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]](\Delta) 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

Unified Semantics for Modality and lambda-terms via Proof Polynomials

It is shown that the modal logic S4, simple -calculus and modal -calculus admit a realization in a very simple propositional logical system LP , which has an exact provability semantics. In LP both modality and -terms become objects of the same nature, namely, proof polynomials. The provability interpretation of modal -terms presented here may be regarded as a system-independent generalization of the Curry-Howard isomorphism of proofs and -terms. 1 Introduction The Logic of Proofs (LP , see Section 2) is a system in the propositional language with an extra basic proposition t : F for "t is a proof of F ". LP is supplied with a formal provability semantics, completeness theorems and decidability algorithms ([3], [4], [5]). In this paper it is shown that LP naturally encompasses -calculi corresponding to intuitionistic and modal logics, and combinatory logic. In addition, LP is strictly more expressive because it admits arbitrary combinations of ":" and propositional connectives. The id..