A Sign-Based Extension to the Lambek Calculus for Discontinuous Constituency

This paper takes as its starting point the work of Moortgat (1991) and aims to provide a linguistically-motivated extension to the basic Lambek calculus that will allow, among other things, for an elegant treatment of various`discontinuous constituency' phenomena, including`tough'-constructions in En-glish, cross-serial agreement in Swiss German and quantiier scoping. The proposal is contrasted favorably with related proposals by Moortgat, Morrill and Solias (1993) and Hepple (1994). 1 Preliminaries This paper takes as its starting point the work of Moortgat (1991) and proposes an alternative extension to the basic Lambek Calculus that will allow, among other things, for the treatment of discontinuous constituents (exempliied herein by tough-class adjective phrases in English and cross-serial dependencies in Swiss German) in terms of operations on headed strings (along the lines of Pollard (1984)) that hitherto could not be accommodated in systems of this type. 2 The Lambek Calculus 2.1 The basic calculus The basic Lambek Calculus (Lambek 1958, 1988), referred to here as L, can be taken to fall under the general rubric of extended categorial grammars, with the key idea being that the category reduction system typically assumed for basic categorial grammars can be viewed as a calculus analogous to the implicational fragment of proposi-tional logic. Categorial parsing, in turn, (i.e., the elimination of categorial connectives by means of functional application) can be viewed as the categorial analogue of Modus Ponens, and a full logic of these connectives can be obtained by adding for them a rule of introduction, analogous to conditionalization in the implicational perspective. More precisely, we can take the set of types to be freely generated from a set of primitive (atomic) types (for example, s, np, and so forth) by a set of binary innx operators (for example, =, and n). 1 As alluded to before, we can associate with each connective two inference rules: (i) a rule of introduction (also called a rule of proof, 1 We ignore the typically-assumed product-connective , for the sake of simplicity.