Intermediate logics and Visser's rules
type: Item
instance of: scholarly article
title: Intermediate logics and Visser's rules
zbMATH Open document ID: 1102.03032
author: Rosalie Iemhoff
published in: Notre Dame Journal of Formal Logic
publication date: 2005-07-06T00:00:00Z
review text: A set \(R\) of admissible rules forms a basis of admissible rules for a logic \(L\) if all the admissible rules of \(L\) can be derived from \(R\). It was proved by the author [J. Symb. Log. 66, 281--294 (2001; Zbl 0986.03013)] that the so-called Visser's rules form a basis of admissible rules for the intuitionistic propositional calculus \textbf{IPC}. This paper discusses bases of admissible rules for other intermediate logics. The paper shows that if all Visser's rules are admissible for an intermediate logic \(L\), then Visser's rules form a basis for \(L\). This implies that if all Visser's rules are derivable in \(L\), then all the admissible rules of \(L\) are derivable. As a result, the author obtains that the Dummett logic \textbf{LC} of linear Kripke frames does not have non-derivable admissible rules. The paper also gives a criterion for admissibility of Visser's rules. Namely, it is shown that Visser's rules are admissible (and therefore, by the result mentioned above, form a basis) for an intermediate logic \(L\) iff \(L\) has the offspring property (which is a special kind of an extension property). As a corollary, the author derives that Visser's rules are admissible for the intermediate logic \textbf{KC} of all directed Kripke frames, and that there exist logics with the disjunction property for which not all Visser's rules are admissible.
reviewed by: Nick Bezhanishvili
Mathematics Subject Classification ID: 03B55
Mathematics Subject Classification ID: 03B35
zbMATH DE Number: 2186751
zbMATH Keywords: intermediate logic
zbMATH Keywords: basis of admissible rules
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1305/ndjfl/1107220674
OpenAlex ID: W1982890778
Wikidata QID: Q56474449
cites work: Hypersequent Calculi for Godel Logics -- a Survey
cites work: Modal logic
cites work: Q3128959
cites work: A propositional calculus with denumerable matrix
cites work: A sequence of decidable finitely axiomatizable intermediate logics with the disjunction property
cites work: Unification in intuitionistic logic
cites work: A Resolution/Tableaux Algorithm for Projective Approximations in IPC
cites work: On the admissible rules of intuitionistic propositional logic
cites work: A(nother) characterization of intuitionistic propositional logic
cites work: Computer Science Logic
cites work: Eine Unableitbarkeitsbeweismethode für den Intuitionistischen Aussagenkalkül
cites work: Admissibility of logical inference rules
DOI: 10.1305/NDJFL/1107220674
Recommended article: On the rules of intermediate logics
Recommended article: Q4487248
Recommended article: Hypersequent Systems for the Admissible Rules of Modal and Intermediate Logics
Recommended article: Admissible Rules of Lukasiewicz Logic
Recommended article: Independent Bases of Admissible Rules
Recommended article: The admissible rules of \(\mathsf{BD}_2\) and \(\mathsf{GSc}\)
Recommended article: Describing a basis in semireduced form for inference rules of intuitionistic logic
Recommended article: Admissible rules in the implication-negation fragment of intuitionistic logic
Recommended article: Rules of inference with parameters for intuitionistic logic
Recommended article: On rules
description: scientific article; zbMATH DE number 2186751