Hostname: page-component-6bf8c574d5-b4m5d Total loading time: 0 Render date: 2025-02-23T14:21:31.491Z Has data issue: false hasContentIssue false
  • English
  • Français

IMPROVING STRONG NEGATION

Part of: General logic

Published online by Cambridge University Press:  02 July 2021

SATORU NIKI
SATORU NIKI*
Affiliation:
DEPARTMENT OF PHILOSOPHY I RUHR UNIVERSITY BOCHUM, UNIVERSITÄTSSTRASSE BOCHUM 150 D-44780, GERMANY
*
E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Strong negation is a well-known alternative to the standard negation in intuitionistic logic. It is defined virtually by giving falsity conditions to each of the connectives. Among these, the falsity condition for implication appears to unnecessarily deviate from the standard negation. In this paper, we introduce a slight modification to strong negation, and observe its comparative advantages over the original notion. In addition, we consider the paraconsistent variants of our modification, and study their relationship with non-constructive principles and connexivity.

Keywords

connexive logicdouble negationintuitionistic logicstrong negationquasi-truth

MSC classification

Primary: 03B20: Subsystems of classical logic (including intuitionistic logic)
Type
Research Article
Information
The Review of Symbolic Logic , Volume 16 , Issue 3 , September 2023 , pp. 951 - 977
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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

Almukdad, A., & Nelson, D. (1984). Constructible falsity and inexact predicates. The Journal of Symbolic Logic, 49(1), 231–233.Google Scholar
Batens, D., & De Clercq, K. (2004). A rich paraconsistent extension of full positive logic. Logique et Analyse, 185–188, 227–257.Google Scholar
Božič, M., & Došen, K. (1983). Axiomatizations of intuitionistic double negation. Bulletin of the Section of Logic, 12(2), 99–102 Google Scholar
Chagrov, A., & Zakharyaschev, M. (1997). Modal Logic. Oxford: Oxford University Press.Google Scholar
Coniglio, M. E., & Silvestrini, L. H. D. C. (2014). An alternative approach for quasi-truth. Logic Journal of IGPL, 22(2), 387–410.Google Scholar
De, M. (2013). Empirical negation. Acta Analytica, 28, 49–69.Google Scholar
De, M., & Omori, H. (2015). Classical negation and expansions of Belnap–Dunn logic. Studia Logica, 103(4), 825–851.Google Scholar
Došen, K. (1986). Negation as a modal operator. Reports on Mathematical Logic, 20, 15–27.Google Scholar
Gabbay, D. M. (1981). Semantical Investigations in Heyting’s Intuitionistic Logic. Synthese Library, Vol. 148. Dordrecht: Springer.Google Scholar
Gurevich, Y. (1977). Intuitionistic logic with strong negation. Studia Logica, 36(1–2), 49–59.Google Scholar
Ishihara, H. (2014). Classical propositional logic and decidability of variables in intuitionistic propositional logic. Logical Methods in Computer Science, 10(3), 1–23.Google Scholar
Kamide, N. (2021). Modal and intuitionistic variants of extended Belnap–Dunn logic with classical negation. Journal of Logic, Language and Information. https://doi.org/10.1007/s10849-021-09330-1 Google Scholar
Kamide, N., & Wansing, H. (2015). Proof Theory of N4-Related Paraconsistent Logics. London: College Publications.Google Scholar
Kapsner, A. (2014). Logics and Falsifications. Trends in logic, Vol. 40. Heidelberg: Springer.Google Scholar
Markov, A. A. (1950). Constructive logic (in Russian). Russian Mathematical Survey, 5(3–37), 187–188.Google Scholar
McCall, S. (1966). Connexive implication. The Journal of Symbolic Logic, 31(3), 415–433.Google Scholar
Mikenberg, I., da Costa, N. C. A., & Chuaqui, R. (1986). Pragmatic truth and approximation to truth. The Journal of Symbolic Logic, 51(1), 201–221.Google Scholar
Nascimento, T., Rivieccio, U., Marcos, J., & Spinks, M. (2018). Algebraic semantics for Nelson’s logic $\mathcal {S}$ . In Moss, L. S., de Queiroz, R., and Martinez, M., editors. International Workshop on Logic, Language, Information, and Computation. Heidelberg: Springer, pp. 271–288.Google Scholar
Nascimento, T., Rivieccio, U., Marcos, J., & Spinks, M. (2020). Nelson’s logic $\mathcal{S}$ . Logic Journal of the IGPL. https://doi.org/10.1093/jigpal/jzaa015 Google Scholar
Nelson, D. (1949). Constructible falsity. The Journal of Symbolic Logic, 14(1), 16–26.Google Scholar
Nelson, D. (1959). Negation and separation of concepts in constructive systems. In Heyting, A., editor. Constructivity in Mathematics. Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Nolland, pp. 205–225.Google Scholar
Odintsov, S. (2008). Constructive Negations and Paraconsistency: A Categorical Approach to L-Fuzzy Relations. Dordrecht: Springer.Google Scholar
Omori, H. (2019). Towards a bridge over two approaches in connexive logic. Logic and Logical Philosophy, 28(3), 553–566.Google Scholar
Omori, H., & Wansing, H. (2020). An extension of connexive logic C. In Olivietti, N., Verbrugge, R., Negri, S., and Sandu, G., editors. Advances in Modal Logic 13. London: College Publications, pp. 503–522.Google Scholar
Ono, H. (2019). Proof Theory and Algebra in Logic. Singapore: Springer.Google Scholar
Rauszer, C. (1974). A formalization of the propositional calculus of HB logic. Studia Logica, 33(1), 23–34.Google Scholar
Sette, A. M. (1973). On The Propositional Calculus $\mathbf{P}^{1}$ . Mathematica Japonica, 18(13).Google Scholar
Sotirov, V. K. (1982). The intuitionistic double negation is a modality. In Seventh International Wittgenstein Symposium. Austrian Ludwig Wittgenstein Society, p. 58.Google Scholar
Thomason, R. H. (1969). A semantical study of constructible falsity. Mathematical Logic Quarterly, 15(16–18), 247–257.Google Scholar
Troelstra, A. S., & van Dalen, D. (1988). Constructivism in Mathematics: An Introduction, Vol. I. Amsterdam: Elsevier.Google Scholar
Vorob’ev, N. N. (1964). A constructive calculus of statements with strong negation (in Russian). Trudy Matematicheskogo Instituta imeni V.A. Steklova, Vol. 72, pp. 195–227.Google 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: College Publications, pp. 387–399.Google Scholar
Wansing, H. (2008). Constructive negation, implication, and co-implication. Journal of Applied Non-Classical Logics, 18(2–3), 341–364.Google Scholar