Abstract
Our aim is to provide a sequent calculus whose external consequence relation coincides with the three-valued paracomplete logic ‘of nonsense’ introduced by Dmitry Bochvar and, independently, presented as the weak Kleene logic \(\textbf{K}_{\textbf{3}}^{\textbf{w}}\) by Stephen C. Kleene. The main features of this calculus are (i) that it is non-reflexive, i.e., Identity is not included as an explicit rule (although a restricted form of it with premises is derivable); (ii) that it includes rules where no variable-inclusion conditions are attached; and (iii) that it is hybrid, insofar as it includes both left and right operational introduction as well as elimination rules.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Avron, A., The semantics and proof theory of linear logic, Theoretical Computer Science 57(2–3):161–184, 1988.
Bochvar, D.A., Ob odnom trechznac̆nom isc̆islenii i ego primenenii k analizu paradoksov klassiceskogo rassirennogo funkcionalnogo iscislenija, Matematiceskij Sbornik 4:287–308, 1938. [English translation by Merrie Bergmann: Bochvar, D.A., On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus, History and Philosophy of Logic 2:87–112, 1981].
Bonzio, S., J. Gil-Férez, F. Paoli, and L. Peruzzi, On paraconsistent weak Kleene logic: Axiomatisation and algebraic analysis, Studia Logica 105(2):253–297, 2017.
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, 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.
Coniglio, M.E., and M.I. Corbalán, Sequent calculi for the classical fragment of Bochvar and Halldén’s nonsense logics, Proceedings of LSFA 2012, 2012, pp. 125–136.
Corbalán, M.I., Conectivos de Restauração Local, MA Thesis, University of Campinas, 2012.
Da Ré, B., F. Pailos, D. Szmuc, and P. Teijeiro, Metainferential duality, Journal of Applied Non-Classical Logics 30(4):312–334, 2020.
Da Ré, B., D. Szmuc, and P. Teijeiro, Derivability and metainferential validity, Journal of Philosophical Logic 51(6):1521–1547, 2022.
Ferguson, T.M., Meaning and Proscription in Formal Logic: Variations on the Propositional Logic of William T. Parry, vol. 49 of Trends in Logic, Springer, Dordrecht, 2017.
Fjellstad, A., Non-classical elegance for sequent calculus enthusiasts, Studia Logica 105(1):93–119, 2017.
Girard, J.-Y., Proof Theory and Logical Complexity. Vol.1, Bibliopolis, Napoli, 1990.
Halldén, S., The Logic of Nonsense, Uppsala Universitet, 1949.
Hösli, B., and G. Jäger, About some symmetries of negation, Journal of Symbolic Logic 59(2):473–485, 1994.
Humberstone, L., Valuational semantics of rule derivability, Journal of Philosophical Logic 25(5):451–461, 1996.
Paoli, F., Regressive analytical entailments. Technical Report n. 33, Konstanzer Berichte zur Logik und Wissenschaftstheorie, 1992.
Paoli, F., Tautological entailments and their rivals, in J.Y. Béziau, W. Carnielli, and D. Gabbay, (eds.), Handbook of Paraconsistency, College Publications, London, 2007, pp. 153–175.
Paoli, F., and M. Pra Baldi, Proof theory of paraconsistent weak Kleene logic, Studia Logica 108(4):779–802, 2020.
Priest, G., Natural Deduction Systems for Logics in the FDE Family, Springer International Publishing, Cham, 2019, pp. 279–292.
Schütte K., Proof Theory, Springer, Berlin, 1977.
Takeuti, G., Proof Theory, Elsevier, Amsterdam, 1987.
Acknowledgements
We would like to thank the members of the Buenos Aires Logic Group (BA-LOGIC), and the audiences of the 22nd Trends in Logic conference (Cagliari, 2022) and the 2nd Workshop between UNAM and BA-LOGIC (Buenos Aires, 2022) for helpful discussion and comments on earlier versions of this material. We are also grateful to two anonymous reviewers of this journal. We also thank the support of the National Scientific and Technical Research Council (CONICET). Corbalán benefited from the financial support of Fondazione di Sardegna, Progetti biennali annualità 2019 through a fellowship hosted by the Department of Pedagogy, Psychology and Philosophy at the University of Cagliari. She wishes to thank Prof. Francesca Ervas and Prof. Francesco Paoli for their personal and academic support during the course of said fellowship. This work was supported by PLEXUS, (Grant Agreement no 101086295) a Marie Sklodowska–Curie action funded by the EU under the Horizon Europe Research and Innovation Programme.
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.
Special Issue: Strong and weak Kleene logics
Edited by Gavin St. John and Francesco Paoli.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Da Ré, B., Szmuc, D. & Corbalán, M.I. Non-Reflexive Nonsense: Proof Theory of Paracomplete Weak Kleene Logic. Stud Logica (2024). https://doi.org/10.1007/s11225-023-10086-x
Published:
DOI: https://doi.org/10.1007/s11225-023-10086-x