 
Summary: Explicit Modal Logic
Sergei N. Artemov \Lambda
May, 1998
Abstract
In 1933 GĻodel 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 GĻodel'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 GĻodel'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 reflection of explicit proofs t : F ! F thus circumventing restrictions imposed on
the provability operator by GĻodel'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 finite 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
