Explicit provability and constructive semantics

Bulletin of Symbolic Logic 7 (1):1-36 (2001)
In 1933 Godel 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 Godel's provability calculus is nothing but the forgetful projection of LP. This also achieves Godel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics 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
Keywords Realism-essay
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DOI 10.2307/2687821
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Sergei Artemov (2008). The Logic of Justification. Review of Symbolic Logic 1 (4):477-513.
Melvin Fitting (2005). The Logic of Proofs, Semantically. Annals of Pure and Applied Logic 132 (1):1-25.
Melvin Fitting (2011). Reasoning About Games. Studia Logica 99 (1-3):143-169.

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