Delusions are explanations of anomalous experiences. A theory of delusion requires an explanation of both the anomalous experience and the apparently irrational explanation generated by the delusional subject. Hence, we require a model of rational belief formation against which the belief formation of delusional subjects can be evaluated. Method. I first describe such a model, distinguishing procedural from pragmatic rationality. Procedural rationality is the use of rules or procedures, deductive or inductive, that produce an inferentially coherent set of propositions. Pragmatic rationality is the use of procedural rationality in context . I then apply the distinction to the explanation of the Capgras and the Cotard delusions. I then argue that delusions are failures of pragmatic rationality. I examine the nature of these failures employing the distinction between performance and competence familiar from Chomskian linguistics. Results. This approach to the irrationality of delusions reconciles accounts in which the explanation of the anomalous experience exhausts the explanation of delusion, accounts that appeal to further deficits within the reasoning processes of delusional subjects, and accounts that argue that delusions are not beliefs at all. (Respectively, one-stage, two-stage, and expressive accounts.) Conclusion. In paradigm cases that concern cognitive neuropsychiatry the irrationality of delusional subjects should be thought of as a performance deficit in pragmatic rationality.