Hi Charlie! Actually I complete agree with Vladimir on this: subjective probabilities are meaningless, meaningful questions are decision theoretic. When the sleeping beauty is asked "what day is it?" the question is meaningless because she is simultaneously in several different days (since identical copies of her are in different days).
Short version: consider Savage's theorem (fulfilling the conditions by offering Sleeping Beauty a bet along with the question "what day is it?", or by having Sleeping Beauty want to watch a sports game on Monday specifically, etc.). Savages theorem requires your agent to have a preference ordering over outcomes, and have things it can do that lead to different outcomes depending on the state of the world (events), and it states that consistent agents have probabilities over the state of the world.
On both days, Sleeping Beauty satisfies these desi...
Vladimir Slepnev (aka cousin_it) gives a popular introduction to logical counterfactuals and modal updateless decision theory in the Tel Aviv LessWrong meetup.
[https://www.youtube.com/watch?v=Ad30JlVh4dM&feature=youtu.be]