On Calculus of Displacement

The calculus of Lambek (1958) did not make much impact until the 1980s, but for more than twenty years now it has constituted the foundation of type logical categorial grammar. It has occupied such a central position because of its good logical properties, but it has also been clear, even from the start of its renaissance, that the Lambek calculus suffers from fundamental shortcomings, which we shall mention below. Certainly it seems that the way ahead in logical categorial grammar is to enrich the Lambek calculus with additional connectives. Thus it was proposed to add intersection and union as far back as Lambek (1961), and such extension is what is meant by type logical categorial grammar. Particular inspiration came from linear logic. Propositional linear connectives divide into additives, multiplicatives, and exponentials. Technically, the Lambek connectives are (noncommutative) linear multiplicatives, so it is natural to consider enrichment of Lambek calculus with (noncommutative) additives and exponentials as well. Quantifiers may also be added (Morrill, 1994, ch. 6) and unary modalities (Morrill 1990, 1992; Moortgat 1995). However, none of these extensions address the essential limitation of the Lambek basis, which is as follows. The Lambek calculus is a sequence logic of concatenation. This is all well and good in that words are arranged sequentially, however natural language exhibits action at a distance: dependencies which are discontinuous. The Lambek calculus can capture some discontinuous dependencies, namely those in which the discontinuous dependency