In the full propositional logic, [formula omitted] of classical tautologies are intuitionistically valid

Abstract

We present a quantitative comparison of classical and intuitionistic logics, based on the notion of density, within the framework of several propositional languages. In the most general case–the language of the “full propositional system”–we prove that the fraction of intuitionistic tautologies among classical tautologies of size n tends to 5/8 when n goes to infinity. We apply two approaches, one with a bounded number of variables, and another, in which formulae are considered “up to the names of variables”. In both cases, we obtain the same results. Our results for both approaches are derived in a unified way based on structural properties of formulae. As a by-product of these considerations, we present a characterization of the structures of almost all random tautologies.