Many variants of categorial grammar assume an underlying logic which is associative and linear. In relation to left extraction, the former property is challenged by island domains, which involve nonassociativity, and the latter property is challenged by parasitic gaps, which involve nonlinearity.  We present a version of type logical grammar including `structural inhibition' for nonassociativity and `structural facilitation' for nonlinearity and we give an account of relativisation including islands and parasitic gaps and their interaction.