Proof Theory for Functional Modal Logic

Abstract

We present some proof-theoretic results for the normal modal logic whose characteristic axiom is \(\mathord {\sim }\mathord {\Box }A\equiv \mathord {\Box }\mathord {\sim }A\). We present a sequent system for this logic and a hypersequent system for its first-order form and show that these are equivalent to Hilbert-style axiomatizations. We show that the question of validity for these logics reduces to that of classical tautologyhood and first-order logical truth, respectively. We close by proving equivalences with a Fitch-style proof system for revision theory.

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

References

  1. 1.

    Anderson, A.R., and N.D. Belnap, Entailment: The Logic of Relevance and Necessity, vol. 1. Princeton University Press, Princeton, 1975.

  2. 2.

    Antonelli, G.A., A revision-theoretic analysis of the arithmetical hierarchy. Notre Dame Journal of Formal Logic 35(2):204–218, 1994.

    Article  Google Scholar 

  3. 3.

    Avron, A., A constructive analysis of RM. Journal of Symbolic Logic 52(4):939–951, 1987.

    Article  Google Scholar 

  4. 4.

    Baillot, P., and D. Mazza, Linear logic by levels and bounded time complexity. Theoretical Computer Science 411(2):470–503, 2010.

    Article  Google Scholar 

  5. 5.

    Belnap, N.D., Tonk, plonk and plink. Analysis 22(6):130–134, 1962.

    Article  Google Scholar 

  6. 6.

    Bimbó, K., Proof theory: Sequent calculi and related formalisms. CRC Press, Boca Raton, 2014.

    Google Scholar 

  7. 7.

    Blackburn, P., M. de Rijke, and Y. Venema, Modal Logic. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge, 2002.

  8. 8.

    Boudes, P., D. Mazza, and L.T. de Falco, An abstract approach to stratification in linear logic. Information and Computation 241:32–61, 2015.

    Article  Google Scholar 

  9. 9.

    Bruni, R., Analytic calculi for circular concepts by finite revision. Studia Logica 101(5):915–932, 2013.

    Article  Google Scholar 

  10. 10.

    Chellas, B.F., Modal Logic: An Introduction. Cambridge University Press, Cambridge, 1980.

    Google Scholar 

  11. 11.

    Curry, H.B., Foundations of Mathematical Logic. Dover Publications, Mineola, 1963.

    Google Scholar 

  12. 12.

    Davies, M., and L. Humberstone, Two notions of necessity. Philosophical Studies 38(1):1–31, 1980.

    Article  Google Scholar 

  13. 13.

    Dunn, J.M., Positive modal logic. Studia Logica 55(2):301–317, 1995.

    Article  Google Scholar 

  14. 14.

    Dunn, M., and G. Restall, Relevance logic, in D. Gabbay, and F. Guenthner, (eds.), Handbook of Philosophical Logic. Kluwer, Alphen aan den Rijn, 2002.

  15. 15.

    Fitting, M., Nested sequents for intuitionistic logics. Notre Dame Journal of Formal Logic 55(1):41–61, 2014.

    Article  Google Scholar 

  16. 16.

    Fitting, M.C., V.W. Marek, and M. Truszczyński, The pure logic of necessitation. Journal of Logic and Computation 2(3):349–373, 1992.

  17. 17.

    French, R., and D. Ripley, Contractions of noncontractive consequence relations. Review of Symbolic Logic 8(3):506–528, 2015.

    Article  Google Scholar 

  18. 18.

    Gabbay D.M., A. Kurucz, F. Wolter, and M. Zakharyaschev, Many-Dimensional Modal Logics: Theory and Applications. Elsevier, London, 2003.

  19. 19.

    Garson, J.W., Modal Logic for Philosophers. 2nd edn, Cambridge University Press, Cambridge, 2013.

    Google Scholar 

  20. 20.

    Gentzen, G., Investigations into logical deduction. American Philosophical Quarterly 1(4):288–306, 1964.

  21. 21.

    Gupta, A., Remarks on definitions and the concept of truth. Proceedings of the Aristotelian Society 89:227–246, 1988–89. Reprinted in Truth, Meaning, Experience. Oxford University Press, 2011, pp. 73–94.

  22. 22.

    Gupta, A., and N. Belnap, The Revision Theory of Truth. MIT Press, Cambridge, 1993.

  23. 23.

    Gupta, A., and S. Standefer, Conditionals in theories of truth. Journal of Philosophical Logic, pp. 1–37. Forthcoming, 2016.

  24. 24.

    Hughes, G.E., and M.J. Cresswell, A New Introduction to Modal Logic. Routledge, Abingdon, 1996.

  25. 25.

    Humberstone, L., Smiley’s distinction between rules of inference and rules of proof, in T.J. Smiley, J. Lear and A. Oliver (eds.), The Force of Argument: Essays in Honor of Timothy Smiley, Routledge, Abingdon, 2010, pp. 107–126.

  26. 26.

    Indrzejczak, A., Linear time in hypersequent framework. Bulletin of Symbolic Logic 22(1):121–144, 2016.

    Article  Google Scholar 

  27. 27.

    Kawai, H., Sequential calculus for a first order infinitary temporal logic. Mathematical Logic Quarterly 33(5):423–432, 1987.

  28. 28.

    Lahav, O., and Y. Zohar, SAT-Based Decision Procedure for Analytic Pure Sequent Calculi, Springer International Publishing, Cham, 2014, pp. 76–90.

  29. 29.

    Lellmann, B., Linear nested sequents, 2-sequents and hypersequents, in H. De Nivelle, (ed.), Automated Reasoning with Analytic Tableaux and Related Methods: 24th International Conference, TABLEAUX 2015, Wroclaw, Poland, September 21-24, Springer International Publishing, Cham, 2015, pp. 135–150.

  30. 30.

    Masini, A., 2-sequent calculus: A proof theory of modalities. Annals of Pure and Applied Logic 58(3):229–246, 1992.

    Article  Google Scholar 

  31. 31.

    Merz, S., Decidability and incompleteness results for first-order temporal logics of linear time. Journal of Applied Non-Classical Logics 2(2):139–156, 1992.

  32. 32.

    Minc, G., On Some Calculi of Modal Logic, in V. Orevkov, (ed.), The Calculi of Symbolic Logic. I., vol. 98, pp. 97–124. American Mathematical Society. Originally published in Russian in 1968 in Proceedings of the Steklov Institute of Mathematics, edited by I.G. Petrovskii and S.M. Nikol’skii, 1971.

  33. 33.

    Negri, S., and J. von Plato, Structural Proof Theory. Cambridge University Press, Cambridge, 2001.

  34. 34.

    Parsons, J., Command and consequence. Philosophical Studies 164(1):61–92, 2013.

  35. 35.

    Poggiolesi, F., Gentzen Calculi for Modal Propositional Logic. Springer, Berlin, 2010.

    Google Scholar 

  36. 36.

    Pottinger, G., Uniform cut-free formulations of T, S4 and S5 (abstract). Journal of Symbolic Logic 48(3):900–901, 1983.

  37. 37.

    Rescher, N., and A. Urquhart, Temporal Logic. Springer, Berlin, 1971.

  38. 38.

    Restall, G., Negation in relevant logics (how I stopped worrying and learned to love the Routley star), in D.M. Gabbay, and H. Wansing, (eds.), What is Negation?, Kluwer Academic Publishers, Dordrecht, 1999, pp. 53–76.

  39. 39.

    Restall, G., An Introduction to Substructural Logics. Routledge, Abingdon, 2000.

    Google Scholar 

  40. 40.

    Restall, G., A cut-free sequent system for two-dimensional modal logic, and why it matters. Annals of Pure and Applied Logic 163(11):1611–1623, 2012.

    Article  Google Scholar 

  41. 41.

    Segerberg, K., Modal logics with functional alternative relations. Notre Dame Journal of Formal Logic 27(4):504–522, 1986.

    Article  Google Scholar 

  42. 42.

    Standefer, S., Solovay-type theorems for circular definitions. Review of Symbolic Logic 8(3):467–487, 2015.

  43. 43.

    Troelstra, A.S., and H. Schwichtenberg, Basic Proof Theory. Cambridge University Press, Cambridge, 2000.

  44. 44.

    Wansing, H., Displaying Modal Logic. Kluwer, Dordrecht, 1998.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Shawn Standefer.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Standefer, S. Proof Theory for Functional Modal Logic. Stud Logica 106, 49–84 (2018). https://doi.org/10.1007/s11225-017-9725-0

Download citation

Keywords

  • Functional modal logic
  • Hypersequents
  • Revision theory
  • Proof theory