I can’t tell you whether it is considered to be sound. In my opinion, it is. But I do know that the issues causing the controversy to continue for 23 years are not a part of the actual problem. And so are unnecessary.
If anyone thinks that sounds odd, they should go back and read the question that Elga posed. It does not mention Monday, Tuesday, or that a Tails-only waking follows a mandatory waking. Those were elements he introduced into the problem for his thirder solution.
In the problem as posed, the subject (I’ll call her SB, even though in the posed problem it is “you”) is woken once, or twice, based on the outcome of a fair coin flip (Heads=once, Tails=twice). Elga enacted that description with a mandatory waking on Monday, and an optional one on Tuesday. This way, could create two partial solutions by revealing two different bits of information that removed one of the three possibilities. The unfortunate side effect of this was that the conditions surrounding Monday and Tuesday are different; thirders require Tuesday to be part of a different outcome, and halfers insist it is the same outcome as Monday+Tails.
And that is what is unnecessary. Instead of that Monday/Tuesday schedule, simply flip two coins (call them C1 and C2) after SB is first put to sleep. Then:
If both coins are showing Heads, skip to step 6.
Wake SB.
Ask SB for her credence that coin C1 is showing Heads.
Wait for her answer.
Put her back to sleep with amnesia.
End this stage of the experiment.
Now turn coin C2 over to show its other face, and repeat these steps.
When SB is woken, she knows that: (A) She is in step 2 of a pass thru these six steps. (B) In step 1, there were four equally-likely states for the two coins: HH, HT, TH, and TT. (C) Since she is awake, the state HH is eliminated but the other three remain equally likely. (D) In only one of those states is C1 showing Heads. So she can confidently state that her credence is 1⁄3.
The difference between this, and how Elga enacted the problem he posed, is that here there is no ambiguity about how she arrived at her current, awake situation.
I originally upvoted your answer as it presented an interesting version of the SB which I didn’t see before.
However, it has similar problems to the original and it’s even more convoluted, you may notice that there is no need to throw the second coin other than to confuse everyone. You can just put it tails and then turn over to get the exact same results.
HH, HT, TH and TT are not four elementary outcomes of the experiment, as there is causal connection between them, even if such formulation makes it less obvious and harder to talk about it.
The “need to throw the second coin” is to make the circumstances underlying any awakening the same. Using a random method is absolutely necessary, although it doesn’t have to be flipped. The director could say that she is choosing her favorite coin face. As long as SB has no reason to think that is more likely to be one result than the other, it works. The reason Elga’s version is debated, is because it essentially flips Tails first for the second coin.
What the coins are showing at the moment are elementary outcomes of the experiment-within-the-experiment. “Causal connection,” whatever you think that means, has nothing to do with it since we are talking about a fixed state of the coins from the examination to the question.
Which statement here to you disagree with?
Just before she was awakened, the possible states of the coins were {HH, HT, TH, TT} with known probabilities {1/4, 1⁄4, 1⁄4, 1⁄4}.
SB knows this with certainty. That is, she did not have to be awake at that moment to know that this was true at that time.
With no change to the coins, they were examined and the next part of the experiment depends on the actual state.
SB knows this with certainty.
Since she is awake, she knows that the possible change is that the state HH is eliminated.
SB knows this with certainty.
The probabilities in step 1 constitute prior probabilities, and her credence in the state is the same as the posterior probabilities
But there are other ways, that have whatever “causal connection” you think is important. That is, they match the way Elga modified the experiment. That’s another point you seem to ignore, that the two-day version differs from the actual question by more than mine does.
Try using four volunteers instead of one, but one coin for all. Each will be wakened on both Monday and Tuesday, except:
SB1 will be left asleep on Tuesday, if the coin landed on Tails. This is Elga’s SB, with the same “causal connection.”
SB2 will be left asleep on Tuesday, if the coin landed on Heads.
SB3 will be left asleep on Monday, if the coin landed on Tails.
SB4 will be left asleep on Monday, if the coin landed on Heads.
This way, on each of Monday and Tuesday, exactly three volunteers will be wakened. And unless you think the specific days or coin faces have differing qualities, the same “causal connections” exist for all.
Each volunteer will be asked for her credence that the coin landed on the result where she would be wakened only once. With the knowledge of this procedure, an awake volunteer knows that she is one of exactly three, and that their credence/probability cannot be different due to symmetry. Since the issue “the coin landed on the result where you would be wakened only once” applies to only one of these three, this credence is 1⁄3.
I can’t tell you whether it is considered to be sound. In my opinion, it is. But I do know that the issues causing the controversy to continue for 23 years are not a part of the actual problem. And so are unnecessary.
If anyone thinks that sounds odd, they should go back and read the question that Elga posed. It does not mention Monday, Tuesday, or that a Tails-only waking follows a mandatory waking. Those were elements he introduced into the problem for his thirder solution.
In the problem as posed, the subject (I’ll call her SB, even though in the posed problem it is “you”) is woken once, or twice, based on the outcome of a fair coin flip (Heads=once, Tails=twice). Elga enacted that description with a mandatory waking on Monday, and an optional one on Tuesday. This way, could create two partial solutions by revealing two different bits of information that removed one of the three possibilities. The unfortunate side effect of this was that the conditions surrounding Monday and Tuesday are different; thirders require Tuesday to be part of a different outcome, and halfers insist it is the same outcome as Monday+Tails.
And that is what is unnecessary. Instead of that Monday/Tuesday schedule, simply flip two coins (call them C1 and C2) after SB is first put to sleep. Then:
If both coins are showing Heads, skip to step 6.
Wake SB.
Ask SB for her credence that coin C1 is showing Heads.
Wait for her answer.
Put her back to sleep with amnesia.
End this stage of the experiment.
Now turn coin C2 over to show its other face, and repeat these steps.
When SB is woken, she knows that: (A) She is in step 2 of a pass thru these six steps. (B) In step 1, there were four equally-likely states for the two coins: HH, HT, TH, and TT. (C) Since she is awake, the state HH is eliminated but the other three remain equally likely. (D) In only one of those states is C1 showing Heads. So she can confidently state that her credence is 1⁄3.
The difference between this, and how Elga enacted the problem he posed, is that here there is no ambiguity about how she arrived at her current, awake situation.
I originally upvoted your answer as it presented an interesting version of the SB which I didn’t see before.
However, it has similar problems to the original and it’s even more convoluted, you may notice that there is no need to throw the second coin other than to confuse everyone. You can just put it tails and then turn over to get the exact same results.
HH, HT, TH and TT are not four elementary outcomes of the experiment, as there is causal connection between them, even if such formulation makes it less obvious and harder to talk about it.
The “need to throw the second coin” is to make the circumstances underlying any awakening the same. Using a random method is absolutely necessary, although it doesn’t have to be flipped. The director could say that she is choosing her favorite coin face. As long as SB has no reason to think that is more likely to be one result than the other, it works. The reason Elga’s version is debated, is because it essentially flips Tails first for the second coin.
What the coins are showing at the moment are elementary outcomes of the experiment-within-the-experiment. “Causal connection,” whatever you think that means, has nothing to do with it since we are talking about a fixed state of the coins from the examination to the question.
Which statement here to you disagree with?
Just before she was awakened, the possible states of the coins were {HH, HT, TH, TT} with known probabilities {1/4, 1⁄4, 1⁄4, 1⁄4}.
SB knows this with certainty. That is, she did not have to be awake at that moment to know that this was true at that time.
With no change to the coins, they were examined and the next part of the experiment depends on the actual state.
SB knows this with certainty.
Since she is awake, she knows that the possible change is that the state HH is eliminated.
SB knows this with certainty.
The probabilities in step 1 constitute prior probabilities, and her credence in the state is the same as the posterior probabilities
Pr(HT|Awake) = Pr(HT)/[Pr(HT)+Pr(TH)+Pr(TT)] = 1⁄3.
But there are other ways, that have whatever “causal connection” you think is important. That is, they match the way Elga modified the experiment. That’s another point you seem to ignore, that the two-day version differs from the actual question by more than mine does.
Try using four volunteers instead of one, but one coin for all. Each will be wakened on both Monday and Tuesday, except:
SB1 will be left asleep on Tuesday, if the coin landed on Tails. This is Elga’s SB, with the same “causal connection.”
SB2 will be left asleep on Tuesday, if the coin landed on Heads.
SB3 will be left asleep on Monday, if the coin landed on Tails.
SB4 will be left asleep on Monday, if the coin landed on Heads.
This way, on each of Monday and Tuesday, exactly three volunteers will be wakened. And unless you think the specific days or coin faces have differing qualities, the same “causal connections” exist for all.
Each volunteer will be asked for her credence that the coin landed on the result where she would be wakened only once. With the knowledge of this procedure, an awake volunteer knows that she is one of exactly three, and that their credence/probability cannot be different due to symmetry. Since the issue “the coin landed on the result where you would be wakened only once” applies to only one of these three, this credence is 1⁄3.