Psychologists have studied syllogistic inferences for more than a century, but no extant theory gives an adequate account of them. Reasoners appear to reason using different strategies. A complete account of syllogisms must explain them and the resulting differences from one individual to another. We propose a dual-process theory that solves these two problems. It is based on the manipulation of mental models, i.e., iconic simulations of possibilities. A computer program implementing the theory, mReasoner, generates initial conclusions by building and scanning models. The theory accounts for individual differences in an early study on syllogisms (Johnson-Laird & Steedman, 1978). The computational model provides an algorithmic account of the different processes on which three subsets of performance relied (Simulation 1). It also simulates the performance of each individual participant in the study (Simulation 2). The theory and its implementation constitute the first robust account of individual differences in syllogistic reasoning.