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.
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.