§3. Inductive Reasoning
167. A generation and a half of evolutionary fashions in philosophy has not sufficed entirely to extinguish the fire of admiration for John Stuart Mill — that very strong but Philistine philosopher whose inconsistencies fitted him so well to be the leader of a popular school — and consequently there will still be those who propose to explain the general principles of formal logic, which are now fully shown to be mathematical principles, by means of induction. Anybody who holds to that view today may be assumed to have a very loose notion of induction; so that all he really means is that the general principles in question are derived from images of the imagination by a process which is, roughly speaking, analogous to induction. Understanding him in that way, I heartily agree with him. But he must not expect me in 1903 to have anything more than a historical admiration for conceptions of induction which shed a brilliant light upon the subject in 1843. Induction is so manifestly inadequate to account for the certainty of these principles that it would be a waste of time to discuss such a theory.
168. However, it is now time for me to pass to the consideration of Inductive Reasoning. When I say that by inductive reasoning I mean a course of experimental investigation, I do not understand experiment in the narrow sense of an operation by which one varies the conditions of a phenomenon almost as one pleases. We often hear students of sciences, which are not in this narrow sense experimental, lamenting that in their departments they are debarred from this aid. No doubt there is much justice in this lament; and yet those persons are by no means debarred from pursuing the same logical method precisely, although not with the same freedom and facility. An experiment, says Stöckhardt, in his excellent School of Chemistry, is a question put to nature.1) Like any interrogatory, it is based on a supposition. If that supposition be correct, a certain sensible result is to be expected under certain circumstances which can be created, or at any rate are to be met with. The question is, Will this be the result? If Nature replies »No!« the experimenter has gained an important piece of knowledge. If Nature says »Yes,« the experimenter's ideas remain just as they were, only somewhat more deeply engrained. If Nature says »Yes« to the first twenty questions, although they were so devised as to render that answer as surprising as possible, the experimenter will be confident that he is on the right track, since 2 to the 20th power exceeds a million.
169. Laplace was of the opinion that the affirmative experiments impart a definite probability to the theory; and that doctrine is taught in most books on probability to this day, although it leads to the most ridiculous results, and is inherently self-contradictory. It rests on a very confused notion of what probability is.
Probability applies to the question whether a specified kind of event will occur when certain predetermined conditions are fulfilled; and it is the ratio of the number of times in the long run in which that specified result would follow upon the fulfillment of those conditions to the total number of times in which those conditions were fulfilled in the course of experience. It essentially refers to a course of experience, or at least of real events; because mere possibilities are not capable of being counted. You can, for example, ask what the probability is that a given kind of object will be red, provided you define red sufficiently. It is simply the ratio of the number of objects of that kind that are red to the total number of objects of that kind. But to ask in the abstract what the probability is that a shade of color will be red is nonsense, because shades of color are not individuals capable of being counted. You can ask what the probability is that the next chemical element to be discovered will have an atomic weight exceeding a hundred. But you cannot ask what the probability is that the law of universal attraction should be that of the inverse square until you can attach some meaning to statistics of the characters of possible universes. When Leibniz said that this world is the best that was possible, he may have had some glimmer of meaning, but when Quételet 1) says that if a phenomenon has been observed on m occasions, the probability that it will occur on the (m + 1) th occasion is (m+1)/(m+2), he is talking downright nonsense. Mr. F.Y. Edgeworth asserts that of all theories that are started one half are correct. That is not nonsense, but it is ridiculously false. For of theories that have enough to recommend them to be seriously discussed, there are more than two on the average to each general phenomenon to be explained. Poincaré, on the other hand, seems to think that all theories are wrong, and that it is only a question of how wrong they are.
170. Induction consists in starting from a theory, deducing from it predictions of phenomena, and observing those phenomena in order to see how nearly they agree with the theory. The justification for believing that an experiential theory which has been subjected to a number of experimental tests will be in the near future sustained about as well by further such tests as it has hitherto been, is that by steadily pursuing that method we must in the long run find out how the matter really stands. The reason that we must do so is that our theory, if it be admissible even as a theory, simply consists in supposing that such experiments will in the long run have results of a certain character. But I must not be understood as meaning that experience can be exhausted, or that any approach to exhaustion can be made. What I mean is that if there be a series of objects, say crosses and circles, this series having a beginning but no end, then whatever may be the arrangement or want of arrangement of these crosses and circles in the entire endless series must be discoverable to an indefinite degree of approximation by examining a sufficient finite number of successive ones beginning at the beginning of the series. This is a theorem capable of strict demonstration. The principle of the demonstration is that whatever has no end can have no mode of being other than that of a law, and therefore whatever general character it may have must be describable, but the only way of describing an endless series is by stating explicitly or implicitly the law of the succession of one term upon another. But every such term has a finite ordinal place from the beginning and therefore, if it presents any regularity for all finite successions from the beginning, it presents the same regularity throughout. Thus the validity of induction depends upon the necessary relation between the general and the singular. It is precisely this which is the support of Pragmatism.