You point out that Elga’s analysis is based on an unproven assertion; that “it is Monday” and “it is Tuesday” are legitimate propositions. As far as I know, there is no definition of what can, or cannot, be used as a proposition. In other words, your analysis is based on the equally unproven assertion that they are not valid. Can remove the need to decide?
On Sunday, the steps of the following experiment are explained to Beauty, and she is put to sleep with a drug that somehow records her memory state. After she is put to sleep, two coins are flipped; a quarter and a nickel.
On Monday, Beauty is awakened. While awake she obtains no information that would help her infer the day, or if she has been previously awakened. If the two coins are not both showing Heads, she is interviewed. An hour after being awakened, she is put back to sleep with a drug that resets her memory to the recorded state.
The nickel is turned over.
On Tuesday, Beauty is awakened. While awake she obtains no information that would help her infer the day, or if she has been previously awakened. If the two coins are not both showing Heads, she is interviewed. An hour after being awakened, she is put back to sleep with a drug that resets her memory to the recorded state.
On Wednesday, Beauty is awakened once more and told that the experiment is over.
In each interview, Beauty is asked for her epistemic probability that the quarter is showing Heads.
Non-consequential option: replace “Beauty is awakened … If the two coins show the same face, she is interviewed” with “If the two coins show the same face, Beauty is awakened and interviewed”.
I hope we can all agree that the order of the potential awakenings changes nothing. She is not asked whether, and her answer cannot depend on if, it is Monday or Tuesday. Or which day she might sleep through. So this Beauty no longer cares about the propositions “it is Monday” and “it is Tuesday.”
Beauty can assess the probability space that describes the two coins before the decision to interview her (or to wake her) was made. Most significantly, it doesn’t change when the nickel has been manually turned over. Yes, the day changes the path it takes to get there, but the states look identical. In that state, there are four possible outcomes: {HH,HT,TH,TT}. The probability distribution is {1/4,1/4,1/4,1/4}.
But when she is interviewed, she knows that {HH} has been ruled out. The probability distribution can be updated to {0,1/3,1/3,1/3}.
+++++
The point of contention in the Sleeping Beauty Problem is whether the probability state at the end of your step 1 is the same state as at the beginning of your steps 2 and 3. The mere fact that the two steps can be distinguished, and can result in different paths, demonstrates that they are not. If your definition of what constitutes a “valid proposition” cannot model this difference, then I suggest that it is that definition that is faulty.
And yes, there are other ways that I can demonstrate that the answer must be 1⁄3.
+++++
Note: The worst red herring in this thread is about how Beauty might be able to tell how she has aged. This is a thought problem in probability, not an exercise in human physiology. We must assume that the mechanisms described in the problem function ideally as described. That includes “While awake she obtains no information that would help her infer the day of the week.” Considering how these mechanisms might not be achievable is not productive.
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.