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We present a theory of how people reason about properties. Such inferences have been studied since Aristotle’s invention of Western logic. But, no previous psychological theory gives an adequate account of them, and most theories do not go beyond syllogistic inferences, such as: All the bankers are architects; Some of the chefs are bankers; What follows? The present theory postulates that such assertions establish relations between properties, which mental models represent in corresponding relations between sets of entities. The theory combines the construction of models with innovative heuristics that scan them to draw conclusions. It explains the processes that generate a conclusion from premises, decides if a given conclusion is necessary or possible, assesses its probability, and evaluates the consistency of a set of assertions. A computer program implementing the theory embodies an intuitive system 1 and a deliberative system 2, and it copes with quantifiers such as more than half the architects. It fit data from over 200 different sorts of inference, including those about the properties of individuals, the properties of a set of individuals, and the properties of several such sets in syllogisms. Another innovation is that the program accounts for differences in reasoning from one individual to another, and from one group of individuals to another: some tend to reason intuitively but some go beyond intuitions to search for alternative models. The theory extends to inferences about disjunctions of properties, about relations rather than properties, and about the properties of properties.