Advertisement

Studia Logica

, Volume 50, Issue 3–4, pp 391–419 | Cite as

Algebraic logic for classical conjunction and disjunction

  • Josep M. Font
  • Ventura Verdú
Article

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R. Balbes and P. Dwinger, Distributive lattices, University of Missouri Press, Columbia (Missouri) 1974.Google Scholar
  2. [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. [3]
    W.J. Blok and D. Pigozzi, Protoalgebraic logics, Studia Logica 45 (1986), pp. 337–369.Google Scholar
  4. [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. [5]
    W.J. blok and D. pigozzi, Algebraizable logics, Memoirs of the American Mathematical Society, 396 (1989).Google Scholar
  6. [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. [7]
    W.J. blok and D. pigozzi, The Deduction Theorem in Algebraic Logic. Preprint, 1989.Google Scholar
  8. [8]
    S.L. Bloom, A note on Ψ-consequences, Reports on Mathematical Logic 8 (1977) pp. 3–9.Google Scholar
  9. [9]
    D.J. Brown and R. Suszko, Abstract Logics, Dissertationes Mathematicae 102 (1973), pp. 9–42.Google Scholar
  10. [10]
    P.M. Cohn, Universal Algebra. Harper and Row, New York 1965.Google Scholar
  11. [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. [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. [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. [14]
    H. B. Enderton, A Mathematical Introduction to Logic. Academic Press, New York, 1972.Google Scholar
  15. [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. [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. [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. [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. [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. [20]
    J.M. Font and V. Verdú, Algebraic study of Belnap's four-valued logic. Manuscript.Google Scholar
  21. [21]
    J.-Y. Girard, Linear logic, Theoretical Computer Science 50 (1987), pp. 1–102.Google Scholar
  22. [22]
    G. Grätzer, Universal Algebra, 2nd Edition, Springer-Verlag, Berlin 1979.Google Scholar
  23. [23]
    B. Jónsson. Topics in Universal Algebra (Lectures Notes in Mathematics, vol. 250) Springer-Verlag, Berlin 1970.Google Scholar
  24. [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. [25]
    J. Łoś and R. Suszko. Remarks on sentential logics, Indagationes Mathematicae 20 (1958), pp. 177–183.Google Scholar
  26. [26]
    W. Pogorzelski and P. Wojtylak, Elements of the Theory of Completeness in Prepositional Logic, The Silesian University, Katowice 1982.Google Scholar
  27. [27]
    W. Rautenberg. 2-element matrices, Studia Logica 40 (1981), pp. 315–353.Google Scholar
  28. [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. [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. [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. [31]
    A. Torrens and V. Verdú. Abstract Łukasiewicz Logics, Manuscript.Google Scholar
  32. [32]
    V. Verdú, Distributive and Boolean logics, (in Catalan) Stochastica 3 (1979) pp. 97–108.Google Scholar
  33. [33]
    V. Verdú, Algebraic logic for the ^, ∨, ⌝ -fragment of the intuitionistic prepositional calculus. Manuscript.Google Scholar
  34. [34]
    R. Wójcicki. Lectures on Propositional Calculi, Ossolineum, Wroclaw 1984.Google Scholar
  35. [35]
    R. Wójcicki. Theory of Logical Calculi. Basic Theory of Consequence Operations (Synthese Library, vol. 199) Reidel, Dordrecht 1988.Google Scholar

Copyright information

© Polish Academy of Sciences 1991

Authors and Affiliations

  • Josep M. Font
    • 1
  • Ventura Verdú
    • 1
  1. 1.Department of Logic, History and Philosophy of Science Faculty of MathematicsUniversity of BarcelonaBarcelonaSpain

Personalised recommendations