The ideas of aleatoric and epistemic uncertainty are widely used to reason about the probabilistic predictions of machine-learning models. We highlight that the common understanding of these ideas ("the standard view") makes a number of spurious associations, including between the data-generating process and a model's predictive uncertainty, between parameter stochasticity and the reducibility of predictive uncertainty, and between subjective uncertainty estimates and objective measures of predictive performance. To address this we derive a simple, principled framework for reasoning about sources of predictive uncertainty.