ABSTRACT
Bar-Hillel et al proved the mutual formal equivalence of unidirectional and bidirectional categorial grammars (CGs), the difference between them in practice for linguistic description being the possibility of 'natural' category assignments in a bidirectional CG. The principal example given by Lambek of the use of associativity is that of a transitive verb which can combine syntactically with either its object or subject first. In the semantic interpretation, the verb is the functor, the order of its arguments indicating which nominal is subject and which is object. The binary rule of function application, as already mentioned, is the first and simplest in any categorial calculus. The set of atomic categories is remarked by Ajdukiewicz and Bar-Hillel themselves to be inadequate, and is universally supplemented, at least by distinguishing common nouns (CN) from proper nouns or noun phrases (PN or NP). The principle can be illustrated with a simpler example, the derivation of a noun phrase with multiple adjectives.
