Abstract A procedure for learning a lexical assignment together with a system of syntactic and semantic categories given a fixed type-logical grammar is briefly described. The logic underlying the grammar can be any cut-free decidable modally enriched extension of the Lambek calculus, but the correspondence between syntactic and semantic categories must be constrained so that no infinite set of categories is ultimately used to generate the language. It is shown that under these conditions various linguistically valuable subsets of the range of the algorithm are classes identifiable in the limit from data consisting of sentences labeled by simply typed lambda calculus meaning terms in normal form. The entire range of the algorithm is shown to be not a learnable class, contrary to a mistaken result reported in a preliminary version of this paper. It is informally argued that, given the right type logic, the learnable classes of grammars include members which generate natural languages, and thus that natural languages are learnable in this way.