MATHEMATICAL FOUNDATIONS OF CONDITIONALS AND THEIR PROBABILISTIC ASSIGNMENTS

Abstract

This paper addresses the mathematical modeling of information as originally expressed in natural language in conditional form. A number of different conditional event algebras—all avoiding the Lewis triviality result—are briefly surveyed, including the main body of those proposed previously, classified as Type I, and the newly expanded Type II product space approach (PS) originally independently offered by Van Fraasen. The issue of higher order conditionals and triviality is also discussed. In addition, this work considers two basic results of McGee: Firstly, this paper provides a new general procedure (M) which specializes to that result of McGee, where in effect, the probability evaluations of logical compounds of conditionals in the PS approach are independently derived from a fair standard betting scheme, though McGee does not obtain any explicit conditional event algebra underpinning for these evaluations. The new procedure, which can act upon any Type I conditional event algebra, is based, in part, upon decision theory principles, and in part, upon a fine-tuned evaluation of the third or indeterminate values of the three-valued logic form of Type I conditionals. It is shown that McGee’s derivation is equivalent to M acting upon the Type I conditional event algebra called GNW in the literature—or equivalently the min, max, 1-() conjunctive, disjunctive, negation fragment of Lukasiewicz’ three-valued logic. On the other hand, it is shown that the action of M upon another Type I conditional event algebra called SAC in the literature—equivalent to a corresponding fragment of a three-valued logic originally proposed by Sobocinski—leads back to the same conditional event algebra. Secondly, this paper provides a new characterization of PS , corresponding to McGee’s second result, and which, unlike McGee, does not assume—nor is compatible, in general, with—the import-export rule. Among other topics discussed, are conditional random variables, deduction and non-monotonic logic applications.