Admissible Rules of Modal Logics

Abstract

We construct explicit bases of admissible rules for a representative class of normal modal logics (including the systems K4, GL, S4, Grz, and GL.3), by extending the methods of S. Ghilardi and R. Iemhoff. We also investigate the notion of admissible multiple conclusion rules.

Unification in modal and description logics

“…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. For the logics K4, S4, GL, Grz, for which we have shown positive results regarding unification in Subsection 3.4, finite admissibility bases actually do not exist [59], but independent infinite bases are exhibited in [44], by refining previous work from [42].…”

“…Note that Proposition 5, Theorem 6, and Corollary 7 can be extended to transitive logics L having finite model property and satisfying a suitable "extensibility" semantic condition [32,42]. The algorithm sketched above (which is based on Proposition 5 and Theorem 6) has a non-elementary complexity, but it can be greatly simplified: in fact, for most applications, there is no need for computing a finite minimal complete set of unifiers, projective approximations can be computed instead.…”

“…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.…”

Admissibility in Finitely Generated Quasivarieties

“…Hence this procedure is terminating. Moreover, the resulting finite set of finite algebras must generate the quasivariety Q (by Lemma 3.3), contain only Q-subdirectly irreducible algebras, and not contain any algebra that embeds into another member of the set (lines [22][23][24][25][26]. Hence by Theorem 3.4, we obtain: Theorem 3.8.…”

An Avron rule for fragments of R-mingle

“…In particular, Iemhoff [16] and Rozière [29] demonstrated independently that an infinite set of "Visser rules" conjectured by De Jongh and Visser provide a basis for the admissible rules of intuitionistic logic. Bases and analytic Gentzen-style proof systems were subsequently obtained for the admissible rules of certain intermediate logics [12,17,18], transitive modal logics [18,19], and temporal logics [6,7]. For substructural logics, however, much less is known.…”