. . if our critics' philosophy of logic entails that induction cannot be valid, validity of induction equally entails that their philosophy of logic is false. Donald Williams, The Ground of Induction The problem of induction has been summarized succinctly by Max Black as: Why, if at all, is it reasonable to accept conclusions of certain inductive arguments as true-or at least probably true? (Black, 1967, 170). It goes without saying that problem of induction is one of big philosophical problems; equally, it goes without saying that there is little consensus among philosophers as to how problem may be solved, or even as to minutiae of problem. But in my view, a great number of arguments have one thing in common: they draw implicitly on a limited philosophical principle of counting. This is principle termed by Georg Cantor, in his Mannigfaltigkeitslehre of 1883, as the first principle of In present essay, however, I shall be arguing that problem of induction can be solved only by means of taking into account considerations pertaining to Cantor's second principle of generation. My strategy will be, firstly, to look at some key arguments associated with problem of induction; secondly, to show that all of these arguments have been framed with first principle of generation implicitly in mind; and, thirdly, to argue that if a more embracing notion of counting-the second principle of generation-is invoked, problem of induction can be seen largely to disappear. One of my conclusions will be that it is Whitehead's philosophy of induction that is closest to providing a solution to problem of induction, but that Whitehead lacks necessary precision of Cantor's system. The Problem of Induction The philosophical ideas of David Hume provide an obvious starting point for any treatment of problem of induction. The vastness of Hume scholarship, however, makes it impractical for me here to address question of extent to Hume's own writing can be said to deal adequately with problem. Nevertheless, following paraphrase of Hume's position is useful: Hume's problem can be put into modem dress without restriction to causal inference, as follows: An inductive inference from an observed association of attributes (A^sub n^-B^sub n^ can justify inference to another case (A^sub n^+ 1-B^sub n^+1) or inference to corresponding generalization (All A are B ) only if association is somehow known to be lawlike, not merely accidental. Yet how can this be known in primary inductions that do not themselves rest upon assumed truth of other laws? Certainly not by immediate experience, nor a priori, nor, without begging question, by appeal to induction. (Black, 1967, 171). There have, of course, been many replies to Hume. Although it may appear trivial, what I wish to emphasize with regard to passage just quoted is fact that a particular problem arises when attempt is made to pass from knowledge of individual cases to knowledge of all such cases. That is, in some sense number-and, by extension, counting or enumeration-seems crucially involved in problem.1 Could it be, indeed, that, so far as induction is concerned, simple enumeration is irrelevant? John Maynard Keynes, in his A Treatise on Probability, is one who feels that success of induction does not depend wholly on enumeration, that is, number of confirming instances: It has often been thought that essence of inductive argument lies in multiplication of instances. Where is that process of reasoning, Hume inquired, which from one draws a conclusion, so different from that it infers from a hundred instances, that are no way different from that single instance I repeat that by emphasizing number of instances Hume obscured real object of method. If it were strictly true that hundred instances are no way different from single instance, Hume would be right to wonder in what manner they can strengthen argument. …
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