Abstract

Answers to two old questions are given in this paper. 1. Modal logic S4, which was informally specified by Gödel in 1933 as a logic for provability, meets its exact provability interpretation. 2. Brouwer-Heyting-Kolmogorov realizing operations (1931-32) for intuitionistic logic Int also get exact interpretation as corresponding propositional operations on proofs; both S4 and Int turn out to be complete with respect to this proof realization. These results are based on operational reading of S4, where a modality is split into three operations. The logic of proofs with these operations is shown to be arithmetically complete with respect to the intended provability semantics and sufficient to realize every operation on proofs admitting propositional specification in arithmetic.

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