Two notions of compactness in Gödel logics
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.
scientific article; zbMATH DE number 5013021
scientific article