This essay argues that there is a conflict between the principle of Countable Additivity and standard views of how we should update centered or de se beliefs. The latter views, this essay argues, entail a general principle, which the essay calls the Generalized Thirder Principle, that conflicts with the principle of Countable Additivity. The essay then argues that the most plausible response to the conflict between these principles is to maintain that they are both valid and that, when an agent cannot satisfy them both, he or she is faced with a rational dilemma.
The essay is divided into five sections. In section 1, it generalizes the one-half and one-third solutions to the Sleeping Beauty problem. In section 2, it argues that while the Generalized Halfer Principle is compatible with Countable Additivity (CA), the Generalized Thirder Principle (GTP) is not. In section 3, it argues that thirders are committed to the GTP since it follows from the premises that underlie each of the main arguments for the one-third solution. In section 4, it discusses the relevance of Dutch book arguments to the conflict between CA and the GTP. And the essay argues that both synchronic and diachronic Dutch book arguments give greater support to CA than to the GTP. In the final section, the essay considers the possible responses to the conflict between CA and the GTP and argues that the most plausible response is to acknowledge the possibility of rational dilemmas.