The Lambek calculus

Johan van Benthem

pp. 35-68

in: Richard T. Oehrle, Emmon Bach, Deirdre Wheeler (eds), Categorial grammars and natural language structures, Berlin, Springer, 1988

Abstract

There is a noticeable revival of categorial grammar these days, as a vehicle for linguistic description. The systems used differ somewhat from the original calculus of Ajdukiewicz and Bar-Hillel, however. In particular, there is a component of rules for "type change' of expressions, making for greater flexibility and elegance. One fundamental system of this kind is the so-called "Lambek Calculus', whose type-change rules show a close analogy with the inference rules of constructive propositional logic. In this paper, we present one calculus of this kind, and survey its theoretical properties as a device in linguistic semantics. Our two main new contributions are a new and complete semantics for this calculus, as well as a modest study of its language-accepting capacity. In this way, we hope to provide a better understanding of the background theory of flexible categorial grammar, in tandem with its descriptive uses.