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.
--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.
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.
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 Inow 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.
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.