Two notions of compactness in Gödel logics
type: Item
instance of: scholarly article
title: Two notions of compactness in Gödel logics
zbMATH Open document ID: 1098.03031
author: Petr Cintula
published in: Studia Logica
publication date: 2006-03-17T00:00:00Z
review text: Compactness is an important property of classical propositional logic. It can be defined in two equivalent ways. The first one states that simultaneous satisfiability of an infinite set of formulae is equivalent to the satisfiability of all its finite subsets. The second one states that if a set of formulae entails a formula, then there is a finite subset entailing this formula as well. In propositional many-valued logic, there are different degrees of satisfiability and different possible definitions of entailment, hence the question of compactness is more complex. In this paper the author deals with compactness of Gödel, Gödel\(_\Delta\), and Gödel\(_\sim\) logic. There are several results (all for the countable set of propositional variables) concerning the compactness (based on satisfiability) of these logic by \textit{P. Cintula} and \textit{M. Navara} in ``Compactness of fuzzy logics'' [Fuzzy Sets Syst. 143, 59--73 (2004; Zbl 1040.03019)]. The question of compactness (based on entailment) for Gödel logic was fully answered by \textit{M. Baaz} and \textit{R. Zach} [``Compact propositional Gödel logics'', Proc. 28th Symp. on Multiple Valued Logic, IEEE, 108--113 (1996)]. In this paper a nearly complete answer is given to the problem of compactness based on both concepts for all three logics and for an arbitrary cardinality of the set of propositional variables. Finally, it is shown that there is a tight correspondence between these two concepts.
Mathematics Subject Classification ID: 03B50
Mathematics Subject Classification ID: 03B22
zbMATH DE Number: 5013021
zbMATH Keywords: Gödel logic
zbMATH Keywords: compactness
zbMATH Keywords: entailment
zbMATH Keywords: satisfiability
reviewed by: Albert Hoogewijs
MaRDI profile type: Publication
full work available at URL: s11225-005-2804-7
OpenAlex ID: W2084875152
cites work: Compactness of fuzzy logics.
cites work: A propositional calculus with denumerable matrix
cites work: Residuated fuzzy logics with an involutive negation
cites work: Metamathematics of fuzzy logic
cites work: Fragments of Many-Valued Statement Calculi
cites work: On triangular norm-based propositional fuzzy logics
DOI: 10.1007/S11225-005-2804-7
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description: scientific article; zbMATH DE number 5013021