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ON CONTRA-CLASSICAL VARIANTS OF NELSON LOGIC N4 AND ITS CLASSICAL EXTENSION

  • HITOSHI OMORI (a1) and HEINRICH WANSING (a2)
Abstract

In two recent articles, Norihiro Kamide introduces unusual variants of Nelson’s paraconsistent logic and its classical extension. Kamide’s systems, IP and CP, are unusual insofar as double negations in these logics behave as intuitionistic and classical negations, respectively. In this article we present Hilbert-style axiomatizations of both IP and CP. The axiom system for IP is shown to be sound and complete with respect to a four-valued Kripke semantics, and the axiom system for CP is characterized by four-valued truth tables. Moreover, we note some properties of IP and CP, and emphasize that these logics are unusual also because they are contra-classical and inconsistent but nontrivial. We point out that Kamide’s approach exemplifies a general method for obtaining contra-classical logics, and we briefly speculate about a linguistic application of Kamide’s logics.

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Corresponding author
*SCHOOL OF INFORMATION SCIENCE JAPAN ADVANCED INSTITUTE OF SCIENCE AND TECHNOLOGY NOMI, JAPAN E-mail: hitoshiomori@gmail.comURL: https://sites.google.com/site/hitoshiomori/home
DEPARTMENT OF PHILOSOPHY I RUHR-UNIVERSITÄT BOCHUM BOCHUM, GERMANY E-mail: Heinrich.Wansing@rub.deURL: http://www.ruhr-uni-bochum.de/philosophy/logic/
References
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Arieli, O. & Avron, A. (1996). Reasoning with logical bilattices. Journal of Logic, Language and Information, 5, 2563.
Arieli, O., Avron, A., & Zamansky, A. (2011). Ideal paraconsistent logics. Studia Logica, 99, 3160.
Avron, A. (1999). On the expressive power of three-valued and four-valued languages. Journal of Logic and Computation, 9, 977994.
Avron, A. (2005). A non-deterministic view on non-classical negations. Studia Logica, 80 (2–3), 159194.
Blanchette, F. (2015). English Negative Concord, Negative Polarity, and Double Negation. New York: Dissertation.
De, M. & Omori, H. (2015). Classical negation and expansions of Belnap-Dunn logic. Studia Logica, 103, 825851.
Horn, L. R. & Wansing, H. (2017). Negation. In Zalta, E. N., editor. The Stanford Encyclopedia of Philosophy (Spring 2017 Edition). Available at: http://plato.stanford.edu/archives/spr2017/entries/negation/.
Humberstone, L. (1995). Negation by iteration. Theoria, 61(1), 124.
Humberstone, L. (2000). Contra-classical logics. Australasian Journal of Philosophy, 78(4), 438474.
Kamide, N. (2016). Paraconsistent double negation that can simulate classical negation. In Yuminaka, Y., editor. Proceedings of the 46th IEEE International Symposium on Multiple-Valued Logic (ISMVL 2016). Los Alamitos, CA: IEEE Computer Society, pp. 131136.
Kamide, N. (2017). Paraconsistent double negations as classical and intuitionistic negations. Studia Logica, 105(6), 11671191.
Kamide, N. & Wansing, H. (2012). Proof theory of Nelson’s paraconsistent logic: A uniform perspective. Theoretical Computer Science, 415, 138.
Kamide, N. & Wansing, H. (2015). Proof Theory of N4-related Paraconsistent Logics. Studies in Logic, Vol. 54. London: College Publications.
Marcos, J. (2005). On negation: Pure local rules. Journal of Applied Logic, 3, 185219.
Odintsov, S. P. (2005). The class of extensions of Nelson paraconsistent logic. Studia Logica, 80, 291320.
Omori, H. (2016). A simple connexive extension of the basic relevant logic BD. IfCoLog Journal of Logics and their Applications, 3(3), 467478.
Omori, H. & Sano, K. (2015). Generalizing functional completeness in Belnap-Dunn logic. Studia Logica, 103(5), 883917.
Ruet, P. (1996). Complete set of connectives and complete sequent calculus for Belnap’s logic. Technical Report Ecole Normale Superieure. Logic Colloquium 96, Document LIENS-96-28.
Słupecki, J. (1972). A criterion of fullness of many-valued systems of propositional logic. Studia Logica, 30, 153157.
Tokarz, M. (1973). Connections between some notions of completeness of structural propositional calculi. Studia Logica, 32(1), 7789.
van der Wouden, T. (1994). Negative Contexts. Dissertation: Rijksuniversiteit Groningen.
Wansing, H. (2001). Negation. In Goble, L., editor. The Blackwell Guide to Philosophical Logic. Cambridge/MA: Basil Blackwell Publishers, pp. 415436
Wansing, H. (2005). Connexive modal logic. In Schmidt, R., Pratt-Hartmann, I., Reynolds, M., and Wansing, H., editors. Advances in Modal Logic, Vol. 5. London: King’s College Publications, pp. 367383.
Wansing, H. (2016). Connexive logic. In Zalta, E. N., editor. The Stanford Encyclopedia of Philosophy (Spring 2016 Edition). Available at: http://plato.stanford.edu/archives/spr2016/entries/logic-connexive/.
Wansing, H. & Odintsov, S. (2016). On the methodology of paraconsistent logic. In Andreas, H. and Verdée, P., editors. Logical Studies of Paraconsistent Reasoning in Science and Mathematics. Trends in Logic, Vol. 45. Cham, Switzerland: Springer, pp. 175204.
Zijlstra, H. (2004). Sentential Negation and Negative Concord. Dissertation, Universiteit van Amsterdam.
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The Review of Symbolic Logic
  • ISSN: 1755-0203
  • EISSN: 1755-0211
  • URL: /core/journals/review-of-symbolic-logic
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