Some may notice I have posted this idea before. Unfortunately it has not sparked much discussion. However I sincerely believe this idea is the key to paradoxes related to anthropic reasoning. So I’m making this post hoping to raise more attention. Feedbacks are very welcome.
1. What is it about.
To summarize my entire argument in one sentence: reasonings from different perspectives should not mix.
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 "I" is defined by my 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 is used to specify me in third person because for an impartial outsider I am just an ordinary person like everybody else. There is no apparent "I" anymore. So some feature must be used to specify me among all people. My argument is that a logic framework must be fully contained within one perspective. It cannot be partially from a first-person reasoning and partially from a third-person reasoning. For example, if my perspective center is used to define "I" then we cannot switch to an outsider's third-person perspective just as we cannot switch to my wife's perspective. Conversely if we reason as an outsider and treat every individual as ordinary, i.e. all in the same reference class, then my first-person perspective’s center cannot be used to define a self explanatory “I”.
A perspective’s center defines not only “I” but also concepts such as “this”, “here” and “now”. So these words are primitively apparent to the first person. No other information is needed to explain them. Only when reasoned from another perspective do they need to be explained by features. Many anthropic arguments do not recognize this difference. So in their logic “I” sometimes refers to the primitively understood first-person center, while sometimes refers to a specific physical entity differentiated from others by some feature. Using these definitions interchangeably inevitably mixes reasoning from different perspectives. This is why many of these arguments give paradoxical conclusions.
2. Using my own existence as evidence
A point of contention in sleeping beauty problem is whether I have new information by waking up in the experiment. Halfers think I can only find myself awake, coupled with the existence of a guaranteed awakening, waking up in the experiment gives no new information. Thirders disagree. The notion I can only find myself awake is inconsequential. There are only two possibilities for today: either there is an awakening or not. Waking up gives the new information that there is an awakening which increases the probability of tails.
Both theories have intuitive appeals. From a first-person perspective it is true that I would always find myself exist. In fact by reasoning from the first-person perspective my existence is already acknowledged. In the case of sleeping beauty problem halfer is right in saying I can only find myself awake thus no new information.
It is also possible to reason from a third-person perspective. From a-third person perspective no one’s existence is inherently guaranteed. In the sleeping beauty problem if we reason from an outsider’s perspective then for a specific day it is possible the experiment subject would just sleep thought it. Thirders think waking up today eliminates that possibility thus is new information. Unfortunately there is a mistake. “Today” like “now” is defined by a perspective’s center. It is only meaningful if reasoned from a first-person perspective. So from the third-person perspective the information available is not the subject is awake on a specific day but rather the subject is awake on an unspecific day. I.e. there is (at least) one awakening in the experiment. This is already known from the experiment setup. The notion of new information is actually caused by mixed reasoning from two perspectives.
3. Are double-halfers un-Bayesian?
Basing on the coin toss result and today’s date halfers typically assigns probability as follows: P(Heads and Monday)=1/2, P(Tails and Monday)=1/4 and P(Tails and Tuesday)=1/4. However this presents a dilemma when she is told today is Monday. By ordinary bayesian update the probability of heads shall rise to 2/3. Since the coin toss can happen after the Monday wakeup I am predicting a fair coin toss yet to happen to deviate from half. The infamous Doomsday argument basically takes the same form. This is highly unpalatable. Alternatively some argue that after learning today is Monday the probability of Heads should remain at 1/2. However this seems un-bayesian. Various rules has been proposed (e.g. Halpern, Meacham, Briggs etc) to try to justify this yet all have very serious drawbacks (e.g. Titelbaum’s embarrassment for double-halfers). So the dilemma remains.
This is because in the context of sleeping beauty problem the probabilities of “today being Monday/Tuesday” do not exist. In another word “what’s the probability of today being Monday/Tuesday” are invalid questions. It is actually asking among the two potential awakenings which one is today’s. The generalize question in anthropic reasoning would be “among all entities in my reference class which one is me?”. These questions mix first and third-person reasoning. At one hand the reference class is only applicable from a third-person perspective while at the other hand it is using my perspective center to specify a self-explanatory “today” or “me”. They are essentially asking from a third-person perspective which one is my first-person perspective center. Such questions/probabilities are therefore invalid. Because these probabilities do not exist in the first place there is no Bayesian update to be performed upon learning today is Monday.
4. Is there a discrepancy between Bayesian and Frequentist probability?
One often seen argument is to repeat the sleeping beauty experiment multiple times. In the long run about 2/3 of the awakenings would be after Tails and 1/3 after Heads. This suggest the frequentist probability of Heads should be 1/3. Halfers are actually arguing the Bayesian and Frequentist probabilities being different. Such a notion is mistaken, once again, due to the mixing of perspectives.
Let’s take a first-person perspective of the experiment subject. For the ease of narratives consider this experiment. (Don’t worry I will get back to sleeping beauty problem momentarily.) When you go to sleep tonight a fair coin would be tossed. Depending on how it falls a highly accurate clone of you may be created and be put into an identical room. So when you wake up there is no way of knowing if you are old or new. Suppose you have fall asleep and wake up in the experiment, i.e. experienced a iteration of the experiment. Now from your first-person perspective what would a repetition of this experiment be like? Quite obviously it would be you going to sleep once more, with another coin toss and potential cloning, and wake up again. This process can be repeated many times up to infinity. Among these repetitions half of them would be Heads. Of course if there exist another copy of you he may go through the same procedure since for outsiders there is no reason to treat you two differently. But that does not concern you so it is irrelevant from a first-person perspective.
Based on the same reasoning the repetitions in the sleeping beauty problem should take the following form. In the first experiment the two potential awakenings are 24 hours apart. After waking up from the first experiment I shall enter another repetition with two potential awakenings 12 hours apart. Another repetition would be 6 hours apart and so on. (this assumes the time taken for actual waking and interviewing is zero for the easy of description etc.) In another word repeating the sleeping beauty experiment from the first-person perspective is a supertask which can be repeated up to infinity within 24 hours. Obviously half of those awakenings would be after Heads and the other half following Tails.
The above process describes first-person perspective. Third-person perspective is way less complicated. For an outsider the identity of the experiment participant, both before and after the memory wipe, are irrelevant. So experiments performed on different people through different times are all considered repetitions. As an outsider the number of repetitions is simply the number of experiments performed. Among which half of the experiments would have Heads tosses and the other half Tails.
From participant’s first-person perspective the frequency can be accounted by the number of awakenings. At the same time only experiments he subjectively experienced are counted as repetitions. Whereas from an outsider’s third-person perspective the frequency shall be accounted by experiments performed. Yet any experiment on any participants are considered repetitions. Only if we mix the two by account the frequency by awakenings and count all experiments performed on different people as repetitions do we get the frequency of Heads as 1/3. This is again a mix of perspectives.
On a side note since many arguments are based on bets and their respective rewards. These rewards should also follow decision maker’s perspective. Meaning when a participant is duplicated his money shall be duplicated as well. Only then would her decision reflect the probability.
5. Is perspective disagreement unreasonable?
John Pittard (2015) pointed out any halfer must affirm robust perspectivism. That is two people in direct communication can give different probabilities. This has been used by some as a counter to halferism. To see this disagreement consider the following example. (Pittard actually used a slightly more complicated example in his paper. However the explanation presented here is applicable to both cases. ) Just like in the previous example when you fall asleep you might be cloned if a coin toss resulted in Tails. The two of you would be put into identical rooms. If the coin fell Heads one of the rooms would be empty. The clone process is highly accurate it retains memory so you have no way of knowing if you are old or new. One of your friend would randomly choose one of the two rooms to enter. Say she saw you in the room. What’s her probability of Heads? What is your answer?
For your friend the question is non-anthropic so it is very easy. If the coin fell Heads one of the room would be empty. If the coin fell Tails then you would be cloned and both rooms would be occupied. Since the room chosen is occupied it is new evidence favouring Tails. Simple Bayesian updating would give her the probability of Heads to be 1/3. For you seeing your friend gives you no new information about the coin toss. Because it doesn’t matter if there is anyone in the other room, you would have the same probability of seeing her (1/2). Therefore whatever probability I assigned at wake up must remain. Meaning as a halfer I should assign the probability of Heads as 1/2.
Here the disagreement is apparent. The two of us are in direct communication and sharing all our information. Yet we are giving different probabilities to the same question. To make the situation weirder, both of us think the other’s reasoning is correct. Some may think it is highly unpalatable suggesting the probability of head cannot be anything other than 1/3.
The explanation for this disagreement is quite simple: we are not actually answering the same question. From my friend’s perspective the person in this room is any copy of me. If I was cloned she has no meaningful way of telling which specific one is in this room. Hence she is answering the question “what’s the probability of Heads given the chosen room is occupied (by any copy of me)”. Whereas from my first-person perspective I am defined by my perspective center. So even if there is another copy of me I am still inherently unique. I.e. nobody else is in my reference class. So I am answering the question “What is the probability of Heads given I, this person specifically, is in the chosen room” Because this specificality is purely based on the first-person perspective it is incommunicable. E.g. I can keep telling her “this is me” but it would mean nothing to her. That’s why we would remain our differences yet agree the other is also correct from his/her perspective.
This disagreement is also valid in the frequentists sense. Suppose I am stuck in this experiment for 1000 days. I would see my friend in about 500 days with half of those following Heads and the other half following Tails. However in these 1000 experiments she would find the room occupied 750 times. The extra 250 are the times when she saw the other copy of me following tails. Of course in reality my friend may be involved in far more repetitions than 1000. Because all the copies produced during the experiments might also want to repeat the experiment by their count. However this higher number of repetitions would not change the relative frequency for her. So the frequentist probability for the two of us are indeed different.
To reason in third-person perspective is to think as some outsider whose center is unrelated to the topic of interest. i.e. uncentered reasoning. First-person perspective uses its center in various aspect of logic. i.e. centered reasoning. Traditionally (e.g. David Lewis) treats centered reasoning as uncentered reasoning with additional information. That uncentered information describe the world and centered information adds my location on top of that. So uncentered reasoning can be subsumed into centered reasoning. In contrast I think they are logics from two distinct perspectives that are actually in parallel. Both centered and uncentered information describes the world just from different perspectives. Therefore the two systems should not mix. If we recognize the importance of it paradoxes in anthropic reasoning can be well explained.