In this paper, I identify and discuss the following feature of our judgments about hypothetical scenarios concerning the identity of persons: with respect to the vast majority of scenarios, both members of a pair of logically complementary propositions about personal identity are conceivable. I consider a number of explanations of this feature that draw on the metaphysics and the epistemology of personal identity, none of which prove to be satisfactory. I then argue that in order to give an adequate explanation, one needs to recognize an important characteristic of our concept of personal identity: it is such that if there are mental substances (or the like), they constitute personal identity. At the same time, there can still be persons if there are no such substances. Since this finding casts doubts on the way that thought experiments about personal identity are usually set up, I end by outlining its potential consequences for the debate over the identity of persons.