We introduce a new approach to semantics which combines the benefits of distributional and formal logical semantics. Distributional models have been successful in modelling the meanings of content words, but logical semantics is necessary to adequately represent many function words. We follow formal semantics in mapping language to logical representations, but differ in that the relational constants used are induced by offline distributional clustering at the level of predicate-argument structure. Our clustering algorithm is highly scalable, allowing us to run on corpora the size of Gigaword. Different senses of a word are disambiguated based on their induced types. We outperform a variety of existing approaches on a wide-coverage question answering task, and demonstrate the ability to make complex multi-sentence inferences involving quantifiers on the FraCaS suite.