# Algebraic logic for classical conjunction and disjunction

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## 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.

## Keywords

Mathematical Logic Distributive Lattice Computational Linguistic Classical Logic Sequent Calculus## Preview

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## References

- [1]R. Balbes and P. Dwinger,
**Distributive lattices**, University of Missouri Press, Columbia (Missouri) 1974.Google Scholar - [2]N.D. Belnap (Jr.),
*A useful four-valued*, in: J.M. Dunn and G. Epstein (eds.),**Modern Uses of Multiple-Valued Logic**(D. Reidel, 1977) 8–37.Google Scholar - [3]W.J. Blok and D. Pigozzi,
*Protoalgebraic logics*,**Studia Logica**45 (1986), pp. 337–369.Google Scholar - [4]W.J. Blok and D. Pigozzi,
*Alfred Tarski's work on general metamathematics*,**The Journal of Symbolic Logic**53 (1988), pp. 36–50.Google Scholar - [5]W.J. blok and D. pigozzi,
*Algebraizable logics*,**Memoirs of the American Mathematical Society**, 396 (1989).Google Scholar - [6]W.J. Blok and D. Pigozzi,
*Local Deduction Theorems in Algebraic Logic*, in: H. Andréka, J.D. Monk and I. Németi, (eds.),**Algebraic Logic. Proceedings of Budapest 1988 Conference**(Colloquia Mathematica Societatis János Bolyai, North-Holland, Amsterdam, 1990) to appear.Google Scholar - [7]
- [8]
- [9]D.J. Brown and R. Suszko,
*Abstract Logics*,**Dissertationes Mathematicae**102 (1973), pp. 9–42.Google Scholar - [10]
- [11]J. Czelakowski and G. Malinowski,
*Key notions of Tarski's methodology of deductive systems*,**Studia Logica**44 (1985), pp. 321–351.Google Scholar - [12]K. Dyrda and T. Prucnal,
*On finitely based consequence determined by a distributive lattice*.**Bulletin of the Section of Logic, Polish Academy of Sciences**9 (1980), pp. 60–66.Google Scholar - [13]W. Dzik,
*On the content of the lattice of logics, part II*,**Reports on Mathematical Logic**14 (1982), pp. 29–47.Google Scholar - [14]
- [15]J.M. font, F. guzmán and V. verdú,
*Characterization of the reduced matrices for the ∧, ∨fragment of classical logic*, Manuscript, to appear.Google Scholar - [16]J.M. font and M. Rius,
*A four-valued modal logic arising from Monteiro's last algebras*.**Proceedings of the 20th International Symposium on Multiple-Valued Logic**(Charlotte, 1990), pp. 85–92.Google Scholar - [17]J.M. Font and V. Verdú,
*Abstract characterization of a four-valued logic*, in:**Proceedings of the 18th International Symposium on Multiple-Valued Logic**(Palma de Mallorca, 1988), pp. 389–396.Google Scholar - [18]J.M. Font and V. Verdú,
*A first approach to abstract modal logics*,**The Journal of Symbolic Logic**54 (1989) pp. 1042–1062.Google Scholar - [19]J.M. Font and V. Verdú,
*On the logic of distributive lattices*,**Bulletin of the Section of Logic, Polish Academy of Sciences**18 (1989), pp. 79–86.Google Scholar - [20]
- [21]
- [22]
- [23]B. Jónsson.
**Topics in Universal Algebra**(Lectures Notes in Mathematics, vol. 250) Springer-Verlag, Berlin 1970.Google Scholar - [24]J. Lambek.
*Logics without structural rules: Another look at cut-elimnination*, in:**Proceedings of the Kleene Conference**(**Chaika 1990**), to appear.Google Scholar - [25]J. Łoś and R. Suszko.
*Remarks on sentential logics*,**Indagationes Mathematicae**20 (1958), pp. 177–183.Google Scholar - [26]W. Pogorzelski and P. Wojtylak,
**Elements of the Theory of Completeness in Prepositional Logic**, The Silesian University, Katowice 1982.Google Scholar - [27]
- [28]G. Sundholm.
*Systems of Deduction*, in: D. Gabbay and F. Guenthner (eds.),**Handbook of Philosophical Logic, vol. I: Elements of Classical Logic**(Reidel, Dordrecht 1983), pp. 133–188.Google Scholar - [29]G. Sundholm.
*Proof Theory and meaning*, in: D. Gabbay and F. Guenthner (eds.),**Handbook of Philosophical Logic, vol. III: Alternatives to Classical Logic**(Reidel, Dordrecht 1986), pp. 471–506.Google Scholar - [30]A. Tarski.
*Über einige fundamentale Begriffe der Metamathematik*,**Comptes Rendus Société Sciences et Lettres Varsovie**, Cl. III, 23 (1930), pp. 22–29.Google Scholar - [31]
- [32]V. Verdú,
*Distributive and Boolean logics*, (in Catalan)**Stochastica**3 (1979) pp. 97–108.Google Scholar - [33]V. Verdú,
*Algebraic logic for the ^, ∨, ⌝ -fragment of the intuitionistic prepositional calculus*. Manuscript.Google Scholar - [34]
- [35]R. Wójcicki.
**Theory of Logical Calculi. Basic Theory of Consequence Operations**(Synthese Library, vol. 199) Reidel, Dordrecht 1988.Google Scholar