Categorical Vector Space Semantics for Lambek Calculus with a Relevant Modality

We develop a categorical compositional distributional semantics for Lambek Calculus with Relevant Modality !L*, which has limited edition of the contraction and permutation rules. The part is monoidal biclosed category coalgebra modality, very similar to structure Differential Category. instantiate this finite dimensional vector spaces linear maps via "quantisation" functors work three concrete interpretations modality. apply model construct semantic motivating example !L*: derivation phrase parasitic gap. effectiveness are evaluated disambiguation task, on an extension sentence dataset gap phrases, using BERT, Word2Vec, FastText vectors Relational tensors.