Note on Contradictions in Francez-Weiss Logics
DOI:
https://doi.org/10.12775/LLP.2024.031Keywords
connexive logic, contradictory logic, relevant logics, sequent calculus, strong negationAbstract
It is an unusual property for a logic to prove a formula and its negation without ending up in triviality. Some systems have nonetheless been observed to satisfy this property: one group of such non-trivial negation inconsistent logics has its archetype in H. Wansing’s constructive connexive logic, whose negation-implication fragment already proves contradictions. N. Francez and Y. Weiss subsequently investigated relevant subsystems of this fragment, and Weiss in particular showed that they remain negation inconsistent. In this note, we take a closer look at this phenomenon in the systems of Francez and Weiss, and point out two types of necessary conditions, one proof-theoretic and one relevant, which any contradictory formula must satisfy. As a consequence, we propose a nine-fold classification of provable contradictions for the logics.
References
Almukdad, A., and D. Nelson, “Constructible falsity and inexact predicates”, The Journal of Symbolic Logic, 49(1), 1984: 231–233. DOI: https://doi.org/10.2307/2274105
Avron, A., “ Relevant entailment—semantics and formal systems”, The Journal of Symbolic Logic, 49(2), 1984: 334–342. DOI: https://doi.org/10.2307/2274169
Bimbó, K., Proof Theory: Sequent Calculi and Related Formalisms, CRC Press, 2014. DOI: https://doi.org/10.1201/b17294
Brady, R. T, “A Routley-Meyer affixing style semantics for logics containing Aristotle’s thesis”, Studia Logica, 48, 1989: 235–241. DOI: https://doi.org/10.1007/BF02770514
Dunn, J M., “R-Mingle is nice, and so is Arnon Arnon”, pages 141–165 in Arnon Avron on Semantics and Proof Theory of Non-Classical Logics, Springer, 2021. DOI: https://doi.org/10.1007/978-3-030-71258-7_7
Dunn, J M., and G. Restall, “Relevance logic”, pages 1–128 in Handbook of Philosophical Logic, vol. 6, Springer, 2002. DOI: https://doi.org/10.1007/978-94-017-0460-1_1
Francez, N., “Relevant connexive logic”, Logic and Logical Philosophy, 28(3), 2019: 409–425. DOI: https://doi.org/10.12775/LLP.2019.007
Kamide, N., “Substructural logics with mingle”, Journal of Logic, Language and Information, 11, 2002: 227–249. DOI: https://doi.org/10.1023/A:1017586008091
Kamide, N., and H. Wansing, “Proof theory of Nelson’s paraconsistent logic: A uniform perspective”, Theoretical Computer Science, 415, 2012: 1–38. DOI: https://doi.org/10.1016/j.tcs.2011.11.001
Kamide, N. and H. Wansing, Proof Theory of N4-related Paraconsistent Logics, College Publications London, 2015.
Mares, E., “Relevance logic”, in E. N. Zalta and U. Nodelman (eds.), The Stanford Encyclopedia of Philosophy, Stanford University, Fall 2022 edition. https://plato.stanford.edu/entries/logic-relevance/
McCall, S., “A history of connexivity”, pages 415–449 in D. M. Gabbay, F. J. Pelletier and J. Woods (eds.), Logic: A History of its Central Concepts, volume 11 of Handbook of the History of Logic, North-Holland, 2012.
Mints, G., A Short Introduction to Intuitionistic Logic, Kluwer Academic Publishers, 2000.
Mortensen, C., “Aristotle’s thesis in consistent and inconsistent logics”, Studia Logica, 43, 1984: 107–116. DOI: https://doi.org/10.1007/BF00935744
Omori, H., “A simple connexive extension of the basic relevant logic BD”, IFCoLog Journal of Logics and their Applications, 3(3), 2016: 467–478.
Omori, H., and H. Wansing, “Connexive logic, connexivity, and connexivism: Remarks on terminology”, Studia Logica 112, 2024: 1–35. DOI: https://doi.org/10.1007/s11225-023-10082-1
Ono, H., Proof Theory and Algebra in Logic, Springer, Singapore, 2019. DOI: https://doi.org/10.1007/978-981-13-7997-0
Sambin, G., “Some points in formal topology”, Theoretical Computer Science, 305(1), 2003: 347–408. DOI: https://doi.org/10.1016/S0304-3975(02)00704-1
Tamura, S., “The implicational fragment of R-mingle”, Proceedings of the Japan Academy, 47(1), 1971: 71–75. DOI: https://doi.org/10.3792/pja/1195520115
Urquhart, A. I. F., “Completeness of weak implication”, Theoria, 37(3), 1971: 274–282. DOI: https://doi.org/10.1111/j.1755-2567.1971.tb00072.x
Urquhart, A., “Semantics for relevant logics”, The Journal of Symbolic Logic, 37(1), 1972: 159–169. DOI: https://doi.org/10.2307/2272559
Urquhart, A., “The semantics of entailment”, PhD thesis, University of Pittsburgh, 1973.
Wansing, H., “Connexive modal logic”, pages 387–399 in R. Schmidt, I. Pratt-Hartmann, M. Reynolds and H. Wansing (eds.), Advances in Modal Logic, vol. 5, College Publications, 2005.
Wansing, H., “Connexive logic”, in E. N. Zalta (ed.), The Stanford Encyclopedia of Philosophy, Stanford University, Summer 2023 edition. https://plato.stanford.edu/entries/logic-connexive/
Wansing, H., and S. Ayhan, “Logical multilateralism”, Journal of Philosophical Logic, 52, 2023: 1603–1636. DOI: https://doi.org/10.1007/s10992-023-09720-9
Weiss, Y., “Semantics for pure theories of connexive implication”, The Review of Symbolic Logic, 15(3), 2022: 591–606. DOI: https://doi.org/10.1017/S1755020320000374
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2024 Satoru Niki
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 153
Number of citations: 0
