The validity degree vectors of formulae in two-valued predicate logic

Abstract

By means of infinite product of uniformly distributed probability spaces of cardinal n, the concept of n-validity degrees and validity degree vectors of formulae in two-valued predicate logic are introduced. It is proved that the validity degree vectors of formulae can preserve the logical relation between formulae. Moreover, a consistency theorem is obtained which says that the n-validity degree τ n (A) of the quantifierfree first-order formula A without any repeated predicate symbols or terms is independent of the natural number n, and is a constant equal to the validity degree τ(A 0 ) of the corresponding proposition A 0 in classical propositional logic.