Type Grammars as Pregroups

At first sight, it seems quite unlikely that mathematics can be applied to the study of natural language. However, upon closer examination, it appears that language itself is a kind of mathematics: producing and recognizing speech involves calculations, albeit at a subconscious level, and the rules of grammar which the speaker has mastered, even if she cannot formulate them, resemble the axioms and rules of inference of mathematical logic. In this article I will present an algebraic model of grammar in the form of a pregroup, which competes with an earlier model which was once proposed by me and is now being developed further by a small but dedicated group of researchers, and took the form of a residuated monoid. I am not fully convinced that either of these models really captures the cognitive processes involved, and I still suspect that rewrite systems, also known as production grammars, do a better job. Yet the algebraic models offer an alternative approach of interest to the more mathematically inclined students of language. Although I have told this story before (Lambek, 1999), the present version is addressed to linguists, hence some mathematical definitions have been deferred to an appendix and proofs have been left out altogether. While some red herrings have been eliminated (protogroups, inflectors), the small fragment of English grammar treated here is essentially the same as in the earlier version.