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Two Notions of Compactness in Gödel Logics

Petr Cintula
Studia Logica: An International Journal for Symbolic Logic
Vol. 81, No. 1 (Oct., 2005), pp. 99-122
Published by: Springer
Stable URL: http://www.jstor.org/stable/20016733
Page Count: 24
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Two Notions of Compactness in Gödel Logics
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Abstract

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, we have different degrees of satisfiability and different possible definitions of entailment, hence the questions of compactness is more complex. In this paper we will deal with compactness of Gödel, $\text{G}\ddot{o}\text{del}_{\Delta}$, and $\text{G}\ddot{o}\text{del}_{\sim}$ logics. There are several results (all for the countable set of propositional variables) concerning the compactness (based on satisfiability) of these logic by Cintula and Navara, and the question of compactness (based on entailment) for Gödel logic was fully answered by Baaz and Zach (see papers [3] and [2]). In this paper we give a nearly complete answer 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, we show a tight correspondence between these two concepts.

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