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Strong Three-Valued Paraconsistent Logics

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New Directions in Paraconsistent Logic

Part of the book series: Springer Proceedings in Mathematics & Statistics ((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.

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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.

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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

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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
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  2. Anna Franceschetto
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Correspondence to Jean-Yves Beziau .

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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

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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

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