Skip to main content
SpringerLink
Log in

Implicational (semilinear) logics III: completeness properties

  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

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).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

  2. Botur, M.: A non-associative generalization of Hájek’s BL-algebras. Fuzzy Sets Syst. 178(1), 24–37 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ciabattoni, A., Metcalfe, G.: Density elimination. Theor. Comput. Sci. 403(1–2), 328–346 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Cintula, P., Noguera, C.: Implicational (semilinear) logics I: a new hierarchy. Arch. Math. Logic 49(4), 417–446 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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. Cintula, P., Noguera, C.: The proof by cases property and its variants in structural consequence relations. Studia Logica 101(4), 713–747 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cintula, P., Noguera, C.: Implicational (semilinear) logics II: disjunction and completeness properties. Arch. Math. Logic 53(3), 353–372 (2016)

    Article  MATH  Google Scholar 

  12. Czelakowski, J.: Protoalgebraic Logics, Trends in Logic, vol. 10. Kluwer, Dordrecht (2001)

    Book  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  16. Font, J.M.: Abstract Algebraic Logic. An Introductory Textbook, Studies in Logic, vol. 60. College Publications, London (2016)

    MATH  Google Scholar 

  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. Font, J.M., Jansana, R., Pigozzi, D.L.: A survey of abstract algebraic logic. Studia Logica 74(1–2), 13–97 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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. Lávička, T., Noguera, C.: A new hierarchy of infinitary logics in abstract algebraic logic. Studia Logica 105(3), 521–551 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  22. Metcalfe, G., Montagna, F.: Substructural fuzzy logics. J. Symb. Logic 72(3), 834–864 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Metcalfe, G., Olivetti, N., Gabbay, D.M.: Proof Theory for Fuzzy Logics, Applied Logic Series, vol. 36. Springer, New York (2008)

    MATH  Google Scholar 

  24. Takeuti, G., Titani, S.: Intuitionistic fuzzy logic and intuitionistic fuzzy set theory. J. Symb. Logic 49(3), 851–866 (1984)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Computer Science, Czech Academy of Sciences, Pod Vodárenskou věží 2, 182 07, Prague, Czechia

    Petr Cintula

  2. Institute of Information Theory and Automation, Czech Academy of Sciences, Pod Vodárenskou věží 4, 182 08, Prague, Czechia

    Carles Noguera

Authors
  1. Petr Cintula
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Carles Noguera
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Carles Noguera.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cintula, P., Noguera, C. Implicational (semilinear) logics III: completeness properties. Arch. Math. Logic 57, 391–420 (2018). https://doi.org/10.1007/s00153-017-0577-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-017-0577-0

Keywords

Mathematics Subject Classification

Search

Navigation