# 9

This is the tenth post in my series on Anthropics. The previous one is Beauty and the Bets.

## Introduction

In my previous posts I've been talking about the actual object-level disagreement between halfers and thirders - which of the answers formally is correct and which is not. I've shown that there is one correct model for the Sleeping Beauty problem, that describes it instead of something else, successfully passes the statistical test, has sound mathematical properties and deals with every betting scheme.

But before we can conclude that the issue is fully resolved, there is still a notable semantic disagreement left, as well as several interesting questions. If the thirder answer isn't the correct "probability", then what is it? What are the properties of this entity? And why are people so eager to confuse it with the actual probability?

In this post I'm going to answer these questions and dissolve the last, purely semantic part of the disagreement.

## A Limitation of Probability Theory

Probability theory has an curious limitation. It is unable to talk about events which happen multiple times during the same iteration of an experiment. Instead, when the conditions of the experiment state that something happens multiple times, we are supposed to define "happening multiple times" as an elementary outcome in itself.

This is, actually, a very useful property, that allows probability theory to work as intended, so that the measure of an event properly corresponds to the credence that rational agents are suppose to have in that event. But it may lead to conflicts with our intuitions about the matter.

Let's consider an example:

A coin is tossed. On Heads you are given a blue ball, on Tails you are given a blue ball and then you are given a red ball.

Here we say that there are two well-defined mutually exclusive outcomes. Exactly one of them happens at every iteration of the experiment.

We can go into a bit more details and define separate sample spaces for getting a blue ball and getting a red ball:

We can coherently talk about the probability to get only one ball. Such event happens when the red ball isn't given

And the probability to get any ball at all. Such an event happens on every iteration of the probability experiment, as you at least get a blue ball.

And if we were asked what is the probability that the coin landed Heads, conditional on the fact that we received any ball, the answer is simple:

But for some people it may feel that something is off here. Doesn't the event "Getting a ball" happen twice on Tails? According to probability theory, it doesn't. Only one outcome from the sample space can happen per iteration of a probability experiment, and there is no way the same event can happen twice.

But it may feel that there has to be something that happens once when the coin is Heads and twice when the coin is Tails. Even though it's impossible to formally define as an event in the probability space, it may still feel that we should be able to talk about it!

## Defining Weighted Probability Space

Let's define a new entity indexical sample space, which has to possess all the properties of a sample space, except that it doesn't require its elements to be mutually exclusive.

Red/Blue means the same as previously - getting a red/blue ball during the experiment. Such outcomes couldn't define a regular sample space for our problem, because on Tails both of them happen. But we specifically defined indexical sample space to be irrelevant to this concern.

And now let's enrich our indexical sample space by the sample space of the coin toss.

if we simply take Cartesian product of the two we get:

Here we have a bit of an issue with  - this outcome doesn't really happen. But that's fine: we have two options, either we can just assume that corresponding elementary event has zero measure and thus we can remove this outcome from our enriched indexical sample space beforehand, or we can initially keep it and then update on the fact that it doesn't happen later. Both of these methods eventually lead to the same values of our measure function. Here, for simplicity I'll just remove it, and so we get:

Now we can define  - indexical event space as some sigma-algebra over the indexical sample space  and  - weighted probability - a measure function the domain of which is . It's similar to regular probability function, with the only difference that instead of

We now have

And therefore we get

- weighted probability space.

We can look at it as a generalization of probability space. While in every iteration of a probability experiment there is only one outcome from the sample space that is realized, here we can have multiple outcomes from the indexical sample space, that are realized during the same iteration of the experiment. In our example, on Tails both Tails&Blue and Tails&Red are realized.

## Properties of Weighted Probability Function

The weighted probability function gives us the probability of an event happening, weighted by the number of outcomes from the indexical sample space that can happen during one iteration of the experiment. And so, the weighted probability that the coin is Heads conditionally on the fact that any ball was received is:

This, mind you, in no way contradicts the fact that  and  are, generally speaking, two different functions and, therefore, can produce different results when receiving the same arguments.

### Conservation of Expected Evidence

Neither should we be troubled by the fact that unconditional weighted probability of a fair coin is not 1/2. For a regular probability function that would be a paradoxical situation, because the unconditional probability of a coin being Heads depends fully on the fairness of the coin. But weighted probability also depends on the number of events that can happen during one iteration of experiment.

Instead of following Conservation of Expected Evidence,  follows a new principle which we can call Conservation of Expected Evidence and Weight. According to which, a weighted probability estimate can be changed either due to receiving new evidence, or when the number of the outcomes from the indexical sample space that can be realized per one iteration of experiment changes.

A consequence from this principle is that, if both the number of outcomes changes and new evidence is received in a compensatory way, weighted probability stays the same.

### Relation to Probability

Switching from probability to weighted probability is easy. We simply need to renormalize the measure function so that .

In our example

and

so

Essentially, weighted probability function treats some of the non-mutually exclusive events the way probability function treats mutually exclusive events. So if we, for some reason, confuse weighted probability with probability, we will be talking about a different problem, where events  and  indeed are mutually exclusive:

You are given blue ball 2/3 of time and red ball 1/3 of time. Half the time you get the Blue ball the coin is put Tails, in other situations it's put Heads.

### Domain of Function

As you might have noticed, I've deliberately selected an example without any memory loss. For the sake of simplicity, but also to explicitly show that amnesia is irrelevant to the question whether we can use the framework of weighted probabilities or not.

All we need is the ability to formally define a weighted probability space and, as it has less strict limitations than a regular probability space, we can at least always do it when a probability space is defined.

In the trivial cases, a weighted probability space is the exact same thing as a regular probability space:

In the more interesting cases, when we have something to weight the probabilities by, as in the example above, the situation is different, but whether we have a trivial case or not doesn't depend on the participant of the experiment going through amnesia at all.

As a matter of fact, we can just have a weird betting rule. For example:

You have to bet on the outcome of a fair coin toss. If you bet on Tails - this bet is repeated. What should be your betting odds?

We can deal with this kind of decision theory problem using regular probability space. Or, using weighted probability space. In the latter case, even though the coin is fair, we have

Which, as we remember, is a completely normal situation, as far as values of weighted probability functions go.

### Betting Application

As weighted probability values can be different from probability values and do not follow Conservation of Expected Evidence, they do not represent the credence that a rational agent should have in the event. Despite that, they can be still useful for betting. We just need to define an appropriate weighted utility function:

Such a weighted utility may have weird properties, inherited from the probability function - like changing its values based on the evidence received. But as long as you keep using them in pair with weighted probability, they will be producing the same betting odds as regular utility and probability.

## Weighted Probabilities in Sleeping Beauty

Now when we define weighted probability and understand its properties, we can see that this is what thirdism has been talking about the whole time.

Previously we were confused why thirder's measure for the coin being Heads shifts from 1/2 to 1/3 and back from Sunday to Wednesday, despite receiving no evidence about the state of the coin. But now it all fits into place.

On Sunday we have a trivial case where probability space equals weighted probability space, there is nothing that can happen twice based on the state of the coin:

And likewise on Wednesday.

But during Monday/Tuesday there may be two awakenings by which we can weight the probability. Indexical sample space is different from regular sample space and therefore:

It would be incorrect to claim that thirders generally find weighted probabilities more intuitive than regular ones. In most non-trivial cases thirders are still intuitively using regular probabilities. But specifically when dealing with "anthropic problems", for instance, the ones including memory loss, they switch to the framework of weighted probability, without noticing it.

The addition of amnesia doesn't change the statistical properties of the experiment, nor is it relevant for the definition of a weighted probability space, but it can make weighted probabilities feel more appropriate for our flawed intuitions, despite complete lack of mathematical justification.

Likewise, Lewisian halfism is also talking about weighted probabilities and confuses them with regular ones. It proposes a different way to define a weighted probability function, while keeping the same weighted utility and, therefore, experimentally produces wrong results.

It has the advantage of appealing towards the principle that the measure of a coin being Heads shouldn't change without receiving new evidence. But it's a principle for regular probabilities, not weighted ones. The latter can be affected not only by received evidence but also by changes in the number of possible indexical events.

So, as soon as we cleared the confusion and properly understood that we are talking about weighted probabilities, we can agree that Lewis's model is wrong, while Elga's model is right. In this sense thirders were correct all this time. All the arguments in favor of thirdism compared to Lewisian halfism stay true, they are simply not about probability.

## Dissolving the Semantic Disagreement

So, with that in mind, let's properly dissolve the last disagreement.

As we remember, it is about the way to factorize expected utility and now we can express it like this:

And we can see that this disagreement is purely semantic. According to the correct model:

While according to thirdism:

But these statements mean exactly the same thing. One necessary implies the other.

As soon as we understand that one model is talking about probability while the other about weighted probability, the appearance of disagreement is no more. We have a direct way to translate from thirder language to halfer and back, fully preserving the meaning.

If only we had a time travel machine so that we could introduce the notion of weighted probability before David Lewis came up with "centred possible worlds" and "attitudes de se" and created all this confusion. In this less embarrassing timeline, when told about the Sleeping Beauty problem, people would immediately see that Beauty's probability that the coin is Heads conditionally on the awakening in the experiment is 1/2, while her weighted probability is 1/3. They would likely not even understand what is there to argue about.

Hopefully, our own timeline is not doomed to keep this confusion in perpetuity. It took quite some effort to cut through the decades long argument, but now, finally, we are done. In the next post we will discuss some of the consequences that follow from the Sleeping Beauty problem and develop a general principle to deal with probability theory problems involving memory loss.

# 9

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[-]JBlack4-1

No, introducing the concept of "indexical sample space" does not capture the thirder position, nor language. 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.

If this were true that the concept of „indexical sample space“ does not capture the thirder position, how do you explain that it produces exactly the same probabilities that thirders entertain? Operating with indexicals is a necessary condition (and motivation) for Thirdism, which means assuming indexical sample spaces when it comes to the mathematical formalization of arguments in terms of probability theory. To my knowledge no relevant thirder literature denies that. And within the thirder model, these probabilities indeed hold true. If we assume Monday and Tuesday to be mutually exclusive, then this is mathematically the case. Math is not a judge of our assumptions here, it is merely the executive organ which in this case produces thirder probabilities. The point at issue is whether the theoretical assumptions of the thirder model fit reality and probabilities could be transfered into the real world. Thirders say yes, speaking of regular probabilities, halfers say no speaking of irregular, „weighted“ probabilities.

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?

Probabilities are measures on a sigma-algebra of subsets of some set, obeying the usual mathematical axioms for measures together with the requirement that the measure for the whole set is 1.

Applying this structure to credence reasoning, the elements of the sample space correspond to relevant states of the universe, the elements of the sigma-algebra correspond to relevant propositions about those states, and the measure (usually called credence for this application) corresponds to a degree of rational belief in the associated propositions. This is a standard probability space structure.

In the Sleeping Beauty problem, the participant is obviously uncertain about both what the coin flip was and which day it is. The questions about the coin flip and day are entangled by design, so a sample space that smears whole timelines into one element is inadequate to represent the structure of the uncertainty.

For example, one of the relevant states of the universe may be "the Sleeping Beauty experiment is going on in which the coin flip was Heads and it is Monday morning and Sleeping Beauty is awake and has just been asked her credence for Heads and not answered yet". One of the measurable propositions (i.e. proposition for which Sleeping Beauty may have some rational credence) may be "it is Monday" which includes multiple states of the universe including the previous example.

Within the space of relevant states of the Sleeping Beauty experiment, the proposition "it is Monday xor it is Tuesday" always holds: there are no relevant states where it is neither Monday nor Tuesday, and no relevant states in which it is both Monday and Tuesday. So P(Monday xor Tuesday) = 1, regardless of what values P(Monday) or P(Tuesday) might take.

Probabilities are measures on a sigma-algebra of subsets of some set, obeying the usual mathematical axioms for measures together with the requirement that the measure for the whole set is 1.

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.

Applying this structure to credence reasoning, the elements of the sample space correspond to relevant states of the universe, the elements of the sigma-algebra correspond to relevant propositions about those states

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.

and the measure (usually called credence for this application) corresponds to a degree of rational belief in the associated propositions. This is a standard probability space structure.

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.

In the Sleeping Beauty problem, the participant is obviously uncertain about both what the coin flip was and which day it is. The questions about the coin flip and day are entangled by design, so a sample space that smears whole timelines into one element is inadequate to represent the structure of the uncertainty

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.

For example, one of the relevant states of the universe may be "the Sleeping Beauty experiment is going on in which the coin flip was Heads and it is Monday morning and Sleeping Beauty is awake and has just been asked her credence for Heads and not answered yet".

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.

One of the measurable propositions (i.e. proposition for which Sleeping Beauty may have some rational credence) may be "it is Monday" which includes multiple states of the universe including the previous example.

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.

Not just any set.

Almost any set: only the empty set is excluded. The identities of the elements themselves are irrelevant to the mathematical structure. Any further restrictions are not a part of mathematical definition of probability space, but some particular application you may have in mind.

If elementary event {A} has P(A) = 0, then we can simply not include outcome A into the sample space for simplicity sake.

In some cases this is reasonable, but in others it is impossible. For example, when defining continuous probability distributions you can't eliminate sample set elements having measure zero or you will be left with the empty set.

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.

It is a synonym for probability in the sense that it is a mathematical probability: that is, a measure over a sigma-algebra for which the axioms of a probability space are satisfied. I use a different term here to denote this application of the mathematical concept to a particular real-world purpose. Beside which, the Sleeping Beauty problem explicit uses the word.

I also don't quite know what you mean by the phrase "stable truth value". 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: either restrict the universe states in your set to locations or time regions over which all selected propositions have a definite truth value, or restrict the propositions to those that have a definite truth value over the selected universe states. Either way works.

Semantic statement "Today is Monday" is not a well-defined event in the Sleeping Beauty problem.

Of course it is. I described the structure under which it is, and you can verify that it does in fact satisfy the axioms of a probability space. As you're looking for a crux, this is probably it.

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.

Otherwise you can't model even a non-coin flip form of the Sleeping Beauty problem in which Beauty is always awoken twice. If the problem asks "what should be Beauty's credence that it is Monday" then you can't even model the question without distinguishing universe states by time.

Almost any set: only the empty set is excluded. The identities of the elements themselves are irrelevant to the mathematical structure. Any further restrictions are not a part of mathematical definition of probability space, but some particular application you may have in mind.

Any non-empty set can be a sample space for some problem. But we are interested in a sample space for a very specific problem - Sleeping Beauty. This "applications we have in mind" is the whole point of the discussion.

Was it really not clear? I have a whole post dedicated towards exploring different probability spaces that people proposed for the Sleeping Beauty problem, where I explicitly notice tha these probability spaces are valid in principle but not sound when applied to the Sleeping Beauty problem - they describe some other problems instead. Likewise, in this post I point that indexical sample space for one problem can be a sample space for another problem - the one where elements of the set indeed are mutually exclusive outcomes. How did you manage to miss it?

Anyway, when we fixed the problem that we are talking about, there are some specific conditions according to which we can say whether a set indeed is a sample space for this problem. And one of them is that the elements of it has to be mutually exclusive outcomes, such as in every iteration of the probability experiment one and only one of them is realized. Are we on the same page here?

So if we define a set in such a way that it consist of outcomes that happen multiple times during the same trial, it can't be a sample space for this problem anymore. However we can remove this restriction by defining a new entity: indexical sample space for this problem.

In some cases this is reasonable, but in others it is impossible. For example, when defining continuous probability distributions you can't eliminate sample set elements having measure zero or you will be left with the empty set.

Fair point. No disagreement here. I was talking specifically about discrete case.

It is a synonym for probability in the sense that it is a mathematical probability: that is, a measure over a sigma-algebra for which the axioms of a probability space are satisfied. I use a different term here to denote this application of the mathematical concept to a particular real-world purpose.

Our particulr real world purpose is the Sleeping Beuaty problem. Credence that the Beauty has isn't just a value of measure function from some probability space it's the value of probability function from the probability space specifically for this problem. Which sample space has to consist from mutually exclusive outcomes for this problem.

I also don't quite know what you mean by the phrase "stable truth value"

There is this thing called probability experiment. Which I provided you a wiki link to in a previous comment. According to Kolmogorov, it's some complex of conditions that can be repeated and on repetition outputs different results all of which belong to some set. On any iteration of the experiment exactly one result is output. This is how we merge our mathematical model with the phisical universe. We say that these results are elements of the sample space. And so one outcome of the sample space is realized at every iteration of the experiment. And every element of the event space to which the realized outcome belongs to is also considered to be realized. Thus we can say that for every iteration of probability experiment every well-defined event is either realized or not realized - has a stable truth value.

If something doesn't have a stable truth value during an iteration of the experiment, it's not a well defined event. This is exactly what happens with statement "Today is Monday" in Sleeping Beauty experiment.

either restrict the universe states in your set to locations or time regions over which all selected propositions have a definite truth value, or restrict the propositions to those that have a definite truth value over the selected universe states. Either way works.

What you are saying is that we can either modify the setting of the experiment so that the statement have a stable truth value or use different statements. This is true. But we want to talk about a specific experimental setting - Sleeping Beauty problem - we can't modify it, otherwise we would be talking about something else, like No-Coin-Toss problem. Therefore, the only way is to acknowledge that some statements are not coherent in the current experimental setting and not use them. This is exactly what I do.

I described the structure under which it is, and you can verify that it does in fact satisfy the axioms of a probability space.

The fact that you can describe such mathematical structure means that there is an experiment where statement "Today is Monday" is a well-defined event. Here we are in agreement. More specifically this is the experiment where awakenings on Monday and Tuesday are mutually exclusive during one trial, such as No-Coin-Toss or Single-Awakening.

But these experiments are not Sleeping Beauty experiment. And this is what I've been talking about the whole time that specifically in Sleeping Beauty experiment, from the perspective of the Beauty "Today is Monday" isn't a well-defined event, it doesn't have a stable truth value.

You seem to be thinking that if something is a well-defined event in some experiment, it has to be a well defined event in Sleeping Beauty experiment as well. Is this our crux?

Otherwise you can't model even a non-coin flip form of the Sleeping Beauty problem in which Beauty is always awoken twice. If the problem asks "what should be Beauty's credence that it is Monday" then you can't even model the question without distinguishing universe states by time.

Yes. This is a completely correct conclusion which I'm planning to talk about in the next post. If a person goes through memory loss and the repetition of the same experience, there is no coherent credence/probability that this experience happens the first time that this person can have.

P(Monday) = 1

P(Tuesday) = 1

The vague inuitive feeling that it has to be 1/2 is once again pointing to weighted probability, which renormalizes the measure function.

[-]JBlack-30

More specifically this is the experiment where awakenings on Monday and Tuesday are mutually exclusive during one trial, such as No-Coin-Toss or Single-Awakening

No, I specifically was referring to the Sleeping Beauty experiment. Re-read my comment. Or not. At this point it's quite clear that we are failing to communicate in a fundamental way. I'm somewhat frustrated that you don't even comment on those parts where I try to communicate the structure of the question, but only on the parts which seem tangential or merely about terminology. There is no need to reply to this comment, as I probably won't continue participating in this discussion any further.

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:

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.