Advertisement

Herzberger’s Limit Rule with Labelled Sequent Calculus

  • Andreas FjellstadEmail author
Article

Abstract

Inspired by recent work on proof theory for modal logic, this paper develops a cut-free labelled sequent calculus obtained by imitating Herzberger’s limit rule for revision sequences as a clause in a possible world semantics. With the help of two completeness theorems, one between the labelled sequent calculus and the corresponding possible world semantics, and one between the axiomatic theory of truth PosFS and a neighbourhood semantics, together with the proof of the equivalence between the two semantics, we show that the theory of truth obtained with the labelled sequent calculus based on Herzberger’s limit rule is equivalent to PosFS.

Keywords

Revision theory of truth Labelled sequent calculus Herzberger Possible world semantics Neighbourhood semantics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

For discussions and comments on the material that resulted in this paper, I would like to thank Casper Storm Hansen, Fausto Barbero, Jan von Plato, Maria Hämeen-Anttila, Marianna Girlando, Ole Hjortland, Sara Negri and Tuukka Tanninen. I would also like to thank the anonymous referees for their many helpful suggestions. The research for this paper was supported by a FRIPRO Mobility Fellowship for the project Expressing Validity by Revision: Logical and Philosophical Aspects from the Research Council of Norway (project no 262837/F10, co-funded by the European Union’s Seventh Framework Programme for research, technological development and demonstration under Marie Curie grant agreement no 608695).

References

  1. 1.
    Baaz, M., C. G. Fermüller, G. Salzer, and R. Zach, Labeled calculi and finite-valued logics, Studia Logica 61(1): 7–33, 1998.CrossRefGoogle Scholar
  2. 2.
    Belnap, N. D., Gupta’s rule of revision theory of truth, Journal of Philosophical Logic 11(1): 103–116, 1982.CrossRefGoogle Scholar
  3. 3.
    Friedman, H., and M. Sheard, An axiomatic approach to self-referential truth, Annals of Pure and Applied Logic 33(1): 1–21, 1987.Google Scholar
  4. 4.
    Gentzen, G., Untersuchungen über das logische Schliessen i, ii’ Mathematische Zeitschrift 39: 176–210, 405–431, 1934.Google Scholar
  5. 5.
    Girlando, M., S. Negri, N. Olivetti, and V. Risch, Conditional beliefs: From neighbourhood semantics to sequent calculus, Review of Symbolic Logic 11(4): 736–779, 2018.CrossRefGoogle Scholar
  6. 6.
    Gupta, A., Truth and paradox, Journal of Philosophical Logic 11(1): 1–60, 1982.CrossRefGoogle Scholar
  7. 7.
    Gupta, A., and N. Belnap, The Revision Theory of Truth. MIT Press, 1993.Google Scholar
  8. 8.
    Halbach, V., A system of complete and consistent truth, Notre Dame Journal of Formal Logic 35(1): 311–327, 1994.CrossRefGoogle Scholar
  9. 9.
    Herzberger, H. G., Notes on naive semantics, Journal of Philosophical Logic 11(1): 61–102, 1982.CrossRefGoogle Scholar
  10. 10.
    Horsten, L., G. E. Leigh, H. Leitgeb, and P. Welch, Revision revisited, Review of Symbolic Logic 5(4): 642–664, 2012.CrossRefGoogle Scholar
  11. 11.
    Kremer, M., Kripke and the logic of truth, Journal of Philosophical Logic 17(3): 225–278, 1988.CrossRefGoogle Scholar
  12. 12.
    McGee, V., How truthlike can a predicate be? a negative result, Journal of Philosophical Logic 14(4): 399–410, 1985.CrossRefGoogle Scholar
  13. 13.
    Negri, S., Proof analysis in modal logic, Journal of Philosophical Logic 34: 507–544, 2005.CrossRefGoogle Scholar
  14. 14.
    Negri, S., and G. Sbardolini, Proof analysis for lewis counterfactuals, The Review of Symbolic Logic 9: 44–75, 2016.CrossRefGoogle Scholar
  15. 15.
    Negri, S., and Jan von Plato, Cut elimination in the presence of axioms, Bulletin of Symbolic Logic 4: 418–435, 1998.CrossRefGoogle Scholar
  16. 16.
    Negri, S., and Jan von Plato, Structural Proof Theory. Cambridge University Press, 2001.Google Scholar
  17. 17.
    Negri, S., and Jan von Plato, Proof Analysis: A Contribution to Hilbert’s Last Problem. Cambridge University Press, 2011.Google Scholar
  18. 18.
    Pacuit, E., Neighborhood Semantics for Modal Logic. MIT Press, 2017.Google Scholar
  19. 19.
    Ripley, D., Conservatively extending classical logic with transparent truth, Review of Symbolic Logic 5(2): 354–378, 2012.CrossRefGoogle Scholar
  20. 20.
    Schröder-Heister, P., Restricting initial sequents: The trade-offs between identity, contraction and cut, in R. Kahle, T. Strahm, and T. Studer, (eds.), Advances in Proof Theory,Progress in Computer Science and Applied Logic 28, Birkhäuser, 2016, pp. 339–351.Google Scholar
  21. 21.
    Troelstra, A. S., and H. Schwichtenberg, Basic Proof Theory. Cambridge University Press, 2000.Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of PhilosophyUniversity of BergenBergenNorway

Personalised recommendations