Sleeping Beauty, Countable Additivity, and Rational Dilemmas Jacob Ross University of Southern California Currently, the most popular views about how to update de se or self- locating beliefs entail the one-third solution to the Sleeping Beauty prob- lem. 1 Another widely held view is that an agent’s credences should be countably additive. 2 In what follows, I will argue that there is a deep ten- sion between these two positions. For the assumptions that underlie the one-third solution to the Sleeping Beauty problem entail a more general principle, which I call the Generalized Thirder Principle, and there are situations in which the latter principle and the principle of Countable Additivity cannot be jointly satisfed. The most plausible response to this tension, I argue, is to accept both of these principles and to maintain that when an agent cannot satisfy them both, he or she is faced with a rational dilemma. In writing this essay, I benefted enormously from comments from, and discussions with, Frank Arntzenius, Cian Dorr, Adam Elga, Branden Fitelson, Matthew Kotzen, David Manley, Chris Meacham, Sarah Moss, Mark Schroeder, Teddy Seidenfeld, Mike Titel- baum, Peter Vranas, Brian Weatherson, Jonathan Weisberg, Gideon Yaffe, and an anony- mous referee. My greatest debt is to Kenny Easwaran, for uncountably many invaluable discussions. 1. This solution is defended in Arntzenius 2003, Dorr 2002, Draper and Pust 2008, Elga 2000, Hitchcock 2004, Horgan 2004, Monton 2002, Seminar 2008, Stalnaker 2008a, Titelbaum 2008, and Weintraub 2004. 2. See, for example, Howard 2006, Walley 1991, Weatherson 2005, and Williamson 1999. The principle of Countable Additivity traces back to Kolmogorov 1956 [1933]. Philosophical Review, Vol. 119, No. 4, 2010 DOI 10.1215/00318108-2010-010  2010 by Cornell University 411