The Development of Logical Probability

Abstract

I shall in the course of this paper investigate some notable attempts to provide a theory of a weak consequence relation ostensibly subsisting between pairs of arbitrarily chosen sentences. At various points in their careers James Bernoulli and Leibniz, Keynes and Carnap were preoccupied with this problem. Leibniz was the first, to my knowledge, to articulate it; the remaining three have been preeminent in their attempts to provide a solution. Part of the significance of a solution lies in its furnishing a criterion of verisimilitude; for we know that if ∑ ⊢ σ holds, and all the sentences in ∑ are true, then so is σ. A measure of the strength of a weak relation of consequence between arbitrary ∑and σ could therefore be regarded as a measure of the verisimilitude of σ, given the truth of all the sentences in ∑. The exigencies of life in an uncertain world appear to require, moreover, for their mitigation, such a criterion by which to discriminate between conjectures which transcend experience at some given time. (I am not, by the way, claiming that this is the only sense that can be given to the vague term ‘verisimilitude’ — Popper, for example, has given an entirely distinct explication¹ —; but that term is indeed so vague that it will surely support the gloss I have given it, the history of which reaches back to Leibniz if not before.)