Explicit provability and constructive semantics

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 semantics for Intwhich resisted formalization since the early 1930s. LPmay be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and ë-calculus. §1. A need for a theory of proofs. According to Brouwer, intuitionistic truth means provability. Here is a summary from Constructivism in Mathematics by Troelstra and van Dalen ([105], 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.” In 1931–34 Heyting and Kolmogorov made Brouwer’s definition of intuitionistic truth explicit, though informal, by introducing what is now known as the Brouwer-Heyting-Kolmogorov (BHK) semantics ([53], [54], [58]). The BHK semantics is widely recognized as the intended semantics for intuitionistic logic ([33], [34], [43], [62], [74], [78], [104], [105], [107], [108], [111], [114]). Its description uses the unexplained primitive notions of construction and proof (Kolmogorov used the term problem solution for the latter). It stipulates that • a proof of A ∧ B consists of a proof of A and a proof of B , • a proof of A ∨ B is given by presenting either a proof of A or a proof of B1, Received September 17, 1998; revised October 10, 2000. The research described in this paper was supported in part by ARO under the MURI program “Integrated Approach to Intelligent Systems”, grant DAAH04-96-1-0341, by DARPA under programLPE, project 34145, and by the Russian Foundation for Basic Research, grant 96-01-01395. Neither Heyting’s paper [54] nor Kolmogorov’s [58] contains the well-known extra condition on the disjunction: a proof of a disjunction should also specify which one of the disjuncts c © 2001, Association for Symbolic Logic 1079-8986/01/0701-0001/$4.60