Currently, the most popular views about how to update
de se or selflocating
beliefs entail the one-third solution to the Sleeping Beauty problem.
Another widely held view is that an agent’s credences should be
2 In what follows, I will argue that there is a deep tension
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 satisfied. 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
In writing this essay, I benefited 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 Titelbaum,
Peter Vranas, Brian Weatherson, Jonathan Weisberg, Gideon Yaffe, and an anonymous
referee. My greatest debt is to Kenny Easwaran, for uncountably many invaluable
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 .