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