On Relational Completeness of Multi-Modal Categories Logics

Several recent results show that the Lambek Calculus L and its close relative L1 is sound and complete under (possibly relativized) relational interpretation. The paper transfers these results to L◊, the multi-modal extension of the Lambek Calculus that was proposed in Moortgat 1996. Two natural relational interpretations of L◊ are proposed and shown to be sound and complete. The completeness proofs make heavy use of the method of relational labeling from Kurtonina 1995. Finally, it is demonstrated that relational interpretation provides a semantic justification for the translation from L◊ to L from Versmissen 1996.