On relativizing Kolmogorov's absolute probability functions.
Hugues Leblanc, Peter J. Roeper · Notre Dame Journal of Formal Logic · 1989
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.