“As for the Tuesday problem, that seems to go away if you consider the process that told you at least one of them was born on a Tuesday”—I don’t think we disagree about how the Tuesday problem works. The argument I’m making is that your method of calculating probabilities is calculating b) when we actually care about a).
To bring it back to the Tuesday problem, let’s suppose you’ll meet the first son on Monday and the second on Tuesday, but in between your memory will be wiped. You wake up on a day (not knowing what day it is) and you notice that they are a boy. This observation corresponds to a) meeting a random son and noting that he was born on Tuesday, not b) discovering that one of the two sons (you don’t know which) was born on a Tuesday. Similarly, our observation corresponds to a) not b) for Sleeping Beauty. Admittedly, a) requires indexicals and so isn’t defined in standard probability theory. This doesn’t mean that we should attempt to cram it in, but instead extend the theory.
“As for the Tuesday problem, that seems to go away if you consider the process that told you at least one of them was born on a Tuesday”—I don’t think we disagree about how the Tuesday problem works. The argument I’m making is that your method of calculating probabilities is calculating b) when we actually care about a).
To bring it back to the Tuesday problem, let’s suppose you’ll meet the first son on Monday and the second on Tuesday, but in between your memory will be wiped. You wake up on a day (not knowing what day it is) and you notice that they are a boy. This observation corresponds to a) meeting a random son and noting that he was born on Tuesday, not b) discovering that one of the two sons (you don’t know which) was born on a Tuesday. Similarly, our observation corresponds to a) not b) for Sleeping Beauty. Admittedly, a) requires indexicals and so isn’t defined in standard probability theory. This doesn’t mean that we should attempt to cram it in, but instead extend the theory.
I’m not sure that can be done: https://www.lesswrong.com/posts/iNi8bSYexYGn9kiRh/paradoxes-in-all-anthropic-probabilities