Keynes vs. Kolmogorov: Two Axiomatics of Probability

The paper examines the logical theory of probability formulated by John Maynard Keynes in A Treatise on Probability (1921) as an axiomatic project competing with the measure-theoretic approach codified by Andrei Kolmogorov in Grundbegriffe der Wahrscheinlichkeitsrechnung (1933). Beyond a synthetic comparison of both systems and their contemporary successors, the paper makes two original contributions. First, we identify and formally prove the Boolean presupposition of Ramsey's Dutch book argument: the argument establishes only a conditional result — if the event domain is a Boolean algebra, then coherence requires classical probability — whose antecedent is precisely what Keynes's partial order postulate denies. Ramsey therefore did not refute Keynesian logicism unconditionally; he refuted it within a domain whose universality Keynes contested. This formalises, using the Wroński–Godziszewski (2016) theorem, a gap in the standard history of the debate that has gone unnamed in the literature. Second, we propose the imprecision index of a credal set as the first candidate formal analogue of Keynes's weight of evidence within the established framework of imprecise probabilities.