Archive for Mathematical Logic

, Volume 57, Issue 3–4, pp 391–420 | Cite as

Implicational (semilinear) logics III: completeness properties

Article

Abstract

This paper presents an abstract study of completeness properties of non-classical logics with respect to matricial semantics. Given a class of reduced matrix models we define three completeness properties of increasing strength and characterize them in several useful ways. Some of these characterizations hold in absolute generality and others are for logics with generalized implication or disjunction connectives, as considered in the previous papers. Finally, we consider completeness with respect to matrices with a linear dense order and characterize it in terms of an extension property and a syntactical metarule. This is the final part of the investigation started and developed in the papers (Cintula and Noguera in Arch Math Logic 49(4):417–446, 2010; Arch Math Logic 53(3):353–372, 2016).

Keywords

Abstract algebraic logic Protoalgebraic logics Implicational logics Disjunctional logics Semilinear logics Non-classical logics Completeness theorems Rational completeness 

Mathematics Subject Classification

03B22 03B47 03B52 03G99 

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References

  1. 1.
    Bergman, C.: Structural completeness in algebra and logic. In: Andréka, H., Monk, J., Németi, I. (eds.) Algebraic Logic (Proceedings of Conference, Budapest, 8–14 August 1988), Colloquia Mathematica Societatis János Bolyai, vol. 54, pp. 59–73. North-Holland, Amsterdam (1991)Google Scholar
  2. 2.
    Botur, M.: A non-associative generalization of Hájek’s BL-algebras. Fuzzy Sets Syst. 178(1), 24–37 (2011)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Ciabattoni, A., Metcalfe, G.: Density elimination. Theor. Comput. Sci. 403(1–2), 328–346 (2008)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cintula, P., Esteva, F., Gispert, J., Godo, L., Montagna, F., Noguera, C.: Distinguished algebraic semantics for t-norm based fuzzy logics: methods and algebraic equivalencies. Ann. Pure Appl. Logic 160(1), 53–81 (2009)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cintula, P., Fermüller, C.G., Hájek, P., Noguera, C. (eds.): Handbook of Mathematical Fuzzy Logic (in Three Volumes), Studies in Logic, Mathematical Logic and Foundations, vols. 37, 38, and 58. College Publications (2011, 2015)Google Scholar
  6. 6.
    Cintula, P., Horčík, R., Noguera, C.: Non-associative substructural logics and their semilinear extensions: axiomatization and completeness properties. Rev. Symb. Logic 6(3), 394–423 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cintula, P., Horčík, R., Noguera, C.: The quest for the basic fuzzy logic. In: Montagna, F. (ed.) Petr Hájek on Mathematical Fuzzy Logic, Outstanding Contributions to Logic, vol. 6, pp. 245–290. Springer, New York (2014)Google Scholar
  8. 8.
    Cintula, P., Noguera, C.: Implicational (semilinear) logics I: a new hierarchy. Arch. Math. Logic 49(4), 417–446 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cintula, P., Noguera, C.: A general framework for mathematical fuzzy logic. In: Cintula, P., Hájek, P., Noguera, C. (eds.) Handbook of Mathematical Fuzzy Logic—Volume 1, Studies in Logic, Mathematical Logic and Foundations, vol. 37, pp. 103–207. College Publications, London (2011)Google Scholar
  10. 10.
    Cintula, P., Noguera, C.: The proof by cases property and its variants in structural consequence relations. Studia Logica 101(4), 713–747 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Cintula, P., Noguera, C.: Implicational (semilinear) logics II: disjunction and completeness properties. Arch. Math. Logic 53(3), 353–372 (2016)CrossRefMATHGoogle Scholar
  12. 12.
    Czelakowski, J.: Protoalgebraic Logics, Trends in Logic, vol. 10. Kluwer, Dordrecht (2001)CrossRefMATHGoogle Scholar
  13. 13.
    Di Nola, A., Leuştean, I.: Łukasiewicz logic and MV-algebras. In: Cintula, P., Hájek, P., Noguera, C. (eds.) Handbook of Mathematical Fuzzy Logic—Volume 2, Studies in Logic, Mathematical Logic and Foundations, vol. 38, pp. 469–583. College Publications, London (2011)Google Scholar
  14. 14.
    Esteva, F., Gispert, J., Godo, L., Montagna, F.: On the standard and rational completeness of some axiomatic extensions of the monoidal t-norm logic. Studia Logica 71(2), 199–226 (2002)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Esteva, F., Gispert, J., Godo, L., Noguera, C.: Adding truth-constants to logics of continuous t-norms: axiomatization and completeness results. Fuzzy Sets Syst. 158(6), 597–618 (2007)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Font, J.M.: Abstract Algebraic Logic. An Introductory Textbook, Studies in Logic, vol. 60. College Publications, London (2016)MATHGoogle Scholar
  17. 17.
    Font, J.M., Jansana, R.: A General Algebraic Semantics for Sentential Logics, Lecture Notes in Logic, vol. 7, 2nd edn. Association for Symbolic Logic, Ithaca. http://projecteuclid.org/euclid.lnl/1235416965 (2009)
  18. 18.
    Font, J.M., Jansana, R., Pigozzi, D.L.: A survey of abstract algebraic logic. Studia Logica 74(1–2), 13–97 (2003)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Horčík, R.: Algebraic semantics: semilinear FL-algebras. In: Cintula, P., Hájek, P., Noguera, C. (eds.) Handbook of Mathematical Fuzzy Logic—Volume 1, Studies in Logic, Mathematical Logic and Foundations, vol. 37, pp. 283–353. College Publications, London (2011)Google Scholar
  20. 20.
    Lávička, T., Noguera, C.: A new hierarchy of infinitary logics in abstract algebraic logic. Studia Logica 105(3), 521–551 (2017)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Łukasiewicz, J., Tarski, A.: Untersuchungen über den Aussagenkalkül. Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, cl. III 23(iii), 30–50 (1930)MATHGoogle Scholar
  22. 22.
    Metcalfe, G., Montagna, F.: Substructural fuzzy logics. J. Symb. Logic 72(3), 834–864 (2007)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Metcalfe, G., Olivetti, N., Gabbay, D.M.: Proof Theory for Fuzzy Logics, Applied Logic Series, vol. 36. Springer, New York (2008)MATHGoogle Scholar
  24. 24.
    Takeuti, G., Titani, S.: Intuitionistic fuzzy logic and intuitionistic fuzzy set theory. J. Symb. Logic 49(3), 851–866 (1984)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Institute of Computer ScienceCzech Academy of SciencesPragueCzechia
  2. 2.Institute of Information Theory and AutomationCzech Academy of SciencesPragueCzechia

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