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This paper addresses the problem of learning to map sentences to logical form, given training data consisting of natural language sentences paired with logical representations of their meaning. Previous approaches have been designed for particular natural languages or specific meaning representations; here we present a more general method. The approach induces a probabilistic CCG grammar that represents the meaning of individual words and defines how these meanings can be combined to analyze complete sentences. We use higher-order unification to define a hypothesis space containing all grammars consistent with the training data, and develop an online learning algorithm that efficiently searches this space while simultaneously estimating the parameters of a log-linear parsing model. Experiments demonstrate high accuracy on benchmark data sets in four languages with two different meaning representations.