the almost infinitely small chances with which we were concerned were rightly neglected from all practical consideration, however proper it might be, on speculative grounds, to keep our minds open to their actual existence. But it has often occurred to me that there is a common error in neglecting to take them into account when they may, though individually small, make up for their minuteness by their number. As the mathematician would express it, they may occasionally be capable of being integrated into a finite or even considerable magnitude.
For instance, we may be confronted with a difficulty out of which there appears to be only one appreciably possible mode of escape. The attempt is made to force us into accepting this, however great the odds apparently are against it, on the ground that improbable as it may seem, it is at any rate vastly more probable than any of the others. I can quite admit that, on practical grounds, we may often find it reasonable to adopt this course; for we can only act on one supposition, and we naturally and rightly choose, out of a quantity of improbabilities, the least improbable. But when we are not forced to act, no such decisive preference is demanded of us. It is then perfectly reasonable to refuse assent to the proposed explanation; even to say distinctly that we do not believe it, and at the same time to decline, at present, to accept any other explanation. We remain, in fact, in a state of suspense of judgment, a state perfectly right and reasonable so long as no action demanding a specific choice is forced upon us. One alternative may be decidedly probable as compared with any other individually, but decidedly improbable as compared with all others collectively. This in itself is intelligible enough; what people often fail to see is that there is no necessary contradiction between saying and feeling this, and yet being prepared vigorously to act, when action is forced upon us, as though this alternative were really the true one.
§ 32. To take a specific instance, this way of regarding the matter has often occurred to me in disputes upon ‘Spiritualist’ manifestations. Assent is urged upon us because, it is said, no other possible solution can be suggested. It may be quite true that apparently overwhelming difficulties may lie as against each separate alternative solution; but is it always sufficiently realized how numerous such solutions may be? No matter that each individually may be almost incredible: they ought all to be massed together and thrown into the scale against the proffered solution, when the only question asked is, Are we to accept this solution? There is no unfairness in such a course. We are perfectly ready to adopt the same plan against any other individual alternative, whenever any person takes to claiming this as the solution of the difficulty. We are looking at the matter from a purely logical point of view, and are quite willing, so far, to place every solution, spiritualist or otherwise, upon the same footing. The partisans of every alternative are in somewhat the same position as the members of a deliberative assembly, in which no one will support the motion of any other member. Every one can aid effectively in rejecting every other motion, but no one can succeed in passing his own. Pressure of urgent necessity may possibly force them out of this state of practical inaction, by, so to say, breaking through the opposition at some point of least resistance; but unless aided by some such pressure they are left in a state of hopeless dead-lock.
§ 33. Assuming that the spiritualistic solution admits of, and is to receive, scientific treatment, this, it seems to me, is the conclusion to which one might sometimes be led in the face of the evidence offered. We might have to say to every individual explanation, It is incredible, I cannot accept it; and unless circumstances should (which it is hardly possible that they should) force us to a hasty decision,—a decision, remember, which need indicate no preference of the judgment beyond what is just sufficient to turn the scale in its favour as against any other single alternative,—we leave the matter thus in abeyance. It will very likely be urged that one of the explanations (assuming that all the possible ones had been included) must be true; this we readily admit. It will probably also be urged that (on the often-quoted principle of Butler) we ought forthwith to accept the one which, as compared with the others, is the most plausible, whatever its absolute worth may be. This seems distinctly an error. To say that such and such an explanation is the one we should accept, if circumstances compelled us to anticipate our decision, is quite compatible with its present rejection. The only rational position surely is that of admitting that the truth is somewhere amongst the various alternatives, but confessing plainly that we have no such preference for one over another as to permit our saying anything else than that we disbelieve each one of them.
§ 34. (VIII.) The very common fallacy of ‘judging by the event,’ as it is generally termed, deserves passing notice here, as it clearly belongs to Probability rather than to Logic; though its nature is so obvious to those who have grasped the general principles of our science, that a very few words of remark will suffice. In one sense every proposition must consent to be judged by the event, since this is merely, in other words, submitting it to the test of experience. But there is the widest difference between the test appropriate to a universal proposition and that appropriate to a merely proportional or statistical one. The former is subverted by a single exception; the latter not merely admits exceptions, but implies them. Nothing, however, is more common than to blame advice (in others) because it has happened to turn out unfortunately, or to claim credit for it (in oneself) because it has happened to succeed. Of course if the conclusion was avowedly one of a probable kind we must be prepared with complacency to accept a hostile event, or even a succession of them; it is not until the succession shows a disposition to continue over long that suspicion and doubt should arise, and then only by a comparison of the degree of the assigned probability, and the magnitude of the departure from it which experience exhibits. For any single failure the reply must be, ‘the advice was sound’ (supposing, that is, that it was to be justified in the long run), ‘and I shall offer it again under the same circumstances.’
§ 35. The distinction drawn in the above instance deserves careful consideration; for owing to the wide difference between the kind of propositions dealt with in Probability and in ordinary Logic, and the consequent difference in the nature of the proof offered, it is quite possible for arguments of the same general appearance to be valid in the former and fallacious in the latter, and conversely.
For instance, take the well-known fallacy which consists in simply converting a universal affirmative, i.e.
in passing from All A is B to All B is A. When, as in common Logic, the conclusion is to be as certain as the premise, there is not a word to be said for such a step. But if we look at the process with the more indulgent eye of Induction or Probability we see that a very fair case may sometimes be made out for it. The mere fact that ‘Some B is A’ raises a certain presumption that any particular B taken at random will be an A. There is some reason, at any rate, for the belief, though in the absence of statistics as to the relative frequency of A and B we are unable to assign a value to this belief. I suspect that there may be many cases in which a man has inferred that some particular B is an A on the ground that All A is B, who might justly plead in his behalf that he never meant it to be a necessary, but only a probable inference. The same remarks will of course apply also to the logical fallacy of Undistributed Middle.
Now for a case of the opposite kind, i.e.
one in which Probability fails us, whereas the circumstances seem closely analogous to those in which ordinary inference would be able to make a stand. Suppose that I know that one letter in a million is lost when in charge of the post. I write to a friend and get no answer. Have I any reason to suppose that the fault lies with him? Here is an event (viz.