When Sleeping Beauty wakes up and observes a sequence, they are learning that this sequence occurs on a on a random day
Right here is your error. You are sneaking in an indexical here—Beauty doesn’t know whether “today” is Monday or Tuesday. As I discussed in detail in Part 2, indexicals are not part of classical logic. Either they are ambiguous, which means you don’t have a proposition at all, or the ambiguity can be resolved, which means you can restate your proposition in a form that doesn’t involve indexicals.
What you are proposing is equivalent to adding an extra binary variable d to the model, and replacing the observation R(y, Monday) or R(y, Tuesday) with R(y, d). That in turn is the same as randomly choosing ONE day on which to wake Beauty (in the Tails case) instead of waking her both times.
This kind of oversight is why I really insist on seeing an explicit model and an explicit statement (as a proposition expressible in the language of the original model) of what new information Beauty receives upon awakening.
“What you are proposing is equivalent to adding an extra binary variable d to the model, and replacing the observation R(y, Monday) or R(y, Tuesday) with R(y, d). That in turn is the same as randomly choosing ONE day on which to wake Beauty (in the Tails case) instead of waking her both times”—Yes, that is equivalent to what I’m proposing by saying that only one day “counts”. I’ll explain why this formalism is useful in my next post.
Right here is your error. You are sneaking in an indexical here—Beauty doesn’t know whether “today” is Monday or Tuesday. As I discussed in detail in Part 2, indexicals are not part of classical logic. Either they are ambiguous, which means you don’t have a proposition at all, or the ambiguity can be resolved, which means you can restate your proposition in a form that doesn’t involve indexicals.
What you are proposing is equivalent to adding an extra binary variable d to the model, and replacing the observation R(y, Monday) or R(y, Tuesday) with R(y, d). That in turn is the same as randomly choosing ONE day on which to wake Beauty (in the Tails case) instead of waking her both times.
This kind of oversight is why I really insist on seeing an explicit model and an explicit statement (as a proposition expressible in the language of the original model) of what new information Beauty receives upon awakening.
“What you are proposing is equivalent to adding an extra binary variable d to the model, and replacing the observation R(y, Monday) or R(y, Tuesday) with R(y, d). That in turn is the same as randomly choosing ONE day on which to wake Beauty (in the Tails case) instead of waking her both times”—Yes, that is equivalent to what I’m proposing by saying that only one day “counts”. I’ll explain why this formalism is useful in my next post.
But randomly awakening Beauty on only one day is a different scenario than waking her both days. A priori you can’t just replace one with the other.