Skip to main content
SpringerLink
Log in

Paraconsistent Double Negations as Classical and Intuitionistic Negations

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

A classical paraconsistent logic (CP), which is regarded as a modified extension of first-degree entailment logic, is introduced as a Gentzen-type sequent calculus. This logic can simulate the classical negation in classical logic by paraconsistent double negation in CP. Theorems for syntactically and semantically embedding CP into a Gentzen-type sequent calculus LK for classical logic and vice versa are proved. The cut-elimination and completeness theorems for CP are also shown using these embedding theorems. Similar results are also obtained for an intuitionistic paraconsistent logic (IP), and several versions of Glivenko and Gödel-Gentzen translation theorems are proved for CP and IP.

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. Almukdad, A., and D. Nelson, Constructible falsity and inexact predicates, Journal of Symbolic Logic 49(1):231–233, 1984.

    Article  Google Scholar 

  2. Belnap, N. D., How a computer should think, in: G. Ryle (ed.), Contemporary Aspects of Philosophy, Oriel Press, Stocksfield, 1977, pp. 30–56.

    Google Scholar 

  3. Belnap, N. D., A useful four-valued logic, in: J. M. Dunn and G. Epstein (eds.), Modern Uses of Multiple-Valued Logic, Reidel, Dordrecht, 1977, pp. 5–37.

    Chapter  Google Scholar 

  4. Béziau, J.-Y., Paraconsistent logic from a modal viewpoint, Journal of Applied Logic 3:7–14, 2005.

    Article  Google Scholar 

  5. Béziau, J.-Y., A new four-valued approach to modal logic, Logique et Analyse 54(213):109–121, 2011.

    Google Scholar 

  6. De, M., and H. Omori, Classical negation and expansions of Belnap-Dunn logic, Studia Logica 103(4):825–851, 2015.

    Article  Google Scholar 

  7. Došen, K., Negative modal operators in intuitionistic logic, Publications de L’institut Mathématique, Nouvelle série 35(49):3–14, 1984.

    Google Scholar 

  8. Dunn, J. M., Intuitive semantics for first-degree entailment and ‘coupled trees’, Philosophical Studies 29(3):149–168, 1976.

    Article  Google Scholar 

  9. Dunn, J. M., Star and perp: Two treatments of negation, in J. Tomberlin, (ed.), Philosophical Perspectives (Philosophy of Languages and Logic), vol. 7, 1993, pp. 331–357.

  10. Gurevich, Y., Intuitionistic logic with strong negation, Studia Logica 36:49–59, 1977.

    Article  Google Scholar 

  11. Horn L. R., and H. Wansing, Negation, Stanford Encyclopedia of Philosophy. First published in 7 January 2015. http://plato.stanford.edu/entries/negation/.

  12. Kamide, N., A hierarchy of weak double negations, Studia Logica 101(6):1277–1297, 2013.

    Article  Google Scholar 

  13. Kamide, N., Trilattice logic: an embedding-based approach, Journal of Logic and Computation 25(3):581–611, 2015.

    Article  Google Scholar 

  14. Kamide, N., Paraconsistent double negation that can simulate classical negation, in Proceedings of the 46th IEEE International Symposium on Multiple-Valued Logic (ISMVL 2016), 2016, pp. 131–136.

  15. Kamide, N., Paraconsistent double negation as a modal operator, Mathematical Logic Quarterly 62(6):552–562, 2016.

    Article  Google Scholar 

  16. Kamide, N., and Y. Shramko, Embedding from multilattice logic into classical logic and vice versa, Journal of Logic and Computation, 2016. doi:10.1093/logcom/exw015.

  17. Nelson, D., Constructible falsity, Journal of Symbolic Logic 14:16–26, 1949.

    Article  Google Scholar 

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

    Article  Google Scholar 

  19. Rautenberg, W., Klassische und nicht-klassische Aussagenlogik, Vieweg, Braunschweig, 1979.

    Book  Google Scholar 

  20. Shramko, Y., Truth, falsehood, information and beyond: The American plan generalized, in: K. Bimbó (ed.), J. Michael Dunn on Information Based Logics, Outstanding Contributions to Logic, Springer, Dordrecht, 2016, pp. 191–212.

    Chapter  Google Scholar 

  21. Vorob’ev, N. N., A constructive propositional calculus with strong negation (in Russian), Doklady Akademii Nauk SSR 85:465–468, 1952.

    Google Scholar 

  22. Zaitsev, D., Generalized relevant logic and models of reasoning, Moscow State Lomonosov University doctoral dissertation, 2012.

Download references

Author information

Authors and Affiliations

  1. Department of Information and Electronic Engineering Faculty of Science and Engineering, Teikyo University, Toyosatodai 1-1, Utsunomiya, Tochigi, 320-8551, Japan

    Norihiro Kamide

Authors
  1. Norihiro Kamide
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Norihiro Kamide.

Additional information

The results of this paper include the result of the conference presentation [14].

Edited by Hitoshi Omori and Heinrich Wansing

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kamide, N. Paraconsistent Double Negations as Classical and Intuitionistic Negations. Stud Logica 105, 1167–1191 (2017). https://doi.org/10.1007/s11225-017-9731-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-017-9731-2

Keywords

Search

Navigation