Skip to main content

Lambek Calculus and its relational semantics: Completeness and incompleteness

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

The problem of whether Lambek Calculus is complete with respect to (w.r.t.) relational semantics, has been raised several times, cf. van Benthem (1989a) and van Benthem (1991). In this paper, we show that the answer is in the affirmative. More precisely, we will prove that that version of the Lambek Calculus which does not use the empty sequence is strongly complete w.r.t. those relational Kripke-models where the set of possible worlds,W, is a transitive binary relation, while that version of the Lambek Calculus where we admit the empty sequence as the antecedent of a sequent is strongly complete w.r.t. those relational models whereW=U×U for some setU. We will also look into extendability of this completeness result to various fragments of Girard's Linear Logic as suggested in van Benthem (1991), p. 235, and investigate the connection between the Lambek Calculus and language models.

This is a preview of subscription content, access via your institution.

References

  1. Andréka, H., 1988, “On the representation problem of distributive semilattice-ordered semigroups”, Mathematical Institute of the Hungarian Academy of Sciences.

  2. Andréka, H., 1991, “Representations of distributive lattice-ordered semigroups with binary relations”,Algebra Universalis 28, 12–25.

    Google Scholar 

  3. Andréka, H., Németi, I., Sain, I., 1991, “On the strength of temporal proofs”,Theoretical Computer Science 80, 125–151.

    Google Scholar 

  4. van Benthem, J., 1989a, “Semantic parallels in natural language and computation” inLogic Colloquium. Granada 1987, Ebbinghaus, H.-D.et al., (eds.), North-Holland, Amsterdam, 331–375.

    Google Scholar 

  5. van Benthem, J., 1989b, “Language in action”,Journal of Philosophical Logic 20, 225–263.

    Google Scholar 

  6. van Benthem, J., 1991,Language in Action, North-Holland, Amsterdam.

    Google Scholar 

  7. van Benthem, J., 1992, “A note on dynamic arrow logic”, to appear in:Logic and Information Flow, van Eyck, J. and Visser, A., (eds.), Studies in Logic, Language and Information, Kluwer, Dordrecht.

    Google Scholar 

  8. Bredihin, D. and Schein, B., 1978, “Representations of ordered semigroups and lattices by binary relations”,Colloquium Mathematicum 39, 1, 1–12.

    Google Scholar 

  9. Brown, C. and Gurr, D., 1991, “Relations and non-commutative linear logic”, University of Aarhus, Technical Report DAIMI PB-372.

  10. Buszkowski, W., 1986, “Completeness results for Lambek syntactic calculus”,Zeitschr. f. math. Logik und Grundlagen d. Math. 32, 13–28.

    Google Scholar 

  11. Došen, K., 1990, “A brief survey of frames for the Lambek calculus”,Zeitschr. f. math. Logik und Grundlagen d. Math. 38, 179–187.

    Google Scholar 

  12. Gabbay, D.M., 1992, “Free semigroup frame semantics for the Lambek calculus and concatenation Logic”, submitted.

  13. Goldblatt, R., 1987,Logics of Time and Computation, CSLI (Stanford), Lecture Notes vol. 7.

    Google Scholar 

  14. Henkin, L., Monk, J. D. and Tarski, A., 1985,Cylindric Algebras, Part II, North-Holland, Amsterdam.

    Google Scholar 

  15. Jipsen, P., 1992, “Some results about r-algebras”, manuscript, Vanderbilt University, Nashville, TN.

    Google Scholar 

  16. Jipsen, P., Jónsson, B. and Rafter, J., 1992, “Adjoining units to residuated Boolean algebras”, manuscript, Vanderbilt University, Nashville, TN.

    Google Scholar 

  17. Jónsson, B. and Tsinakis, C., 1992, “Relation algebras as residuated Boolean algebras”, manuscript, Vanderbilt University, Nashville, TN.

    Google Scholar 

  18. Lambek, J., 1958 “The mathematics of sentence structure”,American Mathematical Monthly 65, 154–170.

    Google Scholar 

  19. Marx, M., 1992, “Dynamic arrow logic with pairs”, inLogic at Work, Volume of Conference, Amsterdam.

  20. Marx, M., Mikulás, Sz., Németi, I. and Sain, I., 1992, “Investigations in arrow logic”, inLogic at Work, Volume of Conference, Amsterdam.

  21. Mikulás, Sz., 1991, “The completeness of the Lambek calculus w.r.t. relational semantics”, Lecture in the Banach Centre, Warsaw, 17th October 1991.

  22. Mikulás, Sz., 1992a, “The completeness of the Lambek Calculus w.r.t. relational semantics”, Institute for Language, Logic and Information, University of Amsterdam.

  23. Mikulás, Sz., 1992b, “Complete calculus for conjugated arrow logic”, inLogic at Work, Volume of Conference, Amsterdam.

  24. Németi, I., 1991, “Algebraizations of quantifier logics, an introductory overview”,Studia Logica 50, 485–569, updated and extended (with proofs) versions available from Németi, I. Mathematical Institute, Budapest Pf 127, H-1364 Budapest, Hungary.

    Google Scholar 

  25. Pentus, M., 1993, “Lambek calculus is L-complete”, preprint, Moscow University.

  26. Pratt, V.R., 1985, “Dynamic algebra as a well-behaved fragment of relation algebras”, inAlgebraic Logic and Universal Algebra in Computer Science, Bergman, C.H.,et al., (eds.), Lecture Notes in Computer Science, Springer Verlag, 77–110.

  27. Pratt, V.R., 1992, “A roadmap of some two-dimensional logics”, inAlgebraic and Categorical Methods in Computer Science, Proc. of TEMPUS Summer School, Prague.

  28. Roorda, D., 1991, “Dyadic modalities and Lambek calculus”, inColloquium of Modal Logic, de Rijke, M., (ed.), Dutch Network for Language, Logic and Information, Amsterdam, 187–210.

    Google Scholar 

  29. Sain, I., 1991, “Beth's and Craig's properties via epimorphisms and amalgamation in algebraic logic”,in Algebraic Logic and Universal Algebra in Computer Science, Bergman C.H.,et al., (eds.), Lecture Notes in Computer Science, Springer Verlag, 209–226.

  30. Simon, A., 1992, “Arrow logic does not have deduction theorem”, inLogic at Work, Volume of Conference, Amsterdam.

  31. Venema, Y., 1992,Many-dimensonal Modal Logic, Ph.D. Thesis, University of Amsterdam.

Download references

Author information

Affiliations

Authors

Additional information

Supported by Hungarian National Foundation for Scientific Research grant Nos. 1911, 2258, and by TEMPUS JEP No. 1941-92/2.

The first report on the first part of this work was in the Banach Centre, Warsaw, 17th October 1991.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Andréka, H., Mikulás, S. Lambek Calculus and its relational semantics: Completeness and incompleteness. J Logic Lang Inf 3, 1–37 (1994). https://doi.org/10.1007/BF01066355

Download citation

Key words

  • Lambek Calculus
  • relational semantics
  • language models
  • algebraic logic
  • residuated semigroups
  • representation theorems