Studia Logica

, Volume 107, Issue 1, pp 85–107 | Cite as

Logicality, Double-Line Rules, and Modalities

  • Norbert Gratzl
  • Eugenio OrlandelliEmail author


This paper deals with the question of the logicality of modal logics from a proof-theoretic perspective. It is argued that if Dos̆en’s analysis of logical constants as punctuation marks is embraced, it is possible to show that all the modalities in the cube of normal modal logics are indeed logical constants. It will be proved that the display calculus for each displayable modality admits a purely structural presentation based on double-line rules which, following Dos̆en’s analysis, allows us to claim that the corresponding modal operators are logical constants.


Normal modalities Logicality Double-line rules Proof-theoretic semantics Display calculi 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



Thanks are due to two anonymous referees and to the audience at the 7th Conference on Non-Classical Logic in Toruń and at the conference General Proof Theory in Tübingen.


  1. 1.
    Avron, A., Simple consequence relations, Inf. Comput. 92(1):105–139, 1991.CrossRefGoogle Scholar
  2. 2.
    Belnap, N.D., Display logic, Journal of Philosophical Logic 11(4):375–417, 1982.CrossRefGoogle Scholar
  3. 3.
    Belnap, N.D., The display problem, in H. Wansing, (ed.), Proof Theory of Modal Logic, Springer Netherlands, Dordrecht, 1996, pp. 79–92.CrossRefGoogle Scholar
  4. 4.
    Binder, D., and T. Piecha, Popper’s notion of duality and his theory of negations, History and Philosophy of Logic 38(2):154–189, 2017.CrossRefGoogle Scholar
  5. 5.
    Bonnay, D., and B. Simmenauer, Tonk strikes back, Australasian Journal of Logic 3:33–44, 2005.CrossRefGoogle Scholar
  6. 6.
    Ciabattoni, A., and R. Ramanayake, Power and limits of structural display rules, ACM Trans. Comput. Logic 17(3):17:1–17:39, 2016.CrossRefGoogle Scholar
  7. 7.
    Dalla Chiara, M. L., and R. Giuntini, Quantum logics, in D.M. Gabbay, and F. Guenthner, (eds.), Philosophical Logic, vol. VI, Springer, Netherlands, Dordrecht, 2002, pp. 129–228.Google Scholar
  8. 8.
    Dicher, B., Weak disharmony: Some lessons for proof-theoretic semantics, Review of Symbolic Logic 9(3):583–602, 2016.CrossRefGoogle Scholar
  9. 9.
    Dos̆en, K., Sequent-systems for modal logic, Journal of Symbolic Logic 50(1):149–168, 1985.CrossRefGoogle Scholar
  10. 10.
    Dos̆en, K., Logical constants as punctuation marks, Notre Dame Journal of Formal Logic 30:362–381, 1989.CrossRefGoogle Scholar
  11. 11.
    Garson, J., Modal logic, in E.N. Zalta, (ed.), The Stanford Encyclopedia of Philosophy, spring 2016 edn., Metaphysics Research Lab, Stanford University, 2016.Google Scholar
  12. 12.
    Goré, R., Gaggles, Gentzen and Galois: How to display your favourite substructural logic, Logic Journal of the IGPL 6(5):669–694, 1998.CrossRefGoogle Scholar
  13. 13.
    Gratzl, N., and E. Orlandelli, Double-line harmony in sequent calculi, in P. Arazim, and T. Lavicka, (eds.), The Logica Yearbook 2016, College Publications, 2017, pp. 157–171.Google Scholar
  14. 14.
    Hacking, I., What is logic?, The Journal of Philosophy 76:285–319, 1979.CrossRefGoogle Scholar
  15. 15.
    Kürbis, N., Proof-theoretic semantics, a problem with negation and prospects for modalities, Journal of Philosophical Logic 44:713–727, 2015.CrossRefGoogle Scholar
  16. 16.
    Kracht, M., Power and weakness of the modal display calculus, in H. Wansing, (ed.), Proof Theory of Modal Logic, Kluwer, 1996, pp. 93–121.Google Scholar
  17. 17.
    MacFarlane, J., Logical constants, in E.N. Zalta, (ed.), The Stanford Encyclopedia of Philosophy, fall 2015 edn., Metaphysics Research Lab, Stanford University, 2015.Google Scholar
  18. 18.
    Martin-Löf, P., Hauptsatz for the intuitionistic theory of iterated inductive definitions, in J.E. Fenstad, (ed.), Proceedings of the Second Scandinavian Logic Symposium, vol. 63, Elsevier, 1971, pp. 179–216.Google Scholar
  19. 19.
    Naibo, A., and M. Petrolo, Are uniqueness and deducibility of identicals the same?, Theoria 81:143–181, 2015.CrossRefGoogle Scholar
  20. 20.
    Negri, S., Proof analysis in modal logics, Journal of Philosophical Logic 33:507–544, 2005.CrossRefGoogle Scholar
  21. 21.
    Negri, S., and J. von Plato, Structural Proof Theory, Cambridge University Press, 2001.Google Scholar
  22. 22.
    Poggiolesi, F., Display calculi and other modal calculi: A comparison, Synthese 173(3):259–279, 2010.CrossRefGoogle Scholar
  23. 23.
    Popper, K.R., Functional logic without axioms or primitive rules of inference, Koninklijke Nederlandsche Akademie van Wetenschappen, Proceedings of the Section of Sciences 50:1214–1224, 1947.Google Scholar
  24. 24.
    Popper, K.R., Logic without assumptions, Proceedings of the Aristotelian Society 47:251–292, 1947.CrossRefGoogle Scholar
  25. 25.
    Popper, K.R., New foundations for logic, Mind 56:193–235, 1947.CrossRefGoogle Scholar
  26. 26.
    Prawitz, D., Natural Deduction: A Proof-Theoretical Study, Almqvist & Wiskell, 1965.Google Scholar
  27. 27.
    Read, S., Harmony and modality, in C. Dégremont, (ed.), Dialogues, Logics and Other Strange Things: Essays in Honour of Shahid Rahman, College Publications, 2008, pp. 285–303.Google Scholar
  28. 28.
    Sambin, G., G. Battilotti, and C. Faggian, Basic logic, Journal of Symbolic Logic 65:979–1013, 2000.CrossRefGoogle Scholar
  29. 29.
    Schroeder-Heister, P., Popper’s theory of deductive inference and the concept of a logical constant, History and Philosophy of Logic 5:79–110, 1984.CrossRefGoogle Scholar
  30. 30.
    Schroeder-Heister, P., Proof-theoretic semantics, self-contradiction, and the format of deductive reasoning, Topoi 31(1):77–85, 2012.CrossRefGoogle Scholar
  31. 31.
    Wansing, H., Sequent calculi for normal modal propositional logics, Journal of Logic and Computation 4(2):125–142, 1994.CrossRefGoogle Scholar
  32. 32.
    Wansing, H., Displaying Modal Logic, Kluwer, 1998.Google Scholar
  33. 33.
    Wolenski, J., First-order logic: (Philosophical) pro and contra, in V.F. Hendricks, (ed.), First-Order Logic Revisited, Logos, 2004, pp. 369–398.Google Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Munich Center for Mathematical PhilosophyLudwig-Maximilians-UniversitätMünchenGermany
  2. 2.Dipartimento di Filosofia e ComunicazioneUniversità di BolognaBolognaItaly

Personalised recommendations