Eh, I’m not doing anything else important right now, so let’s beat this dead horse further.
“As defined, a universe state either satisfies or does not satisfy a proposition. If you’re referring to propositions that may vary over space or time, then when modelling a given situation you have two choices”
Which I neither disagree nor have any interesting to add.
This is the whole point! That’s why I pointed it out as the likely crux, and you’re saying that’s fine, no disagreement there. Then you reject one of the choices.
You agree that any non-empty set can be the sample space for some probability space. I described a set: those of universe states labelled by time of being asked about a credence.
I chose my set to be a cartesian product of the two relevant properties that Beauty is uncertain of on any occasion when she awakens and is asked for her credence: what day it is on that occasion of being asked (Monday or Tuesday), and what the coin flip result was (Heads or Tails). On any possible occasion of being asked, it is either Monday or Tuesday (but not both), and either Heads or Tails (but not both). I can set the credence for (Tuseday,Heads) to zero since Beauty knows that’s impossible by the setup of the experiment.
If Beauty knew which day it was on any occasion when she is asked, then she should give one of two different answers for credences. These correspond to the conditional credences P(Heads | Monday) and P(Heads | Tuesday). Likewise, knowing what the coin flip was would give different conditional credences P(Monday | Heads) and P(Monday | Tails).
All that is mathematically required of these credences is that they obey the axioms of a measure space with total measure 1, because that’s exactly the definition of a probability space. My only claim in this thread—in contrast to your post—is that they can.
Eh, I’m not doing anything else important right now, so let’s beat this dead horse further.
This is the whole point! That’s why I pointed it out as the likely crux, and you’re saying that’s fine, no disagreement there. Then you reject one of the choices.
You agree that any non-empty set can be the sample space for some probability space. I described a set: those of universe states labelled by time of being asked about a credence.
I chose my set to be a cartesian product of the two relevant properties that Beauty is uncertain of on any occasion when she awakens and is asked for her credence: what day it is on that occasion of being asked (Monday or Tuesday), and what the coin flip result was (Heads or Tails). On any possible occasion of being asked, it is either Monday or Tuesday (but not both), and either Heads or Tails (but not both). I can set the credence for (Tuseday,Heads) to zero since Beauty knows that’s impossible by the setup of the experiment.
If Beauty knew which day it was on any occasion when she is asked, then she should give one of two different answers for credences. These correspond to the conditional credences P(Heads | Monday) and P(Heads | Tuesday). Likewise, knowing what the coin flip was would give different conditional credences P(Monday | Heads) and P(Monday | Tails).
All that is mathematically required of these credences is that they obey the axioms of a measure space with total measure 1, because that’s exactly the definition of a probability space. My only claim in this thread—in contrast to your post—is that they can.