Abstract
Much has been said about intuitionistic and classical logical systems since Gentzen’s seminal work. Recently, Prawitz and others have been discussing how to put together Gentzen’s systems for classical and intuitionistic logic in a single unified system. We call Prawitz’ proposal the Ecumenical System, following the terminology introduced by Pereira and Rodriguez. In this work we present an Ecumenical sequent calculus, as opposed to the original natural deduction version, and state some proof theoretical properties of the system. We reason that sequent calculi are more amenable to extensive investigation using the tools of proof theory, such as cut-elimination and rule invertibility, hence allowing a full analysis of the notion of Ecumenical entailment. We then present some extensions of the Ecumenical sequent system and show that interesting systems arise when restricting such calculi to specific fragments. This approach of a unified system enabling both classical and intuitionistic features sheds some light not only on the logics themselves, but also on their semantical interpretations as well as on the proof theoretical properties that can arise from combining logical systems.
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Notes
It turns out that Gentzen had a normalisation result for intuitionistic logic. This original proof was found and published in von Plato (2008).
References
Avigad, J. (2001). Algebraic proofs of cut elimination. The Journal of Logic and Algebraic Programming, 49(1–2), 15–30. https://doi.org/10.1016/S1567-8326(01)00009-1.
Avron, A. (1991). Hypersequents, logical consequence and intermediate logics for concurrency. Annals of Mathematics and Artificial Intelligence, 4, 225–248. https://doi.org/10.1007/BF01531058.
Avron, A. (1996). The method of hypersequents in the proof theory of propositional non-classical logics. In W. Hodges, M. Hyland, C. Steinhorn, & J. Truss (Eds.), Logic: From Foundations to Applications. New York: Clarendon Press.
Blasio, C., ao Marcos, J., & Wansing, H. (2017). An inferentially many-valued two dimensional notion of entailment. Bulletin of the Section of Logic, 46(3), 233–262.
Boudard, M., & Hermant, O. (2013). Polarizing double-negation translations. In 19th International conference on logic for programming, artificial intelligence, and reasoning, LPAR-19, Stellenbosch, South Africa, December 14–19, 2013, pp. 182–197.
Brünnler, K. (2009). Deep sequent systems for modal logic. Archive for Mathematical Logic, 48, 551–577.
Bull, R. A. (1992). Cut elimination for propositional dynamic logic wihout *. Zeitschr f math Logik und Grundlagen d Math, 38, 85–100.
de Paiva, V., & Pereira, L. C. (2005). A short note on intuitionistic propositional logic with multiple conclusions. Manuscrito, 28(2), 317–329.
Dowek, G. (2016). On the definition of the classical connectives and quantifiers. Why is this a proof? Festschrift for Luiz Carlos Pereira, 27, 228–238.
Dyckhoff, R. (1992). Contraction-free sequent calculi for intuitionistic logic. Journal of Symbolic Logic, 57(3), 795–807.
Dyckhoff, R., & Negri, S. (2012). Proof analysis in intermediate logics. Archive for Mathematical Logic, 51(1–2), 71–92. https://doi.org/10.1007/s00153-011-0254-7.
Englander, C., Dowek, G., & Haeusler, E. H. (2015). Yet another bijection between sequent calculus and natural deduction. Electronic Notes in Theoretical Computer Science, 312, 107–124. https://doi.org/10.1016/j.entcs.2015.04.007.
Fitting, M. (2014). Nested sequents for intuitionistic logics. Notre Dame Journal of Formal Logic, 55(1), 41–61. https://doi.org/10.1215/00294527-2377869.
Gabbay, D. (1996). Labelled deductive systems. Oxford: Clarendon Press.
Gentzen, G. (1969). The collected papers of gerhard gentzen. Amsterdam: North-Holland Pub. Co.
Girard, J. (1993). On the unity of logic. Annals of Pure and Applied Logic, 59(3), 201–217. https://doi.org/10.1016/0168-0072(93)90093-S.
Girard, J. Y. (1987). Linear logic. Theoretical Computer Science, 50, 1–102.
Ilik, D., Lee, G., & Herbelin, H. (2010). Kripke models for classical logic. Annals of Pure and Applied Logic, 161(11), 1367–1378. https://doi.org/10.1016/j.apal.2010.04.007.
Kashima, R. (1994). Cut-free sequent calculi for some tense logics. Studia Logica, 53(1), 119–136. https://doi.org/10.1007/BF01053026.
Krauss, P. (1992). A constructive interpretation of classical mathematics, Mathematische Schriften Kassel, preprint No. 5/92.
Lellmann, B. (2015). Linear nested sequents, 2-sequents and hypersequents. In TABLEAUX 2015, LNAI, Vol .9323, Springer, pp. 135–150.
Liang, C., & Miller, D. (2011). A focused approach to combining logics. Annals of Pure and Applied Logic, 162(9), 679–697. https://doi.org/10.1016/j.apal.2011.01.012.
Maehara, S. (1954). Eine darstellung der intuitionistischen logik in der klassischen. Nagoya Mathematical Journal, 7, 45–64.
Miller, D., Nadathur, G., Pfenning, F., & Scedrov, A. (1991). Uniform proofs as a foundation for logic programming. Annals of Pure and Applied Logic, 51, 125–157.
Negri, S., & von Plato, J. (2001). Structural proof theory. Cambridge: Cambridge University Press.
Pereira, L. C., & Rodriguez, R. O. (2017). Normalization, soundness and completeness for the propositional fragment of Prawitz’ ecumenical system. Revista Portuguesa de Filosofia, 73(3–3), 1153–1168.
Pimentel, E. (2018). A semantical view of proof systems. In 25th International workshop on logic, language, information, and computation, WoLLIC 2018, Bogota, Colombia, July 24–27, pp. 61–76.
Poggiolesi, F. (2009). The method of tree-hypersequents for modal propositional logic. In Towards mathematical philosophy, trends in logic, Vol. 28, Springer, pp. 31–51.
Prawitz, D. (1965). Natural deduction, In Stockholm studies in philosophy, Vol. 3, Almqvist and Wiksell.
Prawitz, D. (2015). Classical versus intuitionistic logic. Why is this a proof? Festschrift for Luiz Carlos Pereira, 27, 15–32.
Troelstra, A. S., & Schwichtenberg, H. (1996). Basic proof theory. Cambridge: Cambridge University Press.
Viganò, L. (2000). Labelled non-classical logics. Dordrecht: Kluwer.
von Plato, J. (2003). Translations from natural deduction to sequent calculus. Mathematical Logic Quarterly, 49(5), 435–443. https://doi.org/10.1002/malq.200310047.
von Plato, J. (2008). Gentzen’s proof of normalization for intuitionistic natural deduction. Bulletin of Symbolic Logic, 14, 240–244.
Acknowledgements
The authors would like to thank the anonymous reviewers for their valuable comments on earlier drafts of this paper. We would like to thank also Björn Lellmann for the interesting discussions. The work of Pimentel was supported by CNPq, CAPES (via the STIC AmSud project “EPIC: EPistemic Interactive Concurrency”, Proc. No 88881.117603/2016-01) and the project FWF START Y544-N23. The work of Pereira was supported by CNPq and CAPES/COFECUB.
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In memoriam. This work is dedicated to and inspired by the work of Carolina Blasio, who developed, together with João Marcos and Heinrich Wansing, a two-dimensional notion of entailment (Blasio et al. 2017).
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Pimentel, E., Pereira, L.C. & de Paiva, V. An ecumenical notion of entailment. Synthese 198 (Suppl 22), 5391–5413 (2021). https://doi.org/10.1007/s11229-019-02226-5
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DOI: https://doi.org/10.1007/s11229-019-02226-5