On the admissible rules of intuitionistic propositional logic

Abstract

We present a basis for the admissible rules of intuitionistic propositional logic. Thereby a conjecture by de Jongh and Visser is proved. We also present a proof system for the admissible rules, and give s emantic criteria for admissibility.

Admissible Rules of Modal Logics

“…R. Iemhoff [13] discovered how to use these results to construct explicit bases of admissible rules; specifically, she proved completeness of a basis for IP C earlier conjectured by D. de Jongh and A. Visser.…”

Unification in modal and description logics

“…9 The latter approach has been used in [39,42] to investigate admissibility bases for intuitionistic logic, K4, S4, GL, Grz (bases for S4 were built also in [62] following the former approach). By definition, an admissibility basis for a logic L is a set S of admissible rules such that any other admissible rule r is derivable from S (in the sense that the conclusion of r is L-entailed from the assumption, if L-deduction is enlarged so that it can apply also substitution instances of rules in S, besides axioms from L and standard rules like modus ponens and necessitation); an independent admissibility basis is an admissibility basis which is minimal.…”

Admissible Bases Via Stable Canonical Rules

“…An alternative solution via projectivity and unification was supplied in [11,12]. Explicit bases for admissible rules were built in [15,17,22,23,25]. We refer to Goudsmit [14] for a modern historic account of the admissibility problem.…”