On Logic of Formal Provability and Explicit Proofs

Abstract

In 1933, G\"odel considered two modal approaches to describing provability. One captured formal provability and resulted in the logic GL and Solovay's Completeness Theorem. The other was based on the modal logic S4 and led to Artemov's Logic of Proofs LP. In this paper, we study introduced by the author logic GLA, which is a fusion of GL and LP in the union of their languages. GLA is supplied with a Kripke-style semantics and the corresponding completeness theorem. Soundness and completeness of GLA with respect to the arithmetical provability semantics is established.