Well, idk. My opinion here is that you bite some weird bullet, which I’m very ambivalent too. I think “now” question makes total sense and you factor it out into some separate parts from your model.
The counter-intuitiveness comes from us not being accustomed to reasoning under amnesia and repetition of the same experience. It’s understandable that initially we would think that question about “now”/”today” makes sense as we are used to situation where it indeed does. But then we can clearly see that in such situations there is no problem with formally defining what event we mean by it. Contrary to SB, where such event is ill-defined.
Like, can you add to the sleeping beauty some additional decision problems including the calendar? Will it work seamlessly?
Oh absolutely.
Suppose that on every awakening the Beauty is proposed to bet that “Today is Monday” What odds is she supposed to take?
“Today is Monday” is ill-defined, but she can construct a corresponding betting scheme using events “Monday awakening happens” and “Tuesday awakening happens” like this:
So, she shakes the box contemplatively. There is mechanical calendar. She knows the betting odds of it displaying “Monday” but not the credence. She thinks it’s really really weird
The counter-intuitiveness comes from us not being accustomed to reasoning under amnesia and repetition of the same experience. It’s understandable that initially we would think that question about “now”/”today” makes sense as we are used to situation where it indeed does. But then we can clearly see that in such situations there is no problem with formally defining what event we mean by it. Contrary to SB, where such event is ill-defined.
Oh absolutely.
Suppose that on every awakening the Beauty is proposed to bet that “Today is Monday” What odds is she supposed to take?
“Today is Monday” is ill-defined, but she can construct a corresponding betting scheme using events “Monday awakening happens” and “Tuesday awakening happens” like this:
E(Monday) = P(Monday)U(Monday) - P(Tuesday)U(Tuesday)
P(Monday) = 1; P(Tuesday) = 1⁄2, therefore
E(Monday) = U(Monday) − 1/2U(Tuesday)
solving E(Monday)=0 for U(Monday):
U(Monday) = 1/2U(Tuesday)
Which means 2:1 betting odds
As you see everything is quite seamless.
So, she shakes the box contemplatively. There is mechanical calendar. She knows the betting odds of it displaying “Monday” but not the credence. She thinks it’s really really weird