Admissibility and refutation: some characterisations of intermediate logics

Abstract

Refutation systems are formal systems for inferring the falsity of formulae. These 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.

Refutation Systems: An Overview and Some Applications to Philosophical Logics

“…• Essential connections between refutations and admissibility of rules have been analysed by Goudsmit in the context of IPL and intermediate logics in [Goudsmit, 2014] and [Goudsmit, 2015], where using some techniques for admissible rules he constructs refutation systems (involving structural refutation rules) for the Gabbay -de Jongh logics.…”

Hybrid Deduction–Refutation Systems

“…Furthermore, unifiability of formulae is closely related to admissibility of rules. On the connections of these with refutation rules and systems, see [21].…”

On Rules

“…Another example of an intermediate logic with a basis for admissibility different from the Visser rules is the logic of frames of depth at most 2, as shown by Goudsmit [14]. Goudsmit also provides a nice relation between admissibility and refutation systems in [15].…”