Imaging and Sleeping Beauty: A Case for Double Halfers

Conference: Theoretical Aspects of Rationality and Knowledge (TARK)(Elga 2000) introduced philosophers to the troubling scenario of Sleeping Beauty. On Sunday evening (t0), Sleeping Beauty is put to sleep by an experimental philosopher. She is awaken on Monday morning and at this moment (t1), the experimenter doesn't tell her which day it is. Some time later (t2), she is told that it is actually Monday. At this point, what follows depends on the toss of a fair coin that took place on Sunday evening - Sleeping Beauty is not aware of the outcome. If the coin landed heads (HEADS), then Sleeping Beauty is put to sleep until the end of the week. If the coin landed tails (TAILS), then Sleeping Beauty is awaken on Tuesday morning. The crucial fact is that a drug that is given to her is such that she cannot distinguish her awakening on Monday from her awakening on Tuesday. Of course, Sleeping Beauty is perfectly informed of every detail of the protocol before the experiment. The question that has drawn so much attention since (Elga 2000) is the following: what should be Sleeping Beauty's degree of belief that HEADS? Actually, the question will be asked at two different times: at t1 - when Sleeping Beauty is just awaken on Monday - and t2 - when Sleeping Beauty has been told that it is Monday. Let us call the first question Q1 and the second Q2. In the sequel, Pi (i ε {0, 1, 2}) will denote Sleeping Beauty's credence at ti, that is, her beliefs concerning the relevant propositions