Discontinuous Lambek Calculus

The search for a full treatment of wrapping in type logical grammar has been a task of long-standing. In this paper we present a calculus for discontinuity addressing this challenge, ω-DL. The calculus allows an unbounded number of points of discontinuity (hence the prefix ω-) and includes both deterministic and nondeterministic discontinuous connectives. We believe that it constitutes a general and natural extension of the Lambek calculus L. Like the Lambek calculus it has a sequent calculus which is a sequence logic without structural rules, and it enjoys such properties as Cut-elimination, the subformula property and decidability. By n-DL we refer to ω-DL restricted to at most n points of discontinuity. 0-DL is the original Lambek calculus L. Of particular interest is 1-DL in which the unicity of the point of discontinuity means that the deterministic and nondeterministic discontinuous connectives coincide. We illustrate 1-DL with linguistic applications to medial extraction, discontinuous idioms, parentheticals, gapping, VP ellipsis, reflexivization, quantification, pied-piping, appositive relativisation, comparative subdeletion, and cross-serial dependencies. We further illustrate deterministic 2-DL with linguistic application to anaphora, and nondeterministic 2-DL with linguistic application to particle shift and complement alternation. That is we address, so far as we are aware, all the constructions for which some version of discontinuity has been proposed in the type-logical literature.