A Logic for Categorial Grammars: Lambek’s Syntactic Calculus

  • Richard Moot
  • Christian Retoré
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6850)


Our second chapter is a rather complete study of the Lambek calculus, which enables a completely logical treatment of categorial grammar.

We first present its syntax in full detail, both with sequent calculus and natural deduction, and explain the relationship between these two presentations. Then we turn our attention to the normal forms for such proofs. Normalization and its dual namely interpolation are not only pleasant mathematical properties; they also are key properties for the correspondence between Lambek grammars and more familiar phrase structure grammars; we give a detailed proof of the theorem of Pentus establishing the weak equivalence between context-free grammars and Lambek grammars.

In addition, we prove completeness for the Lambek calculus with respect to linguistically natural models: in these models categories are interpreted as subsets of a free monoid (eg. as strings of words or lexical items). Providing such a simple and natural interpretation provides another strong justification for the categorial approach.


Natural Deduction Sequent Calculus Elimination Rule Primitive Type Introduction Rule 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  • Richard Moot
  • Christian Retoré

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