Abstract
Strict-Tolerant Logic (\(\textrm{ST}\)) underpins naïve theories of truth and vagueness (respectively including a fully disquotational truth predicate and an unrestricted tolerance principle) without jettisoning any classically valid laws. The classical sequent calculus without Cut is sometimes advocated as an appropriate proof-theoretic presentation of \(\textrm{ST}\). Unfortunately, there is only a partial correspondence between its derivability relation and the relation of local metainferential \(\textrm{ST}\)-validity – these relations coincide only upon the addition of elimination rules and only within the propositional fragment of the calculus, due to the non-invertibility of the quantifier rules. In this paper, we present two calculi for first-order \(\textrm{ST}\) with an eye to recapturing this correspondence in full. The first calculus is close in spirit to the Epsilon calculus. The other calculus includes rules for the discharge of sequent-assumptions; moreover, it is normalisable and admits interpolation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Availability of data and materials
Not appicable
References
Aguilera, J. P., & Baaz, M. (2019). Unsound inferences make proofs shorter. Journal of Symbolic Logic, 84(1), 102–122.
Avron, A. (1988). The semantics and proof theory of linear logic. Theoretical Computer Science, 57(2–3), 161–184.
Barrio, E., & Pailos, F. (2021). Why a logic is not only its set of valid inferences. Análisis Filosófico, 41(2), 261–272.
Barrio, E., Pailos, F., & Calderon, J. T. (2021). Anti-exceptionalism, truth and the BA-plan. Synthese, 199, 12561–12586.
Barrio, E., Rosenblatt, L., & Tajer, D. (2015). The logics of strict-tolerant logic. Journal of Philosophical Logic, 44(5), 551–571.
Birkhoff, G. (1944). Subdirect unions in universal algebra. Bulletin of the American Mathematical Society, 50, 764–768.
Cobreros, P., Egré, P., Ripley, D., & van Rooij, R. (2012). Tolerant, classical, strict. Journal of Philosophical Logic, 41, 347–85.
Cobreros, P., Egré, P., Ripley, D., & van Rooij, R. (2013). Reaching transparent truth. Mind, 122(488), 841–866.
Cobreros, P., Egré, P., Ripley, D., & van Rooij, R. (2020). Inferences and metainferences in ST. Journal of Philosophical Logic, 49, 1057–1077.
Da Ré, B., Szmuc, D., & Teijeiro, P. (2022). Derivability and metainferential validity. Journal of Philosophical Logic, 51, 1521–1547.
Dicher, B., & Paoli, F. (2019). ST, LP, and tolerant metainferences. In C. Başkent & G. Ferguson (Eds.), Graham Priest on Dialetheism and Paraconsistency (pp. 383–407). Berlin: Springer.
Ferguson, T. M., Ramírez-Cámara, E., & Deep, S. T. (2022). Journal of Philosophical Logic, 51, 1261–1293.
French, R. (2022). Metasequents and tetravaluations. Journal of Philosophical Logic, 51, 1453–1476.
Girard, J. -Y. (1976). Three-valued logic and cut-elimination: The actual meaning of Takeuti’s conjecture. Dissertationes Mathematicae, 136.
Golan, R. (2021). There is no tenable notion of global metainferential validity. Analysis, 81(3), 411–420.
Golan, R. (2023). A simple sequent system for minimally inconsistent LP. Review of Symbolic Logic, 16(4), 1296–1311.
Hlobil, U. (2019). Faithfulness for naive validity. Synthese, 196, 4759–4774.
Kripke, S. (1975). Outline of a theory of truth. Journal of Philosophy, 72(19), 690–716.
Leisenring, A. (1969). Mathematical Logic and Hilbert’s\(\epsilon \)-Symbol. London: MacDonald Technical and Scientific.
Pailos, F., & Da Ré, B. (2023). Metainferential Logics. Berlin: Springer.
Powell, T. (2019). Computational interpretations of classical reasoning: From the Epsilon calculus to stateful programs. In S. Centrone, et al. (Eds), Mathesis Universalis, Computability and Proof, Synthese Library (vol. 412). Springer, Cham. https://doi.org/10.1007/978-3-030-20447-1-14
Přenosil, A. (2017). Cut elimination, identity elimination, and interpolation in Super-Belnap logics. Studia Logica, 105(6), 1255–1289.
Priest, G. (2021). Substructural solutions to the semantic paradoxes: A dialetheic perspective, talk given at the Logic and Metaphysics Seminar. CUNY.
Pynko, A. P. (2010). Gentzen’s cut-free calculus versus the logic of paradox. Bulletin of the Section of Logic, 39(1–2), 35–42.
Ripley, D. (2013). Revising up: Strengthening classical logic in the face of paradox. Philosophers’ Imprint, 13(5), 1–13.
Ripley, D. (2013). Paradoxes and failures of cut. Australasian Journal of Philosophy, 91(1), 139–164.
Schroeder-Heister, P. (1984). A natural extension of natural deduction. Journal of Symbolic Logic, 49, 1284–1300.
Teijeiro, P. (2021). Strength and stability. Análisis Filosófico, 41(2), 337–349.
Troelstra, A. S., & Schwichtenberg, H. (2000). Basic Proof Theory. Cambridge: Cambridge University Press.
Zach, R. (2017). Semantics and proof theory of the Epsilon calculus. In: S. Ghosh, & S. Prasad (Eds.), Logic and Its Applications, ICLA 2017. Lecture Notes in Computer Science (vol. 10119). Springer, Berlin. https://doi.org/10.1007/978-3-662-54069-5-4
Acknowledgements
We are extremely grateful to Elio La Rosa for providing numerous suggestions and a wealth of bibliographical material concerning the Epsilon calculus, and to Pierluigi Graziani, Rosalie Iemhoff and Dave Ripley for their comments. Finally, we thank two anonymous reviewers for their detailed and insightful feedback.
Funding
Open access funding provided by Università degli Studi di Cagliari within the CRUI-CARE Agreement. 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. We also acknowledge the support of the Italian Ministry of University and Research, under the PRIN project DeKLA: Developing Kleene Logics and their Applications (2022SM4XC8), and of Fondazione di Sardegna, under the project Ubiquitous Quantum Reality (UQR): understanding the natural processes under the light of quantum-like structures (F73C22001360007). The second author’s work was funded by the grant 2021 BP 00212 of the grant agency AGAUR of the Generalitat de Catalunya.
Author information
Authors and Affiliations
Contributions
The authors’ contribution to the paper is equal.
Corresponding author
Ethics declarations
Ethical Approval
Not applicable
Competing interests
The authors have no competing interests of a financial or personal nature.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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
Paoli, F., Přenosil, A. Sequent Calculi for First-order \(\textrm{ST}\). J Philos Logic (2024). https://doi.org/10.1007/s10992-024-09766-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10992-024-09766-3