Abstract
In [14] we used the term finitely algebraizable for algebraizable logics in the sense of Blok and Pigozzi [2] and we introduced possibly infinitely algebraizable, for short, p.i.-algebraizable logics. In the present paper, we characterize the hierarchy of protoalgebraic, equivalential, finitely equivalential, p.i.-algebraizable, and finitely algebraizable logics by properties of the Leibniz operator. A Beth-style definability result yields that finitely equivalential and finitely algebraizable as well as equivalential and p.i.-algebraizable logics can be distinguished by injectivity of the Leibniz operator. Thus, from a characterization of equivalential logics we obtain a new short proof of the main result of [2] that a finitary logic is finitely algebraizable iff the Leibniz operator is injective and preserves unions of directed systems. It is generalized to nonfinitary logics. We characterize equivalential and, by adding injectivity, p.i.-algebraizable logics.
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References
BLOK, W. J., D. PIGOZZI, 1986, ‘Protoalgebraic Logics’, Studia Logica 45, 337-369.
BLOK, W. J., D. PIGOZZI, 1989, ‘Algebraizable Logics’, Memoirs of the Am. Math. Soc. 396.
BLOK, W. J., D. PIGOZZI, 1992, ‘Algebraic Semantics for Universal Horn Logic without Equality’, in A. Romanowska, J. D. H. Smith (eds.) Universal Algebra and Quasigroup Theory, Heldermann, Berlin, 1-56.
BURRIS, S., H.-P. SANKAPPANAVAR, 1981, A Course in Universal Algebra, Berlin.
CZELAKOWSKI, J., 1981, ‘Equivalential Logics (I), (II)’, Studia Logica 40, 227-236, 355–372.
CZELAKOWSKI, J., 1992, ‘Consequence Operations Foundational Studies’, Reports of the Research Project Theories, Models, Cognitive Schemata, Polish Academy of Science, (Prepublication).
CZELAKOWSKI, J., Beyond Protoalgebraic Logics. The Suszko Operator, Manuscript.
CZELAKOWSKI, J., 1994, Logic, Algebra, Consequence Operations, preliminary version.
CZELAKOWSKI, J., W. DZIOBIAK, 1991, ‘A Deduction Theorem Scheme for Deductive Systems of Propositional Logics’, Studia Logica 50, 385-390.
FONT, J. M., R. JJANSANA, 1993, A general algebraic semantics for deductive systems, Preliminary version, University Barcelona.
Font, J. M., V. VerdÚ, 1991, ‘Algebraic Logic for Classical Conjunction and Disjunction’, Studia Logica 50, 391-419.
Font, J. M., V. VerdÚ, 1991, ‘Algebraic Logic for some Non-protoalgebraizable Logics’, in H. Andréka, J. D. Monk, I. Németi (eds.), Algebraic Logic, North-Holland, Amsterdam. 183-188.
HERRMANN, B., 1993, Equivalential Logics and Definability of Truth, Ph. D. Dissertation, Freie Universität Berlin.
HERRMANN, B., 1996, ‘Equivalential and Algebraizable Logics’, Studia Logica 57, 419-436.
HERRMANN, B., F. WOLTER, 1994, ‘Representations of Algebraic Lattices’, Algebra Universalis 31, 612-613.
HERRMANN, B., W. RAUTENBERG, 1992, ‘Finite Replacement and Finite Axiomatizability in Logic’, Zeitschrift für math. Logik und Grundlagen der Math. 38, 327-344.
KEISLER, H. J., 1971, Model Theory for Infinitary Logic, North-Holland, Amsterdam.
Prucnal, T., A. WroŃski, 1974, ‘An Algebraic Characterization of the Notion of Structural Completeness’, Bulletin of the Section of Logic of the Polish Academy of Sciences 3, 30-33.
RAUTENBERG, W., 1981, ‘2-Element Matrices’, Studia Logica 40, 315-353.
RAUTENBERG, W., 1993, ‘On Reduced Matrices’, Studia Logica 52, 63-72.
WÓjcicki, R., 1988, Theory of Logical Calculi, Kluwer, Dordrecht.
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Herrmann, B. Characterizing Equivalential and Algebraizable Logics by the Leibniz Operator. Studia Logica 58, 305–323 (1997). https://doi.org/10.1023/A:1004979825733
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DOI: https://doi.org/10.1023/A:1004979825733