Studia Logica

, Volume 103, Issue 4, pp 825–851 | Cite as

Classical Negation and Expansions of Belnap–Dunn Logic

Article

Abstract

We investigate the notion of classical negation from a non-classical perspective. In particular, one aim is to determine what classical negation amounts to in a paracomplete and paraconsistent four-valued setting. We first give a general semantic characterization of classical negation and then consider an axiomatic expansion BD+ of four-valued Belnap–Dunn logic by classical negation. We show the expansion complete and maximal. Finally, we compare BD+ to some related systems found in the literature, specifically a four-valued modal logic of Béziau and the logic of classical implication and a paraconsistent de Morgan negation of Zaitsev.

Keywords

First-degree entailment Belnap–Dunn logic Classical negation Many-valued logic Paraconsistency Paracompleteness Maximality 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arieli O., Avron A., Zamansky A.: Ideal paraconsistent logics. Studia Logica 99, 31–60 (2011)CrossRefGoogle Scholar
  2. 2.
    Arieli O., Avron A., Zamansky A.: Maximal and premaximal paraconsistency in the framework of three-valued semantics. Studia Logica 97, 31–60 (2011)CrossRefGoogle Scholar
  3. 3.
    Avron A.: Natural 3-valued logics–characterization and proof theory. Journal of Symbolic Logic 56, 276–294 (1991)CrossRefGoogle Scholar
  4. 4.
    Béziau J.-Y.: Classical negation can be expressed by one of its halves. Logic Journal of the IGPL 7(2), 145–151 (1999)CrossRefGoogle Scholar
  5. 5.
    Béziau J.-Y.: A new four-valued approach to modal logic. Logique et Analyse 54(213), 109–121 (2011)Google Scholar
  6. 6.
    Carnielli W., Marcos J., de Amo S.: Formal inconsistency and evolutionary databases. Logic and Logical Philosophy 8, 115–152 (2000)CrossRefGoogle Scholar
  7. 7.
    Copeland B. J.: What is a semantics for classical negation?. Mind 95(380), 478–490 (1986)CrossRefGoogle Scholar
  8. 8.
    Da Costa N. C. A.: On the theory of inconsistent formal systems. Notre Dame Journal of Formal Logic 15, 497–510 (1974)CrossRefGoogle Scholar
  9. 9.
    De, M., Negation in context, Ph.D. thesis, University of St Andrews, Scotland, 2011.Google Scholar
  10. 10.
    Hanazawa M.: A characterization of axiom schema playing the rôle of tertium non datur in intuitionistic logic. Proceedings of the Japan Academy 42, 1007–1010 (1966)CrossRefGoogle Scholar
  11. 11.
    Horn, L. R., and H. Wansing, Negation, The Stanford Encyclopedia of Philosophy. Forthcoming.Google Scholar
  12. 12.
    Jaśkowski S.: A propositional calculus for inconsistent deductive systems. Logic and Logical Philosophy 7, 35–56 (2000)CrossRefGoogle Scholar
  13. 13.
    Kamide N., Wansing H.: Proof theory of Nelson’s paraconsistent logic: A uniform perspective. Theoretical Computer Science 415, 1–38 (2012)CrossRefGoogle Scholar
  14. 14.
    Mendelson, E., Introduction to Mathematical Logic, 4 edn., Chapman and Hall/CRC, Boca Raton, 1997.Google Scholar
  15. 15.
    Meyer, R. K., Proving semantical completeness ‘relevantly’ for R, Australian National University Research School of Social Sciences Logic Group Research Paper, 23 1985.Google Scholar
  16. 16.
    Meyer R. K., Routley R.: Classical relevant logics I. Studia Logica 32(1), 51–66 (1973)CrossRefGoogle Scholar
  17. 17.
    Meyer R. K., Routley R.: Classical relevant logics II.. Studia Logica 33(2), 183–194 (1974)CrossRefGoogle Scholar
  18. 18.
    Odintsov S. P.: The class of extensions of Nelson paraconsistent logic. Studia Logica 80, 291–320 (2005)CrossRefGoogle Scholar
  19. 19.
    Odintsov, S. P., Constructive Negations and Paraconsistency, Springer-Verlag, Dordrecht, 2008.Google Scholar
  20. 20.
    Omori, H., Remarks on naive set theory based on LP, The Review of Symbolic Logic. Forthcoming.Google Scholar
  21. 21.
    Omori, H., and T. Waragai, Some observations on the systems LFI1 and LFI1*, in Proceedings of Twenty-Second International Workshop on Database and Expert Systems Applications (DEXA2011), 2011, pp. 320–324.Google Scholar
  22. 22.
    Priest, G., Can contradictions be true?, Proceedings of the Aristotelian Society, Supplementary Volumes 67:34–54, 1993.Google Scholar
  23. 23.
    Priest, G., Doubt Truth to be a Liar, Oxford University Press, New York, 2006.Google Scholar
  24. 24.
    Priest, G., In Contradiction: A Study of the Transconsistent, 2nd edn., Oxford University Press, Oxford, 2006.Google Scholar
  25. 25.
    Sano K., Omori H.: An expansion of first-order Belnap–Dunn logic. Logic Journal of the IGPL 22(3), 458–481 (2014)CrossRefGoogle Scholar
  26. 26.
    Scroggs S. J.: Extensions of the Lewis system S5.. The Journal of Symbolic Logic 16(2), 112–120 (1951)CrossRefGoogle Scholar
  27. 27.
    Sette A.: On the propositional calculus P1. Mathematica Japonicae 16, 173–180 (1973)Google Scholar
  28. 28.
    Slater B. H.: Paraconsistent logics?. Journal of Philosophical Logic 24(4), 451–454 (1995)CrossRefGoogle Scholar
  29. 29.
    Smiley, T., Can contradictions be true?, Proceedings of the Aristotelian Society, Supplementary Volumes 67:17–33, 1993.Google Scholar
  30. 30.
    Zaitsev, D., Generalized relevant logic and models of reasoning, Moscow State Lomonosov University doctoral (Doctor of Science) dissertation, 2012.Google Scholar

Copyright information

© Springer 2015

Authors and Affiliations

  1. 1.Department of PhilosophyUniversität KonstanzKonstanzGermany
  2. 2.The Graduate Center, City University of New YorkNew YorkUSA

Personalised recommendations