Many valued probability theory

Abstract

The apparent conflict between many valued logic and probability theory is resolved if we treat the probability of a sentence as the probability that the sentence has some specified truth value. The classical probability of a sentence is the probability that the sentence is classically true. In an analogous way, we develop a class of probability theories appropriate for any finite valued logics; the probability of a sentence is interpreted as the probability that the sentence takes some value in a specified subset of the semantic range. We show that for any finite valued logic, there is an appropriate many valued probability theory providing a characteristic probabilistic semantics for which the logic is both sound and complete.