Joachim Lambek - Bilinear logic in linguistics

At least three versions of bilinear (=noncommutative linear) logic have been applied to linguistics. The intuitionistic version (also known as the syntactic calculus), the classical version (recently proposed by Claudia Casadio) and the compact version (proposed by me). Poset models of these deductive systems have been described as residuated monoids, Grishin algerbas and pregroups respectively. A pregroup is a partially ordered group in which each element a has both a left adjoint and right adjoint. In a first approximation to English grammar one works with the free pregroup generated by an ordered set of basic types expressing person, tense, case etc. Words are assigned types which are elements of the free pregroups. To check that strings of words are well-formed sentences, only contractions are required. The fact that two left adjoints don't cancel is exploited for sentences such as "whom did she see?" which otherwise require a Chomskian trace. Information is processed from left to right; thus a hearer will calculate the type of ``whom did'' before hearing the rest of the sentence. Constraints on WH-transformations are explained by showing that the computation becomes too complicated. For a closer approximation to English grammar one may have to abandon the assumption that the pregroups is free, which is to say that all grammatical rules can be recorporated in the dictionary.