Classical Negation and Expansions of Belnap–Dunn Logic


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.

This is a preview of subscription content, access via your institution.


  1. 1.

    Arieli O., Avron A., Zamansky A.: Ideal paraconsistent logics. Studia Logica 99, 31–60 (2011)

    Article  Google 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)

    Article  Google Scholar 

  3. 3.

    Avron A.: Natural 3-valued logics–characterization and proof theory. Journal of Symbolic Logic 56, 276–294 (1991)

    Article  Google 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)

    Article  Google 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)

    Article  Google Scholar 

  7. 7.

    Copeland B. J.: What is a semantics for classical negation?. Mind 95(380), 478–490 (1986)

    Article  Google 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)

    Article  Google Scholar 

  9. 9.

    De, M., Negation in context, Ph.D. thesis, University of St Andrews, Scotland, 2011.

  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)

    Article  Google Scholar 

  11. 11.

    Horn, L. R., and H. Wansing, Negation, The Stanford Encyclopedia of Philosophy. Forthcoming.

  12. 12.

    Jaśkowski S.: A propositional calculus for inconsistent deductive systems. Logic and Logical Philosophy 7, 35–56 (2000)

    Article  Google Scholar 

  13. 13.

    Kamide N., Wansing H.: Proof theory of Nelson’s paraconsistent logic: A uniform perspective. Theoretical Computer Science 415, 1–38 (2012)

    Article  Google Scholar 

  14. 14.

    Mendelson, E., Introduction to Mathematical Logic, 4 edn., Chapman and Hall/CRC, Boca Raton, 1997.

  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.

  16. 16.

    Meyer R. K., Routley R.: Classical relevant logics I. Studia Logica 32(1), 51–66 (1973)

    Article  Google Scholar 

  17. 17.

    Meyer R. K., Routley R.: Classical relevant logics II.. Studia Logica 33(2), 183–194 (1974)

    Article  Google Scholar 

  18. 18.

    Odintsov S. P.: The class of extensions of Nelson paraconsistent logic. Studia Logica 80, 291–320 (2005)

    Article  Google Scholar 

  19. 19.

    Odintsov, S. P., Constructive Negations and Paraconsistency, Springer-Verlag, Dordrecht, 2008.

  20. 20.

    Omori, H., Remarks on naive set theory based on LP, The Review of Symbolic Logic. Forthcoming.

  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.

  22. 22.

    Priest, G., Can contradictions be true?, Proceedings of the Aristotelian Society, Supplementary Volumes 67:34–54, 1993.

  23. 23.

    Priest, G., Doubt Truth to be a Liar, Oxford University Press, New York, 2006.

  24. 24.

    Priest, G., In Contradiction: A Study of the Transconsistent, 2nd edn., Oxford University Press, Oxford, 2006.

  25. 25.

    Sano K., Omori H.: An expansion of first-order Belnap–Dunn logic. Logic Journal of the IGPL 22(3), 458–481 (2014)

    Article  Google Scholar 

  26. 26.

    Scroggs S. J.: Extensions of the Lewis system S5.. The Journal of Symbolic Logic 16(2), 112–120 (1951)

    Article  Google 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)

    Article  Google Scholar 

  29. 29.

    Smiley, T., Can contradictions be true?, Proceedings of the Aristotelian Society, Supplementary Volumes 67:17–33, 1993.

  30. 30.

    Zaitsev, D., Generalized relevant logic and models of reasoning, Moscow State Lomonosov University doctoral (Doctor of Science) dissertation, 2012.

Download references

Author information



Corresponding author

Correspondence to Michael De.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

De, M., Omori, H. Classical Negation and Expansions of Belnap–Dunn Logic. Stud Logica 103, 825–851 (2015).

Download citation


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