Skip to main content
Log in

The Lambek Calculus Extended with Intuitionistic Propositional Logic

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

We present sound and complete semantics and a sequent calculus for the Lambek calculus extended with intuitionistic propositional logic.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Buszkowski W.: Some decision problems in the theory of syntactic categories. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 28, 539–548 (1982)

    Article  Google Scholar 

  2. Buszkowski W.: Interpolation and FEP for logics of residuated algebras. Logic Journal of the IGPL 19, 437–454 (2011)

    Article  Google Scholar 

  3. Buszkowski, W., and M. Farulewski, Nonassociative Lambek calculus with additives and context-free languages, in O. Grumberg, M. Kaminski, S. Katz, and S. Wintner (eds.), Languages: From Formal to Natural, Essays Dedicated to Nissim Francez on the Occasion of His 65th Birthday, vol. 5533 of Lecture Notes in Computer Science, Springer, Heidelberg, 2009, pp. 45–58.

  4. Došen K.: A brief survey of frames for the Lambek calculus. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 38, 179–187 (1992)

    Article  Google Scholar 

  5. Dunn, M., A “Gentzen system” for positive relevant implication, Journal of Synmolic Logic 38:356–357, 1974 (Abstract).

  6. Gentzen G.: Untersuchungen über das logische Schliessen I. Mathematische Zeitschrift 39, 176–210 (1935)

    Article  Google Scholar 

  7. Gentzen G.: Untersuchungen über das logische Schliessen II. Mathematische Zeitschrift 39, 405–431 (1935)

    Article  Google Scholar 

  8. Kaminski M., Francez N.: Relational semantics of the Lambek calculus extended with classical propositional logic. Studia Logica 102, 479–497 (2014)

    Article  Google Scholar 

  9. Kanazawa M.: The Lambek calculus enriched with additional connectives. Journal of Logic, Language, and Information 1, 141–171 (1992)

    Article  Google Scholar 

  10. Kleene, S. C., Introduction to Metamathematics, North-Holland, Amsterdam, 1962.

  11. Kozak M.: Distributive full Lambek calculus has the finite model property. Studia Logica 91, 201–216 (2009)

    Article  Google Scholar 

  12. Kripke, S., Semantical analysis of intuitionistic logic I, in J. Crossley and M. Dummett (eds.), Formal Systems and Recursive Function, Series in Logic and Foundations of Mathematics, North Holland, Amsterdam, 1965, pp. 92–130.

  13. Kurtonina, N., and M. Moortgat, Relational semantics for the Lambek–Grishin calculus, in C. Ebert, G. Jäger, and J. Michaelis (eds.), The Mathematics of Language, 10th and 11th Biennial Conference, MOL 10 and MOL 11, vol. 6149 of Lecture Notes in Computer Science, Springer, Heidelberg, 2009, pp. 210–222.

  14. Lambek, J., The mathematics of sentence structure, American Mathematical Monthly 65:154–170, 1958. (Also in W. Buszkowski, W. Marciszewski, and J. van Benthem (eds.) Categorial Grammars, John Benjamins, Amsterdam, 1988.)

  15. Mints G.E.: Cut-elimination theorem in relevant logics. Journal of Soviet Mathematics 6, 422–428 (1976)

    Article  Google Scholar 

  16. O’Hearn, P. W., and D. J. Pym, The logic of bunched implications, Bulletin of Symbolic Logic 5:215–244, 1999.

  17. Pym, D. J., The Semantics and Proof Theory of the Logic of Bunched Implications, vol. 26 of Applied Logic Series, Springer, Heidelberg, 2002.

  18. Restall, G., On Logics Without Contraction, PhD Thesis, The University of Queensland, 1994.

  19. Segerberg, K., An Essay in Classical Modal Logic, Uppsala: Filosofiska Studier 13, 1971.

  20. Thomason R.H.: On the strong semantical completeness of the intuitionistic predicate calculus. Logic Journal of the IGPL 15, 271–286 (2007)

    Article  Google Scholar 

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

  22. Zimmermann, E., Full Lambek calculus in natural deduction, Mathematical Logic Quarterly 56:85–88, 2010.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Michael Kaminski.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kaminski, M., Francez, N. The Lambek Calculus Extended with Intuitionistic Propositional Logic. Stud Logica 104, 1051–1082 (2016). https://doi.org/10.1007/s11225-016-9665-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-016-9665-0

Keywords

Navigation