Completeness of Full Lambek Calculus for Syntactic Concept Lattices

  • Christian Wurm
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8036)


Syntactic concept lattices are residuated structures which arise from the distributional analysis of a language. We show that these structures form a complete class of models with respect to the logic FL  ⊥ ; furthermore, its reducts are complete with respect to FL and L1.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Buszkowski, W.: Completeness results for Lambek syntactic calculus. Mathematical Logic Quarterly 32(1-5), 13–28 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Buszkowski, W.: Algebraic structures in categorial grammar. Theor. Comput. Sci. 1998(1-2), 5–24 (1998)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Clark, A.: A learnable representation for syntax using residuated lattices. In: de Groote, P., Egg, M., Kallmeyer, L. (eds.) FG 2009. LNCS (LNAI), vol. 5591, pp. 183–198. Springer, Heidelberg (2011)Google Scholar
  4. 4.
    Clark, A.: Learning context free grammars with the syntactic concept lattice. In: Sempere, J.M., García, P. (eds.) ICGI 2010. LNCS, vol. 6339, pp. 38–51. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  5. 5.
    Clark, A.: Logical grammars, logical theories. In: Béchet, D., Dikovsky, A. (eds.) LACL 2012. LNCS, vol. 7351, pp. 1–20. Springer, Heidelberg (2012)Google Scholar
  6. 6.
    Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, Cambridge (1991)Google Scholar
  7. 7.
    Farulewski, M.: On Finite Models of the Lambek Calculus. Studia Logica 80(1), 63–74 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Elsevier (2007)Google Scholar
  9. 9.
    Harris, Z.S.: Structural Linguistics. The University of Chicago Press (1963)Google Scholar
  10. 10.
    Kanazawa, M.: The Lambek calculus enriched with additional connectives. Journal of Logic, Language, and Information 1, 141–171 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Lambek, J.: The Mathematics of Sentence Structure. The American Mathematical Monthly 65, 154–169 (1958)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Lambek, J.: On the calculus of syntactic types. In: Jakobson, R. (ed.) Structure of Language and its Mathematical Aspects, pp. 166–178. American Mathematical Society, Providence (1961)CrossRefGoogle Scholar
  13. 13.
    Morrill, G., Valentín, O., Fadda, M.: The displacement calculus. Journal of Logic, Language and Information 20(1), 1–48 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Okada, M., Terui, K.: The finite model property for various fragments of intuitionistic linear logic. J. Symb. Log. 64(2), 790–802 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Pentus, M.: Lambek grammars are context free. In: Proceedings of the 8th Annual IEEE Symposium on Logic in Computer Science, Los Alamitos, California, pp. 429–433. IEEE Computer Society Press (1993)Google Scholar
  16. 16.
    Sestier, A.: Contributions à une théorie ensembliste des classifications linguistiques (Contributions to a set–theoretical theory of classifications). In: Actes du Ier Congrès de l’AFCAL, Grenoble, pp. 293–305 (1960)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Christian Wurm
    • 1
  1. 1.Fakultät für Linguistik und LiteraturwissenschaftenCITEC Universität BielefeldGermany

Personalised recommendations