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Grammar induction, the task of learning a set of syntactic rules from minimally annotated training data, provides a means of exploring the longstanding question of whether humans rely on innate knowledge to acquire language. Of the various formalisms available for grammar induction, categorial grammars provide an appealing option due to their transparent interface between syntax and semantics. However, to obtain competitive results, previous categorial grammar inducers have relied on shortcuts such as part-of-speech annotations or an ad hoc bias term in the objective function to ensure desirable branching behavior. We present a categorial grammar inducer that eliminates both shortcuts: it learns from raw data, and does not rely on a biased objective function. This improvement is achieved through a novel stochastic process used to select the set of available syntactic categories. On a corpus of English child-directed speech, the model attains a recall-homogeneity of 0.48, a large improvement over previous categorial grammar inducers.