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Proof Realization of Intuitionistic and Modal Logics Sergei N. Artemov \Lambda

Summary: Proof Realization of Intuitionistic and Modal Logics
Sergei N. Artemov \Lambda
Technical Report MSI 96­06, Cornell University, 1996
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 involv­
ing the notion of proof, e.g. the intuitionistic logic and its Brauwer­Heyting­Kolmogorov
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.
1 Introduction
Logic of Proofs (LP) incorporates proof terms directly into the propositional language using
new atomic formulas [[t]]F with the intended reading ``t is a proof of F'' (cf. [2]). A func­
tional completeness theorem from [2] (cf. Section 5 below) demonstrates, that three basic
operations on proofs: application, proof checker, and choice constitute a basis for all oper­
ations on proofs, expressible in the propositional language with the labeled connectives of
the type [[term]](formula). These three operations are explicitely incorporated into LP . The


Source: Artemov, Sergei N. - Faculty of Mechanics and Mathematics, Moscow State University


Collections: Computer Technologies and Information Sciences