Correspondence between modal Hilbert axioms and sequent rules with an application to S5

Abstract

Which modal logics can be 'naturally' captured by a sequent system? Clearly, this question hinges on what one believes to be natural, i.e. which format of sequent rules one is willing to accept. This paper studies the relationship between the format of sequent rules and the corresponding syntactical shape of axioms in an equivalent Hilbert-system. We identify three different such formats, the most general of which captures most logics in the S5-cube. The format is based on restricting the context in rule premises and the correspondence is established by translating axioms into rules of our format and vice versa. As an application we show that there is no set of sequent rules of this format which is sound and cut-free complete for S5 and for which cut elimination can be shown by the standard permutation-of-rules argument. Show more