Algebraic logic for classical conjunction and disjunction

Studia Logica 50 (3-4):391 - 419 (1991)
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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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References found in this work

Linear Logic.Jean-Yves Girard - 1987 - Theoretical Computer Science 50:1–102.
Protoalgebraic logics.W. J. Blok & Don Pigozzi - 1986 - Studia Logica 45 (4):337 - 369.
Lectures on propositional calculi.Ryszard Wójcicki - 1984 - Ossolineum [Poland]: Pub. House of the Polish Academy of Sciences.
2-element matrices.Wolfgang Rautenberg - 1981 - Studia Logica 40 (4):315 - 353.

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