I think what Halfer misses is that probability of coin P is not the same that probability that the last toss of the coin was P
I think what you are trying to say is that there is a difference between unconditional, or “a priori” probability, and conditional, or “a posteriori” probability. The latter is based on evidence that is a direct result of that “last toss of the coin.”
Halfers will argue that there is no such evidence, since Beauty knew she would be wakened during the experiment. That is, that nothing “new” happened in her frame of reference. But that is an incorrect assessment of the experiment. There is an “internal” view, based on what Beauty experiences, and an “external” view, based on what an unbiased observer would see.
My point is that Beauty knows that she is seeing only a portion of the experiment, what I called “internal.” It applies to only one day. But the coin flip exists in the external view, not her internal view. She knows that Tuesday will still happen whether or not she can observe it, and that she can eliminate it as a possibility when she is awake. And the point of the second version of my experiment is that this separation of “internal” and “external” becomes necessary. And it is the valid way to look at the first version even if it doesn’t seem as necessary.
Internal probability can be manipulated by increasing the number of copies. For example, on Heads is created one copy and on Tails – two copies, which later merges. On Tails they have higher “reality fluid” or measure and thus can expect to see Tails as 2⁄3. But objective probability is 1⁄2.
I think what you are trying to say is that there is a difference between unconditional, or “a priori” probability, and conditional, or “a posteriori” probability. The latter is based on evidence that is a direct result of that “last toss of the coin.”
Halfers will argue that there is no such evidence, since Beauty knew she would be wakened during the experiment. That is, that nothing “new” happened in her frame of reference. But that is an incorrect assessment of the experiment. There is an “internal” view, based on what Beauty experiences, and an “external” view, based on what an unbiased observer would see.
My point is that Beauty knows that she is seeing only a portion of the experiment, what I called “internal.” It applies to only one day. But the coin flip exists in the external view, not her internal view. She knows that Tuesday will still happen whether or not she can observe it, and that she can eliminate it as a possibility when she is awake. And the point of the second version of my experiment is that this separation of “internal” and “external” becomes necessary. And it is the valid way to look at the first version even if it doesn’t seem as necessary.
Internal probability can be manipulated by increasing the number of copies. For example, on Heads is created one copy and on Tails – two copies, which later merges. On Tails they have higher “reality fluid” or measure and thus can expect to see Tails as 2⁄3. But objective probability is 1⁄2.