Hostname: page-component-8448b6f56d-gtxcr Total loading time: 0 Render date: 2024-04-20T00:04:59.878Z Has data issue: false hasContentIssue false
  • English
  • Français

SEMANTICS FOR PURE THEORIES OF CONNEXIVE IMPLICATION

Part of: General logic

Published online by Cambridge University Press:  21 October 2020

YALE WEISS
YALE WEISS*
Affiliation:
THE SAUL KRIPKE CENTER THE GRADUATE CENTER, CUNY 365 FIFTH AVE., ROOM 7118 NEW YORK, NY 10016, USAE-mail: yweiss@gradcenter.cuny.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article, I provide Urquhart-style semilattice semantics for three connexive logics in an implication-negation language (I call these “pure theories of connexive implication”). The systems semantically characterized include the implication-negation fragment of a connexive logic of Wansing, a relevant connexive logic recently developed proof-theoretically by Francez, and an intermediate system that is novel to this article. Simple proofs of soundness and completeness are given and the semantics is used to establish various facts about the systems (e.g., that two of the systems have the variable sharing property). I emphasize the intuitive content of the semantics and discuss how natural informational considerations underly each of the examined systems.

Keywords

connexive logicsemilattice semanticscontra-classical logic

MSC classification

Primary: 03B53: Paraconsistent logics
Secondary: 03B60: Other nonclassical logic
Type
Research Article
Information
The Review of Symbolic Logic , Volume 15 , Issue 3 , September 2022 , pp. 591 - 606
Copyright
© The Author(s), 2020. 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. Journal of Symbolic Logic, 49(1), 231–233.CrossRefGoogle Scholar
Anderson, A. R., & Belnap, N. D. Jr. (1975). Entailment: The Logic of Relevance and Necessity, Vol. I. Princeton, NJ: Princeton University Press.Google Scholar
Anderson, A. R., Belnap, N. D. Jr., & Dunn, J. M. (1992). Entailment: The Logic of Relevance and Necessity, Vol. II. Princeton, NJ: Princeton University Press.Google Scholar
Charlwood, G., & Daniels, C. B. (1981). Semilattice relevance logic with classical negation. Unpublished typescript, Victoria, B.C.Google Scholar
Chellas, B. F. (1980). Modal Logic: An Introduction. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Dunn, J. M., & Restall, G. (2002). Relevance logic. In Gabbay, D., and Guenthner, F., editors. Handbook of Philosophical Logic, Vol. 6. Dordrecht, The Netherlands: Kluwer Academic Publishers, pp. 1–128.Google Scholar
Estrada-González, L. (2020). The Bochum plan and the foundations of contra-classical logics. CLE e-Prints, 19(1), 1–22.Google Scholar
Francez, N. (2019). Relevant connexive logic. Logic and Logical Philosophy, 28(3), 409–425.Google Scholar
Kamide, N., & Wansing, H. (2011). Connexive modal logic based on positive S4. In Beziau, J.-Y., & Conigli, M., editors. Logic Without Frontiers: Festschrift for Walter Alexandre Carnielli on the Occasion of His 60th Birthday. London, UK: College Publications, pp. 389–409.Google Scholar
Kapsner, A. (2012). Strong connexivity. Thought, 1(2), 141–145.CrossRefGoogle Scholar
McCall, S. (1966). Connexive implication. Journal of Symbolic Logic, 31(3), 415–433.CrossRefGoogle Scholar
Meyer, R. K. (1977). S5–the poor man’s connexive implication. Relevance Logic Newsletter, 2(2), 117–124.Google 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., & Wansing, H. (2018). On contra-classical variants of Nelson logic N4 and its classical extension. Review of Symbolic Logic, 11(4), 805–820.CrossRefGoogle Scholar
Priest, G. (1999). Negation as cancellation, and connexive logic. Topoi, 18(2), 141–148.CrossRefGoogle Scholar
Priest, G. (2008). An Introduction to Non-Classical Logic second edition. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Routley, R. (1984). The American plan completed: Alternative classical-style semantics, without stars, for relevant and paraconsistent logics. Studia Logica, 43(1-2), 131–158.CrossRefGoogle Scholar
Urquhart, A. (1972). Semantics for relevant logics. Journal of Symbolic Logic, 37(1), 159–169.CrossRefGoogle Scholar
Urquhart, A. I. F. (1971). Completeness of weak implication. Theoria, 37(3), 274–282.CrossRefGoogle Scholar
Urquhart, A. I. F. (1973). The Semantics of Entailment. PhD Thesis, University of Pittsburgh.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, UK: King’s College Publications, pp. 367–383.Google Scholar
Wansing, H. (2007). A note on negation in categorial grammar. Logic Journal of the IGPL, 15(3), 271–286.CrossRefGoogle Scholar
Wansing, H., & Skurt, D. (2018). Negation as cancellation, connexive logic, and qLPm. Australasian Journal of Logic, 15(2), 476–488.CrossRefGoogle Scholar
Weiss, Y. (2019). A note on the relevance of semilattice relevance logic. Australasian Journal of Logic, 16(6), 177–185.CrossRefGoogle Scholar