The paper investigates the application of quantale semantics to non-classical substructural logics, specifically Lambek calculi extended with cyclic negation. We introduce models based on involutive quantales and explore their correspondence with cyclic linear logic frameworks. The approach provides a unified perspective on structural rules and dualities in categorial grammars. Particular attention is given to the interpretation of non-associativity and non-commutativity in linguistic and proof-theoretic contexts.