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

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should be P within the intended limit of approximation. Moreover, should there be any serious reason to suspect that any identifiable S presenting itself for admission to the sample was so connected with any S already admitted as to have a special exceptional liability, whether to being like or to being unlike that already admitted instance in respect to being or not being P (under the same limitation that is not to be repeated,) then our "suitable" method would have to exclude that instance from the sample. And once again, should there happen to be any reason to suspect that an instance

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had attracted our attention because of owing to causes connected, whether directly or indirectly, with its being P, or to such causes as should be connected with its not being P, then our suitable method must exclude that instance. Furthermore, our suitable method must so operate that the sample shall contain a sufficient number of instances to give the intended degree of approximation. For instance, if it will suffice that the figure next following the decimal point in the decimal expression of the ratio proportion among all the Ss of such as are P should be exact, 9 instances may first be taken, and if these make the ratio less than

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0.05 or greater than 0.95, they will suffice. If not, 14 more instance may be collected; and if the whole 23 make the ratio less than 0.15 or greater than 0.85, they will suffice. If not, then if 11 more instance be taken, if the whole 34 make the ratio less than 0.25 or greater than 0.75, they will suffice. If not, add 7 more, and if the ratio appears as less than 0.35 or more than 0.65, the 41 will suffice. If not, take 4 more and if the ratio then appears as less than 0.45 or greater than 0.55, the 45 will suffice. If not one more instance will in any case be enough. If the first two figures of the decimal fraction must be correct, a hundred times as many instances will be requisite. It is sufficiently evident that all the above precautions are requisite.

Every person of common-sense must, upon reflexion, acknowledge, what is familiar to everybody habituated to inductive reasoning, that

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all the above precautions are requisite, except that the concluding rule need not have been so detailed, and that, if the instances were sufficiently multiplied, it would suffice that the other rules should not be too frequently and grossly violated and that they should not always prevailingly be violated in the same direction. Let all such diminutions from them be made, and it still remains true for sound reason, that such an induction does not follow merely from the the fact that P is true of such and such of the Ss of a collection of Ss, but that it is necessary to take account of the manner in which these Ss were brought to the reason inquirer's attention

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This fixes a great gulf between Induction and Deduction. It is quite true that we may describe the general conditions of a valid quantitative induction, and may convince ourselves that if the sample be drawn strictly at random from among the Ss and be made sufficiently numerous, then, the general conditions remaining unchanged, it necessarily follows that future experience, under the same general conditions, will, must on the average, of an indefinite multitude of such inductions, bear out the Inductive conclusion. Still, this in no wise suffices to reduce the quantitative induction to any kind of induction. For even