No, introducing the concept of “indexical sample space” does not capture the thirder position, nor language.
And what does it not capture in thirder position, in your opinion?
You do not need to introduce a new type of space, with new definitions and axioms. The notion of credence (as defined in the Sleeping Beauty problem) already uses standard mathematical probability space definitions and axioms.
So thirder think. But they are mistaken, as I show in the previous posts.
Thirder credence fits the No-Coin-Toss problem where Monday and Tuesday don’t happen during the same iteration of the experiment and on awakening the person indeed learns that “they are awaken Today”, which can be formally expressed as an event .
Not so with Sleeping Beauty, where the participant completely aware that Monday awakening on Tails is followed by Tuesday awakening, therefore, event doesn’t happen in 50% cases, so instead of learning that the Beauty is awakened today she can only learn that she is awakened at least once.
In Sleeping Beauty problem being awakened Today isn’t a thing you can express via probability space. It’s something that can happen twice in the same iteration of the experiment, just like getting a ball in the example from the post. And so we need a new mathematical model to formally talk about this sort of things, therefore weighted probability space.
I suppose you’ve read all my posts on the topic. What is the crux of our disagreement here?
Not just any set. A sample space. And one of its conditions is that its elements are mutually exclusive, so that one and only one happens in any iteration of probability experiment.
That’s why I need to define a new mathematical entity indexical sample space, for which I’m explicitly lifting this restriction to formally talk about thirdism.
A minor point is that outcomes and events can both very well be about map not the territory. If elementary event {A} has P(A) = 0, then we can simply not include outcome A into the sample space for simplicity sake.
There is a potential source of confusion in the “credence” category. Either you mean it as a synonym for probability, and then it follows all the properties of probability, including the fact that it can only measure formally defined events from the event space, which have stable truth value during an iteration of probability experiment. Or you mean “intuitive feeling about semantic statements which has some relation to betting”, which may or may not be formally definable as probability measure because the statement doesn’t have stable truth value.
People tend to implicitly assume that having a vague feeling about a semantic statements has to mean that there is a way to formally define a probability space where this statement is a coherent event, but it doesn’t actually has to be true. Sleeping Beauty problem is an example of such situation.
I’m not sure what you mean by “smears whole timelines into one element”. We of course should use the appropriate granularity for our outcomes and events. The problem is that we may find ourselves in a situation where we intuitively feel that events has to be even more granular then they formally can.
The fact that something is a semantic statement about the universe doesn’t necessary mean that it’s well-defined event in a probability space.
No it can’t. Semantic statement “Today is Monday” is not a well-defined event in the Sleeping Beauty problem. People can have credence about it in the sense of “vague feeling”, but not in the sense of actual probability value.
You can easily observe yourself that there is no formal way to define “Today” in Sleeping Beauty if you actually engage with the mathematical formalism.
Consider No-Coin-Toss or Single-Awakening problems. If Monday means “Monday awakening happens during this iteration of probability experiment” and, likewise, for Tuesday, we can formally define Today as:
Today = Monday xor Tuesday
On every iteration of probability experiment either Monday or Tuesday awakenings happen. So we can say that the participant knows that “she is awakened Today”, meaning that she knows to be awakened either on Monday or on Tuesday.
P(Today) = P(Monday xor Tuesday) = 1
We can express credence in being awakened on Monday, conditionally on being awakened “Today” as:
P(Monday|Today) = P(Monday|Monday xor Tuesday) = P(Monday)
This is a clear case where Beauty’s uncertainty about which day it is can be expressed via probability theory. Statement “Today is Monday” has stable truth values throughout any iteration of probability experiment
Now consider No-Memory-Loss problem where Sleeping Beauty is completely aware which day it is.
Now statement “Today is Monday” doesn’t have a stable truth value throughout the whole experiment. It’s actually two different statement: Monday is Monday and Tuesday is Tuesday. The first one is always True, the second one is always False. So Beauty’s uncertainty about the question which day it is can’t be expressed via probability theory. Thankfully, she doesn’t have any uncertainty about the day of the week.
So we can do a trick. We can describe No-Memory-Loss problem as two different non-independent probability experiments in a sequential order. First one is Monday-No-Memory-Loss, where the Beauty is sure that it’s is Monday and uncertain about the coin. The second is Tuesday-Tails-No-Memory-Loss where the Beauty is sure that it’s Tuesday and the coin is Tails. The second happens only if the coin was Tails in the first.
In Monday-No-Memory-Loss, Today simply means Monday:
Today = Monday
And statement “Today is Monday” is a well defined event with trivial probability measure:
P(Monday|Today) = P(Monday|Monday) = 1
Similarly with Tuesday-Tails-No-Memory-Loss:
Today = Tuesday
P(Monday|Today) = P(Monday|Tuesday) = 0
And now when we consider regular Sleeping Beauty problem the issue should be clear. If we define Today = Monday xor Tuesday, the Beauty can’t be sure that this event happens, because on Tails both Monday and Tuesday are realized.
And we can’t take advantage of Beauty’s lack of uncertainty about the day as before, because now she has no idea what day it is. And so the statement “Today is Monday” is not a well-defined event of the probability space. It doesn’t have a coherent truth value during the experiment—it’s True xor ( True and False).
We can still talk about events “Monday/Tuesday awakening happens during this iteration of probability experiment”.
P(Monday) = 1
P(Tuesday) = 1⁄2
And we can use them in betting schemes. If the Beauty is proposed to bet on the statement “Today is Monday” she can calculate her optimal odds the standard way:
E(Monday) = P(Monday)U(Monday) - P(Tuesday)U(Tuesday)
Solving E(Monday) = 0:
U(Monday) = U(Tuesday)/2
So 1:2 odds.
And the last question is: What was then this intuitive feeling about the semantic statement “Today is Monday”? For which the answer is—it was about weighted probability that Monday happens in the experiment.