Your whole analysis rests on the idea that “it is Monday” is a legitimate proposition. I’ve responded to this many other places in the comments, so I’ll just say here that a legitimate proposition needs to maintain the same truth value throughout the entire analysis (Sunday, Monday, Tuesday, and Wednesday). Otherwise it’s a predicate. The point of introducing R(y,d) is that it’s as close as we can get to what you want “it is Monday” to mean.
Well, I never checked back to see replies, and just tripped back across this.
The error made by halfers is in thinking “the entire analysis” spans four days. Beauty is asked for her assessment, based on her current state of knowledge, that the coin landed Heads. In this state of knowledge, the truth value of the proposition “it is Monday” does not change.
But there is another easy way to find the answer, that satisfies your criterion. Use four Beauties to create an isomorphic problem. Each will be told all of the details on Sunday; that each will be wakened at least once, and maybe twice, over the next two days based on the same coin flip and the day. But only three will be wakened on each day. Each is assigned a different combination of a coin face, and a day, for the circumstances where she will not be wakened. That is, {H,Mon}, {T,Mon}, {H,Tue}, and {T,Tue}.
On each of the two days during the experiment, each awake Beauty is asked for the probability that she will be wakened only once. Note that the truth value of this proposition is the same throughout the experiment. It is only the information a Beauty has that changes. On Sunday or Wednesday, there is no additional information and the answer is 1⁄2. On Monday or Tuesday, an awake Beauty knows that there are three awake Beauties, that the proposition is true for exactly one of them, and that there is no reason for any individual Beauty to be more, or less, likely than the others to be that one. The answer with this knowledge is 1⁄3.
Your whole analysis rests on the idea that “it is Monday” is a legitimate proposition. I’ve responded to this many other places in the comments, so I’ll just say here that a legitimate proposition needs to maintain the same truth value throughout the entire analysis (Sunday, Monday, Tuesday, and Wednesday). Otherwise it’s a predicate. The point of introducing R(y,d) is that it’s as close as we can get to what you want “it is Monday” to mean.
Well, I never checked back to see replies, and just tripped back across this.
The error made by halfers is in thinking “the entire analysis” spans four days. Beauty is asked for her assessment, based on her current state of knowledge, that the coin landed Heads. In this state of knowledge, the truth value of the proposition “it is Monday” does not change.
But there is another easy way to find the answer, that satisfies your criterion. Use four Beauties to create an isomorphic problem. Each will be told all of the details on Sunday; that each will be wakened at least once, and maybe twice, over the next two days based on the same coin flip and the day. But only three will be wakened on each day. Each is assigned a different combination of a coin face, and a day, for the circumstances where she will not be wakened. That is, {H,Mon}, {T,Mon}, {H,Tue}, and {T,Tue}.
On each of the two days during the experiment, each awake Beauty is asked for the probability that she will be wakened only once. Note that the truth value of this proposition is the same throughout the experiment. It is only the information a Beauty has that changes. On Sunday or Wednesday, there is no additional information and the answer is 1⁄2. On Monday or Tuesday, an awake Beauty knows that there are three awake Beauties, that the proposition is true for exactly one of them, and that there is no reason for any individual Beauty to be more, or less, likely than the others to be that one. The answer with this knowledge is 1⁄3.