# zbMATH — the first resource for mathematics

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.

##### MSC:
 03B22 Abstract deductive systems 03B20 Subsystems of classical logic (including intuitionistic logic) 03G10 Logical aspects of lattices and related structures
Full Text:
##### References:
  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)  Bloom S.L., Brown D.J. Classical abstract logics. Diss. Math. 102: 43–51 (1973) · Zbl 0317.02072  Brown D.J., Suszko R. Abstract logics. Diss. Math. 102: 9–42 (1973) · Zbl 0317.02071  Brunner, A.B.M., Lewitzka, S.: Representations of some Abstract Logics via Spectral Spaces, preprint (2007)  Cleave J.P. A Study of Logics. Oxford University Press, Oxford (1991) · Zbl 0763.03003  Cohn P.M. Universal Algebra. Harper and Row, New York (1965)  Czelakowski J. The Suszko operator, Part I. Studia Logica (Special Issue on Algebraic Logic II) 74: 181–231 (2003) · Zbl 1043.03050  Davey B.A., Priestley H.A. Introduction to Lattices and Order, 2nd edn. Cambridge University Press, Cambridge (2002) · Zbl 1002.06001  Dunn J.M., Hardegree G.H. Algebraic Methods in Philosophical Logic. Clarendon Press, Oxford (2001) · Zbl 1014.03002  Font, J.M., Jansana, R.: A General Algebraic Semantics for Sentential Logics. Lecture Notes in Logic, vol. 7. Springer, Berlin (1996) · Zbl 0865.03054  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  Font J.M., Verdú V. A first approach to abstract modal logics. J. Symb. Logic 54: 1042–1062 (1989) · Zbl 0687.03008  Isbell J.R. Directed union and chains. Proc. AMS 17: 1467–1468 (1966) · Zbl 0168.26402  Jansana R., Palmigiano A. Referential semantics: duality and applications. Rep. Math. Log. 41: 63–93 (2006) · Zbl 1136.03010  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  Lewitzka, S.: : A 4-valued Truth Theory and Metalogic, preprint (2007) · Zbl 1131.03006  Martin N.M., Pollard S. Closure Spaces and Logic. Kluwer, Dordrecht (1996) · Zbl 0855.54001  van Fraassen B.C. Formal Semantics and Logic. Macmillan, New York (1971) · Zbl 0253.02002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.