Explicit Provability: The Intended Semantics for Intuitionistic and Modal Logic
CORNELL UNIV ITHACA NY CENTER FOR FOUNDATIONS OF INTELLIGENT SYSTEMS
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The intended meaning of intuitionistic logic is given by the Brouwer-Heyting-Kolmogorov BHK semantics which informally defines intuitionistic truth as provability and specifies the intuitionistic connectives via operations on proofs. The natural problem of formalizing the BHK semantics and establishing the completeness of propositional intuitionistic logic Int with respect to this semantics remained open until recently. This question turned out to be a part of the more general problem of the intended semantics for Godels modal logic of provability S4 with the atoms F is provable which was open since 1933. In this paper we present complete solutions to both of these problems. We find the logic of explicit provability LP with the atoms t is a proof of F and establish that every theorem of S4 admits a reading in LP as the statement about explicit provability. This provides the adequate provability semantics for S4 along the lines of a suggestion made by Godel in 1938. The explicit provability reading of Godels embedding of Znt into S4 gives the desired formalization of the BHK semantics Int is shown to be complete with respect to this semantics. In addition, LP has revealed the relationship between proofs and types, and subsumes the lambda-calculus, modal lambda-calculus and combinatory logic.
- Theoretical Mathematics