Summary: Logic of Proofs: a Unified Semantics
for Modality and terms \Lambda
Sergei N. Artemov y
In 1933 GĻodel introduced a modal logic of provability (S4) and left open the problem of
a formal provability semantics for this logic. We give a complete solution to this problem.
We find a sound and complete axiom system for the logic (LP) with atomic propositions
for explicit proofs ``t is a proof of F '', and construct an exact realization of S4 in LP . In
LP the reflection principle is valid, which circumvents some of the restrictions imposed on
the provability operator by GĻodel's second incompleteness theorem.
LP formalizes the Kolmogorov calculus of problems and proves the Kolmogorov con
jecture (1932) that intuitionistic logic coincides with the classical calculus of problems.
LP has revealed the relationship between proofs and types, and subsumes the calculus,
modal calculus and combinatory logic.
The intended meaning of the intuitionistic logic was informally explained first in terms of
operations on proofs due to Brouwer, Heyting and Kolmogorov (cf. ,,). This in
terpretation is widely known as the BHK semantics of intuitionistic logic. However, despite
some similarities in the informal description of the functions assigned to the intuitionistic con