Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-22T21:12:41.475Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  24 October 2018

HITOSHI OMORI  and
HEINRICH WANSING
HITOSHI OMORI*
Affiliation:
School of Information Science, Japan Advanced Institute of Science and Technology
HEINRICH WANSING*
Affiliation:
Department of Philosophy I, Ruhr-Universität Bochum
*
*SCHOOL OF INFORMATION SCIENCE JAPAN ADVANCED INSTITUTE OF SCIENCE AND TECHNOLOGY NOMI, JAPAN E-mail: [email protected]URL: https://sites.google.com/site/hitoshiomori/home
†DEPARTMENT OF PHILOSOPHY I RUHR-UNIVERSITÄT BOCHUM BOCHUM, GERMANY E-mail: [email protected]URL: http://www.ruhr-uni-bochum.de/philosophy/logic/
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

03B5303B6003B65double negationNelson’s paraconsistent logicHilbert-style axiomatizationscontra-classical logicsfour-valued semanticsnegative concord
Type
Research Article
Information
The Review of Symbolic Logic , Volume 11 , Issue 4 , December 2018 , pp. 805 - 820
Copyright
Copyright © Association for Symbolic Logic 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Arieli, O. & Avron, A. (1996). Reasoning with logical bilattices. Journal of Logic, Language and Information, 5, 25–63.CrossRefGoogle Scholar
Arieli, O., Avron, A., & Zamansky, A. (2011). Ideal paraconsistent logics. Studia Logica, 99, 31–60.CrossRefGoogle Scholar
Avron, A. (1999). On the expressive power of three-valued and four-valued languages. Journal of Logic and Computation, 9, 977–994.CrossRefGoogle Scholar
Avron, A. (2005). A non-deterministic view on non-classical negations. Studia Logica, 80 (2–3), 159–194.CrossRefGoogle Scholar
Blanchette, F. (2015). English Negative Concord, Negative Polarity, and Double Negation. New York: Dissertation.Google Scholar
De, M. & Omori, H. (2015). Classical negation and expansions of Belnap-Dunn logic. Studia Logica, 103, 825–851.CrossRefGoogle Scholar
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/.Google Scholar
Humberstone, L. (1995). Negation by iteration. Theoria, 61(1), 1–24.CrossRefGoogle Scholar
Humberstone, L. (2000). Contra-classical logics. Australasian Journal of Philosophy, 78(4), 438–474.CrossRefGoogle Scholar
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. 131–136.CrossRefGoogle Scholar
Kamide, N. (2017). Paraconsistent double negations as classical and intuitionistic negations. Studia Logica, 105(6), 1167–1191.CrossRefGoogle Scholar
Kamide, N. & Wansing, H. (2012). Proof theory of Nelson’s paraconsistent logic: A uniform perspective. Theoretical Computer Science, 415, 1–38.CrossRefGoogle Scholar
Kamide, N. & Wansing, H. (2015). Proof Theory of N4-related Paraconsistent Logics. Studies in Logic, Vol. 54. London: College Publications.Google Scholar
Marcos, J. (2005). On negation: Pure local rules. Journal of Applied Logic, 3, 185–219.CrossRefGoogle Scholar
Odintsov, S. P. (2005). The class of extensions of Nelson paraconsistent logic. Studia Logica, 80, 291–320.CrossRefGoogle Scholar
Omori, H. (2016). A simple connexive extension of the basic relevant logic BD. IfCoLog Journal of Logics and their Applications, 3(3), 467–478.Google Scholar
Omori, H. & Sano, K. (2015). Generalizing functional completeness in Belnap-Dunn logic. Studia Logica, 103(5), 883–917.CrossRefGoogle Scholar
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.Google Scholar
Słupecki, J. (1972). A criterion of fullness of many-valued systems of propositional logic. Studia Logica, 30, 153–157.CrossRefGoogle Scholar
Tokarz, M. (1973). Connections between some notions of completeness of structural propositional calculi. Studia Logica, 32(1), 77–89.CrossRefGoogle Scholar
van der Wouden, T. (1994). Negative Contexts. Dissertation: Rijksuniversiteit Groningen.Google Scholar
Wansing, H. (2001). Negation. In Goble, L., editor. The Blackwell Guide to Philosophical Logic. Cambridge/MA: Basil Blackwell Publishers, pp. 415–436Google Scholar
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. 367–383.Google Scholar
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/.Google Scholar
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. 175–204.CrossRefGoogle Scholar
Zijlstra, H. (2004). Sentential Negation and Negative Concord. Dissertation, Universiteit van Amsterdam.Google Scholar