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INDUCTION, means sometimes generalization ; sometimes the whole series of steps by which a generalization is discovered and established (that is, the processes of observation, formation of hypothesis, verification of hypothesis by further observations or experiments, and formulation of the generalization in a man ner that will best fit all the observations and experiments) ; and sometimes the term is used in a still wider sense for any attempt to discover any kind of order or connection between certain facts, whether it results in a generalization or not. Now the various methods by the aid of which the attempt is usually made to trace order and connection in phenomena are known as the methods of science, and are described in the article SCIENTIFIC METHOD (q.v.). Here it is only necessary to discuss some of the more general aspects and problems of induction, more especially its logical basis or justification.

Logical Basis of Induction.

Generalization is probably as old as human thought. In fact, the tendency to rash generaliza tion would seem to be one of the original sins of mankind. It is one side of the formation of habits. When a certain number of things have been observed.to have certain attributes, uncritical people form the habit of expecting all such things to have those attributes. Witness the reckless way in which such people general ize about the characters of whole peoples, of ter a superficial ac quaintance with a very few representatives. Similarly, where 'certain events have been observed to occur at the same time, or in rapid succession, they are apt to be regarded as uniformly con nected. It is in this way that the numerous superstitions have originated. Mankind did not have to wait for the growth of the scientific spirit to learn to generalize ; it only had to wait, and still waits, for the growth of the scientific frame of mind in order to learn to check that rash tendency, and to tread warily in order to reach legitimate generalizations.

Early thinkers like Aristotle attempted to check the tendency to rash generalization by setting up a severe standard, and insist ing that the ideal of generalization is what is still known as "per fect induction," that is, generalization based upon an exhaustive examination of the whole group or class of facts concerned. No doubt it would be a great boon to mankind if people refrained from generalizing about whole countries or communities until they knew every citizen or member thereof. But then the ideal of perfect induction has made no impression on practical people, and has proved to be worthless as a guide for scientific people. In the vast majority of cases the classes of objects and events with which science is concerned are far too numerous to permit anything even distantly approaching exhaustive individual ex amination of all the members. All the important inductions of science are what used to be called imperfect inductions, that is to say, generalizations based on the examination of a bare sample of the whole class under investigation. And its great problem has been, and still is, how to excuse, or to justify, such extensive generalizations after the study of but a few instances or speci mens. To this question various answers have been attempted, and the most important of them may now be considered briefly.

One answer, which is rather in favour among some of the more philosophical of contemporary men of science, is to the effect that there is really no justification for induction. All inductions, and all forecasts based on them, are just more or less sanguine adventures, or speculations. And the fact that they do not always disappoint us is nothing short of marvellous. It is just like drawing a cheque on a bank and finding it honoured, although one has no reason for thinking that he has a balance there. This kind of agnostic solution, if it may be called a solution, is not really satisfactory. It practically amounts to giving up the prob lem as hopeless. After all, in this as in other matters, it is the business of the investigator to interpret nature in the light of the clues she affords. If a bank honours one's cheque, one does not marvel at it, but draws certain conclusions about one's balance or one's credit. If nature fulfils anticipations based on inductions, some inference might be drawn about her character or consti tution.

Law of Succession.

Another answer given, and one that is much in favour among certain statisticians and other mathe matically minded people, is based on what is known as the law of succession. Assuming that all generalization is essentially of the artless type mentioned above, and commonly known as in duction by simple enumeration, the solution now under considera tion bases itself on the calculus of probability, and correlates the reliability of the generalization with the number and kind of ob servations made. Each observed occurrence of an event in certain circumstances is treated as a point in favour of expecting its recur rence in similar circumstances. However, the probability dimin ishes for every increase in the number of recurrences contem plated. Thus if the event in question has been observed m times and has failed to occur x times, then the probability of r recur rences will be expressed by the formula (m+ I)/(m+x+r+ I), which is known as Laplace's Law of Succession. If the number of occurrences observed (rn) is very large, and if no exceptions are known (i.e., x=o), and if one contemplates just one recur rence (i.e., r= 1), then the probability may approximate certainty, for (m+ I)/(m+r+ I) will in that case nearly = I, i.e., certainty (see PROBABILITY). But now in a real scientific general ization (say, e.g., the Law of Gravitation, or the Laws of Motion), the number of recurrences contemplated (r) is practically infinite, so that its probability, according to the above formula, cannot really be high, even at the best. Moreover, on what rational grounds can a previous occurrence be regarded as a reason for its recurrence, as distinguished from our spontaneous expectation of it? This solution, too, cannot therefore be regarded as satis factory.

J. S. Mill based all induction on the principle of the uniformity of nature, but his conception of this was not very satisfactory. For, on the one hand, he regarded this assumed objective uni formity, in the character and connections of natural phenomena, as the ground of all induction, and, on the other hand, he regarded it as being itself a very comprehensive induction based upon numerous other inductions each much more limited in scope. This ambiguous attempt to make the same principle at once the foun dation and the roof of this whole structure of science has not been received with favour. But it may be reasonably interpreted, perhaps, as meaning that we start with the assumption of the existence of uniformities among natural phenomena, that we jus tify all actual generalizations on the strength of this assumption, but that, on the other hand, the very success of the numerous generalizations made may be regarded as a kind of verification (and, in that sense, as a ground) of the principle itself. This, at all events, would afford some explanation of the unexplained mar vel referred to in the agnostic answer already stated above. And, in any case, it seems impossible to dispense entirely with some such postulate as that of the uniformity of nature, even if we also admit that by itself it does not help us to discover, or to test, any actual generalization. In this respect we need also the postu late or principle of fair samples, that is, the assumption that, with reasonable care, it is possible to judge the character of a large group, or of a whole class, of phenomena by means of a sample selected with discrimination.

Perhaps the least unsatisfactory way of answering the general question as to the logical ground of induction, using this term in its widest sense for every attempt to trace order in nature, is on the following lines. The scientific search for order among natural phenomena would seem to assume the existence of order there. Science does not propose to invent it and impose upon nature, only to discover it, if possible. This search does not neces sarily presuppose a definite conviction that what is sought is actually there. One may look for what is hoped for, or for what is deemed probable, as well as for what is definitely expected to be there. Moreover, to assume that there is some order in nature is not the same thing as to suppose that nature is orderly through and through.

After all the world is vast, and the field of actual scientific investigation is comparatively limited, so it is always open to the man of science to select for his field of research some class of facts in which the discovery of order looks fairly promising. On the whole, experience has shown that there is some order in nature, indeed, sufficiently so to justify and encourage the con tinued search for more. Turning to the question of the ground of generalization more particularly, one must, in the first place, distinguish between those which rest on induction by simple enumeration only, and those which are based ultimately on one of the induction methods, especially when these can be applied with some rigour, and not rather loosely. Inductions based on simple enumeration, and even statistical generalizations must always be regarded with a measure of diffidence. They may indi cate temporary or partial conjunctions rather than general con nections. It is rather different in those cases in which the inductive methods have been applied (whether in the form in which J. S. Mill has formulated them, or in some similar form) .

Even in such cases, it is true, what the method applied really proves is that in the particular instance observed, or experimented with, a certain phenomenon was causally connected with a cer tain other phenomenon. It does not by itself prove the generaliza tion that those two kinds, or classes, of phenomena are causally connected. But we are, so constituted that, wittingly or un wittingly, we assume what has been called the principle of the uniformity of reasons, which states that "whatever is regarded as a sufficient reason in any one instance is regarded as a suffi cient reason in all instances of that type." If in a particular case it appears, through the application of one of the inductive methods, that d as such was causally connected with z, then d must be assumed to be always causally connected with z, and in this way we arrive at the generalization. The extreme sceptic or agnostic may dismiss this principle as a mere prejudice or defect of human intelligence. But is it so unreasonable to suppose that human intelligence has gradually and painfully been shaped so as to fit the constitution of nature? See SCIENTIFIC METHOD and the bibliography given there. For induction in mathematics see MATHEMATICAL INDUCTION.

(A. Wo.)

generalization, nature, regarded, based, scientific, science and phenomena