This paper describes a system of categorial inference based on insights from Lexicalized Tree Adjoining Grammar (LTAG). LTAG is a tree-rewriting system and therefore deals with structural, not string, adjacency. When looked at from the logic perspective, the nodes of the trees become types as in a categorial grammar, with corresponding deductive connections between parent and daughter nodes. The resulting system is based on a hybrid logic, with one logic for building Partial Proof Trees, and the other for composing the partial proofs. We reexamine the use of structural modalities in categorial grammar from this perspective, concluding that the use of structural modalities can be considerably simplified, or even eliminated in some cases. The generative power of the hybrid logic system is beyond context-free, as we demonstrate with a derivation of the cross-serial dependencies in Dutch. The system also inherits polynomial parsing from LTAG.