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, GödelΔ, and Gödel∼ 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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References
Baaz, M., ‘Infinite-valued Gödel logic with 0-1 projector and relativisations’, in P. Hájek, (ed.), Gödel'96: Logical foundations of mathematics, computer science and physics, Lecture notes in logic 6, Springer-Verlag, 1996, pp. 23–33.
Baaz, M., and R. Zach, ‘Compact propositional Gödel logics’, in Proceedings of 28th International Symposium on Multiple Valued Logic, IEEE Computer Society Press, Los Alamitos, CA, 1998, pp. 108–113.
Cintula, P., and M. Navara, ‘Compactness of fuzzy logics’, Fuzzy Sets and Systems, 143/1:59-73, 2004.
Dummett, M., ‘A propositional calculus with denumerable matrix’, Journ. Symb. Logic, 27:97–106, 1959.
Esteva, F., L. Godo, P. Hájek, and M. Navara, ‘Residuated fuzzy logics with an involutive negation’, Archive of mathematical logic, 40:103–124, 2000.
Gödel, K., ‘Zum intuitionistischen Aussagenkalkül’, Anzieger Akademie der Wissenschaften Wien, Math. – naturwissensch. Klasse, 69:65–66, 1932.
Hájek, P., Metamathematics of fuzzy logic, Kluwer, Dordrecht, 1998.
Navara, M., and U. Bodenhofer, ‘Compactness of fuzzy logic’, in ICSC Congress on computational intelligence: methods and applications (CIMA 2001), Bangor, Wales, UK., 2001, pp. 654–657.
Rose, A., and J. B. Rosser, ‘Fragments on many-valued statement calculi’, Trans. A.M.S., 87:1–53, 1958.
Zafrany, S., D. Butnariu, and E. P. Klement ‘On triangular norm-based propositional fuzzy logics’, Fuzzy Sets and Systems, 69:241–255, 1995.
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Cintula, P. Two notions of compactness in Gödel logics. Stud Logica 81, 99–123 (2005). https://doi.org/10.1007/s11225-005-2804-7
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DOI: https://doi.org/10.1007/s11225-005-2804-7