This paper starts out with a misrepresentation. “As a reminder, this is the Sleeping Beauty problem:”… and then it proceeds to describe the problem as Adam Elga modified it to enable his thirder solution. The actual problem that Elga presented was:
Some researchers are going to put you to sleep. During the two days[1] that your sleep will last, they will briefly wake you up either once or twice, depending on the toss of a fair coin (Heads: once; Tails: twice). After each waking, they will put you to back to sleep with a drug that makes you forget that waking.2 When you are first awakened[2], to what degree ought you believe that the outcome of the coin toss is Heads?
There are two hints of the details Elga will add, but these hints do not impact the problem as stated. At [1], Elga suggests that the two potential wakings occur on different days; all that is really important is that they happen at different times. At [2], the ambiguous “first awakened” clause is added. It could mean that SB is only asked the first time she is awakened; but that renders the controversy moot. With Elga’s modifications, only asking on the first awakening is telling SB that it is Monday. He appears to mean “before we reveal some information,” which is how Elga eliminates one of the three possible events he uses.
Elga’s implementation of this problem was to always wake SB on Monday, and only wake her on Tuesday if the coin result was Tails. After she answers the question, Elga then reveals either that it is Monday, or that the coin landed on Tails. Elga also included DAY=Monday or DAY=Tuesday as a random variable, which creates the underlying controversy. If that is proper, the answer is 1⁄3. If, as Neal argues, it is indexical information, it cannot be used this way and the answer is 1⁄2.
So the controversy was created by Elga’s implementation. And it was unnecessary. There is another implementation of the same problem that does not rely on indexicals.
Once SB is told the details of the experiment and put to sleep, we flip two coins: call then C1 and C2. Then we perform this procedure:
If both coins are showing Heads, we end the procedure now with SB still asleep.
Otherwise, we wake SB and ask for her degree of belief that coin C1 landed on Heads.
After she gives an answer, we put her back to sleep with amnesia.
After these steps are concluded, whether it happened in step 1 or step 3, we turn coin C2 over to show the opposite side. And then repeat the same procedure.
SB will thus be wakened once if coin C1 landed on Heads, and twice if Tails. Either way, she will not recall another waking. But that does not matter. She knows all of the details that apply to the current waking. Going into step 1, there were four possible, equally-likely combinations of (C1,C2); specifically, (H,H), (H,T), (T,H), and (T,T). But since she is awake, she knows that (H,H) was eliminated in step 1. In only one of the remaining, still equally-likely combinations, did coin C1 land on Heads.
The answer is 1⁄3. No indexical information was used to determine this. No reference the other potential waking, whether it occurs before or after this one, is needed. This implements Elga’s question exactly; this only possible issue that remains is if Elga’s implementation does.
This paper starts out with a misrepresentation. “As a reminder, this is the Sleeping Beauty problem:”… and then it proceeds to describe the problem as Adam Elga modified it to enable his thirder solution. The actual problem that Elga presented was:
There are two hints of the details Elga will add, but these hints do not impact the problem as stated. At [1], Elga suggests that the two potential wakings occur on different days; all that is really important is that they happen at different times. At [2], the ambiguous “first awakened” clause is added. It could mean that SB is only asked the first time she is awakened; but that renders the controversy moot. With Elga’s modifications, only asking on the first awakening is telling SB that it is Monday. He appears to mean “before we reveal some information,” which is how Elga eliminates one of the three possible events he uses.
Elga’s implementation of this problem was to always wake SB on Monday, and only wake her on Tuesday if the coin result was Tails. After she answers the question, Elga then reveals either that it is Monday, or that the coin landed on Tails. Elga also included DAY=Monday or DAY=Tuesday as a random variable, which creates the underlying controversy. If that is proper, the answer is 1⁄3. If, as Neal argues, it is indexical information, it cannot be used this way and the answer is 1⁄2.
So the controversy was created by Elga’s implementation. And it was unnecessary. There is another implementation of the same problem that does not rely on indexicals.
Once SB is told the details of the experiment and put to sleep, we flip two coins: call then C1 and C2. Then we perform this procedure:
If both coins are showing Heads, we end the procedure now with SB still asleep.
Otherwise, we wake SB and ask for her degree of belief that coin C1 landed on Heads.
After she gives an answer, we put her back to sleep with amnesia.
After these steps are concluded, whether it happened in step 1 or step 3, we turn coin C2 over to show the opposite side. And then repeat the same procedure.
SB will thus be wakened once if coin C1 landed on Heads, and twice if Tails. Either way, she will not recall another waking. But that does not matter. She knows all of the details that apply to the current waking. Going into step 1, there were four possible, equally-likely combinations of (C1,C2); specifically, (H,H), (H,T), (T,H), and (T,T). But since she is awake, she knows that (H,H) was eliminated in step 1. In only one of the remaining, still equally-likely combinations, did coin C1 land on Heads.
The answer is 1⁄3. No indexical information was used to determine this. No reference the other potential waking, whether it occurs before or after this one, is needed. This implements Elga’s question exactly; this only possible issue that remains is if Elga’s implementation does.