. In so far as his theory
is sound, it is an abstruse and technical way of saying that we may
consider the consistency of propositions without considering whether
or not they are true, and that consistency is the peculiar business of
syllogistic logic.
V. _That the ultimate subject of every judgment is reality._
This is the form in which Mr. Bradley and Mr. Bosanquet deny the
Ultra-Conceptualist position. The same view is expressed by Mill when
he says that "propositions are concerned with things and not with our
ideas of them".
The least consideration shows that there is justice in the view thus
enounced. Take a number of propositions:--
The streets are wet.
George has blue eyes.
The Earth goes round the Sun.
Two and two make four.
Obviously, in any of these propositions, there is a reference beyond
the conceptions in the speaker's mind, viewed merely as incidents
in his mental history. They express beliefs about things and the
relations among things _in rerum natura_: when any one understands
them and gives his assent to them, he never stops to think of the
speaker's state of mind, but of what the words represent. When states
of mind are spoken of, as when we say that our ideas are confused, or
that a man's conception of duty influences his conduct, those states
of mind are viewed as objective facts in the world of realities. Even
when we speak of things that have in a sense no reality, as when we
say that a centaur is a combination of man and horse, or that centaurs
were fabled to live in the vales of Thessaly, it is not the passing
state of mind expressed by the speaker as such that we attend to or
think of; we pass at once to the objective reference of the words.
Psychologically, then, the theory is sound: what is its logical
value? It is sometimes put forward as if it were inconsistent with
the Class-reference theory or the theory that judgment consists in
a comparison of concepts. Historically the origin of its formal
statement is its supposed opposition to those theories. But really
it is only a misconception of them that it contradicts. It is
inconsistent with the Class-reference view only if by a class we
understand an arbitrary subjective collection, not a collection of
things on the ground of common attributes. And it is inconsistent with
the Conceptualist theory only if by a concept we understand not the
objective reference of a general name, but what we have distinguished
as a conception or a conceptual image. The theory that the ultimate
subject is reality is assumed in both the other theories, rightly
understood. If every proposition is the utterance of a judgment, and
every proposition implies a general name, and every general name has
a meaning or connotation, and every such meaning is an attribute of
things and not a mental state, it is implied that the ultimate subject
of every proposition is reality. But we may consider whether or not
propositions are consistent without considering whether or not they
are true, and it is only their mutual consistency that is considered
in the syllogistic formulæ. Thus, while it is perfectly correct to say
that every proposition expresses either truth or falsehood, or that
the characteristic quality of a judgment is to be true or false, it
is none the less correct to say that we may temporarily suspend
consideration of truth or falsehood, and that this is done in what is
commonly known as Formal Logic.
VI. _That every proposition may be regarded as expressing relations
between phenomena._
Bain follows Mill in treating this as the final import of Predication.
But he indicates more accurately the logical value of this view in
speaking of it as important for laying out the divisions of Inductive
Logic. They differ slightly in their lists of Universal Predicates
based upon Import in this sense--Mill's being Resemblance,
Coexistence, Simple Sequence, and Causal Sequence, and Bain's being
Coexistence, Succession, and Equality or Inequality. But both lay
stress upon Coexistence and Succession, and we shall find that the
distinctions between Simple Sequence and Causal Sequence, and between
Repeated and Occasional Coexistence, are all-important in the Logic of
Investigation. But for syllogistic purposes the distinctions have no
relevance.
CHAPTER II.
THE "OPPOSITION" OF PROPOSITIONS.--THE INTERPRETATION OF "NO".
Propositions are technically said to be "opposed" when, having the
same terms in Subject and Predicate, they differ in Quantity, or in
Quality, or in both.[1]
The practical question from which the technical doctrine has been
developed was how to determine the significance of contradiction.
What is meant by giving the answer "No" to a proposition put
interrogatively? What is the interpretation of "No"? What is the
respondent committed to thereby?
"Have all ratepayers a vote?" If you answer "No," you are bound to
admit that some ratepayers have not. O is the CONTRADICTORY of A. If A
is false, O must be true. So if you deny O, you are bound to admit A:
one or other must be true: either Some ratepayers have not a vote or
All have.
Is it the case that no man can live without sleep? Deny this, and you
commit yourself to maintaining that Some man, one at least, can live
without sleep. I is the Contradictory of E; and _vice versâ_.
Contradictory opposition is distinguished from CONTRARY, the
opposition of one Universal to another, of A to E and E to A. There is
a natural tendency to meet a strong assertion with the very reverse.
Let it be maintained that women are essentially faithless or that "the
poor in a lump is bad," and disputants are apt to meet this extreme
with another, that constancy is to be found only in women or true
virtue only among the poor. Both extremes, both A and E, may be false:
the truth may lie between: Some are, Some not.
Logically, the denial of A or E implies only the admission of O or I.
You are not committed to the full contrary. But the implication of the
Contradictory is absolute; there is no half-way house where the
truth may reside. Hence the name of EXCLUDED MIDDLE is applied to the
principle that "Of two Contradictories one or other must be true: they
cannot both be false".
While both CONTRARIES may be false, they cannot both be true.
It is sometimes said that in the case of Singular propositions, the
Contradictory and the Contrary coincide. A more correct doctrine is
that in the case of Singular propositions, the distinction is not
needed and does not apply. Put the question "Is Socrates wise?" or
"Is this paper white?" and the answer "No" admits of only one
interpretation, provided the terms remain the same. Socrates may
become foolish, or this paper may hereafter be coloured differently,
but in either case the subject term is not the same about which the
question was asked. Contrary opposition belongs only to general terms
taken universally as subjects. Concerning individual subjects an
attribute must be either affirmed or denied simply: there is no middle
course. Such a proposition as "Socrates is sometimes not wise," is
not a true Singular proposition, though it has a Singular term as
grammatical subject. Logically, it is a Particular proposition, of
which the subject-term is the actions or judgments of Socrates.[2]
Opposition, in the ordinary sense, is the opposition of incompatible
propositions, and it was with this only that Aristotle concerned
himself. But from an early period in the history of Logic, the word
was extended to cover mere differences in Quantity and Quality
among the four forms A E I O, which differences have been named
and exhibited symmetrically in a diagram known as: The Square of
Opposition.
A______Contraries______E
|\ /|
| \ s |
| C e |
| o i |
| n r |
| t o |
S r t S
u a c u
b \ i b
a \ d a
l \/ l
t /\ t
e / d e
r / i r
n a c n
s r t s
| t o |
| n r |
| o i |
| C e |
| / s |
|/ \|
I____Sub-contraries____O
The four forms being placed at the four corners of the Square, and
the sides and diagonals representing relations between them thus
separated, a very pretty and symmetrical doctrine is the result.
_Contradictories_, A and O, E and I, differ both in Quantity and in
Quality.
_Contraries_, A and E, differ in Quality but not in Quantity, and are
both Universal.
_Sub-contraries_, I and O, differ in Quality but not in Quantity, and
are both Particular.
_Subalterns_, A and I, E and O, differ in Quantity but not in Quality.
Again, in respect of concurrent truth and falsehood there is a certain
symmetry.
Contradictories cannot both be true, nor can they both be false.
Contraries may both be false, but cannot both be true.
Sub-contraries may both be true, but cannot both be false.
Subalterns may both be false and both true. If the Universal is true,
its subalternate Particular is true: but the truth of the Particular
does not similarly imply the truth of its Subalternating Universal.
This last is another way of saying that the truth of the Contrary
involves the truth of the Contradictory, but the truth of the
Contradictory does not imply the truth of the Contrary.
There, however, the symmetry ends. The sides and the diagonals of the
Square do not symmetrically represent degrees of incompatibility, or
opposition in the ordinary sense.
There is no incompatibility between two Sub-contraries or a Subaltern
and its Subalternant. Both may be true at the same time. Indeed, as
Aristotle remarked of I and O, the truth of the one commonly implies
the truth of the other: to say that some of the crew were drowned,
implies that some were not, and _vice versâ_. Subaltern and
Subalternant also are compatible, and something more. If a man has
admitted A or E, he cannot refuse to admit I or O, the Particular of
the same Quality. If All poets are irritable, it cannot be denied
that some are so; if None is, that Some are not. The admission of the
Contrary includes the admission of the Contradictory.
Consideration of Subalterns, however, brings to light a nice ambiguity
in Some. It is only when I is regarded as the Contradictory of E,
that it can properly be said to be Subalternate to A. In that case the
meaning of Some is "not none," _i.e._, "Some at least". But when Some
is taken as the sign of Particular quantity simply, _i.e._, as meaning
"not all," or "some at most," I is not Subalternate to A, but opposed
to it in the sense that the truth of the one is incompatible with the
truth of the other.
Again, in the diagram Contrary opposition is represented by a side and
Contradictory by the diagonal; that is to say, the stronger form of
opposition by the shorter line. The Contrary is more than a denial: it
is a counter-assertion of the very reverse, [Greek: to enantion].
"Are good administrators always good speakers?" "On the contrary, they
never are." This is a much stronger opposition, in the ordinary sense,
than a modest contradictory, which is warranted by the existence of
a single exception. If the diagram were to represent incompatibility
accurately, the Contrary ought to have a longer line than the
Contradictory, and this it seems to have had in the diagram that
Aristotle had in mind (_De Interpret._, c. 10).
It is only when Opposition is taken to mean merely difference in
Quantity and Quality that there can be said to be greater opposition
between Contradictories than between Contraries. Contradictories
differ both in Quantity and in Quality: Contraries, in Quality only.
There is another sense in which the Particular Contradictory may be
said to be a stronger opposite than the Contrary. It is a stronger
position to take up argumentatively. It is easier to defend than a
Contrary. But this is because it offers a narrower and more limited
opposition.
We deal with what is called Immediate Inference in the next chapter.
Pending an exact definition of the process, it is obvious that two
immediate inferences are open under the above doctrines, (1) Granted
the truth of any proposition, you may immediately infer the falsehood
of its Contradictory. (2) Granted the truth of any Contrary, you may
immediately infer the truth of its Subaltern.[3]
[Footnote 1: This is the traditional definition of Opposition
from an early period, though the tradition does not start from
Aristotle. With him opposition ([Greek: antikeisthai]) meant,
as it still means in ordinary speech, incompatibility. The
technical meaning of Opposition is based on the diagram (given
afterwards in the text) known as the Square of Opposition, and
probably originated in a confused apprehension of the reason
why it received that name. It was called the Square of
Opposition, because it was intended to illustrate the doctrine
of Opposition in Aristotle's sense and the ordinary sense of
repugnance or incompatibility. What the Square brings out is
this. If the four forms A E I O are arranged symmetrically
according as they differ in quantity, or quality, or both, it
is seen that these differences do not correspond symmetrically
to compatibility and incompatibility: that propositions may
differ in quantity or in quality without being incompatible,
and that they may differ in both (as Contradictories) and be
less violently incompatible than when they differ in one only
(as Contraries). The original purpose of the diagram was to
bring this out, as is done in every exposition of it. Hence
it was called the Square of Opposition. But as a descriptive
title this is a misnomer: it should have been the Square of
Differences in Quantity or Quality. This misnomer has been
perpetuated by appropriating Opposition as a common name for
difference in Quantity or Quality when the terms are the same
and in the same order, and distinguishing it in this sense
from Repugnance or Incompatibility (Tataretus in Summulas,
_De Oppositionibus_ [1501], Keynes, _The Opposition of
Propositions_ [1887]). Seeing that there never is occasion to
speak of Opposition in the limited sense except in connexion
with the Square, there is no real risk of confusion. A common
name is certainly wanted in that connexion, if only to say
that Opposition (in the limited or diagrammatic sense) does
not mean incompatibility.]
[Footnote 2: Cp. Keynes, pt. ii. ch. ii. s. 57. Aristotle laid
down the distinction between Contrary and Contradictory to
meet another quibble in contradiction, based on taking
the Universal as a whole and indivisible subject like an
Individual, of which a given predicate must be either affirmed
or denied.]
[Footnote 3: I have said that there is little risk of
confusion in using the word Opposition in its technical or
limited sense. There is, however, a little. When it is
said that these Inferences are based on Opposition, or
that Opposition is a mode of Immediate Inference, there is
confusion of ideas unless it is pointed out that when this is
said, it is Opposition in the ordinary sense that is meant.
The inferences are really based on the rules of Contrary and
Contradictory Opposition; Contraries cannot both be true, and
of Contradictories one or other must be.]
CHAPTER III.
THE IMPLICATION OF PROPOSITIONS.--IMMEDIATE FORMAL INFERENCE.
--EDUCATION.
The meaning of Inference generally is a subject of dispute, and
to avoid entering upon debatable ground at this stage, instead of
attempting to define Inference generally, I will confine myself
to defining what is called Formal Inference, about which there is
comparatively little difference of opinion.
FORMAL INFERENCE then is the apprehension of what is implied in a
certain datum or admission: the derivation of one proposition,
called the CONCLUSION, from one or more given, admitted, or assumed
propositions, called the PREMISS or PREMISSES.
When the conclusion is drawn from one proposition, the inference is
said to be IMMEDIATE; when more than one proposition is necessary to
the conclusion, the inference is said to be MEDIATE.
Given the proposition, "All poets are irritable," we can immediately
infer that "Nobody that is not irritable is a poet"; and the one
admission implies the other. But we cannot infer immediately that "all
poets make bad husbands". Before we can do this we must have a second
proposition conceded, that "All irritable persons make bad husbands".
The inference in the second case is called Mediate.[1]
The modes and conditions of valid Mediate Inference constitute
Syllogism, which is in effect the reasoning together of separate
admissions. With this we shall deal presently. Meantime of Immediate
Inference.
To state all the implications of a certain form of proposition, to
make explicit all that it implies, is the same thing with showing
what immediate inferences from it are legitimate. Formal inference, in
short, is the eduction of all that a proposition implies.
Most of the modes of Immediate Inference formulated by logicians are
preliminary to the Syllogistic process, and have no other practical
application. The most important of them technically is the process
known as Conversion, but others have been judged worthy of attention.
ÆQUIPOLLENT OR EQUIVALENT FORMS--OBVERSION.
Æquipollence or Equivalence ([Greek: Isodynamia]) is defined as the
perfect agreement in sense of two propositions that differ somehow in
expression.[2]
The history of Æquipollence in logical treatises illustrates two
tendencies. There is a tendency on the one hand to narrow a theme
down to definite and manageable forms. But when a useful exercise is
discarded from one place it has a tendency to break out in another
under another name. A third tendency may also be said to be specially
well illustrated--the tendency to change the traditional application
of logical terms.
In accordance with the above definition of Æquipollence or
Equivalence, which corresponds with ordinary acceptation, the term
would apply to all cases of "identical meaning under difference of
expression". Most examples of the reduction of ordinary speech into
syllogistic form would be examples of æquipollence; all, in fact,
would be so were it not that ordinary speech loses somewhat in the
process, owing to the indefiniteness of the syllogistic symbol for
particular quality, Some. And in truth all such transmutations
of expression are as much entitled to the dignity of being called
Immediate Inferences as most of the processes so entitled.
Dr. Bain uses the word with an approach to this width of application
in discussing all that is now most commonly called Immediate Inference
under the title of Equivalent Forms. The chief objection to this usage
is that the Converse _per accidens_ is not strictly equivalent. A
debater may want for his argument less than the strict equivalent, and
content himself with educing this much from his opponent's admission.
(Whether Dr. Bain is right in treating the Minor and Conclusion of
a Hypothetical Syllogism as being equivalent to the Major, is not so
much a question of naming.)
But in the history of the subject, the traditional usage has been
to confine Æquipollence to cases of equivalence between positive and
negative forms of expression. "Not all are," is equivalent to "Some
are not": "Not none is," to "Some are". In Pre-Aldrichian text-books,
Æquipollence corresponds mainly to what it is now customary to call
(_e.g._, Fowler, pt. iii. c. ii., Keynes, pt. ii. c. vii.) Immediate
Inference based on Opposition. The denial of any proposition involves
the admission of its contradictory. Thus, if the negative particle
"Not" is placed before the sign of Quantity, All or Some, in
a proposition, the resulting proposition is equivalent to the
Contradictory of the original. Not all S is P = Some S is not P.
Not any S is P = No S is P. The mediæval logicians tabulated these
equivalents, and also the forms resulting from placing the negative
particle after, or both before and after, the sign of Quantity. Under
the title of Æquipollence, in fact, they considered the interpretation
of the negative particle generally. If the negative is placed after
the universal sign, it results in the Contrary: if both before and
after, in the Subaltern. The statement of these equivalents is a
puzzling exercise which no doubt accounts for the prominence given it
by Aristotle and the Schoolmen. The latter helped the student with the
following Mnemonic line: _Præ Contradic., post Contrar., præ postque
Subaltern._[3]
To Æquipollence belonged also the manipulation of the forms known
after the _Summulæ_ as _Exponibiles_, notably _Exclusive_ and
_Exceptive propositions_, such as None but barristers are eligible,
The virtuous alone are happy. The introduction of a negative particle
into these already negative forms makes a very trying problem in
interpretation. The æquipollence of the Exponibiles was dropped from
text-books long before Aldrich, and it is the custom to laugh at them
as extreme examples of frivolous scholastic subtlety: but most modern
text-books deal with part of the doctrine of the _Exponibiles_ in
casual exercises.
Curiously enough, a form left unnamed by the scholastic logicians
because too simple and useless, has the name Æquipollent appropriated
to it, and to it alone, by Ueberweg, and has been adopted under
various names into all recent treatises.
Bain calls it the FORMAL OBVERSE,[4] and the title of OBVERSION (which
has the advantage of rhyming with CONVERSION) has been adopted by
Keynes, Miss Johnson, and others.
Fowler (following Karslake) calls it PERMUTATION. The title is not a
happy one, having neither rhyme nor reason in its favour, but it is
also extensively used.
This immediate inference is a very simple affair to have been honoured
with such a choice of terminology. "This road is long: therefore, it
is not short," is an easy inference: the second proposition is the
Obverse, or Permutation, or Æquipollent, or (in Jevons's title) the
Immediate Inference by Privative Conception, of the first.
The inference, such as it is, depends on the Law of Excluded Middle.
Either a term P, or its contradictory, not-P, must be true of any
given subject, S: hence to affirm P of all or some S, is equivalent
to denying not-P of the same: and, similarly, to deny P, is to affirm
not-P. Hence the rule of Obversion;--Substitute for the predicate term
its Contrapositive,[5] and change the Quality of the proposition.
All S is P = No S is not-P.
No S is P = All S is not-P.
Some S is P = Some S is not not-P.
Some S is not P = Some S is not-P.
CONVERSION.
The process takes its name from the interchange of the terms. The
Predicate-term becomes the Subject-term, and the Subject-term the
Predicate-term.
When propositions are analysed into relations of inclusion or
exclusion between terms, the assertion of any such relation between
one term and another, implies a Converse relation between the second
term and the first. The statement of this implied assertion is
technically known as the CONVERSE of the original proposition, which
may be called the _Convertend_.
Three modes of Conversion are commonly recognised:--(_a_) SIMPLE
CONVERSION; (_b_) CONVERSION _per accidens_ or by limitation; (_c_)
CONVERSION BY CONTRAPOSITION.
(_a_) E and I can be simply converted, only the terms being
interchanged, and Quantity and Quality remaining the same.
If S is wholly excluded from P, P must be wholly excluded from S. If
Some S is contained in P, then Some P must be contained in S.
(_b_) A cannot be simply converted. To know that All S is contained in
P, gives you no information about that portion of P which is outside
S. It only enables you to assert that Some P is S; that portion of P,
namely, which coincides with S.
O cannot be converted either simply or _per accidens_. Some S is not P
does not enable you to make any converse assertion about P. All P
may be S, or No P may be S, or Some P may be not S. All the three
following diagrams are compatible with Some S being excluded from P.
[Illustration:
Concentric circles of S and P - P in centre
S in one circle and P in another circle.
S and P each in a circle, overlapping circle.
]
(_c_) Another mode of Conversion, known by mediæval logicians
following Boethius as _Conversio per contra positionem terminorum_, is
useful in some syllogistic manipulations. This Converse is obtained
by substituting for the predicate term its Contrapositive or
Contradictory, not-P, making the consequent change of Quality, and
simply converting. Thus All S is P is converted into the equivalent No
not-P is S.[6]
Some have called it "Conversion by Negation," but "negation"
is manifestly too wide and common a word to be thus arbitrarily
restricted to the process of substituting for one term its opposite.
Others (and this has some mediæval usage in its favour, though not the
most intelligent) would call the form All not-P is not-S (the Obverse
or Permutation of No not-P is S), the Converse by Contraposition. This
is to conform to an imaginary rule that in Conversion the Converse
must be of the same Quality with the Convertend. But the essence of
Conversion is the interchange of Subject and Predicate: the Quality is
not in the definition except by a bungle: it is an accident. No
not-P is S, and Some not-P is S are the forms used in Syllogism,
and therefore specially named. Unless a form had a use, it was left
unnamed, like the Subalternate forms of Syllogism: Nomen habent
nullum: nec, si bene colligis, usum.
TABLE OF CONTRAPOSITIVE CONVERSES.
Con. Con.
All S is P No not-P is S
No S is P Some not-P is S
Some S is not P Some not-P is S
Some S is P None.
When not-P is substituted for P, Some S is P becomes Some S is not
not-P, and this form is inconvertible.
OTHER FORMS OF IMMEDIATE INFERENCE.
I have already spoken of the Immediate Inferences based on the rules
of Contradictory and Contrary Opposition (see p. 145 - Part III, Ch.
II).
Another process was observed by Thomson, and named _Immediate
Inference by Added Determinants_. If it is granted that "A negro is
a fellow-creature," it follows that "A negro in suffering is a
fellow-creature in suffering". But that this does not follow for every
attribute[7] is manifest if you take another case:--"A tortoise is
an animal: therefore, a fast tortoise is a fast animal". The form,
indeed, holds in cases not worth specifying: and is a mere handle for
quibbling. It could not be erected into a general rule unless it
were true that whatever distinguishes a species within a class, will
equally distinguish it in every class in which the first is included.
MODAL CONSEQUENCE has also been named among the forms of Immediate
Inference. By this is meant the inference of the lower degrees of
certainty from the higher. Thus _must be_ is said to imply _may be_;
and _None can be_ to imply _None is_.
Dr. Bain includes also _Material Obversion_, the analogue of _Formal
Obversion_ applied to a Subject. Thus Peace is beneficial to commerce,
implies that War is injurious to commerce. Dr. Bain calls this
Material Obversion because it cannot be practised safely without
reference to the matter of the proposition. We shall recur to the
subject in another chapter.
[Footnote 1: I purposely chose disputable propositions to
emphasise the fact that Formal Logic has no concern with the
truth, but only with the interdependence of its propositions.]
[Footnote 2: Mark Duncan, _Inst. Log._, ii. 5, 1612.]
[Footnote 3: There can be no doubt that in their doctrine of
Æquipollents, the Schoolmen were trying to make plain a real
difficulty in interpretation, the interpretation of the force
of negatives. Their results would have been more obviously
useful if they had seen their way to generalising them.
Perhaps too they wasted their strength in applying it to the
artificial syllogistic forms, which men do not ordinarily
encounter except in the manipulation of syllogisms. Their
results might have been generalised as follows:--
(1) A "not" placed before the sign of Quantity contradicts
the whole proposition. Not "All S is P," not "No S is P,"
not "Some S is P," not "Some S is not P," are equivalent
respectively to contradictories of the propositions thus
negatived.
(2) A "not" placed after the sign of Quantity affects the
copula, and amounts to inverting its Quality, thus denying
the predicate term of the same quantity of the subject term of
which it was originally affirmed, and _vice versâ_.
All S is "not" P = No S is P.
No S is "not" P = All S is P.
Some S is "not" P = Some S is not P.
Some S is "not" not P = Some S is P.
(3) If a "not" is placed before as well as after, the
resulting forms are obviously equivalent (under Rule 1) to the
assertion of the contradictories of the forms on the right (in
the illustration of Rule 2).
Not | All S is "not" P = No S is P | = Some S is P.
Not | No S is "not" P = All S is P | = Some S is not P.
Not | Some S is "not" P = Some S is not P | = All S is P.
Not | Some S is "not" not P = Some S is P | = No S is P.
]
[Footnote 4: _Formal_ to distinguish it from what he called
the _Material Obverse_, about which more presently.]
[Footnote 5: The mediæval word for the opposite of a term, the
word Contradictory being confined to the propositional form.]
[Footnote 6: It is to be regretted that a practice has
recently crept in of calling this form, for shortness, the
Contrapositive simply. By long-established usage, dating from
Boethius, the word Contrapositive is a technical name for a
terminal form, not-A, and it is still wanted for this use.
There is no reason why the propositional form should not be
called the Converse by Contraposition, or the Contrapositive
Converse, in accordance with traditional usage.]
[Footnote 7: _Cf._ Stock, part iii. c. vii.; Bain,
_Deduction_, p. 109.]
CHAPTER IV.
THE COUNTER-IMPLICATION OF PROPOSITIONS.
In discussing the Axioms of Dialectic, I indicated that the
propositions of common speech have a certain negative implication,
though this does not depend upon any of the so-called Laws of Thought,
Identity, Contradiction, and Excluded Middle. Since, however, the
counter-implicate is an important guide in the interpretation of
propositions, it is desirable to recognise it among the modes of
Immediate Inference.
I propose, then, first, to show that people do ordinarily infer at
once to a counter-sense; second, to explain briefly the Law of Thought
on which such an inference is justified; and, third, how this law
may be applied in the interpretation of propositions, with a view to
making subject and predicate more definite.
Every affirmation about anything is an implicit negation about
something else. Every say is a gainsay. That people ordinarily
act upon this as a rule of interpretation a little observation is
sufficient to show: and we find also that those who object to having
their utterances interpreted by this rule often shelter themselves
under the name of Logic.
Suppose, for example, that a friend remarks, when the conversation
turns on children, that John is a good boy, the natural inference is
that the speaker has in his mind another child who is not a good boy.
Such an inference would at once be drawn by any actual hearer, and the
speaker would protest in vain that he said nothing about anybody but
John. Suppose there are two candidates for a school appointment, A
and B, and that stress is laid upon the fact that A is an excellent
teacher. A's advocate would at once be understood to mean that B was
not equally excellent as a teacher.
The fairness of such inferences is generally recognised. A reviewer,
for example, of one of Mrs. Oliphant's historical works, after
pointing out some small errors, went on to say that to confine himself
to censure of small points, was to acknowledge by implication that
there were no important points to find fault with.
Yet such negative implications are often repudiated as illogical.
It would be more accurate to call them extra-logical. They are not
condemned by any logical doctrine: they are simply ignored. They are
extra-logical only because they are not legitimated by the Laws of
Identity, Contradiction, and Excluded Middle: and the reason why
Logic confines itself to those laws is that they are sufficient for
Syllogism and its subsidiary processes.
But, though extra-logical, to infer a counter-implicate is not
unreasonable: indeed, if Definition, clear vision of things in their
exact relations, is our goal rather than Syllogism, a knowledge of
the counter-implicate is of the utmost consequence. Such an implicate
there must always be under an all-pervading Law of Thought which has
not yet been named, but which may be called tentatively the law of
Homogeneous Counter-relativity. The title, one hopes, is sufficiently
technical-looking: though cumbrous, it is descriptive. The law itself
is simple, and may be thus stated and explained.
_The Law of Homogeneous Counter-relativity._
Every positive in thought has a contrapositive, and the
positive and contrapositive are of the same kind.
The first clause of our law corresponds with Dr. Bain's law of
Discrimination or Relativity: it is, indeed, an expansion and
completion of that law. Nothing is known absolutely or in isolation;
the various items of our knowledge are inter-relative; everything is
known by distinction from other things. Light is known as the opposite
of darkness, poverty of riches, freedom of slavery, in of out; each
shade of colour by contrast to other shades. What Dr. Bain lays stress
upon is the element of difference in this inter-relativity. He bases
this law of our knowledge on the fundamental law of our sensibility
that change of impression is necessary to consciousness. A long
continuance of any unvaried impression results in insensibility to it.
We have seen instances of this in illustrating the maxim that
custom blunts sensibility (p. 74). Poets have been beforehand with
philosophers in formulating this principle. It is expressed with the
greatest precision by Barbour in his poem of "The Bruce," where he
insists that men who have never known slavery do not know what freedom
is.
Thus contrar thingis evermare
Discoverings of t' other are.
Since, then, everything that comes within our consciousness comes as
a change or transition from something else, it results that our
knowledge is counter-relative. It is in the clash or conflict of
impressions that knowledge emerges: every item of knowledge has its
illuminating foil, by which it is revealed, over against which it is
defined. Every positive in thought has its contrapositive.
So much for the element of difference. But this is not the whole of
the inter-relativity. The Hegelians rightly lay stress on the common
likeness that connects the opposed items of knowledge.
"Thought is not only _distinction_; it is, at the same time,
_relation_.[1] If it marks off one thing from another, it, at
the same time, connects one thing with another. Nor can either
of these functions of thought be separated from the other: as
Aristotle himself said, the knowledge of opposites is one. A
thing which has nothing to distinguish it is unthinkable, but
equally unthinkable is a thing which is so separated from all
other things as to have no community with them. If then the
law of contradiction be taken as asserting the self-identity
of things or thoughts in a sense that excludes their
community--in other words, if it be not taken as limited by
another law which asserts the _relativity_ of the things or
thoughts distinguished--it involves a false abstraction....
If, then, the world, as an intelligible world, is a world of
distinction, differentiation, individuality, it is equally
true that in it as an intelligible world there are no absolute
separations or oppositions, no antagonisms which cannot be
reconciled."[2]
In the penultimate sentence of this quotation Dr. Caird
_differentiates_ his theory against a Logical counter-theory of
the Law of Identity, and in the last sentence against an Ethical
counter-theory: but the point here is that he insists on the relation
of likeness among opposites. Every impression felt is felt as a change
or transition from something else: but it is a variation of the same
impression--the something else, the contrapositive, is not entirely
different. Change itself is felt as the opposite of sameness,
difference of likeness, and likeness of difference. We do not
differentiate our impression against the whole world, as it were,
but against something nearly akin to it--upon some common ground. The
positive and the contrapositive are of the same kind.
Let us surprise ourselves in the act of thinking and we shall find
that our thoughts obey this law. We take note, say, of the colour
of the book before us: we differentiate it against some other colour
actually before us in our field of vision or imagined in our minds.
Let us think of the blackboard as black: the blackness is defined
against the whiteness of the figures chalked or chalkable upon it, or
against the colour of the adjacent wall. Let us think of a man as a
soldier; the opposite in our minds is not the colour of his hair,
or his height, or his birthplace, or his nationality, but some other
profession--soldier, sailor, tinker, tailor. It is always by means of
some contrapositive that we make the object of our thoughts definite;
it is not necessarily always the same opposite, but against
whatever opposite it is, they are always homogeneous. One colour is
contradistinguished from another colour, one shade from another shade:
colour may be contradistinguished from shape, but it is within the
common genus of sensible qualities.
A curious confirmation of this law of our thinking has been pointed
out by Mr. Carl Abel.[3] In Egyptian hieroglyphics, the oldest
extant language, we find, he says, a large number of symbols with
two meanings, the one the exact opposite of the other. Thus the
same symbol represents _strong_ and _weak_; _above_--_below_;
_with_--_without_; _for_--_against_. This is what the Hegelians mean
by the reconciliation of antagonisms in higher unities. They do not
mean that black is white, but only that black and white have something
in common--they are both colours.
I have said that this law of Homogeneous Counter-relativity has not
been recognised by logicians. This, however, is only to say that it
has not been explicitly formulated and named, as not being required
for Syllogism; a law so all-pervading could not escape recognition,
tacit or express. And accordingly we find that it is practically
assumed in Definition: it is really the basis of definition _per
genus et differentiam_. When we wish to have a definite conception
of anything, to apprehend what it is, we place it in some genus and
distinguish it from species of the same. In fact our law might be
called the Law of Specification: in obeying the logical law of what
we ought to do with a view to clear thinking, we are only doing with
exactness and conscious method what we all do and cannot help doing
with more or less definiteness in our ordinary thinking.
It is thus seen that logicians conform to this law when they are
not occupied with the narrow considerations proper to Syllogism. And
another unconscious recognition of it may be found in most logical
text-books. Theoretically the not-A of the Law of Contradiction--(A is
not not-A)--is an infinite term. It stands for everything but A. This
is all that needs to be assumed for Conversion and Syllogism. But take
the examples given of the Formal Obverse or Permutation, "All men are
fallible". Most authorities would give as the Formal Obverse of this,
"No men are infallible". But, strictly speaking, "infallible" is
of more limited and definite signification than not-fallible.
Not-fallible, other than fallible, is brown, black, chair, table,
and every other nameable thing except fallible. Thus in Obversion and
Conversion by Contraposition, the homogeneity of the negative term is
tacitly assumed; it is assumed that A and not-A are of the same kind.
Now to apply this Law of our Thought to the interpretation of
propositions. Whenever a proposition is uttered we are entitled to
infer at once (or _immediately_) that the speaker has in his mind
some counter-proposition, in which what is overtly asserted of the
ostensible subject is covertly denied of another subject. And we must
know what this counter-proposition, the counter-implicate is, before
we can fully and clearly understand his meaning. But inasmuch as
any positive may have more than one contrapositive, we cannot tell
immediately or without some knowledge of the circumstances or context,
what the precise counter-implicate is. The peculiar fallacy incident
to this mode of interpretation is, knowing that there must be some
counter-implicate, to jump rashly or unwarily to the conclusion that
it is some definite one.
Dr. Bain applies the term Material Obverse to the form, Not-S is not
P, as distinguished from the form S is not not-P, which he calls
the Formal Obverse, on the ground that we can infer the
Predicate-contrapositive at once from the form, whereas we cannot tell
the Subject-contrapositive without an examination of the matter.
But in truth we cannot tell either Predicate-contrapositive or
Subject-contrapositive as it is in the mind of the speaker from
the bare utterance. We can only tell that if he has in his mind a
proposition definitely analysed into subject and predicate, he must
have contrapositives in his mind of both, and that they must be
homogeneous. Let a man say, "This book is a quarto". For all that we
know he may mean that it is not a folio or that it is not an
octavo: we only know for certain, under the law of Homogeneous
Counter-relativity, that he means some definite other size. Under
the same law, we know that he has a homogeneous contrapositive of the
subject, a subject that admits of the same predicate, some other book
in short. What the particular book is we do not know.
It would however be a waste of ingenuity to dwell upon the
manipulation of formulæ founded on this law. The practical concern is
to know that for the interpretation of a proposition, a knowledge of
the counter-implicate, a knowledge of what it is meant to deny, is
essential.
The manipulation of formulæ, indeed, has its own special snare. We are
apt to look for the counterparts of them in the grammatical forms of
common speech. Thus, it might seem to be a fair application of our law
to infer from the sentence, "Wheat is dear," that the speaker had in
his mind that Oats or Sugar or Shirting or some other commodity is
cheap. But this would be a rash conclusion. The speaker may mean this,
but he _may_ also mean that wheat is dear now as compared with some
other time: that is, the Positive subject in his mind may be "Wheat as
now," and the Contrapositive "Wheat as then". So a man may say, "All
men are mortal," meaning that the angels never taste death, "angels"
being the contrapositive of his subject "men". Or he may mean merely
that mortality is a sad thing, his positive subject being men as
they are, and his contrapositive men as he desires them to be. Or his
emphasis may be upon the _all_, and he may mean only to deny that some
one man in his mind (Mr. Gladstone, for example) is immortal. It would
be misleading, therefore, to prescribe propositions as exercises in
Material Obversion, if we give that name to the explicit expression
of the Contrapositive Subject: it is only from the context that we can
tell what this is. The man who wishes to be clearly understood gives
us this information, as when the epigrammatist said: "We are all
fallible--even the youngest of us".
But the chief practical value of the law is as a guide in studying the
development of opinions. Every doctrine ever put forward has been
put forward in opposition to a previous doctrine on the same subject.
Until we know what the opposed doctrine is, we cannot be certain of
the meaning. We cannot gather it with precision from a mere study of
the grammatical or even (in the narrow sense of the word) the logical
content of the words used. This is because the framers of doctrines
have not always been careful to put them in a clear form of subject
and predicate, while their impugners have not moulded their denial
exactly on the language of the original. No doubt it would have been
more conducive to clearness if they had done so. But they have not,
and we must take them as they are. Thus we have seen that the Hegelian
doctrine of Relativity is directed against certain other doctrines
in Logic and in Ethics; that Ultra-Nominalism is a contradiction of
a certain form of Ultra-Realism; and that various theories of
Predication each has a backward look at some predecessor.
I quote from Mr. A.B. Walkley a very happy application of this
principle of interpretation:--
"It has always been a matter for speculation why so sagacious
an observer as Diderot should have formulated the wild paradox
that the greatest actor is he who feels his part the least.
Mr. Archer's bibliographical research has solved this riddle.
Diderot's paradox was a protest against a still wilder one. It
seems that a previous eighteenth century writer on the
stage, a certain Saint-Albine, had advanced the fantastic
propositions that none but a magnanimous man can act
magnanimity, that only lovers can do justice to a love scene,
and kindred assertions that read like variations on the
familiar 'Who drives fat oxen must himself be fat'. Diderot
saw the absurdity of this; he saw also the essentially
artificial nature of the French tragedy and comedy of his own
day; and he hastily took up the position which Mr. Archer has
now shown to be untenable."
This instance illustrates another principle that has to be borne in
mind in the interpretation of doctrines from their historical context
of counter-implication. This is the tendency that men have to
put doctrines in too universal a form, and to oppose universal to
universal, that is, to deny with the flat contrary, the very reverse,
when the more humble contradictory is all that the truth admits of. If
a name is wanted for this tendency, it might be called the tendency to
Over-Contradiction. Between "All are" and "None are," the sober
truth often is that "Some are" and "Some are not," and the process of
evolution has often consisted in the substitution of these sober forms
for their more violent predecessors.
[Footnote 1: It is significant of the unsuitableness of
the vague unqualified word Relativity to express a logical
distinction that Dr. Bain calls his law the Law of Relativity
simply, having regard to the relation of difference, _i.e._,
to Counter-Relativity, while Dr. Caird applies the name
Relativity simply to the relation of likeness, _i.e._, to
Co-relativity. It is with a view to taking both forms
of relation into account that I name our law the Law of
Homogeneous Counter-relativity. The Protagorean Law of
Relativity has regard to yet another relation, the relation of
knowledge to the knowing mind: these other logical laws are
of relations among the various items of knowledge. Aristotle's
category of Relation is a fourth kind of relation not to be
confused with the others. "Father--son," "uncle--nephew,"
"slave--master," are _relata_ in Aristotle's sense: "father,"
"uncle" are homogeneous counter-relatives, varieties of
kinship; so "slave," "freeman" are counter-relatives in social
status.]
[Footnote 2: Dr. Caird's _Hegel_, p. 134.]
[Footnote 3: See article on Counter-Sense, _Contemporary
Review_, April, 1884.]
PART IV.
THE INTERDEPENDENCE OF PROPOSITIONS.--MEDIATE INFERENCE.
--SYLLOGISM.
CHAPTER I.
THE SYLLOGISM.
We have already defined mediate inference as the derivation of a
conclusion from more than one proposition. The type or form of a
mediate inference fully expressed consists of three propositions so
related that one of them is involved or implied in the other two.
Distraction is exhausting.
Modern life is full of distraction
[.'.] Modern life is exhausting.
We say nothing of the truth of these propositions. I purposely choose
questionable ones. But do they hang together? If you admit the first
two, are you bound in consistency to admit the third? Is the truth of
the conclusion a necessary consequence of the truth of the premisses?
If so, it is a valid mediate inference from them.
When one of the two premisses is more general than the conclusion, the
argument is said to be Deductive. You lead down from the more general
to the less general. The general proposition is called the Major
Premiss, or Grounding Proposition, or Sumption: the other premiss the
Minor, or Applying Proposition, or Subsumption.
Undue haste makes waste.
This is a case of undue hasting.
[.'.] It is a case of undue wasting.
We may, and constantly do, apply principles and draw conclusions
in this way without making any formal analysis of the propositions.
Indeed we reason mediately and deductively whenever we make any
application of previous knowledge, although the process is not
expressed in propositions at all and is performed so rapidly that we
are not conscious of the steps.
For example, I enter a room, see a book, open it and begin to read. I
want to make a note of something: I look round, see a paper case,
open it, take a sheet of paper and a pen, dip the pen in the ink
and proceed to write. In the course of all this, I act upon certain
inferences which might be drawn out in the form of Syllogisms. First,
in virtue of previous knowledge I recognise what lies before me as a
book. The process by which I reach the conclusion, though it passes in
a flash, might be analysed and expressed in propositions.
Whatever presents certain outward appearances,
contains readable print.
This presents such appearances.
[.'.]It contains readable print.
So with the paper case, and the pen, and the ink. I infer from
peculiar appearances that what I see contains paper, that the liquid
will make a black mark on the white sheet, and so forth.
We are constantly in daily life subsuming particulars under known
universals in this way. "Whatever has certain visible properties, has
certain other properties: this has the visible ones: therefore, it has
the others" is a form of reasoning constantly latent in our minds.
The Syllogism may be regarded as the explicit expression of this type
of deductive reasoning; that is, as the analysis and formal expression
of this every-day process of applying known universals to particular
cases. Thus viewed it is simply the analysis of a mental process, as a
psychological fact; the analysis of the procedure of all men when they
reason from signs; the analysis of the kind of assumptions they make
when they apply knowledge to particular cases. The assumptions may be
warranted, or they may not: but as a matter of fact the individual who
makes the confident inference has such assumptions and subsumptions
latent in his mind.
But practically viewed, that is _logically_ viewed, if you regard
Logic as a practical science, the Syllogism is a contrivance to
assist the correct performance of reasoning together or syllogising in
difficult cases. It applies not to mental processes but to results of
such expressed in words, that is, to propositions. Where the Syllogism
comes in as a useful form is when certain propositions are delivered
to you _ab extra_ as containing a certain conclusion; and the
connexion is not apparent. These propositions are analysed and thrown
into a form in which it is at once apparent whether the alleged
connexion exists. This form is the Syllogism: it is, in effect, an
analysis of given arguments.
It was as a practical engine or organon that it was invented by
Aristotle, an organon for the syllogising of admissions in Dialectic.
The germ of the invention was the analysis of propositions into terms.
The syllogism was conceived by Aristotle as a reasoning together
of terms. His prime discovery was that whenever two propositions
necessarily contain or imply a conclusion, they have a common term,
that is, only three terms between them: that the other two terms which
differ in each are the terms of the conclusion; and that the relation
asserted in the conclusion between its two terms is a necessary
consequence of their relations with the third term as declared in the
premisses.
Such was Aristotle's conception of the Syllogism and such it has
remained in Logic. It is still, strictly speaking, a syllogism of
terms: of propositions only secondarily and after they have been
analysed. The conclusion is conceived analytically as a relation
between two terms. In how many ways may this relation be established
through a third term? The various moods and figures of the Syllogism
give the answer to that question.
The use of the very abstract word "relation" makes the problem appear
much more difficult than it really is. The great charm of Aristotle's
Syllogism is its simplicity. The assertion of the conclusion is
reduced to its simplest possible kind, a relation of inclusion or
exclusion, contained or not contained. To show that the one term is
or is not contained in the other we have only to find a third which
contains the one and is contained or not contained in the other.
The practical difficulties, of course, consist in the reduction of
the conclusions and arguments of common speech to definite terms thus
simply related. Once they are so reduced, their independence or the
opposite is obvious. Therein lies the virtue of the Syllogism.
Before proceeding to show in how many ways two terms may be Syllogised
through a third, we must have technical names for the elements.
The third term is called the MIDDLE (M) ([Greek: to meson]): the other
two the Extremes ([Greek: akra]).
The EXTREMES are the Subject (S) and the Predicate (P) of the
conclusion.
In an affirmative proposition (the normal form) S is contained in P:
hence P is called the MAJOR[1] term ([Greek: to meixon]), and S the
MINOR ([Greek: to elatton]), being respectively larger and smaller in
extension. All difficulty about the names disappears if we remember
that in bestowing them we start from the conclusion. That was the
problem ([Greek: problêma]) or thesis in dialectic, the question in
dispute.
The two Premisses, or propositions giving the relations between the
two Extremes and the Middle, are named on an equally simple ground.
One of them gives the relation between the Minor Term, S, and the
Middle, M. S, All or Some, is or is not in M. This is called the Minor
Premiss.
The other gives the relation between the Major Term and the Middle. M,
All or Some, is or is not in P. This is called the Major Premiss.[2]
[Footnote 1: Aristotle calls the Major the First ([Greek:
to prôton]) and the Minor the last ([Greek: to eschaton]),
probably because that was their order in the conclusion when
stated in his most usual form, "P is predicated of S," or "P
belongs to S".]
[Footnote 2: When we speak of the Minor or the Major simply,
the reference is to the terms. To avoid a confusion into
which beginners are apt to stumble, and at the same time to
emphasise the origin of the names, the Premisses might be
spoken of at first as the Minor's Premiss and the Major's
Premiss. It was only in the Middle Ages when the origin of
the Syllogism had been forgotten, that the idea arose that the
terms were called Major and Minor because they occurred in the
Major and the Minor Premiss respectively.]
CHAPTER II.
FIGURES AND MOODS OF THE SYLLOGISM.
I.--The First Figure.
The forms (technically called MOODS, _i.e._, modes) of the First
Figure are founded on the simplest relations with the Middle that will
yield or that necessarily involve the disputed relation between the
Extremes.
The simplest type is stated by Aristotle as follows: "When three terms
are so related that the last (the Minor) is wholly in the Middle, and
the Middle wholly either in or not in the first (the Major) there must
be a perfect syllogism of the Extremes".[1]
When the Minor is partly in the Middle, the Syllogism holds equally
good. Thus there are four possible ways in which two terms ([Greek:
oroi], plane enclosures) may be connected or disconnected through a
third. They are usually represented by circles as being the neatest
of figures, but any enclosing outline answers the purpose, and the
rougher and more irregular it is the more truly will it represent the
extension of a word.
Conclusion A.
All M is in P.
All S is in M.
All S is in P.
[Illustration: concentric circles of P, M and S - S in centre]
Conclusion E.
No M is in P.
All S is in M.
No S is in P.
[Illustration: Concentric circles of M and S,
S in centre and separate circle of P]
Conclusion I.
All M is in P.
Some S is in M.
Some S is in P.
[Illustration: Concentric circles of P and M,
M in centre, both overlapped by circle of S]
Conclusion O.
No M is in P.
Some S is in M.
Some S is not in P.
[Illustration: Circles of M and P touching,
each overlapped by circle of S]
These four forms constitute what are known as the moods of the First
Figure of the Syllogism. Seeing that all propositions may be reduced
to one or other of the four forms, A, E, I, or O, we have in these
premisses abstract types of every possible valid argument from
general principles. It is all the same whatever be the matter of the
proposition. Whether the subject of debate is mathematical, physical,
social or political, once premisses in these forms are conceded, the
conclusion follows irresistibly, _ex vi formæ, ex necessitate formæ_.
If an argument can be analysed into these forms, and you admit
its propositions, you are bound in consistency to admit the
conclusion--unless you are prepared to deny that if one thing is in
another and that other in a third, the first is in the third, or if
one thing is in another and that other wholly outside a third, the
first is also outside the third.
This is called the AXIOM OF SYLLOGISM. The most common form of it in
Logic is that known as the _Dictum_, or _Regula de Omni et Nullo:_
"Whatever is predicated of All or None of a term, is predicated of
whatever is contained in that term". It has been expressed with many
little variations, and there has been a good deal of discussion as to
the best way of expressing it, the relativity of the word best being
often left out of sight. _Best_ for what purpose? Practically that
form is the best which best commands general assent, and for this
purpose there is little to choose between various ways of expressing
it. To make it easy and obvious it is perhaps best to have two
separate forms, one for affirmative conclusions and one for negative.
Thus: "Whatever is affirmed of all M, is affirmed of whatever is
contained in M: and whatever is denied of all M, is denied of whatever
is contained in M". The only advantage of including the two forms in
one expression, is compendious neatness. "A part of a part is a part
of the whole," is a neat form, it being understood that an individual
or a species is part of a genus. "What is said of a whole, is said
of every one of its parts," is really a sufficient statement of the
principle: the whole being the Middle Term, and the Minor being a
part of it, the Major is predicable of the Minor affirmatively or
negatively if it is predicable similarly of the Middle.
This Axiom, as the name imports, is indemonstrable. As Aristotle
pointed out in the case of the Axiom of Contradiction, it can be
vindicated, if challenged, only by reducing the challenger to a
practical absurdity. You can no more deny it than you can deny that
if a leaf is in a book and the book is in your pocket, the leaf is in
your pocket. If you say that you have a sovereign in your purse and
your purse is in your pocket, and yet that the sovereign is not in
your pocket: will you give me what is in your pocket for the value of
the purse?
II.--THE MINOR FIGURES OF THE SYLLOGISM, AND THEIR REDUCTION TO THE
FIRST.
The word Figure ([Greek: schêma]) applies to the form or figure of
the premisses, that is, the order of the terms in the statement of the
premisses, when the Major Premiss is put first, and the Minor second.
In the First Figure the order is
M P
S M
But there are three other possible orders or figures, namely:--
Fig. ii. Fig. iii. Fig. iv.
PM MP PM
SM MS MS.
It results from the doctrines of Conversion that valid arguments may
be stated in these forms, inasmuch as a proposition in one order of
terms may be equivalent to a proposition in another. Thus No M is in P
is convertible with No P is in M: consequently the argument
No P is in M
All S is in M,
in the Second Figure is as much valid as when it is stated in the
First--
No M is in P
All S is in M.
Similarly, since All M is in S is convertible into Some S is in M, the
following arguments are equally valid:--
Fig. iii. Fig. i.
All M is in P All M is in P
=
All M is in S Some S is in M.
Using both the above Converses in place of their Convertends, we
have--
Fig. iv. Fig. i.
No P is in M No M is in P
=
All M is in S Some S is in M.
It can be demonstrated (we shall see presently how) that altogether
there are possible four valid forms or moods of the Second Figure, six
of the Third, and five of the Fourth. An ingenious Mnemonic of
these various moods and their reduction to the First Figure by the
transposition of terms and premisses has come down from the thirteenth
century. The first line names the moods of the First, Normal, or
Standard Figure.
BA_rb_A_r_A, CE_l_A_r_E_nt_, DA_r_II, FE_r_IO_que_ prioris;
CE_s_A_r_E, CA_m_E_str_E_s_, FE_st_I_n_O, BA_r_O_k_O, secundæ;
Tertia DA_r_A_pt_I, DI_s_A_m_I_s_, DA_t_I_s_I, FE_l_A_pt_O_n_,
BO_k_A_rd_O, FE_r_I_s_O_que_, habet; quarta insuper addit,
B_r_A_m_A_nt_IP, CA_m_E_n_E_s_, DI_m_A_r_I_s_, FE_s_A_p_O,
F_r_E_s_I_s_O_n_.
The vowels in the names of the Moods indicate the propositions of the
Syllogism in the four forms, A E I O. To write out any Mood at length
you have only to remember the Figure, and transcribe the propositions
in the order of Major Premiss, Minor Premiss, and Conclusion. Thus,
the Second Figure being
PM
SM
FE_st_I_n_O is written--
No P is in M.
Some S is in M.
Some S is not in P.
The Fourth Figure being
PM
MS
DI_m_A_r_I_s_ is
Some P is in M.
All M is in S.
Some S is in P.
The initial letter in a Minor Mood indicates that Mood of the First to
which it may be reduced. Thus Festino is reduced to Ferio, and Dimaris
to Darii. In the cases of Baroko and Bokardo, B indicates that you
may employ Barbara to bring any impugner to confusion, as shall be
afterwards explained.
The letters _s_, _m_, and _p_ are also significant. Placed after a
vowel, _s_ indicates that the proposition has to be simply converted.
Thus, FE_st_I_n_O:--
No P is in M.
Some S is in M.
Some S is not in P.
Simply convert the Major Premiss, and you get FE_r_IO, of the First.
No M is in P.
Some S is in M.
Some S is not in P.
_m_ (_muta_, or _move_) indicates that the premisses have to be
transposed. Thus, in CA_m_E_str_E_s_, you have to transpose the
premisses, as well as simply convert the Minor Premiss before reaching
the figure of CE_l_A_r_E_nt_.
All P is in M No M is in S
=
No S is in M All P is in M.
From this it follows in CE_l_A_r_E_nt_ that No P is in S, and this
simply converted yields No S is in P.
A simple transposition of the premisses in DI_m_A_r_I_s_ of the Fourth
Some P is in M
All M is in S
yields the premisses of DA_r_II
All M is in S
Some P is in M,
but the conclusion Some P is in S has to be simply converted.
Placed after a vowel, _p_ indicates that the proposition has to be
converted _per accidens_. Thus in FE_l_A_pt_O_n_ of the Third (MP, MS)
No M is in P
All M is in S
Some S is not in P
you have to substitute for All M is in S its converse by limitation to
get the premisses of FE_r_IO.
Two of the Minor Moods, Baroko of the Second Figure, and Bokardo
of the Third, cannot be reduced to the First Figure by the ordinary
processes of Conversion and Transposition. It is for dealing with
these intractable moods that Contraposition is required. Thus in
BA_r_O_k_O of the Second (PM, SM)
All P is in M.
Some S is not in M.
Substitute for the Major Premiss its Converse by Contraposition, and
for the Minor its Formal Obverse or Permutation, and you have FE_r_IO
of the First, with not-M as the Middle.
No not-M is in P.
Some S is in not-M,
Some S is not in P.
The processes might be indicated by the Mnemonic FA_cs_O_c_O, with
_c_ indicating the contraposition of the predicate term or Formal
Obversion.
The reduction of BO_k_A_rd_O,
Some M is not in P
All M is in S
Some S is not in P,
is somewhat more intricate. It may be indicated by DO_cs_A_m_O_sc_.
You substitute for the Major Premiss its Converse by Contraposition,
transpose the Premisses and you have DA_r_II.
All M is in S.
Some not-P is in M.
Some not-P is in S.
Convert now the conclusion by Contraposition, and you have Some S is
not in P.
The author of the Mnemonic apparently did not recognise
Contraposition, though it was admitted by Boethius; and, it being
impossible without this to demonstrate the validity of Baroko and
Bokardo by showing them to be equivalent with valid moods of the First
Figure, he provided for their demonstration by the special process
known as _Reductio ad absurdum_. B indicates that Barbara is the
medium.
The rationale of the process is this. It is an imaginary opponent that
you reduce to an absurdity or self-contradiction. You show that it
is impossible with consistency to admit the premisses and at the same
time deny the conclusion. For, let this be done; let it be admitted as
in BA_r_O_k_O that,
All P is in M
Some S is not in M,
but denied that Some S is not in P. The denial of a proposition
implies the admission of its Contradictory. If it is not true that
Some S is not in P, it must be true that All S is in P. Take this
along with the admission that All P is in M, and you have a syllogism
in BA_rb_A_r_A,
All P is in M
All S is in P,
yielding the conclusion All S is in M. If then the original conclusion
is denied, it follows that All S is in M. But this contradicts the
Minor Premiss, which has been admitted to be true. It is thus shown
that an opponent cannot admit the premisses and deny the conclusion
without contradicting himself.
The same process may be applied to Bokardo.
Some M is not in P.
All M is in S.
Some S is not in P.
Deny the conclusion, and you must admit that All S is in P. Syllogised
in Barbara with All M is in S, this yields the conclusion that All M
is in P, the contradictory of the Major Premiss.
The beginner may be reminded that the argument _ad absurdum_ is not
necessarily confined to Baroko and Bokardo. It is applied to them
simply because they are not reducible by the ordinary processes to the
First Figure. It might be applied with equal effect to other Moods,
DI_m_A_r_I_s_, _e.g._, of the Third.
Some M is in P.
All M is in S.
Some S is in P.
Let Some S is in P be denied, and No S is in P must be admitted. But
if No S is in P and All M is in S, it follows (in Celarent) that No M
is in P, which an opponent cannot hold consistently with his admission
that Some M is in P.
The beginner sometimes asks: What is the use of reducing the Minor
Figures to the First? The reason is that it is only when the relations
between the terms are stated in the First Figure that it is at once
apparent whether or not the argument is valid under the Axiom or
_Dictum de Omni_. It is then undeniably evident that if the Dictum
holds the argument holds. And if the Moods of the First Figure hold,
their equivalents in the other Figures must hold too.
Aristotle recognised only two of the Minor Figures, the Second and
Third, and thus had in all only fourteen valid moods.
The recognition of the Fourth Figure is attributed by Averroes to
Galen. Averroes himself rejects it on the ground that no arguments
expressed naturally, that is, in accordance with common usage, fall
into that form. This is a sufficient reason for not spending time upon
it, if Logic is conceived as a science that has a bearing upon the
actual practice of discussion or discursive thought. And this was
probably the reason why Aristotle passed it over.
If however the Syllogism of Terms is to be completed as an abstract
doctrine, the Fourth Figure must be noticed as one of the forms of
premisses that contain the required relation between the extremes.
There is a valid syllogism between the extremes when the relations
of the three terms are as stated in certain premisses of the Fourth
Figure.
III.--THE SORITES.
A chain of Syllogisms is called a Sorites. Thus:--
All A is in B.
All B is in C.
All C is in D.
:
:
:
:
All X is in Z.
[.'.] All A is in Z.
A Minor Premiss can thus be carried through a series of Universal
Propositions each serving in turn as a Major to yield a conclusion
which can be syllogised with the next. Obviously a Sorites may contain
one particular premiss, provided it is the first; and one universal
negative premiss, provided it is the last. A particular or a negative
at any other point in the chain is an insuperable bar.
[Footnote 1: [Greek: Hotan oun horoi treis autôs echôsi pros
allêlous ôste ton eschaton en holô einai tô mesô, kai ton meson
en holô tô krôtô ê einai ê mê einai, anankê tôn
akrôn einai syllogismon teleion.] (Anal. Prior., i. 4.)]
CHAPTER III.
THE DEMONSTRATION OF THE SYLLOGISTIC MOODS.--THE CANONS OF THE
SYLLOGISM.
How do we know that the nineteen moods are the only possible forms of
valid syllogism?
Aristotle treated this as being self-evident upon trial and simple
inspection of all possible forms in each of his three Figures.
Granted the parity between predication and position in or out of
a limited enclosure (term, [Greek: horos]), it is a matter of the
simplest possible reasoning. You have three such terms or enclosures,
S, P and M; and you are given the relative positions of two of them to
the third as a clue to their relative positions to one another. Is
S in or out of P, and is it wholly in or wholly out or partly in or
partly out? You know how each of them lies toward the third: when can
you tell from this how S lies towards P?
We have seen that when M is wholly in or out of P, and S wholly or
partly in M, S is wholly or partly in or out of P.
Try any other given positions in the First Figure, and you find that
you cannot tell from them how S lies relatively to P. Unless the Major
Premiss is Universal, that is, unless M lies wholly in or out of
P, you can draw no conclusion, whatever the Minor Premiss may give.
Given, _e.g._, All S is in M, it may be that All S is in P, or that No
S is in P, or that Some S is in P, or that Some S is not in P.
[Illustration:
Circles of M and P, overlapping,
with 3 instances of a circle of S:
1. S in M, but not in P;
2. S in the overlap of M and P;
3. S in M, some S in P.
]
Again, unless the Minor Premiss is affirmative, no matter what the
Major Premiss may be, you can draw no conclusion. For if the Minor
Premiss is negative, all that you know is that All S or Some S lies
somewhere outside M; and however M may be situated relatively to P,
that knowledge cannot help towards knowing how S lies relatively to P.
All S may be P, or none of it, or part of it. Given all M is in P; the
All S (or Some S) which we know to be outside of M may lie anywhere in
P or out of it.
[Illustration:
Concentric circles of P and M, M in center,
with 5 instances of circle of S:
1. S wholly outside P and M;
2. S partly overlapping both P and M, and partly outside both;
3. S overlapping P, but outside M;
4. S wholly within P, but wholly outside M;
5. S touching circle of P, but outside both circles.
]
Similarly, in the Second Figure, trial and simple inspection of all
possible conditions shows that there can be no conclusion unless the
Major Premiss is universal, and one of the premisses negative.
Another and more common way of eliminating the invalid forms,
elaborated in the Middle Ages, is to formulate principles applicable
irrespective of Figure, and to rule out of each Figure the moods that
do not conform to them. These regulative principles are known as The
Canons of the Syllogism.
_Canon I._ In every syllogism there should be three, and not more than
three, terms, and the terms must be used throughout in the same sense.
It sometimes happens, owing to the ambiguity of words, that there seem
to be three terms when there are really four. An instance of this is
seen in the sophism:--
He who is most hungry eats most.
He who eats least is most hungry.
[.'.] He who eats least eats most.
This Canon, however, though it points to a real danger of error in the
application of the syllogism to actual propositions, is superfluous
in the consideration of purely formal implication, it being a primary
assumption that terms are univocal, and remain constant through any
process of inference.
Under this Canon, Mark Duncan says (_Inst. Log._, iv. 3, 2), is
comprehended another commonly expressed in this form: There should be
nothing in the conclusion that was not in the premisses: inasmuch as
if there were anything in the conclusion that was in neither of the
premisses, there would be four terms in the syllogism.
The rule that in every syllogism there must be three, and only three,
propositions, sometimes given as a separate Canon, is only a corollary
from Canon I.
_Canon II._ The Middle Term must be distributed once at least in the
Premisses.
The Middle Term must either be wholly in, or wholly out of, one or
other of the Extremes before it can be the means of establishing a
connexion between them. If you know only that it is partly in both,
you cannot know from that how they lie relatively to one another: and
similarly if you know only that it is partly outside both.
The Canon of Distributed Middle is a sort of counter-relative
supplement to the _Dictum de Omni_. Whatever is predicable of a whole
distributively is predicable of all its several parts. If in neither
premiss there is a predication about the whole, there is no case for
the application of the axiom.
_Canon III._ No term should be distributed in the conclusion that was
not distributed in the premisses.
If an assertion is not made about the whole of a term in the
premisses, it cannot be made about the whole of that term in the
conclusion without going beyond what has been given.
The breach of this rule in the case of the Major term is technically
known as the Illicit Process of the Major: in the case of the Minor
term, Illicit Process of the Minor.
Great use is made of this canon in cutting off invalid moods. It
must be remembered that the Predicate term is "distributed" or taken
universally in O (Some S is not in P) as well as in E (No S is in P);
and that P is never distributed in affirmative propositions.
_Canon IV._ No conclusion can be drawn from two negative premisses.
Two negative premisses are really tantamount to a declaration that
there is no connexion whatever between the Major and Minor (as
quantified in the premisses) and the term common to both premisses; in
short, that this is not a Middle term--that the condition of a valid
Syllogism does not exist.
There is an apparent exception to this when the real Middle in an
argument is a contrapositive term, not-M. Thus:--
Nobody who is not thirsty is suffering from fever.
This person is not thirsty.
[.'.] He is not suffering from fever.
But in such cases it is really the absence of a quality or rather
the presence of an opposite quality on which we reason; and the Minor
Premiss is really Affirmative of the form S is in not-M.
_Canon V._ If one premiss is negative, the conclusion must be
negative.
If one premiss is negative, one of the Extremes must be excluded in
whole or in part from the Middle term. The other must therefore (under
Canon IV.) declare some coincidence between the Middle term and the
other extreme; and the conclusion can only affirm exclusion in whole
or in part from the area of this coincidence.
_Canon VI._ No conclusion can be drawn from two particular premisses.
This is evident upon a comparison of terms in all possible positions,
but it can be more easily demonstrated with the help of the preceding
canons. The premisses cannot both be particular and yield a conclusion
without breaking one or other of those canons.
Suppose both are affirmative, II, the Middle is not distributed in
either premiss.
Suppose one affirmative and the other negative, IO, or OI. Then,
whatever the Figure may be, that is, whatever the order of the terms,
only one term can be distributed, namely, the predicate of O. This
(Canon II.) must be the Middle. But in that case there must be Illicit
Process of the Major (Canon III.), for one of the premisses being
negative, the conclusion is negative (Canon V.), and P its predicate
is distributed. Briefly, in a negative mood, both Major and Middle
must be distributed, and if both premisses are particular this cannot
be.
_Canon VII._ If one Premiss is particular the conclusion is
particular.
This canon is sometimes combined with what we have given as Canon V.,
in a single rule: "The conclusion follows the weaker premiss".
It can most compendiously be demonstrated with the help of the
preceding canons.
Suppose both premisses affirmative, then, if one is particular, only
one term can be distributed in the premisses, namely, the subject
of the Universal affirmative premiss. By Canon II., this must be the
Middle, and the Minor, being undistributed in the Premisses, cannot
be distributed in the conclusion. That is, the conclusion cannot be
Universal--must be particular.
Suppose one Premiss negative, the other affirmative. One premiss being
negative, the conclusion must be negative, and P must be distributed
in the conclusion. Before, then, the conclusion can be universal, all
three terms, S, M, and P, must, by Canons II. and III., be distributed
in the premisses. But whatever the Figure of the premisses, only two
terms can be distributed. For if one of the Premisses be O, the other
must be A, and if one of them is E, the other must be I. Hence the
conclusion must be particular, otherwise there will be illicit process
of the Minor, or of the Major, or of the Middle.
The argument may be more briefly put as follows:
In an affirmative mood, with one premiss particular, only one term can
be distributed in the premisses, and this cannot be the Minor without
leaving the Middle undistributed. In a negative mood, with one premiss
particular, only two terms can be distributed, and the Minor cannot
be one of them without leaving either the Middle or the Major
undistributed.
Armed with these canons, we can quickly determine, given any
combination of three propositions in one of the Figures, whether it is
or is not a valid Syllogism.
Observe that though these canons hold for all the Figures, the Figure
must be known, in all combinations containing A or O, before we can
settle a question of validity by Canons II. and III., because
the distribution of terms in A and O depends on their order in
predication.
Take AEE. In Fig. I.--
All M is in P
No S is in M
No S is in P--
the conclusion is invalid as involving an illicit process of the
Major. P is distributed in the conclusion and not in the premisses.
In Fig. II. AEE--
All P is in M
No S is in M
No S is in P--
the conclusion is valid (Camestres).
In Fig. III. AEE--
All M is in P
No M is in S
No S is in P--
the conclusion is invalid, there being illicit process of the Major.
In Fig. IV. AEE is valid (Camenes).
Take EIO. A little reflection shows that this combination is valid in
all the Figures if in any, the distribution of the terms in both cases
not being affected by their order in predication. Both E and I are
simply convertible. That the combination is valid is quickly seen
if we remember that in negative moods both Major and Middle must be
distributed, and that this is done by E.
EIE is invalid, because you cannot have a universal conclusion with
one premiss particular.
AII is valid in Fig. I. or Fig. III., and invalid in Figs. II. and
IV., because M is the subject of A in I. and III. and predicate in II.
and IV.
OAO is valid only in Fig. III., because only in that Figure would this
combination of premisses distribute both M and P.
Simple exercises of this kind may be multiplied till all possible
combinations are exhausted, and it is seen that only the recognised
moods stand the test.
If a more systematic way of demonstrating the valid moods is desired,
the simplest method is to deduce from the Canons special rules for
each Figure. Aristotle arrived at these special rules by simple
inspection, but it is easier to deduce them.
I. In the First Figure, the Major Premiss must be Universal, and the
Minor Premiss affirmative.
To make this evident by the Canons, we bear in mind the Scheme or
Figure--
M in P
S in M--
and try the alternatives of Affirmative Moods and Negative Moods.
Obviously in an affirmative mood the Middle is undistributed unless
the Major Premiss is Universal. In a negative mood, (1) If the Major
Premiss is O, the Minor must be affirmative, and M is undistributed;
(2) if the Major Premiss is I, M may be distributed by a negative
Minor Premiss, but in that case there would be an illicit process of
the Major--P being distributed in the conclusion (Canon V.) and not in
the Premisses. Thus the Major Premiss can neither be O nor I, and must
therefore be either A or E, _i.e._, must be Universal.
That the Minor must be affirmative is evident, for if it were
negative, the conclusion must be negative (Canon V.) and the Major
Premiss must be affirmative (Canon IV.), and this would involve
illicit process of the Major, P being distributed in the conclusion
and not in the Premisses.
These two special rules leave only four possible valid forms in the
First Figure. There are sixteen possible combinations of premisses,
each of the four types of proposition being combinable with itself and
with each of the others.
AA EA IA OA
AE EE IE OE
AI EI II OI
AO EO IO OO
Special Rule I. wipes out the columns on the right with the particular
major premisses; and AE, EE, AO, and EO are rejected by Special Rule
II., leaving BA_rb_A_r_A, CE_l_A_r_E_nt_, DA_r_II and FE_r_IO.
II. In the Second Figure, only Negative Moods are possible, and the
Major Premiss must be universal.
Only Negative moods are possible, for unless one premiss is negative,
M being the predicate term in both--
P in M
S in M--
is undistributed.
Only negative moods being possible, there will be illicit process of
the Major unless the Major Premiss is universal, P being its subject
term.
These special rules reject AA and AI, and the two columns on the
right.
To get rid of EE and EO, we must call in the general Canon IV.;
which leaves us with EA, AE, EI, and AO--CE_s_A_r_E, CA_m_E_str_E_s_,
FE_st_I_n_O BA_r_O_k_O.
III. In the Third Figure, the Minor Premiss must be affirmative.
Otherwise, the conclusion would be negative, and the Major Premiss
affirmative, and there would be illicit process of the Major, P being
the predicate term in the Major Premiss.
M in P
M in S.
This cuts off AE, EE, IE, OE, AO, EO, IO, OO,--the second and fourth
rows in the above list.
II and OI are inadmissible by Canon VI.; which leaves AA, IA, AI,
EA, OA, EI--DA_r_A_pt_I, DI_s_A_m_I_s_, DA_t_I_s_I, FE_l_A_pt_O_n_,
BO_k_A_rd_O, FE_r_I_s_O--three affirmative moods and three negative.
IV. The Fourth Figure is fenced by three special rules. (1) In
negative moods, the Major Premiss is universal. (2) If the Minor
is negative, both premisses are universal. (3) If the Major is
affirmative, the Minor is universal.
(1) Otherwise, the Figure being
P in M
M in S,
there would be illicit process of the Major.
(2) The Major must be universal by special rule (1), and if the Minor
were not also universal, the Middle would be undisturbed.
(3) Otherwise M would be undistributed.
Rule (1) cuts off the right-hand column, OA, OE, OI, and OO; also IE
and IO.
Rule (2) cuts off AO, EO.
Rule (3) cuts off AI, II.
EE goes by general Canon IV.; and we are left with AA, AE, IA, EA,
EI--B_r_A_m_A_nt_I_p_, CA_m_E_n_E_s_, DI_m_A_r_I_s_, FE_s_A_p_O,
F_r_E_s_I_s_O_n_.
CHAPTER IV.
THE ANALYSIS OF ARGUMENTS INTO SYLLOGISTIC FORMS.
Turning given arguments into syllogistic form is apt to seem as
trivial and useless as it is easy and mechanical. In most cases the
necessity of the conclusion is as apparent in the plain speech form as
in the artificial logical form. The justification of such exercises
is that they give familiarity with the instrument, serving at the same
time as simple exercises in ratiocination: what further uses may be
made of the instrument once it is mastered, we shall consider as we
proceed.
I.--FIRST FIGURE.
Given the following argument to be put into Syllogistic form: "No war
is long popular: for every war increases taxation; and the popularity
of anything that touches the pocket is short-lived".
The simplest method is to begin with the conclusion--"No war is long
popular"--No S is P--then to examine the argument to see whether it
yields premisses of the necessary form. Keeping the form in mind,
Celarent of Fig. I.--
No M is P
All S is M
No S is P--
we see at once that "Every war increases taxation" is of the form All
S is M. Does the other sentence yield the Major Premiss No M is P,
when M represents the increasing of taxation, _i.e._, a class bounded
by that attribute? We see that the last sentence of the argument is
equivalent to saying that "Nothing that increases taxation is long
popular"; and this with the Minor yields the conclusion in Celarent.
Nothing that increases taxation is long popular.
Every war increases taxation.
No war is long popular.
Observe, now, what in effect we have done in thus reducing the
argument to the First Figure. In effect, a general principle being
alleged as justifying a certain conclusion, we have put that principle
into such a form that it has the same predicate with the conclusion.
All that we have then to do in order to inspect the validity of the
argument is to see whether the subject of the conclusion is contained
in the subject of the general principle. Is war one of the things that
increase taxation? Is it one of that class? If so, then it cannot long
be popular, long popularity being an attribute that cannot be affirmed
of any of that class.
Reducing to the first figure, then, amounts simply to making the
predication of the proposition alleged as ground uniform with the
conclusion based upon it. The minor premiss or applying proposition
amounts to saying that the subject of the conclusion is contained in
the subject of the general principle. Is the subject of the conclusion
contained in the subject of the general principle when the two have
identical predicates? If so, the argument falls at once under the
_Dictum de Omni et Nullo_.
Two things may be noted concerning an argument thus simplified.
1. It is not necessary, in order to bring an argument under the
_dictum de omni_, to reduce the predicate to the form of an extensive
term. In whatever form, abstract or concrete, the predication is made
of the middle term, it is applicable in the same form to that which is
contained in the middle term.
2. The quantity of the Minor Term does not require special attention,
inasmuch as the argument does not turn upon it. In whatever quantity
it is contained in the Middle, in that quantity is the predicate of
the Middle predicable of it.
These two points being borne in mind, the attention may be
concentrated on the Middle Term and its relations with the extremes.
That the predicate may be left unanalysed without affecting the
simplicity of the argument or in any way obscuring the exhibition
of its turning-point, has an important bearing on the reduction of
Modals. The modality may be treated as part of the predicate without
in any way obscuring what it is the design of the syllogism to make
clear. We have only to bear in mind that however the predicate may be
qualified in the premisses, the same qualification must be transferred
to the conclusion. Otherwise we should have the fallacy of Four Terms,
_quaternio terminorum_.
To raise the question: What is the proper form for a Modal of
Possibility, A or I? is to clear up in an important respect our
conceptions of the Universal proposition, "Victories may be gained
by accident". Should this be expressed as A or I? Is the predicate
applicable to All victories or only to Some? Obviously the meaning is
that of any victory it may be true that it was gained by accident, and
if we treat the "mode" as part of the predicate term "things that may
be gained by accident," the form of the proposition is All S is in P.
But, it may be asked, does not the proposition that victories may be
gained by accident rest, as a matter of fact, on the belief that some
victories have been gained in this way? And is not, therefore, the
proper form of proposition Some S is P?
This, however, is a misunderstanding. What we are concerned with is
the formal analysis of propositions as given. And Some victories have
been gained by accident is not the formal analysis of Victories may be
gained by accident. The two propositions do not give the same meaning
in different forms: the meaning as well as the form is different.
The one is a statement of a matter of fact: the other of an inference
founded on it. The full significance of the Modal proper may be stated
thus: In view of the fact that some victories have been gained by
accident, we are entitled to say of any victory, in the absence of
certain knowledge, that it may be one of them.
A general proposition, in short, is a proposition about a genus, taken
universally.
II.--SECOND FIGURE.
For testing arguments from general principles, the First Figure is the
simplest and best form of analysis.
But there is one common class of arguments that fall naturally,
as ordinarily expressed, into the Second Figure, namely, negative
conclusions from the absence of distinctive signs or symptoms, or
necessary conditions.
Thirst, for example, is one of the symptoms of fever: if a patient is
not thirsty, you can conclude at once that his illness is not fever,
and the argument, fully expressed, is in the Second Figure.
All fever-stricken patients are thirsty.
This patient is not thirsty.
[.'.] He is not fever-stricken.
Arguments of this type are extremely common.
Armed with the general principle that ill-doers are ill-dreaders,
we argue from a man's being unsuspicious that he is not guilty.
The negative diagnosis of the physician, as when he argues from the
absence of sore throat or the absence of a white speck in the throat
that the case before him is not one of scarlatina or diphtheria,
follows this type: and from its utility in making such arguments
explicit, the Second Figure may be called the Figure of Negative
Diagnosis.
It is to be observed, however, that the character of the argument is
best disclosed when the Major Premiss is expressed by its Converse by
Contraposition. It is really from the absence of a symptom that the
physician concludes; as, for example: "No patient that has not a sore
throat is suffering from scarlatina". And the argument thus expressed
is in the First Figure. Thus the reduction of Baroko to the First
Figure by contraposition of the Middle is vindicated as a really
useful process. The real Middle is a contrapositive term, and the form
corresponds more closely to the reasoning when the argument is put in
the First Figure.
The truth is that if the positive term or sign or necessary condition
is prominent as the basis of the argument, there is considerable risk
of fallacy. Sore throat being one of the symptoms of scarlatina, the
physician is apt on finding this symptom present to jump to a positive
conclusion. This is equivalent technically to drawing a positive
conclusion from premisses of the Second Figure.
All scarlatina patients have sore throat.
This patient has sore throat.
A positive conclusion is technically known as a Non-Sequitur (Doesn't
follow). So with arguments from the presence of a necessary condition
which is only one of many. Given that it is impossible to pass without
working at the subject, or that it is impossible to be a good marksman
without having a steady hand, we are apt to argue that given also the
presence of this condition, a conclusion is implicated. But really the
premisses given are only two affirmatives of the Second Figure.
"It is impossible to pass without working at the subject."
This, put into the form No not-M is P, is to say that "None who
have not worked can pass". This is equivalent, as the converse by
contraposition, with--
All capable of passing have worked at the subject.
But though Q has worked at the subject, it does not follow that he is
capable of passing. Technically the middle is undistributed. On the
other hand, if he has not worked at the subject, it follows that he
is not capable of passing. We can draw a conclusion at once from the
absence of the necessary condition, though none can be drawn from its
presence alone.
THIRD FIGURE.
Arguments are sometimes advanced in the form of the Third Figure. For
instance: Killing is not always murder: for tyrannicide is not murder,
and yet it is undoubtedly killing. Or again: Unpleasant things
are sometimes salutary: for afflictions are sometimes so, and no
affliction can be called pleasant.
These arguments, when analysed into terms, are, respectively, Felapton
and Disamis.
No tyrannicide is murder;
All tyrannicide is killing;
Some killing is not murder.
Some afflictions are salutary things;
All afflictions are unpleasant things;
Some unpleasant things are salutary things.
The syllogistic form cannot in such cases pretend to be a
simplification of the argument. The argument would be equally
unmistakable if advanced in this form: Some S is not P, for example,
M. Some killing is not murder, _e.g.,_ tyrannicide. Some unpleasant
things are salutary, _e.g.,_ some afflictions.
There is really no "deduction" in the third figure, no leading down
from general to particular. The middle term is only an example of the
minor. It is the syllogism of Contradictory Examples.
In actual debate examples are produced to disprove a universal
assertion, affirmative or negative. Suppose it is maintained that
every wise man has a keen sense of humour. You doubt this: you
produce an instance of the opposite, say Milton. The force of your
contradictory instance is not increased by exhibiting the argument in
syllogistic form: the point is not made clearer.
The Third Figure was perhaps of some use in Yes and No Dialectic.
When you had to get everything essential to your conclusion definitely
admitted, it was useful to know that the production of an example to
refute a generality involved the admission of two propositions. You
must extract from your opponent both that Milton was a wise man, and
that Milton had not a keen sense of humour, before you could drive him
from the position that all wise men possess that quality.
_Examples for Analysis._
Scarlet flowers have no fragrance: this flower has no fragrance: does
it follow that this flower is of a scarlet colour?
Interest in the subject is an indispensable condition of learning
easily; Z is interested in the subject: he is bound, therefore, to
learn easily.
It is impossible to be a good shot without having a steady hand: John
has a steady hand: he is capable, therefore, of becoming a good shot.
Some victories have been won by accident; for example, Maiwand.
Intemperance is more disgraceful than cowardice, because people have
more opportunities of acquiring control of their bodily appetites.
"Some men are not fools, yet all men are fallible." What follows?
"Some men allow that their memory is not good: every man believes in
his own judgment." What is the conclusion, and in what Figure and Mood
may the argument be expressed?
"An honest man's the noblest work of God: Z is an honest man":
therefore, he is--what?
Examine the logical connexion between the following "exclamation"
and "answer": "But I hear some one exclaiming that wickedness is not
easily concealed. To which I answer, Nothing great is easy."
"If the attention is actively aroused, sleep becomes impossible: hence
the sleeplessness of anxiety, for anxiety is a strained attention upon
an impending disaster."
"To follow truth can never be a subject of regret: free inquiry does
lead a man to regret the days of his childish faith; therefore it is
not following truth."--_J. H. Newman._
He would not take the crown: Therefore 'tis certain he was not
ambitious.
As he was valiant, I honour him; as he was ambitious, I slew him.
The Utopians learned the language of the Greeks with more readiness
because they were originally of the same race with them.
Nothing which is cruel can be expedient, for cruelty is most revolting
to the nature of man.
"The fifth century saw the foundation of the Frank dominion in Gaul,
and the first establishment of the German races in Britain. The former
was effected in a single long reign, by the energy of one great ruling
tribe, which had already modified its traditional usages, and now, by
the adoption of the language and religion of the conquered, prepared
the way for a permanent amalgamation with them." In the second of the
above sentences a general proposition is assumed. Show in syllogistic
form how the last proposition in the sentence depends upon it.
"I do not mean to contend that active benevolence may not hinder a
man's advancement in the world: for advancement greatly depends upon
a reputation for excellence in some one thing of which the world
perceives that it has present need: and an obvious attention to other
things, though perhaps not incompatible with the excellence itself,
may easily prevent a person from obtaining a reputation for it." Pick
out the propositions here given as interdependent. Examine whether the
principle alleged is sufficiently general to necessitate a conclusion.
In what form would it be so?
CHAPTER V.
ENTHYMEMES.
There is a certain variety in the use of the word Enthymeme among
logicians. In the narrowest sense, it is a valid formal syllogism,
with one premiss suppressed. In the widest sense it is simply an
argument, valid or invalid, formal in expression or informal, with
only one premiss put forward or hinted at, the other being held in the
mind ([Greek: en thymô]). This last is the Aristotelian sense.
It is only among formal logicians of the straitest sect that the
narrowest sense prevails. Hamilton divides Enthymemes into three
classes according as it is the Major Premiss, the Minor Premiss, or
the Conclusion that is suppressed. Thus, a full syllogism being:--
All liars are cowards:
Caius is a liar:
[.'.] Caius is a coward:--
this may be enthymematically expressed in three ways.
I. Enthymeme of the First Order (_Major understood_).
Caius is a coward; for Caius is a liar.
II. Enthymeme of the Second Order (_Minor understood_).
Caius is a coward; for all liars are cowards.
III. Enthymeme of the Third Order (_Conclusion understood_).
All liars are cowards, and Caius is a liar.
The Third Order is a contribution of Hamilton's own. It is
superfluous, inasmuch as the conclusion is never suppressed except as
a rhetorical figure of speech. Hamilton confines the word Enthymeme
to valid arguments, in pursuance of his view that Pure Logic has no
concern with invalid arguments.
Aristotle used Enthymeme in the wider sense of an elliptically
expressed argument. There has been some doubt as to the meaning of his
definition, but that disappears on consideration of his examples.
He defines an Enthymeme (Prior Analyt., ii. 27) as "a syllogism from
probabilities or signs" ([Greek: syllogismos ex eikotôn ê sêmeiôn]).
The word syllogism in this connexion is a little puzzling. But it is
plain from the examples he gives that he meant here by syllogism not
even a correct reasoning, much less a reasoning in the explicit form
of three terms and three propositions. He used syllogism, in fact, in
the same loose sense in which we use the words reasoning and argument,
applying without distinction of good and bad.
The sign, he says, is taken in three ways, in as many ways as there
are Syllogistic Figures.
(1) A sign interpreted in the First Figure is conclusive. Thus: "This
person has been drowned, for he has froth in the trachea". Taken in
the First Figure with "All who have froth in the trachea have been
drowned" as a major premiss, this argument is valid. The sign is
conclusive.
(2) "This patient is fever-stricken, for he is thirsty." Assumed that
"All fever-stricken patients are thirsty," this is an argument in the
Second Figure, but it is not a valid argument. Thirst is a sign or
symptom of fever, but not a conclusive sign, because it is indicative
of other ailments also. Yet the argument has a certain probability.
(3) "Wise men are earnest ([Greek: spoudaioi]), for Pittacus is
earnest." Here the suppressed premiss is that "Pittacus is wise".
Fully expressed, the argument is in the Third Figure:--
Pittacus is earnest.
Pittacus is wise.
[.'.] Wise men are earnest.
Here again the argument is inconclusive and yet it has a certain
probability. The coincidence of wisdom with earnestness in one notable
example lends a certain air of probability to the general statement.
Such are Aristotle's examples or strict parallels to them. The
examples illustrate also what he says in his _Rhetoric_ as to the
advantages of enthymemes. For purposes of persuasion enthymemes are
better than explicit syllogisms, because any inconclusiveness there
may be in the argument is more likely to pass undetected. As we shall
see, one main use of the Syllogism is to force tacit assumptions into
light and so make their true connexion or want of connexion apparent.
In Logic enthymemes are recognised only to be shown up: the elliptical
expression is a cover for fallacy, which it is the business of the
logician to strip off.
In Aristotle's examples one of the premisses is expressed. But often
the arguments of common speech are even less explicit than this. A
general principle is vaguely hinted at: a subject is referred to a
class the attributes of which are assumed to be definitely known.
Thus:--
He was too ambitious to be scrupulous in his choice of means.
He was too impulsive not to have made many blunders.
Each of these sentences contains a conclusion and an enthymematic
argument in support of it. The hearer is understood to have in his
mind a definite idea of the degree of ambition at which a man ceases
to be scrupulous, or the degree of impulsiveness that is incompatible
with accuracy.
One form of enthymeme is so common in modern rhetoric as to deserve
a distinctive name. It may be called the ENTHYMEME OF THE ABSTRACTLY
DENOMINATED PRINCIPLE. A conclusion is declared to be at variance with
the principles of Political Economy, or contrary to the doctrine of
Evolution, or inconsistent with Heredity, or a violation of the sacred
principle of Freedom of Contract. It is assumed that the hearer is
familiar with the principles referred to. As a safeguard against
fallacy, it may be well to make the principle explicit in a
proposition uniform with the conclusion.
CHAPTER VI.
THE UTILITY OF THE SYLLOGISM.
The main use of the Syllogism is in dealing with incompletely
expressed or elliptical arguments from general principals. This may be
called Enthymematic argument, understanding by Enthymeme an argument
with only one premiss put forward or hinted at, the other being held
in the mind. In order to test whether such reasoning is sound
or unsound, it is of advantage to make the argument explicit in
Syllogistic form.
There have been heaps and mazes of discussion about the use of the
Syllogism, much of it being profitable as a warning against the
neglect of Formal Logic. Again and again it has been demonstrated that
the Syllogism is useless for certain purposes, and from this it has
been concluded that the Syllogism is of no use at all.
The inventor of the Syllogism had a definite practical purpose, to get
at the simplest, most convincing, undeniable and irresistible way
of putting admitted or self-evident propositions so that their
implication should be apparent. His ambition was to furnish a method
for the Yes and No Dialectician, and the expounder of science from
self-evident principles. A question being put up for discussion, it
was an advantage to analyse it, and formulate the necessary premisses:
you could then better direct your interrogations or guard your
answers. The analysis is similarly useful when you want to construct
an argument from self-evident principles.
All that the Syllogism could show was the consistency of the premisses
with the conclusion. The conclusion could not go beyond the premisses,
because the questioner could not go beyond the admissions of the
respondent. There is indeed an advance, but not an advance upon the
two premisses taken together. There is an advance upon any one of
them, and this advance is made with the help of the other. Both must
be admitted: a respondent may admit one without being committed to
the conclusion. Let him admit both and he cannot without
self-contradiction deny the conclusion. That is all.
Dialectic of the Yes and No kind is no longer practised. Does any
analogous use for the Syllogism remain? Is there a place for it as a
safeguard against error in modern debate? As a matter of fact it
is probably more useful now than it was for its original purpose,
inasmuch as modern discussion, aiming at literary grace and spurning
exact formality as smacking of scholasticism and pedantry, is
much more flabby and confused. In the old dialectic play there was
generally a clear question proposed. The interrogative form forced
this much on the disputants. The modern debater of the unpedantic,
unscholastic school is not so fettered, and may often be seen
galloping wildly about without any game in sight or scent, his maxim
being to--
Spur boldly on, and dash through thick and thin,
Through sense and nonsense, never out nor in.
Now the syllogistic analysis may often be of some use in helping us
to keep a clear head in the face of a confused argument. There is a
brilliant defence of the syllogism as an analysis of arguments in the
_Westminster Review_ for January, 1828. The article was a notice of
Whately's Logic: it was written by J. S. Mill. For some reason it
has never been reprinted, but it puts the utility of the Syllogism on
clearer ground than Mill afterwards sought for it.
Can a fallacy in argument be detected at once? Is common-sense
sufficient? Common-sense would require some inspection. How would
it proceed? Does common-sense inspect the argument in a lump or
piecemeal? All at once or step by step? It analyses. How? First, it
separates out the propositions which contribute to the conclusion from
those which do not, the essential from the irrelevant. Then it
states explicitly all that may have been assumed tacitly. Finally, it
enumerates the propositions in order.
Some such procedure as this would be adopted by common-sense in
analysing an argument. But when common-sense has done this, it has
exhibited the argument in a series of syllogisms.
Such is Mill's early defence of the Syllogism. It is weak only in one
point, in failing to represent how common-sense would arrive at the
peculiar syllogistic form. It is the peculiar form of logical analysis
that is the distinction of the syllogism. When you have disentangled
the relevant propositions you have not necessarily put them in this
form. The arguments given in text-books to be cast into syllogistic
form, consist only as a rule of relevant propositions, but they are
not yet formal syllogisms. But common-sense had only one other step to
make to reach the distinctive form. It had only to ask after analysing
the argument, Is there any form of statement specially suitable
for exhibiting the connexion between a conclusion and the general
principle on which it is alleged to depend? Ask yourself the question,
and you will soon see that there would be an obvious advantage in
making the conclusion and the general principle uniform, in stating
them with the same predicate. But when you do this, as I have already
shown (p. 197) you state the argument in the First Figure of the
Syllogism.
It must, however, be admitted that it is chiefly for exhibiting, or
forcing into light, tacit or lurking assumptions that the Syllogistic
form is of use. Unless identity of meaning is disguised or distorted
by puzzling difference of language, there is no special illuminative
virtue in the Syllogism. The argument in a Euclidean demonstration
would not be made clearer by being cast into formal Syllogisms.
Again, when the subject matter is simple, the Syllogistic form is not
really required for protection against error. In such enthymemes as
the following for example:--
She must be clever: she is so uncompromisingly ugly.
Romeo must be in love: for is he not seventeen?
it is plain to the average intelligence without any knowledge of
Syllogism that the argument takes for granted a general proposition
and what the general proposition is.
Another thing is plain to the average intelligence, perhaps plainer
than to a proficient in the use of the Syllogism. Clearly we cannot
infer with certainty that a woman is clever because she is ugly,
unless it is the case that all ugly women are clever. But a
Syllogiser, seeing that no certain conclusion can be drawn except
upon this condition, is apt to dismiss the argument as altogether
worthless. This may be specified as an error incident to the practice
of the Syllogism, that it inclines us to look for necessarily
conclusive premisses, and to deny all weight to anything short of
this. Now in ordinary life it is comparatively seldom that such
premisses can be found. We are obliged to proceed on maxims that are
not of universal scope, and which lend only a more or less strong
colour of probability to cases that can be brought under them. "A
little learning is a dangerous thing;" "Haste makes waste;" "Slowness
of speech is a sign of depth of thought;" "Vivacity is a sign of
shallowness:" such are the "endoxes" or commonplaces of popular
knowledge that men bring to bear in daily life. They are not true for
all cases, but some of them are true for most or for a good many, and
they may be applied with a certain probability though they are not
rigidly conclusive. The plain man's danger is that he apply them
unthinkingly as universals: the formal logician's danger is that,
seeing them to be inapplicable as universals, he dismisses them as
being void of all argumentative force.
It helps to fix the limits of Formal Logic to remember that it lies
outside its bounds to determine the degree of probability attaching
to the application of approximate truths, such as are the staple of
arguments in ordinary affairs. Formal Logic, we may repeat, is
not concerned with degrees of truth or falsehood, probability
or improbability. It merely shows the interdependency of certain
arguments, the consistency of conclusion with premisses.
This, however, is a function that might easily be underrated. Its
value is more indirect than direct. In showing what is required for a
certain conclusion, it puts us on the road to a more exact estimate of
the premisses alleged, a sounder judgment of their worth. Well begun
is half done: in undertaking the examination of any argument from
authority, a formal syllogism is a good beginning.
CHAPTER VII.
CONDITIONAL ARGUMENTS.--HYPOTHETICAL SYLLOGISM, DISJUNCTIVE SYLLOGISM,
AND DILEMMA.
The justification of including these forms of argument in Logic is
simply that they are sometimes used in debate, and that confusion
may arise unless the precise meaning of the premisses employed
is understood. Aristotle did not include them as now given in his
exposition of the Syllogism, probably because they have no connexion
with the mode of reasoning together to which he appropriated the
title. The fallacies connected with them are of such a simple kind
that to discuss as a question of method the precise place they should
occupy in a logical treatise is a waste of ingenuity.[1]
I.--HYPOTHETICAL SYLLOGISMS.
If A is B, C is D |
A is B } MODUS
[.'.]C is D | PONENS.
If A is B, C is D |
C is not D } MODUS
[.'.]A is not B | TOLLENS.
A so-called Hypothetical Syllogism is thus seen to be a Syllogism in
which the major premiss is a HYPOTHETICAL PROPOSITION, that is to
say, a complex proposition in which two propositions are given as so
related that the truth of one follows necessarily from the truth of
the other.
Two propositions so related are technically called the ANTECEDENT or
Reason, and the CONSEQUENT.
The meaning and implication of the form, If A is B, C is D, is
expressed in what is known as the LAW OF REASON AND CONSEQUENT:--
"_When two propositions are related as Reason and Consequent, the
truth of the Consequent follows from the truth of the Antecedent,
and the falsehood of the Antecedent, from the falsehood of the
Consequent_".
If A is B, C is D, implies that If C is not D, A is not B. If this
subject is educative, it quickens the wits; if it does not quicken the
wits, it is not educative.
Admitted, then, that the law of Reason and Consequent holds
between two propositions--that If A is B, C is D: admitted also the
Antecedent, the truth of the Consequent follows. This is the MODUS
PONENS or Positive Mode, where you reach a conclusion by obtaining the
admission of the Antecedent. Admit the Antecedent and the truth of the
Consequent follows.
With the same Major Premiss, you may also, under the Law of Reason
and Consequent reach a conclusion by obtaining the denial of the
Consequent. This is the MODUS TOLLENS or Negative Mode. Deny the
Consequent and one is bound to deny the Antecedent.
But to guard against the fallacy technically known as FALLACIA
CONSEQUENTIS, we must observe what the relation of Reason and
Consequent does not imply. The truth of the Consequent does not
involve the truth of the Antecedent, and the falsehood of the
Antecedent does not involve the falsehood of the Consequent.
"If the harbour is frozen, the ships cannot come in." If the harbour
is not frozen, it does not follow that the ships can come in: they may
be excluded by other causes. And so, though they cannot come in, it
does not follow that the harbour is frozen.
QUESTIONS CONNECTED WITH HYPOTHETICAL SYLLOGISMS.
(1) _Are they properly called Syllogisms?_ This is purely a question
of Method and Definition. If we want a separate technical name for
forms of argument in which two terms are reasoned together by means of
a third, the Hypothetical Syllogism, not being in such a form, is
not properly so called. The fact is that for the purposes of the
Hypothetical Argument, we do not require an analysis into terms at
all: it is superfluous: we are concerned only with the affirmation or
denial of the constituent propositions as wholes.
But if we extend the word Syllogism to cover all arguments in which
two propositions necessarily involve a third, the Hypothetical
Argument is on this understanding properly enough called a Syllogism.
(2) _Is the inference in the Hypothetical Syllogism Mediate or
Immediate?_
To answer this question we have to consider whether the Conclusion
can be drawn from either of the two premisses without the help of the
other. If it is possible immediately, it must be educible directly
either from the Major Premiss or from the Minor.
(_a_) Some logicians argue as if the Conclusion were immediately
possible from the Major Premiss. The Minor Premiss and the Conclusion,
they urge, are simply equivalent to the Major Premiss. But this is a
misunderstanding. "If A is B, C is D," is not equivalent to "A is B,
_therefore_ C is D". "If the harbour is frozen, the ships cannot come
in" is not to say that "the harbour is frozen, and therefore," etc.
The Major Premiss merely affirms the existence of the relation of
Reason and Consequent between the two propositions. But we cannot
thereupon assert the Conclusion unless the Minor Premiss is also
conceded; that is, the inference of the Conclusion is Mediate, as
being from two premisses and not from one alone.
(_b_) Similarly with Hamilton's contention that the Conclusion
is inferrible immediately from the Minor Premiss, inasmuch as the
Consequent is involved in the Reason. True, the Consequent is involved
in the Reason: but we cannot infer from "A is B" to "C is D," unless
it is conceded that the relation of Reason and Consequent holds
between them; that is, unless the Major Premiss is conceded as well as
the Minor.
(3) _Can Hypothetical Syllogism be reduced to the Categorical Form?_
To oppose Hypothetical Syllogisms to Categorical is misleading, unless
we take note of the precise difference between them. It is only in
the form of the Major Premiss that they differ: Minor Premiss and
Conclusion are categorical in both. And the meaning of a Hypothetical
Major Premiss (unless it is a mere arbitrary convention between two
disputants, to the effect that the Consequent will be admitted if the
Antecedent is proved, or that the Antecedent will be relinquished
if the Consequent is disproved), can always be put in the form of a
general proposition, from which, with the Minor Premiss as applying
proposition, a conclusion identical with the original can be drawn in
regular Categorical form.
Thus:--
If the harbour is frozen, the ships cannot come in.
The harbour is frozen.
[.'.] The ships cannot come in.
This is a Hypothetical Syllogism, _Modus Ponens_. Express the
Hypothetical Major in the form of the general proposition which it
implies, and you reach a conclusion (in _Barbara_) which is only
grammatically different from the original.
All frozen harbours exclude ships.
The harbour is frozen.
[.'.] It excludes ships.
Again, take an example of the _Modus Tollens_--
If rain has fallen, the streets are wet.
The streets are not wet.
[.'.] Rain has not fallen.
This is reducible, by formulating the underlying proposition, to
_Camestres_ or _Baroko_ of the Second Figure.
All streets rained upon are wet.
The streets are not wet.
[.'.] They are not streets rained upon.
Hypothetical Syllogisms are thus reducible, by merely grammatical
change[2], or by the statement of self-evident implications, to the
Categorical form. And, similarly, any Categorical Syllogism may be
reduced to the Hypothetical form. Thus:--
All men are mortal.
Socrates is a man.
[.'.] Socrates is mortal.
This argument is not different, except in the expression of the Major
and the Conclusion, from the following:--
If Socrates is a man, death will overtake him.
Socrates is a man.
[.'.] Death will overtake him.
The advantage of the Hypothetical form in argument is that it is
simpler. It was much used in Mediæval Disputation, and is still more
popular than the Categorical Syllogism. Perhaps the prominence
given to Hypothetical Syllogisms as syllogisms in Post-Renaissance
text-books is due to the use of them in the formal disputations of
graduands in the Universities. It was the custom for the Disputant to
expound his argument in this form:--
If so and so is the case, such and such follows.
So and so is the case.
[.'.] Such and such follows.
To which the Respondent would reply: _Accipio antecedentem, nego
consequentiam_, and argue accordingly. Petrus Hispanus does not give
the Hypothetical Syllogism as a Syllogism: he merely explains the
true law of Reason and Consequent in connexion with the Fallacia
Consequentis in the section on Fallacies. (_Summulæ. Tractatus
Sextus._)
II.--DISJUNCTIVE SYLLOGISMS.
A Disjunctive Syllogism is a syllogism in which the Major Premiss is
a DISJUNCTIVE PROPOSITION, _i.e._, one in which two propositions are
declared to be mutually incompatible. It is of the form Either A is B,
or C is D.[3]
If the disjunction between the alternatives is really complete, the
form implies four hypothetical propositions:--
(1) If A is B, C is not D.
(2) If A is not B, C is D.
(3) If C is D, A is not B.
(4) If C is not D, A is B.
Suppose then that an antagonist has granted you a Disjunctive
Proposition, you can, using this as a Major Premiss, extract from
him four different Conclusions, if you can get him also to admit the
requisite Minors. The Mode of two of these is technically called MODUS
PONENDO TOLLENS, the mode that denies the one alternative by granting
the other--A is B, _therefore_ C is not D; C is D, _therefore_ A is
not B. The other Mode is also twice open, the MODUS TOLLENDO PONENS--A
is not B, _therefore_ C is D; C is not D, _therefore_ A is B.
Fallacy is sometimes committed through the Disjunctive form owing to
the fact that in common speech there is a tendency to use it in place
of a mere hypothetical, when there are not really two incompatible
alternatives. Thus it may be said "Either the witness is perjured,
or the prisoner is guilty," when the meaning merely is that if the
witness is not perjured the prisoner is guilty. But really there is
not a valid disjunction and a correct use of the disjunctive form,
unless four hypotheticals are implied, that is, unless the concession
of either involves the denial of the other, and the denial of either
the concession of the other. Now the prisoner may be guilty and
yet the witness be perjured; so that two of the four hypotheticals,
namely--
If the witness is perjured, the prisoner is not guilty,
If the prisoner is guilty, the witness is not perjured--
do not necessarily hold. If, then, we would guard against fallacy, we
must always make sure before assenting to a disjunctive proposition
that there is really a complete disjunction or mutual incompatibility
between the alternatives.
III.--THE DILEMMA.
A Dilemma is a combination of Hypothetical and Disjunctive
propositions.
The word has passed into common speech, and its ordinary use is a clue
to the logical structure. We are said to be in a dilemma when we
have only two courses open to us and both of them are attended by
unpleasant consequences. In argument we are in this position when we
are shut into a choice between two admissions, and either admission
leads to a conclusion which we do not like. The statement of the
alternatives as the consequences hypothetically of certain conditions
is the major premiss of the dilemma: once we admit that the relations
of Antecedent and Consequent are as stated, we are in a trap, if trap
it is: we are on the horns of the dilemma, ready to be tossed from one
to the other.
For example:--
If A is B, A is C, and if A is not B, A is D. But A either is
or is not B. Therefore, A either is C or is D.
If A acted of his own motive, he is a knave; if A did not act
of his own motive, he is a catspaw. But A either acted of his
own motive or he did not. Thereupon A is either a knave or a
catspaw.
This is an example of the _Constructive_ Dilemma, the form of it
corresponding to the common use of the word as a choice between
equally unpleasant alternatives. The standard example is the dilemma
in which the custodians of the Alexandrian Library are said to have
been put by the Caliph Omar in 640 A.D.
If your books are in conformity with the Koran, they are
superfluous; if they are at variance with it, they are
pernicious. But they must either be in conformity with the
Koran or at variance with it. Therefore they are either
superfluous or pernicious.
Where caution has to be exercised is in accepting the clauses of the
Major. We must make sure that the asserted relations of Reason and
Consequent really hold. It is there that fallacy is apt to creep in
and hide its head. The Alexandrian Librarians were rash in accepting
the first clause of the conqueror's Major: it does not follow that the
books are superfluous unless the doctrines of the Koran are not merely
sound but contain all that is worth knowing. The propounder of the
dilemma covertly assumes this. It is in the facility that it affords
for what is technically known as _Petitio Principii_ that the Dilemma
is a useful instrument for the Sophist. We shall illustrate it further
under that head.
What is known as the _Destructive_ Dilemma is of a somewhat different
form. It proceeds upon the denial of the Consequent as involving the
denial of the Antecedent. In the Major you obtain the admission that
if a certain thing holds, it must be followed by one or other of
two consequences. You then prove by way of Minor that neither of the
alternatives is true. The conclusion is that the antecedent is false.
We had an example of this in discussing whether the inference in the
Hypothetical Syllogism is Immediate. Our argument was in this form:--
If the inference is immediate, it must be drawn either from
the Major alone or from the Minor alone. But it cannot be
drawn from the Major alone, neither can it be drawn from the
Minor alone. Therefore, it is not immediate.
In this form of Dilemma, which is often serviceable for clearness
of exposition, we must as in the other make sure of the truth of the
Major: we must take care that the alternatives are really the only two
open. Otherwise the imposing form of the argument is a convenient mask
for sophistry. Zeno's famous dilemma, directed to prove that motion is
impossible, covers a _petitio principii_.
If a body moves, it must move either where it is or where it
is not. But a body cannot move where it is: neither can it
move where it is not. Conclusion, it cannot move at all,
_i.e._, Motion is impossible.
The conclusion is irresistible if we admit the Major, because the
Major covertly assumes the point to be proved. In truth, _if_ a body
moves, it moves neither where it is nor where it is not, but from
where it is to where it is not. Motion consists in change of place:
the Major assumes that the place is unchanged, that is, that there is
no motion.
[Footnote 1: For the history of Hypothetical Syllogism see
Mansel's _Aldrich_, Appendix I.]
[Footnote 2: It may be argued that the change is not merely
grammatical, and that the implication of a general proposition
in a hypothetical and _vice versâ_ is a strictly logical
concern. At any rate such an implication exists, whether it is
the function of the Grammarian or the Logician to expound it.]
[Footnote 3: Some logicians prefer the form Either A is, or B
is. But the two alternatives are propositions, and if "A is"
represents a proposition, the "is" is not the Syllogistic
copula. If this is understood it does not matter: the analysis
of the alternative propositions is unessential.]
CHAPTER VIII.
FALLACIES IN DEDUCTIVE ARGUMENT.--PETITIO PRINCIPII AND
IGNORATIO ELENCHI.
The traditional treatment of Fallacies in Logic follows Aristotle's
special treatise [Greek: Peri sophistikôn elenchôn]--Concerning
Sophistical Refutations--Pretended Disproofs--Argumentative Tricks.
Regarding Logic as in the main a protection against Fallacies, I have
been going on the plan of taking each fallacy in connexion with its
special safeguard, and in accordance with that plan propose to deal
here with the two great types of fallacy in deductive argument. Both
of them were recognised and named by Aristotle: but before explaining
them it is worth while to indicate Aristotle's plan as a whole.
Some of his Argumentative Tricks were really peculiar to Yes-and-No
Dialectic in its most sportive forms: but his leading types, both
Inductive and Deductive, are permanent, and his plan as a whole has
historical interest. Young readers would miss them from Logic: they
appeal to the average argumentative boy.
He divides Fallacies broadly into Verbal Fallacies ([Greek: para tên
lexin], _in dictione_), and Non-Verbal Fallacies ([Greek: exô tês
lexeôs], _extra dictionem_).
The first class are mere Verbal Quibbles, and hardly deserve serious
treatment, still less minute subdivision. The world was young when
time was spent upon them. Aristotle names six varieties, but they
all turn on ambiguity of word or structure, and some of them, being
dependent on Greek syntax, cannot easily be paralleled in another
tongue.
(1) Ambiguity of word ([Greek: homônymia]). As if one were to argue:
"All cold can be expelled by heat: John's illness is a cold: therefore
it can be expelled by heat". Or: "Some afflictions are cheering, for
afflictions are sometimes light, and light is always cheering".
The serious confusion of ambiguous words is met by Definition, as
explained at length in pt. ii. c i.
(2) _Ambiguity of structure_ ([Greek: amphibolia]).
"What he was beaten with was what I saw him beaten with: what I saw
him beaten with was my eye: therefore, what he was beaten with was my
eye."
"How do you do?" "Do? Do what?" "I mean, how do you feel?" "How do I
feel? With my fingers, of course; but I can see very well." "No, no; I
mean, how do you find yourself?" "Then why did you not say so? I never
exactly noticed, but I will tell you next time I lose myself."
(3) _Illicit conjunction_ ([Greek: synthesis]).
Socrates is good. Socrates is a musician. Therefore Socrates is a good
musician.
(4) _Illicit disjunction_ ([Greek: diairesis]).
Socrates is a good musician. Therefore he is a good man.
(5) _Ambiguity of pronunciation_ ([Greek: prosôdia], _fallacia
accentus_).
Analogies to words that differ only in accent, such as [Greek: ou-with
accents {psili and persipomeni}] and [Greek: ou-with accents {psili
and oxia}], may be found in differences of pronunciation. "Hair very
thick, sir," said a barber to a customer, whose hair was bushy, but
beginning to turn grey. "Yes, I daresay. But I would rather have it
thick than thin." "Ah, too thick to-day, sir." "But I don't want
to dye it." "Excuse me, sir, I mean the hair of the hatmosphere,
t-o-d-a-y, to-day."
"He said, saddle me the ass. And they saddled _him_."
(6) _Ambiguity of inflexion_ ([Greek: schêma tês lexeôs], _Figura
dictionis_).
This is not easy to make intelligible in English. The idea is that
a termination may be ambiguously interpreted, a neuter participle,
_e.g._, taken for an active. Thus: "George is ailing". "Doing what,
did you say? Ailing? What is he ailing? Ginger-aleing?"
Non-Verbal Fallacies, or Fallacies in thought, are a more important
division. Aristotle distinguishes seven.
Of these, three are comparatively unimportant and trifling. One of
them, known to the Schoolmen as _Fallacia Plurium Interrogationum_,
was peculiar to Interrogative disputation. It is the trick of putting
more than one question as one, so that a simple Yes commits the
respondent to something implied. "Have you left off beating your
father?" If you answer Yes, that implies that you have been in the
habit of beating him. "Has the practice of excessive drinking ceased
in your part of the country?" Such questions were unfair when the
Respondent could answer only Yes or No. The modern disputant who
demands a plain answer Yes or No, is sometimes guilty of this trick.
Two others, the fallacies known as _A dicto simpliciter ad dictum
secundum quid_, and _A dicto secundum quid ad dictum simpliciter_,
are as common in modern dialectic as they were in ancient. The trick,
conscious or unconscious, consists in getting assent to a statement
with a qualification and proceeding to argue as if it had been
conceded without qualification, and _vice versâ_. For example, it
being admitted that culture is good, a disputant goes on to argue
as if the admission applied to some sort of culture in special,
scientific, æsthetic, philosophical or moral. The fallacy was also
known as _Fallacia Accidentis_. Proving that the Syllogism is useless
for a certain purpose, and then claiming to have proved that it
is useless for any purpose is another example. Getting a limited
admission and then extending it indefinitely is perhaps the more
common of the two forms. It is common enough to deserve a shorter
name.
The _Fallacia Consequentis_, or _Non-Sequitur_, which consists
specially in ignoring the possibility of a plurality of causes, has
already been partly explained in connexion with the Hypothetical
Syllogism, and will be explained further in the Logic of Induction.
_Post hoc ergo proper hoc_ is a purely Inductive Fallacy, and will be
explained in connexion with the Experimental Methods.
There remain the two typical Deductive Fallacies, PETITIO PRINCIPII
(Surreptitious Assumption) and IGNORATIO ELENCHI (Irrelevant Argument)
about which we must speak more at length.
The phrase of which Petitio Principii or Begging the Question is a
translation--[Greek: to en archê aiteisthai]--was applied by Aristotle
to an argumentative trick in debate by Question and Answer. The
trick consisted in taking for granted a proposition necessary to
the refutation without having obtained the admission of it. Another
expression for the same thing--[Greek: to en archê lambanein]--taking
the principle for granted--is more descriptive.
Generally speaking, Aristotle says, Begging the Question consists in
not demonstrating the theorem. It would be in accordance with this
general description to extend the name to all cases of tacitly or
covertly, unwittingly to oneself or to one's opponent, assuming
any premiss necessary to the conclusion. It is the fallacy of
Surreptitious Assumption, and all cases of Enthymematic or Elliptical
argument, where the unexpressed links in the chain of argument are not
fully understood, are examples of it. By contrast, the articulate and
explicit Syllogism is an _Expositio Principii_. The only remedy for
covert assumptions is to force them into the light.[1]
_Ignoratio Elenchi_, ignoring the refutation ([Greek: tou elenchou
agnoia]), is simply arguing beside the point, distracting the
attention by irrelevant considerations. It often succeeds by proving
some other conclusion which is not the one in dispute, but has a
superficial resemblance to it, or is more or less remotely connected
with it.
It is easier to explain what these fallacies consist in than to
illustrate them convincingly. It is chiefly in long arguments that the
mischief is done. "A Fallacy," says Whately, "which when stated barely
in a few sentences would not deceive a child, may deceive half the
world if diluted in a quarto volume." Very rarely is a series of
propositions put before us in regular form and order, all bearing on
a definite point. A certain conclusion is in dispute, not very
definitely formulated perhaps, and a mixed host of considerations
are tumbled out before us. If we were perfectly clear-headed persons,
capable of protracted concentration of attention, incapable
of bewilderment, always on the alert, never in a hurry, never
over-excited, absolutely without prejudice, we should keep our
attention fixed upon two things while listening to an argument, the
point to be proved, and the necessary premisses. We should hold
the point clearly in our minds, and watch indefatigably for the
corroborating propositions. But none of us being capable of this, all
of us being subject to bewilderment by a rapid whirl of statements,
and all of us biased more or less for or against a conclusion, the
sophist has facilities for doing two things--taking for granted
that he has stated the required premisses (_petitio principii_), and
proving to perfect demonstration something which is not the point
in dispute, but which we are willing to mistake for it (_ignoratio
elenchi_).
It is chiefly in the heat of argument that either Petitio or Ignoratio
succeeds. When a fallacy continues to perplex us in cold blood, it
must have in its favour either some deeply-rooted prejudice or some
peculiar intricacy in the language used, or some abstruseness in the
matter. If we are not familiar with the matter of the argument, and
have but a vague hold of the words employed, we are, of course, much
more easily imposed upon.
The famous Sophisms of antiquity show the fascination exercised over
us by proving something, no matter how irrelevant. If certain steps
in an argument are sound, we seem to be fascinated by them so that we
cannot apply our minds to the error, just as our senses are fascinated
by an expert juggler. We have seen how plausibly Zeno's argument
against the possibility of motion hides a Petitio: the Fatalist
Dilemma is another example of the same sort.
If it is fated that you die, you will die whether you call in
a doctor or not, and if it is fated that you will recover, you
will recover whether you call in a doctor or not. But it must
be fated either that you die or that you recover. _Therefore_,
you will either die or recover whether you call in a doctor or
not.
Here it is tacitly assumed in the Major Premiss that the calling in
of a doctor cannot be a link in the fated chain of events. In the
statement of both the alternative conditions, it is assumed that
Fate does not act through doctors, and the conclusion is merely a
repetition of this assumption, a verbal proposition after an imposing
show of argument. "If Fate does not act through doctors, you will die
whether you call in a doctor or not."
The fallacy in this case is probably aided by our veneration for the
grand abstraction of Fate and the awful idea of Death, which absorbs
our attention and takes it away from the artful _Petitio_.
The Sophism of Achilles and the Tortoise is the most triumphant of
examples of _Ignoratio Elenchi_.
The point that the Sophism undertakes to prove is that Achilles can
never overtake a Tortoise once it has a certain start: what it really
proves, and proves indisputably, is that he cannot overtake the
Tortoise within a certain space or time.
For simplicity of exposition, let us assume that the Tortoise has 100
yards start and that Achilles runs ten times as fast. Then, clearly,
Achilles will not come up with it at the end of 100 yards, for while
he has run 100, the Tortoise has run 10; nor at the end of 110, for
then the Tortoise has run 1 more; nor at the end of 111, for then the
Tortoise has run 1/10 more; nor at the end of 111-1/10, for then the
Tortoise has gained 1/100 more. So while Achilles runs this 1/100, the
Tortoise runs 1/1000; while he runs the 1/1000, it runs 1/10000. Thus
it would seem that the Tortoise must always keep ahead: he can never
overtake it.
But the conclusion is only a confusion of ideas: all that is really
proved is that Achilles will not overtake the Tortoise while running
100 + 10 + 1 + 1/10 + 1/100 + 1/1000 + 1/10000, etc.
That is, that he will not overtake it till he has completed the sum of
this series, 111-1/9 yards. To prove this is an _ignoratio elenchi_;
what the Sophist undertakes to prove is that Achilles will never
overtake it, and he really proves that Achilles passes it between the
111th and 112th yards.
The exposure of this sophism is an example also of the value of a
technical term. All attempts to expose it without using the term
_Ignoratio Elenchi_ or something equivalent to it, succeed only in
bewildering the student. It is customary to say that the root of the
fallacy lies in assuming that the sum of an infinite series is equal
to infinity. This profound error may be implied: but if any assumption
so hard to understand were really required, the fallacy would have
little force with the generality.
It has often been argued that the Syllogism involves a _petitio
principii_, because the Major Premiss contains the Conclusion, and
would not be true unless the Conclusion were true. But this is really
an _Ignoratio Elenchi_. The fact adduced, that the Major Premiss
contains the Conclusion, is indisputable; but this does not prove the
Syllogism guilty of Petitio. _Petitio principii_ is an argumentative
trick, a conscious or unconscious act of deception, a covert
assumption, and the Syllogism, so far from favouring this, is an
_expositio principii_, an explicit statement of premisses such that,
if they are true, the conclusion is true. The Syllogism merely shows
the interdependence of premisses and conclusion; its only tacit
assumption is the _Dictum de Omni_.
If, indeed, an opponent challenges the truth of the conclusion, and
you adduce premisses necessarily containing it as a refutation, that
is an _ignoratio elenchi_ unless your opponent admits those premisses.
If he admits them and denies the conclusion, you convict him of
inconsistency, but you do not prove the truth of the conclusion.
Suppose a man to take up the position: "I am not mortal, for I have
procured the _elixir vitæ_". You do not disprove this by saying, "All
men are mortal, and you are a man". In denying that he is mortal, he
denies that all men are mortal. Whatever is sufficient evidence that
he is not mortal, is sufficient evidence that all men are not mortal.
Perhaps it might be said that in arguing, "All men are mortal, and
you are a man," it is not so much _ignoratio elenchi_ as _petitio
principii_ that you commit. But be it always remembered that you may
commit both fallacies at once. You may both argue beside the point and
beg the question in the course of one and the same argument.
[Footnote 1: Cp. Mr. Sidgwick's instructive treatise on
Fallacies, International Scientific Series, p. 199.]
CHAPTER IX.
FORMAL OR ARISTOTELIAN INDUCTION.--INDUCTIVE ARGUMENT.
The distinction commonly drawn between Deduction and Induction is
that Deduction is reasoning from general to particular, and Induction
reasoning from particular to general.
But it is really only as modes of argumentation that the two processes
can be thus clearly and fixedly opposed. The word Induction is used in
a much wider sense when it is the title of a treatise on the Methods
of Scientific Investigation. It is then used to cover all the
processes employed in man's search into the system of reality; and in
this search deduction is employed as well as induction in the narrow
sense.
We may call Induction in the narrow sense Formal Induction or
Inductive Argument, or we may simply call it Aristotelian Induction
inasmuch as it was the steps of Inductive argument that Aristotle
formulated, and for which he determined the conditions of validity.
Let us contrast it with Deductive argument. In this the questioner's
procedure is to procure the admission of a general proposition with a
view to forcing the admission of a particular conclusion which is in
dispute. In Inductive argument, on the other hand, it is a general
proposition that is in dispute, and the procedure is to obtain the
admission of particular cases with a view to forcing the admission of
this general proposition.
Let the question be whether All horned animals ruminate. You engage to
make an opponent admit this. How do you proceed? You ask him whether
he admits it about the various species. Does the ox ruminate? The
sheep? The goat? And so on. The bringing in of the various particulars
is the induction ([Greek: epagôgê]).
When is this inductive argument complete? When is the opponent bound
to admit that all horned animals ruminate? Obviously, when he has
admitted it about every one. He must admit that he has admitted it
about every one, in other words, that the particulars enumerated
constitute the whole, before he can be held bound in consistency to
admit it about the whole.
The condition of the validity of this argument is ultimately the
same with that of Deductive argument, the identity for purposes of
predication of a generic whole with the sum of its constituent parts.
The Axiom of Inductive Argument is, _What is predicated of every one
of the parts is predicable of the whole._ This is the simple converse
of the Axiom of Deductive argument, the _Dictum de Omni_, "What is
predicated of the whole is predicable about every one of the parts".
The Axiom is simply convertible because for purposes of predication
generic whole and specific or individual parts taken all together are
identical.
Practically in inductive argument an opponent is worsted when he
cannot produce an instance to the contrary. Suppose he admits the
predicate in question to be true of this, that and the other, but
denies that this, that and the other constitute the whole class in
question, he is defeated in common judgement if he cannot instance
a member of the class about which the predicate does not hold. Hence
this mode of induction became technically known as _Inductio per
enumerationem simplicem ubi non reperitur instantia contradictoria_.
When this phrase is applied to a generalisation of fact, Nature or
Experience is put figuratively in the position of a Respondent unable
to contradict the inquirer.
Such in plain language is the whole doctrine of Inductive Argument.
Aristotle's Inductive Syllogism is, in effect, an expression of this
simple doctrine tortuously in terms of the Deductive Syllogism. The
great master was so enamoured of his prime invention that he desired
to impress its form upon everything: otherwise, there was no reason
for expressing the process of Induction syllogistically. Here is his
description of the Inductive Syllogism:--
"Induction, then, and the Inductive Syllogism, consists in
syllogising one extreme with the middle through the other
extreme. For example, if B is middle to A and C, to prove
through C that A belongs to B."[1]
This may be interpreted as follows: Suppose a general proposition is
in dispute, and that you wish to make it good by obtaining severally
the admission of all the particulars that it sums up. The type of a
general proposition in Syllogistic terminology is the Major Premiss,
All M is P. What is the type of the particulars that it sums up?
Obviously, the Conclusion, S is P. This particular is contained in the
Major Premiss, All M is P; its truth is accepted as contained in the
truth of All M is P. S is one of the parts of the generic whole M; one
of the individuals or species contained in the class M. If you wish,
then, to establish P of All M by Induction, you must establish P of
all the parts, species, or individuals contained in M, that is, of all
possible S_s:_ you must make good that this, that and the other S is
P, and also that this, that and the other S constitute the whole of M.
You are then entitled to conclude that All M is P: you have syllogised
one Extreme with the Middle through the other Extreme. The formal
statement of these premisses and conclusion is the Inductive
Syllogism.
This, that and the other S is P, _Major_.
This, that and the other S is all M, _Minor_.
[.'.] All M is P, _Conclusion_.
This, that and the other magnet (_i.e._, magnets individually)
attract iron.
This, that and the other magnet (_i.e._, the individuals
separately admitted) are all magnets.
[.'.] All magnets attract iron.
This, that and the other S being simply convertible with All M, you
have only to make this conversion and you have a syllogism in Barbara
where this, that and the other S figures as the Middle Term.
The practical value of this tortuous expression is not obvious.
Mediæval logicians shortened it into what was known as the Inductive
Enthymeme: "This, that and the other, therefore all," an obvious
conclusion when this, that and the other constitute all. It is
merely an evidence of the great master's intoxication with his
grand invention. It is a proof also that Aristotle really looked
at Induction from the point of view of Interrogative Dialectic.
His question was, When is a Respondent bound to admit a general
conclusion? And his answer was, When he has admitted a certain number
of particulars, and cannot deny that those particulars constitute the
whole whose predicate is in dispute. He was not concerned primarily
with the analysis of the steps of an inquirer generalising from
Nature.
[Footnote 1: [Greek: epagôgê men oun esti kai ho ex epagôgês
syllogismos to dia tou eterou thateron akron tô mesô
syllogisasthai; Oion ei tôn A G meson to B, dia tou G deixai
to A tô B hyparchon.] (An. Prior., ii. 23.)]
BOOK II.
INDUCTIVE LOGIC, OR THE LOGIC OF SCIENCE.
INTRODUCTION.
Perhaps the simplest way of disentangling the leading features of
the departments of Logic is to take them in relation to historical
circumstances. These features are writ large, as it were, in history.
If we recognise that all bodies of doctrine have their origin in
practical needs, we may conceive different ages as controlled each by
a distinctive spirit, which issues its mandate to the men of the age,
assigning to them their distinctive work.
The mandate issued to the age of Plato and Aristotle was _Bring your
beliefs into harmony one with another_. The Aristotelian Logic
was framed in response to this order: its main aim was to devise
instruments for making clear the coherence, the concatenation, the
mutual implication of current beliefs.
The mandate of the Mediæval Spirit was _Bring your beliefs into
harmony with dogma_. The mediæval logic was contracted from
Aristotle's under this impulse. Induction as conceived by him was
neglected, allowed to dwindle, almost to disappear from Logic. Greater
prominence was given to Deduction.
Then as Dogmatic Authority became aggressive, and the Church through
its officials claimed to pronounce on matters outside Theology, a new
spirit was roused, the mandate of which was, _Bring your beliefs
into harmony with facts_. It was under this impulse that a body of
methodical doctrine vaguely called Induction gradually originated.
In dealing with the genesis of the Old Logic, we began with Aristotle.
None can dispute his title to be called its founder. But who was the
founder of the New Logic? In what circumstances did it originate?
The credit of founding Induction is usually given to Francis Bacon,
Lord Verulam. That great man claimed it for himself in calling his
treatise on the Interpretation of Nature the _Novum Organum_. The
claim is generally conceded. Reid's account of the matter represents
the current belief since Bacon's own time.
"After man had laboured in the search of truth near two
thousand years by the help of Syllogisms, [Lord] Bacon
proposed the method of INDUCTION as a more effectual engine
for that purpose. His _Novum Organum_ gave a new turn to the
thoughts and labours of the inquisitive, more remarkable and
more useful than that which the _Organon_ of Aristotle had
given before, and may be considered as a second grand era in
the progress of human nature.... Most arts have been reduced
to rules after they had been brought to a considerable degree
of perfection by the natural sagacity of artists; and the
rules have been drawn from the best examples of the art
that had been before exhibited; but the art of philosophical
induction was delineated by [Lord] Bacon in a very ample
manner before the world had seen any tolerable example of
it."[1]
There is a radical misconception here, which, for reasons that I hope
to make plain, imperatively needs to be cleared up. It obscures the
very essence of "philosophical induction".
There are three ways in which movement in any direction may be helped
forward, Exhortation, Example, and Precept. Exhortation: a man may
exhort to the practice of an art and thereby give a stimulus. Example:
he may practise the art himself, and show by example how a thing
should be done. Precept: he may formulate a clear method, and so make
plain how to do it. Let us see what was Bacon's achievement in each of
those three ways.
Undoubtedly Bacon's powerful eloquence and high political position
contributed much to make the study of Nature fashionable. He was high
in place and great in intellect, one of the commanding personalities
of his time. Taking "all knowledge for his province," though study
was really but his recreation, he sketched out a plan of universal
conquest with a clearness and confidence that made the mob eager
to range themselves under his leadership. He was the magnificent
demagogue of science. There had been champions of "Induction" before
him, but they had been comparatively obscure and tongue-tied.
While, however, we admit to the full the great services of this mighty
advocate in making an "Inductive" method popular, we should not
forget that he had pioneers even in hortatory leadership. His happiest
watchword, the Interpretation of Nature, as distinguished from the
Interpretation of Authoritative Books, was not of his invention. If we
read Whewell's _History of the Inductive Sciences_, we shall find that
many before him had aspired to "give a new turn to the labors of
the inquisitive," and in particular to substitute inquisition for
disquisition.
One might compile from Whewell a long catalogue of eminent men before
Bacon who held that the study of Nature was the proper work of the
inquisitive: Leonardo da Vinci (1452-1519), one of the wonders of
mankind for versatility, a miracle of excellence in many things,
painter, sculptor, engineer, architect, astronomer, and physicist;
Copernicus (1473-1543), the author of the Heliocentric theory;
Telesius (1508-1588), a theoretical reformer, whose _De Rerum Natura_
(1565) anticipated not a little of the _Novum Organum_; Cesalpinus
(1520-1603), the Botanist; Gilbert (1540-1603), the investigator of
Magnetism. By all these men experiment and observation were advocated
as the only way of really increasing knowledge. They all derided mere
book-learning. The conception of the world of sense as the original
MS. of which systems of philosophy are but copies, was a familiar
image with them. So also was Bacon's epigrammatic retort to those
who wish to rest on the wisdom of the ancients, that antiquity is the
youth of the world and that we are the true ancients. "We are older,"
said Giordano Bruno, "and have lived longer than our predecessors."
This last argument, indeed, is much older than the sixteenth century.
It was used by the Doctor Mirabilis of the thirteenth, the Franciscan
Friar, Roger Bacon (1214-1292). "The later men are, the more
enlightened they are; and wise men now are ignorant of much the world
will some day know." The truth is that if you are in search of a
Father for Inductive Philosophy, the mediæval friar has better claims
than his more illustrious namesake. His enthusiasm for the advancement
of learning was not less nobly ambitious and far-reaching, and he
was himself an ardent experimenter and inventor. His _Opus Majus_--an
eloquent outline of his projects for a new learning, addressed in 1265
to Pope Clement IV., through whom he offered to give to the Church the
empire of the world as Aristotle had given it to Alexander--was almost
incredibly bold, comprehensive and sagacious. Fixing upon Authority,
Custom, Popular Opinion, and the Pride of Supposed Knowledge, as the
four causes of human ignorance, he urged a direct critical study of
the Scriptures, and after an acute illustration of the usefulness
of Grammar and Mathematics (widely interpreted), concluded with
Experimental Science as the great source of human knowledge. I have
already quoted (p. 15) the Friar's distinction between the two modes
of Knowing, Argument and Experience, wherein he laid down that it is
only experience that makes us feel certain. It were better, he cried
in his impatience, to burn Aristotle and make a fresh start than to
accept his conclusions without inquiry.
Experimental Science, the sole mistress of Speculative
Science, has three great Prerogatives among other parts
of Knowledge. First, she tests by experiment the noblest
conclusions of all other sciences. Next, she discovers
respecting the notions which other sciences deal with,
magnificent truths to which these sciences can by no means
attain. Her third dignity is that she by her own power and
without respect to other sciences investigates the secret of
Nature.
So far, then, as Exhortation goes, King James's great lawyer and
statesman was not in advance of Pope Clement's friar. Their first
principle was the same. It is only by facts that theories can be
tested. Man must not impose his own preconceptions (_anticipationes
mentis_) on nature. Man is only the interpreter of nature. Both
were also at one in holding that the secrets of nature could not be
discovered by discussion, but only by observation and experiment.
Francis Bacon, however, went beyond all his predecessors in furnishing
an elaborate Method for the interpretation of Nature. When he
protested against the intellect's being left to itself (_intellectus
sibi permissus_), he meant more than speculation left unchecked by
study of the facts. He meant also that the interpreter must have a
method. As man, he says, cannot move rocks by the mere strength of his
hands without instruments, so he cannot penetrate to the secrets of
Nature by mere strength of his intellect without instruments. These
instruments he undertakes to provide in his Inductive Method or _Novum
Organum_. And it is important to understand precisely what his methods
were, because it is on the ground of them that he is called the
founder of Inductive Philosophy, and because this has created a
misapprehension of the methods actually followed by men of science.
Ingenious, penetrating, wide-ranging, happy in nomenclature, the
_Novum Organum_ is a wonderful monument of the author's subtle wit
and restless energy; but, beyond giving a general impulse to testing
speculative fancies by close comparison with facts, it did nothing
for science. His method--with its Tables of Preliminary Muster for
the Intellect (_tabulæ comparentiæ primæ instantiarum ad intellectum_,
facts collected and methodically arranged for the intellect to work
upon); its Elimination upon first inspection of obviously accidental
concomitants (_Rejectio sive Exclusiva naturarum_); its Provisional
Hypothesis (_Vindemiatio Prima sive Interpretatio Inchoata_); its
advance to a true Induction or final Interpretation by examination
of special instances (he enumerates twenty-seven, 3 × 3 × 3,
_Prerogativas Instantiarum_, trying to show the special value of each
for the inquirer)[2]--was beautifully regular and imposing, but it was
only a vain show of a method. It was rendered so chiefly by the end
or aim that Bacon proposed for the inquirer. In this he was not in
advance of his age; on the contrary, he was probably behind Roger
Bacon, and certainly far behind such patient and concentrated thinkers
as Copernicus, Gilbert, and Galileo--no discredit to the grandeur of
his intellect when we remember that science was only his recreation,
the indulgence of his leisure from Law and State.
In effect, his method came to this. Collect as many instances as you
can of the effect to be investigated, and the absence of it where you
would expect it, arrange them methodically, then put aside guesses
at the cause which are obviously unsuitable, then draw up a probably
explanation, then proceed to make this exact by further comparison
with instances. It is when we consider what he directed the inquirer
to search for that we see why so orderly a method was little likely to
be fruitful.
He starts from the principle that the ultimate object of all knowledge
is use, practice (_scimus ut operemur_). We want to know how Nature
produces things that we may produce them for ourselves, if we can.
The inquirer's first aim, therefore, should be to find out how the
qualities of bodies are produced, to discover the _formæ_ or formal
causes of each quality. An example shows what he meant by this. Gold
is a crowd or conjugation of various qualities or "natures"; it is
yellow, it has a certain weight, it is malleable or ductile to a
certain degree, it is not volatile (loses nothing under fire), it can
be melted, it is soluble. If we knew the _forma_ or formal cause of
each of those qualities, we could make gold, provided the causes were
within our control. The first object, then, of the investigator of
Nature is to discover such _formæ_, in order to be able to effect the
transformation of bodies. It may be desirable also to know the _latens
processus_, any steps not apparent to the senses by which a body grows
from its first germs or rudiments, and the _schematismus_ or ultimate
inner constitution of the body. But the discovery of the _formæ_ of
the constituent qualities (_naturæ singulæ_), heat, colour, density or
rarity, sweetness, saltness, and so forth, is the grand object of the
Interpreter of Nature; and it is for this that Bacon prescribed his
method.
The _Sylva Sylvarum_, or Natural History, a miscellaneous collection
of facts and fictions, observations and traditions, with guesses
at the explanation of them, affords us a measure of Bacon's own
advancement as an interpreter of Nature. It was a posthumous work, and
the editor, his secretary, tells us that he often said that if he had
considered his reputation he would have withheld it from the world,
because it was not digested according to his own method: yet he
persuaded himself that the causes therein assigned were far more
certain than those rendered by others, "not for any excellence of his
own wit, but in respect of his continual conversation with Nature and
Experience," and mankind might stay upon them till true Axioms were
more fully discovered. When, however, we examine the causes assigned,
we find that in practice Bacon could not carry out his own precepts:
that he did not attempt to creep up to an explanation by slow and
patient ascent, but jumped to the highest generalisations: and that
his explanatory notions were taken not from nature, but from the
ordinary traditions of mediæval physical science. He deceived himself,
in short, in thinking that he could throw aside tradition and start
afresh from observation.
For example. He is struck by the phenomenon of bubbles on water: "It
seemeth somewhat strange that the air should rise so swiftly, while it
is in the water, and when it cometh to the top should be stayed by so
weak a cover as that of the bubble is". The swift ascent of the air he
explains as a "motion of percussion," the water descending and forcing
up the air, and not a "motion of levity" in the air itself. "The cause
of the enclosure of the bubble is for that the appetite to resist
separation or discontinuance, which is strong in solids, is also
in liquors, though fainter and weaker." "The same reason is of the
roundness of the bubble, as well for the skin of water as for the air
within. For the air likewise avoideth discontinuance, and therefore
casteth itself into a round figure. And for the stop and arrest of the
air a little while, it showeth that the air of itself hath little or
no appetite of ascending."[3] These notions were not taken direct from
the facts: they descended from Aristotle. He differs from Aristotle,
however, in his explanation of the colours of birds' feathers.
"Aristotle giveth the cause vainly" that birds are more in the beams
of the sun than beasts. "But that is manifestly untrue; for cattle are
more in the sun than birds, that live commonly in the woods or in some
covert. The true cause is that the excrementitious moisture of living
creatures, which maketh as well the feathers in birds as the hair in
beasts, passeth in birds through a finer and more delicate strainer
than it doth in beasts. For feathers pass through quills, and hair
through skin." It is an instance of percolation or filtering: other
effects of the same cause being the gums of trees, which are but a
fine passage or straining of the juice through the wood and bark,
and Cornish Diamonds and Rock Rubies, which are in like manner "fine
exudations of stone".[4]
These examples of Bacon's Inductions are taken from the _Sylva_ at
random. But the example which best of all illustrates his attitude
as a scientific investigator is the remark he makes in the _Novum
Organum_ about the Copernican theory. Elsewhere he says that there
is nothing to choose between it and the Ptolemaic; and in the _Novum
Organum_ (lib. ii. 5) he remarks that "no one can hope to terminate
the question whether in diurnal motion it is really the earth or the
sky that rotates, unless he shall first have comprehended the nature
of spontaneous rotation". That is, we must first find out the
_forma_ or formal cause of spontaneous rotation. This is a veritable
_instantia crucis_, as fixing Bacon's place in the mediæval and not in
the new world of scientific speculation.
Bacon, in short, in the practice of induction did not advance an inch
beyond Aristotle. Rather he retrograded, inasmuch as he failed to draw
so clear a line between the respective spheres of Inductive
collection of facts and Explanation. There are two sources of general
propositions, according to Aristotle, Induction and Nous. By Induction
he meant the generalisation of facts open to sense, the summation of
observed particulars, the _inductio per enumerationem simplicem_ of
the schoolmen. By Nous he meant the Reason or Speculative Faculty,
as exercised with trained sagacity by experts. Thus by Induction we
gather that all horned animals ruminate. The explanation of this is
furnished by the Nous, and the explanation that commended itself to
the trained sagacity of his time was that Nature having but a limited
amount of hard material and having spent this on the horns, had none
left for teeth, and so provided four stomachs by way of compensation.
Bacon's guesses at causes are on the same scientific level with this,
only he rather confused matters by speaking of them as if they were
inductions from fact, instead of being merely fancies superinduced
upon fact. His theory of interpretation, it is true, was so far
an advance that he insisted on the necessity of verifying every
hypothesis by further appeal to facts, though in practice he himself
exercised no such patience and never realised the conditions of
verification. Against this, again, must be set the fact that by
calling his method induction, and laying so much stress on the
collection of facts, he fostered, and, indeed, fixed in the public
mind the erroneous idea that the whole work of science consists in
observation. The goal of science, as Herschel said, is Explanation,
though every explanation must be made to conform to fact,
and explanation is only another term for attaining to higher
generalisations, higher unities.
The truth is that Induction, if that is the name we use for scientific
method, is not, as Reid conceived, an exception to the usual rule of
arts in being the invention of one man. Bacon neither invented nor
practised it. It was perfected gradually in the practice of men
of science. The birthplace of it as a conscious method was in the
discussions of the Royal Society of London, as the birthplace of the
Aristotelian Logic was in the discussions of the Athenian schools. Its
first great triumph was Newton's law of Gravitation. If we are to
name it after its first illustrious practitioner, we must call it
the Newtonian method, not the Baconian. Newton really stands to the
Scientific Method of Explanation as Aristotle stands to the Method
of Dialectic and Deduction. He partly made it explicit in his _Regulæ
Philosophandi_ (1685). Locke, his friend and fellow-member of the
Royal Society, who applied the method to the facts of Mind in his
_Essay Concerning Human Understanding_ (1691), made it still further
explicit in the Fourth Book of that famous work.
It was, however, a century and a half later that an attempt was first
made to incorporate scientific method with Logic under the name of
Induction, and add it as a new wing to the old Aristotelian building.
This was the work of John Stuart Mill, whose System of Logic,
Deductive and Inductive, was first published in 1843.
The genesis of Mill's System of Logic, as of other things, throws
light upon its character. And in inquiries into the genesis of
anything that man makes we may profitably follow Aristotle's division
of causes. The Efficient Cause is the man himself, but we have also to
find out the Final Cause, his object or purpose in making the thing,
the Material Cause, the sources of his material, and the Formal Cause,
the reason why he shaped it as he did. In the case of Mill's system
we have to ask: What first moved him to formulate the methods of
scientific investigation? Whence did he derive his materials? Why
did he give his scientific method the form of a supplement to the old
Aristotelian Logic? We cannot absolutely separate the three inquiries,
but motive, matter and form each had a traceable influence on the
leading features of his System.
First, then, as to his motive. It is a mistake to suppose that Mill's
object was to frame an organon that might assist men of science as
ordinarily understood in making discoveries. Bacon, his secretary
tells us, was wont to complain that he should be forced to be a
Workman and a Labourer in science when he thought he deserved to be an
Architect in this building. And men of science have sometimes rebuked
Mill for his presumption in that, not being himself an investigator
in any department of exact science, he should volunteer to teach them
their business. But Mill was really guilty of no such presumption. His
object, on the contrary, was to learn their method with a view to
its application to subjects that had not yet undergone scientific
treatment. Briefly stated, his purpose was to go to the practical
workers in the exact sciences, Astronomy, Chemistry, Heat, Light,
Electricity, Molar and Molecular Physics; ascertain, not so much how
they made their discoveries as how they assured themselves and others
that their conclusions were sound; and having ascertained their tests
of truth and principles of proof, to formulate these tests so that
they might be applied to propositions outside the range of the exact
sciences, propositions in Politics, Ethics, History, Psychology.
More particularly he studied how scientific men verify, and when
they accept as proved, propositions of causation, explanations of
the causes of things. In effect, his survey of scientific method was
designed to lead up to the Sixth Book in his System, the Logic of the
Moral Sciences. There are multitudes of floating endoxes or current
opinions concerning man and his concerns, assigning causes for the
conduct and character of individuals and of communities. Mill showed
himself quite aware that the same modes of investigation may not be
practicable, and that it may not be possible, though men are always
ready to assign causes with confidence, to ascertain causes with
the same degree of certainty: but at least the conditions of exact
verification should be the same, and it is necessary to see what they
are in order to see how far they can be realised.
That such was Mill's design in the main is apparent on internal
evidence, and it was the internal evidence that first struck me. But
there is external evidence as well. We may first adduce some essays
on the Spirit of the Age, published in the _Examiner_ in 1831, essays
which drew from Carlyle the exclamation, "Here is a new Mystic!" These
essays have never been republished, but they contain Mill's first
public expression of the need for a method in social inquiries. He
starts from the Platonic idea that no state can be stable in which
the judgment of the wisest in political affairs is not supreme. He
foresees danger in the prevalent anarchy of opinion. How is it to be
averted? How are men to be brought to accept loyally the judgment of
the expert in public affairs? They accept at once and without question
the decisions of the specially skilled in the physical sciences. Why
is this? For one reason, because there is complete agreement among
experts. And why is there this complete agreement? Because all accept
the same tests of truth, the same conditions of proof. Is it not
possible to obtain among political investigators similar unanimity as
to their methods of arriving at conclusions, so as to secure similar
respect for their authority?
We need not stop to ask whether this was a vain dream, and whether it
must not always be the case that to ensure confidence in a political
or moral adviser more is needed than faith in his special knowledge
and trained sagacity. Our point is that in 1831 Mill was in search
of a method of reasoning in social questions. Opportunely soon after,
early in 1832, was published Herschel's _Discourse on the Study of
Natural Philosophy_, the first attempt by an eminent man of science to
make the methods of science explicit. Mill reviewed this book in the
_Examiner_, and there returns more definitely to the quest on which
he was bent. "The uncertainty," he says, "that hangs over the very
elements of moral and social philosophy proves that the means
of arriving at the truth in those sciences are not yet properly
understood. And whither can mankind so advantageously turn, in order
to learn the proper means and to form their minds to the proper
habits, as to that branch of knowledge in which by universal
acknowledgment the greatest number of truths have been ascertained and
the greatest possible degree of certainty arrived at?"
We learn from Mill himself that he made an attempt about this
time, while his mind was full of Herschel's Discourse, to connect a
scientific method with the body of the Old Logic. But he could not
make the junction to his satisfaction, and abandoned the attempt in
despair. A little later, in 1837, upon the appearance of Whewell's
_History of the Inductive Sciences_, he renewed it, and this time with
happier results. Whewell's _Philosophy of the Inductive Sciences_
was published in 1840, but by that time Mill's system was definitely
shaped.
It was, then, to Herschel and Whewell, but especially to the former,
that Mill owed the raw materials of his Inductive Method. But why did
he desire to concatenate this with the old Logic? Probably because he
considered that this also had its uses for the student of society, the
political thinker. He had inherited a respect for the old Logic from
his father. But it was the point at which he sought to connect the
new material with the old, the point of junction between the two, that
determined the form of his system. We find the explanation of this in
the history of the old Logic. It so happened that Whately's Logic was
in the ascendant, and Whately's treatment of Induction gives the key
to Mill's.
Towards the end of the first quarter of this century there was a
great revival of the study of Logic at Oxford. The study had become
mechanical, Aldrich's Compendium, an intelligent but exceedingly brief
abstract of the Scholastic Logic, being the text-book beyond which
no tutor cared to go. The man who seems to have given new life to the
study was a tutor who subsequently became Bishop of Llandaff, Edward
Copleston. The first public fruits of the revival begun by him was
Whately's article on Logic in the _Encyclopædia Metropolitana_,
published as a separate book in 1827. Curiously enough, one of
Whately's most active collaborators in the work was John Henry Newman,
so that the common room of Oriel, which Mr. Froude describes as the
centre from which emanated the High Church Movement, may also be
said to have been the centre from which emanated the movement that
culminated in the revolution of Logic.
The publication of Whately's Logic made a great stir. It was reviewed
by Mill, then a young man of twenty-one, in the _Westminster Review_
(1828), and by Hamilton, then forty-five years of age, in the
_Edinburgh_ (1833). There can be no doubt that it awakened Mill's
interest in the subject. A society formed for the discussion of
philosophical questions, and called the Speculative Society, met at
Grote's house in 1825, and for some years following. Of this society
young Mill was a member, and their continuous topic in 1827 was Logic,
Whately's treatise being used as a sort of text-book.
It is remarkable that Mill's review of Whately, the outcome of these
discussions, says very little about Induction. At that stage Mill's
chief concern seems to have been to uphold the usefulness of Deductive
Logic, and he even goes so far as to scoff at its eighteenth century
detractors and their ambition to supersede it with a system of
Induction. The most striking feature of the article is the brilliant
defence of the Syllogism as an analysis of arguments to which I have
already referred. He does not deny that an Inductive Logic might be
useful as a supplement, but apparently he had not then formed the
design of supplying such a supplement. When, however, that design
seriously entered his mind, consequent upon the felt need of a method
for social investigations, it was Whately's conception of Induction
that he fell back upon. Historically viewed, his System of Logic was
an attempt to connect the practical conditions of proof set forth in
Herschel's discourse with the theoretic view of Induction propounded
in Whately's. The tag by which he sought to attach the new material
to the old system was the Inductive Enthymeme of the Schoolmen as
interpreted by Whately.
Whately's interpretation--or misinterpretation--of this Enthymeme,
and the conception of Induction underlying it, since it became Mill's
ruling conception of Induction, and virtually the formative principle
of his system, deserves particular attention.
"This, that and the other horned animal, ox, sheep, goat,
ruminate; _therefore_, all horned animals ruminate."
The traditional view of this Enthymeme I have given in my chapter on
Formal Induction (p. 238). It is that a Minor Premiss is suppressed:
"This, that and the other constitute the whole class". This is the
form of the Minor in Aristotle's Inductive Syllogism.
But, Whately argued, how do we know that this, that and the other--the
individuals we have examined--constitute the whole class? Do we not
assume that what belongs to the individuals examined belongs to the
whole class? This tacit assumption, he contended, is really at the
bottom of the Enthymeme, and its proper completion is to take this as
the Major Premiss, with the enumeration of individuals as the Minor.
Thus:--
What belongs to the individuals examined belongs to
the whole class.
The property of the ruminating belongs to the individuals
examined, ox, sheep, goat, etc.
_Therefore_, it belongs to all.
In answer to this, Hamilton repeated the traditional view, treating
Whately's view merely as an instance of the prevailing ignorance of
the history of Logic. He pointed out besides that Whately's Major was
the postulate of a different kind of inference from that contemplated
in Aristotle's Inductive Syllogism, Material as distinguished from
Formal inference. This is undeniable if we take this syllogism purely
as an argumentative syllogism. The "all" of the conclusion simply
covers the individuals enumerated and admitted to be "all" in the
Minor Premiss. If a disputant admits the cases produced to be all and
can produce none to the contrary, he is bound to admit the conclusion.
Now the inference contemplated by Whately was not inference from
an admission to what it implies, but inference from a series of
observations to all of a like kind, observed and unobserved.
It is not worth while discussing what historical justification Whately
had for his view of Induction. It is at least arguable that the word
had come to mean, if it did not mean with Aristotle himself, more
than a mere summation of particulars in a general statement. Even
Aristotle's respondent in the concession of his Minor admitted that
the individuals enumerated constituted all in the truly general sense,
not merely all observed but all beyond the range of observation. The
point, however, is insignificant. What really signifies is that
while Hamilton, after drawing the line between Formal Induction and
Material, fell back and entrenched himself within that line, Mill
caught up Whately's conception of Induction, pushed forward, and made
it the basis of his System of Logic.
In Mill's definition, the mere summation of particulars, _Inductio per
enumerationem simplicem ubi non reperitur instantia contradictoria_,
is Induction improperly so called. The only process worthy of the name
is Material Induction, inference to the unobserved. Here only is
there an advance from the known to the unknown, a veritable "inductive
hazard".
Starting then with this conception of inference to the unobserved
as the only true inference, and with an empirical law--a generality
extended from observed cases to unobserved--as the type of such
inference, Mill saw his way to connecting a new Logic with the old. We
must examine this junction carefully, and the brilliant and plausible
arguments by which he supported it; we shall find that, biased by this
desire to connect the new with the old, he gave a misleading dialectic
setting to his propositions, and, in effect, confused the principles
of Argumentative conclusion on the one hand and of Scientific
Observation and Inference on the other. The conception of Inference
which he adopted from Whately was too narrow on both sides for the
uses to which he put it. Be it understood that in the central methods
both of Syllogistic and of Science, Mill was substantially in accord
with tradition; it is in his mode of junction, and the light thereby
thrown upon the ends and aims of both, that he is most open to
criticism.
As regards the relation between Deduction and Induction, Mill's chief
proposition was the brilliant paradox that all inference is at bottom
Inductive, that Deduction is only a partial and accidental stage in
a process the whole of which may be called Induction. An opinion was
abroad--fostered by the apparently exclusive devotion of Logic to
Deduction--that all inference is essentially Deductive. Not so,
answered Mill, meeting this extreme with another: all inference is
essentially Inductive. He arrives at this through the conception
that Induction is a generalisation from observed particulars, while
Deduction is merely the extension of the generalisation to a new
case, a new particular. The example that he used will make his meaning
plain.
Take a common Syllogism:--
All men are mortal.
Socrates is a man.
Socrates is mortal.
"The proposition," Mill says, "that Socrates is mortal is evidently an
inference. It is got at as a conclusion from something else. But do we
in reality conclude it from the proposition, All men are mortal?" He
answers that this cannot be, because if it is not true that Socrates
is mortal it cannot be true that all men are mortal. It is clear that
our belief in the mortality of Socrates must rest on the same ground
as our belief in the mortality of men in general. He goes on to ask
whence we derive our knowledge of the general truth, and answers: "Of
course from observation. Now all which man can observe are individual
cases.... A general truth is but an aggregate of particular truths.
But a general proposition is not merely a compendious form for
recording a number of particular facts.... It is also a process of
inference. From instances which we have observed we feel warranted in
concluding that what we have found true in those instances, holds in
all similar ones, past, present, and future. We then record all that
we have observed together with what we infer from our observations,
in one concise expression." A general proposition is thus at once a
summary of particular facts and a memorandum of our right to
infer from them. And when we make a deduction we are, as it were,
interpreting this memorandum. But it is upon the particular facts
that the inference really rests, and Mill contends that we might if we
chose infer to the particular conclusion at once without going through
the form of a general inference. Thus Mills seeks to make good his
point that all inference is essentially Inductive, and that it is
only for convenience that the word Induction has been confined to the
general induction, while the word Deduction is applied to the process
of interpreting our memorandum.
Clear and consecutive as this argument is, it is fundamentally
confusing. It confuses the nature of Syllogistic conclusion or
Deduction, and at the same time gives a partial and incomplete account
of the ground of Material inference.
The root of the first confusion lies in raising the question of the
ground of material inference in connexion with the Syllogism. As
regards the usefulness of the Syllogism, this is an IGNORATIO ELENCHI.
That the Major and the conclusion rest upon the same ground as
matters of belief is indisputable: but it is irrelevant. In so far
as "Socrates is mortal" is an inference from facts, it is not the
conclusion of a Syllogism. This is implicitly and with unconscious
inconsistency recognised by Mill when he represents Deduction as
the interpretation of a memorandum. To represent Deduction as the
interpretation of a memorandum--a very happy way of putting it and
quite in accordance with Roger Bacon's view--is really inconsistent
with regarding Deduction as an occasional step in the process of
Induction. If Deduction is the interpretation of a memorandum, it
is no part of the process of inference from facts. The conditions of
correct interpretation as laid down in Syllogism are one thing,
and the methods of correct inference from the facts, the methods of
science that he was in search of, are another.
Let us emphasise this view of Deduction as the interpretation of a
memorandum. It corresponds exactly with the view that I have taken
in discussing the utility of the Syllogism. Suppose we want to know
whether a particular conclusion is consistent with our memorandum,
what have we to look to? We have to put our memorandum into such
a form that it is at once apparent whether or not it covers our
particular case. The Syllogism aspires to be such a form. That is
the end and aim of it. It does not enable us to judge whether the
memorandum is a legitimate memorandum or not. It only makes clear that
if the memorandum is legitimate, so is the conclusion. How to
make clear and consistent memoranda of our beliefs in words is a
sufficiently complete description of the main purpose of Deductive
Logic.
Instead, then, of trying to present Deduction and Induction as parts
of the same process, which he was led to do by his desire to connect
the new and the old, Mill ought rather, in consistency as well as
in the interests of clear system, to have drawn a line of separation
between the two as having really different ends, the conditions of
correct conclusion from accepted generalities on the one hand, and the
conditions of correct inference from facts on the other. Whether the
first should be called inference at all is a question of naming that
ought to have been considered by itself. We may refuse to call it
inference, but we only confuse ourselves and others if we do not
acknowledge that in so doing we are breaking with traditional usage.
Perhaps the best way in the interests of clearness is to compromise
with tradition by calling the one Formal Inference and the other
Material Inference.
It is with the latter that the Physical Sciences are mainly concerned,
and it was the conditions and methods of its correct performance that
Mill desired to systematise in his Inductive Logic. We have next
to see how his statement of the grounds of Material Inference was
affected by his connexion of Deduction and Induction. Here also
we shall find a reason for a clearer separation between the two
departments of Logic.
In his antagonism to a supposed doctrine that all reasoning is
from general to particular, Mill maintained _simpliciter_ that all
reasoning is from particulars to particulars. Now this is true only
_secundum quid_, and although in the course of his argument Mill
introduced the necessary qualifications, the unqualified thesis was
confusing. It is perfectly true that we may infer--we can hardly be
said to reason--from observed particulars to unobserved. We may even
infer, and infer correctly, from a single case. The village matron,
called in to prescribe for a neighbour's sick child, infers that
what cured her own child will cure the neighbour's, and prescribes
accordingly. And she may be right. But it is also true that she may
be wrong, and that no fallacy is more common than reasoning from
particulars to particulars without the requisite precautions. This
is the moral of one of the fables of Camerarius. Two donkeys were
travelling in the same caravan, the one laden with salt, the other
with hay. The one laden with salt stumbled in crossing a stream, his
panniers dipped in the stream, the salt melted, and his burden was
lightened. When they came to another stream, the donkey that was laden
with hay dipped his panniers in the water, expecting a similar
result. Mill's illustrations of correct inference from particulars
to particulars were really irrelevant. What we are concerned with
in considering the grounds of Inference, is the condition of correct
inference, and no inference to an unobserved case is sound unless
it is of a like kind with the observed case or cases on which it is
founded, that is to say, unless we are entitled to make a general
proposition. We need not go through the form of making a general
proposition, but if a general proposition for all particulars of
a certain description is not legitimate, no more is the particular
inference. Mill, of course, did not deny this, he was only betrayed
by the turn of his polemic into an unqualified form of statement that
seemed to ignore it.
But this was not the worst defect of Mill's attempt at a junction of
old and new through Whately's conception of Induction. A more serious
defect was due to the insufficiency of this conception to represent
all the modes of scientific inference. When a certain attribute has
been found in a certain connexion in this, that, and the other, to the
extent of all observed instances, we infer that it will be found in
all, that the connexion that has obtained within the range of our
actual experience has obtained beyond that range and will obtain
in the future. Call this an observed uniformity of nature: we hold
ourselves justified in expecting that the observed uniformities of
nature will continue. Such an observed uniformity--that All animals
have a nervous system, that All animals die, that Quinine cures
ague--is also called an Empirical Law.
But while we are justified in extending an empirical law beyond the
limits within which it has been observed to hold good, it is a
mistake to suppose that the main work of science is the collection
of empirical laws, and that the only scientific inference is the
inference from the observed prevalence of an empirical law to its
continuance. With science the collection of empirical laws is only
a preliminary: "the goal of science," in Herschel's phrase, "is
explanation". In giving such prominence to empirical laws in his
theory, Mill confined Induction to a narrower scope than science
ascribes to it. Science aims at reaching "the causes of things": it
tries to penetrate behind observed uniformities to the explanation
of them. In fact, as long as a science consists only of observed
uniformities, as long as it is in the empirical stage, it is a science
only by courtesy. Astronomy was in this stage before the discovery of
the Law of Gravitation. Medicine is merely empirical as long as its
practice rests upon such generalisations as that Quinine cures ague,
without knowing why. It is true that this explanation may consist only
in the discovery of a higher or deeper uniformity, a more recondite
law of connexion: the point is that these deeper laws are not always
open to observation, and that the method of reaching them is not
merely observing and recording.
In the body of his Inductive Logic, Mill gave a sufficient account of
the Method of Explanation as practised in scientific inquiry. It was
only his mode of approaching the subject that was confusing, and
made it appear as if the proper work of science were merely extending
observed generalities, as when we conclude that all men will die
because all men have died, or that all horned animals ruminate because
all hitherto observed have had this attribute. A minor source of
confusion incident to the same controversy was his refusing the
title of Induction proper to a mere summary of particulars. He seemed
thereby to cast a slight upon the mere summation of particulars. And
yet, according to his theory, it was those particulars that were the
basis of the Induction properly so called. That all men will die is an
inference from the observation summed up in the proposition that
all men have died. If we refuse the name of Induction to the general
proposition of fact, what are we to call it? The truth is that the
reason why the word Induction is applied indifferently to the general
proposition of fact and the general proposition applicable to all
time is that, once we are sure of the facts, the transition to the
inference is so simple an affair that it has not been found necessary
in practice to distinguish them by different names.
Our criticism of Mill would itself mislead if it were taken to mean
that the methods of science which he formulated are not the methods
of science or that his system of those methods is substantially
incomplete. His Inductive Logic as a system of scientific method was
a great achievement in organisation, a veritable _Novum Organum_ of
knowledge. What kept him substantially right was that the methods
which he systematised were taken from the practice of men of science.
Our criticism amounts only to this, that in correlating the new system
with the old he went upon a wrong track. For more than two centuries
Deduction had been opposed to Induction, the _ars disserendi_ to the
_ars inveniendi_. In trying to reconcile them and bring them under one
roof, Mill drew the bonds too tight. In stating the terms of the union
between the two partners, he did not separate their spheres of work
with sufficient distinctness.
Mill's theory of Deduction and Induction and the voluminous criticism
to which in its turn it has been subjected have undoubtedly been of
great service in clearing up the foundations of reasoning. But the
moral of it is that if we are to make the methods of Science a part of
Logic, and to name this department Induction, it is better to discard
altogether the questions of General and Particular which are pertinent
to Syllogism, and to recognise the new department simply as being
concerned with a different kind of inference, inference from facts to
what lies beyond them, inference from the observed to the unobserved.
That this is the general aim and proper work of Science is evident
from its history. Get at the secrets of Nature by the study of Nature,
penetrate to what is unknown and unexperienced by help of what is
known and has been experienced, was the cry of the early reformers of
Science. Thus only, in Roger Bacon's phrase, could certainty--assured,
well grounded, rational belief--be reached. This doctrine, like every
other, can be understood only by what it was intended to deny. The
way of reaching certainty that Roger Bacon repudiated was argument,
discussion, dialectic. This "concludes a question but does not make us
feel certain, or acquiesce in the contemplation of truth that is not
also found in Experience". Argument is not necessarily useless; the
proposition combated is only that by it alone--by discussion that does
not go beyond accepted theories or conceptions--rational belief about
the unknown cannot be reached. The proposition affirmed is that to
this end the conclusions of argument must be tested by experience.
Observation of facts then is a cardinal part of the method of Science.
The facts on which our inferences are based, by which our conclusions
are tested, must be accurate. But in thus laying emphasis on the
necessity of accurate observation, we must beware of rushing to the
opposite extreme, and supposing that observation alone is enough.
Observation, the accurate use of the senses (by which we must
understand inner as well as outer sense), is not the whole work of
Science. We may stare at facts every minute of our waking day without
being a whit the wiser unless we exert our intellects to build upon
them or under them. To make our examination fruitful, we must have
conceptions, theories, speculations, to bring to the test. The
comparison of these with the facts is the inductive verification of
them. Science has to exercise its ingenuity both in making hypotheses
and in contriving occasions for testing them by observation. These
contrived occasions are its artificial experiments, which have come to
be called experiments simply by contrast with conclusive observations
for which Nature herself furnishes the occasion. The observations of
Science are not passive observations. The word experiment simply means
trial, and every experiment, natural or artificial, is the trial of
a hypothesis. In the language of Leonardo da Vinci, "Theory is the
general, Experiments are the soldiers".
Observation and Inference go hand in hand in the work of Science, but
with a view to a methodical exposition of its methods, we may divide
them broadly into Methods of Observation and Methods of Inference.
There are errors specially incident to Observation, and errors
specially incident to Inference. How to observe correctly and how to
make correct inferences from our observations are the two objects of
our study in Inductive Logic: we study the examples of Science because
they have been successful in accomplishing those objects.
That all inference to the unobserved is founded on facts, on the
data of experience, need not be postulated. It is enough to say that
Inductive Logic is concerned with inference in so far as it is founded
on the data of experience. But inasmuch as all the data of experience
are not of equal value as bases of inference, it is well to begin with
an analysis of them, if we wish to take a comprehensive survey of the
various modes of inference and the conditions of their validity.
[Footnote 1: Hamilton's _Reid_, p. 712.]
[Footnote 2: The _Novum Organum_ was never completed. Of
the nine heads of special aids to the intellect in the final
interpretation he completed only the first, the list of
Prerogative Instances.]
[Footnote 3: _Sylva Sylvarum_, Century I, 24.]
[Footnote 4: _Sylva Sylvarum_, Century I, 5.]
CHAPTER I.
THE DATA OF EXPERIENCE AS GROUNDS OF INFERENCE OR RATIONAL BELIEF.
If we examine any of the facts or particulars on which an inference
to the unobserved is founded, we shall find that they are not isolated
individuals or attributes, separate objects of perception or thought,
but relations among things and their qualities, constituents, or
ingredients.
Take the "particular" from which Mill's village matron inferred, the
fact on which she based her expectation of a cure for her neighbour's
child. It is a relation between things. We have the first child's
ailment, the administration of the drug, and the recovery, a series of
events in sequence. This observed sequence is the fact from which she
is said to infer, the datum of experience. She expects this sequence
to be repeated in the case of her neighbour's child.
Similarly we shall find that, in all cases where we infer, the
facts are complex, are not mere isolated things, but relations among
things--using the word thing in its widest sense--relations which we
expect to find repeated, or believe to have occurred before, or to be
occurring now beyond the range of our observation. These relations,
which we may call coincidences or conjunctions, are the data of
experience from which we start in our beliefs or inferences about the
unexperienced.
The problem of Inductive Logic being to determine when or on what
conditions such beliefs are rational, we may begin by distinguishing
the data of coincidence or conjunction accordingly. There are certain
coincidences that we expect to find repeated beyond the occasions on
which we have observed them, and others that we do not expect to find
repeated. If it is a sound basis of inference that we are in search
of, it is evidently to these first, the coincidences that we are
assured of finding again, that we must direct our study. Let us see
whether they can be specified.
(1) If there is no causal connexion between A and B, using these
as symbols for the members of a coincidence--the objects that are
presented together--we do not expect the coincidence to be repeated.
If A and B are connected as cause and effect, we expect the effect
to recur in company with the cause. We expect that when the cause
reappears in similar circumstances, the effect also will reappear.
You are hit, _e.g._, by a snowball, and the blow is followed by a
feeling of pain. The sun, we shall say, was shining at the moment of
the impact of the snowball on your body. The sunshine preceded your
feeling of pain as well as the blow. But you do not expect the pain
to recur next time that the sun shines. You do expect it to recur next
time you are hit by a snowball.
The taking of food and a certain feeling of strength are causally
connected. If we go without food, we are not surprised when faintness
or weariness supervenes.
Suppose that when our village matron administered her remedy to her
own child, a dog stood by the bedside and barked. The barking in that
case would precede the cure. Now, if the matron were what we should
call a superstitious person, and believed that this concomitant had
a certain efficacy, that the dog's barking and the cure were causally
connected, she would take the dog with her when she went to cure her
neighbour's child. Otherwise she would not. She would say that the
barking was an accidental, casual, fortuitous coincidence, and would
build no expectation upon it.
These illustrations may serve to remind us of the familiar fact that
the causal nexus is at least one of the things that we depend on in
our inferences to the unobserved. To a simple sequence we attach
no importance, but a causal sequence or consequence that has been
observed is a mainstay of inference.
Whether the causal sequence holds or not as a matter of fact, we
depend upon it if we believe in it as a matter of fact. But unless
it does hold as a matter of fact, it is valueless as a guide to the
unknown, and our belief is irrational. Clearly, therefore, if rational
belief is what we aim at, it is of importance that we should make sure
of cause and effect as matter of fact in the sequence of events.
One large department of Inductive Logic, the so-called Experimental
Methods, is designed to help us in thus making sure, _i.e._, in
ascertaining causal sequence as a matter of fact. It is assumed
that by careful observation of the circumstances, we can distinguish
between mere simple sequence and causal sequence or consequence,
and methods are recommended of observing with the proper precautions
against error.
Observe that these methods, though called Inductive, are not concerned
with arriving at general propositions. The principle we go upon is
simply this, that if it can be ascertained as matter of fact that a
certain thing is related to another as cause and effect, we may count
upon the same relation as holding in unobserved Nature, on the general
ground that like causes produce like effects in like circumstances.
Observe, also, that I deliberately speak of the causal relation as
a relation among phenomena. Whether this use of the words cause and
effect is philosophically justifiable, is a question that will be
raised and partly discussed later on. Here I simply follow the common
usage, in accordance with which objects of perception, _e.g._, the
administration of a drug and the recovery of a patient, are spoken of
as cause and effect. Such observable sequences are causal sequences in
the ordinary sense, and it is part of the work of Science to observe
them. I do not deny that the _true_ cause, of the cause that
science aims ultimately at discovering, is to be found in the latent
constitution or composition of the things concerned. Only that, as we
shall see more precisely, is a cause of another description. Meantime,
let us take the word to cover what it undoubtedly covers in ordinary
speech, the perceptible antecedent of a perceptible consequent.
Strictly speaking, as we shall find, Science has only one method of
directly observing when events are in causal sequence. But there are
various indirect methods, which shall be described in some sort of
order.
For the practical purposes of life, a single ascertained causal
sequence is of little value as a basis of inference, because we can
infer only to its repetition in identical circumstances. Suppose our
village matron had been able to ascertain as a matter of fact--a feat
as we shall find not to be achieved by direct observation--that the
drug did cure her child, this knowledge by itself would have been
practically valueless, because the only legitimate inference would
have been that an exactly similar dose would have the same effect
in exactly similar circumstances. But, as we shall find, though
practically valueless, a single ascertained causal sequence is of
supreme value in testing scientific speculations as to the underlying
causes.
(2) We have next to see whether there are any other rational
expectations based on observed facts. We may lay down as a principle
the following:--
_If a conjunction or coincidence has constantly been repeated within
our experience, we expect it to recur and believe that it has recurred
outside our experience._
How far such expectations are rational, and with what degrees of
confidence they should be entertained, are the questions for the Logic
of Inference, but we may first note that we do as a matter of habit
found expectations on repeated coincidence, and indeed guide our
daily life in this way. If we meet a man repeatedly in the street at
a certain hour, we go out expecting to meet him: it is a shock to our
expectations, a surprise, when we do not. If we are walking along a
road and find poles set up at regular intervals, we continue our walk
expecting to find a pole coincident with the end of each interval.
What Mill calls the uniformities of Nature, the uniformities expressed
in general propositions, are from the point of view of the observer,
examples of repeated coincidence. Birth, growth, decay, death, are not
isolated or variable coincidences with organised being: all are born,
all grow, all decay, and all die. These uniformities constitute
the order of Nature: the coincidences observed are not occasional,
occurring once in a way or only now and then; they turn up again and
again. Trees are among the uniformities on the varied face of Nature:
certain relations between the soil and the plant, between trunk,
branches, and leaves are common to them. For us who observe, each
particular tree that comes under our observation is a repetition of
the coincidence. And so with animals: in each we find certain tissues,
certain organs, conjoined on an invariable plan.
Technically these uniformities have been divided into uniformities
of Sequence and uniformities of Coexistence. Thus the repeated
alternation of day and night is a uniformity of Sequence: the
invariable conjunction of inertia with weight is a uniformity of
Coexistence. But the distinction is really immaterial to Logic.
What Logic is concerned with is the observation of the facts and the
validity of any inference based on them: and in these respects it
makes no difference whether the uniformity that we observe and found
upon is one of Sequence or of Coexistence.
It was exclusively to such inferences, inferences from observed facts
of repeated coincidence, that Mill confined himself in his theory of
Induction, though not in his exposition of the methods. These are the
inferences for which we must postulate what he calls the Uniformity
of Nature. Every induction, he says, following Whately, may be thrown
into the form of a Syllogism, in which the principle of the Uniformity
of Nature is the Major Premiss, standing to the inference in the
relation in which the Major Premiss of a Syllogism stands to the
conclusion. If we express this abstractly denominated principle in
propositional form, and take it in connexion with Mill's other saying
that the course of Nature is not a uniformity but uniformities, we
shall find, I think, that this postulated Major Premiss amounts to an
assumption that the observed Uniformities of Nature continue. Mill's
Inductive Syllogism thus made explicit would be something like this:--
All the observed uniformities of Nature continue.
That all men have died is an observed uniformity.
_Therefore_, it continues; _i.e._, all men will die and did die
before the beginning of record.
There is no doubt that this is a perfectly sound postulate. Like all
ultimate postulates it is indemonstrable; Mill's derivation of it
from Experience did not amount to a demonstration. It is simply an
assumption on which we act. If any man cares to deny it, there is
no argument that we can turn against him. We can only convict him of
practical inconsistency, by showing that he acts upon this assumption
himself every minute of his waking day. If we do not believe in the
continuance of observed uniformities, why do we turn our eyes to the
window expecting to find it in its accustomed order of place? Why do
we not look for it in another wall? Why do we dip our pens in ink,
and expect the application of them to white paper to be followed by a
black mark?
The principle is sound, but is it our only postulate in inference to
the unobserved, and does the continuance of empirical laws represent
all that Science assumes in its inferences? Mill was not satisfied
about this question. He pointed out a difficulty which a mere belief
in empirical continuity does not solve. Why do we believe more
confidently in some uniformities than in others? Why would a reported
breach of one be regarded with more incredulity than that of another?
Suppose a traveller to return from a strange country and report that
he had met men with heads growing beneath their shoulders, why would
this be pronounced more incredible than a report that he had seen a
grey crow? All crows hitherto observed have been black, and in all men
hitherto observed the heads have been above the shoulders: if the
mere continuity of observed uniformities is all that we go upon in
our inferences, a breach of the one uniformity should be just as
improbable as a breach of the other, neither more nor less. Mill
admitted the difficulty, and remarked that whoever could solve it
would have solved the problem of Induction. Now it seems to me that
this particular difficulty may be solved, and yet leave another
behind. It may be solved within the limits of the principle of
emperical--meaning by that observational--continuity. The uniform
blackness of the crow is an exception within a wider uniformity: the
colour of animals is generally variable. Hence we are not so much
surprised at the reported appearance of a grey crow: it is in
accordance with the more general law. On the other hand, the uniform
position of the head relative to other parts of the body is a
uniformity as wide as the animal kingdom: it is a coincidence repeated
as often as animals have been repeated, and merely on the principle
that uniformities continue, it has an absolutely uncontradicted series
in its favour.
But is this principle really all that we assume? Do we not also assume
that behind the observed fact uniformity, there is a cause for it, a
cause that does not appear on the surface of the observation, but
must be sought outside of its range? And do not the various degrees
of confidence with which we expect a repetition of the coincidence,
depend upon the extent of our knowledge of the producing causes and
the mode of their operation? At bottom our belief in the continuance
of the observed uniformities rests on a belief in the continuance of
the producing causes, and till we know what these are our belief has
an inferior warrant: there is less reason for our confidence.
To go back to the illustrations with which we started. If we have met
a man every day for months at a certain place at a certain hour, it
is reasonable to expect to meet him there to-morrow, even if
our knowledge does not go beyond the observed facts of repeated
coincidence. But if we know also what brings him there, and that this
cause continues, we have a stronger reason for our expectation. And so
with the case of poles at regular intervals on a road. If we know why
they are placed there, and the range of the purpose, we expect their
recurrence more confidently within the limits of that purpose. This
further knowledge is a warrant for stronger confidence, because if
we know the producing causes, we are in a better position for knowing
whether anything is likely to defeat the coincidence. A uniformity is
said to be explained when its cause is known, and an inference from an
explained uniformity is always more certain than an inference from
a uniformity that is merely empirical in the sense of being simply
observed.
Now, the special work of Science is to explain, in the sense of
discovering the causes at work beneath what lies open to observation.
In so doing it follows a certain method, and obeys certain conditions
of satisfactory explanation. Its explanations are inferences from
facts, inasmuch as it is conformity with observed facts, with outward
signs of underlying causal nexus, that is the justification of them.
But they are not inferences from facts in the sense above described
as empirical inference. In its explanations also Science postulates
a principle that may be called the Uniformity of Nature. But this
principle is not merely that observed uniformities continue. It may
be expressed rather as an assumption that the underlying causes
are uniform in their operation, that as they have acted beneath the
recorded experiences of mankind, so they have acted before and will
continue to act.
The foregoing considerations indicate a plan for a roughly systematic
arrangement of the methods of Induction. Seeing that all inference
from the data of experience presupposes causal connexion among the
data from which we infer, all efforts at establishing sound bases of
inference, or rational ground for expectation fall, broadly speaking,
under two heads: (1) Methods of ascertaining causal connexion among
phenomena as a matter of fact, that is, Methods of Observation; and
(2) Methods of ascertaining what the causal connexion is, that is,
Methods of Explanation.
These constitute the body of Inductive Logic. But there is a
preliminary and a pendant. Without raising the question of causal
connexion, we are liable to certain errors in ascertaining in what
sequence and with what circumstances events really occurred. These
tendencies to error deserve to be pointed out by way of warning, and
this I shall attempt in a separate chapter on observation of facts
of simple sequence. This is preliminary to the special methods of
observing causal sequence. Then, by way of pendant, I shall consider
two modes of empirical inference from data in which the causal
connexion has not been ascertained or explained--Inference from
approximate generalisations to particular cases, and Inference from
Analogy.
Most of these methods in one form or another were included by Mill
in his system of Inductive Logic, and the great merit of his work was
that he did include them, though at some sacrifice of consistency
with his introductory theory. With regard to the kind of empirical
inference which that theory, following the lead of Whately, took as
the type of all inference, Logic has really little to say. It was this
probably that was in Mill's mind when he said that there is no Logic
of Observation, ignoring the fact that the Experimental Methods are
really methods of observation, as well as the Methods of Eliminating
Chance by calculation of Probability. There is no method of observing
uniformities except simply observing them. Nor indeed is there any
"method" of inferring from them: we can only point out that in
every particular inference from them we assume or postulate their
continuance generally. As regards their observation, we may point
out further that a special fallacy is incident to it, the fallacy of
ignoring exceptions. If we are prepossessed or prejudiced in favour of
a uniformity, we are apt to observe only the favourable instances, and
to be blind to cases where the supposed invariable coincidence does
not occur. Thus, as Bacon remarked among his _Idola_, we are apt to
remember when our dreams come true, and to forget when they do
not. Suppose we take up the notion that a new moon on a Saturday is
invariably followed by twenty days of unsettled weather, one or two
or a few cases in which this notably holds good are apt to be borne in
mind, while cases where the weather is neither conspicuously good
nor bad are apt to be overlooked. But when a warning has been given
against this besetting fallacy, Logic has nothing further to say about
empirical uniformities, except that we may infer from them with some
degree of reasonable probability, and that if we want ground for a
more certain inference we should try to explain them.
CHAPTER II.
ASCERTAINMENT OF SIMPLE FACTS IN THEIR ORDER.--PERSONAL OBSERVATION.
--HEARSAY EVIDENCE--METHOD OF TESTING TRADITIONAL EVIDENCE.
All beliefs as to simple matter of fact must rest ultimately on
observation. But, of course, we believe many things to have happened
that we have never seen. As Chaucer says:--
But God forbedë but men shouldë 'lieve
Wel morë thing than men han seen with eye.
Man shall not weenen everything a lie
But if himself it seeth or elsë doth.
For the great bulk of matters of fact that we believe we are
necessarily dependent on the observations of others. And if we are
to apply scientific method to the ascertainment of this, we must know
what errors we are liable to in our recollections of what we have
ourselves witnessed, and what errors are apt to arise in the tradition
of what purports to be the evidence of eye-witnesses.
I.--PERSONAL OBSERVATION.
It is hard to convince anybody that he cannot trust implicitly to his
memory of what he has himself seen. We are ready enough to believe
that others may be deceived: but not our own senses. Seeing is
believing. It is well, however, that we should realise that all
observation is fallible, even our own.
Three great besetting fallacies or tendencies to error may be
specified:--
1. Liability to have the attention fastened on special incidents, and
so diverted from other parts of the occurrence.
2. Liability to confuse and transpose the sequence of events.
3. Liability to substitute inference for fact.
It is upon the first of these weaknesses in man as an observing
machine that jugglers chiefly depend on working their marvels. Sleight
of hand counts for much, but diverting the spectator's eyes for a good
deal more. That is why they have music played and patter incessantly
as they operate. Their patter is not purposeless: it is calculated to
turn our eyes away from the movements of their nimble hands.
It must be borne in mind that in any field of vision there are many
objects, and that in any rapid succession of incidents much more
passes before the eyes than the memory can retain in its exact
order. It is of course in moments of excitement and hurry, when our
observation is distracted, that we are most subject to fallacious
illusions of memory. Unconsciously we make a coherent picture of what
we have seen, and very often it happens that the sequence of events
is not what actually passed, but what we were prejudiced in favour of
seeing. Hence the unlikelihood of finding exact agreement among the
witnesses of any exciting occurrence, a quarrel, a railway accident, a
collision at sea, the incidents of a battle.
"It commonly happens," says Mr. Kinglake,[1] "that incidents occurring
in a battle are told by the most truthful bystanders with differences
more or less wide." In the attack on the Great Redoubt in the Battle
of the Alma, a young officer, Anstruther, rushed forward and planted
the colours of the Royal Welsh--but where? Some distinctly remembered
seeing him dig the butt-end of the flagstaff into the parapet: others
as distinctly remembered seeing him fall several paces before he
reached it. Similarly with the incidents of the death of the Prince
Imperial near the Italezi Hills in the Zulu War. He was out as a
volunteer with a reconnoitring party. They had off-saddled at a kraal
and were resting, when a band of Zulus crept up through the long
grass, and suddenly opened fire and made a rush forward. Our scouts
at once took horse, as a reconnoitring party was bound to do, and
scampered off, but the Prince was overtaken and killed. At the
Court-Martial which ensued, the five troopers gave the most
conflicting accounts of particulars which an unskilled investigator
would think could not possibly have been mistaken by eye-witnesses of
the same event. One said that the Prince had given the order to mount
before the Zulus fired: another that he gave the order directly after:
a third was positive that he never gave the order at all, but that it
was given after the surprise by the officer in command. One said that
he saw the Prince vault into the saddle as he gave the order: another
that his horse bolted as he laid hold of the saddle, and that he ran
alongside trying to get up.
The evidence before any Court of Inquiry into an exciting occurrence
is almost certain to reveal similar discrepancies. But what we find it
hard to realise is that we ourselves can possibly be mistaken in what
we have a distinct and positive recollection of having seen. It once
happened to myself in a London street to see a drunken woman
thrown under a cab by her husband. Two cabs were running along, a
four-wheeler and a hansom: the woman staggered almost under the first,
and was thrown under the second. As it happened the case never got
beyond the police station to which the parties were conveyed after
fierce opposition from some neighbours, who sympathised entirely with
the man. The woman herself, when her wounds were dressed, acknowledged
the justice of her punishment, and refused to charge her husband. I
was all the more willing to acquiesce in this because I found
that while I had the most distinct impression of having seen the
four-wheeler run over the woman's body, and should have been obliged
to swear accordingly, there could be no doubt that it was really
the hansom that had done so. This was not only the evidence of the
neighbours, which I suspected at the time of being a trick, but of the
cabdriver, who had stopped at the moment to abide the results of the
accident. I afterwards had the curiosity to ask an eminent police
magistrate, Sir John Bridge, whether this illusion of memory on my
part--which I can only account for by supposing that my eyes had
been fixed on the sufferer and that I had unconsciously referred her
injuries to the heavier vehicle--would have entirely discredited my
testimony in his Court. His answer was that it would not; that he was
constantly meeting with such errors, and that if he found a number of
witnesses of the same occurrence exactly agreed in every particular,
he would suspect that they had talked the matter over and agreed upon
what they were to say. This was the opinion of an experienced judge,
a skilled critic of the defects of personal observation. An Old Bailey
counsel for the defence, who is equally acquainted with the weakness
of human memory, takes advantage of the fact that it is not generally
understood by a Jury, and makes the fallacious assumption that glaring
discrepancies are irreconcilable with the good faith of the witnesses
who differ.[2]
II.--TRADITION.--HEARSAY EVIDENCE.
Next in value to personal observation, we must place the report, oral
or written, of an eye-witness. This is the best evidence we can get
if we have not witnessed an occurrence ourselves. Yet Courts of Law,
which in consideration of the defects of personal observation require
more than one witness to establish the truth, exclude hearsay evidence
altogether in certain cases, and not without reason.
In hearing a report we are in the position of observers of a series
of significant sounds, and we are subject to all the fallacies of
observation already mentioned. In an aggravated degree, for words are
harder to observe than visible things. Our attention is apt to be more
listless than in presence of the actual events. Our minds dwell upon
parts of the narrative to the neglect of other parts, and in the
coherent story or description that we retain in our memories,
sequences are apt to be altered and missing links supplied in
accordance with what we were predisposed to hear. Thus hearsay
evidence is subject to all the imperfections of the original observer,
in addition to the still more insidious imperfections of the second
observer.
How quickly in the course of a few such transmissions hearsay loses
all evidentiary value is simply illustrated by the game known as
Russian Scandal. One of a company, A, writes down a short tale or
sketch, and reads it to B. B repeats it to C, C to D, and so on. When
it has thus gone the round of the company, the last hearer writes
down his version, and it is compared with the original. With every
willingness to play fair, the changes are generally considerable and
significant.
Sometimes it is possible to compare an oral tradition with a
contemporary written record. In one of Mr. Hayward's Essays--"The
Pearls and Mock Pearls of History"--there are some examples of this
disenchanting process. There is, for instance, a pretty story of an
exchange of courtesies between the leaders of the French and English
Guards at the battle of Fontenoy. The tradition runs that Lord Charles
Hay stepped in front of his men and invited the French Guards to fire,
to which M. d'Auteroche with no less chivalry responded: "Monsieur, we
never fire first; you fire". What really passed we learn from a
letter from Lord Charles Hay to his mother, which happens to have
been preserved. "I advanced before our regiment, and drank to the
Frenchmen, and told them we were the English Guards, and hoped they
would stand till we came, and not swim the Scheldt as they did the
Maine at Dettingen." Tradition has changed this lively piece of
buffoonery into an act of stately and romantic courtesy. The change
was probably made quite unconsciously by some tenth or hundredth
transmitter, who remembered only part of the story, and dressed the
remainder to suit his own fancy.
The question has been raised, For how long can oral tradition be
trusted? Newton was of opinion that it might be trusted for eighty
years after the event. Others have named forty years. But if this
means that we may believe a story that we find in circulation forty
years after the alleged events, it is wildly extravagant. It does
injustice to the Mythop[oe]ic Faculty of man. The period of time that
suffices for the creation of a full-blown myth, must be measured
by hours rather than by years. I will give an instance from my own
observation, if that has not been entirely discredited by my previous
confessions. The bazaars of the East are generally supposed to be
the peculiar home of myth, hotbeds in which myths grow with the most
amazing speed, but the locality of my myth is Aberdeen. In the summer
of 1887 our town set up in one of its steeples a very fine carillon
of Belgian bells. There was much public excitement over the event: the
descriptions of enthusiastic promoters had prepared us to hear silvery
music floating all over the town and filling the whole air. On the day
fixed for the inauguration, four hours after the time announced for
the first ceremonial peal, not having heard the bells, I was in a shop
and asked if anything had happened to put off the ceremony. "Yes,"
I was told; "there had been an accident; they had not been properly
hung, and when the wife of the Lord Provost had taken hold of a string
to give the first pull, the whole machinery had come down." As a
matter of fact all that had happened was that the sound of the bells
was faint, barely audible a hundred yards from the belfry, and not at
all like what had been expected. There were hundreds of people in the
streets, and the myth had originated somehow among those who had
not heard what they went out to hear. The shop where it was repeated
circumstantially to me was in the main street, not more than a quarter
of a mile from where the carillon had been played in the hearing of
a large but disappointed crowd. I could not help reflecting that if I
had been a mediæval chronicler, I should have gone home and recorded
the story, which continued to circulate for some days in spite of
the newspapers: and two hundred years hence no historian would have
ventured to challenge the truth of the contemporary evidence.
III.--METHOD OF TESTING TRADITIONAL EVIDENCE.
It is obvious that the tests applied to descriptive testimony in
Courts of Law cannot be applied to the assertions of History. It is
a supreme canon of historical evidence that only the statements of
contemporaries can be admitted: but most even of their statements must
rest on hearsay, and even when the historian professes to have been an
eye-witness, the range of his observation is necessarily limited, and
he cannot be put into the witness-box and cross-examined. Is there
then no way of ascertaining historical fact? Must we reject history as
altogether unworthy of credit?
The rational conclusion only is that very few facts can be established
by descriptive testimony such as would satisfy a Court of Law. Those
who look for such ascertainment are on a wrong track, and are doomed
to disappointment. It is told of Sir Walter Raleigh that when he was
writing his History of the World, he heard from his prison in the
Tower a quarrel outside, tried to find out the rights and the wrongs
and the course of it, and failing to satisfy himself after careful
inquiry, asked in despair how he could pretend to write the history
of the world when he could not find out the truth about what occurred
under his own windows. But this was really to set up an impossible
standard of historical evidence.
The method of testing historical evidence follows rather the lines
of the Newtonian method of Explanation, which we shall afterwards
describe. We must treat any historical record as being itself in the
first place a fact to be explained. The statement at least is extant:
our first question is, What is the most rational way of accounting
for it? Can it be accounted for most probably by supposing the event
stated to have really occurred with all the circumstances alleged? Or
is it a more probable hypothesis that it was the result of an illusion
of memory on the part of the original observer, if it professes to
be the record of an eye-witness, or on the part of some intermediate
transmitter, if it is the record of a tradition? To qualify ourselves
to answer the latter kind of question with reasonable probability
we must acquaint ourselves with the various tendencies to error in
personal observation and in tradition, and examine how far any of
them are likely to have operated in the given case. We must study the
operation of these tendencies within our experience, and apply the
knowledge thus gained. We must learn from actual observation of facts
what the Mythop[oe]ic Faculty is capable of in the way of creation
and transmutation, and what feats are beyond its powers, and then
determine with as near a probability as we can how far it has been
active in the particular case before us.
[Footnote 1: _The Invasion of the Crimea_, iii. 124]
[Footnote 2: The truth is, that we see much less than is
commonly supposed. Not every impression is attended to that
is made on the retina, and unless we do attend we cannot,
properly speaking, be said to see. Walking across to college
one day, I was startled by seeing on the face of a clock in
my way that it was ten minutes to twelve, whereas I generally
passed that spot about twenty minutes to twelve. I hurried
on, fearing to be late, and on my arrival found myself in very
good time. On my way back, passing the clock again, I looked
up to see how much it was fast. It marked ten minutes to
eight. It had stopped at that time. When I passed before I
had really seen only the minute hand. The whole dial must have
been on my retina, but I had looked at or attended to only
what I was in doubt about, taking the hour for granted. I am
bound to add that my business friends hint that it is only
absorbed students that are capable of such mistakes, and that
alert men of business are more circumspect. That can only be
because they are more alive to the danger of error.]
CHAPTER III.
ASCERTAINMENT OF FACTS OF CAUSATION.
I.--_POST HOC ERGO PROPTER HOC_.
One of the chief contributions of the Old Logic to Inductive Method
was a name for a whole important class of misobservations. The fallacy
entitled _Post Hoc ergo Propter Hoc_--"After, therefore, Because
of"--consisted in alleging mere sequence as a proof of consequence or
causal sequence. The sophist appeals to experience, to observed facts:
the sequence which he alleges has been observed. But the appeal is
fallacious: the observation on which he relies amounts only to this,
that the one event has followed upon the other. This much must be
observable in all cases of causal sequence, but it is not enough for
proof. _Post hoc ergo propter hoc_ may be taken as a generic name for
imperfect proof of causation from observed facts of succession.
The standard example of the fallacy is the old Kentish peasant's
argument that Tenterden Steeple was the cause of Goodwin Sands. Sir
Thomas More (as Latimer tells the story in one of his Sermons to
ridicule incautious inference) had been sent down into Kent as a
commissioner to inquire into the cause of the silting up of Sandwich
Haven. Among those who came to his court was the oldest inhabitant,
and thinking that he from his great age must at least have seen more
than anybody else, More asked him what he had to say as to the cause
of the sands. "Forsooth, sir," was the greybeard's answer, "I am an
old man: I think that Tenterden Steeple is the cause of Goodwin Sands.
For I am an old man, and I may remember the building of Tenterden
Steeple, and I may remember when there was no steeple at all there.
And before that Tenterden Steeple was in building, there was no
manner of speaking of any flats or sands that stopped the haven;
and, therefore, I think that Tenterden Steeple is the cause of the
destroying and decaying of Sandwich Haven."
This must be taken as Latimer meant it to be, as a ridiculous example
of a purely imbecile argument from observation, but the appeal to
experience may have more show of reason and yet be equally fallacious.
The believers in Kenelm Digby's "Ointment of Honour" appealed to
experience in support of its efficacy. The treatment was to apply the
ointment, not to the wound, but to the sword that had inflicted it, to
dress this carefully at regular intervals, and, meantime, having bound
up the wound, to leave it alone for seven days. It was observed that
many cures followed upon this treatment. But those who inferred that
the cure was due to the bandaging of the sword, failed to observe
that there was another circumstance that might have been instrumental,
namely, the exclusion of the air and the leaving of the wound
undisturbed while the natural healing processes went on. And it
was found upon further observation that binding up the wound alone
answered the purpose equally well whether the sword was dressed or
not.
In cases where _post hoc_ is mistaken for _propter hoc_, simple
sequence for causal sequence, there is commonly some bias of prejudice
or custom which fixes observation on some one antecedent and diverts
attention from other circumstances and from what may be observed to
follow in other cases. In the minds of Digby and his followers there
was probably a veneration for the sword as the weapon of honour, and a
superstitious belief in some secret sympathy between the sword and its
owner. So when the practice of poisoning was common, and suspicion was
flurried by panic fear, observation was often at fault. Pope Clement
VIII. was said to have been killed by the fumes of a poisoned candle
which was placed in his bedroom. Undoubtedly candles were there, but
those who attributed the Pope's death to them took no notice of the
fact that a brazier of burning charcoal was at the same time in the
apartment with no sufficient outlet for its fumes. Prince Eugene
is said to have received a poisoned letter, which he suspected and
immediately threw from him. To ascertain whether his suspicions were
well founded the letter was administered to a dog, which, to make
assurance doubly sure, was fortified by an antidote. The dog died, but
no inquiry seems to have been made into the character of the antidote.
Hotspur's retort to Glendower showed a sound sense of the true value
to be attached to mere priority.
_Glendower_. At my nativity
The front of heaven was full of fiery shapes,
Of burning cressets: and at my birth
The frame and huge foundation of the earth
Shaked like a coward.
_Hotspur_. Why so it would have done at the same season, if
your mother's cat had but kittened, though yourself had never
been born.
1 Hen. IV., 3, 1, 13.
We all admit at once that the retort was just. What principle of sound
conclusion was involved in it? It is the business of Inductive Logic
to make such principles explicit.
Taking _Post Hoc ergo Propter Hoc_ as a generic name for fallacious
arguments of causation based on observed facts, for the fallacious
proof of causation from experience, the question for Logic is, What
more than mere _sequence_ is required to prove _consequence?_ When do
observations of _Post Hoc_ warrant the conclusion _Propter Hoc?_
II.--MEANING OF "CAUSE".--METHODS OF OBSERVATION--MILL'S EXPERIMENTAL
METHODS.
The methods formulated by Mill under the name of Experimental Methods
are methods actually practised by men of science with satisfactory
results, and are perfectly sound in principle. They were, indeed, in
substance, taken by him from the practice of the scientific laboratory
and study as generalised by Herschel. In effect what Mill did was to
restate them and fit them into a system. But the controversies into
which he was tempted in so doing have somewhat obscured their exact
function in scientific inquiry. Hostile critics, finding that they did
not serve the ends that he seemed to claim for them, have jumped to
the conclusion that they are altogether illusory and serve no purpose
at all.
First, we must dismiss the notion, encouraged by Mill's general theory
of Inference, that the Experimental Methods have anything special to
do with the observation and inferential extension of uniformities such
as that death is common to all organised beings. One of the Methods,
as we shall see, that named by Mill the Method of Agreement, does
incidentally and collaterally establish empirical laws in the course
of its observations, and this probably accounts for the prominence
given to it in Mill's system. But this is not its end and aim, and
the leading Method, that named by him the Method of Difference,
establishes as fact only a particular case of causal coincidence.
It is with the proof of theories of causation that the Experimental
Methods are concerned: they are methods of observing with a view to
such proof.[1]
The next point to be made clear is that the facts of causation with
which the Methods are concerned are observable facts, relations among
phenomena, but that the causal relations or conditions of which they
are the proof are not phenomena, in the meaning of being manifest
to the senses, but rather noumena, inasmuch as they are reached by
reasoning from what is manifest.
Take, for example, what is known as the _quaquaversus_ principle in
Hydrostatics, that pressure upon a liquid is propagated equally in
all directions. We cannot observe this extension of pressure among the
liquid particles directly. It cannot be traced among the particles
by any of our senses. But we can assume that it is so, consider what
ought to be visible if it is so, and then observe whether the visible
facts are in accordance with the hypothesis. A box can be made, filled
with water, and so fitted with pistons on top and bottom and on each
of its four sides that they will indicate the amount of pressure on
them from within. Let pressure then be applied through a hole in
the top, and the pistons show that it has been communicated to them
equally. The application of the pressure and the yielding of the
pistons are observable facts, facts in causal sequence: what happens
among the particles of the liquid is not observed but reasonably
conjectured, is not _phenomenal_ but _noumenal_.
This distinction, necessary to an understanding of the scope of the
Methods, was somewhat obscured by Mill in his preliminary
discussion of the meaning of "cause". Very rightly, though somewhat
inconsistently with his first theory of Induction, he insists that
"the notion of Cause being the root of the whole theory of Induction,
it is indispensable that this idea should at the very outset of our
inquiry be, with the utmost practicable degree of precision, fixed
and determined". But in this determination, not content with simply
recognising that it is with phenomena that the Experimental Methods
primarily deal, it being indeed only phenomena that can be the
subjects of experimental management and observation, he starts by
declaring that science has not to do with any causes except such as
are phenomenal--"when I speak of the cause of any phenomenon, I do not
mean a cause which is not itself a phenomenon"--and goes on to define
as the only correct meaning of cause "the sum total of conditions,"
including among them conditions which are not phenomenal, in the sense
of being directly open to observation.
When Mill protested that he had regard only to phenomenal causes, he
spoke as the partisan of a philosophical tradition. It would have been
well if he had acted upon his own remark that the proper understanding
of the scientific method of investigating cause is independent of
metaphysical analysis of what cause means. Curiously enough, this
remark is the preface to an analysis of cause which has but slight
relevance to science, and is really the continuation of a dispute
begun by Hume. This is the key to his use of the word phenomenon: it
must be interpreted with reference to this: when he spoke of causes
as phenomenal, he opposed the word to "occult" in some supposed
metaphysical sense.[2] And this irrelevant discussion, into the
vortex of which he allowed himself to be carried, obscured the fact,
elsewhere fully recognised by Mill himself, that science does attempt
to get beyond phenomena at ultimate laws which are not themselves
phenomena though they bind phenomena together. The "colligation"
of the facts, to use Whewell's phrase, is not a phenomenon, but a
noumenon.
The truth is that a very simple analysis of "cause" is sufficient for
the purposes of scientific inquiry. It is enough to make sure that
causal sequence or consequence shall not be confounded with simple
sequence. Causal sequence is simple sequence and something more, that
something more being expressed by calling it causal. What we call a
cause is not merely antecedent or prior in time to what we call its
effect: it is so related to the effect that if it or an equivalent
event had not happened the effect would not have happened. Anything
in the absence of which a phenomenon would not have come to pass as it
did come to pass is a cause in the ordinary sense. We may describe it
as an indispensable antecedent, with this reservation (which will
be more fully understood afterwards), that if we speak of a
general effect, such as death, the antecedents must be taken with
corresponding generality.
It is misleading to suggest, as Mill does, by defining cause as "the
sum total of conditions"--a definition given to back up his conception
of cause as phenomenal--that science uses the word cause in a
different meaning from that of ordinary speech. It is quite true
that "the cause, philosophically speaking, is the sum total of the
conditions, positive and negative, taken together: the whole of
the contingencies of every description, which being realised,
the consequent invariably follows". But this does not imply any
discrepancy between the scientific or philosophical meaning and the
ordinary meaning. It is only another way of saying that the business
of science or philosophy is to furnish a complete explanation of an
event, an account of all its indispensable antecedents. The plain
man would not refuse the name of cause to anything that science or
philosophy could prove to be an indispensable antecedent, but his
interest in explanation is more limited. It is confined to what he
wants to know for the purpose he has in hand. Nor could the man of
science consistently refuse the name of cause to what the plain man
applies it to, if it really was something in consequence of which the
event took place. Only his interest in explanation is different. The
indispensable antecedents that he wants to know may not be the same.
Science or philosophy applies itself to the satisfaction of a wider
curiosity: it wants to know all the causes, the whole why, the sum
total of conditions. To that end the various departments of science
interest themselves in various species of conditions. But all
understand the word cause in the ordinary sense.
We must not conclude from accidental differences in explanation or
statement of cause, dependent on the purpose in view, that the word
Cause is used in different senses. In answering a question as to
the cause of anything, we limit ourselves to what we suppose our
interrogator to be ignorant of and desirous of knowing. If asked why
the bells are ringing, we mention a royal marriage, or a victory, or a
church meeting, or a factory dinner hour, or whatever the occasion
may be. We do not consider it necessary to mention that the bells
are struck by a clapper. Our hearer understands this without our
mentioning it. Nor do we consider it necessary to mention the acoustic
condition, that the vibration of the bells is communicated to our ears
through the air, or the physiological condition, that the vibrations
in the drums of our ears are conveyed by a certain mechanism of bone
and tissue to the nerves. Our hearer may not care to know this,
though quite prepared to admit that these conditions are indispensable
antecedents. Similarly, a physiographer, in stating the cause of the
periodical inundation of the Nile, would consider it enough to mention
the melting of snow on the mountains in the interior of Africa,
without saying anything of such conditions as the laws of gravity
or the laws of liquefaction by heat, though he knows that these
conditions are also indispensable. Death is explained by the doctor
when referred to a gunshot wound, or a poison, or a virulent disease.
The Pathologist may inquire further, and the Moral Philosopher further
still. But all inquiries into indispensable conditions are inquiries
into cause. And all alike have to be on their guard against mistaking
simple sequence for consequence.
To speak of the sum total of conditions, as the Cause in a
distinctively scientific sense, is misleading in another direction. It
rather encourages the idea that science investigates conditions in
the lump, merely observing the visible relations between sets of
antecedents and their consequents. Now this is the very thing
that science must avoid in order to make progress. It analyses the
antecedent situation, tries to separate the various coefficients, and
finds out what they are capable of singly. It must recognise that
some of the antecedents of which it is in search are not open to
observation. It is these, indeed, for the most part that constitute
the special subject-matter of the sciences in Molar as well as in
Molecular Physics. For practical every-day purposes, it is chiefly
the visible succession of phenomena that concerns us, and we are
interested in the latent conditions only in as far as they provide
safer ground for inference regarding such visible succession. But to
reach the latent conditions is the main work of science.
It is, however, only through observation of what is open to the senses
that science can reach the underlying conditions, and, therefore,
to understand its methods we must consider generally what is open to
observation in causal succession. What can be observed when phenomena
follow one another as cause and effect, that is, when the one happens
in consequence of the happening of the other? In Hume's theory,
which Mill formally adopted with a modification,[3] there is nothing
observable but the constancy or invariability of the connexion. When
we say that Fire burns, there is nothing to be observed except that a
certain sensation invariably follows upon close proximity to fire. But
this holds good only if our observation is arbitrarily limited to the
facts enounced in the expression. If this theory were sound, science
would be confined to the observation of empirical laws. But that there
is something wrong with it becomes apparent when we reflect that it
has been ascertained beyond doubt that in many observed changes, and
presumably in all, there is a transference of energy from one form to
another. The paralogism really lies in the assumption from which
Hume deduced his theory, namely, that every idea is a copy of some
impression. As a matter of fact, we have ideas that are not copies
of any one impression, but a binding together, colligation, or
intellection of several impressions. Psychological analysis shows us
that even when we say that things exist with certain qualities, we are
expressing not single impressions or mental phenomena, but supposed
causes and conditions of such, _noumena_ in short, which connect our
recollections of many separate impressions and expectations of more.
The Experimental Methods proceed on the assumption that there is other
outward and visible evidence of causal connexion than invariability of
sequence. In the leading Method it is assumed that when events may be
observed to follow one another in a certain way, they are in causal
sequence. If we can make sure that an antecedent change is the only
change that has occurred in an antecedent situation, we have proof
positive that any immediately subsequent change in the situation is a
consequent, that the successive changes are in causal sequence. Thus
when Pascal's barometer was carried to the top of Puy le Dome, and
the mercury in it fell, the experimenters argued that the fall of the
mercury was causally connected with the change of elevation, all the
other circumstances remaining the same. This is the foundation of the
so-called Method of Difference. To determine that the latent condition
was a difference in the weight of the atmosphere, needed other
observations, calculations and inferences; but if it could be shown
that the elevation was the only antecedent changed in a single
instance, causal connexion was established between this and the
phenomenon of the fall of the barometer.
It is obvious that in coming to this conclusion we assume what cannot
be demonstrated but must simply be taken as a working principle to be
confirmed by its accordance with experience, that nothing comes into
being without some change in the antecedent circumstances. This is the
assumption known as the Law of Causation--_ex nihilo nihil fit_.
Again, certain observable facts are taken as evidence that there is
no causal connexion. On the assumption that any antecedent in whose
absence a phenomenon takes place is not causally connected with it,
we set aside or eliminate various antecedents as fortuitous or
non-causal. This negative principle, as we shall see, is the
foundation of what Mill called the Method of Agreement.
Be it remarked, once for all, that before coming to a conclusion on
the Positive Method or Method of Difference, we may often have to make
many observations on the Negative Method. Thus Pascal's experimenters,
before concluding that the change of altitude was the only influential
change, tried the barometer in exposed positions and in sheltered,
when the wind blew and when it was calm, in rain and in fog, in order
to prove that these circumstances were indifferent. We must expound
and illustrate the methods separately, but every method known to
science may have in practice to be employed in arriving at a single
conclusion.
[Footnote 1: This is implied, as I have already remarked, in
the word Experimental. An experiment is a proof or trial: of
what? Of a theory, a conjecture.]
[Footnote 2: If we remember, as becomes apparent on exact
psychological analysis, that things and their qualities are as
much _noumena_ and not, strictly speaking, _phenomena_ as the
attraction of gravity or the quaquaversus principle in liquid
pressure, the prejudice against occultism is mitigated.]
[Footnote 3: The modification was that causation is not only
"invariable" but also "unconditional" sequence. This addition
of unconditionality as part of the meaning of cause, after
defining cause as the sum total of the conditions, is very
much like arguing in a circle. After all, the only point
recognised in the theory as observable is the invariability of
the sequence. But this is less important than the fact that
in his canons of the Experimental Methods Mill recognised that
more is observable.]
CHAPTER IV.
METHODS OF OBSERVATION.--SINGLE DIFFERENCE.
I.--THE PRINCIPLE OF SINGLE DIFFERENCE.--MILL'S "CANON".
On what principle do we decide, in watching a succession of phenomena,
that they are connected as cause and effect, that one happened
in consequence of the happening of another? It may be worded as
follows:--
_When the addition of an agent is followed by the appearance
or its subtraction by the disappearance of a certain effect,
no other influential circumstance having been added or
subtracted at the same time or in the meantime, and no change
having occurred among the original circumstances, that agent
is a cause of the effect._
On this principle we would justify our belief in the causal properties
of common things--that fire burns, that food appeases hunger, that
water quenches thirst, that a spark ignites gunpowder, that taking
off a tight shoe relieves a pinched foot. We have observed the
effect following when there was no other change in the antecedent
circumstances, when the circumstance to which we refer it was simply
added to or subtracted from the prior situation.
Suppose we doubt whether a given agent is or is not capable of
producing a certain effect in certain circumstances, how do we put it
to the proof? We add it singly or subtract it singly, taking care that
everything else remains as before, and watch the result. If we wish
to know whether a spoonful of sugar can sweeten a cup of tea, we taste
the tea without the sugar, then add the sugar, and taste again. The
isolated introduction of the agent is the proof, the experiment. If we
wish to know whether a pain in the foot is due to a tight lacing,
we relax the lacing and make no other change: if the pain then
disappears, we refer it to the lacing as the cause. The proof is the
disappearance of the pain on the subtraction of the single antecedent.
The principle on which we decide that there is causal connexion is
the same whether we make the experimental changes ourselves or merely
watch them as they occur--the only course open to us with the great
forces of nature which are beyond the power of human manipulation. In
any case we have proof of causation when we can make sure that there
was only one difference in the antecedent circumstances corresponding
to the difference of result.
Mill's statement of this principle, which he calls the Canon of the
Method of Difference, is somewhat more abstract, but the proof relied
upon is substantially the same.
_If an instance in which the phenomenon under investigation
occurs, and an instance in which it does not occur, have every
circumstance in common save one, that one occurring only in
the former, the circumstance in which alone the two instances
differ is [the effect, or][1] the cause, or an indispensable
part of the cause, of the phenomenon._
Mill's statement has the merit of exactness, but besides being too
abstract to be easy of application, the canon is apt to mislead in one
respect. The wording of it suggests that the two instances required
must be two separate sets of circumstances, such as may be put side
by side and compared, one exhibiting the phenomenon and the other
not. Now in practice it is commonly one set of circumstances that we
observe with a special circumstance introduced or withdrawn: the two
instances, the data of observation, are furnished by the scene before
and the scene after the experimental interference. In the case, for
example, of a man shot in the head and falling dead, death being the
phenomenon in question, the instance where it does not occur is the
man's condition before he received the wound, and the instance where
it does occur is his condition after, the single circumstance of
difference being the wound, a difference produced by the addition or
introduction of a new circumstance. Again, take the common coin and
feather experiment, contrived to show that the resistance of the air
is the cause of the feather's falling to the ground more slowly than
the coin. The phenomenon under investigation is the retardation of the
feather. When the two are dropped simultaneously in the receiver of
an air-pump, the air being left in, the feather flutters to the ground
after the coin. This is the instance where the phenomenon occurs. Then
the air is pumped out of the receiver, and the coin and the feather
being dropped at the same instant reach the ground together. This
is the instance where the phenomenon does not occur. The single
circumstances of difference is the presence of air in the former
instance, a difference produced by the subtraction of a circumstance.
Mill's Canon is framed so as to suit equally whether the significant
difference is produced by addition to or subtraction from an existing
sum of circumstances. But that is misleading in so far as it suggests
that the two instances must be separate sets of circumstances,
is shown by the fact that it misled himself when he spoke of the
application of the method in social investigations, such as the effect
of Protection on national wealth. "In order," he says, "to apply to
the case the most perfect of the methods of experimental inquiry, the
Method of Difference, we require to find two instances which tally in
every particular except the one which is the subject of inquiry.
We must have two nations alike in all natural advantages and
disadvantages; resembling each other in every quality physical and
moral; habits, usages, laws, and institutions, and differing only in
the circumstance that the one has a prohibitory tariff and the other
has not." It being impossible ever to find two such instances, he
concluded that the Method of Difference could not be applied in
social inquiries. But really it is not necessary in order to have two
instances that we should have two different nations: the same nation
before and after a new law or institution fulfils that requirement.
The real difficulty, as we shall see, is to satisfy the paramount
condition that the two instances shall differ in a single
circumstance. Every new enactment would be an experiment after the
Method of Difference, if all circumstances but it remained the same
till its results appeared. It is because this seldom or never occurs
that decisive observation is difficult or impossible, and the simple
method of difference has to be supplemented by other means.
To introduce or remove a circumstance singly is the typical
application of the principle; but it may be employed also to compare
the effects of different agents, each added alone to exactly similar
circumstances. A simple example is seen in Mr. Jamieson's agricultural
experiments to determine the effects of different manures, such as
coprolite and superphosphate, on the growth of crops. Care is taken
to have all the antecedent circumstances as exactly alike as possible,
except as regards the agency whose effects are to be observed. A field
is chosen of uniform soil and even exposure and divided into plots:
it is equally drained so as to have the same degree of moisture
throughout; the seed is carefully selected for the whole sowing.
Between the sowing and the maturing of the crop all parts of the
field are open to the same weather. Each plot may thus be regarded as
practically composing the same set of conditions, and any difference
in the product may with reasonable probability be ascribed to the
single difference in the antecedents, the manures which it is desired
to compare.
II.--APPLICATION OF THE PRINCIPLE.
The principle of referring a phenomenon to the only immediately
preceding change in antecedent circumstances that could possibly have
affected it, is so simple and so often employed by everybody every
day, that at first we do not see how there can be any difficulty
about it or any possibility of error. And once we understand how many
difficulties there are in reaching exact knowledge even on this simple
principle, and what care has to be taken, we are apt to overrate its
value, and to imagine that it carries us further than it really does.
The scientific expert must know how to apply this principle, and a
single application of it with the proper precautions may take him
days or weeks, and yet all that can be made good by it may carry but a
little way towards the knowledge of which he is in search.
When the circumstances are simple and the effect follows at once, as
when hot water scalds, or a blow with a stick breaks a pane of glass,
there can be no doubt of the causal connexion so far, though plenty
of room for further inquiry into the why. But the mere succession of
phenomena may be obscure. We may introduce more than one agent
without knowing it, and if some time elapses between the experimental
interference and the appearance of the effect, other agents may come
in without our knowledge.
We must know exactly what it is that we introduce and all the
circumstances into which we introduce it. We are apt to ignore the
presence of antecedents that are really influential in the result. A
man heated by work in the harvest field hastily swallows a glass of
water, and drops down dead. There is no doubt that the drinking of the
water was a causal antecedent, but the influential circumstance
may not have been the quantity or the quality of the liquid but its
temperature, and this was introduced into the situation as well as a
certain amount of the liquid components. In making tea we put in so
much tea and so much boiling water. But the temperature of the pot
is also an influential circumstance in the resulting infusion. So in
chemical experiments, where one might expect the result to depend only
upon the proportions of the ingredients, it is found that the quantity
is also influential, the degree of heat evolved entering as a
factor into the result. Before we can apply the principle of single
difference, we must make sure that there is really only a single
difference between the instances that we bring into comparison.
The air-pump was invented shortly before the foundation of the Royal
Society, and its members made many experiments with this new means
of isolating an agent and thus discovering its potentialities.
For example, live animals were put into the receiver, and the air
exhausted, with the result that they quickly died. The absence of the
air being the sole difference, it was thus proved to be indispensable
to life. But air is a composite agent, and when means were contrived
of separating its components, the effects of oxygen alone and of
carbonic acid alone were experimentally determined.
A good example of the difficulty of excluding agencies other than
those we are observing, of making sure that none such intrude,
is found in the experiments that have been made in connexion with
spontaneous generation. The question to be decided is whether life
ever comes into existence without the antecedent presence of living
germs. And the method of determining this is to exclude all germs
rigorously from a compound of inorganic matter, and observe whether
life ever appears. If we could make sure in any one case that no germs
were antecedently present, we should have proved that in that case at
least life was spontaneously generated.
The difficulty here arises from the subtlety of the agent under
observation. The notion that maggots are spontaneously generated in
putrid meat, was comparatively easy to explode. It was found that when
flies were excluded by fine wire-gauze, the maggots did not appear.
But in the case of microscopic organisms proof is not so easy. The
germs are invisible, and it is difficult to make certain of their
exclusion. A French experimenter, Pouchet, thought he had obtained
indubitable cases of spontaneous generation. He took infusions of
vegetable matter, boiled them to a pitch sufficient to destroy all
germs of life, and hermetically sealed up the liquid in glass
flasks. After an interval, micro-organisms appeared. Doubts as to the
conclusion that they had been spontaneously generated turned upon two
questions: whether all germs in the liquid had been destroyed by the
preliminary boiling, and whether germs could have found access in the
course of the interval before life appeared. At a certain stage in
Pouchet's process he had occasion to dip the mouths of the flasks
in mercury. It occurred to Pasteur in repeating the experiments that
germs might have found their way in from the atmospheric dust on the
surface of this mercury. That this was so was rendered probable by his
finding that when he carefully cleansed the surface of the mercury no
life appeared afterwards in his flasks.
The application of the principle in human affairs is rendered
uncertain by the immense complication of the phenomena, the difficulty
of experiment, and the special liability of our judgments to
prejudice. That men and communities of men are influenced by
circumstances is not to be denied, and the influence of circumstances,
if it is to be traced at all, must be traced through observed facts.
Observation of the succession of phenomena must be part at least of
any method of tracing cause and effect. We must watch what follows
upon the addition of new agencies to a previously existing sum. But
we can seldom or never get a decisive observation from one pair of
instances, a clear case of difference of result preceded by a single
difference in the antecedents. The simple Method of Experimental
Addition or Subtraction is practically inapplicable. We can do nothing
with a man analogous to putting him into a hermetically sealed retort.
Any man or any community that is the subject of our observations
must be under manifold influences. Each of them probably works some
fraction of the total change observable, but how are they to be
disentangled? Consider, for example, how impossible it would be
to prove in an individual case, on the strict principle of Single
Difference, that Evil communications corrupt good manners. Moral
deterioration may be observed following upon the introduction of
an evil companion, but how can we make sure that no other degrading
influence has operated, and that no original depravity has developed
itself in the interval? Yet such propositions of moral causation can
be proved from experience with reasonable probability. Only it must
be by more extended observations than the strict Method of Difference
takes into account. The method is to observe repeated coincidences
between evil companionship and moral deterioration, and to account for
this in accordance with still wider observations of the interaction of
human personalities.
For equally obvious reasons the simple Method of Difference is
inapplicable to tracing cause and effect in communities. Every new law
or repeal of an old law is the introduction of a new agency, but the
effects of it are intermixed with the effects of other agencies that
operate at the same time. Thus Professor Cairnes remarks, concerning
the introduction of a high Protective Tariff into the United States
in 1861, that before its results could appear in the trade and
manufacture of the States, there occurred (1) The great Civil War,
attended with enormous destruction of capital; (2) Consequent upon
this the creation of a huge national debt, and a great increase of
taxation; (3) The issue of an inconvertible paper currency, deranging
prices and wages; (4) The discovery of great mineral resources and
oil-springs; (5) A great extension of railway enterprise. Obviously in
such circumstances other methods than the Method of Difference must
be brought into play before there can be any satisfactory reasoning on
the facts observed. Still what investigators aim at is the isolation
of the results of single agencies.
[Footnote 1: Prof. Bain, who adopts Mill's Canon, silently
drops the words within brackets. They seem to be an
inadvertence. The "circumstance," in all the examples
that Mill gives, is an antecedent circumstance. Herschel's
statement, of which Mill's is an adaptation, runs as follows:
"If we can either find produced by nature, or produce
designedly for ourselves, two instances which agree exactly in
all but one particular and differ in that one, its influence
in producing the phenomenon, if it have any, must thereby be
rendered apparent".]
CHAPTER V.
METHODS OF OBSERVATION.--ELIMINATION.--SINGLE AGREEMENT.
I.--THE PRINCIPLE OF ELIMINATION.
The essence of what Mill calls the Method of Agreement is really the
elimination[1] of accidental, casual, or fortuitous antecedents. It
is a method employed when we are given an effect and set to work to
discover the cause. It is from the effect that we start and work back.
We make a preliminary analysis of the antecedents; call the roll, as
it were, of all circumstances present before the effect appeared. Then
we proceed to examine other instances of the same effect, and other
instances of the occurrence of the various antecedents, and bring to
bear the principle that any antecedent in the absence of which the
effect has appeared or on the presence of which it has not appeared
may be set aside as fortuitous, as being not an indispensable
antecedent. This is really the guiding principle of the method as a
method of observation.
Let the inquiry, for example, be into the cause of Endemic Goitre.
Instances of the disease have been collected from the medical
observations of all countries over many years. Why is it endemic in
some localities and not in others? We proceed on the assumption that
the cause, whatever it is, must be some circumstance common to all
localities where it is endemic. If any such circumstance is obvious
at once, we may conclude on the mere principle of repeated coincidence
that there is causal connexion between it and the disease, and
continue our inquiry into the nature of the connexion. But if no such
circumstance is obvious, then in the course of our search for it we
eliminate, as fortuitous, conditions that are present in some cases
but absent in others. One of the earliest theories was that endemic
goitre was connected with the altitude and configuration of the
ground, some notorious centres of it being deeply cleft mountain
valleys, with little air and wind and damp marshy soil. But wider
observation found it in many valleys neither narrower nor deeper than
others that were exempt, and also in wide exposed valleys such as
the Aar. Was it due to the geological formation? This also had to
be abandoned, for the disease is often incident within very narrow
limits, occurring in some villages and sparing others though the
geological formation is absolutely the same. Was it due to the
character of the drinking-water? Especially to the presence of lime
or magnesia? This theory was held strongly, and certain springs
characterised as goitre-springs. But the springs in some goitre
centres show not a trace of magnesia. The comparative immunity of
coast regions suggested that it might be owing to a deficiency of
iodine in the drinking-water and the air, and many instances were
adduced in favour of this. But further inquiries made out the presence
of iodine in considerable quantities, in the air, the water, and the
vegetation of districts where goitre was widely prevalent; while in
Cuba it is said that not a trace of iodine is discoverable either in
the air or the water, and yet it is quite free from goitre. After
a huge multiplication of instances, resulting in the elimination of
every local condition that had been suggested as a possible cause,
Hirsch came to the conclusion that the true cause must be a morbid
poison, and that endemic goitre has to be reckoned among the
infectious diseases.[2]
On this negative principle, that if a circumstance comes and goes
without bringing the phenomenon in its train, the phenomenon
is causally independent of it, common-sense is always at work
disconnecting events that are occasionally coincident in time. A
bird sings at our window, for example, and the clock ticks on the
mantelpiece. But the clock does not begin to tick when the bird begins
to sing, nor cease to tick when the bird flies away. Accordingly, if
the clock should stop at any time, and we wished to inquire into the
cause, and anybody were to suggest that the stoppage of the clock was
caused by the stoppage of a bird's song outside, we should dismiss
the suggestion at once. We should eliminate this circumstance from our
inquiry, on the ground that from other observations we knew it to be
a casual or fortuitous concomitant. Hotspur's retort to Glendover
(p. 297) was based on this principle. When poetic sentiment or
superstition rejects a verdict of common-sense or science, it is
because it imagines a causal connexion to exist that is not open to
observation, as in the case of the grandfather's clock which stopped
short never to go again when the old man died.
II.--THE PRINCIPLE OF SINGLE AGREEMENT.
The procedure in Mill's "Method of Agreement" consists in thus
eliminating fortuitous antecedents or concomitants till only one
remains. We see the nature of the proof relied upon when we ask, How
far must elimination be carried in order to attain proof of causal
connexion? The answer is that we must go on till we have eliminated
all but one. We must multiply instances of the phenomenon, till we
have settled of each of the antecedents except one that it is not the
cause. We must have taken account of all the antecedents, and we
must have found in our observations that all but one have been only
occasionally present.
_When all the antecedents of an effect except one can be
absent without the disappearance of the effect, that one is
causally connected with the effect, due precautions being
taken that no other circumstances have been present besides
those taken account of._
Mill's Canon of the Method of Agreement is substantially identical
with this:--
When two or more instances of the phenomenon under
investigation have only one circumstance in common, the
circumstance in which alone all the instances agree is the
cause (or effect) of the given phenomenon.
Herschel's statement, on which this canon is founded, runs as follows:
"Any circumstance in which all the facts without exception agree, may
be the cause in question, or if not, at least a collateral effect
of the same cause: if there be but one such point of agreement, the
possibility becomes a certainty".
All the instances examined must agree in one circumstance--hence the
title Method of Agreement. But it is not in the agreement merely that
the proof consists, but the agreement in one circumstance combined
with difference in all the other circumstances, when we are certain
that every circumstance has come within our observation. It is the
singleness of the agreement that constitutes the proof just as it is
the singleness of the difference in the Method of Difference.[3]
It has been said that Mill's Method of Agreement amounts after all
only to an uncontradicted _Inductio per enumerationem simplicem_,
which he himself stigmatised as Induction improperly so called. But
this is not strictly correct. It is a misunderstanding probably caused
by calling the method that of agreement simply, instead of calling it
the Method of Single Agreement, so as to lay stress upon the process
of elimination by which the singleness is established. It is true that
in the course of our observations we do perform an induction by simple
enumeration. In eliminating, we at the same time generalise. That is
to say, in multiplying instances for the elimination of non-causes, we
necessarily at the same time multiply instances where the true causal
antecedent, if there is only one possible, is present. An antecedent
containing the true cause must always be there when the phenomenon
appears, and thus we may establish by our eliminating observations a
uniformity of connexion between two facts.
Take, for example, Roger Bacon's inquiry into the cause of the colours
of the rainbow. His first notion seems to have been to connect the
phenomenon with the substance crystal, probably from his thinking of
the crystal firmament then supposed to encircle the universe. He found
the rainbow colours produced by the passage of light through hexagonal
crystals. But on extending his observations, he found that the passage
of light through other transparent mediums was also attended by the
phenomenon. He found it in dewdrops, in the spray of waterfalls, in
drops shaken from the oar in rowing. He thus eliminated the substance
crystal, and at the same time established the empirical law that
the passage of light through transparent mediums of a globular or
prismatic shape was a causal antecedent of the rainbow colours.[4]
Ascertainment of invariable antecedents may thus proceed side by side
with that of variable antecedents, the use of the elimination being
simply to narrow the scope of the inquiry. But the proof set forth in
Mill's Canon does not depend merely on one antecedent or concomitant
being invariably present, but also on the assumption that all the
influential circumstances have been within our observation. Then only
can we be sure that the instances have _only one_ circumstance in
common.
The truth is that owing to the difficulty of fulfilling this
condition, proof of causation in accordance with Mill's Canon is
practically all but impossible. It is not attained in any of the
examples commonly given. The want of conclusiveness is disguised
by the fact that both elimination and positive observation of mere
agreement or uniform concomitance are useful and suggestive in the
search for causes, though they do not amount to complete proof such as
the Canon describes. Thus in the inquiry into the cause of goitre, the
elimination serves some purpose though the result is purely negative.
When the inquirer is satisfied that goitre is not originated by any
directly observable local conditions, altitude, temperature, climate,
soil, water, social circumstances, habits of exertion, his search
is profitably limited. And mere frequency, much more constancy of
concomitance, raises a presumption of causal connexion, and looking
out for it is valuable as a mode of reconnoitring. The first thing
that an inquirer naturally asks when confronted by numerous instances
of a phenomenon is, What have they in common? And if he finds that
they have some one circumstance invariably or even frequently present,
although he cannot prove that they have no other circumstance in
common as the Cannon of Single Agreement requires, the presumption of
causal connexion is strong enough to furnish good ground for further
inquiry. If an inquirer finds an illness with marked symptoms in a
number of different households, and finds also that all the households
get their milk supply from the same source, this is not conclusive
proof of causation, but it is a sufficient presumption to warrant him
in examining whether there is any virulent ingredient in the milk.
Thus varying the circumstances so as to bring out a common antecedent,
though it does not end in exact proof, may indicate causal connexion
though it does not prove what the nature of the connexion is. Roger
Bacon's observations indicated that the production of rainbow colours
was connected with the passage of light through a transparent globe or
prism. It was reserved for Newton to prove by other methods that
white light was composed of rays, and that those rays were differently
refracted in passing through the transparent medium. We have
another example of how far mere agreement, revealed by varying the
circumstances, carries us towards discovery of the cause, in Wells's
investigation of the cause of dew. Comparing the numerous instances of
dew appearing without visible fall of moisture, Wells found that they
all agreed in the comparative coldness of the surface dewed. This was
all the agreement that he established by observation; he did not carry
observation to the point of determining that there was absolutely
no other common circumstance: when he had simply discovered dewed
surfaces, he tried next to show by reasoning from other knows facts
how the coldness of the surface affected the aqueous vapour of the
neighbouring air. He did not establish his Theory of Dew by the Method
of Agreement: but the observation of an agreement or common feature in
a number of instances was a stage in the process by which he reached
his theory.
III.--MILL'S "JOINT METHOD OF AGREEMENT AND DIFFERENCE".
After examining a variety of instances in which an effect appears,
and finding that they all agree in the antecedent presence of some one
circumstance, we may proceed to examine instances otherwise similar
(_in pari materia_, as Prof. Fowler puts it) where the effect does not
appear. If these all agree in the absence of the circumstance that is
uniformly present with the effect, we have corroborative evidence that
there is causal connexion between this circumstance and the effect.
The principle of this method seems to have been suggested to Mill by
Wells's investigations into Dew. Wells exposed a number of polished
surfaces of various substances, and compared those in which there was
a copious deposit of dew with those in which there was little or
none. If he could have got two surfaces, one dewed and the other
not, identical in every concomitant but one, he would have attained
complete proof on the principle of Single Difference. But this being
impracticable, he followed a course which approximated to the method
of eliminating every circumstance but one from instances of dew, and
every circumstance but one in the instances of no-dew. Mill sums up as
follows the results of his experiments: "It appears that the instances
in which much dew is deposited, which are very various, agree in
this, and, _so far as we are able to observe, in this only_, that they
either radiate heat rapidly or conduct it slowly: qualities between
which there is no other circumstance of agreement than that by virtue
of either, the body tends to lose heat from the surface more rapidly
than it can be restored from within. The instances, on the contrary,
in which no dew, or but a small quantity of it, is formed, and which
are also extremely various, agree (_as far as we can observe_) _in
nothing except_ in _not_ having this same property. We seem therefore
to have detected the characteristic difference between the substances
on which the dew is produced, and those on which it is not produced.
And thus have been realised the requisitions of what we have termed
the Indirect Method of Difference, or the Joint Method of Agreement
and Difference." The Canon of this Method is accordingly stated by
Mill as follows:--
If two or more instances in which the phenomenon occurs have
only one circumstance in common, while two or more instances
in which it does not occur have nothing in common save the
absence of that circumstance; the circumstance in which alone
the two sets of instances differ, is the effect, or the cause,
or an indispensable part of the cause, of the phenomenon.
In practice, however, this theoretical standard of proof is never
attained. What investigators really proceed upon is the presumption
afforded, to use Prof. Bain's terms, by Agreement in Presence combined
with Agreement in Absence. When it is found that all substances which
have a strong smell agree in being readily oxidisable, and that the
marsh gas or carbonetted hydrogen which has no smell is not oxidisable
at common temperatures, the presumption that oxidation is one of the
causal circumstances in smell is strengthened, even though we have not
succeeded in eliminating every circumstance but this one from either
the positive or the negative instances. So in the following examples
given by Prof. Fowler there is not really a compliance with the
theoretical requirements of Mill's Method: there is only an increased
presumption from the double agreement. "The Joint Method of Agreement
and Difference (or the Indirect Method of Difference, or, as I
should prefer to call it, the Double Method of Agreement) is being
continually employed by us in the ordinary affairs of life. If when I
take a particular kind of food, I find that I invariably suffer from
some particular form of illness, whereas, when I leave it off, I cease
to suffer, I entertain a double assurance that the food is the cause
of my illness. I have observed that a certain plant is invariably
plentiful on a particular soil; if, with a wide experience, I fail to
find it growing on any other soil, I feel confirmed in my belief that
there is in this particular soil some chemical constituent, or
some peculiar combination of chemical constituents, which is highly
favourable, if not essential, to the growth of the plant."
[Footnote 1: Elimination, or setting aside as being of no
concern, must not be confounded with the exclusion of agents
practised in applying the Method of Difference. We use the
word in its ordinary sense of putting outside the sphere of
an argument. By a curious slip, Professor Bain follows Mill in
applying the word sometimes to the process of singling out or
disentangling a causal circumstance. This is an inadvertent
departure from the ordinary usage, according to which
elimination means discarding from consideration as being
non-essential.]
[Footnote 2: Hirsch's _Geographical and Historical Pathology_,
Creighton's translation, vol. ii. pp. 121-202.]
[Footnote 3: The bare titles Difference and Agreement, though
they have the advantage of simplicity, are apt to puzzle
beginners inasmuch as in the Method of Difference the
agreement among the instances is at a maximum, and the
difference at a minimum, and _vice versâ_ in the Method of
Agreement. In both Methods it is really the isolation of the
connexion between antecedent and sequent that constitutes the
proof.]
[Footnote 4: That rainbows in the sky are produced by the
passage of light through minute drops in the clouds was an
inference from this observed uniformity.]
CHAPTER VI.
METHODS OF OBSERVATION.--MINOR METHODS.
I.--CONCOMITANT VARIATIONS.
_Whatever phenomenon varies in any manner whenever another
phenomenon varies in some particular manner, is either a
cause or an effect of that phenomenon, or is connected with it
through some fact of causation._
This simple principle is constantly applied by us in connecting and
disconnecting phenomena. If we hear a sound which waxes and wanes with
the rise and fall of the wind, we at once connect the two phenomena.
We may not know what the causal connexion is, but if they uniformly
vary together, there is at once a presumption that the one is causally
dependent on the other, or that both are effects of the same cause.
This principle was employed by Wells in his researches into Dew. Some
bodies are worse conductors of heat than others, and rough surfaces
radiate heat more rapidly than smooth. Wells made observations on
conductors and radiators of various degrees, and found that the
amount of dew deposited was greater or less according as the objects
conducted heat slowly or radiated heat rapidly. He thus established
what Herschel called a "scale of intensity" between the conducting and
radiating properties of the bodies bedewed, and the amount of the dew
deposit. The explanation was that in bad conductors the surface cools
more quickly than in good conductors because heat is more slowly
supplied from within. Similarly in rough surfaces there is a more
rapid cooling because heat is given off more quickly. But whatever
the explanation might be, the mere concomitant variation of the
dew deposit with these properties showed that there was some causal
connexion between them.
It must be remembered that the mere fact of concomitant variation is
only an index that some causal connexion exists. The nature of
the connexion must be ascertained by other means, and may remain
a problem, one of the uses of such observed facts being indeed to
suggest problems, for inquiry. Thus a remarkable concomitance has been
observed between spots on the sun, displays of Aurora Borealis, and
magnetic storms. The probability is that they are causally connected,
but science has not yet discovered how. Similarly in the various
sciences properties are arranged in scales of intensity, and any
correspondence between two scales becomes a subject for investigation
on the assumption that it points to a causal connexion. We shall see
afterwards how in social investigations concomitant variations in
averages furnish material for reasoning.
When two variants can be precisely measured, the ratio of the
variation may be ascertained by the Method of Single Difference. We
may change an antecedent in degree, and watch the corresponding change
in the effect, taking care that no other agent influences the effect
in the meantime. Often when we cannot remove an agent altogether,
we may remove it in a measurable amount, and observe the result. We
cannot remove friction altogether, but the more it is diminished, the
further will a body travel under the impulse of the same force.
Until a concomitant variation has been fully explained, it is merely
an empirical law, and any inference that it extends at the same
rate beyond the limits of observation must be made with due caution.
"Parallel variation," says Professor Bain, "is sometimes interrupted
by critical points, as in the expansion of bodies by heat, which
suffers a reverse near the point of cooling. Again, the energy of a
solution does not always follow the strength; very dilute solutions
occasionally exercise a specific power not possessed in any degree
by stronger. So, in the animal body, food and stimulants operate
proportionally up to a certain point, at which their further operation
is checked by the peculiarities in the structure of the living
organs.... We cannot always reason from a few steps in a series to the
whole series, partly because of the occurrence of critical points,
and partly from the development at the extremes of new and unsuspected
powers. Sir John Herschel remarks that until very recently 'the
formulæ empirically deduced for the elasticity of steam, those for
the resistance of fluids, and on other similar subjects, have almost
invariably failed to support the theoretical structures that have been
erected upon them'."[1]
II.--SINGLE RESIDUE.
_Subduct from any phenomenon such part as previous induction
has shown to be the effect of certain antecedents, and the
residue of the phenomenon is the effect of the remaining
antecedents._
"Complicated phenomena, in which several causes concurring, opposing,
or quite independent of each other, operate at once, so as to produce
a compound effect, may be simplified by subducting the effect of all
the known causes, as well as the nature of the case permits, either by
deductive reasoning or by appeal to experience, and thus leaving as it
were a _residual phenomenon_ to be explained. It is by this process,
in fact, that science, in its present advanced state, is chiefly
promoted. Most of the phenomena which nature presents are very
complicated; and when the effects of all known causes are estimated
with exactness, and subducted, the residual facts are constantly
appearing in the form of phenomena altogether new, and leading to the
most important conclusions."[2]
It is obvious that this is not a primary method of observation, but
a method that may be employed with great effect to guide observation
when a considerable advance has been made in accurate knowledge
of agents and their mode of operation. The greatest triumph of the
method, the discovery of the planet Neptune, was won some years after
the above passage from Herschel's Discourse was written. Certain
perturbations were observed in the movements of the planet Uranus:
that is to say, its orbit was found not to correspond exactly with
what it should be when calculated according to the known influences
of the bodies then known to astronomers. These perturbations were a
residual phenomenon. It was supposed that they might be due to
the action of an unknown planet, and two astronomers, Adams and Le
Verrier, simultaneously calculated the position of a body such as
would account for the observed deviations. When telescopes were
directed to the spot thus indicated, the planet Neptune was
discovered. This was in September, 1846: before its actual discovery,
Sir John Herschel exulted in the prospect of it in language that
strikingly expresses the power of the method. "We see it," he said,
"as Columbus saw America from the shores of Spain. Its movements have
been felt, trembling along the far-reaching line of our analysis, with
a certainty hardly inferior to that of ocular demonstration."[3]
Many of the new elements in Chemistry have been discovered in this
way. For example, when distinctive spectrums had been observed for all
known substances, then on the assumption that every substance has a
distinctive spectrum, the appearance of lines not referable to any
known substance indicated the existence of hitherto undiscovered
substances and directed search for them. Thus Bunsen in 1860
discovered two new alkaline metals, Cæsium and Rubidium. He was
examining alkalies left from the evaporation of a large quantity of
mineral water from Durkheim. On applying the spectroscope to the flame
which this particular salt or mixture of salts gave off, he found that
some bright lines were visible which he had never observed before, and
which he knew were not produced either by potash or soda. He then set
to work to analyse the mixture, and ultimately succeeded in separating
two new alkaline substances. When he had succeeded in getting them
separate, it was of course by the Method of Difference that he
ascertained them to be capable of producing the lines that had excited
his curiosity.
[Footnote 1: Bain's _Logic_, vol. ii. p. 64.]
[Footnote 2: Herschel's _Discourse_, § 158.]
[Footnote 3: De Morgan's _Budget of Paradoxes_, p. 237.]
CHAPTER VII.
THE METHOD OF EXPLANATION.
Given perplexity as to the cause of any phenomenon, what is our
natural first step? We may describe it as searching for a clue: we
look carefully at the circumstances with a view to finding some
means of assimilating what perplexes us to what is already within our
knowledge. Our next step is to make a guess, or conjecture, or, in
scientific language, a hypothesis. We exercise our Reason or _Nous_,
or Imagination, or whatever we choose to call the faculty, and try to
conceive some cause that strikes us as sufficient to account for the
phenomenon. If it is not at once manifest that this cause has really
operated, our third step is to consider what appearances ought to
present themselves if it did operate. We then return to the facts in
question, and observe whether those appearances do present themselves.
If they do, and if there is no other way of accounting for the effect
in all its circumstances, we conclude that our guess is correct,
that our hypothesis is proved, that we have reached a satisfactory
explanation.
These four steps or stages may be distinguished in most protracted
inquiries into cause. They correspond to the four stages of what
Mr. Jevons calls the Inductive Method _par excellence_, Preliminary
Observation, Hypothesis, Deduction and Verification. Seeing that the
word Induction is already an overloaded drudge, perhaps it would be
better to call these four stages the Method of Explanation. The word
Induction, if we keep near its original and most established meaning,
would apply strictly only to the fourth stage, the Verification, the
bringing in of the facts to confirm our hypothesis. We might call the
method the Newtonian method, for all four stages are marked in the
prolonged process by which he made good his theory of Gravitation.
To give the name of Inductive Method simply to all the four stages
of an orderly procedure from doubt to a sufficient explanation is
to encourage a widespread misapprehension. There could be no greater
error than to suppose that only the senses are used in scientific
investigation. There is no error that men of science are so apt to
resent in the mouths of the non-scientific. Yet they have partly
brought it on themselves by their loose use of the word Induction,
which they follow Bacon in wresting from the traditional meaning of
Induction, using it to cover both Induction or the bringing in of
facts--an affair mainly of Observation--and Reasoning, the exercise of
Nous, the process of constructing satisfactory hypotheses. In reaction
against the popular misconception which Bacon encouraged, it is
fashionable now to speak of the use of Imagination in Science. This is
well enough polemically. Imagination as commonly understood is akin to
the constructive faculty in Science, and it is legitimate warfare to
employ the familiar word of high repute to force general recognition
of the truth. But in common usage Imagination is appropriated to
creative genius in the Fine Arts, and to speak of Imagination
in Science is to suggest that Science deals in fictions, and has
discarded Newton's declaration _Hypotheses non fingo_. In a fight for
popular respect, men of science may be right to claim for themselves
Imagination; but in the interests of clear understanding, the logician
must deplore that they should defend themselves from a charge due
to their abuse of one word by making an equally unwarrantable and
confusing extension of another.
Call it what we will, the faculty of likely guessing, of making
probable hypotheses, of conceiving in all its circumstances the past
situation or the latent and supramicroscopical situation out of
which a phenomenon has emerged, is one of the most important of
the scientific man's special gifts. It is by virtue of it that the
greatest advancements of knowledge have been achieved, the cardinal
discoveries in Molar and Molecular Physics, Biology, Geology, and
all departments of Science. We must not push the idea of stages in
explanatory method too far: the right explanation may be reached in
a flash. The idea of stages is really useful mainly in trying to make
clear the various difficulties in investigation, and the fact that
different men of genius may show different powers in overcoming them.
The right hypothesis may occur in a moment, as if by simple intuition,
but it may be tedious to prove, and the gifts that tell in proof,
such as Newton's immense mathematical power in calculating what
a hypothesis implies, Darwin's patience in verifying, Faraday's
ingenuity in devising experiments, are all great gifts, and may be
serviceable at different stages. But without originality and fertility
in probable hypothesis, nothing can be done.
The dispute between Mill and Whewell as to the place and value of
hypotheses in science was in the main a dispute about words. Mill
did not really undervalue hypothesis, and he gave a most luminous and
accurate account of the conditions of proof. But here and there he
incautiously spoke of the "hypothetical method" (by which he meant
what we have called the method of Explanation) as if it were a
defective kind of proof, a method resorted to by science when the
"experimental methods" could not be applied. Whether his language
fairly bore this construction is not worth arguing, but this was
manifestly the construction that Whewell had in his mind when he
retorted, as if in defence of hypotheses, that "the inductive process
consists in framing successive hypotheses, the comparison of these
with the ascertained facts of nature, and the introduction into them
of such modifications as the comparison may render necessary". This is
a very fair description of the whole method of explanation. There
is nothing really inconsistent with it in Mill's account of his
"hypothetical method"; only he erred himself or was the cause of error
in others in suggesting, intentionally or unintentionally, that
the Experimental Methods were different methods of proof. The
"hypothetical method," as he described it, consisting of Induction,
Ratiocination, and Verification, really comprehends the principles
of all modes of observation, whether naturally or artificially
experimental. We see this at once when we ask how the previous
knowledge is got in accordance with which hypotheses are framed. The
answer must be, by Observation. However profound the calculations,
it must be from observed laws, or supposed analogues of them, that
we start. And it is always by Observation that the results of these
calculations are verified.
Both Mill and Whewell, however, confined themselves too exclusively
to the great hypotheses of the Sciences, such as Gravitation and the
Undulatory Theory of Light. In the consideration of scientific method,
it is a mistake to confine our attention to these great questions,
which from the multitude of facts embraced can only be verified by
prolonged and intricate inquiry. Attempts at the explanation of the
smallest phenomena proceed on the same plan, and the verification
of conjectures about them is subject to the same conditions, and the
methods of investigation and the conditions of verification can be
studied most simply in the smaller cases. Further, I venture to think
it a mistake to confine ourselves to scientific inquiry in the narrow
sense, meaning thereby inquiry conducted within the pale of the
exact sciences. For not merely the exact sciences but all men in the
ordinary affairs of life must follow the same methods or at least
observe the same principles and conditions, in any satisfactory
attempt to explain.
Tares appear among the wheat. Good seed was sown: whence, then, come
the tares? "An enemy has done this." If an enemy has actually been
observed sowing the tares, his agency can be proved by descriptive
testimony. But if he has not been seen in the act, we must resort to
what is known in Courts of Law as circumstantial evidence. This is the
"hypothetical method" of science. That the tares are the work of an
enemy is a hypothesis: we examine all the circumstances of the case in
order to prove, by inference from our knowledge of similar cases,
that thus, and thus only, can those circumstances be accounted for.
Similarly, when a question is raised as to the authorship of an
anonymous book. We first search for a clue by carefully noting the
diction, the structure of the sentences, the character and sources of
the illustration, the special tracks of thought. We proceed upon the
knowledge that every author has characteristic turns of phrase and
imagery and favourite veins of thought, and we look out for such
internal evidence of authorship in the work before us. Special
knowledge and acumen may enable us to detect the authorship at once
from the general resemblance to known work. But if we would have clear
proof, we must show that the resemblance extends to all the details of
phrase, structure and imagery: we must show that our hypothesis of the
authorship of XYZ explains all the circumstances. And even this is
not sufficient, as many erroneous guesses from internal evidence
may convince us. We must establish further that there is no other
reasonable way of accounting for the matter and manner of the book;
for example, that it is not the work of an imitator. An imitator may
reproduce all the superficial peculiarities of an author with such
fidelity that the imitation can hardly be distinguished from the
original: thus few can distinguish between Fenton's work and Pope's
in the translation of the Odyssey. We must take such known facts into
account in deciding a hypothesis of authorship. Such hypotheses can
seldom be decided on internal evidence alone: other circumstantial
evidence--other circumstances that ought to be discoverable if the
hypothesis is correct--must be searched for.
The operation of causes that are manifest only in their effects must
be proved by the same method as the operation of past causes that
have left only their effects behind them. Whether light is caused by
a projection of particles from a luminous body or by an agitation
communicated through an intervening medium cannot be directly
observed. The only proof open is to calculate what should occur on
either hypothesis, and observe whether this does occur. In such a
case there is room for the utmost calculating power and experimental
ingenuity. The mere making of the general hypothesis or guess is
simple enough, both modes of transmitting influence, the projection
of moving matter and the travelling of an undulation or wave movement,
being familiar facts. But it is not so easy to calculate exactly how
a given impulse would travel, and what phenomena of ray and shadow,
of reflection, refraction and diffraction ought to be visible in
its progress. Still, no matter how intricate the calculation, its
correspondence with what can be observed is the only legitimate proof
of the hypothesis.
II.--OBSTACLES TO EXPLANATION.--PLURALITY OF CAUSES AND INTERMIXTURE
OF EFFECTS.
There are two main ways in which explanation may be baffled. There may
exist more than one cause singly capable of producing the effect in
question, and we may have no means of determining which of the equally
sufficient causes has actually been at work. For all that appears
the tares in our wheat may be the effect of accident or of malicious
design: an anonymous book may be the work of an original author or
of an imitator. Again, an effect may be the joint result of several
co-operating causes, and it may be impossible to determine their
several potencies. The bitter article in the _Quarterly_ may have
helped to kill John Keats, but it co-operated with an enfeebled
constitution and a naturally over-sensitive temperament, and we cannot
assign its exact weight to each of these coefficients. Death may be
the result of a combination of causes; organic disease co-operating
with exposure, over-fatigue co-operating with the enfeeblement of the
system by disease.
The technical names for these difficulties, Plurality of Causes and
Intermixture of Effects, are apt to confuse without some clearing up.
In both kinds of difficulty more causes than one are involved: but
in the one kind of case there is a plurality of possible or equally
probable causes, and we are at a loss to decide which: in the other
kind of case there is a plurality of co-operating causes; the effect
is the result or product of several causes working conjointly, and we
are unable to assign to each its due share.
It is with a view to overcoming these difficulties that Science
endeavours to isolate agencies and ascertain what each is capable of
singly. Mill and Bain treat Plurality of Causes and Intermixture of
Effects in connexion with the Experimental Methods. It is better,
perhaps, to regard them simply as obstacles to explanation, and the
Experimental Methods as methods of overcoming those obstacles. The
whole purpose of the Experimental Methods is to isolate agencies and
effects: unless they can be isolated, the Methods are inapplicable.
In situations where the effects observable may be referred with equal
probability to more than one cause, you cannot eliminate so as to
obtain a single agreement. The Method of Agreement is frustrated. And
an investigator can get no light from mixed effects, unless he
knows enough of the causes at work to be able to apply the Method
of Residues. If he does not, he must simply look out for or devise
instances where the agencies are at work separately, and apply the
principle of Single Difference.
Great, however, as the difficulties are, the theory of Plurality and
Intermixture baldly stated makes them appear greater than they are in
practice. There is a consideration that mitigates the complication,
and renders the task of unravelling it not altogether hopeless. This
is that different causes have distinctive ways of operating, and leave
behind them marks of their presence by which their agency in a given
case may be recognised.
An explosion, for example, occurs. There are several explosive
agencies, capable of causing as much destruction as meets the eye at
the first glance. The agent in the case before us may be gunpowder or
it may be dynamite. But the two agents are not so alike in their mode
of operation as to produce results identical in every circumstance.
The expert inquirer knows by previous observation that when gunpowder
acts the objects in the neighbourhood are blackened; and that an
explosion of dynamite tears and shatters in a way peculiar to
itself. He is thus able to interpret the traces, to make and prove a
hypothesis.
A man's body is found dead in water. It may be a question whether
death came by drowning or by previous violence. He may have been
suffocated and afterwards thrown into the water. But the circumstances
will tell the true story. Death by drowning has distinctive symptoms.
If drowning was the cause, water will be found in the stomach and
froth in the trachea.
Thus, though there may be a plurality of possible causes, the
causation in the given case may be brought home to one by distinctive
accompaniments, and it is the business of the scientific inquirer to
study these. What is known as the "ripple-mark" in sandstone surfaces
may be produced in various ways. The most familiar way is by the
action of the tides on the sand of the sea-shore, and the interpreter
who knows this way only would ascribe the marks at once to this
agency. But ripple-marks are produced also by the winds on drifting
sands, by currents of water where no tidal influence is felt, and
in fact by any body of water in a state of oscillation. Is it, then,
impossible to decide between these alternative possibilities of
causation? No: wind-ripples and current-ripples and tidal-ripples have
each their own special character and accompanying conditions, and the
hypothesis of one rather than another may be made good by means of
these. "In rock-formations," Mr. Page says,[1] "there are many things
which at first sight seem similar, and yet on more minute examination,
differences are detected and conditions discovered which render
it impossible that these appearances can have arisen from the same
causation."
The truth is that generally when we speak of plurality of causes, of
alternative possibilities of causation, we are not thinking of
the effect in its individual entirety, but only of some general or
abstract aspect of it. When we say, _e.g._, that death may be produced
by a great many different causes, poison, gunshot wounds, disease of
this or that organ, we are thinking of death in the abstract, not of
the particular case under consideration, which as an individual case,
has characters so distinctive that only one combination of causes is
possible.
The effort of science is to become less and less abstract in this
sense, by observing agencies or combinations of agencies apart and
studying the special characters of their effects. That knowledge
is then applied, on the assumption that where those characters are
present, the agent or combination of agencies has been at work. Given
an effect to be explained, it is brought home to one out of several
possible alternatives by _circumstantial evidence_.
Bacon's phrase, _Instantia Crucis_,[2] or Finger-post Instance, might
be conveniently appropriated as a technical name for a circumstance
that is decisive between rival hypotheses. This was, in effect,
proposed by Sir John Herschel,[3] who drew attention to the importance
of these crucial instances, and gave the following example: "A curious
example is given by M. Fresnel, as decisive, in his mind, of the
question between the two great opinions on the nature of light, which,
since the time of Newton and Huyghens, have divided philosophers.
When two very clean glasses are laid one on the other, if they be
not perfectly flat, but one or both in an almost imperceptible degree
convex or prominent, beautiful and vivid colours will be seen between
them; and if these be viewed through a red glass, their appearance
will be that of alternate dark and bright stripes.... Now, the
coloured stripes thus produced are explicable on both theories, and
are appealed to by both as strong confirmatory facts; but there is a
difference in one circumstance according as one or the other theory is
employed to explain them. In the case of the Huyghenian doctrine,
the intervals between the bright stripes ought to appear _absolutely
black_; in the other, _half bright_, when viewed [in a particular
manner] through a prism. This curious case of difference was tried as
soon as the opposing consequences of the two theories were noted by M.
Fresnel, and the result is stated by him to be decisive in favour
of that theory which makes light to consist in the vibrations of an
elastic medium."
III.--THE PROOF OF A HYPOTHESIS.
The completest proof of a hypothesis is when that which has been
hypothetically assumed to exist as a means of accounting for certain
phenomena is afterwards actually observed to exist or is proved by
descriptive testimony to have existed. Our argument, for example, from
internal evidence that Mill in writing his Logic aimed at furnishing
a method for social investigations is confirmed by a letter to Miss
Caroline Fox, in which he distinctly avowed that object.
The most striking example of this crowning verification in Science
is the discovery of the planet Neptune, in which case an agent
hypothetically assumed was actually brought under the telescope as
calculated. Examples almost equally striking have occurred in the
history of the Evolution doctrine. Hypothetical ancestors with certain
peculiarities of structure have been assumed as links between living
species, and in some cases their fossils have actually been found in
the geological register.
Such triumphs of verification are necessarily rare. For the most part
the hypothetical method is applied to cases where proof by actual
observation is impossible, such as prehistoric conditions of the earth
or of life upon the earth, or conditions in the ultimate constitution
of matter that are beyond the reach of the strongest microscope.
Indeed, some would confine the word hypothesis to cases of this kind.
This, in fact, was done by Mill: hypothesis, as he defined it, was a
conjecture not completely proved, but with a large amount of evidence
in its favour. But seeing that the procedure of investigation is the
same, namely, conjecture, calculation and comparison of facts with the
calculated results, whether the agency assumed can be brought to the
test of direct observation or not, it seems better not to restrict the
word hypothesis to incompletely proved conjectures, but to apply it
simply to a conjecture made at a certain stage in whatever way it may
afterwards be verified.
In the absence of direct verification, the proof of a hypothesis is
exclusive sufficiency to explain the circumstances. The hypothesis
must account for all the circumstances, and there must be no other way
of accounting for them. Another requirement was mentioned by Newton
in a phrase about the exact meaning of which there has been some
contention. The first of his Regulæ Philosophandi laid down that the
cause assumed must be a _vera causa_. "We are not," the Rule runs, "to
admit other causes of natural things than such as both are true, and
suffice for explaining their phenomena."[4]
It has been argued that the requirement of "verity" is superfluous;
that it is really included in the requirement of sufficiency; that if
a cause is sufficient to explain the phenomena it must _ipso facto_ be
the true cause. This may be technically arguable, given a sufficient
latitude to the word sufficiency: nevertheless, it is convenient
to distinguish between mere sufficiency to explain the phenomena
in question, and the proof otherwise that the cause assigned really
exists _in rerum natura_, or that it operated in the given case. The
frequency with which the expression _vera causa_ has been used since
Newton's time shows that a need is felt for it, though it may be hard
to define "verity" precisely as something apart from "sufficiency". If
we examine the common usage of the expression we shall probably find
that what is meant by insisting on a _vera causa_ is that we must
have some evidence for the cause assigned outside the phenomena in
question. In seeking for verification of a hypothesis we must extend
our range beyond the limited facts that have engaged our curiosity and
that demand explanation.
There can be little doubt that Newton himself aimed his rule at the
Cartesian hypothesis of Vortices. This was an attempt to explain the
solar system on the hypothesis that cosmic space is filled with a
fluid in which the planets are carried round as chips of wood in a
whirlpool, or leaves or dust in a whirlwind. Now this is so far a
_vera causa_ that the action of fluid vortices is a familiar one: we
have only to stir a cup of tea with a bit of stalk in it to get an
instance. The agency supposed is sufficient also to account for the
revolution of a planet round the sun, given sufficient strength in the
fluid to buoy up the planet. But if there were such a fluid in space
there would be other phenomena: and in the absence of these other
phenomena the hypothesis must be dismissed as imaginary. The fact that
comets pass into and out of spaces where the vortices must be assumed
to be in action without exhibiting any perturbation is an _instantia
crucis_ against the hypothesis.
If by the requirement of a _vera causa_ were meant that the cause
assigned must be one directly open to observation, this would
undoubtedly be too narrow a limit. It would exclude such causes as the
ether which is assumed to fill interstellar space as a medium for
the propagation of light. The only evidence for such a medium and
its various properties is sufficiency to explain the phenomena. Like
suppositions as to the ultimate constitution of bodies, it is of the
nature of what Professor Bain calls a "Representative Fiction": the
only condition is that it must explain all the phenomena, and that
there must be no other way of explaining all. When it is proved
that light travels with a finite velocity, we are confined to two
alternative ways of conceiving its transmission, a projection of
matter from the luminous body and the transference of vibrations
through an intervening medium. Either hypothesis would explain many of
the facts: our choice must rest with that which best explains all.
But supposing that all the phenomena of light were explained by
attributing certain properties to this intervening medium, it would
probably be held that the hypothesis of an ether had not been fully
verified till other phenomena than those of light had been shown to
be incapable of explanation on any other hypothesis. If the properties
ascribed to it to explain the phenomena of light sufficed at the same
time to explain otherwise inexplicable phenomena connected with Heat,
Electricity, or Gravity, the evidence of its reality would be greatly
strengthened.
Not only must the circumstances in hand be explained, but other
circumstances must be found to be such as we should expect if the
cause assigned really operated. Take, for example, the case of Erratic
blocks or boulders, huge fragments of rock found at a distance from
their parent strata. The lowlands of England, Scotland, and Ireland,
and the great central plain of Northern Europe contain many such
fragments. Their composition shows indubitably that they once formed
part of hills to the northward of their present site. They must
somehow have been detached and transported to where we now find them.
How? One old explanation is that they were carried by witches, or
that they were themselves witches accidentally dropped and turned
into stone. Any such explanation by supernatural means can neither
be proved nor disproved. Some logicians would exclude such hypotheses
altogether on the ground that they cannot be rendered either more
or less probable by subsequent examination.[5] The proper scientific
limit, however, is not to the making of hypotheses, but to the proof
of them. The more hypotheses the merrier: only if such an agency as
witchcraft is suggested, we should expect to find other evidence
of its existence in other phenomena that could not otherwise be
explained. Again, it has been suggested that the erratic boulders may
have been transported by water. Water is so far a _vera causa_ that
currents are known to be capable of washing huge blocks to a great
distance. But blocks transported in this way have the edges worn off
by the friction of their passage: and, besides, currents strong enough
to dislodge and force along for miles blocks as big as cottages must
have left other marks of their presence. The explanation now received
is that glaciers and icebergs were the means of transport. But this
explanation was not accepted till multitudes of circumstances were
examined all tending to show that glaciers had once been present in
the regions where the erratic blocks are found. The minute habits of
glaciers have been studied where they still exist: how they slowly
move down carrying fragments of rock; how icebergs break off when they
reach water, float off with their load, and drop it when they melt;
how they grind and smooth the surfaces of rocks over which they pass
or that are frozen into them: how they undercut and mark the faces
of precipices past which they move; how moraines are formed at the
melting ends of them, and so forth. When a district exhibits all the
circumstances that are now observed to attend the action of glaciers
the proof of the hypothesis that glaciers were once there is complete.
[Footnote 1: Page's _Philosophy of Geology_, p. 38.]
[Footnote 2: Crux in this phrase means a cross erected at the
parting of ways, with arms to tell whither each way leads.]
[Footnote 3: _Discourse_, § 218.]
[Footnote 4: Causas rerum naturalium non plures admitti
debere quam quæ et veriæ sint et carum phenomenis explicandis
sufficiant.]
[Footnote 5: See Prof. Fowler on the Conditions of Hypotheses,
_Inductive Logic_, pp. 100-115.]
CHAPTER VIII.
SUPPLEMENTARY METHODS OF INVESTIGATION.
I.--THE MAINTENANCE OF AVERAGES.--SUPPLEMENT TO THE METHOD OF
DIFFERENCE.
A certain amount of law obtains among events that are usually spoken
of as matters of chance or accident in the individual case. Every kind
of accident recurs with a certain uniformity. If we take a succession
of periods, and divide the total number of any kind of event by the
number of periods, we get what is called the average for that period:
and it is observed that such averages are maintained from period to
period. Over a series of years there is a fixed proportion between
good harvests and bad, between wet days and dry: every year nearly
the same number of suicides takes place, the same number of crimes, of
accidents to life and limb, even of suicides, crimes, or injuries
by particular means: every year in a town nearly the same number of
children stray from their parents and are restored by the police:
every year nearly the same number of persons post letters without
putting an address on them.
This maintenance of averages is simple matter of observation, a datum
of experience, an empirical law. Once an average for any kind of event
has been noted, we may count upon its continuance as we count upon
the continuance of any other kind of observed uniformity. Insurance
companies proceed upon such empirical laws of average in length of
life and immunity from injurious accidents by sea or land: their
prosperity is a practical proof of the correctness and completeness
of the observed facts and the soundness of their inference to the
continuance of the average.
The constancy of averages is thus a guide in practice. But in
reasoning upon them in investigations of cause, we make a further
assumption than continued uniformity. We assume that the maintenance
of the average is due to the permanence of the producing causes. We
regard the average as the result of the operation of a limited sum
of forces and conditions, incalculable as regards their particular
incidence, but always pressing into action, and thus likely to operate
a certain number of times within a limited period.
Assuming the correctness of this explanation, it would follow that
_any change in the average is due to some change in the producing
conditions_; and this derivative law is applied as a help in the
observation and explanation of social facts. Statistics are collected
and classified: averages are struck: and changes in the average are
referred to changes in the concomitant conditions.
With the help of this law, we may make a near approach to the
precision of the Method of Difference. A multitude of unknown or
unmeasured agents may be at work on a situation, but we may accept the
average as the result of their joint operation. If then a new agency
is introduced or one of the known agents is changed in degree, and
this is at once followed by a change in the average, we may with
fair probability refer the change in the result to the change in the
antecedents.
The difficulty is to find a situation where only one antecedent has
been changed before the appearance of the effect. This difficulty may
be diminished in practice by eliminating changes that we have reason
to know could not have affected the circumstances in question.
Suppose, for example, our question is whether the Education Act
of 1872 had an influence in the decrease of juvenile crime. Such a
decrease took place _post hoc_; was it _propter hoc_? We may at once
eliminate or put out of account the abolition of Purchase in the Army
or the extension of the Franchise as not having possibly exercised any
influence on juvenile crime. But with all such eliminations, there may
still remain other possible influences, such as an improvement in
the organisation of the Police, or an expansion or contraction in
employment. "Can you tell me in the face of chronology," a leading
statesman once asked, "that the Crimes Act of 1887 did not diminish
disorder in Ireland?" But chronological sequence alone is not a proof
of causation as long as there are other contemporaneous changes of
condition that may also have been influential.
The great source of fallacy is our proneness to eliminate or isolate
in accordance with our prejudices. This has led to the gibe that
anything can be proved by statistics. Undoubtedly statistics may be
made to prove anything if you have a sufficiently low standard of
proof and ignore the facts that make against your conclusion. But
averages and variations in them are instructive enough if handled with
due caution. The remedy for rash conclusions from statistics is not no
statistics, but more of them and a sound knowledge of the conditions
of reasonable proof.
II.--THE PRESUMPTION FROM EXTRA-CASUAL COINCIDENCE.
We have seen that repeated coincidence raises a presumption of causal
connexion between the coinciding events. If we find two events going
repeatedly together, either abreast or in sequence, we infer that the
two are somehow connected in the way of causation, that there is a
reason for the coincidence in the manner of their production. It may
not be that the one produces the other, or even that their causes are
in any way connected: but at least, if they are independent one of the
other, both are tied down to happen at the same place and time,--the
coincidence of both with time and place is somehow fixed.
But though this is true in the main, it is not true without
qualification. We expect a certain amount of repeated coincidence
without supposing causal connexion. If certain events are repeated
very often within our experience, if they have great positive
frequency, we may observe them happening together more than once
without concluding that the coincidence is more than fortuitous.
For example, if we live in a neighbourhood possessed of many black
cats, and sally forth to our daily business in the morning, a
misfortune in the course of the day might more than once follow upon
our meeting a black cat as we went out without raising in our minds
any presumption that the one event was the result of the other.
Certain planets are above the horizon at certain periods of the year
and below the horizon at certain other periods. All through the year
men and women are born who afterwards achieve distinction in various
walks of life, in love, in war, in business, at the bar, in the
pulpit. We perceive a certain number of coincidences between
the ascendancy of certain planets and the birth of distinguished
individuals without suspecting that planetary influence was concerned
in their superiority.
Marriages take place on all days of the year: the sun shines on a good
many days at the ordinary time for such ceremonies; some marriages are
happy, some unhappy; but though in the case of many happy marriages
the sun has shone upon the bride, we regard the coincidence as merely
accidental.
Men often dream of calamities and often suffer calamities in real
life: we should expect the coincidence of a dream of calamity followed
by a reality to occur more than once as a result of chance. There are
thousands of men of different nationalities in business in London,
and many fortunes are made: we should expect more than one man of any
nationality represented there to make a fortune without arguing any
connexion between his nationality and his success.
We allow, then, for a certain amount of repeated coincidence without
presuming causal connexion: can any rule be laid down for determining
the exact amount?
Prof. Bain has formulated the following rule: "Consider the positive
frequency of the phenomena themselves, how great frequency of
coincidence must follow from that, supposing there is neither
connexion nor repugnance. If there be greater frequency, there is
connexion; if less, repugnance."
I do not know that we can go further definite in precept. The number
of casual coincidences bears a certain proportion to the positive
frequency of the coinciding phenomena: that proportion is to be
determined by common-sense in each case. It may be possible, however,
to bring out more clearly the principle on which common-sense proceeds
in deciding what chance will and will not account for, although our
exposition amounts only to making more clear what it is that we mean
by chance as distinguished from assignable reason. I would suggest
that in deciding what chance will not account for, we make regressive
application of a principle which may be called the principle of Equal
and Unequal Alternatives, and which may be worded as follows:--
Of a given number of possible alternatives, all equally
possible, one of which is bound to occur at a given time, we
expect each to have its turn an equal number of times in the
long run. If several of the alternatives are of the same
kind, we expect an alternative of that kind to recur with a
frequency proportioned to their greater number. If any of the
alternatives has an advantage, it will recur with a frequency
proportioned to the strength of that advantage.
Situations in which alternatives are absolutely equal are rare in
nature, but they are artificially created for games "of chance," as in
tossing a coin, throwing dice, drawing lots, shuffling and dealing a
pack of cards. The essence of all games of chance is to construct a
number of equal alternatives, making them as nearly equal as possible,
and to make no prearrangement which of the number shall come off. We
then say that this is determined by chance. If we ask why we believe
that when we go on bringing off one alternative at a time, each will
have its turn, part of the answer undoubtedly is that given by De
Morgan, namely, that we know no reason why one should be chosen rather
than another. This, however, is probably not the whole reason for our
belief. The rational belief in the matter is that it is only in the
long run or on the average that each of the equal alternatives will
have its turn, and this is probably founded on the experience of
actual trial. The mere equality of the alternatives, supposing them
to be perfectly equal, would justify us as much in expecting that
each would have its turn in a single revolution of the series, in one
complete cycle of the alternatives. This, indeed, may be described
as the natural and primitive expectation which is corrected by
experience. Put six balls in a wicker bottle, shake them up, and roll
one out: return this one, and repeat the operation: at the end of six
draws we might expect each ball to have had its turn of being drawn
if we went merely on the abstract equality of the alternatives. But
experience shows us that in six successive draws the same ball may
come out twice or even three or four times, although when thousands of
drawings are made each comes out nearly an equal number of times.
So in tossing a coin, heads may turn up ten or twelve times in
succession, though in thousands of tosses heads and tails are nearly
equal. Runs of luck are thus within the rational doctrine of chances:
it is only in the long run that luck is equalised supposing that the
events are pure matter of chance, that is, supposing the fundamental
alternatives to be equal.
If three out of six balls are of the same colour, we expect a ball of
that colour to come out three times as often as any other colour on
the average of a long succession of tries. This illustrates the second
clause of our principle. The third is illustrated by a loaded coin or
die.
By making regressive application of the principle thus ascertained by
experience, we often obtain a clue to special causal connexion. We are
at least enabled to isolate a problem for investigation. If we find
one of a number of alternatives recurring more frequently than the
others, we are entitled to presume that they are not equally possible,
that there is some inequality in their conditions.
The inequality may simply lie in the greater possible frequency of
one of the coinciding events, as when there are three black balls in
a bottle of six. We must therefore discount the positive frequency
before looking for any other cause. Suppose, for example, we find that
the ascendancy of Jupiter coincides more frequently with the birth of
men afterwards distinguished in business than with the birth of men
otherwise distinguished, say in war, or at the bar, or in scholarship.
We are not at liberty to conclude planetary influence till we have
compared the positive frequency of the different modes of distinction.
The explanation of the more frequently repeated coincidence may simply
be that more men altogether are successful in business than in war
or law or scholarship. If so, we say that chance accounts for the
coincidence, that is to say, that the coincidence is casual as far as
planetary influence is concerned.
So in epidemics of fever, if we find on taking a long average that
more cases occur in some streets of a town than in others, we are not
warranted in concluding that the cause lies in the sanitary conditions
of those streets or in any special liability to infection without
first taking into account the number of families in the different
streets. If one street showed on the average ten times as many cases
as another, the coincidence might still be judged casual if there were
ten times as many families in it.
Apart from the fallacy of overlooking the positive frequency, certain
other fallacies or liabilities to error in applying this doctrine of
chances may be specified.
1. We are apt, under the influence of prepossession or prejudice, to
remember certain coincidences better than others, and so to imagine
extra-casual coincidence where none exists. This bias works in
confirming all kinds of established beliefs, superstitious and other,
beliefs in dreams, omens, retributions, telepathic communications, and
so forth. Many people believe that nobody who thwarts them ever comes
to good, and can produce numerous instances from experience in support
of this belief.
2. We are apt, after proving that there is a residuum beyond what
chance will account for on due allowance made for positive frequency,
to take for granted that we have proved some particular cause for
this residuum. Now we have not really explained the residuum by the
application of the principle of chances: we have only isolated a
problem for explanation. There may be more than chance will account
for: yet the cause may not be the cause that we assign off-hand. Take,
for example, the coincidence that has been remarked between race and
different forms of Christianity in Europe. If the distribution of
religious systems were entirely independent of race, it might be
said that you would expect one system to coincide equally often
with different races in proportion to the positive number of their
communities. But the Greek system is found almost solely among
Slavonic peoples, the Roman among Celtic, and the Protestant among
Teutonic. The coincidence is greater than chance will account for. Is
the explanation then to be found in some special adaptability of the
religious system to the character of the people? This may be the right
explanation, but we have not proved it by merely discounting chance.
To prove this we must show that there was no other cause at work, that
character was the only operative condition in the choice of system,
that political combinations, for example, had nothing to do with it.
The presumption from extra-casual coincidence is only that there is
a special cause: in determining what that is we must conform to the
ordinary conditions of explanation.
So coincidence between membership of the Government and a classical
education may be greater than chance would account for, and yet the
circumstance of having been taught Latin and Greek at school may have
had no special influence in qualifying the members for their duties.
The proportion of classically educated in the Government may be
greater than the proportion of them in the House of Commons, and
yet their eminence may be in no way due to their education. Men of
a certain social position have an advantage in the competition for
office, and all those men have been taught Latin and Greek as a matter
of course. Technically speaking, the coinciding phenomena may be
independent effects of the same cause.
3. Where the alternative possibilities are very numerous, we are apt
not to make due allowance for the number, sometimes overrating it,
sometimes underrating it.
The fallacy of underrating the number is often seen in games of
chance, where the object is to create a vast number of alternatives,
all equally possible, equally open to the player, without his being
able to affect the advent of one more than another. In whist, for
example, there are some six billions of possible hands. Yet it is a
common impression that, one night with another, in the course of a
year, a player will have dealt to him about an equal number of good
and bad hands. This is a fallacy. A very much longer time is required
to exhaust the possible combinations. Suppose a player to have
2000 hands in the course of a year: this is only one "set," one
combination, out of thousands of millions of such sets possible. Among
those millions of sets, if there is nothing but chance in the matter,
there ought to be all proportions of good and bad, some sets all good,
some all bad, as well as some equally divided between good and bad.[1]
Sometimes, however, the number of possible alternatives is overrated.
Thus, visitors to London often remark that they never go there without
meeting somebody from their own locality, and they are surprised at
this as if they had the same chance of meeting their fellow-visitors
and any other of the four millions of the metropolis. But really the
possible alternatives of rencounter are far less numerous. The places
frequented by visitors to London are filled by much more limited
numbers: the possible rencounters are to be counted by thousands
rather than by millions.
[Footnote 1: See De Morgan's _Essay on Probabilities_, c. vi.,
"On Common Notions of Probability".]
CHAPTER IX.
PROBABLE INFERENCE TO PARTICULARS--THE MEASUREMENT OF PROBABILITY.
Undoubtedly there are degrees of probability. Not only do we expect
some events with more confidence than others: we may do so, and our
confidence may be misplaced: but we have reason to expect some with
more confidence than others. There are different degrees of rational
expectation. Can those degrees be measured numerically?
The question has come into Logic from the mathematicians. The
calculation of Probabilities is a branch of Mathematics. We have seen
how it may be applied to guide investigation by eliminating what is
due to chance, and it has been vaguely conceived by logicians that
what is called the calculus of probabilities might be found useful
also in determining by exact numerical measurement the probability
of single events. Dr. Venn, who has written a separate treatise on the
Logic of Chance, mentions "accurate quantitative apportionment of our
belief" as one of the goals which Logic should strive to attain. The
following passage will show his drift.[1]
A man in good health would doubtless like to know whether he
will be alive this time next year. The fact will be settled
one way or the other in due time, if he can afford to wait,
but if he wants a present decision, Statistics and the Theory
of Probability can alone give him any information. He learns
that the odds are, say five to one that he will survive, and
this is an answer to his question as far as any answer can be
given. Statisticians are gradually accumulating a vast mass of
data of this general character. What they may be said to aim
at is to place us in the position of being able to say, in any
given time or place, what are the odds for or against any at
present indeterminable fact which belongs to a class admitting
of statistical treatment.
Again, outside the regions of statistics proper--which
deal, broadly speaking, with events which can be numbered or
measured, and which occur with some frequency--there is still
a large field as to which some better approach to a reasoned
intensity of belief can be acquired. What will be the issue of
a coming war? Which party will win in the next election? Will
a patient in the crisis of a given disease recover or not?
That statistics are lying here in the background, and are thus
indirectly efficient in producing and graduating our belief, I
fully hold; but there is such a large intermediate process
of estimating, and such scope for the exercise of a practised
judgment, that no direct appeal to statistics in the common
sense can directly help us. In sketching out therefore the
claims of an Ideal condition of knowledge, we ought clearly to
include a due apportionment of belief to every event of such
a class as this. It is an obvious defect that one man should
regard as almost certain what another man regards as almost
impossible. Short, therefore, of certain prevision of the
future, we want complete agreement as to the degree of
probability of every future event: and for that matter of
every past event as well.
Technically speaking, if we extend the name Modality (see p. 78) to
any qualification of the certainty of a statement of belief, what Dr.
Venn here desiderates, as he has himself suggested, is a more exact
measurement of the Modality of propositions. We speak of things as
being certain, possible, impossible, probable, extremely probable,
faintly probable, and so forth: taking certainty as the highest
degree of probability[2] shading gradually down to the zero of
the impossible, can we obtain an exact numerical measure for the
gradations of assurance?
To examine the principles of all the cases in which chances for and
against an occurrence have been calculated from real or hypothetical
data, would be to trespass into the province of Mathematics, but a few
simple cases will serve to show what it is that the calculus attempts
to measure, and what is the practical value of the measurement as
applied to the probability of a single event.
Suppose there are 100 balls in a box, 30 white and 70 black, all being
alike except in respect of colour, we say that the chances of drawing
a black ball as against a white are as 7 to 3, and the probability of
drawing black is measured by the fraction 7/10. In believing this we
proceed on the principle already explained (p. 356) of Proportional
Chances. We do not know for certain whether black or white will
emerge, but knowing the antecedent situation we expect black rather
than white with a degree of assurance corresponding to the proportions
of the two in the box. It is our degree of rational assurance that
we measure by this fraction, and the rationality of it depends on the
objective condition of the facts, and is the same for all men, however
much their actual degree of confidence may vary with individual
temperament. That black will be drawn seven times out of every ten
on an average if we go on drawing to infinity, is as certain as any
empirical law: it is the probability of a single draw that we measure
by the fraction 7/10.
When we build expectations of single events on statistics of observed
proportions of events of that kind, it is ultimately on the same
principle that rational expectation rests. That the proportion will
obtain on the average we regard as certain: the ratio of favourable
cases to the whole number of possible alternatives is the measure
of rational expectation or probability in regard to a particular
occurrence. If every year five per cent. of the children of a town
stray from their guardians, the probability of this or that child's
going astray is 1/20. The ratio is a correct measure only on the
assumption that the average is maintained from year to year.
Without going into the combination of probabilities, we are now in a
position to see the practical value of such a calculus as applied to
particular cases. There has been some misunderstanding among logicians
on the point. Mr. Jevons rebuked Mill for speaking disrespectfully
of the calculus, eulogised it as one of the noblest creations of the
human intellect, and quoted Butler's saying that "Probability is the
guide of life". But when Butler uttered this famous saying he was
probably not thinking of the mathematical calculus of probabilities
as applied to particular cases, and it was this special application to
which Mill attached comparatively little value.
The truth is that we seldom calculate or have any occasion to
calculate individual chances except as a matter of curiosity. It is
true that insurance offices calculate probabilities, but it is not the
probability of this or that man dying at a particular age. The precise
shade of probability for the individual, in so far as this depends on
vital statistics, is a matter of indifference to the company as long
as the average is maintained. Our expectations about any individual
life cannot be measured by a calculation of the chances because a
variety of other elements affect those expectations. We form beliefs
about individual cases, but we try to get surer grounds for them than
the chances as calculable from statistical data. Suppose a person were
to institute a home for lost dogs, he would doubtless try to ascertain
how many dogs were likely to go astray, and in so doing would be
guided by statistics. But in judging of the probability of the
straying of a particular dog, he would pay little heed to statistics
as determining the chances, but would proceed upon empirical knowledge
of the character of the dog and his master. Even in betting on the
field against a particular horse, the bookmaker does not calculate
from numerical data such as the number of horses entered or the
number of times the favourite has been beaten: he tries to get at
the pedigree and previous performances of the various horses in the
running. We proceed by calculation of chances only when we cannot do
better.
[Footnote 1: _Empirical Logic_, p. 556.]
[Footnote 2: Mr. Jevons held that all inference is merely
probable and that no inference is certain. But this is a
purposeless repudiation of common meaning, which he cannot
himself consistently adhere to. We find him saying that if a
penny is tossed into the air it will _certainly_ come down
on one side or the other, on which side being a matter of
probability. In common speech probability is applied to a
degree of belief short of certainty, but to say that certainty
is the highest degree of probability does no violence to the
common meaning.]
CHAPTER X.
INFERENCE FROM ANALOGY.
The word Analogy was appropriated by Mill, in accordance with the
usage of the eighteenth century, to designate a ground of inference
distinct from that on which we proceed in extending a law, empirical
or scientific, to a new case. But it is used in various other senses,
more or less similar, and in order to make clear the exact logical
sense, it is well to specify some of these. The original word
[Greek: analogia], as employed by Aristotle, corresponds to the word
Proportion in Arithmetic: it signified an equality of ratios, [Greek:
isotês logôn]: two compared with four is analogous to four compared
with eight. There is something of the same meaning in the technical
use of the word in Physiology, where it is used to signify similarity
of function as distinguished from similarity of structure, which is
called homology: thus the tail of a whale is analogous to the tail
of a fish, inasmuch as it is similarly used for motion, but it
is homologous with the hind legs of a quadruped; a man's arms are
homologous with a horse's fore legs, but they are not analogous
inasmuch as they are not used for progression. Apart from these
technical employments, the word is loosely used in common speech for
any kind of resemblance. Thus De Quincey speaks of the "analogical"
power in memory, meaning thereby the power of recalling things by
their inherent likeness as distinguished from their casual connexions
or their order in a series. But even in common speech, there is a
trace of the original meaning: generally when we speak of analogy we
have in our minds more than one pair of things, and what we call
the analogy is some resemblance between the different pairs. This
is probably what Whately had in view when he defined analogy as
"resemblance of relations".
In a strict logical sense, however, as defined by Mill, sanctioned
by the previous usage of Butler and Kant, analogy means more than
a resemblance of relations. It means a preponderating resemblance
between two things such as to warrant us in inferring that the
resemblance extends further. This is a species of argument distinct
from the extension of an empirical law. In the extension of an
empirical law, the ground of inference is a coincidence frequently
repeated within our experience, and the inference is that it has
occurred or will occur beyond that experience: in the argument from
analogy, the ground of inference is the resemblance between two
individual objects or kinds of objects in a certain number of points,
and the inference is that they resemble one another in some other
point, known to belong to the one, but not known to belong to the
other. "Two things go together in many cases, therefore in all,
including this one," is the argument in extending a generalisation:
"Two things agree in many respects, therefore in this other," is the
argument from analogy.
The example given by Reid in his _Intellectual Powers_ has become the
standard illustration of the peculiar argument from analogy.
We may observe a very great similitude between this earth
which we inhabit, and the other planets, Saturn, Jupiter,
Mars, Venus and Mercury. They all revolve round the sun,
as the earth does, although at different distances and in
different periods. They borrow all their light from the sun,
as the earth does. Several of them are known to revolve
round their axis like the earth, and by that means have like
succession of day and night. Some of them have moons, that
serve to give them light in the absence of the sun, as our
moon does to us. They are all, in their motions, subject to
the same law of gravitation as the earth is. From all this
similitude it is not unreasonable to think that these planets
may, like our earth, be the habitation of various orders of
living creatures. There is some probability in this conclusion
from analogy.[1]
The argument from analogy is sometimes said to range through all
degrees of probability from certainty to zero. But this is true only
if we take the word analogy in its loosest sense for any kind of
resemblance. If we do this, we may call any kind of argument an
argument from analogy, for all inferences turn upon resemblance. I
believe that if I throw my pen in the air it will come down again,
because it is like other ponderable bodies. But if we use the word in
its limited logical sense, the degree of probability is much nearer
zero than certainty. This is apparent from the conditions that
logicians have formulated of a strict argument from analogy.
1. The resemblance must be preponderating. In estimating the value of
an argument from analogy, we must reckon the points of difference
as counting against the conclusion, and also the points in regard
to which we do not know whether the two objects agree or differ. The
numerical measure of value is the ratio of the points of resemblance
to the points of difference _plus_ the unknown points. Thus, in the
argument that the planets are inhabited because they resemble the
earth in some respects and the earth is inhabited, the force of the
analogy is weakened by the fact that we know very little about the
surface of the planets.
2. In a numerical estimate all circumstances that hang together as
effects of one cause must be reckoned as one. Otherwise, we might make
a fallaciously imposing array of points of resemblance. Thus in Reid's
enumeration of the agreements between the earth and the planets, their
revolution round the sun and their obedience to the law of gravitation
should count as one point of resemblance. If two objects agree in _a_,
_b_, _c_, _d_, _e_, but _b_ follows from _a_, and _d_ and _e_ from
_c_, the five points count only as two.
3. If the object to which we infer is known to possess some property
incompatible with the property inferred, the general resemblance
counts for nothing. The moon has no atmosphere, and we know that air
is an indispensable condition of life. Hence, however much the moon
may resemble the earth, we are debarred from concluding that there are
living creatures on the moon such as we know to exist on the earth. We
know also that life such as it is on the earth is possible only within
certain limits of temperature, and that Mercury is too hot for life,
and Saturn too cold, no matter how great the resemblance to the earth
in other respects.
4. If the property inferred is known or presumed to be a concomitant
of one or more of the points of resemblance, any argument from analogy
is superfluous. This is, in effect, to say that we have no occasion to
argue from general resemblance when we have reason to believe that a
property follows from something that an object is known to possess.
If we knew that any one of the planets possessed all the conditions,
positive and negative, of life, we should not require to reckon up
all the respects in which it resembles the earth in order to create
a presumption that it is inhabited. We should be able to draw the
conclusion on other grounds than those of analogy. Newton's famous
inference that the diamond is combustible is sometimes quoted as an
argument from analogy. But, technically speaking, it was rather,
as Professor Bain has pointed out, of the nature of an extended
generalisation. Comparing bodies in respect of their densities and
refracting powers, he observed that combustible bodies refract more
than others of the same density; and observing the exceptionally high
refracting power of the diamond, he inferred from this that it was
combustible, an inference afterwards confirmed by experiment. "The
concurrence of high refracting power with inflammability was an
empirical law; and Newton, perceiving the law, extended it to the
adjacent case of the diamond. The remark is made by Brewster that had
Newton known the refractive powers of the minerals _greenockite_ and
_octohedrite_, he would have extended the inference to them, and would
have been mistaken."[2]
From these conditions it will be seen that we cannot conclude with any
high degree of probability from analogy alone. This is not to deny, as
Mr. Jevons seems to suppose, that analogies, in the sense of general
resemblances, are often useful in directing investigation. When
we find two things very much alike, and ascertain that one of them
possesses a certain property, the presumption that the other has
the same is strong enough to make it worth while trying whether as a
matter of fact it has. It is said that a general resemblance of the
hills near Ballarat in Australia to the Californian hills where gold
had been found suggested the idea of digging for gold at Ballarat.
This was a lucky issue to an argument from analogy, but doubtless many
have dug for gold on similar general resemblances without finding that
the resemblance extended to that particular. Similarly, many of
the extensions of the Pharmacopeia have proceeded upon general
resemblances, the fact that one drug resembles another in certain
properties being a sufficient reason for trying whether the
resemblance goes further. The lucky guesses of what is known as
natural sagacity are often analogical. A man of wide experience in any
subject-matter such as the weather, or the conduct of men in war, in
business, or in politics, may conclude to the case in hand from some
previous case that bears a general resemblance to it, and very often
his conclusions may be perfectly sound though he has not made a
numerical estimate of the data.
The chief source of fallacy in analogical argument is ignoring the
number of points of difference. It often happens that an amount of
resemblance only sufficient for a rhetorical simile is made to do duty
as a solid argument. Thus the resemblance between a living body
and the body politic is sometimes used to support inferences from
successful therapeutic treatment to State policy. The advocates of
annual Parliaments in the time of the Commonwealth based their case on
the serpent's habit of annually casting its skin.
Wisest of beasts the serpent see,
Just emblem of eternity,
And of a State's duration;
Each year an annual skin he takes,
And with fresh life and vigour wakes
At every renovation.
Britain! that serpent imitate.
Thy Commons House, that skin of State,
By annual choice restore;
So choosing thou shall live secure,
And freedom to thy sons inure,
Till Time shall be no more.
Carlyle's saying that a ship could never be taken round Cape Horn if
the crew were consulted every time the captain proposed to alter
the course, if taken seriously as an analogical argument against
Representative Government, is open to the objection that the
differences between a ship and a State are too great for any argument
from one to the other to be of value. It was such fallacious analogies
as these that Heine had in view in his humorous prayer, "Heaven defend
us from the Evil One and from metaphors".
[Footnote 1: Hamilton's _Reid_, p. 236.]
[Footnote 2: Bain's _Logic_, ii. 145.]
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Transcriber's Note:
page 113: 'aneo symplokês' corrected to 'aneu symplokês'
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