Absolute Probability Functions for Intuitionistic Propositional Logic

Abstract

Provided here is a characterisation of absolute probability functions for intuitionistic (propositional) logic L, i.e. a set of constraints on the unary functions P from the statements of L to the reals, which insures that (i) if a statement A of L is provable in L, then P(A) = 1 for every P, L's axiomatisation being thus sound in the probabilistic sense, and (ii) if P(A) = 1 for every P, then A is provable in L, L's axiomatisation being thus complete in the probabilistic sense. As there are theorems of classical (propositional) logic that are not intuitionistic ones, there are unary probability functions for intuitionistic logic that are not classical ones. Provided here because of this is a means of singling out the classical probability functions from among the intuitionistic ones.