CHAPTER VI.[1]

_THE SUBJECTIVE SIDE OF PROBABILITY. MEASUREMENT OF BELIEF._

1. Originally written in somewhat of a spirit of protest against what

seemed to me the prevalent disposition to follow De Morgan in taking

too subjective a view of the science. In reading it through now I

cannot find any single sentence to which I could take distinct

objection, though I must admit that if I were writing it entirely

afresh I should endeavour to express myself with less emphasis, and

I have made alterations in that direction. The reader who wishes to

see a view not substantially very different from mine, but expressed

with a somewhat opposite emphasis, can refer to Mr F. Y. Edgeworth's

article on "The Philosophy of Chance" (_Mind_, Vol. IX.)

1. Having now obtained a clear conception of a

certain kind of series, the next enquiry is, What is to be

done with this series? How is it to be employed as a means

of making inferences? The general step that we are now

about to take might be described as one from the objective

to the subjective, from the things themselves to the state of

our minds in contemplating them.

The reader should observe that a substitution has, in a

great number of cases, already been made as a first stage

towards bringing the things into a shape fit for calculation.

This substitution, as described in former chapters, is, in a

measure, a process of _idealization_. The series we actually

meet with are apt to show a changeable type, and the individuals

of them will sometimes transgress their licensed irregularity.

Hence they have to be pruned a little into shape, as

natural objects almost always have before they are capable of

being accurately reasoned about. The form in which the

series emerges is that of a series with a fixed type. This

imaginary or ideal series is the basis of our calculation.

2. It must not be supposed that this is at all at variance

with the assertion previously made, that Probability is a

science of inference about real things; it is only by a substitution

of the above kind that we are enabled to reason about

the things. In nature nearly all phenomena present themselves

in a form which departs from that rigorously accurate

one which scientific purposes mostly demand, so we have to

introduce an imaginary series, which shall be free from any

such defects. The only condition to be fulfilled is, that the

substitution is to be as little arbitrary, that is, to vary from

the truth as slightly, as possible. This kind of substitution

generally passes without notice when natural objects of any

kind are made subjects of exact science. I direct distinct

attention to it here simply from the apprehension that want

of familiarity with the subject-matter might lead some readers

to suppose that it involves, in this case, an exceptional deflection

from accuracy in the formal process of inference.

It may be remarked also that the adoption of this imaginary

series offers no countenance whatever to the doctrine

criticised in the last chapter, in accordance with which it was

supposed that our series possessed a fixed unchangeable type

which was merely the "development of the probabilities" of

things, to use Laplace's expression. It differs from anything

contemplated on that hypothesis by the fact that it is to be

recognized as a necessary substitution of our own for the

actual series, and to be kept in as close conformity with facts as

possible. It is a mere fiction or artifice necessarily resorted

to for the purpose of calculation, and for this purpose only.

This caution is the more necessary, because in the example

that I shall select, and which belongs to the most favourite

class of examples in this subject, the substitution becomes

accidentally unnecessary. The things, as has been repeatedly

pointed out, may sometimes need no trimming, because in

the form in which they actually present themselves they _are_

almost idealized. In most cases a good deal of alteration is

necessary to bring the series into shape, but in some--prominently

in the case of games of chance--we find the alterations,

for all practical purposes, needless.

3. We start then, from such a series as this, upon the

enquiry, What kind of inference can be made about it? It

may assist the logical reader to inform him that our first step

will be analogous to one class of what are commonly known

as _immediate_ inferences,--inferences, that is, of the type,--'All

men are mortal, therefore any particular man or men

are mortal.' This case, simple and obvious as it is in Logic,

requires very careful consideration in Probability.

It is obvious that we must be prepared to form an opinion

upon the propriety of taking the step involved in making

such an inference. Hitherto we have had as little to do as

possible with the irregular individuals; we have regarded

them simply as fragments of a regular series. But we cannot

long continue to neglect all consideration of them. Even if

these events in the gross be tolerably certain, it is not only

in the gross that we have to deal with them; they constantly

come before us a few at a time, or even as individuals,

and we have to form some opinion about them in this

state. An insurance office, for instance, deals with numbers

large enough to obviate most of the uncertainty, but

each of their transactions has another party interested in

it--What has the man who insures to say to their proceedings?

for to him this question becomes an individual one.

And even the office itself receives its cases singly, and would

therefore like to have as clear views as possible about these

single cases. Now, the remarks made in the preceding chapters

about the subjects which Probability discusses might seem to

preclude all enquiries of this kind, for was not ignorance of

the individual presupposed to such an extent that even (as

will be seen hereafter) causation might be denied, within

considerable limits, without affecting our conclusions? The

answer to this enquiry will require us to turn now to the

consideration of a totally distinct side of the question, and

one which has not yet come before us. Our best introduction

to it will be by the discussion of a special example.

4. Let a penny be tossed up a very great many

times; we may then be supposed to know for certain this

fact (amongst many others) that in the long run head and

tail will occur about equally often. But suppose we consider

only a moderate number of throws, or fewer still, and so

continue limiting the number until we come down to three

or two, or even one? We have, as the extreme cases, certainty

or something undistinguishably near it, and utter

uncertainty. Have we not, between these extremes, all

gradations of belief? There is a large body of writers, including

some of the most eminent authorities upon this

subject, who state or imply that we are distinctly conscious of

such a variation of the amount of our belief, and that this

state of our minds can be measured and determined with

almost the same accuracy as the external events to which

they refer. The principal mathematical supporter of this

view is De Morgan, who has insisted strongly upon it in all

his works on the subject. The clearest exposition of his

opinions will be found in his _Formal Logic_, in which work he

has made the view which we are now discussing the basis of

his system. He holds that we have a certain amount of belief

of every proposition which may be set before us, an amount

which in its nature admits of determination, though we may

practically find it difficult in any particular case to determine

it. He considers, in fact, that Probability is a sort of sister

science to Formal Logic,[1] speaking of it in the following

words: "I cannot understand why the study of the effect,

which partial belief of the premises produces with respect to

the conclusion, should be separated from that of the consequences

of supposing the former to be absolutely true."[2]

In other words, there is a science--Formal Logic--which investigates

the rules according to which one proposition can be

necessarily inferred from another; in close correspondence

with this there is a science which investigates the rules according

to which the amount of our belief of one proposition

varies with the amount of our belief of other propositions

with which it is connected.

The same view is also supported by another high authority,

the late Prof. Donkin, who says (_Phil. Mag._ May, 1851), "It

will, I suppose, be generally admitted, and has often been

more or less explicitly stated, that the subject-matter of

calculation in the mathematical theory of Probabilities is

_quantity of belief_."

5. Before proceeding to criticise this opinion, one remark

may be made upon it which has been too frequently

overlooked. It should be borne in mind that, even were this

view of the subject not actually incorrect, it might be objected

to as insufficient for the purpose of a definition, on the ground

that variation of belief is not confined to Probability. It is

a property with which that science is concerned, no doubt,

but it is a property which meets us in other directions as

well. In every case in which we extend our inferences by

Induction or Analogy, or depend upon the witness of others,

or trust to our own memory of the past, or come to a conclusion

through conflicting arguments, or even make a long and

complicated deduction by mathematics or logic, we have a

result of which we can scarcely feel as certain as of the premises

from which it was obtained. In all these cases then

we are conscious of varying quantities of belief, but are the

laws according to which the belief is produced and varied the

same? If they cannot be reduced to one harmonious scheme,

if in fact they can at best be brought to nothing but a number

of different schemes, each with its own body of laws and rules,

then it is vain to endeavour to force them into one science.

This opinion is strengthened by observing that most of

the writers who adopt the definition in question do practically

dismiss from consideration most of the above-mentioned

examples of diminution of belief, and confine their attention

to classes of events which have the property discussed in

Chap I., viz. 'ignorance of the few, knowledge of the many.'

It is quite true that considerable violence has to be done to

some of these examples, by introducing exceedingly arbitrary

suppositions into them, before they can be forced to assume

a suitable form. But still there is little doubt that, if we

carefully examine the language employed, we shall find that

in almost every case assumptions are made which virtually

imply that our knowledge of the individual is derived from

propositions given in the typical form described in Chap I.

This will be more fully proved when we come to consider

some common misapplications of the science.

6. Even then, if the above-mentioned view of the

subject were correct, it would yet, I consider, be insufficient

for the purpose of a definition; but it is at least very doubtful

whether it is correct. Before we could properly assign to

the belief side of the question the prominence given to it by

De Morgan and others, certainly before the science could be

defined from that side, it would be necessary, it appears, to

establish the two following positions, against both of which

strong objections can be brought.

(1) That our belief of every proposition is a thing which

we can, strictly speaking, be said to measure; that

there must be a certain amount of it in every case,

which we can realize somehow in consciousness and

refer to some standard so as to pronounce upon its

value.

(2) That the value thus apprehended is the correct one

according to the theory, viz. that it is the exact

fraction of full conviction that it should be. This

statement will perhaps seem somewhat obscure at

first; it will be explained presently.

7. (I.) Now, in the first place, as regards the difficulty

of obtaining any measure of the amount of our belief. One

source of this difficulty is too obvious to have escaped notice;

this is the disturbing influence produced on the quantity of

belief by any strong emotion or passion. A deep interest in

the matter at stake, whether it excite hope or fear, plays great

havoc with the belief-meter, so that we must assume the

mind to be quite unimpassioned in weighing the evidence.

This is noticed and acknowledged by Laplace and others;

but these writers seem to me to assume it to be the only

source of error, and also to be of comparative unimportance.

Even if it were the only source of error I cannot see that it

would be unimportant. We experience hope or fear in so

very many instances, that to omit such influences from consideration

would be almost equivalent to saying that whilst

we profess to consider the whole quantity of our belief we

will in reality consider only a portion of it. Very strong

feelings are, of course, exceptional, but we should nevertheless

find that the emotional element, in some form or other,

makes itself felt on almost every occasion. It is very seldom

that we cannot speak of our surprise or expectation in reference

to any particular event. Both of these expressions, but

especially the former, seem to point to something more than

mere belief. It is true that the word 'expectation' is generally

defined in treatises on Probability as equivalent to

belief; but it seems doubtful whether any one who attends

to the popular use of the terms would admit that they were

exactly synonymous. Be this however as it may, the emotional

element is present upon almost every occasion, and its

disturbing influence therefore is constantly at work.

8. Another cause, which co-operates with the former,

is to be found in the extreme complexity and variety of the

evidence on which our belief of any proposition depends.

Hence it results that our actual belief at any given moment

is one of the most fugitive and variable things possible, so

that we can scarcely ever get sufficiently clear hold of it to

measure it. This is not confined to the times when our

minds are in a turmoil of excitement through hope or fear.

In our calmest moments we shall find it no easy thing to

give a precise answer to the question, How firmly do I hold

this or that belief? There may be one or two prominent

arguments in its favour, and one or two corresponding objections

against it, but this is far from comprising all the

causes by which our state of belief is produced. Because

such reasons as these are all that can be practically introduced

into oral or written controversies, we must not conclude

that it is by these only that our conviction is influenced.

On the contrary, our conviction generally rests upon a sort

of chaotic basis composed of an infinite number of inferences

and analogies of every description, and these moreover distorted

by our state of feeling at the time, dimmed by the

degree of our recollection of them afterwards, and probably

received from time to time with varying force according to

the way in which they happen to combine in our consciousness

at the moment. To borrow a striking illustration from

Abraham Tucker, the substructure of our convictions is not

so much to be compared to the solid foundations of an ordinary

building, as to the piles of the houses of Rotterdam

which rest somehow in a deep bed of soft mud. They bear

their weight securely enough, but it would not be easy to

point out accurately the dependence of the different parts

upon one another. Directly we begin to think of the amount

of our belief, we have to think of the arguments by which it

is produced--in fact, these arguments will intrude themselves

without our choice. As each in turn flashes through the

mind, it modifies the strength of our conviction; we are like

a person listening to the confused hubbub of a crowd, where

there is always something arbitrary in the particular sound

we choose to listen to. There may be reasons enough to

suffice abundantly for our ultimate choice, but on examination

we shall find that they are by no means apprehended

with the same force at different times. The belief produced

by some strong argument may be very decisive at the moment,

but it will often begin to diminish when the argument

is not actually before the mind. It is like being dazzled by

a strong light; the impression still remains, but begins almost

immediately to fade away. I think that this is the

case, however we try to limit the sources of our conviction.

9. (II.) But supposing that it were possible to strike

a sort of average of this fluctuating state, should we find this

average to be of the amount assigned by theory? In other

words, is our natural belief in the happening of two different

events in direct proportion to the frequency with which those

events happen in the long run? There is a lottery with 100

tickets and ten prizes; is a man's belief that he will get a

prize fairly represented by one-tenth of certainty? The mere

reference to a lottery should be sufficient to disprove this.

Lotteries have flourished at all times, and have never failed

to be abundantly supported, in spite of the most perfect conviction,

on the part of many, if not of most, of those who put

into them, that in the long run all will lose. Deductions

should undoubtedly be made for those who act from superstitious

motives, from belief in omens, dreams, and so on.

But apart from these, and supposing any one to come fortified

by all that mathematics can do for him, it is difficult to

believe that his natural impressions about single events would

be always what they should be according to theory. Are

there many who can honestly declare that they would have

no desire to buy a single ticket? They would probably say

to themselves that the sum they paid away was nothing

worth mentioning to lose, and that there was a chance of

gaining a great deal; in other words, they are not apportioning

their belief in the way that theory assigns.

What bears out this view is, that the same persons who

would act in this way in single instances would often not

think of doing so in any but single instances. In other

words, the natural tendency here is to attribute too great an

amount of belief where it is or should be small; i.e. to depreciate

the risk in proportion to the contingent advantage.

They would very likely, when argued with, attach disparaging

epithets to this state of feeling, by calling it an unaccountable

fascination, or something of that kind, but of

its existence there can be little doubt. We are speaking

now of what is the natural tendency of our minds, not of

that into which they may at length be disciplined by education

and thought. If, however, educated persons have succeeded

for the most part in controlling this tendency in

games of chance, the spirit of reckless speculation has

scarcely yet been banished from commerce. On examination,

this tendency will be found so prevalent in all ages, ranks,

and dispositions, that it would be inadmissible to neglect it in

order to bring our supposed instincts more closely into accordance

with the commonly received theories of Probability.

10. There is another aspect of this question which has

been often overlooked, but which seems to deserve some

attention. Granted that we have an instinct of credence,

why should it be assumed that this must be just of that intensity

which subsequent experience will justify? Our

instincts are implanted in us for good purposes, and are intended

to act immediately and unconsciously. They are,

however, subject to control, and have to be brought into

accordance with what we believe to be true and right. In

other departments of psychology we do not assume that

every spontaneous prompting of nature is to be left just as

we find it, or even that on the average, omitting individual

variations, it is set at that pitch that will be found in the

end to be the best when we come to think about it and assign

its rules. Take, for example, the case of resentment. Here

we have an instinctive tendency, and one that on the whole

is good in its results. But moralists are agreed that almost

all our efforts at self-control are to be directed towards subduing

it and keeping it in its right direction. It is assumed

to be given as a sort of rough protection, and to be set, if

one might so express oneself, at too high a pitch to be

deliberately and consciously acted on in society. May not

something of this kind be the case also with our belief?

I only make a passing reference to this point here, as on

the theory of Probability adopted in this work it does not

appear to be at all material to the science. But it seems

a strong argument against the expediency of commencing

the study of the science from the subjective side, or even of

assigning any great degree of prominence to this side.

That men _do_ not believe in exact accordance with this

theory must have struck almost every one, but this has

probably been considered as mere exception and irregularity;

the assumption being made that on the average, and in far

the majority of cases, they do so believe. As stated above,

it is very doubtful whether the tendency which has just

been discussed is not so widely prevalent that it might with

far more propriety be called the rule than the exception.

And it may be better that this should be so: many good

results may follow from that cheerful disposition which induces

a man sometimes to go on trying after some great

good, the chance of which he overvalues. He will keep on

through trouble and disappointment, without serious harm

perhaps, when the cool and calculating bystander sees plainly

that his 'measure of belief' is much higher than it should

be. So, too, the tendency also so common, of underrating

the chance of a great evil may also work for good. By many

men death might be looked upon as an almost infinite evil,

at least they would so regard it themselves; suppose they

kept this contingency constantly before them at its right

value, how would it be possible to get through the practical

work of life? Men would be stopping indoors because if

they went out they might be murdered or bitten by a mad

dog. To say this is not to advocate a return to our instincts;

indeed when we have once reached the critical and conscious

state, it is hardly possible to do so; but it should be noticed

that the advantage gained by correcting them is at best but

a balanced one.[3] What is most to our present purpose, it

suggests the inexpediency of attempting to found an exact

theory on what may afterwards prove to be a mere instinct,

unauthorized in its full extent by experience.

11. It may be replied, that though people, as a matter

of fact, do not apportion belief in this exact way, yet they

_ought_ to do so. The purport of this remark will be examined

presently; it need only be said here that it grants all that

is now contended for. For it admits that the degree of our

belief is capable of modification, and may need it. But in

accordance with what is the belief to be modified? obviously

in accordance with experience; it cannot be trusted to by

itself, but the fraction at which it is to be rated must be

determined by the comparative frequency of the events to

which it refers. Experience then furnishing the standard, it

is surely most reasonable to start from this experience, and to

found the theory of our processes upon it.

If we do not do this, it should be observed that we are

detaching Probability altogether from the study of things

external to us, and making it nothing else in effect than a

portion of Psychology. If we refuse to be controlled by

experience, but confine our attention to the laws according

to which belief is naturally or instinctively compounded and

distributed in our minds, we have no right then to appeal to

experience afterwards even for illustrations, unless under the

express understanding that we do not guarantee its accuracy.

Our belief in some single events, for example, might be correct,

and yet that in a compound of several (if derived merely

from our instinctive laws of belief) very possibly might not

be correct, but might lead us into practical mistakes if we

determined to act upon it. Even if the two were in accordance,

this accordance would have to be proved, which would

lead us round, by what I cannot but think a circuitous

process, to the point which has been already chosen for

commencing with.

12. De Morgan seems to imply that the doctrine

criticised above finds a justification from the analogy of

Formal Logic. If the laws of necessary inference can be

studied apart from all reference to external facts (except

by way of illustration), why not those of probable inference?

There does not, however, seem to be much force in any such

analogy. Formal Logic, at any rate under its modern or

Kantian mode of treatment, is based upon the assumption

that there are laws of thought as distinguished from laws of

things, and that these laws of thought can be ascertained and

studied without taking into account their reference to any

particular object. Now so long as we are confined to necessary

or irreversible laws, as is of course the case in ordinary

Formal Logic, this assumption leads to no special difficulties.

We mean by this, that no conflict arises between these subjective

and objective necessities. The two exist in perfect

harmony side by side, the one being the accurate counterpart

of the other. So precise is the correspondence between

them, that few persons would notice, until study of metaphysics

had called their attention to such points, that there

were these two sides to the question. They would make

their appeal to either with equal confidence, saying indifferently,

'the thing must be so,' or, 'we cannot conceive its being

otherwise.' In fact it is only since the time of Kant that

this mental analysis has been to any extent appreciated and

accepted. And even now the dominant experience school of

philosophy would not admit that there are here two really

distinct sides to the phenomenon; they maintain either that

the subjective necessity is nothing more than the consequence

by inveterate association of the objective uniformity,

or else that this so-called necessity (say in the Law of Contradiction)

is after all merely verbal, merely a different way of

saying the same thing over again in other words. Whatever

the explanation adopted, the general result is that fallacies,

as real acts of thought, are impossible within the domain of

pure logic; error within that province is only possibly by a

momentary lapse of attention, that is of consciousness.

13. But though this perfect harmony between subjective

and objective uniformities or laws may exist within

the domain of pure logic, it is far from existing within that

of probability. The moment we make the _quantity_ of our

belief an integral part of the subject to be studied, any such

invariable correspondence ceases to exist. In the former

case, we could not consciously think erroneously even though

we might try to do so; in the latter, we not only can believe

erroneously but constantly do so. Far from the quantity of

our belief being so exactly adjusted in conformity with the

facts to which it refers that we cannot even in imagination

go astray, we find that it frequently exists in excess or defect

of that which subsequent judgment will approve. Our instincts

of credence are unquestionably in frequent hostility

with experience; and what do we do then? We simply

modify the instincts into accordance with the things. We

are constantly performing this practice, and no cultivated

mind would find it possible to do anything else. No man

would think of divorcing his belief from the things on which

it was exercised, or would suppose that the former had anything

else to do than to follow the lead of the latter. Hence

it results that that separation of the subjective necessity from

the objective, and that determination to treat the former

as a science apart by itself, for which a plausible defence

could be made in the case of pure logic, is entirely inadmissible

in the case of probability. However we might

contrive to '_think_' aright without appeal to facts, we cannot

_believe_ aright without incessantly checking our proceedings

by such appeals. Whatever then may be the

claims of Formal Logic to rank as a separate science, it

does not appear that it can furnish any support to the

theory of Probability at present under examination.

14. The point in question is sometimes urged as

follows. Suppose a man with two, and only two, alternatives

before him, one of which he knows must involve

success and the other failure. He knows nothing more

about them than this, and he is forced to act. Would he

not regard them with absolutely similar and equal feelings

of confidence, without the necessity of referring them to any

real or imaginary series? If so, is not this equivalent to

saying that his belief of either, since one of them must

come to pass, is equal to that of the other, and therefore that

his belief of each is one-half of full confidence? Similarly

if there are more than two alternatives: let it be supposed

that there are any number of them, amongst which no

distinctions whatever can be discerned except in such particulars

as we know for certain will not affect the result;

should we not feel equally confident in respect of each of

them? and so here again should we riot have a fractional

estimate of our absolute amount of belief? It is thus

attempted to lay the basis of a pure science of Probability,

determining the distribution and combination of our belief

hypothetically; viz. _if_ the contingencies are exactly alike,

then our belief is so apportioned, the question whether the

contingencies are equal being of course decided as the objective

data of Logic or Mathematics are decided.

To discuss this question fully would require a statement

at some length of the reasons in favour of the objective or

material view of Logic, as opposed to the Formal or Conceptualist.

I shall have to speak on this subject in another

chapter, and will not therefore enter upon it here. But one

conclusive objection which is applicable more peculiarly to

Probability may be offered at once. To pursue the line of

enquiry just indicated, is, as already remarked, to desert

the strictly logical ground, and to take up that appropriate

to psychology; the proper question, in all these cases, being

not what _do_ men believe, but what ought they to believe?

Admitting, as was done above, that in the case of Formal

Logic these two enquiries, or rather those corresponding to

them, practically run into one, owing to the fact that men

cannot consciously 'think' wrongly; it cannot be too strongly

insisted on that in Probability the two are perfectly separable

and distinct. It is of no use saying what men do or

will believe, we want to know what they will be right in

believing; and this can never be settled without an appeal

to the phenomena themselves.

15. But apart from the above considerations, this way

of putting the case does not seem to me at all conclusive.

Take the following example. A man[4] finds himself on the

sands of the Wash or Morecambe Bay, in a dense mist, when

the spring-tide is coming in; and knows therefore that to

be once caught by the tide would be fatal. He hears a

church-bell at a distance, but has no means of knowing

whether it is on the same side of the water with himself or

on the opposite side. He cannot tell therefore whether by

following its sound he will be led out into the mid-stream

and be lost, or led back to dry land and safety. Here there

can be no repetition of the event, and the cases are indistinguishably

alike, to him, in the only circumstances which

can affect the issue: is not then his prospect of death, it

will be said, necessarily equal to one-half? A proper analysis

of his state of mind would be a psychological rather than

a logical enquiry, and in any case, as above remarked, the

decision of this question does not touch our logical position.

But according to the best introspection I can give I should

say that what really passes through the mind in such a case

is something of this kind: In most doubtful positions and

circumstances we are accustomed to decide our conduct by

a consideration of the relative advantages and disadvantages

of each side, that is by the observed or inferred frequency

with which one or the other alternative has succeeded. In

proportion as these become more nearly balanced, we are

more frequently mistaken in the individual cases; that is, it

becomes more and more nearly what would be called 'a

mere toss up' whether we are right or wrong. The case

in question seems merely the limiting case, in which it has

been contrived that there shall be no appreciable difference

between the alternatives, by which to decide in favour of

one or other, and we accordingly feel no confidence in the

particular result. Having to decide, however, we decide according

to the precedent of similar cases which have occurred

before. To stand still and wait for better information is

certain death, and we therefore appeal to and employ the

only rule we know of; or rather we feel, or endeavour to

feel, as we have felt before when acting in the presence of

alternatives as nearly balanced as possible. But I can

neither perceive in my own case, nor feel convinced in that

of others, that this appeal, in a case which cannot be repeated,[5]

to a rule acted on and justified in cases which can be

and are repeated, at all forces us to admit that our state of

mind is the same in each case.

16. This example serves to bring out very clearly a point

which has been already mentioned, and which will have to be

insisted upon again, viz. that all which Probability discusses

is the statistical frequency of events, or, if we prefer so to

put it, the quantity of belief with which any one of these

events should be individually regarded, but leaves all the

subsequent conduct dependent upon that frequency, or that

belief, to the choice of the agents. Suppose there are two

travellers in the predicament in question: shall they keep

together, or separate in opposite directions? In either case

alike the chance of safety to each is the same, viz. one-half,

but clearly their circumstances must decide which course it

is preferable to adopt. If they are husband and wife, they will

probably prefer to remain together; if they are sole depositaries

of an important state secret, they may decide to part.

In other words, we have to select here between the two alternatives

of the certainty of a single loss, and the even chance

of a double loss; alternatives which the common mathematical

statement of their chances has a decided tendency

to make us regard as indistinguishable from one another.

But clearly the decision must be grounded on the desires,

feelings, and conscience of the agents. Probability cannot

say a word upon this question. As I have pointed out elsewhere,

there has been much confusion on this matter in

applications of the science to betting, and in the discussion

of the Petersburg problem.

We have thus examined the doctrine in question with

a minuteness which may seem tedious, but in consequence of

the eminence of its supporters it would have been presumptuous

to have rejected it without the strongest grounds. The

objections which have been urged might be summarised as

follows:--the amount of our belief of any given proposition,

supposing it to be in its nature capable of accurate determination

(which does not seem to be the case), depends upon a

great variety of causes, of which statistical frequency--the

subject of Probability--is but one. That even if we confine

our attention to this one cause, the natural amount of our

belief is not necessarily what theory would assign, but has to

be checked by appeal to experience. The subjective side of

Probability therefore, though very interesting and well deserving

of examination, seems a mere appendage of the objective,

and affords in itself no safe ground for a science of

inference.

17. The conception then of the science of Probability

as a science of the laws of belief seems to break down at

every point. We must not however rest content with such

merely negative criticism. The degree of belief we entertain

of a proposition may be hard to get at accurately, and

when obtained may be often wrong, and may need therefore

to be checked by an appeal to the objects of belief. Still in

popular estimation we do seem to be able with more or less

accuracy to form a graduated scale of intensity of belief.

What we have to examine now is whether this be possible,

and, if so, what is the explanation of the fact?

That it is generally believed that we can form such a

scale scarcely admits of doubt. There is a whole vocabulary

of common expressions such as, 'I feel almost sure,' 'I do not

feel quite certain,' 'I am less confident of this than of that,'

and so on. When we make use of any one of these phrases

we seldom doubt that we have a distinct meaning to convey

by means of it. Nor do we feel much at a loss, under any

given circumstances, as to which of these expressions we

should employ in preference to the others. If we were asked

to arrange in order, according to the intensity of the belief

with which we respectively hold them, things broadly marked

off from one another, we could do it from our consciousness

of belief alone, without a fresh appeal to the evidence upon

which the belief depended. Passing over the looser propositions

which are used in common conversation, let us take but

one simple example from amongst those which furnish numerical

data. Do I not feel more certain that some one will die

this week in the whole town, than in the particular street in

which I live? and if the town is known to contain a population

one hundred times greater than that in the street, would

not almost any one be prepared to assert on reflection that

he felt a hundred times more sure of the first proposition

than of the second? Or to take a non-numerical example, are

we not often able to say unhesitatingly which of two propositions

we believe the most, and to some rough degree how

much more we believe one than the other, at a time when all

the evidence upon which each rests has faded from the

mind, so that each has to be judged, as we may say, solely on

its own merits?

Here then a problem proposes itself. If popular opinion,

as illustrated in common language, be correct,--and very

considerable weight must of course be attributed to it,--there

does exist something which we call partial belief in reference

to any proposition of the numerical kind described

above. Now what we want to do is to find some test or

justification of this belief, to obtain in fact some intelligible

answer to the question, Is it correct? We shall find incidentally

that the answer to this question will throw a good deal

of light upon another question nearly as important and far

more intricate, viz. What is the meaning of this partial belief?

18. We shall find it advisable to commence by ascertaining

how such enquiries as the above would be answered

in the case of ordinary full belief. Such a step would not

offer the slightest difficulty. Suppose, to take a simple

example, that we have obtained the following proposition,--whether

by induction, or by the rules of ordinary deductive

logic, does not matter for our present purpose,--that a certain

mixture of oxygen and hydrogen is explosive. Here we have

an inference, and consequent belief of a proposition. Now

suppose there were any enquiry as to whether our belief

were correct, what should we do? The simplest way of

settling the matter would be to find out by a distinct appeal

to experience whether the proposition was true. Since we

are reasoning about things, the justification of the belief, that

is, the test of its correctness, would be most readily found in

the truth of the proposition. If by any process of inference

I have come to believe that a certain mixture will explode, I

consider my belief to be justified, that is to be correct, if

under proper circumstances the explosion always does occur;

if it does not occur the belief was wrong.

Such an answer, no doubt, goes but a little way, or rather

no way at all, towards explaining what is the nature of belief

in itself; but it is sufficient for our present purpose, which is

merely that of determining what is meant by the correctness

of our belief, and by the test of its correctness. In all inferences

about things, in which the _amount_ of our belief is not

taken into account, such an explanation as the above is quite

sufficient; it would be the ordinary one in any question of

science. It is moreover perfectly intelligible, whether the

conclusion is particular or universal. Whether we believe

that 'some men die', or that 'all men die', our belief may with

equal ease be tested by the appropriate train of experience.

19. But when we attempt to apply the same test to

_partial_ belief, we shall find ourselves reduced to an awkward

perplexity. A difficulty now emerges which has been singularly

overlooked by those who have treated of the subject.

As a simple example will serve our purpose, we will take the

case of a penny. I am about to toss one up, and I therefore

half believe, to adopt the current language, that it will give

head. Now it seems to be overlooked that if we appeal to

the event, as we did in the case last examined, our belief

must inevitably be wrong, and therefore the test above mentioned

will fail. For the thing must either happen or not

happen: i.e. in this case the penny must either give head, or

not give it; there is no third alternative. But whichever

way it occurs, our half-belief, so far as such a state of mind

admits of interpretation, must be wrong. If head does come,

I am wrong in not having expected it enough; for I only half

believed in its occurrence. If it does not happen, I am

equally wrong in having expected it too much; for I half

believed in its occurrence, when in fact it did not occur at all.

The same difficulty will occur in every case in which we

attempt to justify our state of partial belief in a single contingent

event. Let us take another example, slightly differing

from the last. A man is to receive £1 if a die gives six,

to pay 1_s_. if it gives any other number. It will generally be

admitted that he ought to give 2_s_. 6_d_. for the chance, and

that if he does so he will be paying a fair sum. This example

only differs from the last in the fact that instead of

simple belief in a proposition, we have taken what mathematicians

call 'the _value_ of the expectation'. In other words,

we have brought into a greater prominence, not merely the belief,

but the conduct which is founded upon the belief. But

precisely the same difficulty recurs here. For appealing to the

event,--the single event, that is,--we see that one or other

party must lose his money without compensation. In what

sense then can such an expectation be said to be a fair one?

20. A possible answer to this, and so far as appears the

only possible answer, will be, that what we really mean by

saying that we half believe in the occurrence of head is to

express our conviction that head will certainly happen on

the average every other time. And similarly, in the second

example, by calling the sum a fair one it is meant that in

the long run neither party will gain or lose. As we shall

recur presently to the point raised in this form of answer, the

only notice that need be taken of it at this point is to call

attention to the fact that it entirely abandons the whole

question in dispute, for it admits that this partial belief does

not in any strict sense apply to the individual event, since it

clearly cannot be justified there. At such a result indeed we

cannot be surprised; at least we cannot on the theory adopted

throughout this Essay. For bearing in mind that the employment

of Probability postulates ignorance of the single

event, it is not easy to see how we are to justify any other

opinion or statement about the single event than a confession

of such ignorance.

21. So far then we do not seem to have made the

slightest approximation to a solution of the particular

question now under examination. The more closely we have

analysed special examples, the more unmistakeably are we

brought to the conclusion that in the individual instance no

justification of anything like quantitative belief is to be

found; at least none is to be found in the same sense in

which we expect it in ordinary scientific conclusions, whether

Inductive or Deductive. And yet we have to face and

account for the fact that common impressions, as attested by

a whole vocabulary of common phrases, are in favour of the

existence of this quantitative belief. How are we to account

for this? If we appeal to an example again, and analyse it

somewhat more closely, we may yet find our way to some

satisfactory explanation.

In our previous analysis (§18) we found it sufficient to stop

at an early stage, and to give as the justification of our belief

the fact of the proposition being true. Stopping however at

that stage, we have found this explanation fail altogether to

give a justification of partial belief; fail, that is, when applied

to the individual instance. The two states of belief and disbelief

correspond admirably to the two results of the event

happening and not happening respectively, and unless for

psychological purposes we saw no reason to analyse further;

but to partial belief there is nothing corresponding in the result,

for the event cannot partially happen in such cases as we

are concerned with. Suppose then we advance a step further

in the analysis, and ask again what is meant by the proposition

being true? This introduces us, of course, to a very long

and intricate path; but in the short distance along it which

we shall advance, we shall not, it is to be hoped, find any

very serious difficulty. As before, we will illustrate the

analysis by first applying it to the case of ordinary full belief.

22. Whatever opinion then may be held about the

essential nature of belief, it will probably be admitted that a

readiness to act upon the proposition believed is an inseparable

accompaniment of that state of mind. There can be no

alteration in our belief (at any rate in the case of sane persons)

without a possible alteration in our conduct, nor anything in

our conduct which is not connected with something in our

belief. We will first take an example in connection with the

penny, in which there is full belief; we will analyse it a step

further than we did before, and then attempt to apply the

same analysis to an example of a similar kind, but one in

which the belief is partial instead of full.

Suppose that I am about to throw a penny up, and contemplate

the prospect of its falling upon one of its sides and

not upon its edge. We feel perfectly confident that it will

do so. Now whatever else may be implied in our belief, we

certainly mean this; that we are ready to stake our conduct

upon its falling thus. All our betting, and everything else

that we do, is carried on upon this supposition. Any risk

whatever that might ensue upon its falling otherwise will be

incurred without fear. This, it must be observed, is equally

the case whether we are speaking of a single throw or of a

long succession of throws.

But now let us take the case of a penny falling, not upon

one side or the other, but upon a given side, _head_. To a

certain extent this example resembles the last. We are perfectly

ready to stake our conduct upon what comes to pass in

the long run. When we are considering the result of a large

number of throws, we are ready to act upon the supposition

that head comes every other time. If e.g. we are betting

upon it, we shall not object to paying £1 every time that

head comes, on condition of receiving £1 every time that

head does not come. This is nothing else than the _translation_,

as we may call it, into practice, of our belief that head

and tail occur equally often.

Now it will be obvious, on a moment's consideration, that

our conduct is capable of being slightly varied: of being

varied, that is, in form, whilst it remains identical in respect

of its results. It is clear that to pay £1 every time we lose,

and to get £1 every time we gain, comes to precisely the same

thing, in the case under consideration, as to pay ten shillings

every time without exception, and to receive £1 every time

that head occurs. It is so, because heads occur, on the

average, every other time. In the long run the two results

coincide; but there is a marked difference between the two

cases, considered individually. The difference is two-fold.

In the first place we depart from the notion of a payment

every other time, and come to that of one made every time.

In the second place, what we pay every time is half of what

we get in the cases in which we do get anything. The difference

may seem slight; but mark the effect when our conduct

is translated back again into the subjective condition

upon which it depends, viz. into our belief. It is in consequence

of such a translation, as it appears to me, that the

notion has been acquired that we have an accurately determinable

amount of belief as to every such proposition. To

have losses and gains of equal amount, and to incur them

equally often, was the experience connected with our belief

that the two events, head and tail, would occur equally often.

This was quite intelligible, for it referred to the long run.

To find that this could be commuted for a payment made

every time without exception, a payment, observe, of half the

amount of what we occasionally receive, has very naturally

been interpreted to mean that there must be a state of half-belief

which refers to each individual throw.

23. One such example, of course, does not go far towards

establishing a theory. But the reader will bear in

mind that almost all our conduct tends towards the same

result; that it is not in betting only, but in every course of

action in which we have to count the events, that such a

numerical apportionment of our conduct is possible. Hence,

by the ordinary principles of association, it would appear

exceedingly likely that, not exactly a numerical condition of

mind, but rather numerical associations, become inseparably

connected with each particular event which we know to occur

in a certain proportion of times. Once in six times a die

gives ace; a knowledge of this fact, taken in combination

with all the practical results to which it leads, produces, one

cannot doubt, an inseparable notion of one-sixth connected

with each _single_ throw. But it surely cannot be called belief

to the amount of one-sixth; at least it admits neither of

justification nor explanation in these single cases, to which

alone the fractional belief, if such existed, ought to apply.

It is in consequence, I apprehend, of such association that

we act in such an unhesitating manner in reference to any

single contingent event, even when we have no expectation

of its being repeated. A die is going to be thrown up once,

and once only. I bet 5 to 1 against ace, not, as is commonly

asserted, because I feel one-sixth part of certainty in the

occurrence of ace; but because I know that such conduct

would be justified in the long run of such cases, and I apply

to the solitary individual the same rule that I should apply

to it if I knew it were one of a long series. This accounts

for my conduct being the same in the two cases; by association,

moreover, we probably experience very similar feelings

in regard to them both.

24. And here, on the view of the subject adopted in

this Essay, we might stop. We are bound to explain the

'measure of our belief' in the occurrence of a single event

when we judge solely from the statistical frequency with

which such events occur, for such a series of events was our

starting-point; but we are not bound to inquire whether in

every case in which persons have, or claim to have, a certain

measure of belief there must be such a series to which to

refer it, and by which to justify it. Those who start from

the subjective side, and regard Probability as the science of

quantitative belief, are obliged to do this, but we are free

from the obligation.

Still the question is one which is so naturally raised in

connection with this subject, that it cannot be altogether

passed by. I think that to a considerable extent such a

justification as that mentioned above will be found applicable

in other cases. The fact is that we are very seldom called

upon to decide and act upon a single contingency which cannot

be viewed as being one of a series. Experience introduces

us, it must be remembered, not merely to a succession

of events neatly arranged in a single series (as we have

hitherto assumed them to be for the purpose of illustration),

but to an infinite number belonging to a vast variety of

different series. A man is obliged to be acting, and therefore

exercising his belief about one thing or another, almost

the whole of every day of his life. Any one person will have

to decide in his time about a multitude of events, each one

of which may never recur again within his own experience.

But by the very fact of there being a multitude, though they

are all of different kinds, we shall still find that order is

maintained, and so a course of conduct can be justified. In

a plantation of trees we should find that there is order of a

certain kind if we measure them in any one direction, the

trees being on an average about the same distance from each

other. But a somewhat similar order would be found if we

were to examine them in any other direction whatsoever. So

in nature generally; there is regularity in a succession of

events of the same kind. But there may also be regularity

if we form a series by taking successively a number out of

totally distinct kinds.

It is in this circumstance that we find an extension of the

practical justification of the measure of our belief. A man,

say, buys a life annuity, insures his life on a railway journey,

puts into a lottery, and so on. Now we may make a series

out of these acts of his, though each is in itself a single event

which he may never intend to repeat. His conduct, and therefore

his belief, measured by the result in each individual

instance, will not be justified, but the reverse, as shown in §19.

Could he indeed repeat each kind of action often

enough it would be justified; but from this, by the conditions

of life, he is debarred. Now it is perfectly conceivable that

in the new series, formed by his successive acts of different

kinds, there should be no regularity. As a matter of fact,

however, it is found that there is regularity. In this way the

equalization of his gains and losses, for which he cannot hope

in annuities, insurances, and lotteries taken separately, may

yet be secured to him out of these events taken collectively.

If in each case he values his chance at its right proportion

(and acts accordingly) he will in the course of his life neither

gain nor lose. And in the same way if, whenever he has the

alternative of different courses of conduct, he acts in accordance

with the estimate of his belief described above, i.e.

chooses the event whose chance is the best, he will in the end

gain more in this way than by any other course. By the existence,

therefore, of these _cross-series_, as we may term them,

there is an immense addition to the number of actions which

may be fairly considered to belong to those courses of conduct

which offer many successive opportunities of equalizing

gains and losses. All these cases then may be regarded as

admitting of justification in the way now under discussion.

25. In the above remarks it will be observed that we

have been giving what is to be regarded as a justification of

his belief from the point of view of the individual agent himself.

If we suppose the existence of an enlarged fellow-feeling,

the applicability of such a justification becomes still more

extensive. We can assign a very intelligible sense to the

assertion that it is 999 to 1 that I shall not get a prize in a

lottery, even if this be stated in the form that my belief in

my so doing is represented by the fraction 1/1000th of certainty.

Properly it means that in a very large number of throws I

should gain once in 1000 times. If we include other contingencies

of the same kind, as described in the last section,

each individual may be supposed to reach to something like

this experience within the limits of his own life. He could

not do it in this particular line of conduct alone, but he could

do it in this line combined with others. Now introduce the

possibility of each man feeling that the gain of others offers

some analogy to his own gains, which we may conceive his

doing except in the case of the gains of those against whom

he is directly competing, and the above justification becomes

still more extensively applicable.

The following would be a fair illustration to test this

view. I know that I must die on some day of the week, and

there are but seven days. My belief, therefore, that I shall

die on a Sunday is one-seventh. Here the contingent event

is clearly one that does not admit of repetition; and yet

would not the belief of every man have the value assigned it

by the formula? It would appear that the same principle

will be found to be at work here as in the former examples.

It is quite true that I have only the opportunity of dying

once myself, but I am a member of a class in which deaths

occur with frequency, and I form my opinion upon evidence

drawn from that class. If, for example, I had insured my

life for £1000, I should feel a certain propriety in demanding

£7000 in case the office declared that it would only pay in

the event of my dying on a Sunday. _I_, indeed, for my own

private part, might not find the arrangement an equitable

one; but mankind at large, in case they acted on such a

principle, might fairly commute their aggregate gains in such

a way, whilst to the Insurance Office it would not make any

difference at all.

26. The results of the last few sections might be summarised

as follows:--the different amounts of belief which

we entertain upon different events, and which are recognized

by various phrases in common use, have undoubtedly some

meaning. But the greater part of their meaning, and certainly

their only justification, are to be sought in the _series_

of corresponding events to which they belong; in regard to

which it may be shown that far more events are capable of

being referred to a series than might be supposed at first

sight. The test and justification of belief are to be found in

conduct; in this test applied to the series as a whole, there

is nothing peculiar, it differs in no way from the similar test

when we are acting on our belief about any single event.

But so applied, from the nature of the case it is applied

successively to each of the individuals of the series; here our

_conduct_ generally admits of being separately considered in

reference to each particular event; and this has been understood

to denote a certain amount of belief which should be a

fraction of certainty. Probably on the principles of association,

a peculiar condition of mind is produced in reference to

each single event. And these associations are not unnaturally

retained even when we contemplate any one of these single

events isolated from any series to which it belongs. When

it is found alone we treat it, and feel towards it, as we do

when it is in company with the rest of the series.

27. We may now see, more clearly than we could

before, why it is that we are free from any necessity of assuming

the existence of causation, in the sense of necessary

invariable sequence, in the case of the events which compose

our series. Against such a view it might very plausibly be

urged, that we constantly talk of the probability of a single

event; but how can this be done, it may reasonably be said,

if we once admit the possibility of that event occurring fortuitously?

Take an instance from human life; the average

duration of the lives of a batch of men aged thirty will be

about thirty-four years. We say therefore to any individual

of them, Your expectation of life is thirty-four years. But

how can this be said if we admit that the train of events

composing his life is liable to be destitute of all regular

sequence of cause and effect? To this it may be replied

that the denial of causation enables us to say neither more

nor less than its assertion, in reference to the length of the

individual life, for of this we are ignorant in each case alike.

By assigning, as above, an expectation in reference to the

individual, we _mean_ nothing more than to make a statement

about the average of his class. Whether there be causation

or not in these individual cases does not affect our knowledge

of the average, for this by supposition rests on independent

experience. The legitimate inferences are the same on either

hypothesis, and of equal value. The only difference is that

on the hypothesis of non-causation we have forced upon our

attention the impropriety of talking of the 'proper' expectation

of the individual, owing to the fact that all knowledge of

its amount is formally impossible; on the other hypothesis

the impropriety is overlooked from the fact of such knowledge

being only practically unattainable. As a matter of

fact the amount of our knowledge is the same in each case;

it is a knowledge of the average, and of that only.[6]

28. We may conclude, then, that the limits within

which we are thus able to justify the amount of our belief

are far more extensive than might appear at first sight.

Whether every case in which persons feel an amount of

belief short of perfect confidence could be forced into the

province of Probability is a wider question. Even, however,

if the belief could be supposed capable of justification on its

principles, its rules could never in such cases be made use of.

Suppose, for example, that a father were in doubt whether

to give a certain medicine to his sick child. On the one

hand, the doctor declared that the child would die unless the

medicine were given; on the other, through a mistake, the

father cannot feel quite sure that the medicine he has is the

right one. It is conceivable that some mathematicians, in

their conviction that everything has its definite numerical

probability, would declare that the man's belief had some

'value' (if they could only find out what it is), say nine-tenths;

by which they would mean that in nine cases out of

ten in which he entertained a belief of that particular value

he proved to be right. So with his belief and doubt on

the other side of the question. Putting the two together,

there is but one course which, as a prudent man and a good

father, he can possibly follow. It may be so, but when (as

here) the identification of an event in a series depends on

purely subjective conditions, as in this case upon the degree

of vividness of his conviction, of which no one else can judge,

no test is possible, and therefore no proof can be found.

29. So much then for the attempts, so frequently

made, to found the science on a subjective basis; they can

lead, as it has here been endeavoured to show, to no satisfactory

result. Still our belief is so inseparably connected with

our action, that something of a defence can be made for the

attempts described above; but when it is attempted, as is

often the case, to import other sentiments besides pure belief,

and to find a justification for them also in the results of our

science, the confusion becomes far worse. The following

extract from Archbishop Thomson's _Laws of Thought_ (§122,

Ed. II.) will show what kind of applications of the science are

contemplated here: "In applying the doctrine of chances to

that subject in connexion with which it was invented--games

of chance,--the principles of what has been happily termed

'moral arithmetic' must not be forgotten. Not only would

it be difficult for a gamester to find an antagonist on terms,

as to fortune and needs, precisely equal, but also it is impossible

that with such an equality the advantage of a

considerable gain should balance the harm of a serious loss.

'If two men,' says Buffon, 'were to determine to play for

their whole property, what would be the effect of this agreement?

The one would only double his fortune, and the

other reduce his to naught. What proportion is there between

the loss and the gain? The same that there is between

all and nothing. The gain of the one is but a moderate

sum,--the loss of the other is numerically infinite, and

morally so great that the labour of his whole life may not

perhaps suffice to restore his property.'"

As moral advice this is all very true and good. But if it

be regarded as a contribution to the science of the subject it

is quite inappropriate, and seems calculated to cause confusion.

The doctrine of chances pronounces upon certain

kinds of events in respect of number and magnitude; it has

absolutely nothing to do with any particular person's feelings

about these relations. We might as well append a corollary

to the rules of arithmetic, to point out that although it is

very true that twice two are four it does not follow that four,

horses will give twice as much pleasure to the owner as two

will. If two men play on equal terms their chances are

equal; in other words, if they were often to play in this

manner each would lose as frequently as he would gain. That

is all that Probability can say; what under the circumstances

may be the determination and opinions of the men in question,

it is for them and them alone to decide. There are

many persons who cannot bear mediocrity of any kind, and

to whom the prospect of doubling their fortune would outweigh

a greater chance of losing it altogether. They alone

are the judges.

If we will introduce such a balance of pleasure and pain

the individual must make the calculation for himself. The

supposition is that total ruin is very painful, partial loss

painful in a less proportion than that assigned by the ratio

of the losses themselves; the inference is therefore drawn

that on the average more pain is caused by occasional great

losses than by frequent small ones, though the money value

of the losses in the long run may be the same in each case.

But if we suppose a country where the desire of spending

largely is very strong, and where owing to abundant production

loss is easily replaced, the calculation might incline the

other way. Under such circumstances it is quite possible

that more happiness might result from playing for high than

for low stakes. The fact is that all emotional considerations

of this kind are irrelevant; they are, at most, mere applications

of the theory, and such as each individual is alone

competent to make for himself. Some more remarks will be

made upon this subject in the chapter upon Insurance and

Gambling.

30. It is by the introduction of such considerations as

these that the Petersburg Problem has been so perplexed.

Having already given some description of this problem we

will refer to it very briefly here. It presents us with a

sequence of sets of throws for each of which sets I am to

receive something, say a shilling, as the minimum receipt.

My receipts increase in proportion to the rarity of each

particular kind of set, and each kind is observed or inferred

to grow more rare in a certain definite but unlimited order.

By the wording of the problem, properly interpreted, I am

supposed never to stop. Clearly therefore, however large a

fee I pay for each of these sets, I shall be sure to make it up

in time. The mathematical expression of this is, that I

ought always to pay an infinite sum. To this the objection

is opposed, that no sensible man would think of advancing

even a large finite sum, say £50. Certainly he would not;

but why? Because neither he nor those who are to pay

him would be likely to live long enough for him to obtain

throws good enough to remunerate him for one-tenth of his

outlay; to say nothing of his trouble and loss of time. We

must not suppose that the problem, as stated in the ideal

form, will coincide with the practical form in which it

presents itself in life. A carpenter might as well object to

Euclid's second postulate, because his plane came to a stop

in six feet on the plank on which he was at work. Many

persons have failed to perceive this, and have assumed that,

besides enabling us to draw numerical inferences about the

members of a series, the theory ought also to be called upon

to justify all the opinions which average respectable men

might be inclined to form about them, as well as the conduct

they might choose to pursue in consequence. It is obvious

that to enter upon such considerations as these is to diverge

from our proper ground. We are concerned, in these cases,

with the actions of men only, as given in statistics; with the

emotions they experience in the performance of these actions

we have no direct concern whatever. The error is the same

as if any one were to confound, in political economy, value in

use with value in exchange, and object to measuring the

value of a loaf by its cost of production, because bread is

worth more to a man when he is hungry than it is just after

his dinner.

31. One class of emotions indeed ought to be excepted,

which, from the apparent uniformity and consistency

with which they show themselves in different persons

and at different times, do really present some better claim to

consideration. In connection with a science of inference

they can never indeed be regarded as more than an accident

of what is essential to the subject, but compared with other

emotions they seem to be inseparable accidents.

The reader will remember that attention was drawn in

the earlier part of this chapter to the compound nature of

the state of mind which we term belief. It is partly intellectual,

partly also emotional; it professes to rest upon

experience, but in reality the experience acts through the

distorting media of hopes and fears and other disturbing

agencies. So long as we confine our attention to the _state

of mind_ of the person who believes, it appears to me that

these two parts of belief are quite inseparable. Indeed, to

speak of them as two parts may convey a wrong impression;

for though they spring from different sources, they so entirely

merge in one result as to produce what might be

called an indistinguishable compound. Every kind of inference,

whether in probability or not, is liable to be disturbed

in this way. A timid man may honestly believe that he will

be wounded in a coming battle, when others, with the same

experience but calmer judgments, see that the chance is

too small to deserve consideration. But such a man's belief,

if we look only to that, will not differ in its nature from

sound belief. His conduct also in consequence of his belief

will by itself afford no ground of discrimination; he will

make his will as sincerely as a man who is unmistakeably on

his death-bed. The only resource is to check and correct

his belief by appealing to past and current experience.[7] This

was advanced as an objection to the theory on which probability

is regarded as concerned primarily with laws of belief.

But on the view taken in this Essay in which we are supposed

to be concerned with laws of inference about things,

error and difficulty from this source vanish. Let us bear clearly

in mind that we are concerned with inferences about things,

and whatever there may be in belief which does not depend

on experience will disappear from notice.

32. These emotions then can claim no notice as an

integral portion of any science of inference, and should in

strictness be rigidly excluded from it. But if any of them

are uniform and regular in their production and magnitude,

they may be fairly admitted as accidental and extraneous

accompaniments. This is really the case to some extent

with our surprise. This emotion does show a considerable

degree of uniformity. The rarer any event is the more am I,

in common with most other men, surprised at it when it does

happen. This surprise may range through all degrees, from

the most languid form of interest up to the condition which

we term 'being startled'. And since the surprise seems

to be pretty much the same, under similar circumstances,

at different times, and in the case of different persons, it is

free from that extreme irregularity which is found in most

of the other mental conditions which accompany the contemplation

of unexpected events. Hence our surprise, though,

as stated above, having no proper claim to admission into

the science of Probability, is such a constant and regular

accompaniment of that which Probability is concerned with,

that notice must often be taken of it. References will occasionally

be found to this aspect of the question in the

following chapters.

It may be remarked in passing, for the sake of further

illustration of the subject, that this emotional accompaniment

of surprise, to which we are thus able to assign something

like a fractional value, differs in two important respects

from the commonly accepted fraction of belief. In the first

place, it has what may be termed an independent existence;

it is intelligible by itself. The belief, as we endeavoured to

show, needs explanation and finds it in our consequent conduct.

Not so with the emotion; this stands upon its own

footing, and may be examined in and by itself. Hence, in

the second place, it is as applicable, and as capable of any kind

of justification, in relation to the _single event_, as to a series of

events. In this respect, as will be remembered, it offers a

complete contrast to our state of belief about any one contingent

event. May not these considerations help to account

for the general acceptance of the doctrine, that we have a

certain definite and measurable amount of belief about these

events? I cannot help thinking that what is so obviously

true of the emotional portion of the belief, has been unconsciously

transferred to the other or intellectual portion of the

compound condition, to which it is not applicable, and where

it cannot find a justification.

33. A further illustration may now be given of the

subjective view of Probability at present under discussion.

An appeal to common language is always of service, as

the employment of any distinct word is generally a proof

that mankind have observed some distinct properties in the

things, which have caused them to be singled out and have

that name appropriated to them. There is such a class of

words assigned by popular usage to the kind of events of

which Probability takes account. If we examine them we

shall find, I think, that they direct us unmistakeably to the

two-fold aspect of the question,--the objective and the subjective,

the quality in the events and the state of our minds

in considering them,--that have occupied our attention

during the former chapters.

The word 'extraordinary', for instance, seems to point to

the observed fact, that events are arranged in a sort of _ordo_

or rank. No one of them might be so exactly placed that

we could have inferred its position, but when we take a great

many into account together, running our eye, as it were,

along the line, we begin to see that they really do for the

most part stand in order. Those which stand away from the

line have this divergence observed, and are called extraordinary,

the rest ordinary, or in the line. So too 'irregular'

and 'abnormal' are doubtless used from the appearance

of things, when examined in large numbers, being that

of an arrangement by rule or measure. This only holds

when there are a good many; we could not speak of the

single events being so arranged. Again the word 'law', in

its philosophical sense, has now become quite popularised.

How the term became introduced is not certain, but

there can be little doubt that it was somewhat in this

way:--The effect of a law, in its usual application to

human conduct, is to produce regularity where it did not

previously exist; when then a regularity began to be perceived

in nature, the same word was used, whether the cause

was supposed to be the same or not. In each case there

was the same generality of agreement, subject to occasional

deflection.[8]

On the other hand, observe the words 'wonderful', 'unexpected',

'incredible'. Their connotation describes states

of mind simply; they are of course not confined to Probability,

in the sense of statistical frequency, but imply simply

that the events they denote are such as from some cause we

did not expect would happen, and at which therefore, when

they do happen, we are surprised.

Now when we bear in mind that these two classes of

words are in their origin perfectly distinct;--the one denoting

simply events of a certain character; the other,

though also denoting events, _connoting_ simply states of

mind;--and yet that they are universally applied to the

same events, so as to be used as perfectly synonymous, we

have in this a striking illustration of the two sides under

which Probability may be viewed, and of the universal recognition

of a close connection between them. The words are

popularly used as synonymous, and we must not press their

meaning too far; but if it were to be observed, as I am

rather inclined to think it could, that the application of the

words which denote mental states is wider than that of the

others, we should have an illustration of what has been

already observed, viz. that the province of Probability is not

so extensive as that over which variation of belief might be

observed. Probability only considers the case in which this

variation is brought about in a certain definite statistical

way.

34. It will be found in the end both interesting and

important to have devoted some attention to this subjective

side of the question. In the first place, as a mere speculative

inquiry the quantity of our belief of any proposition

deserves notice. To study it at all deeply would be to trespass

into the province of Psychology, but it is so intimately

connected with our own subject that we cannot avoid all

reference to it. We therefore discuss the laws under which

our expectation and surprise at isolated events increases or

diminishes, so as to account for these states of mind in any

individual instance, and, if necessary, to correct them when

they vary from their proper amount.

But there is another more important reason than this.

It is quite true that when the subjects of our discussion in

any particular instance lie entirely within the province

of Probability, they may be treated without any reference

to our belief. We may or we may not employ this side of

the question according to our pleasure. If, for example, I

am asked whether it is more likely that A. B. will die this

year, than that it will rain to-morrow, I may calculate the

chance (which really is at bottom the same thing as my

belief) of each, find them respectively, one-sixth and one-seventh,

say, and therefore decide that my 'expectation' of

the former is the greater, viz. that this is the more likely

event. In this case the process is precisely the same whether

we suppose our belief to be introduced or not; our mental

state is, in fact, quite immaterial to the question. But, in

other cases, it may be different. Suppose that we are comparing

two things, of which one is wholly alien to Probability,

in the sense that it is hopeless to attempt to assign

any degree of numerical frequency to it, the only ground

they have in common may be the amount of belief to which

they are respectively entitled. We cannot compare the

frequency of their occurrence, for one may occur too seldom

to judge by, perhaps it may be unique. It has been already

said, that our belief of many events rests upon a very complicated

and extensive basis. My belief may be the product

of many conflicting arguments, and many analogies more or

less remote; these proofs themselves may have mostly faded

from my mind, but they will leave their effect behind them

in a weak or strong conviction. At the time, therefore, I

may still be able to say, with some degree of accuracy,

though a very slight degree, what amount of belief I entertain

upon the subject. Now we cannot compare things that

are heterogeneous: if, therefore, we are to decide between

this and an event determined naturally and properly by

Probability, it is impossible to appeal to chances or frequency

of occurrence. The measure of belief is the only common

ground, and we must therefore compare this quantity in each

case. The test afforded will be an exceedingly rough one,

for the reasons mentioned above, but it will be better than

none; in some cases it will be found to furnish all we want.

Suppose, for example, that one letter in a million is lost

in the Post Office, and that in any given instance I wish to

know which is more likely, that a letter has been so lost, or

that my servant has stolen it? If the latter alternative

could, like the former, be stated in a numerical form, the

comparison would be simple. But it cannot be reduced to

this form, at least not consciously and directly. Still, if we

could feel that our belief in the man's dishonesty was greater

than one-millionth, we should then have homogeneous things

before us, and therefore comparison would be possible.

35. We are now in a position to give a tolerably accurate

definition of a phrase which we have frequently been

obliged to employ, or incidentally to suggest, and of which

the reader may have looked for a definition already, viz. the

probability of an event, or what is equivalent to this, the

chance of any given event happening. I consider that these

terms presuppose a series; within the indefinitely numerous

class which composes this series a smaller class is distinguished

by the presence or absence of some attribute or

attributes, as was fully illustrated and explained in a previous

chapter. These larger and smaller classes respectively

are commonly spoken of as instances of the 'event,' and of

'its happening in a given particular way.' Adopting this

phraseology, which with proper explanations is suitable

enough, we may define the probability or chance (the terms

are here regarded as synonymous) of the event happening

in that particular way as the numerical fraction which represents

the proportion between the two different classes in the

long run. Thus, for example, let the probability be that

of a given infant living to be eighty years of age. The

larger series will comprise all infants, the smaller all who live

to eighty. Let the proportion of the former to the latter be

9 to 1; in other words, suppose that one infant in ten lives

to eighty. Then the chance or probability that any given

infant will live to eighty is the numerical fraction 1/10. This

assumes that the series are of indefinite extent, and of the

kind which we have described as possessing a fixed type.

If this be not the case, but the series be supposed terminable,

or regularly or irregularly fluctuating, as might be the

case, for instance, in a society where owing to sanitary or

other causes the average longevity was steadily undergoing

a change, then in so far as this is the case the series ceases

to be a subject of science. What we have to do under these

circumstances, is to substitute a series of the right kind for

the inappropriate one presented by nature, choosing it, of

course, with as little deflection as possible from the observed

facts. This is nothing more than has to be done, and invariably

is done, whenever natural objects are made subjects

of strict science.

36. A word or two of explanation may be added about

the expression employed above, 'the proportion in the long

run.' The run must be supposed to be very long indeed, in

fact never to stop. As we keep on taking more terms of the

series we shall find the proportion still fluctuating a little,

but its fluctuations will grow less. The proportion, in fact,

will gradually approach towards some fixed numerical value,

what mathematicians term its _limit_. This fractional value

is the one spoken of above. In the cases in which deductive

reasoning is possible, this fraction may be obtained without

direct appeal to statistics, from reasoning about the conditions

under which the events occur, as was explained in

the fourth chapter.

Here becomes apparent the full importance of the distinction

so frequently insisted on, between the actual irregular

series before us and the substituted one of calculation, and

the meaning of the assertion (Ch. I. §13), that it was in the

case of the latter only that strict scientific inferences could

be made. For how can we have a 'limit' in the case of

those series which ultimately exhibit irregular fluctuations?

When we say, for instance, that it is an even chance that

a given person recovers from the cholera, the meaning of

this assertion is that in the long run one half of the persons

attacked by that disease do recover. But if we examined

a sufficiently extensive range of statistics, we might find

that the manners and customs of society had produced such

a change in the type of the disease or its treatment, that we

were no nearer approaching towards a fixed limit than we

were at first. The conception of an ultimate limit in the

ratio between the numbers of the two classes in the series

necessarily involves an absolute fixity of the type. When

therefore nature does not present us with this absolute fixity,

as she seldom or never does except in games of chance (and

not demonstrably there), our only resource is to introduce

such a series, in other words, as has so often been said, to

substitute a series of the right kind.

37. The above, which may be considered tolerably

complete as a definition, might equally well have been

given in the last chapter. It has been deferred however

to the present place, in order to connect with it at once a

proposition involving the conceptions introduced in this

chapter; viz. the state of our own minds, in reference to the

amount of belief we entertain in contemplating any one

of the events whose probability has just been described.

Reasons were given against the opinion that our belief admitted

of any exact apportionment like the numerical one

just mentioned. Still, it was shown that a reasonable explanation

could be given of such an expression as, 'my belief is

1/10th of certainty', though it was an explanation which pointed

unmistakeably to a series of events, and ceased to be intelligible,

or at any rate justifiable, when it was not viewed in

such a relation to a series. In so far, then, as this explanation

is adopted, we may say that our belief is in proportion

to the above fraction. This referred to the purely

intellectual part of belief which cannot be conceived to be

separable, even in thought, from the things upon which it

is exercised. With this intellectual part there are commonly

associated various emotions. These we can to a

certain extent separate, and, when separated, can measure

with that degree of accuracy which is possible in the case of

other emotions. They are moreover intelligible in reference

to the individual events. They will be found to increase

and diminish in accordance, to some extent, with the fraction

which represents the scarcity of the event. The emotion of

surprise does so with some degree of accuracy.

The above investigation describes, though in a very brief

form, the amount of truth which appears to me to be contained

in the assertion frequently made, that the fraction

expressive of the probability represents also the fractional

part of full certainty to which our belief of the individual

event amounts. Any further analysis of the matter would

seem to belong to Psychology rather than to Probability.

1. In the ordinary signification of this term. As De Morgan uses it he

makes Formal Logic _include_ Probability, as one of its branches, as

indicated in his title "Formal Logic, or the Calculus of Inference,

necessary and probable."

2. _Formal Logic_. Preface, page v.

3. An illustration of the points here insisted on has recently [1876]

been given in a quarter where few would have expected it; I allude,

as many readers will readily infer, to J. S. Mill's exceedingly

interesting Essays on Theism. It is not within our province here to

criticise any of their conclusions, but they have expressed in a

very significant way the conviction entertained by him that beliefs

which are not justified by evidence, and possibly may not be capable

of justification (those for instance of immortality and the

existence of the Deity), may nevertheless not only continue to exist

in cultivated minds, but may also be profitably encouraged there, at

any rate in the shape of hopes, for certain supposed advantages

attendant on their retention, irrespective even of their truth.

4. It is necessary to take an example in which the man is forced to

act, or we should not be able to shew that he has any belief on the

subject at all. He may declare that he neither knows nor cares

anything about the matter, and that therefore there is nothing of

the nature of belief to be extracted out of his mental condition. He

very likely would take this ground if we asked him, as De Morgan

does, with a slightly different reference (_Formal Logic_, p. 183),

whether he considers that there are volcanoes on the unseen side of

the moon larger than those on the side turned towards us; or, with

Jevons (_Principles of Science_, Ed. II. p. 212) whether he

considers that a Platythliptic Coefficient is positive. These do not

therefore seem good instances to illustrate the position that we

always entertain a certain degree of belief on every question which

can be stated, and that utter inability to give a reason in favour

of either alternative corresponds to half belief.

5. Except indeed on the principles indicated further on in §§24, 25.

6. For a fuller discussion of this, see the Chapter on Causation.

7. The best example I can recall of the distinction between judging

from the subjective and the objective side, in such cases as these,

occurred once in a railway train. I met a timid old lady who was in

much fear of accidents. I endeavoured to soothe her on the usual

statistical ground of the extreme rarity of such events. She

listened patiently, and then replied, "Yes, Sir, that is all very

well; but I don't see how the real danger will be a bit the less

because I don't believe in it."

8. This would still hold of _empirical_ laws which may be capable of

being broken: we now have very much shifted the word, to denote an

_ultimate_ law which it is supposed cannot be broken.