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New Directions in Paraconsistent Logic pp 91–129Cite as

Three-Valued Paraconsistent Propositional Logics

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

Abstract

Three-valued matrices provide the simplest semantic framework for introducing paraconsistent logics. This paper is a comprehensive study of the main properties of propositional paraconsistent three-valued logics in general, and of the most important such logics in particular. For each logic in the latter group, we also provide a corresponding cut-free Gentzen-type system.

Keywords

  • Paraconsistency
  • 3-valued matrices
  • Proof systems

Mathematics Subject Classification (2000)

  • Primary 03B53
  • 03B50
  • Secondary 03C90
  • 03F05

This work is supported by The Israel Science Foundation under grant agreement No. 817-15.

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Notes

  1. 1.

    When truth functionality is not required, further approaches based on nondeterministic semantics [10] are available. They give rise to another brand of useful three-valued logics, which includes many of the LFIs considered in [16]. We refer the reader to [12, 13] for further information on these logics and references to related papers.

  2. 2.

    The condition of nontriviality is not always demanded in the literature, but we find it very convenient (and natural) to include it here.

  3. 3.

    Note that \(\mathbf{L}^*\) is a propositional logic unless \(C_\mathbf{L}(S)\) contains all the pairs of finite theories in \({\mathcal W}({\mathcal L})\) and formulas in \({\mathcal W}({\mathcal L})\). Moreover, \(\mathbf{L}^*\) is in that case the minimal extension of \(\mathbf{L}\) such that \(\Gamma \vdash ^* {\varphi }\) whenever \({\Gamma /{\varphi }} \in S\).

  4. 4.

    This is a variant of a notion from [16].

  5. 5.

    In [16] the language is extended with a consistency operator \(\circ \), defined by \(\tilde{\circ }t = t\), \(\tilde{\circ }f = t\), and \(\tilde{\circ }\top = f\).

  6. 6.

    Note that in our notations \(\mathbf {P_1}\) is also denoted \(\mathbf{L}_\mathsf {P_1}\).

  7. 7.

    Meyer has shown (see [1]) that Sobociński’s system induces the \(\{\lnot ,\rightarrow ,\otimes \}\)-fragment of the semirelevant logic \(\mathbf{RM}\).

  8. 8.

    We refer to [2, 15, 23, 25] for further motivation and discussions on algebraic structures that combine order relations about truth and knowledge.

  9. 9.

    In [11] a general algorithm has been given for deriving sound and complete, cut-free Gentzen-type systems for finite-valued logics which have sufficiently expressive languages. That algorithm in fact works for all three-valued paraconsistent logics, but we shall not describe it here.

  10. 10.

    Although the notation \(\vdash _\mathsf{G}\) is overloaded in this definition, this should not cause any confusion in what follows.

  11. 11.

    Note that by Proposition 4.66, the four \(\le _k\)-monotonic expansions of \(\mathbf{LP}\) (including \(\mathbf{LP}\) itself) have no implication, and so they cannot have a corresponding Hilbert-type system of the above type. In contrast, by Proposition 4.57 \(\mathbf {SRM_{\mathop {\rightarrow }\limits ^{\sim }}}\) can be defined using such a system, but the resulting system does not look very natural. A natural Hilbert-type system for \(\mathbf {SRM_{\mathop {\rightarrow }\limits ^{\sim }}}\) in its primitive language (but with two inference rules) can be found in [9].

  12. 12.

    As usual, in the formulation of the axioms of the systems the association of nested implications is taken to the right.

  13. 13.

    In such a case we need also the structural rules of Permutation, Contraction, and Expansion that assure that the underlying consequence relation remains the same.

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Authors and Affiliations

  1. School of Computer Science, The Academic College of Tel-Aviv, Tel Aviv, Israel

    Ofer Arieli

  2. School of Computer Science, Tel-Aviv University, Tel Aviv, Israel

    Arnon Avron

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  1. Ofer Arieli
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Correspondence to Ofer Arieli .

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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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Arieli, O., Avron, A. (2015). Three-Valued Paraconsistent Propositional 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_4

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