Standard probability works over a set of possibilities that are exclusive. If the possibilities are non-exclusive, then we can either: a) decide on a way to map the problem to a set of exclusive possibilities b) work with a non-standard probability, in this case one that handles indexicals. Diving too deep into trying to justify the halver solution is outside the scope of this post, which merely attempts to demonstrate that if the halver solution is valid for Sleeping Beauty, it can also be extended to The Beauty and the Prince. As I said, I’ll write another post on the anthropic principle when I’ve had time to do more research, but I thought that this objection was persuasive enough that it deserved to be handled in its own post.
In terms of defining the term “count”: If we want to use the term “you” then in addition to information about the state of the world, we also need to now which person-component “you” refers to. So the version which “counts” is basically just the indexical information.
“I’m Beauty. I’m a real person. I’ve woken up. I can see the Prince sitting over there, though if you like, you can suppose that we can’t talk. The Prince is also a real person. I’m interested in the probability that the result of a flip of a actual, real coin is Heads”—Yes, but regardless of whether you go the halver or thirder route you need a notion of probability that extends standard probability to cover indexicals. You seem to be assuming that going the thirder route doesn’t require extending standard probability?
Right. I see no need to extend standard probability, because the mildly fantastic aspect of Sleeping Beauty does not take it outside the realm of standard probability theory and its applications.
Note that all actual applications of probability and decision theory involve “indexicals”, since whenever I make a decision (often based on probabilities) I am concerned with the effect this decision will have on me, or on things I value. Note all the uses of “I” and “me”. They occur in every application of probability and decision theory that I actually care about. If the occurrence of such indexicals was generally problematic, probability theory would be of no use to me (or anyone).
“If the occurrence of such indexicals was generally problematic, probability theory would be of no use to me (or anyone)”—Except that de-indexicalising is often trivial—“If I eat ice-cream, what is the chance that I will enjoy it” → “If Chris Leong eats ice-cream, what is the probability that Chris Leong will enjoy it”.
Anyway, to the extent that this approach works, it works just as well for Beauty. Beauty has unique experiences all the time. You (or more importantly, Beauty herself) can identify Beauty-at-any-moment by what her recent thoughts and experiences have been, which are of course different on Monday and Tuesday (if she is awake then). There is no difficulty in applying standard probability and decision theory.
At least there’s no problem if you are solving the usual Sleeping Beauty problem. I suspect that you are simply refusing to solve this problem, and instead are insisting on solving only a different problem. You’re not saying exactly what that problem is, but it seems to involve something like Beauty having exactly the same experiences on Monday as on Tuesday, which is of course impossible for any real human.
Standard probability works over a set of possibilities that are exclusive. If the possibilities are non-exclusive, then we can either: a) decide on a way to map the problem to a set of exclusive possibilities b) work with a non-standard probability, in this case one that handles indexicals. Diving too deep into trying to justify the halver solution is outside the scope of this post, which merely attempts to demonstrate that if the halver solution is valid for Sleeping Beauty, it can also be extended to The Beauty and the Prince. As I said, I’ll write another post on the anthropic principle when I’ve had time to do more research, but I thought that this objection was persuasive enough that it deserved to be handled in its own post.
In terms of defining the term “count”: If we want to use the term “you” then in addition to information about the state of the world, we also need to now which person-component “you” refers to. So the version which “counts” is basically just the indexical information.
“I’m Beauty. I’m a real person. I’ve woken up. I can see the Prince sitting over there, though if you like, you can suppose that we can’t talk. The Prince is also a real person. I’m interested in the probability that the result of a flip of a actual, real coin is Heads”—Yes, but regardless of whether you go the halver or thirder route you need a notion of probability that extends standard probability to cover indexicals. You seem to be assuming that going the thirder route doesn’t require extending standard probability?
Right. I see no need to extend standard probability, because the mildly fantastic aspect of Sleeping Beauty does not take it outside the realm of standard probability theory and its applications.
Note that all actual applications of probability and decision theory involve “indexicals”, since whenever I make a decision (often based on probabilities) I am concerned with the effect this decision will have on me, or on things I value. Note all the uses of “I” and “me”. They occur in every application of probability and decision theory that I actually care about. If the occurrence of such indexicals was generally problematic, probability theory would be of no use to me (or anyone).
“If the occurrence of such indexicals was generally problematic, probability theory would be of no use to me (or anyone)”—Except that de-indexicalising is often trivial—“If I eat ice-cream, what is the chance that I will enjoy it” → “If Chris Leong eats ice-cream, what is the probability that Chris Leong will enjoy it”.
What makes you think that you are “Chris Leong”?
Anyway, to the extent that this approach works, it works just as well for Beauty. Beauty has unique experiences all the time. You (or more importantly, Beauty herself) can identify Beauty-at-any-moment by what her recent thoughts and experiences have been, which are of course different on Monday and Tuesday (if she is awake then). There is no difficulty in applying standard probability and decision theory.
At least there’s no problem if you are solving the usual Sleeping Beauty problem. I suspect that you are simply refusing to solve this problem, and instead are insisting on solving only a different problem. You’re not saying exactly what that problem is, but it seems to involve something like Beauty having exactly the same experiences on Monday as on Tuesday, which is of course impossible for any real human.