Notre Dame Journal of Formal Logic

Nested Sequents for Intuitionistic Logics

Melvin Fitting

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Relatively recently nested sequent systems for modal logics have come to be seen as an attractive deep-reasoning extension of familiar sequent calculi. In an earlier paper I showed that there was a strong connection between modal nested sequents and modal prefixed tableaux. In this paper I show that the connection continues to intuitionistic logic, both standard and constant domain, relating nested intuitionistic sequent calculi to intuitionistic prefixed tableaux. Modal nested sequent machinery generalizes one-sided sequent calculi—intuitionistic nested sequents similarly generalize two-sided sequents. It is noteworthy that the resulting system for constant domain intuitionistic logic is particularly simple. It amounts to a combination of intuitionistic propositional rules and classical quantifier rules, a combination that is known to be inadequate when conventional intuitionistic sequent systems are used.

Article information

Source
Notre Dame J. Formal Logic, Volume 55, Number 1 (2014), 41-61.

Dates
First available in Project Euclid: 20 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1390246437

Digital Object Identifier
doi:10.1215/00294527-2377869

Mathematical Reviews number (MathSciNet)
MR3161411

Zentralblatt MATH identifier
1327.03006

Subjects
Primary: 03B20: Subsystems of classical logic (including intuitionistic logic)
Secondary: 03B44: Temporal logic 03B60: Other nonclassical logic 68T15: Theorem proving (deduction, resolution, etc.) [See also 03B35]

Keywords
intuitionistic logic constant domain logic sequent nested sequent tableau prefixed tableau

Citation

Fitting, Melvin. Nested Sequents for Intuitionistic Logics. Notre Dame J. Formal Logic 55 (2014), no. 1, 41--61. doi:10.1215/00294527-2377869. https://projecteuclid.org/euclid.ndjfl/1390246437


Export citation

References

  • [1] Brünnler, K., “Deep sequent systems for modal logic,” Archive for Mathematical Logic, vol. 48 (2009), pp. 551–77.
  • [2] Brünnler, K., “Nested Sequents,” Habilitation thesis, Institut für Informatik und angewandte Mathematik, Universität Bern, 2010.
  • [3] Došen, K., “Sequent-Systems for modal logic,” Journal of Symbolic Logic, vol. 50 (1985), pp. 149–68.
  • [4] Fiorentini, C., and P. Miglioli, “A cut-free sequent calculus for the logic of constant domains with a limited amount of duplications,” Logic Journal of the IGPL, vol. 7 (1999), pp. 733–53.
  • [5] Fitch, F., “Tree proofs in modal logic,” Journal of Symbolic Logic, vol. 31 (1966), p. 152. (abstract).
  • [6] Fitting, M. C., “Tableau methods of proof for modal logics,” Notre Dame Journal of Formal Logic, vol. 13 (1972), pp. 237–47.
  • [7] Fitting, M. C., Proof Methods for Modal and Intuitionistic Logics, vol. 169of Synthese Library, D. Reidel Publishing, Dordrecht, 1983.
  • [8] Fitting, M. C., “A mistake on my part,” pp. 665–69 in We Will Show Them! Essays in honour of Dov Gabbay, Vol. I, edited by S. Artemov, H. Berringer, A. d’Avila Garcez, L. Lamb, and J. Woods, College Publications, London, 2005.
  • [9] Fitting, M. C., “Modal proof theory,” pp. 85–138 in Handbook of Modal Logic, edited by P. Blackburn, J. van Benthem, and F. Wolter, vol. 3 of Studies in Logic and Practical Reasoning, Elsevier, Amsterdam, 2007.
  • [10] Fitting, M. C., “Prefixed tableaus and nested sequents,” Annals of Pure and Applied Logic, vol. 163 (2012), pp. 291–313.
  • [11] Goré, R., L. Postniece, and A. Tiu, “Cut-elimination and proof-search for bi-intuitionistic logic using nested sequents,” pp. 43–66 in Advances in Modal Logic, Vol. 7, College Publications, London, 2008.
  • [12] Görneman, S., “A logic stronger than intuitionism,” Journal of Symbolic Logic, vol. 36 (1971), pp. 249–61.
  • [13] Grzegorczyk, A., “A philosophically plausible formal interpretation of intuitionistic logic,” Indagationes Mathematicae, vol. 26 (1964), pp. 596–601.
  • [14] Kashima, R., “Cut-free sequent calculi for some tense logics,” Studia Logica, vol. 53 (1994), pp. 119–35.
  • [15] Kashima, R., and T. Shimura, “Cut-elimination theorem for the logic of constant domains,” Mathematical Logic Quarterly, vol. 40 (1994), pp. 153–72.
  • [16] Klemke, D., Ein vollständiger Kalkül für die Folgerungsbeziehung der Grzegorcyk-Semantik, Ph.D. dissertation, Freiburg University, Freiburg, Germany, 1969.
  • [17] Mints, G., G. Olkhovikov, and A. Urquhart, “Failure of interpolation in the intuitionistic logic of constant domains,” preprint, arXiv:1202.3519v3 [math.LO].
  • [18] Poggiolesi, F., “The method of tree-hypersequents for modal propositional logic,” pp. 31–51 in Trends in Logic: Towards Mathematical Philosophy, edited by D. Makinson, J. Malinowski, and H. Wansing, vol. 28 of Trends in Logic, Studia Logica Library, Springer, Dordrecht, 2009.
  • [19] Postniece, L., “Deep inference in bi-intuitionistic logic,” pp. 320–34 in Logic, Language, Information, and Computation (Tokyo, 2009), edited by H. Ono, M. Kanazawa, and R. de Queiroz, vol. 5514 of Lecture Notes in Computer Science, Springer, Berlin, 2009.
  • [20] Sambin, G., G. Battilotti, and C. Faggian, “Basic logic: Reflection, symmetry, visibility,” Journal of Symbolic Logic, vol. 65 (2000), pp. 979–1013.
  • [21] Scott, D., “Identity and existence in intuitionistic logic,” pp. 660–96 in Applications of Sheaves, edited by F. Michael, M. Christopher, and S. Dana, vol. 753 of Lecture Notes in Mathematics, Springer, Berlin, 1979.