§1. We are now in a position to explain and justify some important conclusions which, if not direct consequences of the distinctions laid down in the last chapter, will at any rate be more readily appreciated and accepted after that exposition.

In the first place, it will be seen that in Probability time has nothing to do with the question; in other words, it does not matter whether the event, whose probability we are discussing, be past, present, or future. The problem before us, in its simplest form, is this:—Statistics (extended by Induction, and practically often gained by Deduction) inform us that a certain event has happened, does happen, or will happen, in a certain way in a certain proportion of cases. We form a conception of that event, and regard it as possible; but we want to do more; we want to know how much we ought to expect it (under the explanations given in a former chapter about quantity of belief). There is therefore a sort of relative futurity about the event, inasmuch as our knowledge of the fact, and therefore our justification or otherwise of the correctness of our surmise, almost necessarily comes after the surmise was formed; but the futurity is only relative. The evidence by which the question is to be settled may not be forthcoming yet, or we may have it by us but only consult it afterwards. It is from the fact of the futurity being, as above described, only relative, that I have preferred to speak of the conception of the event rather than of the anticipation of it. The latter term, which in some respects would have seemed more intelligible and appropriate, is open to the objection, that it does rather, in popular estimation, convey the notion of an absolute as opposed to a relative futurity.

§2. For example; a die is thrown. Once in six times it gives ace; if therefore we assume, without examination, that the throw is ace, we shall be right once in six times. In so doing we may, according to the usual plan, go forwards in time; that is, form our opinion about the throw beforehand, when no one can tell what it will be. Or we might go backwards; that is, form an opinion about dice that had been cast on some occasion in time past, and then correct our opinion by the testimony of some one who had been a witness of the throws. In either case the mental operation is precisely the same; an opinion formed merely on statistical grounds is afterwards corrected by specific evidence. The opinion may have been formed upon a past, present, or future event; the evidence which corrects it afterwards may be our own eyesight, or the testimony of others, or any kind of inference; by the evidence is merely meant such subsequent examination of the case as is assumed to set the matter at rest. It is quite possible, of course, that this specific evidence should never be forthcoming; the conception in that case remains as a conception, and never obtains that degree of conviction which qualifies it to be regarded as a ‘fact.’ This is clearly the case with all past throws of dice the results of which do not happen to have been recorded.

In discussing games of chance there are obvious advantages in confining ourselves to what is really, as well as relatively, future, for in that case direct information concerning the contemplated result being impossible, all persons are on precisely the same footing of comparative ignorance, and must form their opinion entirely from the known or inferred frequency of occurrence of the event in question. On the other hand, if the event be passed, there is almost always evidence of some kind and of some value, however slight, to inform us what the event really was; if this evidence is not actually at hand, we can generally, by waiting a little, obtain something that shall be at least of some use to us in forming our opinion. Practically therefore we generally confine ourselves, in anticipations of this kind, to what is really future, and so in popular estimation futurity becomes indissolubly associated with probability.

§3. There is however an error closely connected with the above view of the subject, or at least an inaccuracy of expression which is constantly liable to lead to error, which has found wide acceptance, and has been sanctioned by writers of the greatest authority. For instance, both Butler, in his Analogy, and Mill, have drawn attention, under one form of expression or another, to the distinction between improbability before the event and improbability after the event, which they consider to be perfectly different things. That this phraseology indicates a distinction of importance cannot be denied, but it seems to me that the language in which it is often expressed requires to be amended.

Butler's remarks on this subject occur in his Analogy, in the chapter on miracles. Admitting that there is a strong presumption against miracles (his equivalent for the ordinary expression, an ‘improbability before the event’) he strives to obtain assent for them by showing that other events, which also have a strong presumption against them, are received on what is in reality very slight evidence. He says, “There is a very strong presumption against common speculative truths, and against the most ordinary facts, before the proof of them; which yet is overcome by almost any proof. There is a presumption of millions to one against the story of CÆsar, or of any other man. For, suppose a number of common facts so and so circumstanced, of which one had no kind of proof, should happen to come into one's thoughts, every one would without any possible doubt conclude them to be false. And the like may be said of a single common fact.”

§4. These remarks have been a good deal criticized, and they certainly seem to me misleading and obscure in their reference. If one may judge by the context, and by another passage in which the same argument is afterwards referred to,[1] it would certainly appear that Butler drew no distinction between miraculous accounts, and other accounts which, to use any of the various expressions in common use, are unlikely or improbable or have a presumption against them; and concluded that since some of the latter were instantly accepted upon somewhat mediocre testimony, it was altogether irrational to reject the former when similarly or better supported.[2] This subject will come again under our notice, and demand fuller discussion, in the chapter on the Credibility of extraordinary stories. It will suffice here to remark that, however satisfactory such a view of the matter might be to some theologians, no antagonist of miracles would for a moment accept it. He would naturally object that, instead of the miraculous element being (as Butler considers) “a small additional presumption” against the narrative, it involved the events in a totally distinct class of incredibility; that it multiplied, rather than merely added to, the difficulties and objections in the way of accepting the account.

Mill's remarks (Logic, Bk.III. ch.XXV. §4) are of a different character. Discussing the grounds of disbelief he speaks of people making the mistake of “overlooking the distinction between (what may be called) improbability before the fact, and improbability after it, two different properties, the latter of which is always a ground of disbelief, the former not always.” He instances the throwing of a die. It is improbable beforehand that it should turn up ace, and yet afterwards, “there is no reason for disbelieving it if any credible witness asserts it.” So again, “the chances are greatly against A.B.'s dying, yet if any one tells us that he died yesterday we believe it.”

§5. That there is some difficulty about such problems as these must be admitted. The fact that so many people find them a source of perplexity, and that such various explanations are offered to solve the perplexity, are a sufficient proof of this.[3] The considerations of the last chapter, however, over-technical and even scholastic as some of the language in which it was expressed may have seemed to the reader, will I hope guide us to a more satisfactory way of regarding the matter.

When we speak of an improbable event, it must be remembered that, objectively considered, an event can only be more or less rare; the extreme degree of rarity being of course that in which the event does not occur at all. Now, as was shown in the last chapter, our position, when forming judgments of the time in question, is that of entertaining a conception or conjecture (call it what we will), and assigning a certain weight of trustworthiness to it. The real distinction, therefore, between the two classes of examples respectively, which are adduced both by Butler and by Mill, consists in the way in which those conceptions are obtained; they being obtained in one case by the process of guessing, and in the other by that of giving heed to the reports of witnesses.

§6. Take Butler's instance first. In the ‘presumption before the proof’ we have represented to us a man thinking of the story of CÆsar, that is, making a guess about certain historical events without any definite grounds for it, and then speculating as to what value is to be attached to the probability of its truth. Such a guess is of course, as he says, concluded to be false. But what does he understand by the ‘presumption after the proof’? That a story not adopted at random, but actually suggested and supported by witnesses, should be true. The latter might be accepted, whilst the former would undoubtedly be rejected; but all that this proves, or rather illustrates, is that the testimony of almost any witness is in most cases vastly better than a mere guess.[4] We may in both cases alike speak of ‘the event’ if we will; in fact, as was admitted in the last chapter, common language will not readily lend itself to any other way of speaking. But it should be clearly understood that, phrase it how we will, what is really present to the man's mind, and what is to have its probable value assigned to it, is the conception of an event, in the sense in which that expression has already been explained. And surely no two conceptions can have a much more important distinction put between them than that which is involved in supposing one to rest on a mere guess, and the other on the report of a witness. Precisely the same remarks apply to the example given by Mill. Before A.B.'s death our opinion upon the subject was nothing but a guess of our own founded upon life statistics; after his death it was founded upon the evidence of some one who presumably had tolerable opportunities of knowing what the facts really were.

§7. That the distinction before us has no essential connection whatever with time is indeed obvious on a moment's consideration. Conceive for a moment that some one had opportunities of knowing whether A. B. would die or not. If he told us that A.B. would die to-morrow, we should in that case be just as ready to believe him as when he tells us that A.B. has died. If we continued to feel any doubt about the statement (supposing always that we had full confidence about his veracity in matters into which he had duly enquired), it would be because we thought that in his case, as in ours, it was equivalent to a guess, and nothing more. So with the event when past, the fact of its being past makes no difference whatever; until the credible witness informs us of what he knows to have occurred, we should doubt it if it happened to come into our minds, just as much as if it were future.

The distinction, therefore, between probability before the event and probability after the event seems to resolve itself simply into this;—before the event we often have no better means of information than to appeal to statistics in some form or other, and so to guess amongst the various possible alternatives; after the event the guess may most commonly be improved or superseded by appeal to specific evidence, in the shape of testimony or observation. Hence, naturally, our estimate in the latter case is commonly of much more value. But if these characteristics were anyhow inverted; if, that is, we were to confine ourselves to guessing about the past, and if we could find any additional evidence about the future, the respective values of the different estimates would also be inverted. The difference between these values has no necessary connection with time, but depends entirely upon the different grounds upon which our conception or conjecture about the event in question rests.

§8. The following imaginary example will serve to bring out the point indicated above. Conceive a people with very short memories, and who preserved no kind of record to perpetuate their hold upon the events which happened amongst them.[5] The whole region of the past would then be to them what much of the future is to us; viz. a region of guesses and conjectures, one in reference to which they could only judge upon general considerations of probability, rather than by direct and specific evidence. But conceive also that they had amongst them a race of prophets who could succeed in foretelling the future with as near an approach to accuracy and trustworthiness as our various histories, and biographies, and recollections, can attain in respect to the past. The present and usual functions of direct evidence or testimony, and of probability, would then be simply inverted; and so in consequence would the present accidental characteristics of improbability before and after the event. It would then be the latter which would by comparison be regarded as ‘not always a ground of disbelief,’ whereas in the case of the former we should then have it maintained that it always was so.

§9. The origin of the mistake just discussed is worth enquiring into. I take it to be as follows. It is often the case, as above remarked, when we are speculating about a future event, and almost always the case when that future event is taken from a game of chance, that all persons are in precisely the same condition of ignorance in respect to it. The limit of available information is confined to statistics, and amounts to the knowledge that the unknown event must assume some one of various alternative forms. The conjecture, therefore, of any one man about it is as valuable as that of any other. But in regard to the past the case is very different. Here we are not in the habit of relying upon statistical information. Hence the conjectures of different men are of extremely different values; in the case of many they amount to what we call positive knowledge. This puts a broad distinction, in popular estimation, between what may be called the objective certainty of the past and of the future, a distinction, however, which from the standing-point of a science of inference ought to have no existence.

In consequence of this, when we apply to the past and the future respectively the somewhat ambiguous expression ‘the chance of the event,’ it commonly comes to bear very different significations. Applied to the future it bears its proper meaning, namely, the value to be assigned to a conjecture upon statistical grounds. It does so, because in this case hardly any one has more to judge by than such conjectures. But applied to the past it shifts its meaning, owing to the fact that whereas some men have conjectures only, others have positive knowledge. By the chance of the event is now often meant, not the value to be assigned to a conjecture founded on statistics, but to such a conjecture derived from and enforced by any body else's conjecture, that is by his knowledge and his testimony.

§10. There is a class of cases in apparent opposition to some of the statements in this chapter, but which will be found, when examined closely, decidedly to confirm them. I am walking, say, in a remote part of the country, and suddenly meet with a friend. At this I am naturally surprised. Yet if the view be correct that we cannot properly speak about events in themselves being probable or improbable, but only say this of our conjectures about them, how do we explain this? We had formed no conjecture beforehand, for we were not thinking about anything of the kind, but yet few would fail to feel surprise at such an incident.

The reply might fairly be made that we had formed such anticipations tacitly. On any such occasion every one unconsciously divides things into those which are known to him and those which are not. During a considerable previous period a countless number of persons had met us, and all fallen into the list of the unknown to us. There was nothing to remind us of having formed the anticipation or distinction at all, until it was suddenly called out into vivid consciousness by the exceptional event. The words which we should instinctively use in our surprise seem to show this:—‘Who would have thought of seeing you here?’ viz. Who would have given any weight to the latent thought if it had been called out into consciousness beforehand? We put our words into the past tense, showing that we have had the distinction lurking in our minds all the time. We always have a multitude of such ready-made classes of events in our minds, and when a thing happens to fall into one of those classes which are very small we cannot help noticing the fact.

Or suppose I am one of a regiment into which a shot flies, and it strikes me, and me only. At this I am surprised, and why? Our common language will guide us to the reason. ‘How strange that it should just have hit me of all men!’ We are thinking of the very natural two-fold division of mankind into, ourselves, and everybody else; our surprise is again, as it were, retrospective, and in reference to this division. No anticipation was distinctly formed, because we did not think beforehand of the event, but the event, when it has happened, is at once assigned to its appropriate class.

§11. This view is confirmed by the following considerations. Tell the story to a friend, and he will be a little surprised, but less so than we were, his division in this particular case being,—his friends (of whom we are but one), and the rest of mankind. It is not a necessary division, but it is the one which will be most likely suggested to him.

Tell it again to a perfect stranger, and his division being different (viz. we falling into the majority) we shall fail to make him perceive that there is anything at all remarkable in the event.

It is not of course attempted in these remarks to justify our surprise in every case in which it exists. Different persons might be differently affected in the cases supposed, and the examples are therefore given mainly for illustration. Still on principles already discussed (Ch.VI. §32) we might expect to find something like a general justification of the amount of surprise.

§12. The answer commonly given in these cases is confined to attempting to show that the surprise should not arise, rather than to explaining how it does arise. It takes the following form,—‘You have no right to be surprised, for nothing remarkable has really occurred. If this particular thing had not happened something equally improbable must. If the shot had not hit you or your friend, it must have hit some one else who was Àpriori as unlikely to be hit.’

For one thing this answer does not explain the fact that almost every one is surprised in such cases, and surprised somewhat in the different proportions mentioned above. Moreover it has the inherent unsatisfactoriness of admitting that something improbable has really happened, but getting over the difficulty by saying that all the other alternatives were equally improbable. A natural inference from this is that there is a class of things, in themselves really improbable, which can yet be established upon very slight evidence. Butler accepted this inference, and worked it out to the strange conclusion given above. Mill attempts to avoid it by the consideration of the very different values to be assigned to improbability before and after the event. Some further discussion of this point will be found in the chapter on Fallacies, and in that on the Credibility of Extraordinary Stories.

§13. In connection with the subject at present under discussion we will now take notice of a distinction which we shall often find insisted on in works on Probability, but to which apparently needless importance has been attached. It is frequently said that probability is relative, in the sense that it has a different value to different persons according to their respective information upon the subject in question. For example, two persons, A andB, are going to draw a ball from a bag containing 4balls: Aknows that the balls are black and white, but does not know more; Bknows that three are black and one white. It would be said that the probability of a white ball toA is¹/₂, and toB¹/₄.

When however we regard the subject from the material standing point, there really does not seem to me much more in this than the principle, equally true in every other science, that our inferences will vary according to the data we assume. We might on logical grounds with almost equal propriety speak of the area of a field or the height of a mountain being relative, and therefore having one value to one person and another to another. The real meaning of the example cited above is this: Asupposes that he is choosing white at random out of a series which in the long run would give white and black equally often; Bsupposes that he is choosing white out of a series which in the long run would give three black to one white. By the application, therefore, of a precisely similar rule they draw different conclusions; but so they would under the same circumstances in any other science. If two men are measuring the height of a mountain, and one supposes his base to be 1000 feet, whilst the other takes it to be1001, they would of course form different opinions about the height. The science of mensuration is not supposed to have anything to do with the truth of the data, but assumes them to have been correctly taken; why should not this be equally the case with Probability, making of course due allowance for the peculiar character of the data with which it is concerned?

§14. This view of the relativeness of probability is connected, as it appears to me, with the subjective view of the science, and is indeed characteristic of it. It seems a fair illustration of the weak side of that view, that it should lead us to lay any stress on such an expression. As was fully explained in the last chapter, in proportion as we work out the Conceptualist principle we are led away from the fundamental question of the material logic, viz. Is our belief actually correct, or not? and, if the former, to what extent and degree is it correct? We are directed rather to ask, What belief does any one as a matter of fact hold? And, since the belief thus entertained naturally varies according to the circumstances and other sources of information of the person in question, its relativeness comes to be admitted as inevitable, or at least it is not to be wondered at if such should be the case.

On our view of Probability, therefore, its ‘relativeness’ in any given case is a misleading expression, and it will be found much preferable to speak of the effect produced by variations in the nature and amount of the data which we have before us. Now it must be admitted that there are frequently cases in our science in which such variations are peculiarly likely to be found. For instance, I am expecting a friend who is a passenger in an ocean steamer. There are a hundred passengers on board, and the crew also numbers a hundred. I read in the papers that one person was lost by falling overboard; my anticipation that it was my friend who was lost is but small, of course. On turning to another paper, I see that the man who was lost was a passenger, not one of the crew; my slight anxiety is at once doubled. But another account adds that it was an Englishman, and on that line at that season the English passengers are known to be few; I at once begin to entertain decided fears. And so on, every trifling bit of information instantly affecting my expectations.

§15. Now since it is peculiarly characteristic of Probability, as distinguished from Induction, to be thus at the mercy, so to say, of every little fact that may be floating about when we are in the act of forming our opinion, what can be the harm (it may be urged) of expressing this state of things by terming our state of expectation relative?

There seem to me to be two objections. In the first place, as just mentioned, we are induced to reject such an expression on grounds of consistency. It is inconsistent with the general spirit and treatment of the subject hitherto adopted, and tends to divorce Probability from Inductive logic instead of regarding them as cognate sciences. We are aiming at truth, as far as that goal can be reached by our road, and therefore we dislike to regard our conclusions as relative in any other sense than that in which truth itself may be said to be relative.

In the second place, this condition of unstable assent, this constant liability to have our judgment affected, to any degree and at any moment, by the accession of new knowledge, though doubtless characteristic of Probability, does not seem to me characteristic of it in its sounder and more legitimate applications. It seems rather appropriate to a precipitate judgment formed in accordance with the rules, than a strict example of their natural employment. Such precipitate judgments may occur in the case of ordinary deductive conclusions. In the practical exigencies of life we are constantly in the habit of forming a hasty opinion with nearly full confidence, at any rate temporarily, upon the strength of evidence which we must well know at the time cannot be final. We wait a short time, and something else turns up which induces us to alter our opinion, perhaps to reverse it. Here our conclusions may have been perfectly sound under the given circumstances, that is, they may be such as every one else would have drawn who was bound to make up his mind upon the data before us, and they are unquestionably ‘relative’ judgments in the sense now under discussion. And yet, I think, every one would shrink from so terming them who wished systematically to carry out the view that Logic was to be regarded as an organon of truth.

§16. In the examples of Probability which we have hitherto employed, we have for the most part assumed that there was a certain body of statistics set before us on which our conclusion was to rest. It was assumed, on the one hand, that no direct specific evidence could be got, so that the judgment was really to be one of Probability, and to rest on these statistics; in other words, that nothing better than them was available for us. But it was equally assumed, on the other hand, that these statistics were open to the observation of every one, so that we need not have to put up with anything inferior to them in forming our opinion. In other words, we have been assuming that here, as in the case of most other sciences, those who have to draw a conclusion start from the same footing of opportunity and information. This, for instance, clearly is or ought to be the case when we are concerned with games of chance; ignorance or misapprehension of the common data is never contemplated there. So with the statistics of life, or other insurance: so long as our judgment is to be accurate (after its fashion) or justifiable, the common tables of mortality are all that any one has to go by.

§17. It is true that in the case of a man's prospect of death we should each qualify our judgment by what we knew or reasonably supposed as to his health, habits, profession, and so on, and should thus arrive at varying estimates. But no one could justify his own estimate without appealing explicitly or implicitly to the statistical grounds on which he had relied, and if these were not previously available to other persons, he must now set them before their notice. In other words, the judgments we entertain, here as elsewhere, are only relative so long as we rest them on grounds peculiar to ourselves. The process of justification, which I consider to be essential to logic, has a tendency to correct such individualities of judgment, and to set all observers on the same basis as regards their data.

It is better therefore to regard the conclusions of Probability as being absolute and objective, in the same sense as, though doubtless in a far less degree than, they are in Induction. Fully admitting that our conclusions will in many cases vary exceedingly from time to time by fresh accessions of knowledge, it is preferable to regard such fluctuations of assent as partaking of the nature of precipitate judgments, founded on special statistics, instead of depending only on those which are common to all observers. In calling such judgments precipitate it is not implied that there is any blame in entertaining them, but simply that, for one reason or another, we have been induced to form them without waiting for the possession of the full amount of evidence, statistical or otherwise, which might ultimately be looked for. This explanation will suit the facts equally well, and is more consistent with the general philosophical position maintained in this work.