The categorial fine-structure of natural language

Categorial grammar analyzes linguistic syntax and semantics in terms of type theory and lambda calculus. A major attraction of this approach is its unifying power, as its basic function/argument structures occur across the foundations of mathematics, language and computation. This paper considers, in a light example-based manner, where this elegant logical paradigm stands when confronted with the wear and tear of reality. Starting from a brief history of the Lambek tradition since the 1980s, we discuss three main issues: (a) the fit of the lambda calculus engine to characteristic semantic structures in natural language, (b) the coexistence of the original type-theoretic and more recent modal interpretations of categorial logics, and (c) the place of categorial grammars in the complex total architecture of natural language, which involves amongst others mixtures of interpretation and inference. 1 From Montague Grammar to Categorial Grammar Logic and linguistics have had lively connections from Antiquity right until today (GAMUT 1991). A recurrent theme in this history is the categorial structure of language and ontology, from Aristotle's grammatical categories to Russell's theory of types in the foundations of mathematics. Further bridges were thrown as logic and linguistics developed, with Lambek 1958 as a major advance in sophistication, many years ahead of its time. Large-scale logical analysis of natural language really took off around 1970, thanks to Montague's work (Montague 1975, Partee 1997). This system connects linguistic structure and logical semantics in the following architecture. The pivotal deep structure are syntax trees formed by grammatical construction rules. In one direction, these trees project to actual surface expressions by an obvious deletion map shedding all category symbols, brackets, indices, and other theoretical entities put there by the theory. In another, semantic direction, trees constructed in this explicit manner can be translated compositionally into logical formulas of a sufficiently rich language, for which Montague chose an intensional type logic IL (cf. Gallin 1975 for its mathematical theory). In particular, this sends syntactic categories to semantic types. These logical formulas have a standard