Your argument at that link is interesting, but I can see why Halfers would just say it’s a different problem.
For Beauty and the Prince, I start with a version where Beauty and the Prince can talk to each other, which obviously isn’t the same as the usual Sleeping Beauty problem. Supposing that it’s agreed that they should both assess the probability of Heads as 1⁄3 in this version, we then go on to a version where Beauty can see the Prince, but not talk with him. But she knows perfectly well what he would say anyway, so does that matter? And if we then put a curtain between Beauty and the Prince, so she can’t see him, though she knows he is there, does that change anything? If we move the Prince to a different room a thousand miles away, would that change things? Finally, does getting rid of the Prince altogether matter?
If none of these steps from Beauty and the Prince to the usual Sleeping Beauty matters, then the answers should be the same. So Halfers would have to claim that one or more steps does matter (or that the answer is 1⁄2 for the full Beauty and the Prince problem, but I see that as less likely). Perhaps they will claim one of these steps matters, but I see problems with this. For instance, if a Halfer thinks getting rid of the Prince altogether is different from him being in a room a thousand miles away, it seems that they would be committed to the 1⁄2 answer being sensitive to all sorts of details of the world that one would normally consider irrelevant (and which are assumed irrelevant in the usual problem statement).
I suppose the question becomes, “Why can’t Sleeping Beauty copy the Prince’s answer when he doesn’t count on Monday given that he still exists?”. And indeed, the Prince gives an answer of 1⁄3 to regardless of whether or not his answer counts at that point.
I guess the answer is that halvers don’t believe that you can answer: “If Sleeping Beauty is awake, what are the chance that the coin came up heads?” without de-indexicalising the situation first. After de-indexicalising it become, “If the Prince counts and Sleeping Beauty is awake, what are the odds that the coin came up heads?” (which is true 1⁄3 of the time).
Now that the statement has been de-indexicalised, it’s clear that including possibilities where the Prince doesn’t count or Sleeping Beauty isn’t awake doesn’t change the probability as they are filtered out by the “if” clause.
Next we de-indexicalise the question that Sleeping Beauty asks, “If Sleeping Beauty is awake and she counts, what are the odds that the coin came up heads?” It’s now clear that this includes a different set of possibilities than what the Prince asks, so they reach different answers. So even though the original questions is the same, the question becomes different once it is de-indexicalised. So she can’t just go ask the Prince, so he’s answering a different question.
I’ve noticed something that may explain some of the confusion. You say above:
...halvers don’t believe that you can answer: “If Sleeping Beauty is awake, what are the chance that the coin came up heads?” without de-indexicalising the situation first.
But in the Sleeping Beauty problem as usually specified, the question is what probability Beauty should assign to Heads, not what some external observer should think she should be doing. Beauty is in no doubt about who she is (eg, she’s the person who just stubbed her toe on this bedpost here) even though she doesn’t know what day of the week it is.
Well, there’s two kinds of probability that we could calculate: we could roughly call them subjective probability (which gives an answer of 1⁄3) or objective probability (which gives an answer of 1⁄2).
The confusing part is that you can ask an “objective” observer about the subjective probability relative to beauty and they’d say 1⁄3 or you can ask a “subjective” observer like beauty about the objective probability and they’d say 1⁄2.
This is further obscured by subjective probability already having a definition, so I really need to find a different name.
Anyway, I hope that most of the questions will be cleared up when I write up a more comprehensive post dealing the various issues that have already been raised, though I’ll probably wait at least a week (quite possibly two) because I think people need time to digest all the conversation that has already occurred.
Your argument at that link is interesting, but I can see why Halfers would just say it’s a different problem.
For Beauty and the Prince, I start with a version where Beauty and the Prince can talk to each other, which obviously isn’t the same as the usual Sleeping Beauty problem. Supposing that it’s agreed that they should both assess the probability of Heads as 1⁄3 in this version, we then go on to a version where Beauty can see the Prince, but not talk with him. But she knows perfectly well what he would say anyway, so does that matter? And if we then put a curtain between Beauty and the Prince, so she can’t see him, though she knows he is there, does that change anything? If we move the Prince to a different room a thousand miles away, would that change things? Finally, does getting rid of the Prince altogether matter?
If none of these steps from Beauty and the Prince to the usual Sleeping Beauty matters, then the answers should be the same. So Halfers would have to claim that one or more steps does matter (or that the answer is 1⁄2 for the full Beauty and the Prince problem, but I see that as less likely). Perhaps they will claim one of these steps matters, but I see problems with this. For instance, if a Halfer thinks getting rid of the Prince altogether is different from him being in a room a thousand miles away, it seems that they would be committed to the 1⁄2 answer being sensitive to all sorts of details of the world that one would normally consider irrelevant (and which are assumed irrelevant in the usual problem statement).
I suppose the question becomes, “Why can’t Sleeping Beauty copy the Prince’s answer when he doesn’t count on Monday given that he still exists?”. And indeed, the Prince gives an answer of 1⁄3 to regardless of whether or not his answer counts at that point.
I guess the answer is that halvers don’t believe that you can answer: “If Sleeping Beauty is awake, what are the chance that the coin came up heads?” without de-indexicalising the situation first. After de-indexicalising it become, “If the Prince counts and Sleeping Beauty is awake, what are the odds that the coin came up heads?” (which is true 1⁄3 of the time).
Now that the statement has been de-indexicalised, it’s clear that including possibilities where the Prince doesn’t count or Sleeping Beauty isn’t awake doesn’t change the probability as they are filtered out by the “if” clause.
Next we de-indexicalise the question that Sleeping Beauty asks, “If Sleeping Beauty is awake and she counts, what are the odds that the coin came up heads?” It’s now clear that this includes a different set of possibilities than what the Prince asks, so they reach different answers. So even though the original questions is the same, the question becomes different once it is de-indexicalised. So she can’t just go ask the Prince, so he’s answering a different question.
I’ve noticed something that may explain some of the confusion. You say above:
...halvers don’t believe that you can answer: “If Sleeping Beauty is awake, what are the chance that the coin came up heads?” without de-indexicalising the situation first.
But in the Sleeping Beauty problem as usually specified, the question is what probability Beauty should assign to Heads, not what some external observer should think she should be doing. Beauty is in no doubt about who she is (eg, she’s the person who just stubbed her toe on this bedpost here) even though she doesn’t know what day of the week it is.
Speaking very roughly:
Well, there’s two kinds of probability that we could calculate: we could roughly call them subjective probability (which gives an answer of 1⁄3) or objective probability (which gives an answer of 1⁄2).
The confusing part is that you can ask an “objective” observer about the subjective probability relative to beauty and they’d say 1⁄3 or you can ask a “subjective” observer like beauty about the objective probability and they’d say 1⁄2.
This is further obscured by subjective probability already having a definition, so I really need to find a different name.
Anyway, I hope that most of the questions will be cleared up when I write up a more comprehensive post dealing the various issues that have already been raised, though I’ll probably wait at least a week (quite possibly two) because I think people need time to digest all the conversation that has already occurred.