Basing Probabilistic Logic on Gambles

This article presents a probabilistic logic whose sentences can be interpreted as asserting the acceptability of gambles described in terms of an underlying logic. This probabilistic logic has a concrete syntax and a complete inference procedure, and it handles conditional as well as unconditional probabilities. It synthesizes Nilsson’s probabilistic logic and Frisch and Haddawy’s anytime inference procedure with Wilson and Moral’s logic of gambles. Two distinct semantics can be used for our probabilistic logic: (1) the measure-theoretic semantics used by the prior logics already mentioned and also by the more expressive logic of Fagin, Halpern, and Meggido and (2) a behavioral semantics. Under the measure-theoretic semantics, sentences of our probabilistic logic are interpreted as assertions about a probability distribution over interpretations of the underlying logic. Under the behavioral semantics, these sentences are interpreted only as asserting the acceptability of gambles, and this suggests different directions for generalization.