“At any point in the history that Beauty remembers in step 2 of step 3, the proposition has a simple, single truth value.”
No, it doesn’t. This boils down to a question of identity. Absent any means of uniquely identifying the day—such as, “the day in which a black marble is on the dresser”—there is a fundamental ambiguity.
At any point in the history that Beauty remembers when she is in one of those steps, the proposition M, “Today is Monday,” has a simple, single truth value. All day. Either day. If she is in step 2, it is “true.” If she is in step 3, it is “false.”
The properties of “indexicals” that you are misusing apply when, within her current memory state, the value of “today” could change. Not within the context of the overarching experiment.
This has nothing to do with whether she knows what that truth value is. In fact, probability is how we represent the “fundamental ambiguity” that the simple, single truth value belonging to a proposition is unknown to us. If you want to argue this point, I suggest that you try looking for the forest through the trees.
If Beauty’s remembered experiences and mental state are identical at a point in time on Monday and another point in time on Tuesday, then “today” becomes ill-defined for her.
I tell you that I will flip a coin, ask a question, and then repeat the process.
If the question is “What is the probability that the coin is showing Heads?”, and I require an answer before I repeat the flip, then coin’s state has a simple, single truth value that you can represent with a probability.
If the question is “What is the probability that the coin is showing Heads?”, and I require an answer only at after the second flip, the question only applies to the second since it asks about a current state.But it has a simple, single truth value that you can represent with a probability.
If the question is “What is the probability of showing Heads?” then the we have the logical conundrum you describe.
“Showing” is an indexical. It can change over time. But it is only an issue if we refer to it in the context of a range of time where it does change. That’s why indexicals are a problem in general, but maybe not in a specific case.
“Today” is never ill-defined for Beauty.
“To an awake Beauty, the “experiment” she sees consists of Sunday and a single day after it.”
No, it doesn’t. She knows quite well that if the coin lands Tails, she will awaken on two separate days. It doesn’t matter that she can only remember one of them.
The entirety of the experiment includes Sunday, Wednesday, and two other days. She knows that. The portion that exists in her memory state at the time she is asked to provide an answer consists of Sunday (when she learned it all), which cannot be “Today,” and Today, which has a simple, single value.
I do not understand why you are so insistent on using “propositions” that include indexicals
Because the property that defines an indexical is that it can change over the domain where it is evaluated. Beauty is asked for her answer within a domain where “Today” does not change.
You didn’t answer my questions, about the variable Sleeping Beauty Problem.
They’re irrelevant.
I’ve learned from experience that I need halfers to answer them while they seem irrelevant. Otherwise, they argue that there is a difference, but can’t say what that difference is. Yes, this has happened more than once.
Each of the four card outcomes leads to a problem equivalent to the first. But randomly choosing one of four problems equivalent to the first problem doesn’t tell you what the solution to the first problem is.
Not yet, but it does tell you that the same answer applies to the original problem, and to the random-card problem.
So use four Beauties. Deal one card to each, but don’t show them. And flip the coin on Sunday (necessary since we need the result on Monday).
In your step 2, bring the three awake volunteers together to discuss their answers. Tell them, truthfully, what they already know: “One of you was dealt card where the coin value matches the flip we performed on Sunday. Two were dealt a card with the opposite coin result. What probability should you assign the propositions that each of you is the one whose card matches?”
There are three possibilities. Each must have the same probability, since they have no information that distinguishes any one from the other. The probabilities must add up to 1.
[Kinda speaking from my experience as a moderator here, but not actually really doing anything super mod-related]: I haven’t been able to follow the details from this conversation, and I apologize for that, but from the outside it does really look like you two are talking past each other. I don’t know what the best way to fix that is, or even whether I am right, but my guess is that it’s better to retire this thread for now and continue some other time. I am also happy to offer some more moderation if either of you requests that.
Also feel free to ignore this and just continue with your discussion, but it seemed better to give you two an out, if either of you feels like you are wasting time but are forced to continue talking for some reason or another.
At any point in the history that Beauty remembers when she is in one of those steps, the proposition M, “Today is Monday,” has a simple, single truth value. All day. Either day. If she is in step 2, it is “true.” If she is in step 3, it is “false.”
The properties of “indexicals” that you are misusing apply when, within her current memory state, the value of “today” could change. Not within the context of the overarching experiment.
This has nothing to do with whether she knows what that truth value is. In fact, probability is how we represent the “fundamental ambiguity” that the simple, single truth value belonging to a proposition is unknown to us. If you want to argue this point, I suggest that you try looking for the forest through the trees.
I tell you that I will flip a coin, ask a question, and then repeat the process.
If the question is “What is the probability that the coin is showing Heads?”, and I require an answer before I repeat the flip, then coin’s state has a simple, single truth value that you can represent with a probability.
If the question is “What is the probability that the coin is showing Heads?”, and I require an answer only at after the second flip, the question only applies to the second since it asks about a current state.But it has a simple, single truth value that you can represent with a probability.
If the question is “What is the probability of showing Heads?” then the we have the logical conundrum you describe.
“Showing” is an indexical. It can change over time. But it is only an issue if we refer to it in the context of a range of time where it does change. That’s why indexicals are a problem in general, but maybe not in a specific case.
“Today” is never ill-defined for Beauty.
The entirety of the experiment includes Sunday, Wednesday, and two other days. She knows that. The portion that exists in her memory state at the time she is asked to provide an answer consists of Sunday (when she learned it all), which cannot be “Today,” and Today, which has a simple, single value.
Because the property that defines an indexical is that it can change over the domain where it is evaluated. Beauty is asked for her answer within a domain where “Today” does not change.
I’ve learned from experience that I need halfers to answer them while they seem irrelevant. Otherwise, they argue that there is a difference, but can’t say what that difference is. Yes, this has happened more than once.
Not yet, but it does tell you that the same answer applies to the original problem, and to the random-card problem.
So use four Beauties. Deal one card to each, but don’t show them. And flip the coin on Sunday (necessary since we need the result on Monday).
In your step 2, bring the three awake volunteers together to discuss their answers. Tell them, truthfully, what they already know: “One of you was dealt card where the coin value matches the flip we performed on Sunday. Two were dealt a card with the opposite coin result. What probability should you assign the propositions that each of you is the one whose card matches?”
There are three possibilities. Each must have the same probability, since they have no information that distinguishes any one from the other. The probabilities must add up to 1.
They are all 1⁄3.
[Kinda speaking from my experience as a moderator here, but not actually really doing anything super mod-related]: I haven’t been able to follow the details from this conversation, and I apologize for that, but from the outside it does really look like you two are talking past each other. I don’t know what the best way to fix that is, or even whether I am right, but my guess is that it’s better to retire this thread for now and continue some other time. I am also happy to offer some more moderation if either of you requests that.
Also feel free to ignore this and just continue with your discussion, but it seemed better to give you two an out, if either of you feels like you are wasting time but are forced to continue talking for some reason or another.