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An Extended Paradefinite Logic Combining Conflation, Paraconsistent Negation, Classical Negation, and Classical Implication: How to Construct Nice Gentzen-type Sequent Calculi


In this study, an extended paradefinite logic with classical negation (EPLC), which has the connectives of conflation, paraconsistent negation, classical negation, and classical implication, is introduced as a Gentzen-type sequent calculus. The logic EPLC is regarded as a modification of Arieli, Avron, and Zamansky’s ideal four-valued paradefinite logic (4CC) and as an extension of De and Omori’s extended Belnap–Dunn logic with classical negation (BD+) and Avron’s self-extensional four-valued paradefinite logic (SE4). The completeness, cut-elimination, and decidability theorems for EPLC are proved and EPLC is shown to be embeddable into classical logic. The strong equivalence substitution property and the admissibilities of the rules of negative symmetry, contraposition, and involution are shown for EPLC. Some alternative simple Gentzen-type sequent calculi, which are theorem-equivalent to EPLC, are obtained via these characteristic properties.

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

    MathSciNet  Article  Google Scholar 

  2. Angell, R.: A propositional logics with subjunctive conditionals. J. Symb. Logic 27, 327–343 (1962)

    MathSciNet  Article  Google Scholar 

  3. Arieli, O., Avron, A.: Reasoning with logical bilattices. J. Logic, Lang. Info. 5, 25–63 (1996)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    Chapter  Google Scholar 

  6. Arieli, O., Avron, A.: Minimal paradefinite logics for reasoning with incompleteness and inconsistency, Proceedings of the 1st International Conference on Formal Structures for Computation and Deduction (FSCD), Leibniz International Proceedings in Informatics (LIPIcs) 52, pp. 7:1-7:15, (2016)

  7. Arieli, O., Avron, A.: Four-valued paradefinite logics. Stud. Log. 105(6), 1087–1122 (2017)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  9. Asenjo, F.G.: A calculus of antinomies. Notre Dame J. Form. Log. 7, 103–106 (1966)

    MathSciNet  Article  Google Scholar 

  10. Avron, A.: Self-extensional three-valued paraconsistent logics. Log. Univ. 11(3), 297–315 (2017)

    MathSciNet  Article  Google Scholar 

  11. Avron, A.: The normal and self-extensional extension of Dunn-Belnap logic. Log. Univ. 14(3), 281–296 (2020)

    MathSciNet  Article  Google Scholar 

  12. Avron, A., Béziau, J.-Y.: Self-extensional three-valued paraconsistent logics have no implication. Log. J. IGPL 25(2), 183–194 (2017)

    MathSciNet  MATH  Google Scholar 

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

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

  15. Béziau, J.-Y.: Idempotent full paraconsistent negations are not algebraizable. Notre Dame J. Form. Log. 39, 135–139 (1994)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  17. Béziau, J.-Y.: Bivalent semantics for De Morgan logic (The uselessness of four-valuedness), In W.A. Carnieli, M.E. Coniglio, and I.M. D’Ottaviano (eds.), The many sides of logic, pp. 391-402, College Publications, (2009)

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  20. Dunn, J.M.: Partiality and its dual. Stud. Log. 65, 5–40 (2000)

    MathSciNet  Article  Google Scholar 

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

  22. Gentzen, G.: Collected papers of Gerhard Gentzen, M.E. Szabo, ed., Studies in logic and the foundations of mathematics, North-Holland (English translation), (1969)

  23. Gurevich, Y.: Intuitionistic logic with strong negation. Stud. Log. 36, 49–59 (1977)

    MathSciNet  Article  Google Scholar 

  24. Kamide, N.: Extending ideal paraconsistent four-valued logic, Proceedings of the 47th IEEE International Symposium on Multiple-Valued Logic (ISMVL 2017), pp. 49-54, (2017)

  25. Kamide, N.: An extended paradefinte Belnap-Dunn logic that is embeddable into classical logic and vice versa, Proceedings of the 11th International Conference on Agents and Artificial Intelligence (ICAART 2019), 2,377-387 (2019)

  26. Kamide, N.: Gentzen-type sequent calculi for extended Belnap-Dunn logics with classical negation: A general framework. Log. Univ. 13(1), 37–63 (2019)

    MathSciNet  Article  Google Scholar 

  27. Kamide, N.: Kripke-completeness and cut-elimination theorems for intuitionistic paradefinite logics with and without quasi-explosion. J. Philos. Log. 49(6), 1185–1212 (2020)

    MathSciNet  Article  Google Scholar 

  28. Kamide, N.: Notes on Avron’s self-extensional four-valued paradefinite logic, Proceedings of the 51st IEEE International Symposium on Multiple-Valued Logic (ISMVL 2021), pp. 43-49, (2021)

  29. Kamide, N.: Herbrand and contraposition-elimination theorems for extended first-order Belnap–Dunn logic, Relevance logics and other tools for reasoning: Essays in Honor of J. Mchael Dunn (Katalin Bimbo editor), Volume 46 of Tribute Series, pp. 237–260, College Publications (2022)

  30. Kamide, N., Omori, H.: An extended first-order Belnap-Dunn logic with classical negation, Proceedings of the 6th International Workshop on Logic, Rationality, and Interaction (LORI 2017), Lecture Notes in Computer Science 10455, pp. 79-93, (2017)

  31. Kamide, N., Zohar, Y.: Yet another paradefinite logic: The role of conflation. Log. J. IGPL 27(1), 93–117 (2019)

    MathSciNet  MATH  Google Scholar 

  32. Lahav, O., Avron, A.: A unified semantic framework for fully structural propositional sequent systems. ACM Trans. Comput. Log. 14(4), 27:1-27:33 (2013)

    MathSciNet  Article  Google Scholar 

  33. McCall, S.: Connexive implication. J. Symb. Log. 31, 415–433 (1966)

    MathSciNet  Article  Google Scholar 

  34. Méndez, J.M., Robles, G.: A strong and rich 4-valued modal logic without Łukasiewicz-type paradoxes. Log. Univ. 9(4), 501–522 (2015)

    Article  Google Scholar 

  35. Nelson, D.: Constructible falsity. J. Symb. Log. 14, 16–26 (1949)

    MathSciNet  Article  Google Scholar 

  36. Ono, H.: Logic for information science (in Japanese), Nihon hyouron shya, 297 pages, (1994)

  37. Priest, G.: The logic of paradox. J. Philos. Log. 8(1), 219–241 (1979)

    MathSciNet  Article  Google Scholar 

  38. Priest, G.: Paraconsistent logic, Handbook of Philosophical Logic (Second Edition), Vol. 6, D. Gabbay and F. Guenthner (eds.), Kluwer Academic Publishers, Dordrecht, pp. 287-393, (2002)

  39. Rautenberg, W.: Klassische und nicht-klassische Aussagenlogik. Vieweg, Braunschweig (1979)

    Book  Google Scholar 

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

    Google Scholar 

  41. Wansing, H.: The logic of information structures, Lecture Notes in Computer Science 681, 163 pages, Springer (1993)

  42. Wansing, H.: Informational interpretation of substructural propositional logics. J. Log. Lang. Inf. 2(4), 285–308 (1993)

    MathSciNet  Article  Google Scholar 

  43. Wansing, H.: Connexive logic, Stanford Encyclopedia of Philosophy, (2021):

  44. Zaitsev, D.: Generalized relevant logic and models of reasoning, Moscow State Lomonosov University doctoral dissertation, (2012)

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Correspondence to Norihiro Kamide.

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We would like to thank the anonymous referee for his or her valuable comments. This work was supported by JSPS KAKENHI Grant Numbers JP18K11171 and JP16KK0007 and Grant-in-Aid for Takahashi Industrial and Economic Research Foundation.

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Kamide, N. An Extended Paradefinite Logic Combining Conflation, Paraconsistent Negation, Classical Negation, and Classical Implication: How to Construct Nice Gentzen-type Sequent Calculi. Log. Univers. (2022).

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  • Paradefinite logic
  • Gentzen-type sequent calculus
  • Cut-elimination theorem
  • Completeness theorem

Mathematics Subject Classification

  • Primary 03B53
  • Secondary 03B50