 
Summary: SERGEI N. ARTEMOV & LEV D. BEKLEMISHEV
PROVABILITY LOGIC
1 INTRODUCTION
The idea of provability logic seems to originate in a short paper [G¨odel,
1933]. K. G¨odel was motivated by the question of providing Brouwer's
intuitionistic logic, as formalized by Heyting, with an adequate semantics.
According to Brouwer, intuitionistic truth means provability. Here is a
summary from Constructivism in Mathematics ([Troelstra and van Dalen,
1988], p. 4):
"A statement is true if we have a proof of it, and false if we can
show that the assumption that there is a proof for the statement
leads to a contradiction."
An axiom system for intuitionistic logic was introduced by Heyting in 1930;
its full description may be found in fundamental monographs [Kleene, 1952;
Troelstra and van Dalen, 1988]. In 193134 A. Heyting and A.N. Kol
mogorov made Brouwer's definition of intuitionistic truth explicit, though
informal, by introducing what is now known as the BrouwerHeytingKol
mogorov (BHK) semantics ([Heyting, 1931; Heyting, 1934; Kolmogoroff,
1932]). BHK semantics suggests that a formula is called true if it has a
proof. Further, a proof of a compound statement is described in terms of
