Explicit Modal Logic

In 1933 Godel introduced a modal logic of provability (S4) and left open the problem of a formal provability semantics for this logic. Since then numerous attempts have been made to give an adequate provability semantics to Godel's provability logic with only partial success. In this paper we give the complete solution to this problem in the Logic of Proofs (LP). LP implements Godel's suggestion (1938) of replacing formulas \F is provable" by the propositions for explicit proofs \t is a proof of F" (t : F ). LP admits the re ection of explicit proofs t : F ! F thus circumventing restrictions imposed on the provability operator by Godel's second incompleteness theorem. LP formalizes the Kolmogorov calculus of problems and proves the Kolmogorov conjecture that intuitionistic logic coincides with the classical calculus of problems. Introduction In 1932 Kolmogorov ([16]) gave an informal description of the calculus of problems in classical mathematics and conjectured that it coincides with intuitioinistic propositional logic Int. Kleene realizability [15], Medvedev nite problems [23] and its variants ([36], [37]) are regarded (cf. [34],[10],[36],[37]) as formalizations of Kolmogorov's calculus of problems. However, they give only necessary conditions for Int, each of them realizes some formulas not derivable in Int. In 1933 Godel ([12]) de ned Int on the basis of the notion of proof in a classical mathematical system, where \proof" may be regarded as a special case of Kolmogorov's \problem solution". Namely, Godel introduced the logic of provability (coinciding with the modal logic S4) and constructed a conservative embedding of Int into S4. S4 has all axioms and rules of classical logic in the modal propositional language along with the axioms 2F ! F , 2(F!G)!(2F!2G), 2F!22F , and the necessitation rule F ` 2F . In [12] no formal provability semantics for S4 was suggested. The straightforward interpretation of 2F as the arithmetical formula Provable(F ) \there exists a number x which is the code of a proof of F". leads to logics of formal provability incompatible with S4 (cf.[7],[8]). 627 Rhodes Hall, Cornell University, Ithaca NY, 14853 U.S.A. email:[email protected]; Moscow University, Russia.