Studia Logica 50 (3-4):391 - 419 (1991)

Abstract
In this paper we study the relations between the fragment L of classical logic having just conjunction and disjunction and the variety D of distributive lattices, within the context of Algebraic Logic. We prove that these relations cannot be fully expressed either with the tools of Blok and Pigozzi's theory of algebraizable logics or with the use of reduced matrices for L. However, these relations can be naturally formulated when we introduce a new notion of model of a sequent calculus. When applied to a certain natural calculus for L, the resulting models are equivalent to a class of abstract logics (in the sense of Brown and Suszko) which we call distributive. Among other results, we prove that D is exactly the class of the algebraic reducts of the reduced models of L, that there is an embedding of the theories of L into the theories of the equational consequence (in the sense of Blok and Pigozzi) relative to D, and that for any algebra A of type (2,2) there is an isomorphism between the D-congruences of A and the models of L over A. In the second part of this paper (which will be published separately) we will also apply some results to give proofs with a logical flavour for several new or well-known lattice-theoretical properties.
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DOI 10.1007/BF00370680
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References found in this work BETA

Protoalgebraic Logics.W. J. Blok & Don Pigozzi - 1986 - Studia Logica 45 (4):337 - 369.
Lectures on Propositional Calculi.Ryszard Wójcicki - 1984 - Pub. House of the Polish Academy of Sciences.
2-Element Matrices.Wolfgang Rautenberg - 1981 - Studia Logica 40 (4):315 - 353.
Distributive Lattices.Raymond Balbes & Philip Dwinger - 1977 - Journal of Symbolic Logic 42 (4):587-588.

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Citations of this work BETA

Correspondences Between Gentzen and Hilbert Systems.J. G. Raftery - 2006 - Journal of Symbolic Logic 71 (3):903 - 957.
Equivalential and Algebraizable Logics.Burghard Herrmann - 1996 - Studia Logica 57 (2-3):419 - 436.
Algebraic Study of Two Deductive Systems of Relevance Logic.Josep Maria Font & Gonzalo Rodríguez - 1994 - Notre Dame Journal of Formal Logic 35 (3):369-397.

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