CONSEQUENCES OF THE FOREGOING DISTINCTIONS.
§ 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.