MS 842 (1908) - A Neglected Argument - Early Drafts

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capable of inferring the truth of what, in logic, is termed a universal proposition. fo For what is called "complete induction" is not inductive reasoning, but is logistic deduction. We might further say, if we chose, that every crude induction concludes a universal proposition; but this would be merely the expression of a way of regarding matters. For any proposition concerning the general run of future experience may be regarded as universal, even if it be "A pair of dice will, every

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now and then, turn up doublets." The undipped heel of crude induction is that if its conclusion be understood as indefinite, it will be of little use, while if it be taken definitely, it is liable at any moment to be utterly shattered by a single experience; or for a series of experiences, constituting a single one of if the whole constitute but a single one of the instances to which an inductive conclusion refers, is to be regarded as a single experience.

From the weakest class kind of induction let us pass at once to the strongest. This invetigates investigates the interrogative suggestion

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of retroduction, 'What is the "real probability" that an individual member of a certain experiential class, say the Ss, will have the a certain character, of say that of being P?' This it does by first collecting, on scientific principles, a "fair sample" of the Ss, taking due account, in doing so, of the intention of using its proportion of members that possess the predesignate character of being P. This sample will contain none of those Ss that suggested on which the retroduction was founded. The induction then presumes that the value of the proportion,

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among the Ss of the sample, of those that are P, probably approximates, within a certain limit of approximation, to the value of the real probability in question. I propose to term such reasoning Quantitative Induction. Now, if I were writing a treatise on logic, I should here be obliged, not only to explain teach the art of sampling, including all that Dr. Karl Pearson and others have taught us about distributions of specific instances among general ones, and the consequent proper inferences in such cases, but I should

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have to state and expound the exact definitions of "real probability," "independent," "fair sample," "predesignate," etc. As it is, I will limit myself to a single needful explanation that, so far as I know, the reader could not find definitely stated in any of the books. It is that when we say that a certain ratio will have a certain value in "the long run," (Printer: Do not [let] this be hyphened.) we refer to the probability-limit of an endless succession of fractional values; that is, to the only possible value from 0 to ∞, inclusive, about which the values of the endless succession will never