In the Treatise, Hume attempts to explain why we all believe that the self is a single unified entity that persists over time, a belief which Hume calls a fiction. In this paper, I demonstrate how Hume uses a type of functional explanation to account for this belief. After explicating Hume's view, I shall argue that it faces two related problems, which constitute a sort of dilemma. In the final section, I show how one of the horns of this dilemma (...) is plausibly what troubles Hume in the famous Appendix passage where he retracts his positive account of personal identity. (shrink)