Ah, it started so well. And then the numbered list started, and you didn't use any of the things from before the list at all! You assumed some new things (1, 2 and 3) that contained your entire conclusion.
Let me try to redirect you just a little.
Suppose we flip a coin and hide it under a cup without looking at it. We should bet as if the coin has P(Heads)=0.5, because when we are ignorant we can't do better than assigning a probability, even though the reality is fixed. In fact, the same argument applies before flipping the coin if we ignore quantum effects - the universe is already arranged such that the coin will land heads or tails, but because we don't know which, we assign a probability.
Now suppose that you get to look at the coin, while I don't. Now you should assign P(Heads)=1 if it is heads, and P(Heads)=0 if it is tails, but I should still assign P(Heads)=0.5. Different people can assign different probabilities, and that's okay.
The Sleeping Beauty problem has two perspectives - Sleeping Beauty's view, and the experimenter's view (or god's view). In these two views, you face different constraints. To Sleeping Beauty, she is special and she knows that certain logical relationships hold between the allowed day and the state of the coin. To the experimenter, the coin and the day are independent variables, and no instance of Sleeping Beauty is special.
(note: if you think the day being Monday is an "invalid" observable, just suppose that there is a calendar outside the room and Sleeping Beauty is predicting what she will see when she checks the calendar, much like how we predicted what we would see when we looked at the flipped coin.)
Everyone thinks that assigning probabilities from the experimenter's view is easy, but they disagree about Sleeping Beauty's view.
Here's a trick that tells you about what betting odds Sleeping Beauty should assign, using only the easy experimenter's view! Just suppose that the experimenter is betting money against Sleeping Beauty - every time Sleeping Beauty wakes up she makes this bet. Every dollar won by Sleeping Beauty is lost by the experimenter. What is a fair price for Sleeping Beauty to pay, in exchange for the experimenter paying her $1.00 if the day is Monday?
We don't need to use Sleeping Beauty's view to answer this question. We just use the fact that the experimenter's view is easy, and the bet is fair if the experimenter doesn't gain or lose any money on average, from the experimenter's view. With probability 0.5 (for the experimenter) Sleeping Beauty only wakes up on Monday, and with probability 0.5 (for the experimenter) she wakes up on both Monday and Tuesday and makes the bet both times. So with probability 0.5 the experimenter pays a dollar and gets the fair price, and with probability 0.5 the experimenter pays a dollar and gets twice the fair price.
In other words, 3 times the fair price = 2 dollars. The fair price for a bet that pays Sleeping Beauty on Monday is $2/3.
I appreciate the insight. Something I want to clarify. Point 1,2,3 are not new assumptions but rather derived from the notion that reasonings from different perspectives should not mix. The complete deduction is explained in the pdf. It is quite lengthy so I will try to summarize it at the risk of oversimplifying.
Point 1 for example, when woke up in the experiment, from beauty's first-person perspective the information available is "I am awake today". Here "today" is a valid specification of the time since it is from first-person perspective. However from first-person perspective finding oneself awake is a guarantee. So there is no new info. Beauty can also reason from third-person perspective (e.g. by imagine how would an outside observer assign probability with the available info). Here beauty being awake is not a guarantee but "today" is no longer a valid specification of time. So the only info available is "beauty is awake on an unspecific day" i.e. "there is at least one awakening". There's no new info either. That's why "No matter which perspective one choose to reason from there is no new information when beauty wakes up." Only if we mix the two perspectives together: treating beauty awake as a surprise AND treat "today" as a specific day, then new information can be argued.
For point 2. My argument is not that today being Monday is an invalid observation. I'm arguing if you treat one of the days as inherently special then you should no longer treat the twos potential days as equals. Even though "today" is inherently special from first-person perspective and the two days are ordinary like one another from third-person perspective. To comprehend "the probability of today being Monday" requires one to do both. It uses "today" as one specified day, and it treats Monday and Tuesday as two parallel outcomes in the sample space. This is discussed in section 5 in detail. I also explained in length the difference between such probabilities and probabilities of random experiments such as your example of coin toss. One thing to point out: there is no random experiment deciding if "today" is Monday or Tuesday (or if the duplicated instances are created by memory retained cloning, there is nothing determining if "I" am the original or the clone). So the uncertainty cannot be solely explained from first-person perspective. I really wish if you are interested then give it a look. I look forward to you counter argument.
Point 3 is just a natural following of point 1 and 2. Since there's no such thing as "the probability of today being Monday/Tuesday". Then after being told it's Monday keeping the probability of heads unchanged at 1/2 does not violate extant bayesian updating rules.
Also my frequentist argument for 1/2 (point 4) is not contained in point 1,2 and 3. To my knowledge it is entirely new. The key is that, considering the importance of a consistent perspective, the repetitions should not be chronologically linear. It should be a nested structure. That is: i.e. using memory wipes, the first level separate the duration of the experiment into two halves, the next level separate each resulting segment into two halves again, so on and so forth. It would be in a form similar to a supertask. Even if the experiment is repeated near infinite number of times it would still have to happen within two days. From beauty's first-person perspective as the number of repetitions increases, note all these repetitions do not pass though any memory wipe, the relative frequency would approach 1/2. (this is discussed in detail in section 3 with betting arguments follows)
As for your betting argument. Forgive my bluntness, it is a common argument for the thirders' side. Betting argument have the problem of too many underlying assumptions involved. E.g. beauty's motive, accounting rule, how many bets offered per toss etc. This causes halfers and thirders have little common ground in discussion so they are rarely effective at persuading the opposition. As expected there are many things I do not agree with the argument you presented. However I only wish to point out one thing: only considering the outsider's view is not enough. An outsider could be in direct communication with beauty and they should still assign different probability to heads. Even with bets involved. This is what I meant by point 6. It is discussed in detail in section 4.
In hindsight I should probably make clear the point forms are not my assumptions but notable conclusions of my argument. They were intended to spark some interest for the pdf. Sorry for any confusion.
Sorry for the slow reply. I eventually looked though the pdf, but really just wanted to argue one more time against this idea that "what day is it?" is not a valid statement. Today is Thursday. This is a fact that has very strong predictive powers. I can test it by checking the internet or asking someone for the date. It is part of the external world, just as much as "did I go dancing on New Year's Eve 2018?" or "when I flip this coin, will it land Heads?"
It seems to me like you're treating Sleeping Beauty as being some sort of communal consciousness encompassing all of her copies, and there's no fact of the matter about what "today" is for this communal consciousness. But that's not actually what's happening - each instance of Sleeping Beauty is a complete person, and gets to have beliefs about the external world just as good as any other person's.
You might also think of the problem in terms of Turing machines that get fed a tape of observations, trying to predict the next observation. By "Today is Thursday" I mean that the state of my Turing machine is such that I predict I'll see "Thursday" when I click on the time at the bottom-left of my computer screen (there is, after all, no real thursday-ness that I could identify by doing a fundamental physics experiment). The copies of Sleeping Beauty can be thought of as identical Turing machines that have been fed identical observations so far, but can't tell for certain what they will observe when they look at a calendar.
Sorry for the slow reply as well. For some reason replies from this thread is not showing up in my notifications.
If I say "I'm a man" then it is definitely true and meaningful as the body most immediate to my perception is of a human male. To my wife the same statement is obviously false since she is a woman. It is rather trivial if the subject "I" is defined by a first-person perspective's center then the statement may not be valid once switched to someone else's perspective. Of course in real life if I say "I'm a man" my wife would (hopefully) agree. She maybe interpreting the sentence as "My husband is a man". However "my husband" is still defined by her perspective and the sentence might be invalid to others (e.g. I do not have a husband). Or we can employ a third-person's perspective and reason as an impartial outsider. The statement may become "Dadadarren is a man." Here "The person named Dadadarren" is not defined by any perspective center. Notice the name must be used to specify me in third person because for a impartial outsider I am just an ordinary person like everybody else. There is no self apparent "I" anymore. So some feature must be used to specify me among all persons. My argument is that a logic framework must be fully contained within one perspective. E.g. if we use my perspective center to define "I" then we cannot switch to an outsider's third-person perspective just as we cannot switch to my wife's perspective. Consequently if my first-person center is used then we are no longer treating every person as ordinary, or belongs to the same reference class, since I'm inherently different from everybody else in this reasoning
First-person perspective center can be used to define not only "I" but also other concepts such as "this", "here", "now" and subsequently "today".
With the above in mind let's get back to "Today is Tuesday." As my previous reply had said, my argument is Not that "Today's Tuesday" is invalid. It is a perfectly meaningful and valid statement. It can be fully understood from the first person perspective just like the statement "I'm a man." It could simply means on the day that is most immediate to my perception, aka "today", the calendar says Tuesday. We can also take an outsider's perspective and change "today" to a day defined by some feature/event. E.g. if the experimenter randomly assigns the color of the wallpaper to be red or blue during the two days in the experiment then "The red day is Tuesday" may also be a valid statement just like "The person named Dadadarren is a man." I'm arguing that "the probability of today being Monday/Tuesday" is invalid in the setup of the Sleeping Beauty Problem. It's asking out of the two possible awakenings which is THIS one? "This awakening" or "today's awakening" is defined by first-person perspective center. But the questions also treats the two awakenings as elements in the same reference class as a third-person would. The question itself involves a perspective switch. It is essentially asking in the eyes of an outside observer which one is my first-person perspective center. Where from a third-person perspective an awakening must be specified by some feature (e.g. the red awakening) since there is no self-apparent "today" or "this awakening".
It may help to clarify my argument a bit to think about this example. Imagine you are cloned during your sleep last night. The clone is highly accurate and retains your memory well enough such that the clone could not tell he is the new copy. After waking up I argue the question of "The probability of me being the clone/original" is invalid. If the two copies are randomly put into either a red or a blue room. It would be valid to ask "The probability of the copy in the red room being the clone/original." Since this question is entirely from a third-person perspective (it doesn't treat one copy as inherently unique "I" as a result has to specify a copy by its feature).
I also want to clarify that I am not arguing "The probability of today being Tuesday"is invalid simply because of its wording. For example, imagine you are put to sleep on Sunday and the experimenters toss a coin. If it lands head they will inject you with a drug that would make you sleep though Monday and wake up Tuesday morning. Otherwise you would wake up on Monday as usual. In this problem when you wake up "the probability of today being Monday/Tuesday" is perfectly valid. The two possibilities reflects the two outcomes of a coin toss. Which can be well understood from a first-person perspective. Compare it to me being clone/original, or "today being Monday/Tuesday" in the sleeping beauty problem, where a third-person perspective is needed to comprehend the possibilities as elements in the same reference class.
I'm not arguing for a communal consciousness among sleeping beauties. In fact it can be said I argue the exact opposite. So it worries me how I gave you that impression. To me consciousness among the awakenings or among different clones are definitely distinct. Or in your words every beauty is a complete person. That's the reason why there is a self apparent "I" from first-person perspective even without knowing any objective difference between me and others. Whereas from a third-person perspective nobody's consciousness is inherently unique. So it is better to just ignore consciousness altogether as it is unobservable and unmeasurable to anyone except the first-person anyway.
With regard to your clone example, you say:
--After waking up I argue the question of "The probability of me being the clone/original" is invalid.
Perhaps the question is invalid, but only because you specified the question poorly to begin with. You say there are two individuals, but don't specify who is asking the questions. Who is it who is "waking up", and who asks the question "The probability of me being the clone/original?" Are both clones waking up and asking the same question? If your question applies to both clones at the same time, then it is entirely appropriate to say: whatever clone you are talking about, the probability is 1/2. If the question applies only to a specific clone, then in the question you must specify which clone is asking the probability question. But you never specified who was asking the question. Just because you use the word 'me', it doesn't mean you are referring to a specific first person entity (e.g. the person writing this response, or you, or whoever). Your reference to 'me' is fundamentally ambiguous, and that is why the question is invalid--not because you mix first and third person perspectives.
I read your paper. I think your insight revolves around the following invalid question: "What is the probability that I am me." I have no idea how to answer this questions, but it is not problematic because of mixing perspectives. It's problematic because of ambiguous reference. But this question is completely different in character than "What is the probability that today is Monday?". That question is not problematic at all. "Today is Monday" is a benign proposition that is either true or false, and easily confirmed or denied.
Thank you for the reply. Let's take a step back. Thirders argue there is new information when waking up in the sleeping beauty problem. That information is "I am awake today". Similarly for the cloning problem thirders argue the new information is "I exist". Both of which are interpreted as evidence favouring more observers even though not explaining who is this specific I or what is this today's date. It is treated as if they are primitively understood concepts do not need any explanation. By this standard I do not need to explain who is this specific me in the question of "what's the probability of me being a clone?". If you think the question is invalid/ambiguous because by using me I did not specify a particular clone then by the same logic thirders' argument for new information also fails. When waking up in cloning experiment instead of saying a specific I exist thirders are only saying an unspecific copy exists, (i.e. there is at least one copy exists). That is not new information.
The reason why thirder's argument seems convincing is because if reasoned from an observer's first person perspective the indexical concepts such as I, now or here are indeed primitively understood. E.g I do not need to know how I look like to tell this is me. I do not need to know the exact date and hour to tell when is this now (and subsequently today). They do not need any further explanation because they are primitively identified base on their immediacy to the subjective experience. In another word the meaning of these indexicals depends on the thinker's perspective center. If we reason purely objectively (from a third-person perspective) then these indexicals are meaningless.
So in a sense you are correct. The questions are invalid because a particular copy is not specified by using the indexical me. Because in this third-person question it is asking about someone only identifiable from the first-person perspective. To make the question valid, instead of using the indexical me an individual must be specified objectively. e.g. a randomly selected person etc.
Dadadarren, I appreciate the reply. I happen to be a double halfer (triple halfer!) myself. I do not support the thirder arguments, but I am trying to extract the insights you bring to the debate. You seem to agree that 'ambiguous' or 'unspecified' reference has something to do with the problems in these questions. But is this problem distinct from your claim that 'First and Third Person Perspectives' get mixed up/conflated, or is it the same problem?
From a first-person perspective, Today has at least two different meanings. Today might mean "primitively understood' as 'this day' or 'the day I am in now' that is known primitively and only from a first person experience. But Today might also mean (starting from 1st person) that heterophenomenologically shared specific day (e.g. Sunday, or 5/12/2019). You might say "Today is a busy day for me" which mostly implies the former first person perspective. Or you may ask a cowoker: "What day is Today?", where Today is both your first person experience "This day for me", but also implies a third-person shared reference class that your coworker will pick from. There is no ambiguous reference in these propositions, even though the latter example 'mixes' both first and third person perspectives. This is why it seems that ambiguous reference generates the confusion more so than mixing perspectives.
Anyway, I am a double halfer, but I have a specific argument that purposely mixes both perspectives (first and third) that argues for the halfer position:
MIRROR ARGUMENT:
HEADS=the fair coin lands on Heads
MUH = HEADS and you wake on Monday (temporally uncentered)
MCH = HEADS and it is Monday (temporally centered)
Then:
(M1) P(MUH)=P(HEADS)=1/2
Now imagine contemplating the following credences (probabilities) during an awakening:
(M2) P(MCH|MUH)=1, or in words, the credence this is a Monday awakening and Heads, given you wake on Monday and Heads, is one.
(M3) P(MUH|MCH)=1, the credence you wake on Monday and Heads, given this is a Monday awakening and Heads, is one.
(M4) Therefor: P(MUH)=P(MCH)=P(HEADS)=1/2 , where P(MUH)=P(MCH) follows from (M2), (M3), and the definition of conditional probability.
See below for full paper.
http://philsci-archive.pitt.edu/15911/
I also argue that an infinitely repeated Sleeping Beauty Problem will give the 1/3 solution, and that this does not conflict with the double halfer position.
Marc
Hello Marc,
Appreciate the reply. Allow me to explain why the perceived ambiguity is in my opinion caused by mixing of perspectives. When we try to analysis an anthropic related problem, such as sleeping beauty or an equivalent cloning experiment, there are actually two school of thoughts. One is to reason from the participant's first person perspective. The other is to reason purely objectively as an impartial observer (third-person perspective).
Reasoning from the participant's first-person perspective makes concepts such as I, now or here primitively understood. These indexical concepts inherently standout to the first-person as they are defined by their subjective immediacy to the perspective center. (an everyday example would be an identical twin can inherently distinguish himself from his brother. He can do this without knowing any difference between them which is impossible for an impartial outsider.) At the same time, reasoning from the first-person perspective would affirm self-existence/consciousness. It is necessarily true that from first-person perspective "I am, here, now." (cogito ergo sum)
If reasoned objectively then no agent, time or place would be inherently unique. It is uncentered. So technically from the third-person perspective there is no I now or here. e.g. to objectively specify an agent instead of using the indexical I some feature such like the proper name dadadarren have to be used. (In everyday language we tend to use indexical terms even when trying to reason objectively. In such instances words such as I should not be regarded as indexicals pointing to a perspective center but a conventional shorthand representing various objective features defining an agent.) From this objective third-person perspective no agent's existence/consciousness is guaranteed. It is logical to ask about any agent's probability of existence. Furthermore, from this perspective all agents/times are logical equals. So a principle of indifference can be applied and they can belong to the same reference class.
Now if we are to ask the probability density function of today being Monday or Tuesday, or the pdf of I being the original or clone, problems arise. At one end, to use the indexicals I and now to specify an agent or date requires we take the participant's first-person perspective. They cannot be used to specify a particular agent or time if we are reasoning objectively as an impartial observer since these indexicals have no objective significance. This is why using these terms in anthropic problems seems ambiguous. On the other hand if we reason from the first-person perspective of a participant, yes the meaning of I or now are clear. Yet a sample space containing all possible agents (the original and the clone) or all times (Monday and Tuesday) cannot be constructed. Because from a participant's first-person perspective all perspective centres are not logical equals. A principle of indifference among the two days was thrown out of the window the minute a particular day is regarded as inherently special such that it can be specified not by its objective difference from other days but by a simple utterance of of the word "today". Therefore even though it is correct to say that I am either the original or the clone it is impossible to put a probability on either alternative. This is also backed by frequentist analysis. Which btw, if a consistent perspective is used the long run frequency of Heads should be 1/2.
When I ask my coworker "what day is it today?" The today is still primitively defined from the first-person perspective by both me and my coworker. I don't think it is a shared concept. It is just that the time taken of our communication is minuscule compare to the duration of interest (day) such that one's perspective center can be used to approximate the other's perspective center without causing problems in communication. If instead you find a message in a bottle on the beach which says "what day is it today?" Then this question becomes impossible to answer. Since you would have no idea what this today refers to.
Since my entire position is that first-person (centered) and third-person (uncentered) reasoning should not mix I cannot agree with your mirror argument. Although I would say due to its simplicity (this is a compliment) thirders would have a hard time countering it. Yet the current reality of the Sleeping Beauty discussion is less of finding mistakes in other’s argument but more of whose position have less undesirable consequences. So I won’t be surprised if the mirror argument fail to convince many thirders. In your paper you seems to agree with David Lewis that if the coin toss has already happened and beauty is told it is Monday then the probability of Heads should be rightfully 2/3. Only when the coin toss is yet to happen then the probability shall be 1/2. That the time of coin toss has a material influence on the probability. If this is the correct understanding then I feel the argument is quite problematic. I don’t think there is any logical significance to the physical coin toss. You argued the toss is a chancy event whose timing could affect probability. Yet it can also be deemed as a deterministic event: as long as one has the detailed information of the various variables, such as magnitude and direction of force and air resistance and impact surface shape etc, the result can be readily predicted. The randomness could be interpreted as entirely due to the lack of information. Whether it is a truly random event or a pseudo one shouldn’t affect the probability calculation. Yet this version of double-halving argument depends on that.
Imagine an only slightly different problem: You volunteer for an experiment where you may be wakened once, or twice, on Monday and/or Tuesday. The administrators of the experiment will flip a coin to decide whether it will be once, or twice; but they do not tell you what coin result determines that the number of wakings. And they do not tell you whether the day you will be left asleep, if only one waking is to occur, will be Monday or Tuesday. Oh, and you will be given the amnesia drug on Monday, if you are wakened on that day.
Whenever you are awake, you will be asked to assess, from your perspective based on these procedures, the probability that you will be wakened only once during the experiment. Surely (call this assertion A) the answer to this question must be the same as in the original Sleeping Beauty Problem.
But when you are awakened, you are told that you are one of four volunteers undergoing the exact same procedures based on the same coin flip; with one exception. The choices for the coin results, and the days, are different for each of the four volunteers. (Since there are four possible combinations, each is used.) You are not told which choices were made for you, or introduced to the others. Surely (Assertion B) this gives you no information about your own situation, so it cannot affect your answer.
But it does allow you to evaluate your perspective with a bit more clarity. Of the four volunteers, you know that one must still be asleep, and it isn't you. That volunteer will be wakened only once. Of the other three, including you, two will be wakened on both days, and one will be wakened only once. Surely (Assertion C) this means that, from your perspective, the probability that you will be wakened once is 1/3.
Unless you can find a flaw with one of my assertions, this means that the answer to the original problem is also 1/3. There are valid, if unorthodox, mathematical proofs of this. But those who trust their intuition over mathematics want the answer to be 1/2. The so-called "double halfer" approach is just a rationalization for ignoring valid mathematics.
The rationalization: an indexical is an identifier for a value relative to some index. If no way to associate the indexical with actual values is provided, as in "the probability it will rain tomorrow," then the indexical cannot be used to assess your perspective. But if such a means is provided, EVEN AS A PROBABILITY DISTRIBUTION, then it can be assessed. For example, say a different volunteer is told she will be wakened once or twice based on the roll of two standard dice; 1=Monday, 2=Tuesday, etc. If both dice results the same, she will be wakened only once. Upon being awake, she can determine that the probability that it is Tuesday is 1/6. "Tuesday" may ne an indexical, but context is provided to index it.
Hello Jeff. Good seeing you again. Happy new year.
A typical thirder argument follows the Self-Indication Assumption. That this awakening should be regarded as being randomly selected from one of the three possible awakenings. One awakening for Monday-Heads, one for Monday-Tails and another for Tuesday-Tails. The way I understand what your experiment does is, very cleverly, using actually existing participants to represent these possibilities. I.e. today, out of the three awaking participants two of witch would be awake on both days and one would be awake on one day only. So my argument against you is the same as my argument against SIA (or SSA as a matter of fact): in the same logical framework it is wrong to use terms such as [today] or [myself] as self-explanatory concepts while also treat them as in the same reference class of all days and all people. Because [now] or [I] is only meaningful if reasoning from a first-person perspective for they are inherently unique to oneself. If they are treated as unique then it is logically inconsistent to treat them as ordinary time or people. Even though they might be ordinary by objective measures, i.e. in the same reference class as other time and people from a third-person perspective/uncentered reasoning. More specifically applying to your argument, I disagree with assertion C. Just because there are three people awake today does not mean I should regard myself as randomly selected among those three or as if today is randomly selected among Monday or Tuesday.
In this question Monday and Tuesday are not indexical. They are defined by a calendar which could be interpreted as defined by the relative positions of planetary bodies. In this sense the dates are defined by objective events. They can be treated as ordinary compare to one another from a third-person perspective. What is indexical is rather the concept of [today]. Which require a perspective center (first-person) to define. My argument is that first-person and third-person reasoning (or centered and uncentered reasoning) should not be mixed together in the same logic.
I want to present a new argument for double-halving position for the sleeping beauty problem. It is my contention that paradoxes related to the anthropic principle such as the Sleeping Beauty Problem and the Doomsday Argument is caused by mixing logics from different perspectives. To solve these paradoxes we only need to reason from a single consistent perspective.
When thinking about anthropic related paradoxes one perspective we can employ is an observer’s first-person perspective. The center of a perspective is primitively unique in logic and reasoning. E.g. I am fundamentally special to myself. I do not need to know any objective differences between me and every other human being to tell “this is me”. Similarly, the present moment is primitively meaningful. I can inherently tell “now” is not any other time. From first-person perceptive it is possible to specify the center, e.g. myself and now, from other people and time without any additional information.
We can also reason as an outside observer, or employ a God’s view, I call it the third-person perspective. The main point is that the center of that perspective is irrelevant to the topic of interest. From this perspective there is no obvious “me”. Every observer in question is ordinary as everybody else. In order to specify someone from a group one have to know the differences between him and the rest. Similarly there is no “now” that inherently stands out from any other moment. From third-person perspective every observer and everyday are treated as equals.
I argue the reasonings from these two perspectives should not be mixed and used together. So I am either inherently special (such that no information is needed to specify me) or just an ordinary person as everybody else is. Similarly “today” is either a meaningful moment uniquely stands out or everyday in the experiment are equals. We should always stick to one perspective and not use both points in the same logic framework. Mix them up and paradoxes would ensue.
From this starting point I derived the answer to the Sleeping Beauty Problem should be double halving. Several point worth mentioning:
No matter which perspective one choose to reason from there is no new information when beauty wakes up.
Questions such as “the probability of today is Monday” is invalid. Such probabilities do not exist since it requires specifying today from first-person perspective and treats both days in the experiment as equals like third-person perspective does.
This means after being told it is Monday beauty could keep her answer unchanged at 1/2.
Repeating the experiment from any single perspective and the relative frequency of heads can be shown to be 1/2.
This provide new arguments for double halfer position when it involves bets and rewards (such as Dutch book arguments)
It provides a perfect explanation for the perspective disagreement troubling halfers as pointed out by Pittard (2015).
It does not result in the embarrassment as other double halfer position do as pointed out by Titelbaum(2012).
It disproves the Doomsday Argument and the Presumptuous Philosopher base on the same principle.
My complete argument in pdf can be found here.
https://www.sleepingbeautyproblem.com/wp-content/uploads/2018/11/Perspective-Reasoning-and-the-Solution-to-the-Sleeping-Beauty-Problem-Xianda-Gao-19-11-2018.pdf
I apologize in advance for my language skills and possible misuse of terminologies. English is not my native language and philosophy is not my field. I desperately need feedbacks especially counter arguments. Thank you.