I find this situation quite ironic. From my perspective, I painstakingly cited and answered your comments piece by piece even though you didn’t engage much neither with the arguments in my posts nor with any of my replies.
I’m not sure how I could have missed “parts where you try to communicate the structure of the question”. The only things which I haven’t directly cited in your previous comment are:
Beside which, the Sleeping Beauty problem explicit uses the word.
and
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.
Universe states can be distinguished by time information, and in problems like this where time is part of the world-model, they should be. The mathematical structure of a probability space has nothing to do with it, as the mathematical formalism doesn’t care what the elements of a sample space are.
Which does not have any argument in it. You just make statements without explaining why they are supposed to be true. I obviously do not agree that in Sleeping Beauty case our model should be treating time states as individual outcomes. But simply proclaming this disagreement doesn’t serve any purpose after I’ve already presented comprehensive explanation why we can’t lawfully reason about time states in this particular case as if they are mutually exclusive outcomes, which you, once again, failed to engage with.
I would appreciate if you addressed this meta point because I’m honestly confused about your perspective on this failure of our communication.
Meta ended.
No, I specifically was referring to the Sleeping Beauty
If you were specifically referring to Sleeping Beauty problem then your previous comment doesn’t make sense.
Either you are logically pinpointing any sample space for at least some problem, and then you can say that any non-empty set fits this description.
Or you are logically pinpointing sample space specifically for the Sleeping Beauty problem, then you can’t dismiss the condition of mutually exclusivity of outcomes which are relevant to the particular application of the sample space.
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.
Meta:
I find this situation quite ironic. From my perspective, I painstakingly cited and answered your comments piece by piece even though you didn’t engage much neither with the arguments in my posts nor with any of my replies.
I’m not sure how I could have missed “parts where you try to communicate the structure of the question”. The only things which I haven’t directly cited in your previous comment are:
and
Which I neither disagree nor have any interesting to add.
Which does not have any argument in it. You just make statements without explaining why they are supposed to be true. I obviously do not agree that in Sleeping Beauty case our model should be treating time states as individual outcomes. But simply proclaming this disagreement doesn’t serve any purpose after I’ve already presented comprehensive explanation why we can’t lawfully reason about time states in this particular case as if they are mutually exclusive outcomes, which you, once again, failed to engage with.
I would appreciate if you addressed this meta point because I’m honestly confused about your perspective on this failure of our communication.
Meta ended.
If you were specifically referring to Sleeping Beauty problem then your previous comment doesn’t make sense.
Either you are logically pinpointing any sample space for at least some problem, and then you can say that any non-empty set fits this description.
Or you are logically pinpointing sample space specifically for the Sleeping Beauty problem, then you can’t dismiss the condition of mutually exclusivity of outcomes which are relevant to the particular application of the sample space.
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.