You said: “The standard textbook definition of a proposition is a sentence that has a truth value of either true or false.
This is correct. And when a well-defined truth value is not known to an observer, the standard textbook definition of a probability (or confidence) for the proposition, is that there is a probability P that it is “true” and a probability 1-P that it is “false.”
For example, if I flip a coin but keep it hidden from you, the statement “The coin shows Heads on the face-up side” fits your definition of a proposition. But since you do not know whether it is true or false, you can assign a 50% probability to the result where “It shows Heads” is true, and a 50% probability the event where “it shows Heads” is false. This entire debate can be reduced to you confusing a truth value, with the probability of that truth value.
On Monday Beauty is awakened. While awake she obtains no information that would help her infer the day of the week. Later in the day she is put to sleep again.
During this part of the experiment, the statement “today is Monday” has the truth value “true”, and does not have the truth value “false.” So by your definition, it is a valid proposition. But Beauty does not know that it is “true.”
On Tuesday the experimenters flip a fair coin. If it lands Tails, Beauty is administered a drug that erases her memory of the Monday awakening, and step 2 is repeated.
During this part of the experiment, the statement “today is Monday” has the truth value “false”, and does not have the truth value “true.” So by your definition, it is a valid proposition. But Beauty dos not know that it is “false.”
In either case, the statement “today is Monday” is a valid proposition by the standard definition you use. What you refuse to acknowledge, is that it is also a proposition that Beauty can treat as “true” or “false” with probabilities P and 1-P.
You mis-characterize what Elga does. He never directly formulates the state M1, where Beauty is awake. Instead, he formulates two states that are derived from information being added to M1. I’ll call them M2A (Beauty learns the outcome is Tails) and M2B (Beauty learns that it is Monday). While he may not do it as formally as you want, he works backwards to show that three of the four components of a proper description of state M1 must have the same probability. What he skips over, is identifying the fourth component (whose probability is now zero).
What it seems Elga was trying to avoid—as everybody does—is that Beauty still “exists” on Tuesday, after Heads. She just can’t observe it. But it is a component you need to consider in your more formal modeling. To illustrate, here’s a simple re-structuring of your steps that changes nothing relevant to the question she is asked:
On Sunday the steps of the experiment are explained to Beauty, and she is put to sleep.
On Monday Beauty is awakened. She has no information that would help her infer the day of the week. Later in the day she is interviewed. Afterwards, she is administered a drug that resets her memory to its state when she was put to sleep on Sunday, and puts her to sleep again.
On Tuesday Beauty is awakened. She has no information that would help her infer the day of the week. The experimenters flip a fair coin. If it lands Tails, Beauty is interviewed again; if it lands Heads, she is not. In either case, she is then administered a drug that resets her memory to its state when she was put to sleep on Sunday, and puts her to sleep again.
On Wednesday Beauty is awakened once more and told that the experiment is over.
In the interview(s), Beauty is asked to give a probability for her belief that the coin in step 3 lands Heads.
I’m sure you can make this more formal, so I’ll be brief: State M, on Sunday, requires only proposition C describing what Beauty thinks the coin result is (*not* for what it actually is, which becomes deterministic at different times in different versions of the problem). There is no information in state M that favors either result, so the Principle of Indifference applies and the probability for each is 1⁄2.
State M1, when Beauty is first awakened, requires another proposition: D, for what day Beauty thinks it is (*not* for what day it actually is, which is deterministic). Due to the memory-reset drug, the same state M1 applies on both Monday and Tuesday. Since there is no information in state M1 that favors either result, the Principle of Indifference applies and the probability for each is 1⁄2. And (what seems to be overlooked by denying the existence of Tuesday when Beauty sleeps through it) D and H are independent. So M1 comprises four possible combinations of D and H that all have a probability of 1⁄4.
State M2 applies when Beauty is interviewed. The information that takes Beauty from M1 to M2 is that one of the four combinations is ruled out. The remaining three now have probability 1⁄3.
State M1 applies to your version of the problem at the point in time just before Beauty could be wakened, in either step 2 or step 3. It applies, and can be determined later when Beauty is awake, whether or not Beauty is awake at that time. Elga’s solution is essentially the same as mine, except he does it in two parts by adding more information to each. It just avoids identifying the component of the state that Beauty sleeps through.