Geert-Jan Kruijff & Richard T. Oehrle A categorial grammar is both a grammar and a type inference system. As a result of this duality, the categorial framework offers a natural setting in which to study questions of grammatical composition, both empirically and abstractly. There are affinities in this perspective, of course, to basic questions in formal language theory. But the fact that categorial grammars are type in­ ference systems makes possible intrinsic connections among syntactic types, syntactic type inference, semantic types, and semantic type inference, a con­ nection less apparent in the standard constructions of formal language theory. Fixing a system of grammatical type inference T, we may explore what gram­ matical phenomena are compatible with T-and equally, what grammatical phenomena are not. Equally, fixing a class of grammatical phenomena g, we may seek to ascertain what systems of type inference characterize g. This dual perspective is a strong current in the categorial literature, going back to the classical papers of Ajdukiewicz, Bar-Hillel, Curry, and Lambek.