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Three-Valued Paraconsistent Propositional Logics

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Book cover New Directions in Paraconsistent Logic

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

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.

References

  1. Anderson, A., Belnap, N.: Entailment, vol. 1. Princeton University Press (1975)

    Google Scholar 

  2. Arieli, O., Avron, A.: The value of the four values. Artif. Intell. 102(1), 97–141 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arieli, O., Avron, A., Zamansky, A.: Ideal paraconsistent logics. Stud. Logica. 99(1–3), 31–60 (2011)

    MathSciNet  MATH  Google Scholar 

  4. Arieli, O., Avron, A., Zamansky, A.: Maximal and premaximal paraconsistency in the framework of three-valued semantics. Stud. Logica. 97(1), 31–60 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Asenjo, F.G.: A calculus of antinomies. Notre Dame J. Formal Logic 7, 103–106 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  6. Avron, A.: On an implication connective of RM. Notre Dame J. Formal Logic 27, 201–209 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  7. Avron, A.: Natural 3-valued logics: characterization and proof theory. J. Symbolic Logic 56(1), 276–294 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  8. Avron, A.: On the expressive power of three-valued and four-valued languages. J. Logic Comput. 9(6), 977–994 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Avron, A.: Combining classical logic, paraconsistency and relevance. J. Appl. Logic 3, 133–160 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Avron, A., Lev, I.: Non-deterministic multi-valued structures. J. Logic Comput. 15, 241–261 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Avron, A., Ben-Naim, J., Konikowska, B.: Cut-free ordinary sequent calculi for logics having generalized finite-valued semantics. Log. Univers. 1, 41–69 (2006)

    Article  MathSciNet  Google Scholar 

  12. Avron, A., Konikowska, B., Zamansky, A.: Cut-free sequent calculi for C-systems with generalized finite-valued semantics. J. Logic Comput. 23, 517–540 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Avron, A., Zamansky, A.: Non-deterministic semantics for logical systems—a survey. In: Gabbay, D., Guenther, F. (eds) Handbook of Philosophical Logic, vol. 16, pp. 227–304. Springer (2011)

    Google Scholar 

  14. Batens, D.: Paraconsistent extensional propositional logics. Logique et Analyse 90–91, 195–234 (1980)

    MathSciNet  Google Scholar 

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

    Google Scholar 

  16. Carnielli, W., Coniglio, M., Marcos, J.: Logics of formal inconsistency. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, 2nd edn., pp, vol. 14, pp. 1–93. Springer (2007)

    Google Scholar 

  17. Ciucci, D., Dubois, D.: A modal theorem-preserving translation of a class of three-valued logics of incomplete information. J. Appl. Non-Class. Logics 23(4), 321–352 (2013)

    Article  MathSciNet  Google Scholar 

  18. Ciucci, D., Dubois, D.: From possibility theory to paraconsistency (2015). A chapter in this book

    Google Scholar 

  19. da Costa, N.: On the theory of inconsistent formal systems. Notre Dame J. Formal Logic 15, 497–510 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  20. D’Ottaviano, I.: The completeness and compactness of a three-valued first-order logic. Revista Colombiana de Matematicas XIX(1–2), 31–42 (1985)

    Google Scholar 

  21. D’Ottaviano, I., da Costa, N.C.: Sur un problèm de Jakowśki. C. R. Acad Sc. Paris 270(Sèrie A), 1349–1353 (1970)

    Google Scholar 

  22. Epstein, R.L.: The Semantic Foundation of Logic, vol. I, Propositional Logics. Kluwer (1990)

    Google Scholar 

  23. Fitting, M.: Kleene’s three valued logics and their children. Fundamenta Informaticae 20(1–3), 113–131 (1994)

    MathSciNet  MATH  Google Scholar 

  24. Gentzen, G.: Investigations into Logical Deduction (1934). In: Szabo, M.E. (ed.) In German. An English translation appears in ‘The Collected Works of Gerhard Gentzen’. North-Holland (1969)

    Google Scholar 

  25. Ginsberg, M.: Multi-valued logics: a uniform approach to reasoning in AI. Comput. Intell. 4, 256–316 (1988)

    Google Scholar 

  26. Gottwald, S.: A treatise on many-valued logics. In: Studies in Logic and Computation, vol. 9. Research Studies Press, Baldock (2001)

    Google Scholar 

  27. Kleene, S.C.: Introduction to Metamathematics. Van Nostrand (1950)

    Google Scholar 

  28. Łukasiewicz, J.: On 3-valued logic. Ruch Filosoficzny 5, 169–171 (1920). English translation: McCall, S. (ed.) Polish Logic 1920–1939, pp. 15–18. Oxford University Press, Oxford (1967)

    Google Scholar 

  29. Łukasiewicz, J.: Selected Works. North Holland (1970). L. Borkowski (ed)

    Google Scholar 

  30. Malinowski, G.: Many-Valued Logics. Clarendon Press (1993)

    Google Scholar 

  31. Priest, G.: Logic of paradox. J. Philos. Logic 8, 219–241 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  32. Priest, G.: Reasoning about truth. Artif. Intell. 39, 231–244 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  33. Priest, G.: Minimally inconsistent LP. Stud. Logica. 50, 321–331 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  34. Rozoner, L.I.: On interpretation of inconsistent theories. Inf. Sci. 47, 243–266 (1989)

    Article  Google Scholar 

  35. Sette, A.M.: On propositional calculus \(P_1\). Mathematica Japonica 16, 173–180 (1973)

    MathSciNet  Google Scholar 

  36. Shoesmith, D.J., Smiley, T.J.: Deducibility and many-valuedness. J. Symbolic Logic 36, 610–622 (1971)

    Article  MathSciNet  Google Scholar 

  37. Shoesmith, D.J., Smiley, T.J.: Multiple Conclusion Logic. Cambridge University Press (1978)

    Google Scholar 

  38. Sobociński, B.: Axiomatization of a partial system of three-value calculus of propositions. J. Comput. Syst. 1, 23–55 (1952)

    Google Scholar 

  39. Urquhart, A.: Many-valued logic. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, 2nd edn., vol. II, pp. 249–295. Kluwer (2001)

    Google Scholar 

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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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