Skip to main content
SpringerLink
Log in

Constructing Cut Free Sequent Systems with Context Restrictions Based on Classical or Intuitionistic Logic

  • Conference paper
Logic and Its Applications (ICLA 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7750))

Included in the following conference series:

Abstract

We consider a general format for sequent rules for not necessarily normal modal logics based on classical or intuitionistic propositional logic and provide relatively simple local conditions ensuring cut elimination for such rule sets. The rule format encompasses e.g. rules for the boolean connectives and transitive modal logics such as S4 or its constructive version. We also adapt the method of constructing suitable rule sets by saturation to the intuitionistic setting and provide a criterium for translating axioms for intuitionistic modal logics into sequent rules. Examples include constructive modal logics and conditional logic \(\mathbb{VA}\).

Supported by EPSRC-Project EP/H016317/1.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 49.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abadi, M.: Variations in Access Control Logic. In: van der Meyden, R., van der Torre, L. (eds.) DEON 2008. LNCS (LNAI), vol. 5076, pp. 96–109. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  2. Alechina, N., Mendler, M., de Paiva, V., Ritter, E.: Categorical and Kripke Semantics for Constructive S4 Modal Logic. In: Fribourg, L. (ed.) CSL 2001 and EACSL 2001. LNCS, vol. 2142, pp. 292–307. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  3. Avron, A., Lahav, O.: Kripke Semantics for Basic Sequent Systems. In: Brünnler, K., Metcalfe, G. (eds.) TABLEAUX 2011. LNCS (LNAI), vol. 6793, pp. 43–57. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  4. Bellin, G., de Paiva, V., Ritter, E.: Extended Curry-Howard correspondence for a basic contructive modal logic. In: Areces, C., de Rijke, M. (eds.) M4M-2. ILLC Amsterdam (2001)

    Google Scholar 

  5. Benton, P., Bierman, G., de Paiva, V.: Computational types from a logical perspective. J. Funct. Programming 8(2), 177–193 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ciabattoni, A., Galatos, N., Terui, K.: From axioms to analytic rules in nonclassical logics. In: LICS 2008, pp. 229–240. IEEE Computer Society (2008)

    Google Scholar 

  7. Fairtlough, M., Mendler, M.: Propositional lax logic. Inform. and Comput. 137(1), 1–33 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gentzen, G.: Untersuchungen über das logische Schließen. I. Math. Z. 39(2), 176–210 (1934)

    MathSciNet  Google Scholar 

  9. Johansson, I.: Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus. Compos. Math. 4, 119–136 (1937)

    MathSciNet  Google Scholar 

  10. Lellmann, B., Pattinson, D.: Cut Elimination for Shallow Modal Logics. In: Brünnler, K., Metcalfe, G. (eds.) TABLEAUX 2011. LNCS, vol. 6793, pp. 211–225. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  11. Lellmann, B., Pattinson, D.: Sequent Systems for Lewis’ Conditional Logics. In: del Cerro, L.F., Herzig, A., Mengin, J. (eds.) JELIA 2012. LNCS, vol. 7519, pp. 320–332. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  12. Lewis, D.: Counterfactuals. Blackwell (1973)

    Google Scholar 

  13. Mendler, M., Scheele, S.: Cut-free Gentzen calculus for multimodal CK. Inform. and Comput. 209, 1465–1490 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Negri, S.: Proof analysis in modal logic. J. Philos. Logic 34, 507–544 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Negri, S., von Plato, J.: Structural proof theory. Cambridge University Press (2001)

    Google Scholar 

  16. Pattinson, D., Schröder, L.: Generic modal cut elimination applied to conditional logics. Log. Methods Comput. Sci. 7(1) (2011)

    Google Scholar 

  17. Pfenning, F., Davies, R.: A judgmental reconstruction of modal logic. Math. Structures Comput. Sci. 11(4), 511–540 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Troelstra, A.S., Schwichtenberg, H.: Basic Proof Theory, 2nd edn. Cambridge Tracts Theoret. Comput. Sci. Cambridge University Press (2000)

    Google Scholar 

  19. Wansing, H.: Sequent systems for modal logics. In: Gabbay, D.M., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 8. Springer (2002)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computing, Imperial College London, UK

    Björn Lellmann

  2. Research School of Computer Science, The Australian National University, Australia

    Dirk Pattinson

Authors
  1. Björn Lellmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dirk Pattinson
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. The Institute of Mathematical Sciences, CIT Campus, 600113, Chennai, India

    Kamal Lodaya

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Lellmann, B., Pattinson, D. (2013). Constructing Cut Free Sequent Systems with Context Restrictions Based on Classical or Intuitionistic Logic. In: Lodaya, K. (eds) Logic and Its Applications. ICLA 2013. Lecture Notes in Computer Science, vol 7750. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36039-8_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-36039-8_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-36038-1

  • Online ISBN: 978-3-642-36039-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation