In this paper, we present a proof-theoretic analysis of the logic of scope \(\text {NL}_{\lambda }\) introduced by Barker and Shan. We notably introduce a novel calculus of proof nets and prove it is sound and complete with respect to the sequent calculus for the logic. We study decidability and complexity of the logic using this new calculus, proving a new upper bound for complexity of the logic (showing it is in NP) and a new lower bound for the class of formal language generated by the formalism (mildly context-sensitive languages extended with a permutation closure operation). Finally, thanks to this new calculus, we present a novel comparison between \(\text {NL}_{\lambda }\) and the hybrid type-logical grammars of Kubota and Levine. We show there is an unexpected convergence of the natural language analyses proposed in the two formalisms. In addition to studying the proof-theoretic properties of \(\text {NL}_{\lambda }\) , we greatly extends its linguistic coverage.
