We must carefully avoid confusing together inductive investigations which terminate in the establishment of general laws, and those which seem to lead directly to the knowledge of future particular events. That process only can be called induction which gives general laws, and it is by the subsequent employment of deduction that we anticipate particular events. If the observation of a number of cases shows that alloys of metals fuse at lower temperatures than their constituent metals, I may with more or less probability draw a general inference to that effect, and may thence deductively ascertain the probability that the next alloy examined will fuse at a lower temperature than its constituents. It has been asserted, indeed, by Mill,[133] and partially admitted by Mr. Fowler,[134] that we can argue directly from case to case, so that what is true of some alloys will be true of the next. Professor Bain has adopted the same view of reasoning. He thinks that Mill has extricated us from the dead lock of the syllogism and effected a total revolution in logic. He holds that reasoning from particulars to particulars is not only the usual, the most obvious and the most ready method, but that it is the type of reasoning which best discloses the real process.[135] Doubtless, this is the usual result of our reasoning, regard being had to degrees of probability; but these logicians fail entirely to give any explanation of the process by which we get from case to case.
It may be allowed that the knowledge of future particular events is the main purpose of our investigations, and if there were any process of thought by which we could pass directly from event to event without ascending into general truths, this method would be sufficient, and certainly the briefest. It is true, also, that the laws of mental association lead the mind always to expect the like again in apparently like circumstances, and even animals of very low intelligence must have some trace of such powers of association, serving to guide them more or less correctly, in the absence of true reasoning faculties. But it is the purpose of logic, according to Mill, to ascertain whether inferences have been correctly drawn, rather than to discover them.[136] Even if we can, then, by habit, association, or any rude process of inference, infer the future directly from the past, it is the work of logic to analyse the conditions on which the correctness of this inference depends. Even Mill would admit that such analysis involves the consideration of general truths,[137] and in this, as in several other important points, we might controvert Mill’s own views by his own statements. It seems to me undesirable in a systematic work like this to enter into controversy at any length, or to attempt to refute the views of other logicians. But I shall feel bound to state, in a separate publication, my very deliberate opinion that many of Mill’s innovations in logical science, and especially his doctrine of reasoning from particulars to particulars, are entirely groundless and false.
The Grounds of Inductive Inference.
I hold that in all cases of inductive inference we must invent hypotheses, until we fall upon some hypothesis which yields deductive results in accordance with experience. Such accordance renders the chosen hypothesis more or less probable, and we may then deduce, with some degree of likelihood, the nature of our future experience, on the assumption that no arbitrary change takes place in the conditions of nature. We can only argue from the past to the future, on the general principle set forth in this work, that what is true of a thing will be true of the like. So far then as one object or event differs from another, all inference is impossible, particulars as particulars can no more make an inference than grains of sand can make a rope. We must always rise to something which is general or same in the cases, and assuming that sameness to be extended to new cases we learn their nature. Hearing a clock tick five thousand times without exception or variation, we adopt the very probable hypothesis that there is some invariably acting machine which produces those uniform sounds, and which will, in the absence of change, go on producing them. Meeting twenty times with a bright yellow ductile substance, and finding it always to be very heavy and incorrodible, I infer that there was some natural condition which tended in the creation of things to associate these properties together, and I expect to find them associated in the next instance. But there always is the possibility that some unknown change may take place between past and future cases. The clock may run down, or be subject to a hundred accidents altering its condition. There is no reason in the nature of things, so far as known to us, why yellow colour, ductility, high specific gravity, and incorrodibility, should always be associated together, and in other cases, if not in this, men’s expectations have been deceived. Our inferences, therefore, always retain more or less of a hypothetical character, and are so far open to doubt. Only in proportion as our induction approximates to the character of perfect induction, does it approximate to certainty. The amount of uncertainty corresponds to the probability that other objects than those examined may exist and falsity our inferences; the amount of probability corresponds to the amount of information yielded by our examination; and the theory of probability will be needed to prevent us from over-estimating or under-estimating the knowledge we possess.
Illustrations of the Inductive Process.
To illustrate the passage from the known to the apparently unknown, let us suppose that the phenomena under investigation consist of numbers, and that the following six numbers being exhibited to us, we are required to infer the character of the next in the series:—
5, 15, 35, 45, 65, 95.
The question first of all arises, How may we describe this series of numbers? What is uniformly true of them? The reader cannot fail to perceive at the first glance that they all end in five, and the problem is, from the properties of these six numbers, to infer the properties of the next number ending in five. If we test their properties by the process of perfect induction, we soon perceive that they have another common property, namely that of being divisible by five without remainder. May we then assert that the next number ending in five is also divisible by five, and, if so, upon what grounds? Or extending the question, Is every number ending in five divisible by five? Does it follow that because six numbers obey a supposed law, therefore 376,685,975 or any other number, however large, obeys the law? I answer certainly not. The law in question is undoubtedly true; but its truth is not proved by any finite number of examples. All that these six numbers can do is to suggest to my mind the possible existence of such a law; and I then ascertain its truth, by proving deductively from the rules of decimal numeration, that any number ending in five must be made up of multiples of five, and must therefore be itself a multiple.
To make this more plain, let the reader now examine the numbers—
7, 17, 37, 47, 67, 97.