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In this paper, we introduce a variant of the Lambek calculus allowing emptyantecedents. This variant uses two connecives: the left division and a unarymodality that occurs only with negative polarity and allows weakening inantecedents of sequents. We define the notion of a proof net for this calculus,which is similar to those for the ordinary Lambek calculus and multiplicativelinear logic. We prove that a sequent is derivable in the calculus underconsideration if and only if there exists a proof net for it. Thus, weestablish a derivability criterion for this calculus in terms of the existenceof a graph with certain properties. The size of the graph is bounded by thelength of the sequent.

Proof nets for the Lambek calculus with one division and a negative-polarity modality for weakening