PART I

CHANCE AND LOGIC

(ILLUSTRATIONS OF THE LOGIC OF SCIENCE)

CHANCE AND LOGIC

FIRST PAPER

THE FIXATION OF BELIEF[26]

I

Few persons care to study logic, because everybody conceives himself to

be proficient enough in the art of reasoning already. But I observe that

this satisfaction is limited to one’s own ratiocination, and does not

extend to that of other men.

We come to the full possession of our power of drawing inferences the

last of all our faculties, for it is not so much a natural gift as a

long and difficult art. The history of its practice would make a grand

subject for a book. The medieval schoolman, following the Romans, made

logic the earliest of a boy’s studies after grammar, as being very easy.

So it was as they understood it. Its fundamental principle, according to

them, was, that all knowledge rests on either authority or reason; but

that whatever is deduced by reason depends ultimately on a premise

derived from authority. Accordingly, as soon as a boy was perfect in the

syllogistic procedure, his intellectual kit of tools was held to be

complete.

To Roger Bacon, that remarkable mind who in the middle of the thirteenth

century was almost a scientific man, the schoolmen’s conception of

reasoning appeared only an obstacle to truth. He saw that experience

alone teaches anything—a proposition which to us seems easy to

understand, because a distinct conception of experience has been handed

down to us from former generations; which to him also seemed perfectly

clear, because its difficulties had not yet unfolded themselves. Of all

kinds of experience, the best, he thought, was interior illumination,

which teaches many things about Nature which the external senses could

never discover, such as the transubstantiation of bread.

Four centuries later, the more celebrated Bacon, in the first book of

his “Novum Organum,” gave his clear account of experience as something

which must be open to verification and reëxamination. But, superior as

Lord Bacon’s conception is to earlier notions, a modern reader who is

not in awe of his grandiloquence is chiefly struck by the inadequacy of

his view of scientific procedure. That we have only to make some crude

experiments, to draw up briefs of the results in certain blank forms, to

go through these by rule, checking off everything disproved and setting

down the alternatives, and that thus in a few years physical science

would be finished up—what an idea! “He wrote on science like a Lord

Chancellor,”[27] indeed.

The early scientists, Copernicus, Tycho Brahe, Kepler, Galileo and

Gilbert, had methods more like those of their modern brethren. Kepler

undertook to draw a curve through the places of Mars;[28] and his

greatest service to science was in impressing on men’s minds that this

was the thing to be done if they wished to improve astronomy; that they

were not to content themselves with inquiring whether one system of

epicycles was better than another but that they were to sit down by the

figures and find out what the curve, in truth, was. He accomplished this

by his incomparable energy and courage, blundering along in the most

inconceivable way (to us), from one irrational hypothesis to another,

until, after trying twenty-two of these, he fell, by the mere exhaustion

of his invention, upon the orbit which a mind well furnished with the

weapons of modern logic would have tried almost at the outset.[29]

In the same way, every work of science great enough to be remembered for

a few generations affords some exemplification of the defective state of

the art of reasoning of the time when it was written; and each chief

step in science has been a lesson in logic. It was so when Lavoisier and

his contemporaries took up the study of Chemistry. The old chemist’s

maxim had been, “Lege, lege, lege, labora, ora, et relege.” Lavoisier’s

method was not to read and pray, not to dream that some long and

complicated chemical process would have a certain effect, to put it into

practice with dull patience, after its inevitable failure to dream that

with some modification it would have another result, and to end by

publishing the last dream as a fact: his way was to carry his mind into

his laboratory, and to make of his alembics and cucurbits instruments of

thought, giving a new conception of reasoning as something which was to

be done with one’s eyes open, by manipulating real things instead of

words and fancies.

The Darwinian controversy is, in large part, a question of logic. Mr.

Darwin proposed to apply the statistical method to biology. The same

thing has been done in a widely different branch of science, the theory

of gases. Though unable to say what the movement of any particular

molecule of gas would be on a certain hypothesis regarding the

constitution of this class of bodies, Clausius and Maxwell were yet

able, by the application of the doctrine of probabilities, to predict

that in the long run such and such a proportion of the molecules would,

under given circumstances, acquire such and such velocities; that there

would take place, every second, such and such a number of collisions,

etc.; and from these propositions they were able to deduce certain

properties of gases, especially in regard to their heat-relations. In

like manner, Darwin, while unable to say what the operation of variation

and natural selection in every individual case will be, demonstrates

that in the long run they will adapt animals to their circumstances.

Whether or not existing animal forms are due to such action, or what

position the theory ought to take, forms the subject of a discussion in

which questions of fact and questions of logic are curiously interlaced.

II

The object of reasoning is to find out, from the consideration of what

we already know, something else which we do not know. Consequently,

reasoning is good if it be such as to give a true conclusion from true

premises, and not otherwise. Thus, the question of validity is purely

one of fact and not of thinking. A being the premises and B being the

conclusion, the question is, whether these facts are really so related

that if A is B is. If so, the inference is valid; if not, not. It is not

in the least the question whether, when the premises are accepted by the

mind, we feel an impulse to accept the conclusion also. It is true that

we do generally reason correctly by nature. But that is an accident; the

true conclusion would remain true if we had no impulse to accept it; and

the false one would remain false, though we could not resist the

tendency to believe in it.

We are, doubtless, in the main logical animals, but we are not perfectly

so. Most of us, for example, are naturally more sanguine and hopeful

than logic would justify. We seem to be so constituted that in the

absence of any facts to go upon we are happy and self-satisfied; so that

the effect of experience is continually to counteract our hopes and

aspirations. Yet a lifetime of the application of this corrective does

not usually eradicate our sanguine disposition. Where hope is unchecked

by any experience, it is likely that our optimism is extravagant.

Logicality in regard to practical matters is the most useful quality an

animal can possess, and might, therefore, result from the action of

natural selection; but outside of these it is probably of more advantage

to the animal to have his mind filled with pleasing and encouraging

visions, independently of their truth; and thus, upon unpractical

subjects, natural selection might occasion a fallacious tendency of

thought.

That which determines us, from given premises, to draw one inference

rather than another, is some habit of mind, whether it be constitutional

or acquired. The habit is good or otherwise, according as it produces

true conclusions from true premises or not; and an inference is regarded

as valid or not, without reference to the truth or falsity of its

conclusion specially, but according as the habit which determines it is

such as to produce true conclusions in general or not. The particular

habit of mind which governs this or that inference may be formulated in

a proposition whose truth depends on the validity of the inferences

which the habit determines; and such a formula is called a _guiding

principle_ of inference. Suppose, for example, that we observe that a

rotating disk of copper quickly comes to rest when placed between the

poles of a magnet, and we infer that this will happen with every disk of

copper. The guiding principle is, that what is true of one piece of

copper is true of another. Such a guiding principle with regard to

copper would be much safer than with regard to many other

substances—brass, for example.

A book might be written to signalize all the most important of these

guiding principles of reasoning. It would probably be, we must confess,

of no service to a person whose thought is directed wholly to practical

subjects, and whose activity moves along thoroughly beaten paths. The

problems which present themselves to such a mind are matters of routine

which he has learned once for all to handle in learning his business.

But let a man venture into an unfamiliar field, or where his results are

not continually checked by experience, and all history shows that the

most masculine intellect will ofttimes lose his orientation and waste

his efforts in directions which bring him no nearer to his goal, or even

carry him entirely astray. He is like a ship on the open sea, with no

one on board who understands the rules of navigation. And in such a case

some general study of the guiding principles of reasoning would be sure

to be found useful.

The subject could hardly be treated, however, without being first

limited; since almost any fact may serve as a guiding principle. But it

so happens that there exists a division among facts, such that in one

class are all those which are absolutely essential as guiding

principles, while in the other are all those which have any other

interest as objects of research. This division is between those which

are necessarily taken for granted in asking whether a certain conclusion

follows from certain premises, and those which are not implied in that

question. A moment’s thought will show that a variety of facts are

already assumed when the logical question is first asked. It is implied,

for instance, that there are such states of mind as doubt and

belief—that a passage from one to the other is possible, the object of

thought remaining the same, and that this transition is subject to some

rules which all minds are alike bound by. As these are facts which we

must already know before we can have any clear conception of reasoning

at all, it cannot be supposed to be any longer of much interest to

inquire into their truth or falsity. On the other hand, it is easy to

believe that those rules of reasoning which are deduced from the very

idea of the process are the ones which are the most essential; and,

indeed, that so long as it conforms to these it will, at least, not lead

to false conclusions from true premises. In point of fact, the

importance of what may be deduced from the assumptions involved in the

logical question turns out to be greater than might be supposed, and

this for reasons which it is difficult to exhibit at the outset. The

only one which I shall here mention is, that conceptions which are

really products of logical reflections, without being readily seen to be

so, mingle with our ordinary thoughts, and are frequently the causes of

great confusion. This is the case, for example, with the conception of

quality. A quality as such is never an object of observation. We can see

that a thing is blue or green, but the quality of being blue and the

quality of being green are not things which we see; they are products of

logical reflections. The truth is, that common-sense, or thought as it

first emerges above the level of the narrowly practical, is deeply

imbued with that bad logical quality to which the epithet _metaphysical_

is commonly applied; and nothing can clear it up but a severe course of

logic.

III

We generally know when we wish to ask a question and when we wish to

pronounce a judgment, for there is a dissimilarity between the sensation

of doubting and that of believing.

But this is not all which distinguishes doubt from belief. There is a

practical difference. Our beliefs guide our desires and shape our

actions. The Assassins, or followers of the Old Man of the Mountain,

used to rush into death at his least command, because they believed that

obedience to him would insure everlasting felicity. Had they doubted

this, they would not have acted as they did. So it is with every belief,

according to its degree. The feeling of believing is a more or less sure

indication of there being established in our nature some habit which

will determine our actions. Doubt never has such an effect.

Nor must we overlook a third point of difference. Doubt is an uneasy and

dissatisfied state from which we struggle to free ourselves and pass

into the state of belief; while the latter is a calm and satisfactory

state which we do not wish to avoid, or to change to a belief in

anything else.[30] On the contrary, we cling tenaciously, not merely to

believing, but to believing just what we do believe.

Thus, both doubt and belief have positive effects upon us, though very

different ones. Belief does not make us act at once, but puts us into

such a condition that we shall behave in a certain way, when the

occasion arises. Doubt has not the least effect of this sort, but

stimulates us to action until it is destroyed. This reminds us of the

irritation of a nerve and the reflex action produced thereby; while for

the analogue of belief, in the nervous system, we must look to what are

called nervous associations—for example, to that habit of the nerves in

consequence of which the smell of a peach will make the mouth water.

IV

The irritation of doubt causes a struggle to attain a state of belief. I

shall term this struggle _inquiry_, though it must be admitted that this

is sometimes not a very apt designation.

The irritation of doubt is the only immediate motive for the struggle to

attain belief. It is certainly best for us that our beliefs should be

such as may truly guide our actions so as to satisfy our desires; and

this reflection will make us reject any belief which does not seem to

have been so formed as to insure this result. But it will only do so by

creating a doubt in the place of that belief. With the doubt, therefore,

the struggle begins, and with the cessation of doubt it ends. Hence, the

sole object of inquiry is the settlement of opinion. We may fancy that

this is not enough for us, and that we seek not merely an opinion, but a

true opinion. But put this fancy to the test, and it proves groundless;

for as soon as a firm belief is reached we are entirely satisfied,

whether the belief be false or true. And it is clear that nothing out of

the sphere of our knowledge can be our object, for nothing which does

not affect the mind can be a motive for a mental effort. The most that

can be maintained is, that we seek for a belief that we shall _think_ to

be true. But we think each one of our beliefs to be true, and, indeed,

it is mere tautology to say so.

That the settlement of opinion is the sole end of inquiry is a very

important proposition. It sweeps away, at once, various vague and

erroneous conceptions of proof. A few of these may be noticed here.

1. Some philosophers have imagined that to start an inquiry it was only

necessary to utter or question or set it down on paper, and have even

recommended us to begin our studies with questioning everything! But the

mere putting of a proposition into the interrogative form does not

stimulate the mind to any struggle after belief. There must be a real

and living doubt, and without all this discussion is idle.

2. It is a very common idea that a demonstration must rest on some

ultimate and absolutely indubitable propositions. These, according to

one school, are first principles of a general nature; according to

another, are first sensations. But, in point of fact, an inquiry, to

have that completely satisfactory result called demonstration, has only

to start with propositions perfectly free from all actual doubt. If the

premises are not in fact doubted at all, they cannot be more

satisfactory than they are.

3. Some people seem to love to argue a point after all the world is

fully convinced of it. But no further advance can be made. When doubt

ceases, mental action on the subject comes to an end; and, if it did go

on, it would be without a purpose.

V

If the settlement of opinion is the sole object of inquiry, and if

belief is of the nature of a habit, why should we not attain the desired

end, by taking any answer to a question, which we may fancy, and

constantly reiterating it to ourselves, dwelling on all which may

conduce to that belief, and learning to turn with contempt and hatred

from anything which might disturb it? This simple and direct method is

really pursued by many men. I remember once being entreated not to read

a certain newspaper lest it might change my opinion upon free-trade.

“Lest I might be entrapped by its fallacies and misstatements,” was the

form of expression. “You are not,” my friend said, “a special student of

political economy. You might, therefore, easily be deceived by

fallacious arguments upon the subject. You might, then, if you read this

paper, be led to believe in protection. But you admit that free-trade is

the true doctrine; and you do not wish to believe what is not true.” I

have often known this system to be deliberately adopted. Still oftener,

the instinctive dislike of an undecided state of mind, exaggerated into

a vague dread of doubt, makes men cling spasmodically to the views they

already take. The man feels that, if he only holds to his belief without

wavering, it will be entirely satisfactory. Nor can it be denied that a

steady and immovable faith yields great peace of mind. It may, indeed,

give rise to inconveniences, as if a man should resolutely continue to

believe that fire would not burn him, or that he would be eternally

damned if he received his _ingesta_ otherwise than through a

stomach-pump. But then the man who adopts this method will not allow

that its inconveniences are greater than its advantages. He will say, “I

hold steadfastly to the truth and the truth is always wholesome.” And in

many cases it may very well be that the pleasure he derives from his

calm faith overbalances any inconveniences resulting from its deceptive

character. Thus, if it be true that death is annihilation, then the man

who believes that he will certainly go straight to heaven when he dies,

provided he have fulfilled certain simple observances in this life, has

a cheap pleasure which will not be followed by the least disappointment.

A similar consideration seems to have weight with many persons in

religious topics, for we frequently hear it said, “Oh, I could not

believe so-and-so, because I should be wretched if I did.” When an

ostrich buries its head in the sand as danger approaches, it very likely

takes the happiest course. It hides the danger, and then calmly says

there is no danger; and, if it feels perfectly sure there is none, why

should it raise its head to see? A man may go through life,

systematically keeping out of view all that might cause a change in his

opinions, and if he only succeeds—basing his method, as he does, on two

fundamental psychological laws—I do not see what can be said against his

doing so. It would be an egotistical impertinence to object that his

procedure is irrational, for that only amounts to saying that his method

of settling belief is not ours. He does not propose to himself to be

rational, and indeed, will often talk with scorn of man’s weak and

illusive reason. So let him think as he pleases.

But this method of fixing belief, which may be called the method of

tenacity, will be unable to hold its ground in practice. The social

impulse is against it. The man who adopts it will find that other men

think differently from him, and it will be apt to occur to him in some

saner moment that their opinions are quite as good as his own, and this

will shake his confidence in his belief. This conception, that another

man’s thought or sentiment may be equivalent to one’s own, is a

distinctly new step, and a highly important one. It arises from an

impulse too strong in man to be suppressed, without danger of destroying

the human species. Unless we make ourselves hermits, we shall

necessarily influence each other’s opinions; so that the problem becomes

how to fix belief, not in the individual merely, but in the community.

Let the will of the state act, then, instead of that of the individual.

Let an institution be created which shall have for its object to keep

correct doctrines before the attention of the people, to reiterate them

perpetually, and to teach them to the young; having at the same time

power to prevent contrary doctrines from being taught, advocated, or

expressed. Let all possible causes of a change of mind be removed from

men’s apprehensions. Let them be kept ignorant, lest they should learn

of some reason to think otherwise than they do. Let their passions be

enlisted, so that they may regard private and unusual opinions with

hatred and horror. Then, let all men who reject the established belief

be terrified into silence. Let the people turn out and tar-and-feather

such men, or let inquisitions be made into the manner of thinking of

suspected persons, and, when they are found guilty of forbidden beliefs,

let them be subjected to some signal punishment. When complete agreement

could not otherwise be reached, a general massacre of all who have not

thought in a certain way has proved a very effective means of settling

opinion in a country. If the power to do this be wanting, let a list of

opinions be drawn up, to which no man of the least independence of

thought can assent, and let the faithful be required to accept all these

propositions, in order to segregate them as radically as possible from

the influence of the rest of the world.

This method has, from the earliest times, been one of the chief means of

upholding correct theological and political doctrines, and of preserving

their universal or catholic character. In Rome, especially, it has been

practiced from the days of Numa Pompilius to those of Pius Nonus. This

is the most perfect example in history; but wherever there is a

priesthood—and no religion has been without one—this method has been

more or less made use of. Wherever there is aristocracy, or a guild, or

any association of a class of men whose interests depend or are supposed

to depend on certain propositions, there will be inevitably found some

traces of this natural product of social feeling. Cruelties always

accompany this system; and when it is consistently carried out, they

become atrocities of the most horrible kind in the eyes of any rational

man. Nor should this occasion surprise, for the officer of a society

does not feel justified in surrendering the interests of that society

for the sake of mercy, as he might his own private interests. It is

natural, therefore, that sympathy and fellowship should thus produce a

most ruthless power.

In judging this method of fixing belief, which may be called the method

of authority, we must in the first place, allow its immeasurable mental

and moral superiority to the method of tenacity. Its success is

proportionally greater; and in fact it has over and over again worked

the most majestic results. The mere structures of stone which it has

caused to be put together—in Siam, for example, in Egypt, and in

Europe—have many of them a sublimity hardly more than rivaled by the

greatest works of Nature. And, except the geological epochs, there are

no periods of time so vast as those which are measured by some of these

organized faiths. If we scrutinize the matter closely, we shall find

that there has not been one of their creeds which has remained always

the same; yet the change is so slow as to be imperceptible during one

person’s life, so that individual belief remains sensibly fixed. For the

mass of mankind, then, there is perhaps no better method than this. If

it is their highest impulse to be intellectual slaves, then slaves they

ought to remain.

But no institution can undertake to regulate opinions upon every

subject. Only the most important ones can be attended to, and on the

rest men’s minds must be left to the action of natural causes. This

imperfection will be no source of weakness so long as men are in such a

state of culture that one opinion does not influence another—that is, so

long as they cannot put two and two together. But in the most

priest-ridden states some individuals will be found who are raised above

that condition. These men possess a wider sort of social feeling; they

see that men in other countries and in other ages have held to very

different doctrines from those which they themselves have been brought

up to believe; and they cannot help seeing that it is the mere accident

of their having been taught as they have, and of their having been

surrounded with the manners and associations they have, that has caused

them to believe as they do and not far differently. And their candor

cannot resist the reflection that there is no reason to rate their own

views at a higher value than those of other nations and other centuries;

and this gives rise to doubts in their minds.

They will further perceive that such doubts as these must exist in their

minds with reference to every belief which seems to be determined by the

caprice either of themselves or of those who originated the popular

opinions. The willful adherence to a belief, and the arbitrary forcing

of it upon others, must, therefore, both be given up and a new method of

settling opinions must be adopted, which shall not only produce an

impulse to believe, but shall also decide what proposition it is which

is to be believed. Let the action of natural preferences be unimpeded,

then, and under their influence let men conversing together and

regarding matters in different lights, gradually develop beliefs in

harmony with natural causes. This method resembles that by which

conceptions of art have been brought to maturity. The most perfect

example of it is to be found in the history of metaphysical philosophy.

Systems of this sort have not usually rested upon observed facts, at

least not in any great degree. They have been chiefly adopted because

their fundamental propositions seemed “agreeable to reason.” This is an

apt expression; it does not mean that which agrees with experience, but

that which we find ourselves inclined to believe. Plato, for example,

finds it agreeable to reason that the distances of the celestial spheres

from one another should be proportional to the different lengths of

strings which produce harmonious chords. Many philosophers have been led

to their main conclusions by considerations like this; but this is the

lowest and least developed form which the method takes, for it is clear

that another man might find Kepler’s [earlier] theory, that the

celestial spheres are proportional to the inscribed and circumscribed

spheres of the different regular solids, more agreeable to _his_ reason.

But the shock of opinions will soon lead men to rest on preferences of a

far more universal nature. Take, for example, the doctrine that man only

acts selfishly—that is, from the consideration that acting in one way

will afford him more pleasure than acting in another. This rests on no

fact in the world, but it has had a wide acceptance as being the only

reasonable theory.

This method is far more intellectual and respectable from the point of

view of reason than either of the others which we have noticed. But its

failure has been the most manifest. It makes of inquiry something

similar to the development of taste; but taste, unfortunately, is always

more or less a matter of fashion, and accordingly, meta-physicians have

never come to any fixed agreement, but the pendulum has swung backward

and forward between a more material and a more spiritual philosophy,

from the earliest times to the latest. And so from this, which has been

called the _a priori_ method, we are driven, in Lord Bacon’s phrase, to

a true induction. We have examined into this _a priori_ method as

something which promised to deliver our opinions from their accidental

and capricious element. But development, while it is a process which

eliminates the effect of some casual circumstances, only magnifies that

of others. This method, therefore, does not differ in a very essential

way from that of authority. The government may not have lifted its

finger to influence my convictions; I may have been left outwardly quite

free to choose, we will say, between monogamy and polygamy, and

appealing to my conscience only, I may have concluded that the latter

practice is in itself licentious. But when I come to see that the chief

obstacle to the spread of Christianity among a people of as high culture

as the Hindoos has been a conviction of the immorality of our way of

treating women, I cannot help seeing that, though governments do not

interfere, sentiments in their development will be very greatly

determined by accidental causes. Now, there are some people, among whom

I must suppose that my reader is to be found, who, when they see that

any belief of theirs is determined by any circumstance extraneous to the

facts, will from that moment not merely admit in words that that belief

is doubtful, but will experience a real doubt of it, so that it ceases

to be a belief.

To satisfy our doubts, therefore, it is necessary that a method should

be found by which our beliefs may be caused by nothing human, but by

some external permanency—by something upon which our thinking has no

effect. Some mystics imagine that they have such a method in a private

inspiration from on high. But that is only a form of the method of

tenacity, in which the conception of truth as something public is not

yet developed. Our external permanency would not be external, in our

sense, if it was restricted in its influence to one individual. It must

be something which affects, or might affect, every man. And, though

these affections are necessarily as various as are individual

conditions, yet the method must be such that the ultimate conclusion of

every man shall be the same. Such is the method of science. Its

fundamental hypothesis, restated in more familiar language, is this:

There are real things, whose characters are entirely independent of our

opinions about them; whose realities affect our senses according to

regular laws, and, though our sensations are as different as our

relations to the objects, yet, by taking advantage of the laws of

perception, we can ascertain by reasoning how things really are, and any

man, if he have sufficient experience and reason enough about it, will

be led to the one true conclusion. The new conception here involved is

that of reality. It may be asked how I know that there are any

realities. If this hypothesis is the sole support of my method of

inquiry, my method of inquiry must not be used to support my hypothesis.

The reply is this: 1. If investigation cannot be regarded as proving

that there are real things, it at least does not lead to a contrary

conclusion; but the method and the conception on which it is based

remain ever in harmony. No doubts of the method, therefore, necessarily

arise from its practice, as is the case with all the others. 2. The

feeling which gives rise to any method of fixing belief is a

dissatisfaction at two repugnant propositions. But here already is a

vague concession that there is some _one_ thing to which a proposition

should conform. Nobody, therefore, can really doubt that there are

realities, or, if he did, doubt would not be a source of

dissatisfaction. The hypothesis, therefore, is one which every mind

admits. So that the social impulse does not cause me to doubt it. 3.

Everybody uses the scientific method about a great many things, and only

ceases to use it when he does not know how to apply it. 4. Experience of

the method has not led me to doubt it, but, on the contrary, scientific

investigation has had the most wonderful triumphs in the way of settling

opinion. These afford the explanation of my not doubting the method or

the hypothesis which it supposes; and not having any doubt, nor

believing that anybody else whom I could influence has, it would be the

merest babble for me to say more about it. If there be anybody with a

living doubt upon the subject, let him consider it.

To describe the method of scientific investigation is the object of this

series of papers. At present I have only room to notice some points of

contrast between it and other methods of fixing belief.

This is the only one of the four methods which presents any distinction

of a right and a wrong way. If I adopt the method of tenacity and shut

myself out from all influences, whatever I think necessary to doing this

is necessary according to that method. So with the method of authority:

the state may try to put down heresy by means which, from a scientific

point of view, seems very ill-calculated to accomplish its purposes; but

the only test _on that method_ is what the state thinks, so that it

cannot pursue the method wrongly. So with the _a priori_ method. The

very essence of it is to think as one is inclined to think. All

metaphysicians will be sure to do that, however they may be inclined to

judge each other to be perversely wrong. The Hegelian system recognizes

every natural tendency of thought as logical, although it is certain to

be abolished by counter-tendencies. Hegel thinks there is a regular

system in the succession of these tendencies, in consequence of which,

after drifting one way and the other for a long time, opinion will at

last go right. And it is true that metaphysicians get the right ideas at

last; Hegel’s system of Nature represents tolerably the science of that

day; and one may be sure that whatever scientific investigation has put

out of doubt will presently receive _a priori_ demonstration on the part

of the metaphysicians. But with the scientific method the case is

different. I may start with known and observed facts to proceed to the

unknown; and yet the rules which I follow in doing so may not be such as

investigation would approve. The test of whether I am truly following

the method is not an immediate appeal to my feelings and purposes, but,

on the contrary, itself involves the application of the method. Hence it

is that bad reasoning as well as good reasoning is possible; and this

fact is the foundation of the practical side of logic.

It is not to be supposed that the first three methods of settling

opinion present no advantage whatever over the scientific method. On the

contrary, each has some peculiar convenience of its own. The _a priori_

method is distinguished for its comfortable conclusions. It is the

nature of the process to adopt whatever belief we are inclined to, and

there are certain flatteries to one’s vanities which we all believe by

nature, until we are awakened from our pleasing dream by rough facts.

The method of authority will always govern the mass of mankind; and

those who wield the various forms of organized force in the state will

never be convinced that dangerous reasoning ought not to be suppressed

in some way. If liberty of speech is to be untrammeled from the grosser

forms of constraint, then uniformity of opinion will be secured by a

moral terrorism to which the respectability of society will give its

thorough approval. Following the method of authority is the path of

peace. Certain non-conformities are permitted; certain others

(considered unsafe) are forbidden. These are different in different

countries and in different ages; but, wherever you are let it be known

that you seriously hold a tabooed belief, and you may be perfectly sure

of being treated with a cruelty no less brutal but more refined than

hunting you like a wolf. Thus, the greatest intellectual benefactors of

mankind have never dared, and dare not now, to utter the whole of their

thought; and thus a shade of _prima facie_ doubt is cast upon every

proposition which is considered essential to the security of society.

Singularly enough, the persecution does not all come from without; but a

man torments himself and is oftentimes most distressed at finding

himself believing propositions which he has been brought up to regard

with aversion. The peaceful and sympathetic man will, therefore, find it

hard to resist the temptation to submit his opinions to authority. But

most of all I admire the method of tenacity for its strength,

simplicity, and directness. Men who pursue it are distinguished for

their decision of character, which becomes very easy with such a mental

rule. They do not waste time in trying to make up their minds to what

they want, but, fastening like lightning upon whatever alternative comes

first, they hold to it to the end, whatever happens, without an

instant’s irresolution. This is one of the splendid qualities which

generally accompany brilliant, unlasting success. It is impossible not

to envy the man who can dismiss reason, although we know how it must

turn out at last.

Such are the advantages which the other methods of settling opinions

have over scientific investigation. A man should consider well of them;

and then he should consider that, after all, he wishes his opinions to

coincide with the fact, and that there is no reason why the results of

these three methods should do so. To bring about this effect is the

prerogative of the method of science. Upon such considerations he has to

make his choice—a choice which is far more than the adoption of any

intellectual opinion, which is one of the ruling decisions of his life,

to which when once made he is bound to adhere. The force of habit will

sometimes cause a man to hold on to old beliefs, after he is in a

condition to see that they have no sound basis. But reflection upon the

state of the case will overcome these habits, and he ought to allow

reflection full weight. People sometimes shrink from doing this, having

an idea that beliefs are wholesome which they cannot help feeling rest

on nothing. But let such persons suppose an analogous though different

case from their own. Let them ask themselves what they would say to a

reformed Mussulman who should hesitate to give up his old notions in

regard to the relations of the sexes; or to a reformed Catholic who

should still shrink from the Bible. Would they not say that these

persons ought to consider the matter fully, and clearly understand the

new doctrine, and then ought to embrace it in its entirety? But, above

all, let it be considered that what is more wholesome than any

particular belief, is integrity of belief; and that to avoid looking

into the support of any belief from a fear that it may turn out rotten

is quite as immoral as it is disadvantageous. The person who confesses

that there is such a thing as truth, which is distinguished from

falsehood simply by this, that if acted on it will carry us to the point

we aim at and not astray, and then though convinced of this, dares not

know the truth and seeks to avoid it, is in a sorry state of mind,

indeed.

Yes, the other methods do have their merits: a clear logical conscience

does cost something—just as any virtue, just as all that we cherish,

costs us dear. But, we should not desire it to be otherwise. The genius

of a man’s logical method should be loved and reverenced as his bride,

whom he has chosen from all the world. He need not condemn the others;

on the contrary, he may honor them deeply, and in doing so he only

honors her the more. But she is the one that he has chosen, and he knows

that he was right in making that choice. And having made it, he will

work and fight for her, and will not complain that there are blows to

take, hoping that there may be as many and as hard to give, and will

strive to be the worthy knight and champion of her from the blaze of

whose splendors he draws his inspiration and his courage.

Footnote 26:

_Popular Science Monthly_, November, 1877.

Footnote 27:

[This is substantially the dictum of Harvey to John Aubrey. See the

latter’s _Brief Lives_ (Oxford ed. 1898) I 299.]

Footnote 28:

Not quite so, but as nearly so as can be told in a few words.

Footnote 29:

[This modern logic, however, is largely the outcome of Kepler’s work.]

Footnote 30:

I am not speaking of secondary effects occasionally produced by the

interference of other impulses.

SECOND PAPER

HOW TO MAKE OUR IDEAS CLEAR[31]

I

Whoever has looked into a modern treatise on logic of the common sort,

will doubtless remember the two distinctions between _clear_ and

_obscure_ conceptions, and between _distinct_ and _confused_

conceptions. They have lain in the books now for nigh two centuries,

unimproved and unmodified, and are generally reckoned by logicians as

among the gems of their doctrine.

A clear idea is defined as one which is so apprehended that it will be

recognized wherever it is met with, and so that no other will be

mistaken for it. If it fails of this clearness, it is said to be

obscure.

This is rather a neat bit of philosophical terminology; yet, since it is

clearness that they were defining, I wish the logicians had made their

definition a little more plain. Never to fail to recognize an idea, and

under no circumstances to mistake another for it, let it come in how

recondite a form it may, would indeed imply such prodigious force and

clearness of intellect as is seldom met with in this world. On the other

hand, merely to have such an acquaintance with the idea as to have

become familiar with it, and to have lost all hesitancy in recognizing

it in ordinary cases, hardly seems to deserve the name of clearness of

apprehension, since after all it only amounts to a subjective feeling of

mastery which may be entirely mistaken. I take it, however, that when

the logicians speak of “clearness,” they mean nothing more than such a

familiarity with an idea, since they regard the quality as but a small

merit, which needs to be supplemented by another, which they call

_distinctness_.

A distinct idea is defined as one which contains nothing which is not

clear. This is technical language; by the _contents_ of an idea

logicians understand whatever is contained in its definition. So that an

idea is _distinctly_ apprehended, according to them, when we can give a

precise definition of it, in abstract terms. Here the professional

logicians leave the subject; and I would not have troubled the reader

with what they have to say, if it were not such a striking example of

how they have been slumbering through ages of intellectual activity,

listlessly disregarding the enginery of modern thought, and never

dreaming of applying its lessons to the improvement of logic. It is easy

to show that the doctrine that familiar use and abstract distinctness

make the perfection of apprehension, has its only true place in

philosophies which have long been extinct; and it is now time to

formulate the method of attaining to a more perfect clearness of

thought, such as we see and admire in the thinkers of our own time.

When Descartes set about the reconstruction of philosophy, his first

step was to (theoretically) permit skepticism and to discard the

practice of the schoolmen of looking to authority as the ultimate source

of truth. That done, he sought a more natural fountain of true

principles, and professed to find it in the human mind; thus passing, in

the directest way, from the method of authority to that of apriority, as

described in my first paper. Self-consciousness was to furnish us with

our fundamental truths, and to decide what was agreeable to reason. But

since, evidently, not all ideas are true, he was led to note, as the

first condition of infallibility, that they must be clear. The

distinction between an idea _seeming_ clear and really being so, never

occurred to him. Trusting to introspection, as he did, even for a

knowledge of external things, why should he question its testimony in

respect to the contents of our own minds? But then, I suppose, seeing

men, who seemed to be quite clear and positive, holding opposite

opinions upon fundamental principles, he was further led to say that

clearness of ideas is not sufficient, but that they need also to be

distinct, i.e., to have nothing unclear about them. What he probably

meant by this (for he did not explain himself with precision) was, that

they must sustain the test of dialectical examination; that they must

not only seem clear at the outset, but that discussion must never be

able to bring to light points of obscurity connected with them.

Such was the distinction of Descartes, and one sees that it was

precisely on the level of his philosophy. It was somewhat developed by

Leibnitz. This great and singular genius was as remarkable for what he

failed to see as for what he saw. That a piece of mechanism could not do

work perpetually without being fed with power in some form, was a thing

perfectly apparent to him; yet he did not understand that the machinery

of the mind can only transform knowledge, but never originate it, unless

it be fed with facts of observation. He thus missed the most essential

point of the Cartesian philosophy, which is, that to accept propositions

which seem perfectly evident to us is a thing which, whether it be

logical or illogical, we cannot help doing. Instead of regarding the

matter in this way, he sought to reduce the first principles of science

to formulas which cannot be denied without self-contradiction, and was

apparently unaware of the great difference between his position and that

of Descartes. So he reverted to the old formalities of logic, and, above

all, abstract definitions played a great part in his philosophy. It was

quite natural, therefore, that on observing that the method of Descartes

labored under the difficulty that we may seem to ourselves to have clear

apprehensions of ideas which in truth are very hazy, no better remedy

occurred to him than to require an abstract definition of every

important term. Accordingly, in adopting the distinction of _clear_ and

_distinct_ notions, he described the latter quality as the clear

apprehension of everything contained in the definition; and the books

have ever since copied his words. There is no danger that his chimerical

scheme will ever again be over-valued. Nothing new can ever be learned

by analyzing definitions. Nevertheless, our existing beliefs can be set

in order by this process, and order is an essential element of

intellectual economy, as of every other. It may be acknowledged,

therefore, that the books are right in making familiarity with a notion

the first step toward clearness of apprehension, and the defining of it

the second. But in omitting all mention of any higher perspicuity of

thought, they simply mirror a philosophy which was exploded a hundred

years ago. That much-admired “ornament of logic”—the doctrine of

clearness and distinctness—may be pretty enough, but it is high time to

relegate to our cabinet of curiosities the antique _bijou_, and to wear

about us something better adapted to modern uses.

The very first lesson that we have a right to demand that logic shall

teach us is, how to make our ideas clear; and a most important one it

is, depreciated only by minds who stand in need of it. To know what we

think, to be masters of our own meaning, will make a solid foundation

for great and weighty thought. It is most easily learned by those whose

ideas are meagre and restricted; and far happier they than such as

wallow helplessly in a rich mud of conceptions. A nation, it is true,

may, in the course of generations, overcome the disadvantage of an

excessive wealth of language and its natural concomitant, a vast,

unfathomable deep of ideas. We may see it in history, slowly perfecting

its literary forms, sloughing at length its metaphysics, and, by virtue

of the untirable patience which is often a compensation, attaining great

excellence in every branch of mental acquirement. The page of history is

not yet unrolled which is to tell us whether such a people will or will

not in the long run prevail over one whose ideas (like the words of

their language) are few, but which possesses a wonderful mastery over

those which it has. For an individual, however, there can be no question

that a few clear ideas are worth more than many confused ones. A young

man would hardly be persuaded to sacrifice the greater part of his

thoughts to save the rest; and the muddled head is the least apt to see

the necessity of such a sacrifice. Him we can usually only commiserate,

as a person with a congenital defect. Time will help him, but

intellectual maturity with regard to clearness comes rather late, an

unfortunate arrangement of Nature, inasmuch as clearness is of less use

to a man settled in life, whose errors have in great measure had their

effect, than it would be to one whose path lies before him. It is

terrible to see how a single unclear idea, a single formula without

meaning, lurking in a young man’s head, will sometimes act like an

obstruction of inert matter in an artery, hindering the nutrition of the

brain, and condemning its victim to pine away in the fullness of his

intellectual vigor and in the midst of intellectual plenty. Many a man

has cherished for years as his hobby some vague shadow of an idea, too

meaningless to be positively false; he has, nevertheless, passionately

loved it, has made it his companion by day and by night, and has given

to it his strength and his life, leaving all other occupations for its

sake, and in short has lived with it and for it, until it has become, as

it were, flesh of his flesh and bone of his bone; and then he has waked

up some bright morning to find it gone, clean vanished away like the

beautiful Melusina of the fable, and the essence of his life gone with

it. I have myself known such a man; and who can tell how many histories

of circle-squarers, metaphysicians, astrologers, and what not, may not

be told in the old German story?

II

The principles set forth in the first of these papers lead, at once, to

a method of reaching a clearness of thought of a far higher grade than

the “distinctness” of the logicians. We have there found that the action

of thought is excited by the irritation of doubt, and ceases when belief

is attained; so that the production of belief is the sole function of

thought. All these words, however, are too strong for my purpose. It is

as if I had described the phenomena as they appear under a mental

microscope. Doubt and Belief, as the words are commonly employed, relate

to religious or other grave discussions. But here I use them to

designate the starting of any question, no matter how small or how

great, and the resolution of it. If, for instance, in a horse-car, I

pull out my purse and find a five-cent nickel and five coppers, I

decide, while my hand is going to the purse, in which way I will pay my

fare. To call such a question Doubt, and my decision Belief, is

certainly to use words very disproportionate to the occasion. To speak

of such a doubt as causing an irritation which needs to be appeased,

suggests a temper which is uncomfortable to the verge of insanity. Yet,

looking at the matter minutely, it must be admitted that, if there is

the least hesitation as to whether I shall pay the five coppers or the

nickel (as there will be sure to be, unless I act from some previously

contracted habit in the matter), though irritation is too strong a word,

yet I am excited to such small mental activity as may be necessary to

deciding how I shall act. Most frequently doubts arise from some

indecision, however momentary, in our action. Sometimes it is not so. I

have, for example, to wait in a railway-station, and to pass the time I

read the advertisements on the walls, I compare the advantages of

different trains and different routes which I never expect to take,

merely fancying myself to be in a state of hesitancy, because I am bored

with having nothing to trouble me. Feigned hesitancy, whether feigned

for mere amusement or with a lofty purpose, plays a great part in the

production of scientific inquiry. However the doubt may originate, it

stimulates the mind to an activity which may be slight or energetic,

calm or turbulent. Images pass rapidly through consciousness, one

incessantly melting into another, until at last, when all is over—it may

be in a fraction of a second, in an hour, or after long years—we find

ourselves decided as to how we should act under such circumstances as

those which occasioned our hesitation. In other words, we have attained

belief.

In this process we observe two sorts of elements of consciousness, the

distinction between which may best be made clear by means of an

illustration. In a piece of music there are the separate notes, and

there is the air. A single tone may be prolonged for an hour or a day,

and it exists as perfectly in each second of that time as in the whole

taken together; so that, as long as it is sounding, it might be present

to a sense from which everything in the past was as completely absent as

the future itself. But it is different with the air, the performance of

which occupies a certain time, during the portions of which only

portions of it are played. It consists in an orderliness in the

succession of sounds which strike the ear at different times; and to

perceive it there must be some continuity of consciousness which makes

the events of a lapse of time present to us. We certainly only perceive

the air by hearing the separate notes; yet we cannot be said to directly

hear it, for we hear only what is present at the instant, and an

orderliness of succession cannot exist in an instant. These two sorts of

objects, what we are _immediately_ conscious of and what we are

_mediately_ conscious of, are found in all consciousness. Some elements

(the sensations) are completely present at every instant so long as they

last, while others (like thought) are actions having beginning, middle,

and end, and consist in a congruence in the succession of sensations

which flow through the mind. They cannot be immediately present to us,

but must cover some portion of the past or future. Thought is a thread

of melody running through the succession of our sensations.

We may add that just as a piece of music may be written in parts, each

part having its own air, so various systems of relationship of

succession subsist together between the same sensations. These different

systems are distinguished by having different motives, ideas, or

functions. Thought is only one such system; for its sole motive, idea,

and function is to produce belief, and whatever does not concern that

purpose belongs to some other system of relations. The action of

thinking may incidentally have other results. It may serve to amuse us,

for example, and among _dilettanti_ it is not rare to find those who

have so perverted thought to the purposes of pleasure that it seems to

vex them to think that the questions upon which they delight to exercise

it may ever get finally settled; and a positive discovery which takes a

favorite subject out of the arena of literary debate is met with

ill-concealed dislike. This disposition is the very debauchery of

thought. But the soul and meaning of thought, abstracted from the other

elements which accompany it, though it may be voluntarily thwarted, can

never be made to direct itself toward anything but the production of

belief. Thought in action has for its only possible motive the

attainment of thought at rest; and whatever does not refer to belief is

no part of the thought itself.

And what, then, is belief? It is the demi-cadence which closes a musical

phrase in the symphony of our intellectual life. We have seen that it

has just three properties: First, it is something that we are aware of;

second, it appeases the irritation of doubt; and, third, it involves the

establishment in our nature of a rule of action, or, say for short, a

_habit_. As it appeases the irritation of doubt, which is the motive for

thinking, thought relaxes, and comes to rest for a moment when belief is

reached. But, since belief is a rule for action, the application of

which involves further doubt and further thought, at the same time that

it is a stopping-place, it is also a new starting-place for thought.

That is why I have permitted myself to call it thought at rest, although

thought is essentially an action. The _final_ upshot of thinking is the

exercise of volition, and of this thought no longer forms a part; but

belief is only a stadium of mental action, an effect upon our nature due

to thought, which will influence future thinking.

[Illustration: Figure 1.]

[Illustration: Figure 2.]

The essence of belief is the establishment of a habit, and different

beliefs are distinguished by the different modes of action to which they

give rise. If beliefs do not differ in this respect, if they appease the

same doubt by producing the same rule of action, then no mere

differences in the manner of consciousness of them can make them

different beliefs, any more than playing a tune in different keys is

playing different tunes. Imaginary distinctions are often drawn between

beliefs which differ only in their mode of expression;—the wrangling

which ensues is real enough, however. To believe that any objects are

arranged as in Fig. 1, and to believe that they are arranged as in Fig.

2, are one and the same belief; yet it is conceivable that a man should

assert one proposition and deny the other. Such false distinctions do as

much harm as the confusion of beliefs really different, and are among

the pitfalls of which we ought constantly to beware, especially when we

are upon metaphysical ground. One singular deception of this sort, which

often occurs, is to mistake the sensation produced by our own

unclearness of thought for a character of the object we are thinking.

Instead of perceiving that the obscurity is purely subjective, we fancy

that we contemplate a quality of the object which is essentially

mysterious; and if our conception be afterward presented to us in a

clear form we do not recognize it as the same, owing to the absence of

the feeling of unintelligibility. So long as this deception lasts, it

obviously puts an impassable barrier in the way of perspicuous thinking;

so that it equally interests the opponents of rational thought to

perpetuate it, and its adherents to guard against it.

Another such deception is to mistake a mere difference in the

grammatical construction of two words for a distinction between the

ideas they express. In this pedantic age, when the general mob of

writers attend so much more to words than to things, this error is

common enough. When I just said that thought is an _action_, and that it

consists in a _relation_, although a person performs an action but not a

relation, which can only be the result of an action, yet there was no

inconsistency in what I said, but only a grammatical vagueness.

From all these sophisms we shall be perfectly safe so long as we reflect

that the whole function of thought is to produce habits of action; and

that whatever there is connected with a thought, but irrelevant to its

purpose, is an accretion to it, but no part of it. If there be a unity

among our sensations which has no reference to how we shall act on a

given occasion, as when we listen to a piece of music, why we do not

call that thinking. To develop its meaning, we have, therefore, simply

to determine what habits it produces, for what a thing means is simply

what habits it involves. Now, the identity of a habit depends on how it

might lead us to act, not merely under such circumstances as are likely

to arise, but under such as might possibly occur, no matter how

improbable they may be. What the habit is depends on _when_ and _how_ it

causes us to act. As for the _when_, every stimulus to action is derived

from perception; as for the _how_, every purpose of action is to produce

some sensible result. Thus, we come down to what is tangible and

practical, as the root of every real distinction of thought, no matter

how subtile it may be; and there is no distinction of meaning so fine as

to consist in anything but a possible difference of practice.

To see what this principle leads to, consider in the light of it such a

doctrine as that of transubstantiation. The Protestant churches

generally hold that the elements of the sacrament are flesh and blood

only in a tropical sense; they nourish our souls as meat and the juice

of it would our bodies. But the Catholics maintain that they are

literally just that; although they possess all the sensible qualities of

wafer-cakes and diluted wine. But we can have no conception of wine

except what may enter into a belief, either—

1. That this, that, or the other, is wine; or,

2. That wine possesses certain properties.

Such beliefs are nothing but self-notifications that we should, upon

occasion, act in regard to such things as we believe to be wine

according to the qualities which we believe wine to possess. The

occasion of such action would be some sensible perception, the motive of

it to produce some sensible result. Thus our action has exclusive

reference to what affects the senses, our habit has the same bearing as

our action, our belief the same as our habit, our conception the same as

our belief; and we can consequently mean nothing by wine but what has

certain effects, direct or indirect, upon our senses; and to talk of

something as having all the sensible characters of wine, yet being in

reality blood, is senseless jargon. Now, it is not my object to pursue

the theological question; and having used it as a logical example I drop

it, without caring to anticipate the theologian’s reply. I only desire

to point out how impossible it is that we should have an idea in our

minds which relates to anything but conceived sensible effects of

things. Our idea of anything _is_ our idea of its sensible effects; and

if we fancy that we have any other we deceive ourselves, and mistake a

mere sensation accompanying the thought for a part of the thought

itself. It is absurd to say that thought has any meaning unrelated to

its only function. It is foolish for Catholics and Protestants to fancy

themselves in disagreement about the elements of the sacrament, if they

agree in regard to all their sensible effects, here or hereafter.

It appears, then, that the rule for attaining the third grade of

clearness of apprehension is as follows: Consider what effects, which

might conceivably have practical bearings, we conceive the object of our

conception to have. Then, our conception of these effects is the whole

of our conception of the object.

III

Let us illustrate this rule by some examples; and, to begin with the

simplest one possible, let us ask what we mean by calling a thing

_hard_. Evidently that it will not be scratched by many other

substances. The whole conception of this quality, as of every other,

lies in its conceived effects. There is absolutely no difference between

a hard thing and a soft thing so long as they are not brought to the

test. Suppose, then, that a diamond could be crystallized in the midst

of a cushion of soft cotton, and should remain there until it was

finally burned up. Would it be false to say that that diamond was soft?

This seems a foolish question, and would be so, in fact, except in the

realm of logic. There such questions are often of the greatest utility

as serving to bring logical principles into sharper relief than real

discussions ever could. In studying logic we must not put them aside

with hasty answers, but must consider them with attentive care, in order

to make out the principles involved. We may, in the present case, modify

our question, and ask what prevents us from saying that all hard bodies

remain perfectly soft until they are touched, when their hardness

increases with the pressure until they are scratched. Reflection will

show that the reply is this: there would be no _falsity_ in such modes

of speech. They would involve a modification of our present usage of

speech with regard to the words hard and soft, but not of their

meanings. For they represent no fact to be different from what it is;

only they involve arrangements of facts which would be exceedingly

maladroit. This leads us to remark that the question of what would occur

under circumstances which do not actually arise is not a question of

fact, but only of the most perspicuous arrangement of them. For example,

the question of free-will and fate in its simplest form, stripped of

verbiage, is something like this: I have done something of which I am

ashamed; could I, by an effort of the will, have resisted the

temptation, and done otherwise? The philosophical reply is, that this is

not a question of fact, but only of the arrangement of facts. Arranging

them so as to exhibit what is particularly pertinent to my

question—namely, that I ought to blame myself for having done wrong—it

is perfectly true to say that, if I had willed to do otherwise than I

did, I should have done otherwise. On the other hand, arranging the

facts so as to exhibit another important consideration, it is equally

true that, when a temptation has once been allowed to work, it will, if

it has a certain force, produce its effect, let me struggle how I may.

There is no objection to a contradiction in what would result from a

false supposition. The _reductio ad absurdum_ consists in showing that

contradictory results would follow from a hypothesis which is

consequently judged to be false. Many questions are involved in the

free-will discussion, and I am far from desiring to say that both sides

are equally right. On the contrary, I am of opinion that one side denies

important facts, and that the other does not. But what I do say is, that

the above single question was the origin of the whole doubt; that, had

it not been for this question, the controversy would never have arisen;

and that this question is perfectly solved in the manner which I have

indicated.

Let us next seek a clear idea of Weight. This is another very easy case.

To say that a body is heavy means simply that, in the absence of

opposing force, it will fall. This (neglecting certain specifications of

how it will fall, etc., which exist in the mind of the physicist who

uses the word) is evidently the whole conception of weight. It is a fair

question whether some particular facts may not _account_ for gravity;

but what we mean by the force itself is completely involved in its

effects.

This leads us to undertake an account of the idea of Force in general.

This is the great conception which, developed in the early part of the

seventeenth century from the rude idea of a cause, and constantly

improved upon since, has shown us how to explain all the changes of

motion which bodies experience, and how to think about all physical

phenomena; which has given birth to modern science, and changed the face

of the globe; and which, aside from its more special uses, has played a

principal part in directing the course of modern thought, and in

furthering modern social development. It is, therefore, worth some pains

to comprehend it. According to our rule, we must begin by asking what is

the immediate use of thinking about force; and the answer is, that we

thus account for changes of motion. If bodies were left to themselves,

without the intervention of forces, every motion would continue

unchanged both in velocity and in direction. Furthermore, change of

motion never takes place abruptly; if its direction is changed, it is

always through a curve without angles; if its velocity alters, it is by

degrees. The gradual changes which are constantly taking place are

conceived by geometers to be compounded together according to the rules

of the parallelogram of forces. If the reader does not already know what

this is, he will find it, I hope, to his advantage to endeavor to follow

the following explanation; but if mathematics are insupportable to him,

pray let him skip three paragraphs rather than that we should part

company here.

A _path_ is a line whose beginning and end are distinguished. Two paths

are considered to be equivalent, which, beginning at the same point,

lead to the same point. Thus the two paths, _A B C D E_ and _A F G H E_

(Fig. 3), are equivalent. Paths which do _not_ begin at the same point

are considered to be equivalent, provided that, on moving either of them

without turning it, but keeping it always parallel to its original

position, [so that] when its beginning coincides with that of the other

path, the ends also coincide. Paths are considered as geometrically

added together, when one begins where the other ends; thus the path _A

E_ is conceived to be a sum of _A B_, _B C_, _C D_, and _D E_. In the

parallelogram of Fig. 4 the diagonal _A C_ is the sum of _A B_ and _B

C_; or, since _A D_ is geometrically equivalent to _B C_, _A C_ is the

geometrical sum of _A B_ and _A D_.

[Illustration: Figure 3.]

[Illustration: Figure 4.]

All this is purely conventional. It simply amounts to this: that we

choose to call paths having the relations I have described equal or

added. But, though it is a convention, it is a convention with a good

reason. The rule for geometrical addition may be applied not only to

paths, but to any other things which can be represented by paths. Now,

as a path is determined by the varying direction and distance of the

point which moves over it from the starting-point, it follows that

anything which from its beginning to its end is determined by a varying

direction and a varying magnitude is capable of being represented by a

line. Accordingly, _velocities_ may be represented by lines, for they

have only directions and rates. The same thing is true of

_accelerations_, or changes of velocities. This is evident enough in the

case of velocities; and it becomes evident for accelerations if we

consider that precisely what velocities are to positions—namely, states

of change of them—that accelerations are to velocities.

[Illustration: Figure 5.]

The so-called “parallelogram of forces” is simply a rule for compounding

accelerations. The rule is, to represent the accelerations by paths, and

then to geometrically add the paths. The geometers, however, not only

use the “parallelogram of forces” to compound different accelerations,

but also to resolve one acceleration into a sum of several. Let _A B_

(Fig. 5) be the path which represents a certain acceleration—say, such a

change in the motion of a body that at the end of one second the body

will, under the influence of that change, be in a position different

from what it would have had if its motion had continued unchanged, such

that a path equivalent to _A B_ would lead from the latter position to

the former. This acceleration may be considered as the sum of the

accelerations represented by _A C_ and _C B_. It may also be considered

as the sum of the very different accelerations represented by _A D_ and

_D B_, where _A D_ is almost the opposite of _A C_. And it is clear that

there is an immense variety of ways in which _A B_ might be resolved

into the sum of two accelerations.

After this tedious explanation, which I hope, in view of the

extraordinary interest of the conception of force, may not have

exhausted the reader’s patience, we are prepared at last to state the

grand fact which this conception embodies. This fact is that if the

actual changes of motion which the different particles of bodies

experience are each resolved in its appropriate way, each component

acceleration is precisely such as is prescribed by a certain law of

Nature, according to which bodies in the relative positions which the

bodies in question actually have at the moment,[32] always receive

certain accelerations, which, being compounded by geometrical addition,

give the acceleration which the body actually experiences.

This is the only fact which the idea of force represents, and whoever

will take the trouble clearly to apprehend what this fact is, perfectly

comprehends what force is. Whether we ought to say that a force _is_ an

acceleration, or that it _causes_ an acceleration, is a mere question of

propriety of language, which has no more to do with our real meaning

than the difference between the French idiom “_Il fait froid_” and its

English equivalent “_It is cold_.” Yet it is surprising to see how this

simple affair has muddled men’s minds. In how many profound treatises is

not force spoken of as a “mysterious entity,” which seems to be only a

way of confessing that the author despairs of ever getting a clear

notion of what the word means! In a recent admired work on _Analytic

Mechanics_ it is stated that we understand precisely the effect of

force, but what force itself is we do not understand! This is simply a

self-contradiction. The idea which the word force excites in our minds

has no other function than to affect our actions, and these actions can

have no reference to force otherwise than through its effects.

Consequently, if we know what the effects of force are, we are

acquainted with every fact which is implied in saying that a force

exists, and there is nothing more to know. The truth is, there is some

vague notion afloat that a question may mean something which the mind

cannot conceive; and when some hair-splitting philosophers have been

confronted with the absurdity of such a view, they have invented an

empty distinction between positive and negative conceptions, in the

attempt to give their non-idea a form not obviously nonsensical. The

nullity of it is sufficiently plain from the considerations given a few

pages back; and, apart from those considerations, the quibbling

character of the distinction must have struck every mind accustomed to

real thinking.

IV

Let us now approach the subject of logic, and consider a conception

which particularly concerns it, that of _reality_. Taking clearness in

the sense of familiarity, no idea could be clearer than this. Every

child uses it with perfect confidence, never dreaming that he does not

understand it. As for clearness in its second grade, however, it would

probably puzzle most men, even among those of a reflective turn of mind,

to give an abstract definition of the real. Yet such a definition may

perhaps be reached by considering the points of difference between

reality and its opposite, fiction. A figment is a product of somebody’s

imagination; it has such characters as his thought impresses upon it.

That those characters are independent of how you or I think is an

external reality. There are, however, phenomena within our own minds,

dependent upon our thought, which are at the same time real in the sense

that we really think them. But though their characters depend on how we

think, they do not depend on what we think those characters to be. Thus,

a dream has a real existence as a mental phenomenon, if somebody has

really dreamt it; that he dreamt so and so, does not depend on what

anybody thinks was dreamt, but is completely independent of all opinion

on the subject. On the other hand, considering, not the fact of

dreaming, but the thing dreamt, it retains its peculiarities by virtue

of no other fact than that it was dreamt to possess them. Thus we may

define the real as that whose characters are independent of what anybody

may think them to be.

But, however satisfactory such a definition may be found, it would be a

great mistake to suppose that it makes the idea of reality perfectly

clear. Here, then, let us apply our rules. According to them, reality,

like every other quality, consists in the peculiar sensible effects

which things partaking of it produce. The only effect which real things

have is to cause belief, for all the sensations which they excite emerge

into consciousness in the form of beliefs. The question, therefore, is,

how is true belief (or belief in the real) distinguished from false

belief (or belief in fiction). Now, as we have seen in the former paper,

the ideas of truth and falsehood, in their full development, appertain

exclusively to the scientific method of settling opinion. A person who

arbitrarily chooses the propositions which he will adopt can use the

word truth only to emphasize the expression of his determination to hold

on to his choice. Of course, the method of tenacity never prevailed

exclusively; reason is too natural to men for that. But in the

literature of the dark ages we find some fine examples of it. When

Scotus Erigena is commenting upon a poetical passage in which hellebore

is spoken of as having caused the death of Socrates, he does not

hesitate to inform the inquiring reader that Helleborus and Socrates

were two eminent Greek philosophers, and that the latter having been

overcome in argument by the former took the matter to heart and died of

it! What sort of an idea of truth could a man have who could adopt and

teach, without the qualification of a perhaps, an opinion taken so

entirely at random? The real spirit of Socrates, who I hope would have

been delighted to have been “overcome in argument,” because he would

have learned something by it, is in curious contrast with the naïve idea

of the glossist, for whom discussion would seem to have been simply a

struggle. When philosophy began to awake from its long slumber, and

before theology completely dominated it, the practice seems to have been

for each professor to seize upon any philosophical position he found

unoccupied and which seemed a strong one, to intrench himself in it, and

to sally forth from time to time to give battle to the others. Thus,

even the scanty records we possess of those disputes enable us to make

out a dozen or more opinions held by different teachers at one time

concerning the question of nominalism and realism. Read the opening part

of the _Historia Calamitatum_ of Abelard, who was certainly as

philosophical as any of his contemporaries, and see the spirit of combat

which it breathes. For him, the truth is simply his particular

stronghold. When the method of authority prevailed, the truth meant

little more than the Catholic faith. All the efforts of the scholastic

doctors are directed toward harmonizing their faith in Aristotle and

their faith in the Church, and one may search their ponderous folios

through without finding an argument which goes any further. It is

noticeable that where different faiths flourish side by side, renegades

are looked upon with contempt even by the party whose belief they adopt;

so completely has the idea of loyalty replaced that of truth-seeking.

Since the time of Descartes, the defect in the conception of truth has

been less apparent. Still, it will sometimes strike a scientific man

that the philosophers have been less intent on finding out what the

facts are, than on inquiring what belief is most in harmony with their

system. It is hard to convince a follower of the _a priori_ method by

adducing facts; but show him that an opinion he is defending is

inconsistent with what he has laid down elsewhere, and he will be very

apt to retract it. These minds do not seem to believe that disputation

is ever to cease; they seem to think that the opinion which is natural

for one man is not so for another, and that belief will, consequently,

never be settled. In contenting themselves with fixing their own

opinions by a method which would lead another man to a different result,

they betray their feeble hold of the conception of what truth is.

On the other hand, all the followers of science are fully persuaded that

the processes of investigation, if only pushed far enough, will give one

certain solution to every question to which they can be applied. One man

may investigate the velocity of light by studying the transits of Venus

and the aberration of the stars; another by the oppositions of Mars and

the eclipses of Jupiter’s satellites; a third by the method of Fizeau; a

fourth by that of Foucault; a fifth by the motions of the curves of

Lissajoux; a sixth, a seventh, an eighth, and a ninth, may follow the

different methods of comparing the measures of statical and dynamical

electricity. They may at first obtain different results, but, as each

perfects his method and his processes, the results will move steadily

together toward a destined center. So with all scientific research.

Different minds may set out with the most antagonistic views, but the

progress of investigation carries them by a force outside of themselves

to one and the same conclusion. This activity of thought by which we are

carried, not where we wish, but to a fore-ordained goal, is like the

operation of destiny. No modification of the point of view taken, no

selection of other facts for study, no natural bent of mind even, can

enable a man to escape the predestinate opinion. This great law is

embodied in the conception of truth and reality. The opinion which is

fated[33] to be ultimately agreed to by all who investigate, is what we

mean by the truth, and the object represented in this opinion is the

real. That is the way I would explain reality.

But it may be said that this view is directly opposed to the abstract

definition which we have given of reality, inasmuch as it makes the

characters of the real depend on what is ultimately thought about them.

But the answer to this is that, on the one hand, reality is independent,

not necessarily of thought in general, but only of what you or I or any

finite number of men may think about it; and that, on the other hand,

though the object of the final opinion depends on what that opinion is,

yet what that opinion is does not depend on what you or I or any man

thinks. Our perversity and that of others may indefinitely postpone the

settlement of opinion; it might even conceivably cause an arbitrary

proposition to be universally accepted as long as the human race should

last. Yet even that would not change the nature of the belief, which

alone could be the result of investigation carried sufficiently far; and

if, after the extinction of our race, another should arise with

faculties and disposition for investigation, that true opinion must be

the one which they would ultimately come to. “Truth crushed to earth

shall rise again,” and the opinion which would finally result from

investigation does not depend on how anybody may actually think. But the

reality of that which is real does depend on the real fact that

investigation is destined to lead, at last, if continued long enough, to

a belief in it.

But I may be asked what I have to say to all the minute facts of

history, forgotten never to be recovered, to the lost books of the

ancients, to the buried secrets.

“Full many a gem of purest ray serene

The dark, unfathomed caves of ocean bear;

Full many a flower is born to blush unseen,

And waste its sweetness on the desert air.”

Do these things not really exist because they are hopelessly beyond the

reach of our knowledge? And then, after the universe is dead (according

to the prediction of some scientists), and all life has ceased forever,

will not the shock of atoms continue though there will be no mind to

know it? To this I reply that, though in no possible state of knowledge

can any number be great enough to express the relation between the

amount of what rests unknown to the amount of the known, yet it is

unphilosophical to suppose that, with regard to any given question

(which has any clear meaning), investigation would not bring forth a

solution of it, if it were carried far enough. Who would have said, a

few years ago, that we could ever know of what substances stars are made

whose light may have been longer in reaching us than the human race has

existed? Who can be sure of what we shall not know in a few hundred

years? Who can guess what would be the result of continuing the pursuit

of science for ten thousand years, with the activity of the last

hundred? And if it were to go on for a million, or a billion, or any

number of years you please, how is it possible to say that there is any

question which might not ultimately be solved?

But it may be objected, “Why make so much of these remote

considerations, especially when it is your principle that only practical

distinctions have a meaning?” Well, I must confess that it makes very

little difference whether we say that a stone on the bottom of the

ocean, in complete darkness, is brilliant or not—that is to say, that it

_probably_ makes no difference, remembering always that that stone _may_

be fished up to-morrow. But that there are gems at the bottom of the

sea, flowers in the untraveled desert, etc., are propositions which,

like that about a diamond being hard when it is not pressed, concern

much more the arrangement of our language than they do the meaning of

our ideas.

It seems to me, however, that we have, by the application of our rule,

reached so clear an apprehension of what we mean by reality, and of the

fact which the idea rests on, that we should not, perhaps, be making a

pretension so presumptuous as it would be singular, if we were to offer

a metaphysical theory of existence for universal acceptance among those

who employ the scientific method of fixing belief. However, as

metaphysics is a subject much more curious than useful, the knowledge of

which, like that of a sunken reef, serves chiefly to enable us to keep

clear of it, I will not trouble the reader with any more Ontology at

this moment. I have already been led much further into that path than I

should have desired; and I have given the reader such a dose of

mathematics, psychology, and all that is most abstruse, that I fear he

may already have left me, and that what I am now writing is for the

compositor and proofreader exclusively. I trusted to the importance of

the subject. There is no royal road to logic, and really valuable ideas

can only be had at the price of close attention. But I know that in the

matter of ideas the public prefer the cheap and nasty; and in my next

paper I am going to return to the easily intelligible, and not wander

from it again. The reader who has been at the pains of wading through

this paper, shall be rewarded in the next one by seeing how beautifully

what has been developed in this tedious way can be applied to the

ascertainment of the rules of scientific reasoning.

We have, hitherto, not crossed the threshold of scientific logic. It is

certainly important to know how to make our ideas clear, but they may be

ever so clear without being true. How to make them so, we have next to

study. How to give birth to those vital and procreative ideas which

multiply into a thousand forms and diffuse themselves everywhere,

advancing civilization and making the dignity of man, is an art not yet

reduced to rules, but of the secret of which the history of science

affords some hints.

Footnote 31:

_Popular Science Monthly_, January, 1878.

Footnote 32:

Possibly the velocities also have to be taken into account.

Footnote 33:

Fate means merely that which is sure to come true, and can nohow be

avoided. It is a superstition to suppose that a certain sort of events

are ever fated, and it is another to suppose that the word fate can

never be freed from its superstitious taint. We are all fated to die.

THIRD PAPER

THE DOCTRINE OF CHANCES[34]

I

It is a common observation that a science first begins to be exact when

it is quantitatively treated. What are called the exact sciences are no

others than the mathematical ones. Chemists reasoned vaguely until

Lavoisier showed them how to apply the balance to the verification of

their theories, when chemistry leaped suddenly into the position of the

most perfect of the classificatory sciences. It has thus become so

precise and certain that we usually think of it along with optics,

thermotics, and electrics. But these are studies of general laws, while

chemistry considers merely the relations and classification of certain

objects; and belongs, in reality, in the same category as systematic

botany and zoölogy. Compare it with these last, however, and the

advantage that it derives from its quantitative treatment is very

evident.

The rudest numerical scales, such as that by which the mineralogists

distinguish the different degrees of hardness, are found useful. The

mere counting of pistils and stamens sufficed to bring botany out of

total chaos into some kind of form. It is not, however, so much from

_counting_ as from _measuring_, not so much from the conception of

number as from that of continuous quantity, that the advantage of

mathematical treatment comes. Number, after all, only serves to pin us

down to a precision in our thoughts which, however beneficial, can

seldom lead to lofty conceptions, and frequently descends to pettiness.

Of those two faculties of which Bacon speaks, that which marks

differences and that which notes resemblances, the employment of number

can only aid the lesser one; and the excessive use of it must tend to

narrow the powers of the mind. But the conception of continuous quantity

has a great office to fulfill, independently of any attempt at

precision. Far from tending to the exaggeration of differences, it is

the direct instrument of the finest generalizations. When a naturalist

wishes to study a species, he collects a considerable number of

specimens more or less similar. In contemplating them, he observes

certain ones which are more or less alike in some particular respect.

They all have, for instance, a certain S-shaped marking. He observes

that they are not _precisely_ alike, in this respect; the S has not

precisely the same shape, but the differences are such as to lead him to

believe that forms could be found intermediate between any two of those

he possesses. He, now, finds other forms apparently quite dissimilar—say

a marking in the form of a C—and the question is, whether he can find

intermediate ones which will connect these latter with the others. This

he often succeeds in doing in cases where it would at first be thought

impossible; whereas he sometimes finds those which differ, at first

glance, much less, to be separated in Nature by the non-occurrence of

intermediaries. In this way, he builds up from the study of Nature a new

general conception of the character in question. He obtains, for

example, an idea of a leaf which includes every part of the flower, and

an idea of a vertebra which includes the skull. I surely need not say

much to show what a logical engine there is here. It is the essence of

the method of the naturalist.[35] How he applies it first to one

character, and then to another, and finally obtains a notion of a

species of animals, the differences between whose members, however

great, are confined within limits, is a matter which does not here

concern us. The whole method of classification must be considered later;

but, at present, I only desire to point out that it is by taking

advantage of the idea of continuity, or the passage from one form to

another by insensible degrees, that the naturalist builds his

conceptions. Now, the naturalists are the great builders of conceptions;

there is no other branch of science where so much of this work is done

as in theirs; and we must, in great measure, take them for our teachers

in this important part of logic. And it will be found everywhere that

the idea of continuity is a powerful aid to the formation of true and

fruitful conceptions. By means of it, the greatest differences are

broken down and resolved into differences of degree, and the incessant

application of it is of the greatest value in broadening our

conceptions. I propose to make a great use of this idea in the present

series of papers; and the particular series of important fallacies,

which, arising from a neglect of it, have desolated philosophy, must

further on be closely studied. At present, I simply call the reader’s

attention to the utility of this conception.

In studies of numbers, the idea of continuity is so indispensable, that

it is perpetually introduced even where there is no continuity in fact,

as where we say that there are in the United States 10.7 inhabitants per

square mile, or that in New York 14.72 persons live in the average

house.[36] Another example is that law of the distribution of errors

which Quetelet, Galton, and others, have applied with so much success to

the study of biological and social matters. This application of

continuity to cases where it does not really exist illustrates, also,

another point which will hereafter demand a separate study, namely, the

great utility which fictions sometimes have in science.

II

The theory of probabilities is simply the science of logic

quantitatively treated. There are two conceivable certainties with

reference to any hypothesis, the certainty of its truth and the

certainty of its falsity. The numbers _one_ and _zero_ are appropriated,

in this calculus, to marking these extremes of knowledge; while

fractions having values intermediate between them indicate, as we may

vaguely say, the degrees in which the evidence leans toward one or the

other. The general problem of probabilities is, from a given state of

facts, to determine the numerical probability of a possible fact. This

is the same as to inquire how much the given facts are worth, considered

as evidence to prove the possible fact. Thus the problem of

probabilities is simply the general problem of logic.

Probability is a continuous quantity, so that great advantages may be

expected from this mode of studying logic. Some writers have gone so far

as to maintain that, by means of the calculus of chances, every solid

inference may be represented by legitimate arithmetical operations upon

the numbers given in the premises. If this be, indeed, true, the great

problem of logic, how it is that the observation of one fact can give us

knowledge of another independent fact, is reduced to a mere question of

arithmetic. It seems proper to examine this pretension before

undertaking any more recondite solution of the paradox.

But, unfortunately, writers on probabilities are not agreed in regard to

this result. This branch of mathematics is the only one, I believe, in

which good writers frequently get results entirely erroneous. In

elementary geometry the reasoning is frequently fallacious, but

erroneous conclusions are avoided; but it may be doubted if there is a

single extensive treatise on probabilities in existence which does not

contain solutions absolutely indefensible. This is partly owing to the

want of any regular method of procedure; for the subject involves too

many subtilties to make it easy to put its problems into equations

without such an aid. But, beyond this, the fundamental principles of its

calculus are more or less in dispute. In regard to that class of

questions to which it is chiefly applied for practical purposes, there

is comparatively little doubt; but in regard to others to which it has

been sought to extend it, opinion is somewhat unsettled.

This last class of difficulties can only be entirely overcome by making

the idea of probability perfectly clear in our minds in the way set

forth in our last paper.

III

To get a clear idea of what we mean by probability, we have to consider

what real and sensible difference there is between one degree of

probability and another.

The character of probability belongs primarily, without doubt, to

certain inferences. Locke explains it as follows: After remarking that

the mathematician positively knows that the sum of the three angles of a

triangle is equal to two right angles because he apprehends the

geometrical proof, he thus continues: “But another man who never took

the pains to observe the demonstration, hearing a mathematician, a man

of credit, affirm the three angles of a triangle to be equal to two

right ones, _assents_ to it; i.e., receives it for true. In which case

the foundation of his assent is the probability of the thing, the proof

being such as, for the most part, carries truth with it; the man on

whose testimony he receives it not being wont to affirm anything

contrary to, or besides his knowledge, especially in matters of this

kind.” The celebrated _Essay concerning Human Understanding_ contains

many passages which, like this one, make the first steps in profound

analyses which are not further developed. It was shown in the first of

these papers that the validity of an inference does not depend on any

tendency of the mind to accept it, however strong such tendency may be;

but consists in the real fact that, when premises like those of the

argument in question are true, conclusions related to them like that of

this argument are also true. It was remarked that in a logical mind an

argument is always conceived as a member of a _genus_ of arguments all

constructed in the same way, and such that, when their premises are real

facts, their conclusions are so also. If the argument is demonstrative,

then this is always so; if it is only probable, then it is for the most

part so. As Locke says, the probable argument is “_such as_ for the most

part carries truth with it.”

According to this, that real and sensible difference between one degree

of probability and another, in which the meaning of the distinction

lies, is that in the frequent employment of two different modes of

inference, one will carry truth with it oftener than the other. It is

evident that this is the only difference there is in the existing fact.

Having certain premises, a man draws a certain conclusion, and as far as

this inference alone is concerned the only possible practical question

is whether that conclusion is true or not, and between existence and

non-existence there is no middle term. “Being only is and nothing is

altogether not,” said Parmenides; and this is in strict accordance with

the analysis of the conception of reality given in the last paper. For

we found that the distinction of reality and fiction depends on the

supposition that sufficient investigation would cause one opinion to be

universally received and all others to be rejected. That presupposition,

involved in the very conceptions of reality and figment, involves a

complete sundering of the two. It is the heaven-and-hell idea in the

domain of thought. But, in the long run, there is a real fact which

corresponds to the idea of probability, and it is that a given mode of

inference sometimes proves successful and sometimes not, and that in a

ratio ultimately fixed. As we go on drawing inference after inference of

the given kind, during the first ten or hundred cases the ratio of

successes may be expected to show considerable fluctuations; but when we

come into the thousands and millions, these fluctuations become less and

less; and if we continue long enough, the ratio will approximate toward

a fixed limit. We may, therefore, define the probability of a mode of

argument as the proportion of cases in which it carries truth with it.

The inference from the premise, A, to the conclusion, B, depends, as we

have seen, on the guiding principle, that if a fact of the class A is

true, a fact of the class B is true. The probability consists of the

fraction whose numerator is the number of times in which both A and B

are true, and whose denominator is the total number of times in which A

is true, whether B is so or not. Instead of speaking of this as the

probability of the inference, there is not the slightest objection to

calling it the probability that, if A happens, B happens. But to speak

of the probability of the event B, without naming the condition, really

has no meaning at all. It is true that when it is perfectly obvious what

condition is meant, the ellipsis may be permitted. But we should avoid

contracting the habit of using language in this way (universal as the

habit is), because it gives rise to a vague way of thinking, as if the

action of causation might either determine an event to happen or

determine it not to happen, or leave it more or less free to happen or

not, so as to give rise to an _inherent_ chance in regard to its

occurrence.[37] It is quite clear to me that some of the worst and most

persistent errors in the use of the doctrine of chances have arisen from

this vicious mode of expression.[38]

IV

But there remains an important point to be cleared up. According to what

has been said, the idea of probability essentially belongs to a kind of

inference which is repeated indefinitely. An individual inference must

be either true or false, and can show no effect of probability; and,

therefore, in reference to a single case considered in itself,

probability can have no meaning. Yet if a man had to choose between

drawing a card from a pack containing twenty-five red cards and a black

one, or from a pack containing twenty-five black cards and a red one,

and if the drawing of a red card were destined to transport him to

eternal felicity, and that of a black one to consign him to everlasting

woe, it would be folly to deny that he ought to prefer the pack

containing the larger portion of red cards, although, from the nature of

the risk, it could not be repeated. It is not easy to reconcile this

with our analysis of the conception of chance. But suppose he should

choose the red pack, and should draw the wrong card, what consolation

would he have? He might say that he had acted in accordance with reason,

but that would only show that his reason was absolutely worthless. And

if he should choose the right card, how could he regard it as anything

but a happy accident? He could not say that if he had drawn from the

other pack, he might have drawn the wrong one, because an hypothetical

proposition such as, “if A, then B,” means nothing with reference to a

single case. Truth consists in the existence of a real fact

corresponding to the true proposition. Corresponding to the proposition,

“if A, then B,” there may be the fact that _whenever_ such an event as A

happens such an event as B happens. But in the case supposed, which has

no parallel as far as this man is concerned, there would be no real fact

whose existence could give any truth to the statement that, if he had

drawn from the other pack, he might have drawn a black card. Indeed,

since the validity of an inference consists in the truth of the

hypothetical proposition that _if_ the premises be true the conclusion

will also be true, and since the only real fact which can correspond to

such a proposition is that whenever the antecedent is true the

consequent is so also, it follows that there can be no sense in

reasoning in an isolated case, at all.

These considerations appear, at first sight, to dispose of the

difficulty mentioned. Yet the case of the other side is not yet

exhausted. Although probability will probably manifest its effect in,

say, a thousand risks, by a certain proportion between the numbers of

successes and failures, yet this, as we have seen, is only to say that

it certainly will, at length, do so. Now the number of risks, the number

of probable inferences, which a man draws in his whole life, is a finite

one, and he cannot be absolutely _certain_ that the mean result will

accord with the probabilities at all. Taking all his risks collectively,

then, it cannot be certain that they will not fail, and his case does

not differ, except in degree, from the one last supposed. It is an

indubitable result of the theory of probabilities that every gambler, if

he continues long enough, must ultimately be ruined. Suppose he tries

the martingale, which some believe infallible, and which is, as I am

informed, disallowed in the gambling-houses. In this method of playing,

he first bets say $1; if he loses it he bets $2; if he loses that he

bets $4; if he loses that he bets $8; if he then gains he has lost 1 + 2

+ 4 = 7, and he has gained $1 more; and no matter how many bets he

loses, the first one he gains will make him $1 richer than he was in the

beginning. In that way, he will probably gain at first; but, at last,

the time will come when the run of luck is so against him that he will

not have money enough to double, and must, therefore, let his bet go.

This will _probably_ happen before he has won as much as he had in the

first place, so that this run against him will leave him poorer than he

began; some time or other it will be sure to happen. It is true that

there is always a possibility of his winning any sum the bank can pay,

and we thus come upon a celebrated paradox that, though he is certain to

be ruined, the value of his expectation calculated according to the

usual rules (which omit this consideration) is large. But, whether a

gambler plays in this way or any other, the same thing is true, namely,

that if he plays long enough he will be sure some time to have such a

run against him as to exhaust his entire fortune. The same thing is true

of an insurance company. Let the directors take the utmost pains to be

independent of great conflagrations and pestilences, their actuaries can

tell them that, according to the doctrine of chances, the time must

come, at last, when their losses will bring them to a stop. They may

tide over such a crisis by extraordinary means, but then they will start

again in a weakened state, and the same thing will happen again all the

sooner. An actuary might be inclined to deny this, because he knows that

the expectation of his company is large, or perhaps (neglecting the

interest upon money) is infinite. But calculations of expectations leave

out of account the circumstance now under consideration, which reverses

the whole thing. However, I must not be understood as saying that

insurance is on this account unsound, more than other kinds of business.

All human affairs rest upon probabilities, and the same thing is true

everywhere. If man were immortal he could be perfectly sure of seeing

the day when everything in which he had trusted should betray his trust,

and, in short, of coming eventually to hopeless misery. He would break

down, at last, as every good fortune, as every dynasty, as every

civilization does. In place of this we have death.

But what, without death, would happen to every man, with death must

happen to some man. At the same time, death makes the number of our

risks, of our inferences, finite, and so makes their mean result

uncertain. The very idea of probability and of reasoning rests on the

assumption that this number is indefinitely great. We are thus landed in

the same difficulty as before, and I can see but one solution of it. It

seems to me that we are driven to this, that logicality inexorably

requires that our interests shall _not_ be limited. They must not stop

at our own fate, but must embrace the whole community. This community,

again, must not be limited, but must extend to all races of beings with

whom we can come into immediate or mediate intellectual relation. It

must reach, however vaguely, beyond this geological epoch, beyond all

bounds. He who would not sacrifice his own soul to save the whole world,

is, as it seems to me, illogical in all his inferences, collectively.

Logic is rooted in the social principle.

To be logical men should not be selfish; and, in point of fact, they are

not so selfish as they are thought. The willful prosecution of one’s

desires is a different thing from selfishness. The miser is not selfish;

his money does him no good, and he cares for what shall become of it

after his death. We are constantly speaking of _our_ possessions on the

Pacific, and of _our_ destiny as a republic, where no personal interests

are involved, in a way which shows that we have wider ones. We discuss

with anxiety the possible exhaustion of coal in some hundreds of years,

or the cooling-off of the sun in some millions, and show in the most

popular of all religious tenets that we can conceive the possibility of

a man’s descending into hell for the salvation of his fellows.

Now, it is not necessary for logicality that a man should himself be

capable of the heroism of self-sacrifice. It is sufficient that he

should recognize the possibility of it, should perceive that only that

man’s inferences who has it are really logical, and should consequently

regard his own as being only so far valid as they would be accepted by

the hero. So far as he thus refers his inferences to that standard, he

becomes identified with such a mind.

This makes logicality attainable enough. Sometimes we can personally

attain to heroism. The soldier who runs to scale a wall knows that he

will probably be shot, but that is not all he cares for. He also knows

that if all the regiment, with whom in feeling he identifies himself,

rush forward at once, the fort will be taken. In other cases we can only

imitate the virtue. The man whom we have supposed as having to draw from

the two packs, who if he is not a logician will draw from the red pack

from mere habit, will see, if he is logician enough, that he cannot be

logical so long as he is concerned only with his own fate, but that that

man who should care equally for what was to happen in all possible cases

of the sort could act logically, and would draw from the pack with the

most red cards, and thus, though incapable himself of such sublimity,

our logician would imitate the effect of that man’s courage in order to

share his logicality.

But all this requires a conceived identification of one’s interests with

those of an unlimited community. Now, there exist no reasons, and a

later discussion will show that there can be no reasons, for thinking

that the human race, or any intellectual race, will exist forever. On

the other hand, there can be no reason against it;[39] and, fortunately,

as the whole requirement is that we should have certain sentiments,

there is nothing in the facts to forbid our having a _hope_, or calm and

cheerful wish, that the community may last beyond any assignable date.

It may seem strange that I should put forward three sentiments, namely,

interest in an indefinite community, recognition of the possibility of

this interest being made supreme, and hope in the unlimited continuance

of intellectual activity, as indispensable requirements of logic. Yet,

when we consider that logic depends on a mere struggle to escape doubt,

which, as it terminates in action, must begin in emotion, and that,

furthermore, the only cause of our planting ourselves on reason is that

other methods of escaping doubt fail on account of the social impulse,

why should we wonder to find social sentiment presupposed in reasoning?

As for the other two sentiments which I find necessary, they are so only

as supports and accessories of that. It interests me to notice that

these three sentiments seem to be pretty much the same as that famous

trio of Charity, Faith, and Hope, which, in the estimation of St. Paul,

are the finest and greatest of spiritual gifts. Neither Old nor New

Testament is a textbook of the logic of science, but the latter is

certainly the highest existing authority in regard to the dispositions

of heart which a man ought to have.

V

Such average statistical numbers as the number of inhabitants per square

mile, the average number of deaths per week, the number of convictions

per indictment, or, generally speaking, the numbers of _x_’s per _y_,

where the _x_’s are a class of things some or all of which are connected

with another class of things, their _y_’s, I term _relative numbers_. Of

the two classes of things to which a relative number refers, that one of

which it is a number may be called its _relate_, and that one _per_

which the numeration is made may be called its _correlate_.

Probability is a kind of relative number; namely, it is the ratio of the

number of arguments of a certain genus which carry truth with them to

the total number of arguments of that genus, and the rules for the

calculation of probabilities are very easily derived from this

consideration. They may all be given here, since they are extremely

simple, and it is sometimes convenient to know something of the

elementary rules of calculation of chances.

RULE I. _Direct Calculation._—To calculate, directly, any relative

number, say for instance the number of passengers in the average trip of

a street-car, we must proceed as follows:

Count the number of passengers for each trip; add all these numbers, and

divide by the number of trips. There are cases in which this rule may be

simplified. Suppose we wish to know the number of inhabitants to a

dwelling in New York. The same person cannot inhabit two dwellings. If

he divide his time between two dwellings he ought to be counted a

half-inhabitant of each. In this case we have only to divide the total

number of the inhabitants of New York by the number of their dwellings,

without the necessity of counting separately those which inhabit each

one. A similar proceeding will apply wherever each individual relate

belongs to one individual correlate exclusively. If we want the number

of _x_’s per _y_, and no _x_ belongs to more than one _y_, we have only

to divide the whole number of _x_’s of _y_’s by the number of _y_’s.

Such a method would, of course, fail if applied to finding the average

number of street-car passengers per trip. We could not divide the total

number of travelers by the number of trips, since many of them would

have made many passages.

To find the probability that from a given class of premises, A, a given

class of conclusions, B, follow, it is simply necessary to ascertain

what proportion of the times in which premises of that class are true,

the appropriate conclusions are also true. In other words, it is the

number of cases of the occurrence of both the events A and B, divided by

the total number of cases of the occurrence of the event A.

RULE II. _Addition of Relative Numbers._—Given two relative numbers

having the same correlate, say the number of _x_’s per _y_, and the

number of _z_’s per _y_; it is required to find the number of _x_’s and

_z_’s together per _y_. If there is nothing which is at once an _x_ and

a _z_ to the same _y_, the sum of the two given numbers would give the

required number. Suppose, for example, that we had given the average

number of friends that men have, and the average number of enemies, the

sum of these two is the average number of persons interested in a man.

On the other hand, it plainly would not do to add the average number of

persons having constitutional diseases and over military age, to the

average number exempted by each special cause from military service, in

order to get the average number exempt in any way, since many are exempt

in two or more ways at once.

This rule applies directly to probabilities, given the probability that

two different and mutually exclusive events will happen under the same

supposed set of circumstances. Given, for instance, the probability that

if A then B, and also the probability that if A then C, then the sum of

these two probabilities is the probability that if A then either B or C,

so long as there is no event which belongs at once to the two classes B

and C.

RULE III. _Multiplication of Relative Numbers._—Suppose that we have

given the relative number of _x_’s per _y_; also the relative number of

_z_’s per _x_ of _y_; or, to take a concrete example, suppose that we

have given, first, the average number of children in families living in

New York; and, second, the average number of teeth in the head of a New

York child—then the product of these two numbers would give the average

number of children’s teeth in a New York family. But this mode of

reckoning will only apply in general under two restrictions. In the

first place, it would not be true if the same child could belong to

different families, for in that case those children who belonged to

several different families might have an exceptionally large or small

number of teeth, which would affect the average number of children’s

teeth in a family more than it would affect the average number of teeth

in a child’s head. In the second place, the rule would not be true if

different children could share the same teeth, the average number of

children’s teeth being in that case evidently something different from

the average number of teeth belonging to a child.

In order to apply this rule to probabilities, we must proceed as

follows: Suppose that we have given the probability that the conclusion

B follows from the premise A, B and A representing as usual certain

classes of propositions. Suppose that we also knew the probability of an

inference in which B should be the premise, and a proposition of a third

kind, C, the conclusion. Here, then, we have the materials for the

application of this rule. We have, first, the relative number of B’s per

A. We next should have the relative number of C’s per B following from

A. But the classes of propositions being so selected that the

probability of C following from any B in general is just the same as the

probability of C’s following from one of those B’s which is deducible

from an A, the two probabilities may be multiplied together, in order to

give the probability of C following from A. The same restrictions exist

as before. It might happen that the probability that B follows from A

was affected by certain propositions of the class B following from

several different propositions of the class A. But, practically

speaking, all these restrictions are of very little consequence, and it

is usually recognized as a principle universally true that the

probability that, if A is true, B is, multiplied by the probability

that, if B is true, C is, gives the probability that, if A is true, C

is.

There is a rule supplementary to this, of which great use is made. It is

not universally valid, and the greatest caution has to be exercised in

making use of it—a double care, first, never to use it when it will

involve serious error; and, second, never to fail to take advantage of

it in cases in which it can be employed. This rule depends upon the fact

that in very many cases the probability that C is true if B is, is

substantially the same as the probability that C is true if A is.

Suppose, for example, we have the average number of males among the

children born in New York; suppose that we also have the average number

of children born in the winter months among those born in New York. Now,

we may assume without doubt, at least as a closely approximate

proposition (and no very nice calculation would be in place in regard to

probabilities), that the proportion of males among all the children born

in New York is the same as the proportion of males born in summer in New

York; and, therefore, if the names of all the children born during a

year were put into an urn, we might multiply the probability that any

name drawn would be the name of a male child by the probability that it

would be the name of a child born in summer, in order to obtain the

probability that it would be the name of a male child born in summer.

The questions of probability, in the treatises upon the subject, have

usually been such as relate to balls drawn from urns, and games of

cards, and so on, in which the question of the _independence_ of events,

as it is called—that is to say, the question of whether the probability

of C, under the hypothesis B, is the same as its probability under the

hypothesis A, has been very simple; but, in the application of

probabilities to the ordinary questions of life, it is often an

exceedingly nice question whether two events may be considered as

independent with sufficient accuracy or not. In all calculations about

cards it is assumed that the cards are thoroughly shuffled, which makes

one deal quite independent of another. In point of fact the cards seldom

are, in practice, shuffled sufficiently to make this true; thus, in a

game of whist, in which the cards have fallen in suits of four of the

same suit, and are so gathered up, they will lie more or less in sets of

four of the same suit, and this will be true even after they are

shuffled. At least some traces of this arrangement will remain, in

consequence of which the number of “short suits,” as they are

called—that is to say, the number of hands in which the cards are very

unequally divided in regard to suits—is smaller than the calculation

would make it to be; so that, when there is a misdeal, where the cards,

being thrown about the table, get very thoroughly shuffled, it is a

common saying that in the hands next dealt out there are generally short

suits. A few years ago a friend of mine, who plays whist a great deal,

was so good as to count the number of spades dealt to him in 165 hands,

in which the cards had been, if anything, shuffled better than usual.

According to calculation, there should have been 85 of these hands in

which my friend held either three or four spades, but in point of fact

there were 94, showing the influence of imperfect shuffling.

According to the view here taken, these are the only fundamental rules

for the calculation of chances. An additional one, derived from a

different conception of probability, is given in some treatises, which

if it be sound might be made the basis of a theory of reasoning. Being,

as I believe it is, absolutely absurd, the consideration of it serves to

bring us to the true theory; and it is for the sake of this discussion,

which must be postponed to the next number, that I have brought the

doctrine of chances to the reader’s attention at this early stage of our

studies of the logic of science.

Footnote 34:

_Popular Science Monthly_, March, 1878.

Footnote 35:

[Later, pp. 170 ff. and 215 ff., it is shown that continuity is also

at the basis of mathematical generalization. See also article on

Synechism in _Baldwin’s Dictionary of Philosophy_.]

Footnote 36:

This mode of thought is so familiarly associated with all exact

numerical consideration, that the phrase appropriate to it is imitated

by shallow writers in order to produce the appearance of exactitude

where none exists. Certain newspapers which affect a learned tone talk

of “the average man,” when they simply mean _most men_, and have no

idea of striking an average.

Footnote 37:

_Cf._ pp. 179 ff. below.

Footnote 38:

The conception of probability here set forth is substantially that

first developed by Mr. Venn, in his _Logic of Chance_. Of course, a

vague apprehension of the idea had always existed, but the problem was

to make it perfectly clear, and to him belongs the credit of first

doing this.

Footnote 39:

I do not here admit an absolutely unknowable. Evidence could show us

what would probably be the case after any given lapse of time; and

though a subsequent time might be assigned which that evidence might

not cover, yet further evidence would cover it.

FOURTH PAPER

THE PROBABILITY OF INDUCTION[40]

I