On Relativizing Kolmogorov's Absolute Probability Functions

Let 5 be a Boolean algebra; let Π be a set of relative (= conditional) probability functions on S, and IT a set of absolute ones; and let V be A Π A, with A here an arbitrary but fixed member of S. (i) A function P' in IT is then the V-restriction of a function P in Π (= P has P' as its K-restriction) if V(A) = P(,4, V) for each A in S; and (ii) the functions in Π relativize those in IT if each function in Π has one in IT as its Γ-restriction and each function in IT is the K-restriction of one in Π. Considered in the paper are two sets of absolute probability functions (Kolmogorov's and Carnap's, the latter like Kolmogorov's except for P(>1) equaling 1 only when A = V), and ten sets of relative ones (among them Popper's, Renyi's, Carnap's, and Kolmogorov's, the last thus called because of their relationship to Kolmogorov's absolute functions). And it is determined which sets of relative functions relativize which sets of absolute ones. S is then allowed to be an arbitrary set, and Popper's relative probability functions on such a set are shown to relativize his absolute ones. I wish to point out here that I have received considerable encouragement from reading A. Renyi's most interesting paper 'On a new Axiomatic Theory of Probability', Acta Mathematica Acad. Scient. Hungaricae 5, 1955, pp. 286-335. Although I had realized for years that Kolmogorov's system ought *Some of our results were presented at the 1987 Meeting of the Society for Exact Philosophy, at the Memorial University of Newfoundland, St. John's, Newfoundland, and at the 1987-88 Annual Meeting of the Association for Symbolic Logic, in New York City. Thanks are due to Professor Sherry May of the Memorial University for inviting us to address the first meeting, and to Professor Harold Hodes of Cornell for inviting us to address the second. Thanks are also due to Kit Fine for his encouragement, and in particular his help concerning a point in Section 4, and to Professor John H. Serembus of Saint Joseph's University for proofreading the text with us. Received December 8, 1987; revised June 6, 1988 486 HUGUES LEBLANC and PETER ROEPER to be relativized, and although I had on several occasions pointed out some of the mathematical advantages of a relativized system, I only learned from Renyi's paper how fertile this relativization could be. The relative systems published by me since 1955 are more general still than Renyi's system. . . . K. R. Popper [11], p. 346n