Tuples, Discontinuity, and Gapping in Categorial Grammar

This paper solves some puzzles in the formalisation of logic for discontinuity in categorial grammar. A 'tuple' operation introduced in [Solias, 1992] is defined as a mode of prosodic combination which has associated projection functions, and consequently can support a property of unique prosodic decomposability. Discontinuity operators are defined model-theoretically by a residuation scheme which is particularly ammenable proof-theoretically. This enables a formulation which both improves on the logic for wrapping and infixing of [Moortgat, 1988] which is only partial, and resolves some problems of determinacy of insertion point in the application of these proposals to in-situ binding phenomena. A discontinuous product is also defined by the residuation scheme, enabling formulation of rules of both use and proof for a 'substring' product that would have been similarly doomed to partial logic.We show how the apparatus enables characterisation of discontinous functors such as particle verbs and phrasal idioms, and binding phenomena such as quantifier raising and pied piping. We conclude by showing how the apparatus enables a simple categorial analysis of (SVO) gapping using the discontinuity product and the wrapping operator.