Proof Nets for the Multimodal Lambek Calculus Richard Moot and Quintijn Puite January 22, 2001 Abstract We present a novel way of using proof nets for the multimodal Lambek calculus, which provides a general treatment of both the unary and binary connectives. We also introduce a correctness criterion which is valid for a large class of structural rules and prove basic soundness, completeness and cut elimination results. Finally, we will present a correctness criterion for the original Lambek calculus L as an instance of our general correctness criterion. 1 Introduction One of the most important proof theoretic innovations of linear logic has been the introduction of proof nets as a redundancy-free and elegant way to repre- sent proofs. Proof nets are usually presented as members of a larger class of structures, called proof structures. Proof nets are those proof structures which satisfy some condition, a correctness criterion. In the theory of proof nets, correctness criteria usually fall into one of two categories. On the one hand, there are the graph theoretic or geometric cor- rectness criteria, where correct proof structures are characterized by properties of the graphs representing these proof structures. We find this approach in for example [Girard 1987] and [DR 1989]. On the other hand there are the alge- braic correctness criteria, where in addition to the proof structures we have a term algebra and a correspondence between proof structures and terms. In this approach correctness is stated in terms of properties of these terms. This ap- proach has been advocated by for example [Groote 1997] and [Moortgat 1997]. Conceptually, algebraic correctness criteria are more complicated, because they use a two level theory, and part of the soundness and completeness results for these proof net calculi needs to be devoted to the correspondence between the two levels. However, some fragments of linear logic, such as MALL, the Multiplicative Additive fragment [Girard 1996], seem to resist a purely geo- metrical correctness criterion. The main goals of this paper are to present a geometric correctness criterion for NL , which is the non-associative Lambek calculus NL [Lambek 1961] extended with unary connectives and and with a package of structural rules , and to prove basic soundness and completeness results. Though our criterion is geometric, it is closely related to the algebraic criterion presented in [Moortgat 1997] in the sense that our proof nets can be seen as graph theoretical 1