Sleeping Beauty Meets Descartes’s Demon:Thirding Leads to Skepticism.

Sleeping Beauty is put to sleep on Sunday, and a fair coin is flipped. If it lands Heads, she is woken on Monday; if Tails, on Monday and Tuesday, forgetting each waking. When she wakes, what should her credence be that the coin landed Heads? Halfers say ½, since she learns nothing just by waking up. Thirders, Adam Elga among them, say 1⁄3; over many runs, only a third of her wakings follow Heads.

Sleeping Beauty has two forms of uncertainty at once. Her uncertainty about the coin is factual: she does not know which possible world she is in, the world where the coin landed Heads or the one where it landed Tails. Her uncertainty about the day is indexical: she does not know when in that world she is. A Heads world has one epistemically possible center for her, Monday; a Tails world has two, Monday and Tuesday. Halfers and thirders disagree about how she should update her credences on waking, and each answer supports a more general rule. The Generalized Halfer Principle leaves one’s credence in each possible world unchanged, whatever its number of centers. The Generalized Thirder Principle multiplies it by that number and renormalizes, which turns Sleeping Beauty’s ½ and ½ into 1⁄3 and 2⁄3.

I argue that the Generalized Thirder Principle leads to global skepticism for anyone with substantively rational priors about being ‘skeptically tortured’—when one is made to dream, for some number of nights, that one is having a real experience—and that the Generalized Halfer Principle does not. We begin with:

666 Days of Torture. The epistemic terrors of the lab experiment are behind Sleeping Beauty. But they have left some trauma. Sleeping Beauty has a philosophical nightmare—an actual nightmare of Descartes’s Demon! The Demon says that he will cause her to have another 665 consecutive dreams, each seeming veridical, before she finally awakens.

Naturally, Sleeping Beauty doubts that Descartes’s Demon is real; after all, he appeared in a dream! But her credence that his threat is real is not quite zero. Suppose it is 0.01%. If the threat is real, the world she wakes in has 666 centers, so the Generalized Thirder Principle multiplies that credence by 666, and on waking her credence that she is in a skeptical scenario should be about 6%. It is vague whether a 6% credence makes one a skeptic. But fix any threshold past which it does. As the number of threatened nights n grows, her prior in the threat must fall at least as fast as 1/n, or else she should be a global skeptic upon waking up. But even priors falling in such a way is not enough:

St. Petersburg’s Demon. Descartes’s Demon tells Sleeping Beauty that he’s in a gambling mood. He will start with one night of torture and flip a coin; each time it lands Heads, he doubles the torture so far, and the torture ends when it lands Tails. So 2^k nights of torture require k Heads in a row.

Suppose Sleeping Beauty’s prior in being tortured for 2^k falls as 1/2^k. The Generalized Thirder Principle multiplies that prior by 2^k, the number of centers. Every length of torture then weighs equally after updating, and since there are infinitely many lengths, Sleeping Beauty’s credence that her experience is veridical goes to zero. Escaping this requires priors that fall strictly faster than 1/n.

I am the Demon. It’s time to come clean. I am Descartes’s Demon, and you are Sleeping Beauty. I will do exactly what I claimed I would in St. Petersburg’s Demon. Naturally, you doubt me. Cartesian demons typically do not reveal themselves in philosophy papers. But is your credence in my threat zero? If it is only 10⁻¹⁰⁰, and you trust my coin as you would any other, the Generalized Thirder Principle says you should be a global skeptic the next time you wake up.

So a thirder must either hold substantively irrational priors or, having read I am the Demon, rationally become a global skeptic upon waking. But it is not plausible that simply reading an example should make one a skeptic. The Generalized Halfer Principle faces neither horn. On it, Sleeping Beauty’s credences in worlds stay the same when she wakes up, so if her credence in the Demon’s threat was low enough for her not to be a skeptic before she woke up, it is low enough after.

The argument has the structure of a problem Yoaav Isaacs, John Hawthorne, and Jeffrey Sanford Russell raise for scientific confirmation. If Theory A entails one universe and Theory B entails 10¹⁰⁰, then unless our prior in Theory B is below 10⁻¹⁰⁰, we should favor Theory B from the armchair. If it is implausible that we can do armchair cosmology in just this way, it should be implausible for you to rule out veridical experience just by countenancing an indexically laden skeptical scenario. Assuming you, who have just read I am the Demon, are not orders of magnitude more skeptical the next time you wake up than you were this morning, you should reject the Generalized Thirder Principle.

Written for a seminar in Bayesian epistemology (Snow Zhang), UC Berkeley, 2025. An earlier version was given at the Wollheim Society in September 2023.

Questions, objections, or a better example? Write to me: patrick.pan@patrickpan.com.