Abstract
In one of his papers, JC Beall advanced a new and interesting interpretation of Weak Kleene logic, in terms of on-topic/off-topic. In brief, Beall suggests to read the third value as off-topic, whereas the two classical values are read as true and on-topic and false and on-topic. Building on Beall’s new interpretation, the aim of this paper is threefold. First, we discuss two motivations to enrich Beall’s interpretation, and offer an alternative semantic framework that reflects our motivations. Second, by making use of our new framework, we will offer a new interpretation of the logic of Catuskoti which combines Beall’s proposal of having FDE as the correct logic with the on-topic/off-topic divide. Finally, we will offer a general result that will allow us to make sense of a family of infectious logics in terms of Beall’s on-topic/off-topic reading.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Beall, J. C., Off-topic: a new interpretation of weak-Kleene logic, The Australasian Journal of Logic 13(6), 2016.
Beall, J. C., The simple argument for subclassical logic, Philosophical Issues 28(1):30–54, 2018.
Beall, J. C., FDE as the one true logic, in H. Omori, and H. Wansing, (eds.), New Essays on Belnap–Dunn Logic, Springer, 2019, pp. 115–125.
Belikov, A., and Y. Petrukhin, Exactly true and non-falsity logics meeting infectious ones, Journal of Applied Non-Classical Logics 30(2):93–122, 2020.
Bochvar, D., On a three-valued calculus and its application in the analysis of the paradoxes of the extended functional calculus, Matamaticheskii Sbornik 4(46):287–308, 1938.
Bonzio, S., T. Moraschini, and M. Pra Baldi, Logics of left variable inclusion and Płonka sums of matrices, Archive for Mathematical Logic 60(1):49–76, 2021.
Bonzio, S., and M. Pra Baldi, Containment logics: Algebraic completeness and axiomatization, Studia Logica 109(5):969–994, 2021.
Caleiro, C., S. Marcelino, and P. Filipe, Infectious semantics and analytic calculi for even more inclusion logics, in Proceedings of ISMVL 2020, 2020, pp. 224–229.
Ciuni, R., and M. Carrara, Semantical analysis of weak Kleene logics, Journal of Applied Non-Classical Logics 29(1):1–36, 2019.
Ciuni, R., T. M. Ferguson, and D. Szmuc, Modeling the interaction of computer errors by four-valued contaminating logics, in Proceedings of WoLLIC 2019, Springer, 2019, pp. 119–139.
Ciuni, R., T. Macaulay Ferguson, and D. Szmuc, Logics based on linear orders of contaminating values, Journal of Logic and Computation 29(5):631–663, 2019.
Cobreros, P., P. Egré, D. Ripley, and R. van Rooij, Tolerant, classical, strict, Journal of Philosophical Logic 41(2):347–385, 2012.
Cobreros, P., P. Égré, D. Ripley, and R. Van Rooij, Reaching transparent truth, Mind 122(488):841–866, 2013.
Coniglio, M. E., and M. I.Corbalán, Sequent calculi for the classical fragment of Bochvar and Halldén’s Nonsense Logics, in Proceedings of LSFA 2012, 2012, pp. 125–136.
Correia, F., Weak necessity on weak Kleene matrices, in Advances in Modal Logic: Volume 3, World Scientific, 2002, pp. 73–90.
Da Ré, B., F. Pailos, and D. Szmuc, Theories of truth based on four-valued infectious logics, Logic Journal of the IGPL 28(5):712–746, 2020.
Ferguson, T. M., A computational interpretation of conceptivism, Journal of Applied Non-Classical Logics 24(4):333–367, 2014.
Ferguson, T. M., Logics of nonsense and Parry systems, Journal of Philosophical Logic 44(1):65–80, 2015.
Fjellstad, A., Structural proof theory for first-order weak Kleene logics, Journal of Applied Non-Classical Logics 30(3):272–289, 2020.
Frankowski, S., Formalization of a plausible inference, Bulletin of the Section of Logic 33(1):41–52, 2004.
Halldén, S., The logic of nonsense, Uppsala Universitets Årsskrift, 1949.
Herzberger, H. G., Dimensions of truth, Journal of Philosophical Logic 2(4):535–556, 1973.
Kleene, S. C., Introduction to Metamathematics, North-Holland, Amsterdam, 1952.
Oller, C. A., Paraconsistency and analyticity, Logic and Logical Philosophy 7:91–99, 1999.
Omori, H., Halldén’s logic of nonsense and its expansions in view of logics of formal inconsistency, in Proceedings of DEXA 2016, IEEE Computer Society, 2016, pp. 129–133.
Omori, H., and J. R. B. Arenhart, Haack meets herzberger and priest, in 2022 IEEE 52nd International Symposium on Multiple-Valued Logic (ISMVL), IEEE, 2022, pp. 137–144.
Omori, H., and J. R. B. Arenhart, Change of logic, without change of meaning, Theoria 89:414–431, 2023.
Omori, H., and J. R. B. Arenhart, Is the de finetti conditional a conditional? Argumenta (forthcoming).
Omori, H., and D. Szmuc, Conjunction and disjunction in infectious logics, in Proceedings of LORI 2017, Springer, 2017, pp. 268–283.
Omori, H., and H. Wansing, 40 years of FDE: an introductory overview, Studia Logica 105(6):1021–1049, 2017.
Paoli, F. and M. Pra Baldi, Proof theory of paraconsistent weak Kleene logic, Studia Logica 108:779–802, 2019.
Priest, G., Hyper-contradictions, Logique et Analyse 27(107):237–243, 1984.
Priest, G., The logic of the catuskoti, Comparative Philosophy 1(2):51, 2010.
Priest, G., Plurivalent logics, The Australasian Journal of Logic 11(1):1–13, 2014.
Priest, G., Natural deduction systems for logics in the FDE family, in H. Omori, and H. Wansing, editors, New Essays on Belnap-Dunn Logic, Springer, 2019, pp. 279–292.
Song, Y., H. Omori, and S. Tojo, A two-valued semantics for infectious logics, in 2021 IEEE 51st International Symposium on Multiple-Valued Logic (ISMVL), IEEE, 2021, pp. 50–55.
Szmuc, D., Defining LFIs and LFUs in extensions of infectious logics, Journal of Applied Non-Classical Logics 26(4):286–314, 2017.
Szmuc, D., An epistemic interpretation of paraconsistent weak Kleene logic, Logic and Logical Philosophy 28(2):277–330, 2019.
Szmuc, D., and H. Omori, A note on Goddard and Routley’s significance logic, The Australasian Journal of Logic 15(2):431–448, 2018.
Woodruff, P., On constructive nonsense logic, in Modality, Morality, and Other problems of Sense and Nonsense: Essays Dedicated to Sören Halldén, GWK Gleerup Bokforlag, Lund, 1973, pp. 192–205.
Acknowledgements
The research of Y.S. and S.T. was supported by JSPS KAKEN 17H02258. Moreover, the research of H.O. was supported by a Sofja Kovalevskaja Award of the Alexander von Humboldt-Foundation, funded by the German Ministry for Education and Research. The research of J.R.B.A. was also supported by the Alexander von Humboldt-Foundation through the Experienced Researcher Fellowship program, funded by CAPES-Humboldt. We would like to thank the referees for their careful comments, as well as Francesco Paoli for his detailed written comments as our handling editor.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Presented by Francesco Paoli
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Song, Y., Omori, H., Arenhart, J.R.B. et al. A Generalization of Beall’s Off-Topic Interpretation. Stud Logica (2023). https://doi.org/10.1007/s11225-023-10075-0
Received:
Published:
DOI: https://doi.org/10.1007/s11225-023-10075-0