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Consider a directional categorial grammar with the rule of functional application A and, in addition, the two type-shifting rules below, lifting L and division D. For obvious reasons I will call this system L after Lambek's (1958) syntactic calculus, where these reduction rules have the status of theorems. I single out lifting and division among the theorems of Lambek's calculus because of their prominent role in linguistic discussion. As will appear from the format, fractional categories are projected from the vertical into the horizontal mode by giving them a quarter turn clockwise. This gives the fractional categories a domain-range structure, which makes it easy to think of them either as syntactic or semantic types: X | Y is a functor which combines syntactically with an expression of type X to give an expression of type Y; semantically it is interpreted as a function f: type (X) → type(Y). (In what follows, I shall refer to the type of a syntactic or semantic object X as "t(X)'.) Directionality is encoded in the form of the fraction sign, back-slash standing for left-concatenation, slash for right-concatenation. The vertical slant is used to conflate the modes of concatenation, so that for example one interpretation schema can serve for a pair of directional reduction rules.

DOI: 10.1007/978-94-015-6878-4_12

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