CHAPTER XXV.

OF THE GROUNDS OF DISBELIEF.

§ 1. The method of arriving at general truths, or general propositions

fit to be believed, and the nature of the evidence on which they are

grounded, have been discussed, as far as space and the writer's

faculties permitted, in the twenty-four preceding chapters. But the

result of the examination of evidence is not always belief, nor even

suspension of judgment; it is sometimes disbelief. The philosophy,

therefore, of induction and experimental inquiry is incomplete, unless

the grounds not only of belief, but of disbelief, are treated of; and to

this topic we shall devote one, and the final, chapter.

By disbelief is not here to be understood the mere absence of belief.

The ground for abstaining from belief is simply the absence or

insufficiency of proof; and in considering what is sufficient evidence

to support any given conclusion, we have already, by implication,

considered what evidence is not sufficient for the same purpose. By

disbelief is here meant, not the state of mind in which we form no

opinion concerning a subject, but that in which we are fully persuaded

that some opinion is not true; insomuch that if evidence, even of great

apparent strength, (whether grounded on the testimony of others or on

our own supposed perceptions,) were produced in favour of the opinion,

we should believe that the witnesses spoke falsely, or that they, or we

ourselves if we were the direct percipients, were mistaken.

That there are such cases, no one is likely to dispute. Assertions for

which there is abundant positive evidence are often disbelieved, on

account of what is called their improbability, or impossibility. And the

question for consideration is what, in the present case, these words

mean, and how far and in what circumstances the properties which they

express are sufficient grounds for disbelief.

§ 2. It is to be remarked in the first place, that the positive evidence

produced in support of an assertion which is nevertheless rejected on

the score of impossibility or improbability, is never such as amounts to

full proof. It is always grounded on some approximate generalization.

The fact may have been asserted by a hundred witnesses; but there are

many exceptions to the universality of the generalization that what a

hundred witnesses affirm is true. We may seem to ourselves to have

actually seen the fact: but, that we really see what we think we see, is

by no means an universal truth; our organs may have been in a morbid

state; or we may have inferred something, and imagined that we perceived

it. The evidence, then, in the affirmative being never more than an

approximate generalization, all will depend on what the evidence in the

negative is. If that also rests on an approximate generalization, it is

a case for comparison of probabilities. If the approximate

generalizations leading to the affirmative are, when added together,

less strong, or in other words, farther from being universal, than the

approximate generalizations which support the negative side of the

question, the proposition is said to be improbable, and is to be

disbelieved provisionally. If however an alleged fact be in

contradiction, not to any number of approximate generalizations, but to

a completed generalization grounded on a rigorous induction, it is said

to be impossible, and is to be disbelieved totally.

This last principle, simple and evident as it appears, is the doctrine

which, on the occasion of an attempt to apply it to the question of the

credibility of miracles, excited so violent a controversy. Hume's

celebrated doctrine, that nothing is credible which is contradictory to

experience, or at variance with laws of nature, is merely this very

plain and harmless proposition, that whatever is contradictory to a

complete induction is incredible. That such a maxim as this should

either be accounted a dangerous heresy, or mistaken for a great and

recondite truth, speaks ill for the state of philosophical speculation

on such subjects.

But does not (it may be asked) the very statement of the proposition

imply a contradiction? An alleged fact, according to this theory, is not

to be believed if it contradict a complete induction. But it is

essential to the completeness of an induction that it shall not

contradict any known fact. Is it not then a _petitio principii_ to say,

that the fact ought to be disbelieved because the induction opposed to

it is complete? How can we have a right to declare the induction

complete, while facts, supported by credible evidence, present

themselves in opposition to it?

I answer, we have that right whenever the scientific canons of induction

give it to us; that is, whenever the induction _can_ be complete. We

have it, for example, in a case of causation in which there has been an

_experimentum crucis_. If an antecedent A, superadded to a set of

antecedents in all other respects unaltered, is followed by an effect B

which did not exist before, A is, in that instance at least, the cause

of B, or an indispensable part of its cause; and if A be tried again

with many totally different sets of antecedents and B still follows,

then it is the whole cause. If these observations or experiments have

been repeated so often, and by so many persons, as to exclude all

supposition of error in the observer, a law of nature is established;

and so long as this law is received as such, the assertion that on any

particular occasion A took place, and yet B did not follow, _without any

counteracting cause_, must be disbelieved. Such an assertion is not to

be credited on any less evidence than what would suffice to overturn the

law. The general truths, that whatever has a beginning has a cause, and

that when none but the same causes exist, the same effects follow, rest

on the strongest inductive evidence possible; the proposition that

things affirmed by even a crowd of respectable witnesses are true, is

but an approximate generalization; and--even if we fancy we actually saw

or felt the fact which is in contradiction to the law--what a human

being can see is no more than a set of appearances; from which the real

nature of the phenomenon is merely an inference, and in this inference

approximate generalizations usually have a large share. If, therefore,

we make our election to hold by the law, no quantity of evidence

whatever ought to persuade us that there has occurred anything in

contradiction to it. If, indeed, the evidence produced is such that it

is more likely that the set of observations and experiments on which the

law rests should have been inaccurately performed or incorrectly

interpreted, than that the evidence in question should be false, we may

believe the evidence; but then we must abandon the law. And since the

law was received on what seemed a complete induction, it can only be

rejected on evidence equivalent; namely, as being inconsistent not with

any number of approximate generalizations, but with some other and

better established law of nature. This extreme case, of a conflict

between two supposed laws of nature, has probably never actually

occurred where, in the process of investigating both the laws, the true

canons of scientific induction had been kept in view; but if it did

occur, it must terminate in the total rejection of one of the supposed

laws. It would prove that there must be a flaw in the logical process by

which either one or the other was established: and if there be so, that

supposed general truth is no truth at all. We cannot admit a proposition

as a law of nature, and yet believe a fact in real contradiction to it.

We must disbelieve the alleged fact, or believe that we were mistaken in

admitting the supposed law.

But in order that any alleged fact should be contradictory to a law of

causation, the allegation must be, not simply that the cause existed

without being followed by the effect, for that would be no uncommon

occurrence; but that this happened in the absence of any adequate

counteracting cause. Now in the case of an alleged miracle, the

assertion is the exact opposite of this. It is, that the effect was

defeated, not in the absence, but in consequence of a counteracting

cause, namely, a direct interposition of an act of the will of some

being who has power over nature; and in particular of a Being, whose

will being assumed to have endowed all the causes with the powers by

which they produce their effects, may well be supposed able to

counteract them. A miracle (as was justly remarked by Brown[41]) is no

contradiction to the law of cause and effect; it is a new effect,

supposed to be produced by the introduction of a new cause. Of the

adequacy of that cause, if present, there can be no doubt; and the only

antecedent improbability which can be ascribed to the miracle, is the

improbability that any such cause existed.

All, therefore, which Hume has made out, and this he must be considered

to have made out, is, that (at least in the imperfect state of our

knowledge of natural agencies, which leaves it always possible that some

of the physical antecedents may have been hidden from us,) no evidence

can prove a miracle to any one who did not previously believe the

existence of a being or beings with supernatural power; or who believes

himself to have full proof that the character of the Being whom he

recognises, is inconsistent with his having seen fit to interfere on the

occasion in question.

If we do not already believe in supernatural agencies, no miracle can

prove to us their existence. The miracle itself, considered merely as an

extraordinary fact, may be satisfactorily certified by our senses or by

testimony; but nothing can ever prove that it is a miracle: there is

still another possible hypothesis, that of its being the result of some

unknown natural cause: and this possibility cannot be so completely shut

out, as to leave no alternative but that of admitting the existence and

intervention of a being superior to nature. Those, however, who already

believe in such a being, have two hypotheses to choose from, a

supernatural and an unknown natural agency; and they have to judge which

of the two is the most probable in the particular case. In forming this

judgment, an important element of the question will be the conformity of

the result to the laws of the supposed agent, that is, to the character

of the Deity as they conceive it. But, with the knowledge which we now

possess of the general uniformity of the course of nature, religion,

following in the wake of science, has been compelled to acknowledge the

government of the universe as being on the whole carried on by general

laws, and not by special interpositions. To whoever holds this belief,

there is a general presumption against any supposition of divine agency

not operating through general laws, or in other words, there is an

antecedent improbability in every miracle, which, in order to outweigh

it, requires an extraordinary strength of antecedent probability derived

from the special circumstances of the case.

§ 3. It appears from what has been said, that the assertion that a cause

has been defeated of an effect which is connected with it by a

completely ascertained law of causation, is to be disbelieved or not,

according to the probability or improbability that there existed in the

particular instance an adequate counteracting cause. To form an estimate

of this, is not more difficult than of other probabilities. With regard

to all _known_ causes capable of counteracting the given causes, we have

generally some previous knowledge of the frequency or rarity of their

occurrence, from which we may draw an inference as to the antecedent

improbability of their having been present in any particular case. And

neither in respect to known or unknown causes are we required to

pronounce on the probability of their existing in nature, but only of

their having existed at the time and place at which the transaction is

alleged to have happened. We are seldom, therefore, without the means

(when the circumstances of the case are at all known to us) of judging

how far it is likely that such a cause should have existed at that time

and place without manifesting its presence by some other marks, and (in

the case of an unknown cause) without having hitherto manifested its

existence in any other instance. According as this circumstance, or the

falsity of the testimony, appears more improbable, that is, conflicts

with an approximate generalization of a higher order, we believe the

testimony, or disbelieve it; with a stronger or a weaker degree of

conviction, according to the preponderance: at least until we have

sifted the matter further.

So much, then, for the case in which the alleged fact conflicts, or

appears to conflict, with a real law of causation. But a more common

case, perhaps, is that of its conflicting with uniformities of mere

coexistence, not proved to be dependent on causation: in other words,

with the properties of Kinds. It is with these uniformities principally,

that the marvellous stories related by travellers are apt to be at

variance: as of men with tails, or with wings, and (until confirmed by

experience) of flying fish; or of ice, in the celebrated anecdote of the

Dutch travellers and the King of Siam. Facts of this description, facts

previously unheard of but which could not from any known law of

causation be pronounced impossible, are what Hume characterizes as not

contrary to experience, but merely unconformable to it; and Bentham, in

his treatise on Evidence, denominates them facts disconformable _in

specie_, as distinguished from such as are disconformable _in toto_ or

in _degree_.

In a case of this description, the fact asserted is the existence of a

new Kind; which in itself is not in the slightest degree incredible, and

only to be rejected if the improbability that any variety of object

existing at the particular place and time should not have been

discovered sooner, be greater than that of error or mendacity in the

witnesses. Accordingly, such assertions, when made by credible persons,

and of unexplored places, are not disbelieved, but at most regarded as

requiring confirmation from subsequent observers; unless the alleged

properties of the supposed new Kind are at variance with known

properties of some larger kind which includes it; or in other words,

unless, in the new Kind which is asserted to exist, some properties are

said to have been found disjoined from others which have always been

known to accompany them; as in the case of Pliny's men, or any other

kind of animal of a structure different from that which has always been

found to coexist with animal life. On the mode of dealing with any such

case, little needs be added to what has been said on the same topic in

the twenty-second chapter.[42] When the uniformities of coexistence

which the alleged fact would violate, are such as to raise a strong

presumption of their being the result of causation, the fact which

conflicts with them is to be disbelieved; at least provisionally, and

subject to further investigation. When the presumption amounts to a

virtual certainty, as in the case of the general structure of organized

beings, the only question requiring consideration is whether, in

phenomena so little understood, there may not be liabilities to

counteraction from causes hitherto unknown; or whether the phenomena may

not be capable of originating in some other way, which would produce a

different set of derivative uniformities. Where (as in the case of the

flying fish, or the ornithorhynchus) the generalization to which the

alleged fact would be an exception is very special and of limited range,

neither of the above suppositions can be deemed very improbable; and it

is generally, in the case of such alleged anomalies, wise to suspend our

judgment, pending the subsequent inquiries which will not fail to

confirm the assertion if it be true. But when the generalization is very

comprehensive, embracing a vast number and variety of observations, and

covering a considerable province of the domain of nature; then, for

reasons which have been fully explained, such an empirical law comes

near to the certainty of an ascertained law of causation: and any

alleged exception to it cannot be admitted, unless on the evidence of

some law of causation proved by a still more complete induction.

Such uniformities in the course of nature as do not bear marks of being

the results of causation, are, as we have already seen, admissible as

universal truths with a degree of credence proportioned to their

generality. Those which are true of all things whatever, or at least

which are totally independent of the varieties of Kinds, namely, the

laws of number and extension, to which we may add the law of causation

itself, are probably the only ones, an exception to which is absolutely

and permanently incredible. Accordingly, it is to assertions supposed to

be contradictory to these laws, or to some others coming near to them in

generality, that the word impossibility (at least _total_ impossibility)

seems to be generally confined. Violations of other laws, of special

laws of causation for instance, are said, by persons studious of

accuracy in expression, to be impossible _in the circumstances of the

case_; or impossible unless some cause had existed which did not exist

in the particular case.[43] Of no assertion, not in contradiction to

some of these very general laws, will more than improbability be

asserted by any cautious person; and improbability not of the highest

degree, unless the time and place in which the fact is said to have

occurred, render it almost certain that the anomaly, if real, could not

have been overlooked by other observers. Suspension of judgment is in

all other cases the resource of the judicious inquirer; provided the

testimony in favour of the anomaly presents, when well sifted, no

suspicious circumstances.

But the testimony is scarcely ever found to stand that test, in cases in

which the anomaly is not real. In the instances on record in which a

great number of witnesses, of good reputation and scientific

acquirements, have testified to the truth of something which has turned

out untrue, there have almost always been circumstances which, to a keen

observer who had taken due pains to sift the matter, would have rendered

the testimony untrustworthy. There have generally been means of

accounting for the impression on the senses or minds of the alleged

percipients, by fallacious appearances; or some epidemic delusion,

propagated by the contagious influence of popular feeling, has been

concerned in the case; or some strong interest has been

implicated--religious zeal, party feeling, vanity, or at least the

passion for the marvellous, in persons strongly susceptible of it. When

none of these or similar circumstances exist to account for the apparent

strength of the testimony; and where the assertion is not in

contradiction either to those universal laws which know no counteraction

or anomaly, or to the generalizations next in comprehensiveness to them,

but would only amount, if admitted, to the existence of an unknown cause

or an anomalous Kind, in circumstances not so thoroughly explored but

that it is credible that things hitherto unknown may still come to

light; a cautious person will neither admit nor reject the testimony,

but will wait for confirmation at other times and from other unconnected

sources. Such ought to have been the conduct of the King of Siam when

the Dutch travellers affirmed to him the existence of ice. But an

ignorant person is as obstinate in his contemptuous incredulity as he is

unreasonably credulous. Anything unlike his own narrow experience he

disbelieves, if it flatters no propensity; any nursery tale is swallowed

implicitly by him if it does.

§ 4. I shall now advert to a very serious misapprehension of the

principles of the subject, which has been committed by some of the

writers against Hume's Essay on Miracles, and by Bishop Butler before

them, in their anxiety to destroy what appeared to them a formidable

weapon of assault against the Christian religion; and the effect of

which is entirely to confound the doctrine of the Grounds of Disbelief.

The mistake consists in overlooking the distinction between (what may be

called) improbability before the fact, and improbability after it; or

(since, as Mr. Venn remarks, the distinction of past and future is not

the material circumstance) between the improbability of a mere guess

being right, and the improbability of an alleged fact being true.

Many events are altogether improbable to us, before they have happened,

or before we are informed of their happening, which are not in the least

incredible when we are informed of them, because not contrary to any,

even approximate, induction. In the cast of a perfectly fair die, the

chances are five to one against throwing ace, that is, ace will be

thrown on an average only once in six throws. But this is no reason

against believing that ace was thrown on a given occasion, if any

credible witness asserts it; since though ace is only thrown once in six

times, _some_ number which is only thrown once in six times must have

been thrown if the die was thrown at all. The improbability, then, or in

other words, the unusualness, of any fact, is no reason for disbelieving

it, if the nature of the case renders it certain that either that or

something equally improbable, that is, equally unusual, did happen. Nor

is this all: for even if the other five sides of the die were all twos,

or all threes, yet as ace would still on the average come up once in

every six throws, its coming up in a given throw would be not in any way

contradictory to experience. If we disbelieved all facts which had the

chances against them beforehand, we should believe hardly anything. We

are told that A. B. died yesterday: the moment before we were so told,

the chances against his having died on that day may have been ten

thousand to one; but since he was certain to die at some time or other,

and when he died must necessarily die on some particular day, while the

preponderance of chances is very great against every day in particular,

experience affords no ground for discrediting any testimony which may be

produced to the event's having taken place on a given day.

Yet it has been considered, by Dr. Campbell and others, as a complete

answer to Hume's doctrine (that things are incredible which are

_contrary_ to the uniform course of experience), that we do not

disbelieve, merely because the chances were against them, things in

strict _conformity_ to the uniform course of experience; that we do not

disbelieve an alleged fact merely because the combination of causes on

which it depends occurs only once in a certain number of times. It is

evident that whatever is shown by observation, or can be proved from

laws of nature, to occur in a certain proportion (however small) of the

whole number of possible cases, is not contrary to experience; though we

are right in disbelieving it, if some other supposition respecting the

matter in question involves on the whole a less departure from the

ordinary course of events. Yet, on such grounds as this have able

writers been led to the extraordinary conclusion, that nothing supported

by credible testimony ought ever to be disbelieved.

§ 5. We have considered two species of events, commonly said to be

improbable; one kind which are in no way extraordinary, but which,

having an immense preponderance of chances against them, are improbable

until they are affirmed, but no longer; another kind which, being

contrary to some recognised law of nature, are incredible on any amount

of testimony except such as would be sufficient to shake our belief in

the law itself. But between these two classes of events, there is an

intermediate class, consisting of what are commonly termed Coincidences:

in other words, those combinations of chances which present some

peculiar and unexpected regularity, assimilating them, in so far, to the

results of law. As if, for example, in a lottery of a thousand tickets,

the numbers should be drawn in the exact order of what are called the

natural numbers, 1, 2, 3, &c. We have still to consider the principles

of evidence applicable to this case: whether there is any difference

between coincidences and ordinary events, in the amount of testimony or

other evidence necessary to render them credible.

It is certain, that on every rational principle of expectation, a

combination of this peculiar sort may be expected quite as often as any

other given series of a thousand numbers; that with perfectly fair dice,

sixes will be thrown twice, thrice, or any number of times in

succession, quite as often in a thousand or a million throws, as any

other succession of numbers fixed upon beforehand; and that no judicious

player would give greater odds against the one series than against the

other. Notwithstanding this, there is a general disposition to regard

the one as much more improbable than the other, and as requiring much

stronger evidence to make it credible. Such is the force of this

impression, that it has led some thinkers to the conclusion, that nature

has greater difficulty in producing regular combinations than irregular

ones; or in other words, that there is some general tendency of things,

some law, which prevents regular combinations from occurring, or at

least from occurring so often as others. Among these thinkers may be

numbered D'Alembert; who, in an Essay on Probabilities to be found in

the fifth volume of his _Mélanges_, contends that regular combinations,

though equally probable according to the mathematical theory with any

others, are physically less probable. He appeals to common sense, or in

other words, to common impressions; saying, if dice thrown repeatedly in

our presence gave sixes every time, should we not, before the number of

throws had reached ten, (not to speak of thousands of millions,) be

ready to affirm, with the most positive conviction, that the dice were

false?

The common and natural impression is in favour of D'Alembert: the

regular series would be thought much more unlikely than an irregular.

But this common impression is, I apprehend, merely grounded on the fact,

that scarcely anybody remembers to have ever seen one of these peculiar

coincidences: the reason of which is simply that no one's experience

extends to anything like the number of trials, within which that or any

other given combination of events can be expected to happen. The chance

of sixes on a single throw of two dice being 1/36, the chance of sixes

ten times in succession is 1 divided by the tenth power of 36; in other

words, such a concurrence is only likely to happen once in

3,656,158,440,062,976 trials, a number which no dice-player's experience

comes up to a millionth part of. But if, instead of sixes ten times, any

other given succession of ten throws had been fixed upon, it would have

been exactly as unlikely that in any individual's experience that

particular succession had ever occurred; although this does not _seem_

equally improbable, because no one could possibly have remembered

whether it had occurred or not, and because the comparison is tacitly

made, not between sixes ten times and any one particular series of

throws, but between all regular and all irregular successions taken

together.

That (as D'Alembert says) if the succession of sixes was actually thrown

before our eyes, we should ascribe it not to chance, but to unfairness

in the dice, is unquestionably true. But this arises from a totally

different principle. We should then be considering, not the probability

of the fact in itself, but the comparative probability with which, when

it is known to have happened, it may be referred to one or to another

cause. The regular series is not at all less likely than the irregular

one to be brought about by chance, but it is much more likely than the

irregular one to be produced by design; or by some general cause

operating through the structure of the dice. It is the nature of casual

combinations to produce a repetition of the same event, as often and no

oftener than any other series of events. But it is the nature of general

causes to reproduce, in the same circumstances, always the same event.

Common sense and science alike dictate that, all other things being the

same, we should rather attribute the effect to a cause which if real

would be very likely to produce it, than to a cause which would be very

unlikely to produce it. According to Laplace's sixth theorem, which we

demonstrated in a former chapter, the difference of probability arising

from the superior _efficacy_ of the constant cause, unfairness in the

dice, would after a very few throws far outweigh any antecedent

probability which there could be against its existence.

D'Alembert should have put the question in another manner. He should

have supposed that we had ourselves previously tried the dice, and knew

by ample experience that they were fair. Another person then tries them

in our absence, and assures us that he threw sixes ten times in

succession. Is the assertion credible or not? Here the effect to be

accounted for is not the occurrence itself, but the fact of the

witness's asserting it. This may arise either from its having really

happened, or from some other cause. What we have to estimate is the

comparative probability of these two suppositions.

If the witness affirmed that he had thrown any other series of numbers,

supposing him to be a person of veracity, and tolerable accuracy, and to

profess that he took particular notice, we should believe him. But the

ten sixes are exactly as likely to have been really thrown as the other

series. If, therefore, this assertion is less credible than the other,

the reason must be, not that it is less likely than the other to be made

truly, but that it is more likely than the other to be made falsely.

One reason obviously presents itself why what is called a coincidence,

should be oftener asserted falsely than an ordinary combination. It

excites wonder. It gratifies the love of the marvellous. The motives,

therefore, to falsehood, one of the most frequent of which is the desire

to astonish, operate more strongly in favour of this kind of assertion

than of the other kind. Thus far there is evidently more reason for

discrediting an alleged coincidence, than a statement in itself not more

probable, but which if made would not be thought remarkable. There are

cases, however, in which the presumption on this ground would be the

other way. There are some witnesses who, the more extraordinary an

occurrence might appear, would be the more anxious to verify it by the

utmost carefulness of observation before they would venture to believe

it, and still more before they would assert it to others.

§ 6. Independently, however, of any peculiar chances of mendacity

arising from the nature of the assertion, Laplace contends, that merely

on the general ground of the fallibility of testimony, a coincidence is

not credible on the same amount of testimony on which we should be

warranted in believing an ordinary combination of events. In order to do

justice to his argument, it is necessary to illustrate it by the example

chosen by himself.

If, says Laplace, there were one thousand tickets in a box, and one only

has been drawn out, then if an eye-witness affirms that the number drawn

was 79, this, though the chances were 999 in 1000 against it, is not on

that account the less credible; its credibility is equal to the

antecedent probability of the witness's veracity. But if there were in

the box 999 black balls and only one white, and the witness affirms that

the white ball was drawn, the case according to Laplace is very

different: the credibility of his assertion is but a small fraction of

what it was in the former case; the reason of the difference being as

follows.

The witnesses of whom we are speaking must, from the nature of the case,

be of a kind whose credibility falls materially short of certainty: let

us suppose, then, the credibility of the witness in the case in question

to be 9/10; that is, let us suppose that in every ten statements which

the witness makes, nine on an average are correct, and one incorrect.

Let us now suppose that there have taken place a sufficient number of

drawings to exhaust all the possible combinations, the witness deposing

in every one. In one case out of every ten in all these drawings he will

actually have made a false announcement. But in the case of the thousand

tickets these false announcements will have been distributed impartially

over all the numbers, and of the 999 cases in which No. 79 was not

drawn, there will have been only one case in which it was announced. On

the contrary, in the case of the thousand balls, (the announcement being

always either "black" or "white,") if white was not drawn, and there was

a false announcement, that false announcement _must_ have been white;

and since by the supposition there was a false announcement once in

every ten times, white will have been announced falsely in one tenth

part of all the cases in which it was not drawn, that is, in one tenth

part of 999 cases out of every thousand. White, then, is drawn, on an

average, exactly as often as No. 79, but it is announced, without having

been really drawn, 999 times as often as No. 79; the announcement

therefore requires a much greater amount of testimony to render it

credible.[44]

To make this argument valid it must of course be supposed, that the

announcements made by the witness are average specimens of his general

veracity and accuracy; or, at least, that they are neither more nor less

so in the case of the black and white balls, than in the case of the

thousand tickets. This assumption, however, is not warranted. A person

is far less likely to mistake, who has only one form of error to guard

against, than if he had 999 different errors to avoid. For instance, in

the example chosen, a messenger who might make a mistake once in ten

times in reporting the number drawn in a lottery, might not err once in

a thousand times if sent simply to observe whether a ball was black or

white. Laplace's argument therefore is faulty even as applied to his own

case. Still less can that case be received as completely representing

all cases of coincidence. Laplace has so contrived his example, that

though black answers to 999 distinct possibilities, and white only to

one, the witness has nevertheless no bias which can make him prefer

black to white. The witness did not know that there were 999 black balls

in the box and only one white; or if he did, Laplace has taken care to

make all the 999 cases so undistinguishably alike, that there is hardly

a possibility of any cause of falsehood or error operating in favour of

any of them, which would not operate in the same manner if there were

only one. Alter this supposition, and the whole argument falls to the

ground. Let the balls, for instance, be numbered, and let the white ball

be No. 79. Considered in respect of their colour, there are but two

things which the witness can be interested in asserting, or can have

dreamt or hallucinated, or has to choose from if he answers at random,

viz. black and white: but considered in respect of the numbers attached

to them, there are a thousand: and if his interest or error happens to

be connected with the numbers, though the only assertion he makes is

about the colour, the case becomes precisely assimilated to that of the

thousand tickets. Or instead of the balls suppose a lottery, with 1000

tickets and but one prize, and that I hold No. 79, and being interested

only in that, ask the witness not what was the number drawn, but whether

it was 79 or some other. There are now only two cases, as in Laplace's

example; yet he surely would not say that if the witness answered 79,

the assertion would be in an enormous proportion less credible, than if

he made the same answer to the same question asked in the other way. If,

for instance, (to put a case supposed by Laplace himself,) he has staked

a large sum on one of the chances, and thinks that by announcing its

occurrence he shall increase his credit; he is equally likely to have

betted on any one of the 999 numbers which are attached to black balls,

and so far as the chances of mendacity from this cause are concerned,

there will be 999 times as many chances of his announcing black falsely,

as white.

Or suppose a regiment of 1000 men, 999 Englishmen and one Frenchman, and

that of these one man has been killed, and it is not known which. I ask

the question, and the witness answers, the Frenchman. This was not only

as improbable _à priori_, but is in itself as singular a circumstance,

as remarkable a coincidence, as the drawing of the white ball: yet we

should believe the statement as readily, as if the answer had been John

Thompson. Because though the 999 Englishmen were all alike in the point

in which they differed from the Frenchman, they were not, like the 999

black balls, undistinguishable in every other respect; but being all

different, they admitted as many chances of interest or error, as if

each man had been of a different nation; and if a lie was told or a

mistake made, the misstatement was as likely to fall on any Jones or

Thompson of the set, as on the Frenchman.

The example of a coincidence selected by D'Alembert, that of sixes

thrown on a pair of dice ten times in succession, belongs to this sort

of cases rather than to such as Laplace's. The coincidence is here far

more remarkable, because of far rarer occurrence, than the drawing of

the white ball. But though the improbability of its really occurring is

greater, the superior probability of its being announced falsely cannot

be established with the same evidence. The announcement "black"

represented 999 cases, but the witness may not have known this, and if

he did, the 999 cases are so exactly alike, that there is really only

one set of possible causes of mendacity corresponding to the whole. The

announcement "sixes _not_ drawn ten times," represents, and is known by

the witness to represent, a great multitude of contingencies, every one

of which being unlike every other, there may be a different and a fresh

set of causes of mendacity corresponding to each.

It appears to me, therefore, that Laplace's doctrine is not strictly

true of any coincidences, and is wholly inapplicable to most: and that

to know whether a coincidence does or does not require more evidence to

render it credible than an ordinary event, we must refer, in every

instance, to first principles, and estimate afresh what is the

probability that the given testimony would have been delivered in that

instance, supposing the fact which it asserts not to be true.

With these remarks we close the discussion of the Grounds of Disbelief;

and along with it, such exposition as space admits, and as the writer

has it in his power to furnish, of the Logic of Induction.

FOOTNOTES:

[1] _Cours de Philosophie Positive_, ii. 656.

[2] Vide supra, book iii. ch. xi.

[3] _Philosophy of Discovery_, pp. 185 et seqq.

[4] _Philosophie Positive_, ii. 434-437.

[5] As an example of legitimate hypothesis according to the test here

laid down, has been justly cited that of Broussais, who, proceeding on

the very rational principle that every disease must originate in some

definite part or other of the organism, boldly assumed that certain

fevers, which not being known to be local were called constitutional,

had their origin in the mucous membrane of the alimentary canal. The

supposition was indeed, as is now generally admitted, erroneous; but he

was justified in making it, since by deducing the consequences of the

supposition, and comparing them with the facts of those maladies, he

might be certain of disproving his hypothesis if it was ill founded, and

might expect that the comparison would materially aid him in framing

another more conformable to the phenomena.

The doctrine now universally received, that the earth is a natural

magnet, was originally an hypothesis of the celebrated Gilbert.

Another hypothesis, to the legitimacy of which no objection can lie, and

which is well calculated to light the path of scientific inquiry, is

that suggested by several recent writers, that the brain is a voltaic

pile, and that each of its pulsations is a discharge of electricity

through the system. It has been remarked that the sensation felt by the

hand from the beating of a brain, bears a strong resemblance to a

voltaic shock. And the hypothesis, if followed to its consequences,

might afford a plausible explanation of many physiological facts, while

there is nothing to discourage the hope that we may in time sufficiently

understand the conditions of voltaic phenomena to render the truth of

the hypothesis amenable to observation and experiment.

The attempt to localize, in different regions of the brain, the physical

organs of our different mental faculties and propensities, was, on the

part of its original author, a legitimate example of a scientific

hypothesis; and we ought not, therefore, to blame him for the extremely

slight grounds on which he often proceeded, in an operation which could

only be tentative, though we may regret that materials barely sufficient

for a first rude hypothesis should have been hastily worked up into the

vain semblance of a science. If there be really a connexion between the

scale of mental endowments and the various degrees of complication in

the cerebral system, the nature of that connexion was in no other way so

likely to be brought to light as by framing, in the first instance, an

hypothesis similar to that of Gall. But the verification of any such

hypothesis is attended, from the peculiar nature of the phenomena, with

difficulties which phrenologists have not shown themselves even

competent to appreciate, much less to overcome.

Mr. Darwin's remarkable speculation on the Origin of Species is another

unimpeachable example of a legitimate hypothesis. What he terms "natural

selection" is not only a _vera causa_, but one proved to be capable of

producing effects of the same kind with those which the hypothesis

ascribes to it: the question of possibility is entirely one of degree.

It is unreasonable to accuse Mr. Darwin (as has been done) of violating

the rules of Induction. The rules of Induction are concerned with the

conditions of Proof. Mr. Darwin has never pretended that his doctrine

was proved. He was not bound by the rules of Induction, but by those of

Hypothesis. And these last have seldom been more completely fulfilled.

He has opened a path of inquiry full of promise, the results of which

none can foresee. And is it not a wonderful feat of scientific knowledge

and ingenuity to have rendered so bold a suggestion, which the first

impulse of every one was to reject at once, admissible and discussable,

even as a conjecture?

[6] Whewell's _Phil. of Discovery_, pp. 275, 276.

[7] What has most contributed to accredit the hypothesis of a physical

medium for the conveyance of light, is the certain fact that light

_travels_, (which cannot be proved of gravitation,) that its

communication is not instantaneous, but requires time, and that it is

intercepted (which gravitation is not) by intervening objects. These are

analogies between its phenomena and those of the mechanical motion of a

solid or fluid substance. But we are not entitled to assume that

mechanical motion is the only power in nature capable of exhibiting

those attributes.

[8] _Phil. of Disc._ p. 274.

[9] P. 271.

[10] P. 251 and the whole of Appendix G.

[11] In Dr. Whewell's latest version of his theory (_Philosophy of

Discovery_, p. 331) he makes a concession respecting the medium of the

transmission of light, which, taken in conjunction with the rest of his

doctrine on the subject, is not, I confess, very intelligible to me, but

which goes far towards removing, if it does not actually remove, the

whole of the difference between us. He is contending, against Sir

William Hamilton, that all matter has weight. Sir William, in proof of

the contrary, cited the luminiferous ether, and the calorific and

electric fluids, "which," he said, "we can neither denude of their

character of substance, nor clothe with the attribute of weight." "To

which," continues Dr. Whewell, "my reply is, that precisely because I

cannot clothe these agents with the attribute of Weight, I _do_ denude

them of the character of Substance. They are not substances, but

agencies. These Imponderable Agents, are not properly called

Imponderable Fluids. This I conceive that I have proved." Nothing can be

more philosophical. But if the luminiferous ether is not matter, and

fluid matter too, what is the meaning of its undulations? Can an agency

undulate? Can there be alternate motion forward and backward of the

particles of an agency? And does not the whole mathematical theory of

the undulations imply them to be material? Is it not a series of

deductions from the known properties of elastic fluids? _This_ opinion

of Dr. Whewell reduces the undulations to a figure of speech, and the

undulatory theory to the proposition which all must admit, that the

transmission of light takes place according to laws which present a very

striking and remarkable agreement with those of undulations. If Dr.

Whewell is prepared to stand by this doctrine, I have no difference with

him on the subject.

Since this chapter was written, the hypothesis of the luminiferous ether

has acquired a great accession of apparent strength, by being adopted

into the new doctrine of the Conservation of Force, as affording a

mechanism by which to explain the mode of production not of light only,

but of heat, and probably of all the other so-called imponderable

agencies. In the present immature stage of the great speculation in

question, I would not undertake to define the ultimate relation of the

hypothetical fluid to it; but I must remark that the essential part of

the new theory, the reciprocal convertibility and interchangeability of

these great cosmic agencies, is quite independent of the molecular

motions which have been imagined as the immediate causes of those

different manifestations and of their substitutions for one another; and

the former doctrine by no means necessarily carries the latter with it.

I confess that the entire theory of the vibrations of the ether, and the

movements which these vibrations are supposed to communicate to the

particles of solid bodies, seems to me at present the weakest part of

the new system, tending rather to weigh down than to prop up those of

its doctrines which rest on real scientific induction.

[12] Thus, water, of which eight-ninths in weight are oxygen, dissolves

most bodies which contain a high proportion of oxygen, such as all the

nitrates, (which have more oxygen than any others of the common salts,)

most of the sulphates, many of the carbonates, &c. Again, bodies largely

composed of combustible elements, like hydrogen and carbon, are soluble

in bodies of similar composition; rosin, for instance, will dissolve in

alcohol, tar in oil of turpentine. This empirical generalization is far

from being universally true; no doubt because it is a remote, and

therefore easily defeated, result of general laws too deep for us at

present to penetrate; but it will probably in time suggest processes of

inquiry, leading to the discovery of those laws.

[13] Or (according to Laplace's theory) the sun and the sun's rotation.

[14] Supra, book iii. ch. v. § 7.

[15] Supra, book iii. ch. x. § 2.

[16] In the preceding discussion, the _mean_ is spoken of as if it were

exactly the same thing with the _average_. But the mean for purposes of

inductive inquiry, is not the average, or arithmetical mean, though in a

familiar illustration of the theory the difference may be disregarded.

If the deviations on one side of the average are much more numerous than

those on the other (these last being fewer but greater), the effect due

to the invariable cause, as distinct from the variable ones, will not

coincide with the average, but will be either below or above the

average, whichever be the side on which the greatest number of the

instances are found. This follows from a truth, ascertained both

inductively and deductively, that small deviations from the true central

point are greatly more frequent than large ones. The mathematical law

is, "that the most probable determination of one or more invariable

elements from observation is that in which _the sum of the squares_ of

the individual aberrations," or deviations, "_shall be the least

possible_." See this principle stated, and its grounds popularly

explained, by Sir John Herschel, in his review of Quetelet on

Probabilities, _Essays_, pp. 395 _et seq._

[17] _Essai Philosophique sur les Probabilités_, fifth Paris Edition, p.

7.

[18] It even appears to me that the calculation of chances, where there

are no data grounded either on special experience or on special

inference, must, in an immense majority of cases, break down, from sheer

impossibility of assigning any principle by which to be guided in

setting out the list of possibilities. In the case of the coloured balls

we have no difficulty in making the enumeration, because we ourselves

determine what the possibilities shall be. But suppose a case more

analogous to those which occur in nature: instead of three colours, let

there be in the box all possible colours: we being supposed ignorant of

the comparative frequency with which different colours occur in nature,

or in the productions of art. How is the list of cases to be made out?

Is every distinct shade to count as a colour? If so, is the test to be a

common eye, or an educated eye, a painter's for instance? On the answer

to these questions would depend whether the chances against some

particular colour would be estimated at ten, twenty, or perhaps five

hundred to one. While if we knew from experience that the particular

colour occurs on an average a certain number of times in every hundred

or thousand, we should not require to know anything either of the

frequency or of the number of the other possibilities.

[19] _Prospective Review_ for February 1850.

[20] "If this be not so, why do we feel so much more probability added

by the first instance, than by any single subsequent instance? Why,

except that the first instance gives us its possibility (a cause

_adequate_ to it), while every other only gives us the frequency of its

conditions? If no reference to a cause be supposed, possibility would

have no meaning; yet it is clear, that, antecedent to its happening, we

might have supposed the event impossible, _i.e._, have believed that

there was no physical energy really existing in the world equal to

producing it.... After the first time of happening, which is, then, more

important to the whole probability than any other single instance

(because proving the possibility), the _number_ of times becomes

important as an index to the intensity or extent of the cause, and its

independence of any particular time. If we took the case of a tremendous

leap, for instance, and wished to form an estimate of the probability of

its succeeding a certain number of times; the first instance, by showing

its possibility (before doubtful) is of the most importance; but every

succeeding leap shows the power to be more perfectly under control,

greater and more invariable, and so increases the probability; and no

one would think of reasoning in this case straight from one instance to

the next, without referring to the physical energy which each leap

indicated. Is it not then clear that we do not ever" (let us rather say,

that we do not in an advanced state of our knowledge) "conclude directly

from the happening of an event to the probability of its happening

again; but that we refer to the cause, regarding the past cases as an

index to the cause, and the cause as our guide to the future?"--_Ibid._

[21] The writer last quoted says that the valuation of chances by

comparing the number of cases in which the event occurs with the number

in which it does not occur, "would generally be wholly erroneous," and

"is not the true theory of probability." It is at least that which forms

the foundation of insurance, and of all those calculations of chances in

the business of life which experience so abundantly verifies. The reason

which the reviewer gives for rejecting the theory, is that it "would

regard an event as certain which had hitherto never failed; which is

exceedingly far from the truth, even for a very large number of constant

successes." This is not a defect in a particular theory, but in any

theory of chances. No principle of evaluation can provide for such a

case as that which the reviewer supposes. If an event has never once

failed, in a number of trials sufficient to eliminate chance, it really

has all the certainty which can be given by an empirical law: it _is_

certain during the continuance of the same collocation of causes which

existed during the observations. If it ever fails, it is in consequence

of some change in that collocation. Now, no theory of chances will

enable us to infer the future probability of an event from the past, if

the causes in operation, capable of influencing the event, have

intermediately undergone a change.

[22] Pp. 18, 19. The theorem is not stated by Laplace in the exact terms

in which I have stated it; but the identity of import of the two modes

of expression is easily demonstrable.

[23] For a fuller treatment of the many interesting questions raised by

the theory of probabilities, I may now refer to a recent work by Mr.

Venn, Fellow of Caius College, Cambridge, "The Logic of Chance;" one of

the most thoughtful and philosophical treatises on any subject connected

with Logic and Evidence, which have been produced in this or any other

country for many years. Some criticisms contained in it have been very

useful to me in revising the corresponding chapters of the present work.

In several of Mr. Venn's opinions, however, I do not agree. What these

are will be obvious to any reader of Mr. Venn's work who is also a

reader of this.

[24] There was no greater foundation for this than for Newton's

celebrated conjecture that the diamond was combustible. He grounded his

guess on the very high refracting power of the diamond, comparatively to

its density; a peculiarity which had been observed to exist in

combustible substances; and on similar grounds he conjectured that

water, though not combustible, contained a combustible ingredient.

Experiment having subsequently shown that in both instances he guessed

right, the prophecy is considered to have done great honour to his

scientific sagacity; but it is to this day uncertain whether the guess

was, in truth, what there are so many examples of in the history of

science, a farsighted anticipation of a law afterwards to be discovered.

The progress of science has not hitherto shown ground for believing that

there is any real connexion between combustibility and a high refracting

power.

[25] Hartley's _Observations on Man_, vol. i. p. 16. The passage is not

in Priestley's curtailed edition.

[26] I am happy to be able to quote the following excellent passage from

Mr. Baden Powell's _Essay on the Inductive Philosophy_, in confirmation,

both in regard to history and to doctrine, of the statement made in the

text. Speaking of the "conviction of the universal and permanent

uniformity of nature," Mr. Powell says (pp. 98-100),

"We may remark that this idea, in its proper extent, is by no means one

of popular acceptance or natural growth. Just so far as the daily

experience of every one goes, so far indeed he comes to embrace a

certain persuasion of this kind, but merely to this limited extent, that

what is going on around him at present, in his own narrow sphere of

observation, will go on in like manner in future. The peasant believes

that the sun which rose to-day will rise again to-morrow; that the seed

put into the ground will be followed in due time by the harvest this

year as it was last year, and the like; but has no notion of such

inferences in subjects beyond his immediate observation. And it should

be observed that each class of persons, in admitting this belief within

the limited range of his own experience, though he doubt or deny it in

everything beyond, is, in fact, bearing unconscious testimony to its

universal truth. Nor, again, is it only among the _most_ ignorant that

this limitation is put upon the truth. There is a very general

propensity to believe that everything beyond common experience, or

especially ascertained laws of nature, is left to the dominion of chance

or fate or arbitrary intervention; and even to object to any attempted

explanation by physical causes, if conjecturally thrown out for an

apparently unaccountable phenomenon.

"The precise doctrine of the _generalization_ of this idea of the

uniformity of nature, so far from being obvious, natural, or intuitive,

is utterly beyond the attainment of the many. In all the extent of its

universality it is characteristic of the philosopher. It is clearly the

result of philosophic cultivation and training, and by no means the

spontaneous offspring of any primary principle naturally inherent in the

mind, as some seem to believe. It is no mere vague persuasion taken up

without examination, as a common prepossession to which we are always

accustomed; on the contrary, all common prejudices and associations are

against it. It is pre-eminently _an acquired idea_. It is not attained

without deep study and reflection. The best informed philosopher is the

man who most firmly believes it, even in opposition to received notions;

its acceptance depends on the extent and profoundness of his inductive

studies."

[27] Supra, book iii. ch. iii. § 1.

[28] It deserves remark, that these early generalizations did not, like

scientific inductions, presuppose causation. What they did presuppose,

was _uniformity_ in physical facts. But the observers were as ready to

presume uniformity in the coexistences of facts as in the sequences. On

the other hand, they never thought of assuming that this uniformity was

a principle pervading all nature: their generalizations did not imply

that there was uniformity in everything, but only that as much

uniformity as existed within their observation, existed also beyond it.

The induction, Fire burns, does not require for its validity that all

nature should observe uniform laws, but only that there should be

uniformity in one particular class of natural phenomena: the effects of

fire on the senses and on combustible substances. And uniformity to this

extent was not assumed, anterior to the experience, but proved by the

experience. The same observed instances which proved the narrower truth,

proved as much of the wider one as corresponded to it. It is from losing

sight of this fact, and considering the law of causation in its full

extent as necessarily presupposed in the very earliest generalizations,

that persons have been led into the belief that the law of causation is

known _à priori_, and is not itself a conclusion from experience.

[29] Book ii. chap. iii.

[30] One of the most rising thinkers of the new generation in France, M.

Taine (who has given, in the Revue des Deux Mondes, the most masterly

analysis, at least in one point of view, ever made of the present work),

though he rejects, on this and similar points of psychology, the

intuition theory in its ordinary form, nevertheless assigns to the law

of causation, and to some other of the most universal laws, that

certainty beyond the bounds of human experience, which I have not been

able to accord to them. He does this on the faith of our faculty of

abstraction, in which he seems to recognise an independent source of

evidence, not indeed disclosing truths not contained in our experience,

but affording an assurance which experience cannot give, of the

universality of those which it does contain. By abstraction M. Taine

seems to think that we are able, not merely to analyse that part of

nature which we see, and exhibit apart the elements which pervade it,

but to distinguish such of them as are elements of the system of nature

considered as a whole, not incidents belonging to our limited

terrestrial experience. I am not sure that I fully enter into M. Taine's

meaning; but I confess I do not see how any mere abstract conception,

elicited by our minds from our experience, can be evidence of an

objective fact in universal Nature, beyond what the experience itself

bears witness of; or how, in the process of interpreting in general

language the testimony of experience, the limitations of the testimony

itself can be cast off.

[31] Book i. chap. vii.

[32] In some cases, a Kind is sufficiently identified by some one

remarkable property: but most commonly several are required; each

property considered singly, being a joint property of that and of other

Kinds. The colour and brightness of the diamond are common to it with

the paste from which false diamonds are made; its octohedral form is

common to it with alum, and magnetic iron ore; but the colour and

brightness and the form together, identify its Kind; that is, are a mark

to us that it is combustible; that when burnt it produces carbonic acid;

that it cannot be cut with any known substance; together with many other

ascertained properties, and the fact that there exist an indefinite

number still unascertained.

[33] This doctrine of course assumes that the allotropic forms of what

is chemically the same substance are so many different Kinds; and such,

in the sense in which the word Kind is used in this treatise, they

really are.

[34] Mr. De Morgan, in his _Formal Logic_, makes the just remark, that

from two such premises as Most A are B, and Most A are C, we may infer

with certainty that some B are C. But this is the utmost limit of the

conclusions which can be drawn from two approximate generalizations,

when the precise degree of their approximation to universality is

unknown or undefined.

[35] _Rationale of Judicial Evidence_, vol. iii. p. 224.

[36] Supra, vol. i. p. 115.

[37] Supra, book i. ch. v. § 1, and book ii. ch. v. § 5.

[38] The axiom, "Equals subtracted from equals leave equal differences,"

may be demonstrated from the two axioms in the text. If A = _a_ and B =

_b_, A - B = _a - b_. For if not, let A - B = _a - b + c_. Then since B

= _b_, adding equals to equals, A = _a + c_. But A = _a_. Therefore _a =

a + c_, which is impossible.

This proposition having been demonstrated, we may, by means of it,

demonstrate the following: "If equals be added to unequals, the sums are

unequal." If A = _a_ and B not = _b_, A + B is not = _a + b_. For

suppose it be so. Then, since A = _a_ and A + B = _a + b_, subtracting

equals from equals, B = _b_; which is contrary to the hypothesis.

So again, it may be proved that two things, one of which is equal and

the other unequal to a third thing, are unequal to one another. If A =

_a_ and A not = B, neither is _a_ = B. For suppose it to be equal. Then

since A = _a_ and _a_ = B, and since things equal to the same thing are

equal to one another, A = B: which is contrary to the hypothesis.

[39] Geometers have usually preferred to define parallel lines by the

property of being in the same plane and never meeting. This, however,

has rendered it necessary for them to assume, as an additional axiom,

some other property of parallel lines; and the unsatisfactory manner in

which properties for that purpose have been selected by Euclid and

others has always been deemed the opprobrium of elementary geometry.

Even as a verbal definition, equidistance is a fitter property to

characterize parallels by, since it is the attribute really involved in

the signification of the name. If to be in the same plane and never to

meet were all that is meant by being parallel, we should feel no

incongruity in speaking of a curve as parallel to its asymptote. The

meaning of parallel lines is, lines which pursue exactly the same

direction, and which, therefore, neither draw nearer nor go farther from

one another; a conception suggested at once by the contemplation of

nature. That the lines will never meet is of course included in the more

comprehensive proposition that they are everywhere equally distant. And

that any straight lines which are in the same plane and not equidistant

will certainly meet, may be demonstrated in the most rigorous manner

from the fundamental property of straight lines assumed in the text,

viz. that if they set out from the same point, they diverge more and

more without limit.

[40] _Philosophie Positive_, iii. 414-416.

[41] See the two remarkable notes (A) and (F), appended to his _Inquiry

into the Relation of Cause and Effect_.

[42] Supra, pp. 119, 120.

[43] A writer to whom I have several times referred, gives as the

definition of an impossibility, that which there exists in the world no

cause adequate to produce. This definition does not take in such

impossibilities as these--that two and two should make five; that two

straight lines should inclose a space; or that anything should begin to

exist without a cause. I can think of no definition of impossibility

comprehensive enough to include all its varieties, except the one which

I have given: viz. An impossibility is that, the truth of which would

conflict with a complete induction, that is, with the most conclusive

evidence which we possess of universal truth.

As to the reputed impossibilities which rest on no other grounds than

our ignorance of any cause capable of producing the supposed effects;

very few of them are certainly impossible, or permanently incredible.

The facts of travelling seventy miles an hour, painless surgical

operations, and conversing by instantaneous signals between London and

New York, held a high place, not many years ago, among such

impossibilities.

[44] Not, however, as might at first sight appear, 999 times as much. A

complete analysis of the cases shows that (always assuming the veracity

of the witness to be 9/10) in 10,000 drawings, the drawing of No. 79

will occur nine times, and be announced incorrectly once; the

credibility therefore of the announcement of No. 79 is 9/10; while the

drawing of a white ball will occur nine times, and be announced

incorrectly 999 times. The credibility therefore of the announcement of

white is 9/1008, and the ratio of the two 1008:10; the one announcement

being thus only about a hundred times more credible than the other,

instead of 999 times.

BOOK IV.

OF OPERATIONS SUBSIDIARY TO INDUCTION.

"Clear and distinct ideas are terms which, though familiar and frequent

in men's mouths, I have reason to think every one who uses does not

perfectly understand. And possibly it is but here and there one who

gives himself the trouble to consider them so far as to know what he

himself or others precisely mean by them; I have, therefore, in most

places, chose to put determinate or determined, instead of clear and

distinct, as more likely to direct men's thoughts to my meaning in this

matter."--LOCKE'S _Essay on the Human Understanding_; Epistle to the

Reader.

"Il ne peut y avoir qu'une méthode parfaite, qui est la _méthode

naturelle_; on nomme ainsi un arrangement dans lequel les êtres du même

genre seraient plus voisins entre eux que ceux de tous les autres

genres; les genres du même ordre, plus que ceux de tous les autres

ordres; et ainsi de suite. Cette méthode est l'idéal auquel l'histoire

naturelle doit tendre; car il est évident que si l'on y parvenait, l'on

aurait l'expression exacte et complète de la nature entière."--CUVIER,

_Règne Animal_, Introduction.

"Deux grandes notions philosophiques dominent la théorie fondamentale de

la méthode naturelle proprement dite, savoir la formation des groupes

naturels, et ensuite leur succession hiérarchique."--COMTE, _Cours de

Philosophie Positive_, 42me leçon.

CHAPTER I.

OF OBSERVATION AND DESCRIPTION.

§ 1. The inquiry which occupied us in the two preceding books, has

conducted us to what appears a satisfactory solution of the principal

problem of Logic, according to the conception I have formed of the

science. We have found, that the mental process with which Logic is

conversant, the operation of ascertaining truths by means of evidence,

is always, even when appearances point to a different theory of it, a

process of induction. And we have particularized the various modes of

induction, and obtained a clear view of the principles to which it must

conform, in order to lead to results which can be relied on.

The consideration of Induction, however, does not end with the direct

rules for its performance. Something must be said of those other

operations of the mind, which are either necessarily presupposed in all

induction, or are instrumental to the more difficult and complicated

inductive processes. The present book will be devoted to the

consideration of these subsidiary operations: among which our attention

must first be given to those, which are indispensable preliminaries to

all induction whatsoever.

Induction being merely the extension to a class of cases, of something

which has been observed to be true in certain individual instances of

the class; the first place among the operations subsidiary to induction,

is claimed by Observation. This is not, however, the place to lay down

rules for making good observers; nor is it within the competence of

Logic to do so, but of the art of intellectual Education. Our business

with observation is only in its connexion with the appropriate problem

of logic, the estimation of evidence. We have to consider, not how or

what to observe, but under what conditions observation is to be relied

on; what is needful, in order that the fact, supposed to be observed,

may safely be received as true.

§ 2. The answer to this question is very simple, at least in its first

aspect. The sole condition is, that what is supposed to have been

observed shall really have been observed; that it be an observation, not

an inference. For in almost every act of our perceiving faculties,

observation and inference are intimately blended. What we are said to

observe is usually a compound result, of which one-tenth may be

observation, and the remaining nine-tenths inference.

I affirm, for example, that I hear a man's voice. This would pass, in

common language, for a direct perception. All, however, which is really

perception, is that I hear a sound. That the sound is a voice, and that

voice the voice of a man, are not perceptions but inferences. I affirm,

again, that I saw my brother at a certain hour this morning. If any

proposition concerning a matter of fact would commonly be said to be

known by the direct testimony of the senses, this surely would be so.

The truth, however, is far otherwise. I only saw a certain coloured

surface; or rather I had the kind of visual sensations which are usually

produced by a coloured surface; and from these as marks, known to be

such by previous experience, I concluded that I saw my brother. I might

have had sensations precisely similar, when my brother was not there. I

might have seen some other person so nearly resembling him in

appearance, as, at the distance, and with the degree of attention which

I bestowed, to be mistaken for him. I might have been asleep, and have

dreamed that I saw him; or in a state of nervous disorder, which brought

his image before me in a waking hallucination. In all these modes, many

have been led to believe that they saw persons well known to them, who

were dead or far distant. If any of these suppositions had been true,

the affirmation that I saw my brother would have been erroneous; but

whatever was matter of direct perception, namely the visual sensations,

would have been real. The inference only would have been ill grounded;

I should have ascribed those sensations to a wrong cause.

Innumerable instances might be given, and analysed in the same manner,

of what are vulgarly called errors of sense. There are none of them

properly errors of sense; they are erroneous inferences from sense. When

I look at a candle through a multiplying glass, I see what seems a dozen

candles instead of one: and if the real circumstances of the case were

skilfully disguised, I might suppose that there were really that number;

there would be what is called an optical deception. In the kaleidoscope

there really is that deception: when I look through the instrument,

instead of what is actually there, namely a casual arrangement of

coloured fragments, the appearance presented is that of the same

combination several times repeated in symmetrical arrangement round a

point. The delusion is of course effected by giving me the same

sensations which I should have had if such a symmetrical combination had

really been presented to me. If I cross two of my fingers, and bring any

small object, a marble for instance, into contact with both, at points

not usually touched simultaneously by one object, I can hardly, if my

eyes are shut, help believing that there are two marbles instead of one.

But it is not my touch in this case, nor my sight in the other, which is

deceived; the deception, whether durable or only momentary, is in my

judgment. From my senses I have only the sensations, and those are

genuine. Being accustomed to have those or similar sensations when, and

only when, a certain arrangement of outward objects is present to my

organs, I have the habit of instantly, when I experience the sensations,

inferring the existence of that state of outward things. This habit has

become so powerful, that the inference, performed with the speed and

certainty of an instinct, is confounded with intuitive perceptions. When

it is correct, I am unconscious that it ever needed proof; even when I

know it to be incorrect, I cannot without considerable effort abstain

from making it. In order to be aware that it is not made by instinct but

by an acquired habit, I am obliged to reflect on the slow process

through which I learnt to judge by the eye of many things which I now

appear to perceive directly by sight; and on the reverse operation

performed by persons learning to draw, who with difficulty and labour

divest themselves of their acquired perceptions, and learn afresh to see

things as they appear to the eye.

It would be easy to prolong these illustrations, were there any need to

expatiate on a topic so copiously exemplified in various popular works.

From the examples already given, it is seen sufficiently, that the

individual facts from which we collect our inductive generalizations are

scarcely ever obtained by observation alone. Observation extends only to

the sensations by which we recognise objects; but the propositions which

we make use of, either in science or in common life, relate mostly to

the objects themselves. In every act of what is called observation,

there is at least one inference--from the sensations to the presence of

the object; from the marks or diagnostics, to the entire phenomenon. And

hence, among other consequences, follows the seeming paradox, that a

general proposition collected from particulars is often more certainly

true than any one of the particular propositions from which, by an act

of induction, it was inferred. For, each of those particular (or rather

singular) propositions involved an inference, from the impression on the

senses to the fact which caused that impression: and this inference may

have been erroneous in any one of the instances, but cannot well have

been erroneous in all of them, provided their number was sufficient to

eliminate chance. The conclusion, therefore, that is, the general

proposition, may deserve more complete reliance than it would be safe to

repose in any one of the inductive premises.

The logic of observation, then, consists solely in a correct

discrimination between that, in a result of observation, which has

really been perceived, and that which is an inference from the

perception. Whatever portion is inference, is amenable to the rules of

induction already treated of, and requires no further notice here: the

question for us in this place is, when all which is inference is taken

away, what remains. There remains, in the first place, the mind's own

feelings or states of consciousness, namely, its outward feelings or

sensations, and its inward feelings--its thoughts, emotions, and

volitions. Whether anything else remains, or all else is inference from

this; whether the mind is capable of directly perceiving or apprehending

anything except states of its own consciousness--is a problem of

metaphysics not to be discussed in this place. But after excluding all

questions on which metaphysicians differ, it remains true, that for most

purposes the discrimination we are called upon practically to exercise

is that between sensations or other feelings, of our own or of other

people, and inferences drawn from them. And on the theory of Observation

this is all which seems necessary to be said for the purposes of the

present work.

§ 3. If, in the simplest observation, or in what passes for such, there

is a large part which is not observation but something else; so in the

simplest description of an observation, there is, and must always be,

much more asserted than is contained in the perception itself. We cannot

describe a fact, without implying more than the fact. The perception is

only of one individual thing; but to describe it is to affirm a

connexion between it and every other thing which is either denoted or

connoted by any of the terms used. To begin with an example, than which

none can be conceived more elementary: I have a sensation of sight, and

I endeavour to describe it by saying that I see something white. In

saying this, I do not solely affirm my sensation; I also class it. I

assert a resemblance between the thing I see, and all things which I and

others are accustomed to call white. I assert that it resembles them in

the circumstance in which they all resemble one another, in that which

is the ground of their being called by the name. This is not merely one

way of describing an observation, but the only way. If I would either

register my observation for my own future use, or make it known for the

benefit of others, I must assert a resemblance between the fact which I

have observed and something else. It is inherent in a description, to be

the statement of a resemblance, or resemblances.

We thus see that it is impossible to express in words any result of

observation, without performing an act possessing what Dr. Whewell

considers to be characteristic of Induction. There is always something

introduced which was not included in the observation itself; some

conception common to the phenomenon with other phenomena to which it is

compared. An observation cannot be spoken of in language at all without

declaring more than that one observation; without assimilating it to

other phenomena already observed and classified. But this identification

of an object--this recognition of it as possessing certain known

characteristics--has never been confounded with Induction. It is an

operation which precedes all induction, and supplies it with its

materials. It is a perception of resemblances, obtained by comparison.

These resemblances are not always apprehended directly, by merely

comparing the object observed with some other present object, or with

our recollection of an object which is absent. They are often

ascertained through intermediate marks, that is, deductively. In

describing some new kind of animal, suppose me to say that it measures

ten feet in length, from the forehead to the extremity of the tail. I

did not ascertain this by the unassisted eye. I had a two-foot rule

which I applied to the object, and, as we commonly say, measured it; an

operation which was not wholly manual, but partly also mathematical,

involving the two propositions, Five times two is ten, and Things which

are equal to the same thing are equal to one another. Hence, the fact

that the animal is ten feet long is not an immediate perception, but a

conclusion from reasoning; the minor premises alone being furnished by

observation of the object. Nevertheless, this is called an observation

or a description of the animal, not an induction respecting it.

To pass at once from a very simple to a very complex example: I affirm

that the earth is globular. The assertion is not grounded on direct

perception; for the figure of the earth cannot, by us, be directly

perceived, though the assertion would not be true unless circumstances

could be supposed under which its truth could be so perceived. That the

form of the earth is globular is inferred from certain marks, as for

instance from this, that its shadow thrown upon the moon is circular; or

this, that on the sea, or any extensive plain, our horizon is always a

circle; either of which marks is incompatible with any other than a

globular form. I assert further, that the earth is that particular kind

of globe which is termed an oblate spheroid; because it is found by

measurement in the direction of the meridian, that the length on the

surface of the earth which subtends a given angle at its centre,

diminishes as we recede from the equator and approach the poles. But

these propositions, that the earth is globular, and that it is an oblate

spheroid, assert, each of them, an individual fact; in its own nature

capable of being perceived by the senses when the requisite organs and

the necessary position are supposed, and only not actually perceived

because those organs and that position are wanting. This identification

of the earth, first as a globe, and next as an oblate spheroid, which,

if the fact could have been seen, would have been called a description

of the figure of the earth, may without impropriety be so called when,

instead of being seen, it is inferred. But we could not without

impropriety call either of these assertions an induction from facts

respecting the earth. They are not general propositions collected from

particular facts, but particular facts deduced from general

propositions. They are conclusions obtained deductively, from premises

originating in induction: but of these premises some were not obtained

by observation of the earth, nor had any peculiar reference to it.

If, then, the truth respecting the figure of the earth is not an

induction, why should the truth respecting the figure of the earth's

orbit be so? The two cases only differ in this, that the form of the

orbit was not, like the form of the earth itself, deduced by

ratiocination from facts which were marks of ellipticity, but was got at

by boldly guessing that the path was an ellipse, and finding afterwards,

on examination, that the observations were in harmony with the

hypothesis. According to Dr. Whewell, however, this process of guessing

and verifying our guesses is not only induction, but the whole of

induction: no other exposition can be given of that logical operation.

That he is wrong in the latter assertion, the whole of the preceding

book has, I hope, sufficiently proved; and that the process by which the

ellipticity of the planetary orbits was ascertained, is not induction at

all, was attempted to be shown in the second chapter of the same

book.[1] We are now, however, prepared to go more into the heart of the

matter than at that earlier period of our inquiry, and to show, not

merely what the operation in question is not, but what it is.

§ 4. We observed, in the second chapter, that the proposition "the earth

moves in an ellipse," so far as it only serves for the colligation or

connecting together of actual observations, (that is, as it only affirms

that the observed positions of the earth may be correctly represented by

as many points in the circumference of an imaginary ellipse,) is not an

induction, but a description: it is an induction, only when it affirms

that the intermediate positions, of which there has been no direct

observation, would be found to correspond to the remaining points of the

same elliptic circumference. Now, though this real induction is one

thing, and the description another, we are in a very different condition

for making the induction before we have obtained the description, and

after it. For inasmuch as the description, like all other descriptions,

contains the assertion of a resemblance between the phenomenon described

and something else; in pointing out something which the series of

observed places of a planet resembles, it points out something in which

the several places themselves agree. If the series of places correspond

to as many points of an ellipse, the places themselves agree in being

situated in that ellipse. We have, therefore, by the same process which

gave us the description, obtained the requisites for an induction by the

Method of Agreement. The successive observed places of the earth being

considered as effects, and its motion as the cause which produces them,

we find that those effects, that is, those places, agree in the

circumstance of being in an ellipse. We conclude that the remaining

effects, the places which have not been observed, agree in the same

circumstance, and that the _law_ of the motion of the earth is motion in

an ellipse.

The Colligation of Facts, therefore, by means of hypotheses, or, as Dr.

Whewell prefers to say, by means of Conceptions, instead of being, as he

supposes, Induction itself, takes its proper place among operations

subsidiary to Induction. All Induction supposes that we have previously

compared the requisite number of individual instances, and ascertained

in what circumstances they agree. The Colligation of Facts is no other

than this preliminary operation. When Kepler, after vainly endeavouring

to connect the observed places of a planet by various hypotheses of

circular motion, at last tried the hypothesis of an ellipse and found it

answer to the phenomena; what he really attempted, first unsuccessfully

and at last successfully, was to discover the circumstance in which all

the observed positions of the planet agreed. And when he in like manner

connected another set of observed facts, the periodic times of the

different planets, by the proposition that the squares of the times are

proportional to the cubes of the distances, what he did was simply to

ascertain the property in which the periodic times of all the different

planets agreed.

Since, therefore, all that is true and to the purpose in Dr. Whewell's

doctrine of Conceptions might be fully expressed by the more familiar

term Hypothesis; and since his Colligation of Facts by means of

appropriate Conceptions, is but the ordinary process of finding by a

comparison of phenomena, in what consists their agreement or

resemblance; I would willingly have confined myself to those better

understood expressions, and persevered to the end in the same abstinence

which I have hitherto observed from ideological discussions; considering

the mechanism of our thoughts to be a topic distinct from and irrelevant

to the principles and rules by which the trustworthiness of the results

of thinking is to be estimated. Since, however, a work of such high

pretensions, and, it must also be said, of so much real merit, has

rested the whole theory of Induction upon such ideological

considerations, it seems necessary for others who follow, to claim for

themselves and their doctrines whatever position may properly belong to

them on the same metaphysical ground. And this is the object of the

succeeding chapter.

CHAPTER II.

OF ABSTRACTION, OR THE FORMATION OF CONCEPTIONS.

§ 1. The metaphysical inquiry into the nature and composition of what

have been called Abstract Ideas, or in other words, of the notions which

answer in the mind to classes and to general names, belongs not to

Logic, but to a different science, and our purpose does not require that

we should enter upon it here. We are only concerned with the universally

acknowledged fact, that such notions or conceptions do exist. The mind

can conceive a multitude of individual things as one assemblage or

class; and general names do really suggest to us certain ideas or mental

representations, otherwise we could not use the names with consciousness

of a meaning. Whether the idea called up by a general name is composed

of the various circumstances in which all the individuals denoted by the

name agree, and of no others, (which is the doctrine of Locke, Brown,

and the Conceptualists;) or whether it be the idea of some one of those

individuals, clothed in its individualizing peculiarities, but with the

accompanying knowledge that those peculiarities are not properties of

the class, (which is the doctrine of Berkeley, Mr. Bailey,[2] and the

modern Nominalists;) or whether (as held by Mr. James Mill) the idea of

the class is that of a miscellaneous assemblage of individuals belonging

to the class; or whether, finally, (what appears to be the truest

opinion,) it be any one or any other of all these, according to the

accidental circumstances of the case; certain it is, that _some_ idea or

mental conception is suggested by a general name, whenever we either

hear it or employ it with consciousness of a meaning. And this, which we

may call if we please a general idea, _represents_ in our minds the

whole class of things to which the name is applied. Whenever we think or

reason concerning the class, we do so by means of this idea. And the

voluntary power which the mind has, of attending to one part of what is

present to it at any moment, and neglecting another part, enables us to

keep our reasonings and conclusions respecting the class unaffected by

anything in the idea or mental image which is not really, or at least

which we do not really believe to be, common to the whole class.[3]

There are, then, such things as general conceptions, or conceptions by

means of which we can think generally: and when we form a set of

phenomena into a class, that is, when we compare them with one another

to ascertain in what they agree, some general conception is implied in

this mental operation. And inasmuch as such a comparison is a necessary

preliminary to Induction, it is most true that Induction could not go on

without general conceptions.

§ 2. But it does not therefore follow that these general conceptions

must have existed in the mind previously to the comparison. It is not a

law of our intellect, that in comparing things with each other and

taking note of their agreement we merely recognise as realized in the

outward world something that we already had in our minds. The conception

originally found its way to us as the _result_ of such a comparison. It

was obtained (in metaphysical phrase) by _abstraction_ from individual

things. These things may be things which we perceived or thought of on

former occasions, but they may also be the things which we are

perceiving or thinking of on the very occasion. When Kepler compared the

observed places of the planet Mars, and found that they agreed in being

points of an elliptic circumference, he applied a general conception

which was already in his mind, having been derived from his former

experience. But this is by no means universally the case. When we

compare several objects and find them to agree in being white, or when

we compare the various species of ruminating animals and find them to

agree in being cloven-footed, we have just as much a general conception

in our minds as Kepler had in his: we have the conception of "a white

thing," or the conception of "a cloven-footed animal." But no one

supposes that we necessarily bring these conceptions with us, and

_superinduce_ them (to adopt Dr. Whewell's expression) upon the facts:

because in these simple cases everybody sees that the very act of

comparison which ends in our connecting the facts by means of the

conception, may be the source from which we derive the conception

itself. If we had never seen any white object or had never seen any

cloven-footed animal before, we should at the same time and by the same

mental act acquire the idea, and employ it for the colligation of the

observed phenomena. Kepler, on the contrary, really had to bring the

idea with him, and superinduce it upon the facts; he could not evolve it

out of them: if he had not already had the idea, he would not have been

able to acquire it by a comparison of the planet's positions. But this

inability was a mere accident: the idea of an ellipse could have been

acquired from the paths of the planets as effectually as from anything

else, if the paths had not happened to be invisible. If the planet had

left a visible track, and we had been so placed that we could see it at

the proper angle, we might have abstracted our original idea of an

ellipse from the planetary orbit. Indeed, every conception which can be

made the instrument for connecting a set of facts, might have been

originally evolved from those very facts. The conception is a conception

_of_ something; and that which it is a conception of, is really _in_ the

facts, and might, under some supposable circumstances, or by some

supposable extension of the faculties which we actually possess, have

been detected in them. And not only is this always in itself possible,

but it actually happens, in almost all cases in which the obtaining of

the right conception is a matter of any considerable difficulty. For if

there be no new conception required; if one of those already familiar to

mankind will serve the purpose, the accident of being the first to whom

the right one occurs, may happen to almost anybody; at least in the case

of a set of phenomena which the whole scientific world are engaged in

attempting to connect. The honour, in Kepler's case, was that of the

accurate, patient, and toilsome calculations by which he compared the

results that followed from his different guesses, with the observations

of Tycho Brahe; but the merit was very small of guessing an ellipse; the

only wonder is that men had not guessed it before, nor could they have

failed to do so if there had not existed an obstinate _à priori_

prejudice that the heavenly bodies must move, if not in a circle, in

some combination of circles.

The really difficult cases are those in which the conception destined to

create light and order out of darkness and confusion, has to be sought

for among the very phenomena which it afterwards serves to arrange. Why,

according to Dr. Whewell himself, did the ancients fail in discovering

the laws of mechanics, that is, of equilibrium and of the communication

of motion? Because they had not, or at least had not clearly, the ideas

or conceptions of pressure and resistance, momentum, and uniform and

accelerating force. And whence could they have obtained these ideas,

except from the very facts of equilibrium and motion? The tardy

development of several of the physical sciences, for example of optics,

electricity, magnetism, and the higher generalizations of chemistry, he

ascribes to the fact that mankind had not yet possessed themselves of

the Idea of Polarity, that is, the idea of opposite properties in

opposite directions. But what was there to suggest such an idea, until,

by a separate examination of several of these different branches of

knowledge, it was shown that the facts of each of them did present, in

some instances at least, the curious phenomenon of opposite properties

in opposite directions? The thing was superficially manifest only in two

cases, those of the magnet, and of electrified bodies; and there the

conception was encumbered with the circumstance of material poles, or

fixed points in the body itself, in which points this opposition of

properties seemed to be inherent. The first comparison and abstraction

had led only to this conception of poles; and if anything corresponding

to that conception had existed in the phenomena of chemistry or optics,

the difficulty now justly considered so great, would have been extremely

small. The obscurity rose from the fact, that the polarities in

chemistry and optics were distinct species, though of the same genus,

with the polarities in electricity and magnetism: and that in order to

assimilate the phenomena to one another, it was necessary to compare a

polarity without poles, such for instance as is exemplified in the

polarization of light, and the polarity with (apparent) poles, which we

see in the magnet; and to recognise that these polarities, while

different in many other respects, agree in the one character which is

expressed by the phrase, opposite properties in opposite directions.

From the result of such a comparison it was that the minds of scientific

men formed this new general conception: between which, and the first

confused feeling of an analogy between some of the phenomena of light

and those of electricity and magnetism, there is a long interval, filled

up by the labours and more or less sagacious suggestions of many

superior minds.

The conceptions, then, which we employ for the colligation and

methodization of facts, do not develop themselves from within, but are

impressed upon the mind from without; they are never obtained otherwise

than by way of comparison and abstraction, and, in the most important

and the most numerous cases, are evolved by abstraction from the very

phenomena which it is their office to colligate. I am far, however, from

wishing to imply that it is not often a very difficult thing to perform

this process of abstraction well, or that the success of an inductive

operation does not, in many cases, principally depend on the skill with

which we perform it. Bacon was quite justified in designating as one of

the principal obstacles to good induction, general conceptions wrongly

formed, "notiones temerè à rebus abstractæ:" to which Dr. Whewell adds,

that not only does bad abstraction make bad induction, but that in order

to perform induction well, we must have abstracted well; our general

conceptions must be "clear" and "appropriate" to the matter in hand.

§ 3. In attempting to show what the difficulty in this matter really is,

and how it is surmounted, I must beg the reader, once for all, to bear

this in mind; that although in discussing the opinions of a different

school of philosophy, I am willing to adopt their language, and to

speak, therefore, of connecting facts through the instrumentality of a

conception, this technical phraseology means neither more nor less than

what is commonly called comparing the facts with one another and

determining in what they agree. Nor has the technical expression even

the advantage of being metaphysically correct. The facts are not

_connected_, except in a merely metaphorical acceptation of the term.

The _ideas_ of the facts may become connected, that is, we may be led to

think of them together; but this consequence is no more than what may be

produced by any casual association. What really takes place, is, I

conceive, more philosophically expressed by the common word Comparison,

than by the phrases "to connect" or "to superinduce." For, as the

general conception is itself obtained by a comparison of particular

phenomena, so, when obtained, the mode in which we apply it to other

phenomena is again by comparison. We compare phenomena with each other

to get the conception, and we then compare those and other phenomena

_with_ the conception. We get the conception of an animal (for instance)

by comparing different animals, and when we afterwards see a creature

resembling an animal, we compare it with our general conception of an

animal; and if it agrees with that general conception, we include it in

the class. The conception becomes the type of comparison.

And we need only consider what comparison is, to see that where the

objects are more than two, and still more when they are an indefinite

number, a type of some sort is an indispensable condition of the

comparison. When we have to arrange and classify a great number of

objects according to their agreements and differences, we do not make a

confused attempt to compare all with all. We know that two things are as

much as the mind can easily attend to at a time, and we therefore fix

upon one of the objects, either at hazard or because it offers in a

peculiarly striking manner some important character, and, taking this as

our standard, compare it with one object after another. If we find a

second object which presents a remarkable agreement with the first,

inducing us to class them together, the question instantly arises, in

what particular circumstances do they agree? and to take notice of these

circumstances is already a first stage of abstraction, giving rise to a

general conception. Having advanced thus far, when we now take in hand a

third object we naturally ask ourselves the question, not merely whether

this third object agrees with the first, but whether it agrees with it

in the same circumstances in which the second did? in other words,

whether it agrees with the general conception which has been obtained by

abstraction from the first and second? Thus we see the tendency of

general conceptions, as soon as formed, to substitute themselves as

types, for whatever individual objects previously answered that purpose

in our comparisons. We may, perhaps, find that no considerable number of

other objects agree with this first general conception; and that we must

drop the conception, and beginning again with a different individual

case, proceed by fresh comparisons to a different general conception.

Sometimes, again, we find that the same conception will serve, by merely

leaving out some of its circumstances; and by this higher effort of

abstraction, we obtain a still more general conception; as in the case

formerly referred to, the scientific world rose from the conception of

poles to the general conception of opposite properties in opposite

directions; or as those South-Sea islanders, whose conception of a

quadruped had been abstracted from hogs (the only animals of that

description which they had seen), when they afterwards compared that

conception with other quadrupeds, dropped some of the circumstances,

and arrived at the more general conception which Europeans associate

with the term.

These brief remarks contain, I believe, all that is well-grounded in the

doctrine, that the conception by which the mind arranges and gives unity

to phenomena must be furnished by the mind itself, and that we find the

right conception by a tentative process, trying first one and then

another until we hit the mark. The conception is not furnished _by_ the

mind until it has been furnished _to_ the mind; and the facts which

supply it are sometimes extraneous facts, but more often the very facts

which we are attempting to arrange by it. It is quite true, however,

that in endeavouring to arrange the facts, at whatever point we begin,

we never advance three steps without forming a general conception, more

or less distinct and precise; and that this general conception becomes

the clue which we instantly endeavour to trace through the rest of the

facts, or rather, becomes the standard with which we thenceforth compare

them. If we are not satisfied with the agreements which we discover

among the phenomena by comparing them with this type, or with some still

more general conception which by an additional stage of abstraction we

can form from the type; we change our path, and look out for other

agreements: we recommence the comparison from a different

starting-point, and so generate a different set of general conceptions.

This is the tentative process which Dr. Whewell speaks of; and which has

not unnaturally suggested the theory, that the conception is supplied by

the mind itself: since the different conceptions which the mind

successively tries, it either already possessed from its previous

experience, or they were supplied to it in the first stage of the

corresponding act of comparison; so that, in the subsequent part of the

process, the conception manifested itself as something compared with the

phenomena, not evolved from them.

§ 4. If this be a correct account of the instrumentality of general

conceptions in the comparison which necessarily precedes Induction, we

shall easily be able to translate into our own language what Dr. Whewell

means by saying that conceptions, to be subservient to Induction, must

be "clear" and "appropriate."

If the conception corresponds to a real agreement among the phenomena;

if the comparison which we have made of a set of objects has led us to

class them according to real resemblances and differences; the

conception which does this cannot fail to be appropriate, for some

purpose or other. The question of appropriateness is relative to the

particular object we have in view. As soon as, by our comparison, we

have ascertained some agreement, something which can be predicated in

common of a number of objects; we have obtained a basis on which an

inductive process is capable of being founded. But the agreements, or

the ulterior consequences to which those agreements lead, may be of very

different degrees of importance. If, for instance, we only compare

animals according to their colour, and class those together which are

coloured alike, we form the general conceptions of a white animal, a

black animal, &c., which are conceptions legitimately formed; and if an

induction were to be attempted concerning the causes of the colours of

animals, this comparison would be the proper and necessary preparation

for such an induction, but would not help us towards a knowledge of the

laws of any other of the properties of animals: while if, with Cuvier,

we compare and class them according to the structure of the skeleton,

or, with Blainville, according to the nature of their outward

integuments, the agreements and differences which are observable in

these respects are not only of much greater importance in themselves,

but are marks of agreements and differences in many other important

particulars of the structure and mode of life of the animals. If,

therefore, the study of their structure and habits be our object, the

conceptions generated by these last comparisons are far more

"appropriate" than those generated by the former. Nothing, other than

this, can be meant by the appropriateness of a conception.

When Dr. Whewell says that the ancients, or the schoolmen, or any modern

inquirers, missed discovering the real law of a phenomenon because they

applied to it an inappropriate instead of an appropriate conception; he

can only mean that in comparing various instances of the phenomenon, to

ascertain in what those instances agreed, they missed the important

points of agreement; and fastened upon such as were either imaginary,

and not agreements at all, or if real agreements, were comparatively

trifling, and had no connexion with the phenomenon, the law of which was

sought.

Aristotle, philosophizing on the subject of motion, remarked that

certain motions apparently take place spontaneously; bodies fall to the

ground, flame ascends, bubbles of air rise in water, &c.: and these he

called natural motions; while others not only never take place without

external incitement, but even when such incitement is applied, tend

spontaneously to cease; which, to distinguish them from the former, he

called violent motions. Now, in comparing the so-called natural motions

with one another, it appeared to Aristotle that they agreed in one

circumstance, namely, that the body which moved (or seemed to move)

spontaneously, was moving _towards its own place_; meaning thereby the

place from whence it originally came, or the place where a great

quantity of matter similar to itself was assembled. In the other class

of motions, as when bodies are thrown up in the air, they are, on the

contrary, moving _from_ their own place. Now, this conception of a body

moving towards its own place may justly be considered inappropriate;

because, though it expresses a circumstance really found in some of the

most familiar instances of motion apparently spontaneous, yet, first,

there are many other cases of such motion, in which that circumstance is

absent: the motion, for instance, of the earth and planets. Secondly,

even when it is present, the motion, on closer examination, would often

be seen not to be spontaneous: as, when air rises in water, it does not

rise by its own nature, but is pushed up by the superior weight of the

water which presses upon it. Finally, there are many cases in which the

spontaneous motion takes place in the contrary direction to what the

theory considers as the body's own place; for instance, when a fog rises

from a lake, or when water dries up. The agreement, therefore, which

Aristotle selected as his principle of classification, did not extend to

all cases of the phenomenon he wanted to study, spontaneous motion;

while it did include cases of the absence of the phenomenon, cases of

motion not spontaneous. The conception was hence "inappropriate." We may

add that, in the case in question, no conception would be appropriate;

there is no agreement which runs through all the cases of spontaneous or

apparently spontaneous motion and no others: they cannot be brought

under one law: it is a case of Plurality of Causes.[4]

§ 5. So much for the first of Dr. Whewell's conditions, that conceptions

must be appropriate. The second is, that they shall be "clear:" and let

us consider what this implies. Unless the conception corresponds to a

real agreement, it has a worse defect than that of not being clear; it

is not applicable to the case at all. Among the phenomena, therefore,

which we are attempting to connect by means of the conception, we must

suppose that there really is an agreement, and that the conception is a

conception of that agreement. In order, then, that it may be clear, the

only requisite is, that we shall know exactly in what the agreement

consists; that it shall have been carefully observed, and accurately

remembered. We are said not to have a clear conception of the

resemblance among a set of objects, when we have only a general feeling

that they resemble, without having analysed their resemblance, or

perceived in what points it consists, and fixed in our memory an exact

recollection of those points. This want of clearness, or, as it may be

otherwise called, this vagueness, in the general conception, may be

owing either to our having no accurate knowledge of the objects

themselves, or merely to our not having carefully compared them. Thus a

person may have no clear idea of a ship because he has never seen one,

or because he remembers but little, and that faintly, of what he has

seen. Or he may have a perfect knowledge and remembrance of many ships

of various kinds, frigates among the rest, but he may have no clear but

only a confused idea of a frigate, because he has never been told, and

has not compared them sufficiently to have remarked and remembered, in

what particular points a frigate differs from some other kind of ship.

It is not, however, necessary, in order to have clear ideas, that we

should know all the common properties of the things which we class

together. That would be to have our conception of the class complete as

well as clear. It is sufficient if we never class things together

without knowing exactly why we do so,--without having ascertained

exactly what agreements we are about to include in our conception; and

if, after having thus fixed our conception, we never vary from it, never

include in the class anything which has not those common properties, nor

exclude from it anything which has. A clear conception means a

determinate conception; one which does not fluctuate, which is not one

thing to-day and another to-morrow, but remains fixed and invariable,

except when, from the progress of our knowledge, or the correction of

some error, we consciously add to it or alter it. A person of clear

ideas, is a person who always knows in virtue of what properties his

classes are constituted; what attributes are connoted by his general

names.

The principal requisites, therefore, of clear conceptions, are habits of

attentive observation, an extensive experience, and a memory which

receives and retains an exact image of what is observed. And in

proportion as any one has the habit of observing minutely and comparing

carefully a particular class of phenomena, and an accurate memory for

the results of the observation and comparison, so will his conceptions

of that class of phenomena be clear; provided he has the indispensable

habit, (naturally, however, resulting from those other endowments,) of

never using general names without a precise connotation.

As the clearness of our conceptions chiefly depends on the _carefulness_

and _accuracy_ of our observing and comparing faculties, so their

appropriateness, or rather the chance we have of hitting upon the

appropriate conception in any case, mainly depends on the _activity_ of

the same faculties. He who by habit, grounded on sufficient natural

aptitude, has acquired a readiness in accurately observing and comparing

phenomena, will perceive so many more agreements and will perceive them

so much more rapidly than other people, that the chances are much

greater of his perceiving, in any instance, the agreement on which the

important consequences depend.

§ 6. It is of so much importance that the part of the process of

investigating truth, discussed in this chapter, should be rightly

understood, that I think it is desirable to restate the results we have

arrived at, in a somewhat different mode of expression.

We cannot ascertain general truths, that is, truths applicable to

classes, unless we have formed the classes in such a manner that general

truths can be affirmed of them. In the formation of any class, there is

involved a conception of it as a class, that is, a conception of certain

circumstances as being those which characterize the class, and

distinguish the objects composing it from all other things. When we know

exactly what these circumstances are, we have a clear idea (or

conception) of the class, and of the meaning of the general name which

designates it. The primary condition implied in having this clear idea,

is that the class be really a class; that it correspond to a real

distinction; that the things it includes really do agree with one

another in certain particulars, and differ, in those same particulars,

from all other things. A person without clear ideas, is one who

habitually classes together, under the same general names, things which

have no common properties, or none which are not possessed also by other

things; or who, if the usage of other people prevents him from actually

misclassing things, is unable to state to himself the common properties

in virtue of which he classes them rightly.

But it is not the sole requisite of classification that the classes

should be real classes, framed by a legitimate mental process. Some

modes of classing things are more valuable than others for human uses,

whether of speculation or of practice; and our classifications are not

well made, unless the things which they bring together not only agree

with each other in something which distinguishes them from all other

things, but agree with each other and differ from other things in the

very circumstances which are of primary importance for the purpose

(theoretical or practical) which we have in view, and which constitutes

the problem before us. In other words, our conceptions, though they may

be clear, are not _appropriate_ for our purpose, unless the properties

we comprise in them are those which will help us towards what we wish to

understand--_i. e._, either those which go deepest into the nature of

the things, if our object be to understand that, or those which are most

closely connected with the particular property which we are endeavouring

to investigate.

We cannot, therefore, frame good general conceptions beforehand. That

the conception we have obtained is the one we want, can only be known

when we have done the work for the sake of which we wanted it; when we

completely understand the general character of the phenomena, or the

conditions of the particular property with which we concern ourselves.

General conceptions formed without this thorough knowledge, are Bacon's

"notiones temerè à rebus abstractæ." Yet such premature conceptions we

must be continually making up, in our progress to something better. They

are an impediment to the progress of knowledge, only when they are

permanently acquiesced in. When it has become our habit to group things

in wrong classes--in groups which either are not really classes, having

no distinctive points of agreement (absence of _clear_ ideas), or which

are not classes of which anything important to our purpose can be

predicated (absence of _appropriate_ ideas); and when, in the belief

that these badly made classes are those sanctioned by Nature, we refuse

to exchange them for others, and cannot or will not make up our general

conceptions from any other elements; in that case all the evils which

Bacon ascribes to his "notiones temerè abstractæ" really occur. This was

what the ancients did in physics, and what the world in general does in

morals and politics to the present day.

It would thus, in my view of the matter, be an inaccurate mode of

expression to say, that obtaining appropriate conceptions is a condition

precedent to generalization. Throughout the whole process of comparing

phenomena with one another for the purpose of generalization, the mind

is trying to make up a conception; but the conception which it is trying

to make up is that of the really important point of agreement in the

phenomena. As we obtain more knowledge of the phenomena themselves, and

of the conditions on which their important properties depend, our views

on this subject naturally alter; and thus we advance from a less to a

more "appropriate" general conception, in the progress of our

investigations.

We ought not, at the same time, to forget that the really important

agreement cannot always be discovered by mere comparison of the very

phenomena in question, without the aid of a conception acquired

elsewhere; as in the case, so often referred to, of the planetary

orbits.

The search for the agreement of a set of phenomena is in truth very

similar to the search for a lost or hidden object. At first we place

ourselves in a sufficiently commanding position, and cast our eyes round

us, and if we can see the object it is well; if not, we ask ourselves

mentally what are the places in which it may be hid, in order that we

may there search for it: and so on, until we imagine the place where it

really is. And here too we require to have had a previous conception, or

knowledge, of those different places. As in this familiar process, so in

the philosophical operation which it illustrates, we first endeavour to

find the lost object or recognise the common attribute, without

conjecturally invoking the aid of any previously acquired conception, or

in other words, of any hypothesis. Having failed in this, we call upon

our imagination for some hypothesis of a possible place, or a possible

point of resemblance, and then look, to see whether the facts agree with

the conjecture.