On Modal Grzegorczyk Logic

Abstract

In 1964-68 Andrzej Grzegorczyk investigated relational and topological semantics for the intuitionistic logic [15]–[18] . In this connection, following to McKinsey and Tarski, he also considered a semantics for modal logics. It is well known that there is a translation of the intuitionistic logic into the modal S4 logic. This translation was suggested by Gödel in order to find an interpretation of the intuitionistic logic via provability operator.

A. Grzegorczyk found a modal formula G valid in all partially ordered frames with descending chain condition but not in all topological spaces. It follows that this formula is not valid in S4 but one can translate the intuitionistic logic into the calculus arising from S4 by adding Grzegorczyk's formula as a new axiom schema. The resulting logic was called the Grzegorczyk logic in later papers and books on modal logic.

There are a lot of investigations devoted to the Grzegorczyk logic. In this paper we give a short overview of results on the Grzegorczyk logic and its extensions.