A logic for inconsistent mathematics must be strong enough to support reasoning in proofs, while weak enough to avoid paradoxes. We present a substructural logic intended to meet the needs of a working dialetheic mathematician—specifically, by adding a de Morgan negation to light linear logic, and extending the logic with a relevant conditional. The logic satisfies a deduction theorem. Soundness and completeness is established, showing in particular that contraction is invalidated. This opens the way for a robust naive set theory; we conclude by showing how the set theory provides a foundation for induction.