From an agent’s first-person perspective there is no reference class for himself, i.e. he is the only one in its reference class. A reference class containing multiple agents only exists if we employ an outsider view.
When beauty wakes up in the experiment she can tell it is “today” and she’s experiencing “this awakening”. That is not because she knows any objective differences between “today” and “the other day” or between “this awakening” and “the other awakening”. It is because from her perspective “today” and “this awakening” is most immediate to her subjective experience which makes them inherently unique and identifiable. She doesn’t need to consider the other day(s) to specify today. “Today” is in a class of its own to begin with. But if we reason as an objective outsider and not use any perspective center in our logic then none of the two days are inherently unique. To specify one among the two would require a selection process. For example a day can be specified by say “the earlier day of the two”, “the hotter day of the two” or the old fashioned “the randomly selected among of the two”. (an awakening can similarly be specified among all awakenings the same way) It is this selection process from the outsider view that defines the reference class.
Paradoxes happens when we mix reasonings from the first-person perspective and the outsider’s perspective in the same logic framework. “Today” becomes both uniquely identifiable while at the same time also belongs to a reference class of multiple days. The same can be said about “this awakening”. This difference leads to the debate between SIA and SSA.
The importance of perspectives also means when using betting argument we need to repeat the experiment from the perspective of the agent as well. This also means from an agent’s first-person perspective, if his objective is simply to maximize his own utility no other agent’s decision need to be considered.
Ok, imagine a Simple Beauty problem, without the coin toss: she only wakes up on Monday and on Tuesday. When she wakes up, she know, that it is “today”, but “today” is an unknown variable, which could be either Monday or Tuesday, and she doesn’t know the day.
In that case she (or me on her place) will still use the reference class logic to get 0.5 probability of Tuesday.
In this case beauty still shouldn’t use the reference class logic to assign a probability of 0.5. I argue for sleeping beauty problem the probability of “today” being Monday/Tuesday is an incoherent concept so it do not exist. To ask this question we must specify a day from the view of an outsider. E.g. “what’s the probability the hotter day is Monday?” or ” what is the probability the randomly selected day among the two is Monday?”.
Imagine you participate in a cloning experiment. At night when you are sleeping a highly accurate clone of you with indistinguishable memory is created in an identical room. When waking up there is no way to tell if you are old or new. It might be tempted to ask “what’s the probability of “me” being the clone?” I would guess your answer is 0.5 as well. But you can repeat the same experiment as many times as you want by falling asleep let another clone of you be created and wake up again. Each time waking up you can easily tell “this is me”, but the is no reason to expect in all these repetitions the “me” would be the new clone about half the times. In fact there is no reason the relative frequency of me being the clone would converge to any value as the number of repetition increases. However if instead of this first person concept of “me” we use an outsider’s specification then the question is easily answerable. E.g. what is the probability the randomly chosen version among the two is the clone? The answer is obviously 0.5. If we repeat the experiments and each time let an outsider randomly choose a version then the relative frequency would obviously approach 0.5 as well.
On a side note this also explains why double-halving is not unBayesian.
If the original is somehow privileged over its copies, when his “me” statistic will be different of the copies statistic.
Not sure if I’m following. I don’t see in anyway the original is privileged over its copies. In each repetition after waking up I could be the newly created clone just like in the first experiment. The only privileged concepts are due to my first-person perspective such as here, now, this, or the “me” based on my subjective experience.