From Categorial Grammar to Bilinear Logic
Joachim Lambek · 1993
Abstract The syntactic calculus, also known as ‘ bidirectional categorial grammar’, is a kind of logic without any structural rules, other than the obligatory reflexive law and cut-rule. It had been inspired by multilinear algebra and non-commutative ring theory and was developed with applications to linguistics in mind. Here we shall confine attention to the associative version, although a non-associative version has also been studied [L 1961, Kandulski 1988, Došen 1988, 1989]. It differs from a very rudimentary form of Girard’s linear logic [Girard 1987, 1989] by the absence of the interchange rule which licenses commutativity. Because of its roots in non-commutative algebra and syntax, all appearance of commutativity is forbidden. The notation, which goes back to a pre-historic collaboration with George Findlay, was carefully chosen to reflect this absence of commutativity: the order of two letters was never to be wantonly interchanged, yet all rules were to be preserved under left-right symmetry, the guiding slogan being ‘ symmetry without commutativity’. It must be admitted, however, that the notation, proposed for linguistics [L 1958] and ring theory [L 1966], caught on in neither field (with some recent exceptions in linguistics).