Admissibility and Refutation: Some Characterisations of Intermediate Logics

Abstract

Refutation systems are formal systems for inferring the falsity of formulae. ese systems can, in particular, be used to syntactically characterise logics. In this paper we explore the close connection between refutation systems and admissible rules. We develop technical machinery to construct refutation systems, employing techniques from the study of admissible rules. Concretely we provide a refutation system for the intermediate logics of bounded branching, known as the Gabbay–de Jongh logics. We show that this gives a characterisation of these logics in terms of their admissible rules. To illustrate the technique we also provide a refutation system for Medvedev’s logic.