Skip to main content
SpringerLink
Log in
Book cover

New Directions in Paraconsistent Logic pp 131–145Cite as

Strong Three-Valued Paraconsistent Logics

Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 152)

Abstract

After describing the two formulations of the principle of non contradiction in modern logic \(T \vdash \lnot (p \wedge \lnot p)\) (NC) and \(T, p, \lnot p \vdash q\) (EC) and explaining that three-valued matrices can be used to easily prove their independence, we investigate the possibilities to construct strong paraconsistent negations, i.e., for which neither (NC) nor (EC) holds, using three-valued logical matrices.

Keywords

  • Principle of non contradiction
  • Negation
  • Paraconsistent logic
  • Many-valued logic

Mathematics Subject Classification (2000)

  • Primary 03B53
  • Secondary 03B50

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    For a general discussion about how a paraconisstent negation can be defined, see [7, 8].

  2. 2.

    This means that reflexivity, monotonicity, transitivity hold as well as substitution, see [17].

  3. 3.

    We are working in abstract logic, not in proof theory, so we are not considering that these are rules.

  4. 4.

    Same remark as in the previous footnote.

  5. 5.

    Since we are working with truth-tables which are conservative extensions of the classical ones, we omit the classical parts in all tables built to check De Morgan laws hereafter.

References

  1. Asenjo, F.G.: A calculus of antinomies. Notre Dame J. Form. Log. 7, 103–105

    Google Scholar 

  2. Arieli, O., Avron, A.: Three-valued paraconsistent propositional logics. In: Beziau, J.-Y., Chakraborty, M., Dutta, S. (eds.) New Directions in Paraconsistent Logic. Springer India (2015), pp. 91–129

    Google Scholar 

  3. Beziau, J.-Y.: Logiques construites suivant les méthodes de da Costa. Log. Anal. 131–132, 259–272 (1990)

    MathSciNet  Google Scholar 

  4. Beziau, J.-Y.: Nouveaux résultats et nouveau regard sur la logique paraconsistante C1. Log. Anal. 141–142, 45–48 (1993)

    MathSciNet  Google Scholar 

  5. Beziau, J.-Y.: Théorie législative de la négation pure. Log. Anal. 147–148, 209–225 (1994)

    MathSciNet  Google Scholar 

  6. Beziau, J.-Y.: Idempotent full paraconsistent negations are not algebraizable. Notre Dame J. Form. Log. 39, 135–139 (1998)

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Beziau, J.-Y.: What is paraconsistent logic?. In: Batens, D. et al. (eds.) Frontiers of Paraconsistent Logic, pp. 95–111. Research Studies Press, Baldock (2000)

    Google Scholar 

  8. Beziau, J.-Y.: Are paraconsistent negations negations? In: Carnielli, W. et al. (eds.) Paraconsistency: the Logical Way to the Inconsistent, pp. 465–486. Marcel Dekker, New-York (2002)

    Google Scholar 

  9. Beziau, J.-Y.: History of the concept of truth-value. In: Gabbay, D.M., Pelletier, J., Woods, J. (eds.) Handbook of the History of Logic, vol. 11. Elsevier, Amsteradm (2012)

    Google Scholar 

  10. Beziau, J.-Y.: Trivial dialetheism and the logic of paradox. Log. Logic. Philos. 25 (2016)

    Google Scholar 

  11. da Costa, N.C.A.: Calculs propositionnels pour les systémes formels inconsistants. Cr. R. Acad. Sc. Paris 257, 3790–3793 (1963)

    MATH  Google Scholar 

  12. da Costa, N.C.A., Guillaume, M.: Négations composées et Loi de Peirce dans les systémes Cn. Portugalia Mathematica 24, 201–210 (1965)

    MATH  Google Scholar 

  13. D’Ottaviano, I.M.L., da Costa, N.C.A.: Sur un probléme de Jaśkowski. Cr. R. Acad. Sc. Paris 270, 1349–1353 (1970)

    MATH  Google Scholar 

  14. Epstein, R.L.: The Semantic Foundations of Logic. Volume 1: Propositional Logics. Kluwer, Dordrecht (1990)

    Google Scholar 

  15. Humberstone, L.: Beziau’s translation paradox. Theoria 71, 138–181 (2005)

    CrossRef  MathSciNet  Google Scholar 

  16. Kleene, S.: On a notation for ordinal numbers. J. Symbolic Log. 3, 150–155 (1938)

    CrossRef  Google Scholar 

  17. Łoś, J., Suszko, R.: Remarks on sentential logics. Indigationes Math. 10, 177–183 (1958)

    Google Scholar 

  18. Łukasiewicz, J.: O logice trójwartościowej. Ruch Filozoficny 5, 170–171 (1920)

    Google Scholar 

  19. Marcos, J.: 8K solutions and semi-solutions to a problem of da Costa

    Google Scholar 

  20. Priest, G.: The logic of paradox. J. Philos. Log. 8, 219–241 (1979)

    CrossRef  MathSciNet  MATH  Google Scholar 

  21. Sette, A.M.: On the propositional calculus P1. Notas e comunicacões de matemática, 17 (1971)

    Google Scholar 

  22. Urbas, I.: Paraconsistency and the C-ysytems of da Costa. Notre Dame J. Form. Log. 30, 583–597 (1989)

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work starts during a seminar at the Federal University of Rio de Janeiro (August 2013–December 2013) conducted by J.Y. Beziau during the visit of Anne Franceschetto who was visiting Brazil to know more about paraconsistent logic. Other students, in particular Rodrigo de Almeida and Edson Vinicius Bezerra, had an active participation to this seminar

Author information

Authors and Affiliations

  1. UFRJ - Federal University of Rio de Janeiro, CNPq - Brazilian Research Council, Rio de Janeiro, Brazil

    Jean-Yves Beziau

  2. Department of Mathematics, University of Padua, Padua, Italy

    Anna Franceschetto

Authors
  1. Jean-Yves Beziau
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anna Franceschetto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean-Yves Beziau .

Editor information

Editors and Affiliations

  1. Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

    Jean-Yves Beziau

  2. Indian Statistical Institute Kolkat, Kolkata, India

    Mihir Chakraborty

  3. C.I.T Campus, Institute for Mathematical Sciences, Chennai, Tamil Nadu, India

    Soma Dutta

Rights and permissions

Reprints and Permissions

Copyright information

© 2015 Springer India

About this paper

Cite this paper

Beziau, JY., Franceschetto, A. (2015). Strong Three-Valued Paraconsistent Logics. In: Beziau, JY., Chakraborty, M., Dutta, S. (eds) New Directions in Paraconsistent Logic. Springer Proceedings in Mathematics & Statistics, vol 152. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2719-9_5

Download citation

search

Navigation