Skip to main content
SpringerLink
Log in

Combining Intuitionistic and Classical Propositional Logic: Gentzenization and Craig Interpolation

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

This paper studies a combined system of intuitionistic and classical propositional logic from proof-theoretic viewpoints. Based on the semantic treatment of Humberstone (J Philos Log 8:171–196, 1979) and del Cerro and Herzig (Frontiers of combining systems: FroCoS, Springer, 1996), a sequent calculus \(\textsf{G}(\textbf{C}+\textbf{J})\) is proposed. An approximate idea of obtaining \(\textsf{G}(\textbf{C}+\textbf{J})\) is adding rules for classical implication on top of the intuitionistic multi-succedent sequent calculus by Maehara (Nagoya Math J 7:45–64, 1954). However, in the semantic treatment, some formulas do not satisfy heredity, which leads to the necessity of a restriction on the right rule for intuitionistic implication to keep the soundness of the calculus. The calculus \(\textsf{G}(\textbf{C}+\textbf{J})\) enjoys cut elimination and Craig interpolation, whose detailed proofs are described in this paper. Cut elimination enables us to show the decidability of this combination both directly and syntactically. This paper also employs a canonical model argument to establish the strong completeness of Hilbert system \(\mathbf {C+J}\) proposed by del Cerro and Herzig (Frontiers of combining systems: FroCoS, Springer, 1996).

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Humberstone, L.: Interval semantics for tense logic: some remarks. Journal of Philosophical Logic 8, 171–196 (1979). https://doi.org/10.1007/BF00258426

    Article  Google Scholar 

  2. del Cerro, L.F., A. Herzig, Combining classical and intuitionistic logic or: Intuitionistic implication as a conditional. In: Badder, F., and K.U. Schulz, (eds.) Frontiers of Combining Systems: FroCoS 1996, pp. 93–102. Springer, (1996). https://doi.org/10.1007/978-94-009-0349-4_4

  3. Toyooka, M., K. Sano, Analytic multi-succedent sequent calculus for combining intuitionistic and classical propositional logic. In: Ghosh, S., and R. Ramanujam, (eds.) ICLA 2021 Proceedings: 9th Indian Conference on Logic and Its Applications, pp. 128–133 (2021)

  4. Popper, K. On the Theory of Deduction, Part II. The Definitions of Classical and Intuitionist Negation (1948). In: Binder, D., T. Piecha, and P. Schroeder-Heister, (eds.) The Logical Writings of Karl Popper, pp. 181–192. Springer, Cham (2022). https://doi.org/10.1007/978-3-030-94926-6_6.

  5. Williamson, T., Equivocation and existence. In: Proceedings of the Aristotelian Society vol. 88, pp. 109–127 (1988). https://doi.org/10.1093/aristotelian/88.1.109

  6. Gabbay, D.M., An overview of fibered semantics and the combination of logics. In: Badder, F., and K.U. Schulz, (eds.) Frontiers of Combining Systems: FroCoS 1996. Applied Logic Series (APLS), vol. 3, pp. 1–55. Springer, (1996). https://doi.org/10.1007/978-94-009-0349-4_1

  7. Caleiro, C., and J. Ramos, Combining classical and intuitionistic implications. In: Konev, B., and F. Wolter, (eds.) Frontiers of Combining Systems. FroCoS 2007. Lecture Notes in Computer Science, pp. 118–132. Springer, (2007). https://doi.org/10.1007/978-3-540-74621-8_8

  8. Kurokawa, H. Intuitionistic logic with classical atoms. CUNY Ph.D. Program in Computer Science Technical Report: TR-2004003 (2004)

  9. Kurokawa, H.: Hypersequent calculi for intuitionistic logic with classical atoms. Annals of Pure and Applied Logic 161(3):427–446, 2009. https://doi.org/10.1016/j.apal.2009.07.013

    Article  Google Scholar 

  10. Liang, C., and D. Miller, A focused approach to combining logics. Annals of Pure and Applied Logic 162:679–697, 2011. https://doi.org/10.1016/j.apal.2011.01.012

    Article  Google Scholar 

  11. Liang, C., and D. Miller, Kripke semantics and proof systems for combining intuitionistic logic and classical logic. Annals of Pure and Applied Logic 164(2):86–111, 2013. https://doi.org/10.1016/j.apal.2012.09.005

    Article  Google Scholar 

  12. De, M.: Empirical negation. Acta Analytica: International Periodical for Philosophy in the Analytical Tradition 28(1):49–69, 2013. https://doi.org/10.1007/s12136-011-0138-9

    Article  Google Scholar 

  13. De, M., and H. Omori, H.: More on empirical negation. In: Goré, R., Kooi, B., and Kurucz, A. (eds.) Advances in Modal Logic vol. 10, pp. 114–133. College Publications, (2014)

  14. Prawitz, D.: Classical versus intuitionistic logic. In: Haeusler, E.H., E. de Campos Sanz, and B. Lopes, (eds.) Why Is this a Proof?, pp. 15–32. College Publications, 2015

  15. Pereira, L.C., Rodriguez, R.O.: Normalization, soundness and completeness for the propositional fragment of Prawitz’ ecumenical system. Revista Portuguesa de Filosofia 73(3/4):1153–1168 2017

    Article  Google Scholar 

  16. Pimentel, E., L.C. Pereira, and V. de Paiva, A proof theoretical view of ecumenical systems. In: Proceedings of the 13th Workshop on Logical and Semantic Framework with Applications, pp. 109–114 2018

  17. Pimentel, E., L.C. Pereira, and V. de Paiva, An ecumenical notion of entailment. Synthese 2019. https://doi.org/10.1007/s11229-019-02226-5

  18. Dowek, G.: On the definition of the classical connectives and quantifiers. In: Haeusler, E.H., W. de Campos Sanz, and B. Lopes, (eds.) Why Is this a Proof?, pp. 228–238. College Publications, 2015

  19. Niki, S., H. Omori, A note on Humberstone’s constant \(\Omega \). Reports on Mathematical Logic 56:75–99 2021. https://doi.org/10.4467/20842589RM.21.006.14376

    Article  Google Scholar 

  20. Niki, S., H. Omori, Another combination of classical and intuitionistic conditionals. In: Indrzejczak, A., and M. Zawidzki, (eds.) Proceeding of the 10th International Conference on Non-Classical Logics. Theory and Applications. Electronic Proceedings in Theoretical Computer Science (EPTCS), vol. 358, pp. 174–188 (2022). https://doi.org/10.4204/EPTCS.358.13

  21. Blanqui, F., G. Dowek, É., Grienenberger, G. Hondet, F. Thiré, Some axioms for mathematics. In: Kobayashi, N. (ed.) 6th International Conference on Formal Structures for Computation and Deduction, FSCD 2021, Buenos Aires, Argentina (Virtual Conference). Leibniz International Proceedings in Informatics (LIPIcs), vol. 195, pp. 20–12019. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, (2021). https://doi.org/10.4230/LIPIcs.FSCD.2021.20

  22. Krauss, P.: A constructive interpretation of classical mathematics. Mathematische Schriften Kassel, preprint No. 5/92 1992

  23. Moschovakis, J.R. The logic of Brouwer and Heyting. In: Gabbay, D.M., and J. Woods, J. (eds.) Handbook of the History of Logic Volume 5: Logic from Russell to Church, pp. 77–125. North-Holland, (2009)

  24. De, M., and H. Omori, Classical and empirical negation in subintuitionistic logic. In: Beklemishev, L., S. Demri, and A. Máté, (eds.) Advances in Modal Logic, Volume11, pp. 217–235. CSLI Publications, (2016)

  25. Kashima, R.: Sequent calculi of non-classical logics - proofs of completeness theorems by sequent calculi. In: Proceedings of Mathematical Society of Japan Annual Colloquium of Foundations of Mathematics (in Japanese), pp. 49–67 (1999)

  26. Lucio, P.: Structured sequent calculi for combining intuitionistic and classical first-order logic. In: Kirchner, H., and Ringeissen, C. (eds.) Frontiers of Combining Systems. FroCoS 2000, pp. 88–104. Springer, (2000). https://doi.org/10.1007/10720084_7

  27. Maehara, S.: Eine Darstellung der intuitionistischen Logik in der klassischen. Nagoya Mathematical Journal 7:45–64, 1954. https://doi.org/10.1017/S0027763000018055

    Article  Google Scholar 

  28. Maehara, S.: Craig’s interpolation theorem (in Japanese). Sûgaku 12(4):235–237, 1961. https://doi.org/10.11429/sugaku1947.12.235

  29. Chellas, B.F.: Basic conditional logic. Journal of Philosophical Logic 4: 133–153, 1975. https://doi.org/10.1007/BF00693270

    Article  Google Scholar 

  30. Chellas, B.F. Conditional logic. In: Modal Logic: An Introduction, pp. 268–276. Cambridge University Press, (1980). https://doi.org/10.1017/CBO9780511621192.011

  31. Toyooka, M., and K. Sano, Semantic incompleteness of Hilbert system for a combination of classical and intuitionistic propositional logic. arXiv (2022). https://doi.org/10.48550/ARXIV.2207.07416

  32. Restall, G.: Subintuitionistic logics. Notre Dame Journal of Formal Logic 35(1):116–129, 1994. https://doi.org/10.1305/ndjfl/1040609299

    Article  Google Scholar 

  33. van Dalen, D.: Logic and Structure, Fifth edn. Springer, (2013)

    Book  Google Scholar 

  34. Kripke, S.A.: The problem of entailment. The Journal of Symbolic Logic 24(4):324, 1959

    Google Scholar 

  35. Ishigaki, R. and R. Kashima, Sequent calculi for some strict implication logics. Logic Journal of the IGPL 16(2):155–174 (2008). https://doi.org/10.1093/jigpal/jzm058

    Article  Google Scholar 

  36. Kashima, R. Mathematical Logic (in Japanese). Asakura Publishing Company, (2009)

  37. Ono, H. Proof Theory and Algebra in Logic, 1st edn. Springer, (2019)

  38. Ono, H., Y. Komori, Logics without the contraction rule. The Journal of Symbolic Logic 50(1):169–201 (1985). https://doi.org/10.2307/2273798

    Article  Google Scholar 

  39. Troelstra, A.S., H. Schwichtenberg, Basic Proof Theory, 2nd edn. Cambridge University Press, (2012). https://doi.org/10.1017/CBO9781139168717

  40. Ono, H.: Proof-theoretic methods in nonclassical logic- an introduction. Mathematical Society of Japan Memoirs 2:207–254, 1998. https://doi.org/10.2969/msjmemoirs/00201C060

    Article  Google Scholar 

  41. Schütte, K.: Der Interpolationssatz der intuitionistischen Prädikatenlogik. Mathematische Annalen 148:192–200, 1962. https://doi.org/10.1007/BF01470747

    Article  Google Scholar 

  42. Mints, G. Interpolation theorems for intuitionistic predicate logic. Annals of Pure and Applied Logic 113(1-3):225–242, 2001. https://doi.org/10.1016/S0168-0072(01)00060-4

    Article  Google Scholar 

  43. Kowalski, T., H. Ono, Analytic cut and interpolation for bi-intuitionistic logic. The Review of Symbolic Logic 10(2):259–283, 2017. https://doi.org/10.1017/S175502031600040X

    Article  Google Scholar 

  44. Ohnishi, M., K. Matsumoto, Gentzen method in modal calculi. Osaka Mathematical Journal 9(2):113–130, 1957

    Google Scholar 

  45. Parikh, R., L.S. Moss, and C. Steinsvold, Topology and epistemic logic. In: Aiello, M., Pratt-Hartmann, I.E., and van Benthem, J. (eds.) Handbook of Spatial Logics, pp. 299–341. Springer, (2007). Chap. 6

  46. Gabbay, D.M. Intuitionistic basis for non-monotonic logic. In: Loveland, D.W. (ed.) 6th Conference on Automated Deduction: New York, USA, June 7-9, 1982. Lecture Notes in Computer Science (LNCS), vol. 138, pp. 260–273. Springer, (1982). https://doi.org/10.1007/BFb0000064

  47. Pitts, A.M. On an interpretation of second order quantification in first order intuitionistic propositional logic. The Journal of Symbolic Logic 57(1):33–52, 1992. https://doi.org/10.2307/2275175

    Article  Google Scholar 

  48. Ghilardi, S., and M. Zawadowski, Undefinability of propositional quantifiers in the modal system S4. Studia Logica 55:259–271, 1995. https://doi.org/10.1007/BF01061237

    Article  Google Scholar 

  49. Maksimova, L.L. The Lyndon interpolation theorem in modal logics. Matematicheskie Trudy 2:45–55, 1982

    Google Scholar 

  50. Fitting, M. Proof Methods for Modal and Intuitionistic Logics. Synthese Library book series (SYLI), vol. 169. Springer, (1983)

  51. Aboolian, N., and M. Alizadeh, Lyndon’s interpolation property for the logic of strict implication. Logic Journal of the IGPL 30(1):34–70, 2022. https://doi.org/10.1093/jigpal/jzaa029

    Article  Google Scholar 

  52. Toyooka, M., and K. Sano, Combining first-order classical and intuitionistic logic. In: Indrzejczak, A., and M. Zawidzki,(eds.) Proceeding of the 10th International Conference on Non-Classical Logics. Theory and Applications. Electronic Proceedings in Theoretical Computer Science (EPTCS), vol. 358, pp. 25–40 (2022). https://doi.org/10.4204/EPTCS.358.3

Download references

Acknowledgements

We would like to thank Rineke Verbrugge and R. Ramanujam for their comments at ICLA2021: 9th Indian Conference on Logic and its Applications. In addition, two anonymous referees of this paper for giving us very helpful comments. The work of the first author is partially supported by Grant-in-Aid for JSPS Fellows Grant Number JP22J20341. The work of the second author was partially supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (B) Grant Number JP 22H00597 and (C) Grant Number JP 19K12113.

Author information

Authors and Affiliations

  1. Graduate School of Humanities and Human Sciences, Hokkaido University, Sapporo, Japan

    Masanobu Toyooka

  2. Faculty of Humanities and Human Sciences, Hokkaido University, Sapporo, Japan

    Katsuhiko Sano

Authors
  1. Masanobu Toyooka
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Katsuhiko Sano
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Masanobu Toyooka.

Additional information

Publisher's Note

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

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.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Toyooka, M., Sano, K. Combining Intuitionistic and Classical Propositional Logic: Gentzenization and Craig Interpolation. Stud Logica (2024). https://doi.org/10.1007/s11225-023-10067-0

Download citation

  • Received:

  • Published:

  • DOI: https://doi.org/10.1007/s11225-023-10067-0

Keywords

Search

Navigation