Proof Realization of Intuitionistic and Modal Logics

Logic of Proofs (LP) has been introduced in [2] as a collection of all valid formulas in the propositional language with labeled logical connectives [[t]]( ) where t is a proof term with the intended reading of [[t]]F as \t is a proof of F". LP is supplied with a natural axiom system, completeness and decidability theorems. LP may express some constructions of logic which have been formulated or/and interpreted in an informal metalanguage involving the notion of proof, e.g. the intuitionistic logic and its Brauwer-Heyting-Kolmogorov semantics, classical modal logic S4, etc (cf. [2]). In the current paper we demonstrate how the intuitionistic propositional logic Int can be directily realized into the Logic of Proofs. It is shown, that the proof realizability gives a fair semantics for Int.