A Note on Extensions: Admissible Rules via Semantics

Abstract

Any intermediate logic with the disjunction property admits the Visser rules if and only if it has the extension property. This equivalence restricts nicely to the extension property up to n.In this paper we demonstrate that the same goes even when omitting the rule ex falso quod libet, that is, working over minimal rather than intuitionistic logic. We lay the groundwork for providing a basis of admissibility for minimal logic, and tie the admissibility of the Mints-Skura rule to the extension property in a stratified manner.Keywords: admissible rules, minimal logic, disjunction property, extensions of Kripke modelsThe admissible rules of a theory are those rules under which the theory is closed. Derivable rules are admissible. For classical propositional logic, this is the whole story. For intuitionistic propositional logic (IPC) -and minimal logic -it is not.Friedman (1975, Problem 40) conjectured admissibility for IPC to be decidable, as has been confirmed by Rybakov (1984). De Jongh and Visser conjectured that the Visser rules form a basis of admissibility for IPC, that is to say, all admissible rules of IPC become derivable after adjoining the Visser rules. Rozière (1992) and Iemhoff (2001b) independently confirmed this. Again independently, Skura (1989) demonstrated that IPC is the sole intermediate logic that admits a restricted form of the Visser rules.At the Pisa Proof Theory workshop of 2012 George Metcalfe gave a tutorial on admissible rules. As has become standard practice, Metcalfe mentioned Lorenzen (1955) as the first place where admissible rules where studied an sich. Jan von Plato objected that Johansson (1937) already discussed them. Odintsov and Rybakov (2012) proved admissibility for minimal logic to be decidable. In this paper we lay the groundwork for studying all admissible rules of Johansson's minimal logic, with the eventual goal of providing an explicit basis of admissibility. This paper aims to provide uniformity to some of the literature regarding admissible rules for logics above minimal logic. We make several observations, many of which not elsewhere available in the generality stated here. Although this paper contains novel results, most notably the semantic characterization of admissibility for an adaptation of the rules studied by Skura, its main purpose is to provide a unified approach to the study of admissible rules over minimal logic.

Admissibility and refutation: some characterisations of intermediate logics

“…In this paper we focus on multi-conclusion admissible rules, as first suggested by Kracht (1999), and used in the context of modal logic by Jeřábek (2005) and intermediate logics by Goudsmit (2013a). Admissible rules, as defined by the aforementioned authors, are inherently structural.…”

Multiple Conclusion Rules in Logics with the Disjunction Property

“…It was reasonable to ask the same questions regarding m-rules: whether a given logic has admissible, not derived m-rules, whether m-rules have a finite or recursively enumerable basis, or whether the admissibility of m-rules is decidable. The bases of m-rules for a variety of intermediate and normal modal logics were constructed in [19,20,13,11,12]. 2 Using idea from [8], it is not hard to show that if a logic has a recursively enumerable explicit basis of admissible rules, it has a recursive basis.…”