We present a chatbot implementing a novel dialogue management approach based on logical inference. Instead of framing conversation a sequence of response generation tasks, we model conversation as a collaborative inference process in which speakers share information to synthesize new knowledge in real time. Our chatbot pipeline accomplishes this modelling in three broad stages. The first stage translates user utterances into a symbolic predicate representation. The second stage then uses this structured representation in conjunction with a larger knowledge base to synthesize new predicates using efficient graph matching. In the third and final stage, our bot selects a small subset of predicates and translates them into an English response. This approach lends itself to understanding latent semantics of user inputs, flexible initiative taking, and responses that are novel and coherent with the dialogue context.