² J. B. Pratt, “Logical Positivism and Professor Lewis,” Journal of Philosophy, XXXI, No. 26, Dec. 20, 1934, pp. 701–710. William Savery, “Chance and Cosmogony,” The Philosophical Review, XLI, No. 3, March, 1932, pp. 147–179. Donald Williams, “The Argument for Realism,” The Monist, July, 1934.

³ Cf., E. J. Nelson, “A Note on Parsimony,” Philosophy of Science, Vol. 3, No. 1, Jan., 1936, pp. 62 ff.

⁴ This deduction is as follows: The Multiplicative Axiom is: pq/h = p/h·q/ph = q/h·p/qh. Replacing p by C which is a consequence of hypothesis H, and q by H, we get: C/h·H/Ch = H/h·C/Hh. Hence,

H/h represents the antecedent probability in question. Only if the probability of H independent of C be finite will H/Ch have a value or will it be possible for H to increase in probability by a verification of its consequence C. (Though C/Hh is greater than C/h, since H is favorable to C, the common assertion that H implies C or that C/Hh = 1 is seldom if ever true in actual cases of induction unless H is a generalization. E.g., the proposition that the victim was shot by a revolver known to have belonged to the accused is not implied by the hypothesis that the accused committed the crime. It will not help to extend H so that C/Hh will more nearly equal 1, for then H/h would be proportionately decreased.)

If acceptance of this axiom be questioned, I should reply that it is explicative of the notion of probability. Though we may not know the definition of probability, I think we do know this, that no proposed definition not implying it would be acceptable: it is a postulate of probability and so systematically defines in part that notion.

⁵ Discussions of the precise formulation of a Principle of Induction so that the required antecedent probability will be finite may be found in Keynes, A Treatise on Probability, and in Nicod, Le problème logique de l'induction. (The English translation of this latter work is so inaccurate and misleading that philosophers should be warned to avoid it.)

⁶ Apology for mentioning this hackneyed point would be in order were it not that some pragmatists seem not yet to have learned from Mill's blunder on this issue.

⁷ Charles S. Peirce seemed to think that if a chance series be continued indefinitely any given order or sequence would certainly take place. He said, “It is an indubitable result of the theory of probabilities that every gambler, if he continues long enough, must ultimately be ruined.” Collected Papers, vol. III, p. 396. Since Peirce surely does not mean to beg the question by the phrase “long enough,” I cannot see that his assertion is quite correct. The most we can say, it seems to me, is that as a gambler continues the probability that he will ultimately be ruined approaches certainty as a limit but will never reach it.

⁸ These two requirements, (1) of a finite probability for a law connecting the external and phenomenal worlds, and (2) of a finite probability for the existence of an external world, are the demands, respectively, that the value of C/Hh be finite and greater than C/h, and that H/h have a finite value. See footnote on p. 241.

⁹ “Essay on Probabilities,” Cabinet Encyclopaedia, p. 27. Quoted from Keynes, op. cit., p. 178. The Inverse Principle, which follows from the Multiplicative Axiom, is this:

where a ₁ and a ₂ are possible hypotheses having the consequence b. Cf., Keynes, op. cit., pp. 148f.

¹⁰ Even if the Principle of Indifference be acceptable, appeal to it would not help here, for, as Keynes has shown, that principle is not applicable when the alternatives are of the form, p or ±.

¹¹ See footnote on p. 241.

¹² If phenomenal events be grounded in an external world, then it is to be expected that no direct connections can be found between them, for phenomena would then be simply epiphenomena of the external world. In other words, if there be such an external world grounding phenomena, the results of Hume's analysis are to be expected.