Skip to main content
Log in

An ecumenical notion of entailment

  • S.I.: Varieties of Entailment
  • Published:
Synthese Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Bull, R. A. (1992). Cut elimination for propositional dynamic logic wihout *. Zeitschr f math Logik und Grundlagen d Math, 38, 85–100.

    Article  Google Scholar 

  • de Paiva, V., & Pereira, L. C. (2005). A short note on intuitionistic propositional logic with multiple conclusions. Manuscrito, 28(2), 317–329.

    Google Scholar 

  • 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.

    Google Scholar 

  • Dyckhoff, R. (1992). Contraction-free sequent calculi for intuitionistic logic. Journal of Symbolic Logic, 57(3), 795–807.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Gabbay, D. (1996). Labelled deductive systems. Oxford: Clarendon Press.

    Google Scholar 

  • Gentzen, G. (1969). The collected papers of gerhard gentzen. Amsterdam: North-Holland Pub. Co.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Girard, J. Y. (1987). Linear logic. Theoretical Computer Science, 50, 1–102.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Kashima, R. (1994). Cut-free sequent calculi for some tense logics. Studia Logica, 53(1), 119–136. https://doi.org/10.1007/BF01053026.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Maehara, S. (1954). Eine darstellung der intuitionistischen logik in der klassischen. Nagoya Mathematical Journal, 7, 45–64.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Negri, S., & von Plato, J. (2001). Structural proof theory. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Google Scholar 

  • Troelstra, A. S., & Schwichtenberg, H. (1996). Basic proof theory. Cambridge: Cambridge University Press.

    Google Scholar 

  • Viganò, L. (2000). Labelled non-classical logics. Dordrecht: Kluwer.

    Book  Google Scholar 

  • 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.

    Article  Google Scholar 

  • von Plato, J. (2008). Gentzen’s proof of normalization for intuitionistic natural deduction. Bulletin of Symbolic Logic, 14, 240–244.

    Article  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Elaine Pimentel.

Additional information

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).

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11229-019-02226-5

Keywords

Navigation