  Linear logic and the linear {\lambda}-calculus have a long standing tradition
in the study of natural language form and meaning. Among the proof calculi of
linear logic, proof nets are of particular interest, offering an attractive
geometric representation of derivations that is unburdened by the bureaucratic
complications of conventional prooftheoretic formats. Building on recent
advances in set-theoretic learning, we propose a neural variant of proof nets
based on Sinkhorn networks, which allows us to translate parsing as the problem
of extracting syntactic primitives and permuting them into alignment. Our
methodology induces a batch-efficient, end-to-end differentiable architecture
that actualizes a formally grounded yet highly efficient neuro-symbolic parser.
We test our approach on {\AE}Thel, a dataset of type-logical derivations for
written Dutch, where it manages to correctly transcribe raw text sentences into
proofs and terms of the linear {\lambda}-calculus with an accuracy of as high
as 70%.
