The need for distinguishing between SIA and SSA is not needed in the Sleeping Beauty Problem. It was inserted into Adam Elga’s problem when he changed it from the one he posed, to the one he solved. I agree that they should have the same answer, which may help in choosing SIA or SSA, but it is not needed. This is what he posed:
“Some researchers are going to put you to sleep. During the two days 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. When you are first awakened, to what degree ought you believe that the outcome of the coin toss is Heads?”
The need was created when Elga created a schedule for waking, that treated two days differently. So that “existence” became an alleged issue. There is a better way. First, consider this “little experiment”:
Two coins, C1 and C2, will be randomly arranged so that the four possible combinations, HH, HT, TH, and TT, are equally likely.
If either coin is showing Tails, you will be asked for what you believe to be the probability that coin C1 is showing Heads.
If both are showing Heads, you will not be asked a question.
The answer should be obvious: From the fact that you are in step 2, as established by being asked a question, the combination HH is eliminated. What happens in step 3 is irrelevant, since the question is not asked there. In only one of the three remaining combinations is coin C1 showing Heads, so there is a 1⁄3 chance that coin C1 is showing heads.
To implement the experiment Elga proposed—not the one he solved—put SB to sleep on Sunday night, and flip the two coins. On Monday, perform the “little experiment” using the result of the flips. You will need to wake SB if step 2 is executed. What you do in step 3 is still irrelevant, but can include leaving her asleep. Afterwards, if she is awake, put her back to sleep with amnesia. AND THEN TURN COIN C2 OVER. On Tuesday, perform the “little experiment” again, using the modified result of the flips.
SB does not need to consider any other observers than herself to answer the question, because she knows every detail of the “little experiment.” If she is awake, and asked a question, the coins were arranged as described in step 1 and she is in step 2. The answer is 1⁄3.
The need for distinguishing between SIA and SSA is not needed in the Sleeping Beauty Problem. It was inserted into Adam Elga’s problem when he changed it from the one he posed, to the one he solved. I agree that they should have the same answer, which may help in choosing SIA or SSA, but it is not needed. This is what he posed:
“Some researchers are going to put you to sleep. During the two days 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. When you are first awakened, to what degree ought you believe that the outcome of the coin toss is Heads?”
The need was created when Elga created a schedule for waking, that treated two days differently. So that “existence” became an alleged issue. There is a better way. First, consider this “little experiment”:
Two coins, C1 and C2, will be randomly arranged so that the four possible combinations, HH, HT, TH, and TT, are equally likely.
If either coin is showing Tails, you will be asked for what you believe to be the probability that coin C1 is showing Heads.
If both are showing Heads, you will not be asked a question.
The answer should be obvious: From the fact that you are in step 2, as established by being asked a question, the combination HH is eliminated. What happens in step 3 is irrelevant, since the question is not asked there. In only one of the three remaining combinations is coin C1 showing Heads, so there is a 1⁄3 chance that coin C1 is showing heads.
To implement the experiment Elga proposed—not the one he solved—put SB to sleep on Sunday night, and flip the two coins. On Monday, perform the “little experiment” using the result of the flips. You will need to wake SB if step 2 is executed. What you do in step 3 is still irrelevant, but can include leaving her asleep. Afterwards, if she is awake, put her back to sleep with amnesia. AND THEN TURN COIN C2 OVER. On Tuesday, perform the “little experiment” again, using the modified result of the flips.
SB does not need to consider any other observers than herself to answer the question, because she knows every detail of the “little experiment.” If she is awake, and asked a question, the coins were arranged as described in step 1 and she is in step 2. The answer is 1⁄3.