Admissible rules for logics containing S4.3
type: Item
instance of: scholarly article
title: Admissible rules for logics containing S4.3
zbMATH Open document ID: 0582.03009
DOI: 10.1007/BF00968695
published in: Siberian Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: The aim of this article is the investigation of rules of inference in logics containing S4.3 modal propositional logic. We use an algebraic approach to admissibility: the rule A/B is admissible in logic \(\lambda\) iff the quasi-identity \(A=1\Rightarrow B=1\) is true on the free algebra \(F_{\omega}(\lambda)\) of denumerable rank from the variety of modal algebras corresponding to \(\lambda\). Theorem 5. The free algebra \(F_{\omega}(\lambda)\) has a finite basis of quasi-identities which is constructed by adding \(\diamond x\wedge \diamond \neg x=1\Rightarrow y=1\) to basis identities of \(F_{\omega}(\lambda)\) when \(\lambda\) \(\supseteq S4.3.\) From this theorem we obtain that there exists an algorithm for recognizing admissibility rules in a modal logic \(\lambda\) by its axiom set when \(\lambda\) \(\supseteq S4.3.\) Proposition 7. Let f/g be an admissible but not derivable rule in a logic \(\lambda\) \(\supseteq S4.3\). Then for every \(t_ i:\) \(f(t_ i)\not\in \lambda\). That is, each admissible, non derivable rule in \(\lambda\) \(\supseteq S4.3\) has no antecedent feasible in \(\lambda\). Proposition 8. For all \(\lambda\) \(\supseteq S4.3\), the logic \(\lambda\) with additional rule of inference \(\diamond x\wedge \diamond \neg x/y\) is structurally complete. In particular, when \(\lambda =S5\), these propositions give an affirmative solution to the questions of \textit{J. Porte} [J. Philos. Logic 10, 409-422 (1981; Zbl 0475.03005)].
Mathematics Subject Classification ID: 03B45
Mathematics Subject Classification ID: 03G25
zbMATH DE Number: 3930970
zbMATH Keywords: rules of inference
zbMATH Keywords: free algebra
zbMATH Keywords: variety of modal algebras
zbMATH Keywords: modal logic
zbMATH Keywords: S5
author: Vladimir Rybakov
MaRDI profile type: Publication
cites work: The deducibilities of S5
cites work: Q3950540
cites work: The Logics Containing S 4.3
full work available at URL: https://doi.org/10.1007/bf00968695
OpenAlex ID: W2030606453
description: scientific article; zbMATH DE number 3930970