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Minimally generated abstract logics. (English) Zbl 1255.03022
Summary: In this paper we study an alternative approach to the concept of abstract logic and to connectives in abstract logics. The notion of abstract logic was introduced by D. J. Brown and R. Suszko [Diss. Math 102, 9–41 (1973; Zbl 0317.02071)] – nevertheless, similar concepts have been investigated by various authors. Considering abstract logics as intersection structures we extend several notions to their \(\kappa\)-versions ({\(\kappa\)} {\(\omega\)}), introduce a hierarchy of \(\kappa\)-prime theories, which is important for our treatment of infinite connectives, and study different concepts of {\(\kappa\)}-compactness. We are particularly interested in non-topped intersection structures viewed as semi-lattices with a minimal meet-dense subset, i.e., with a minimal generator set. We study a chain condition which is sufficient for a minimal generator set, implies compactness of the logic, and in regular logics is equivalent to (\(\kappa\)-) compactness of the consequence relation together with the existence of a (\(\kappa\)-) inconsistent set, where \(\kappa\) is the cofinality of the cardinality of the logic. Some of these results are known in a similar form in the context of closure spaces, we give extensions to (non-topped) intersection structures and to big cardinals presenting new proofs based on set-theoretical tools. The existence of a minimal generator set is crucial for our way to define connectives. Although our method can be extended to further non-classical connectives we concentrate here on intuitionistic and infinite ones. Our approach leads us to the concept of the set of complete theories which is stable under all considered connectives and gives rise to the definition of the topological space of the logic. Topological representations of (non-classical) abstract logics by means of this space remain to be further investigated.

03B22 Abstract deductive systems
03B20 Subsystems of classical logic (including intuitionistic logic)
03G10 Logical aspects of lattices and related structures
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[1] Abramsky S., Jung A. Domain theory. In: Abramsky S., Gabbay D.M., Maibaum T.S.E. (eds). Handbook for Logic and Computer Science, vol. 3. Clarendon Press, Oxford (1994)
[2] Bloom S.L., Brown D.J. Classical abstract logics. Diss. Math. 102: 43–51 (1973) · Zbl 0317.02072
[3] Brown D.J., Suszko R. Abstract logics. Diss. Math. 102: 9–42 (1973) · Zbl 0317.02071
[4] Brunner, A.B.M., Lewitzka, S.: Representations of some Abstract Logics via Spectral Spaces, preprint (2007)
[5] Cleave J.P. A Study of Logics. Oxford University Press, Oxford (1991) · Zbl 0763.03003
[6] Cohn P.M. Universal Algebra. Harper and Row, New York (1965)
[7] Czelakowski J. The Suszko operator, Part I. Studia Logica (Special Issue on Algebraic Logic II) 74: 181–231 (2003) · Zbl 1043.03050
[8] Davey B.A., Priestley H.A. Introduction to Lattices and Order, 2nd edn. Cambridge University Press, Cambridge (2002) · Zbl 1002.06001
[9] Dunn J.M., Hardegree G.H. Algebraic Methods in Philosophical Logic. Clarendon Press, Oxford (2001) · Zbl 1014.03002
[10] Font, J.M., Jansana, R.: A General Algebraic Semantics for Sentential Logics. Lecture Notes in Logic, vol. 7. Springer, Berlin (1996) · Zbl 0865.03054
[11] Font J.M., Jansana R., Pigozzi D. A survey of abstract algebraic logic. Studia Logica (Special Issue on Algebraic Logic II) 74: 13–97 (2003) · Zbl 1057.03058
[12] Font J.M., Verdú V. A first approach to abstract modal logics. J. Symb. Logic 54: 1042–1062 (1989) · Zbl 0687.03008
[13] Isbell J.R. Directed union and chains. Proc. AMS 17: 1467–1468 (1966) · Zbl 0168.26402
[14] Jansana R., Palmigiano A. Referential semantics: duality and applications. Rep. Math. Log. 41: 63–93 (2006) · Zbl 1136.03010
[15] Lewitzka, S.: Abstract logics, logic maps and logic homomorphisms. In: Logica Universalis, vol. 1(2), pp. 243–276. Birkhäuser Verlag, Basel (2007) · Zbl 1131.03006
[16] Lewitzka, S.: : A 4-valued Truth Theory and Metalogic, preprint (2007) · Zbl 1131.03006
[17] Martin N.M., Pollard S. Closure Spaces and Logic. Kluwer, Dordrecht (1996) · Zbl 0855.54001
[18] van Fraassen B.C. Formal Semantics and Logic. Macmillan, New York (1971) · Zbl 0253.02002
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