Proof-Theoretic Aspects of the Lambek-Grishin Calculus

  • Philippe de GrooteEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9160)


We compare the Lambek-Grishin Calculus (LG) as defined by Moortgat [9, 10] with the non-associative classical Lambek calculus (CNL) introduced by de Groote and Lamarche [4]. We provide a translation of LG into CNL, which allows CNL to be seen as a non-conservative extension of LG. We then introduce a bimodal version of CNL that we call 2-CNL. This allows us to define a faithful translation of LG into 2-CNL. Finally, we show how to accomodate Grishin’s interaction principles by using an appropriate notion of polarity. From this, we derive a new one-sided sequent calculus for LG.


  1. 1.
    Abrusci, V.M., Casadio, C. (eds.): New perspectives in logic and formal linguistics. In: Proceedings of the 5th Roma Workshop. Bulzoni Editore, Roma (2002)Google Scholar
  2. 2.
    Bastenhof, A.: Focalization and phase models for classical extensions of non-associative Lambek calculus. CoRR, abs/1106.0399, 2011Google Scholar
  3. 3.
    Bastenhof, A.: Categorial symmetry. Ph.D. thesis Utrecht University (2013)Google Scholar
  4. 4.
    de Groote, P., Lamarche, F.: Classical non-associative Lambek calculus. Stud. Logica. 71, 355–388 (2002)CrossRefzbMATHGoogle Scholar
  5. 5.
    Girard, J.-Y.: Linear logic. Theoret. Comput. Sci. 50, 1–102 (1987)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Grishin, V.N.: On a generalization of the Ajdukiewicz-Lambek system. In: Mikhailov, A.I. (ed.) Studies in Non-classical Logics and Formal Systems, pp. 315–334, Moscow, Nauka (1983)(In Russian. English translation in [1, pp. 9–27])Google Scholar
  7. 7.
    Kurtonina, N., Moortgat, M.: Relational semantics for the Lambek-Grishin calculus. In: Ebert, C., Jäger, G., Michaelis, J. (eds.) MOL 10. LNCS, vol. 6149, pp. 210–222. Springer, Heidelberg (2010) CrossRefGoogle Scholar
  8. 8.
    Lambek, J.: On the calculus of syntactic types. In: Proceedings of the 12th Symposium Applied Mathematics Studies of Language and its Mathematical Aspects, pp. 166–178, Providence (1961)Google Scholar
  9. 9.
    Moortgat, M.: Symmetries in natural language syntax and semantics: the Lambek-Grishin calculus. In: Leivant, D., de Queiroz, R. (eds.) WoLLIC 2007. LNCS, vol. 4576, pp. 264–284. Springer, Heidelberg (2007) CrossRefGoogle Scholar
  10. 10.
    Moortgat, M.: Symmetric categorial grammar. J. Philos. Logic 38(6), 681–710 (2009)CrossRefMathSciNetzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Inria Nancy - Grand EstNancyFrance

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