Explicit Provability and Constructive Semantics

Abstract

In 1933 Gödel introduced a calculus of provability (also known as modal logicS4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logicLPof propositions and proofs and show that Gödel's provability calculus is nothing but the forgetful projection ofLP. This also achieves Gödel's objective of defining intuitionistic propositional logicIntvia classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics forIntwhich resisted formalization since the early 1930s.LPmay be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and λ-calculus.

Gödel's Functional Interpretation and Its Use in Current Mathematics

“…In contrast to this, the ∃v B 0 (u, v)-translation of the negative translation apparently requires primitive recursion at type 1 (sufficient to define the Ackermann function) for its modified realizability interpretation (see [69]). 2 Strangely enough, functional interpretation of the ∃v B 0 (u, v)-translation (though that step actually is not necessary by our discussion above) again only uses primitive recursion of type 0. So here we have a statement whose modified realizability seems to require more complicated functionals than its functional interpretation.…”

“…a fixed formal system such as Peano arithmetic PA since the S4-axiom A → A would -as a consequence of Gödel's 2nd incompleteness theorem -not be valid under such an interpretation. This, however, can be overcome by refining the interpretation in terms of provability into a logic of proofs which was sketched by Gödel in 1938 (see [34], published only in 1995 in [41]) and fully elaborated by S. Artemov in [2].…”

Negative Operations on Proofs and Labels

“…It is proved in [2] that LP is complete with respect to arithmetical interpretations based on multi-conclusion proof predicates (i.e. those where a proof can prove more than one proposition).…”

“…In [1,2], S. Artemov defined the Logic of Proofs LP. It is formulated in the propositional language enriched by formulas of the form [[t]]F with the intended meaning "t is a proof of F".…”

Justification logics, logics of knowledge, and conservativity

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AbstractSeveral justification logics have been created, starting with the logic LP, [1]. These can be thought of as explicit versions of modal logics, or of logics of knowledge or belief, in which the unanalyzed necessity (knowledge, belief) operator has been replaced with a family of explicit justification terms.

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