Explicit Provability and Constructive Semantics

Sergei N. Artemov
2001 Bulletin of Symbolic Logic  
In 1933 Gödel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that Gödel's provability calculus is nothing but the forgetful projection of LP. This also achieves Gödel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability
more » ... emantics for Int which resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and λ-calculus.
doi:10.2307/2687821 fatcat:twz2ui2ckzabzp6b2f64nlhm6i