This post discusses why any halfer position in the Sleeping Beauty Problem would lead to disagreements between two agents sharing all information. This issue has not been much discussed except by @Katja Grace and John Pittard. Furthermore I would explain why these seemingly absurd disagreements are actually valid. This post is another attempt by me trying to get attention to the important difference between reasoning as the first person versus reasoning as an impartial observer in anthropic problems.
To show any halfer position would lead to disagreements between two communicating agents consider this problem:
Bring a Friend: You and one of your friend are participating in a cloning experiment. After you fall asleep the experimenter would toss a fair coin. If it lands Heads nothing happens. If it lands Tails you would be cloned and the clone would be put into an identical room. The cloning process is highly accurate such that it retains the memory to a level of fidelity that is humanly indistinguishable. As a result, next morning after waking up there is no way to tell if you are physically the original or the clone. Your friend wouldn’t be cloned in any case. The next morning she would choose one of the two rooms to enter. Suppose your friend enters your room. How should she reason the probability of Heads? How should you reason?
For the friend this is not an anthropic problem. So her answer shouldn’t be controversial. If the coin landed Heads she has 50% chance of seeing an occupied room. While if the coin landed Tails both room would be occupied. Therefore seeing me in the room is evidence favouring Tails. She would update the probability of Heads to 1/3.
From my (the participant’s) perspective this is a classical anthropic problem just like Sleeping Beauty. There are two camps. Halfers would say the probability of Heads is 1/2. Reason being I knew I would find myself in this situation. Therefore I haven’t gained any new information about the coin toss. The probability must remain unchanged. The other camp says the probability of Heads should actually be 1/3. Most thirders argue I have gained the new information that I exist which is evidence favouring more copies of the participants exist. Therefore the probability of Tails shall increase from the prior.
Both camps should agree that seeing the friend (or not) would not change their answer. Because the friend is simply choosing one room out of the two. Regardless of coin toss result there is always a 50% chance for her to enter my room.
Now Halfers are in a peculiar situation. My probability of Heads is 1/2 while the friend’s answer is 1/3. We can share our information and communicate any way we like. Nothing I say can change her answer as nothing she says can change mine. To make the matter even more interesting I would have to admit there is no mistake in my friend’s reasoning and she would think I am correct too. Our difference is purely due to the differences in perspectives. This seems to contradict Aumann’s Agreement Theorem.
Thirders do not have any of these problems. The friend and I would be in perfect agreement that the probability is 1/3. As a result this issue is occasionally used as a counter to Halferism (as did so by Katja Grace even though that post targets SSA specifically her argument applies to all halfers). However I would like to argue these disagreements are indeed valid.
Repeating the Experiment as the Friend vs as the Participant
Re-experiencing the experiment as the friend is not the same as re-experiencing it as a participant. From the friend’s perspective repeating it is straightforward. Let another coin toss and potential cloning happen to someone else and then choose a random room again. It is easy to see if the number of repetition is large she would see the chosen room occupied about 3/4 of the times. Out of which about 1/3 of the meetings would be after Heads. The relative frequency agrees with her answer of 1/3.
To repeat the experiment from my first-person perspective is a different story. After waking up from the first experiment (and potentially meeting my friend) I would simply participate in the same process again. I shall fall asleep and let another coin toss and potential cloning take place. I would wake up again not knowing if I’m the same physical person the day before. Suppose I’m told the coin toss result at the end of each day. If this is repeated a large number of times then I would count about 1/2 of the awakenings following Heads. I would also meet my friend 1/2 of the times, with about equal numbers after heads or tails. My relative frequency of Heads would be 1/2, agreeing with my answer.
In tosses involving both my friend and me she may see the other copy of participant instead of me specifically after Tails. This caused our difference in answers. Which leads to our different interpretation of:
Who is in this Meeting
From the friend’s perspective choosing a random room have two possible outcomes. Either the chosen room is empty or it is occupied. The new information she received is simply there is someone in the room. She interprets that person as an unspecific participant.
On the other hand from my perspective the person in the room is specific, i.e. me. The possible outcome for the room selection is either I see the friend or I do not see the friend. There may or may not exist another version of participant who is highly similar to me but that would not affect my observation.
Effectively we are answering two different questions. For my friend “what is the probability of Heads given there is a version of participant in the chosen room”. For me “what is the probability of Heads given I specifically am in the chosen room”. In this sense the disagreement does not violate Aumann’s Agreement Theorem.
A point to note: the specification of I is limited to my first-person perspective. It is incommunicable. I can keep telling my friend “It’s me” yet it would carry no information to her. Because this specification has nothing to do with objective differences between me and other copies of the participant. I refer this person as me only because I’m experiencing the world from its perspective. Because this person is undeniably most immediate to the only subjective experience and consciousness accessible. Identifying me is primitive. Which begs the question:
Can Indexicals Be Used In Anthropics?
Indexicals (or pure indexicals by some definition) such as I, here, now and by extension we, today or even this world is a point of contention between halfers and thirders. Typically thirders think indexicals can be used in anthropic reasoning while halfers disagree. In the Sleeping Beauty Problem this conflict manifest in the debate of new information. Most thirders think there is new information since I learned I am awake on one specific day, i.e. today. Halfers typically argue the indexical today is not an objective specification and it can only be said I am awake on an unknown day, i.e. there is at least one awakening. In the cloning example above thirders think the new information is a specific participant I exist whereas halfers often argue objectively speaking it only shows at least one participant exists.
The disagreement between participant and friend presents another explanation for the indexicals. Indexicals are references to the perspective center. When someone says here he is talking about the location where he is experiencing the world from, the place most immediate to the subjective experience. Similarly I and now points to other aspects of the perspective center. They refer to the agent and time most immediate to the subjective experience respectively. Because it is only related to the perspective a participant can intrinsically identify I and never confuse himself with others. He can do this without knowing any difference between him and other highly similar participants. Also because of this dependency on the perspective this identification is not meaningful to the friend. Which lead to their disagreement.
By this logic the use of indexicals in anthropic problems is valid, that is, as long as we are reasoning from the first-person perspective of a participant. The debate of their usage is a debate between perspectives. When thirders say today is a specific day it requires us to imagine being in beauty’s shoes wakening up in the experiment. Halfers oppose today’s use because they are reasoning as an outsider. In which case some objective measure is required to differentiate the two days. This shows the conflicting logics between an-
Impartial Observer vs the First Person
We often purposely formulate our logics in a way so it is not from any specific perspective. As if the perspective center is irrelevant to the problem at hand, i.e. we reason as (imaginary) impartial observers would. This uncentered reasoning would not treat any agent, time or location as inherently special. It is what we usually meant as thinking objectively. Comparing to first-person reasoning it is different in several aspects.
The obvious difference is the aforementioned use of indexicals. Impartial observers’ uncentered reasoning cannot use indexicals since they are references to the perspective center. For most problems this simply means substituting the indexical I to a third-person identity in logical expressions. Similarly now and here are switched to some objectively identified time and location. For anthropic problems however this has further implications because the ability to inherently identify oneself affects reasoning as shown by the disagreement between the participant and the friend.
Another difference is about one’s uniqueness. The indexicals as references to the perspective center are inherently special. From the first-person perspective I am one of a kind. Other agents, no matter how physically similar, are not its logical equals. This explains why as the first person I can be identified without knowing any difference between I and others. The differentiation is not needed because they are never in the same category to begin with. The same is true for now and here. On the other hand for an impartial outsider no agent, time or location is inherently special, i.e. they are indifferent.
The last difference is the probability of existence. The existence of I is a logical truth. Because to use the indexicals one has to reason from the first-person perspective. Yet reasoning from its perspective could only conclude in its self-existence, i.e. “I think, therefore I am”. It is sometimes presented as “ I can only find myself exist.” Furthermore given reasoned from a consistent perspective, “I am, here, now” would always be true. Because these indexicals refer to different aspects of the same perspective center. On the other hand we can also reason as impartial observers and specify a participant or time by some third-person identifiable measures. In this case it is entirely possible that agent does not exist or not conscious at the specified time. E.g. in the previous cloning example we can identify a participant as the one in the chosen room. It is possible that he does not exist since the chosen room can be potentially empty. In summary, it takes an outsiders’ perspective to think about someone’s nonexistence/unconsciousness.
Given these differences, logics from the first person and impartial outsiders should not mix in anthropic related problems. However most arguments in this field paid no attention to these distinctions. It is my core argument that anthropic paradoxes are caused by arbitrarily switching perspectives in reasoning, mixing the conflicting logics.
Paradoxes and Mixed Reasoning
To recap: the first-person perspective is centered. It can use indexicals because I, here and now are inherently special comparing to other agents, locations or time. Where “I exist, here and now” is a logical truth. On the other hand the impartial observers’ perspective is uncentered. Indexicals cannot be used because impartial observers are indifferent to any agent, time or location. Where the existence of any specific agents at any time or location is not guaranteed. Within a single logical framework we can employ either perspective, but not both. If an argument mixes the two, paradoxes ensue.
Take the Doomsday Argument as an example. It suggests we should take a more pessimistic outlook for human’s future than observed evidence suggests. The argument is simple. First, it recognizes a principle of indifference among all human beings (past, present or future alike). Then it specifically considers my birth rank among all humans (sometimes it is expressed as our birth rank or that of the current generation). As a result it concludes I am more likely to have my birth rank if there are fewer people in total, i.e. doom soon is more likely. This is a classic case of mixed perspectives. On one hand it treats all human beings indifferently as an impartial outsider would. Yet at the same time it uses indexicals by employing a first-person perspective and take a special interest in my or by extension our birth rank. Only by mixing the two it enables the conditional update shifting the probability to doom soon. If we reason as the first person and identify the indexical I by treating it as inherently special then the principle of indifference among all humans no longer applies. Similarly if we reason as an impartial outsider and recognize the principle of indifference then there is no reason to consider my birth rank specifically, in fact there is no way to identify I to begin with. Either way, the outlook of mankind can only be estimated by observed evidence. The probability shift is false.
Interestingly on some level we realize the inherently apparent I and the indifferent principle to all humans are not logically consistent. To reconcile this conflict a conscious step is often added: anthropic assumptions, which suggests treating I as a randomly selected individual among indifferent agents. Even though there is no justification to such assumptions accepting them feels natural. Because they allow two highly intuitive ideas to coexist. However, those two ideas are based on different perspectives which should be kept separate to begin with.
An example is the Self-Sampling Assumption (SSA). It suggests we should reason as if I am randomly selected from all actually existent (past, present or future) observers. This would lead to the infamous Doomsday Argument. An alternative to the SSA is the Self-Indication Assumption (SIA). It suggests we should reason as if I am randomly selected from all potentially existent observers. While it would refute the Doomsday Argument it has its own paradox: the Presumptuous Philosopher. (It concludes the number of intelligent life-forms in the universe should be higher than observed evidence suggests. Due to the fact that I exist is evidence favouring more observers.) The debate between SSA and SIA is about the correct reference class of I, whether it should be all existent or possible observers. Yet if the perspective reasonings are not mixed this problem would never exist in the first place. There is no default reference class for I since right from the start it is never in the same category with other observers, let it be actual or potential.
No default reference class also means any notions of the probability distribution of me being members of the said reference class are false. Such probabilities do not exist. Consider the paradox related to Boltzmann Brain. Some arguments suggest that under current theories of universe Boltzmann brains would vastly outnumber human brains in the universe. Then the probability of me being a Boltzmann brain is almost 100%. Essential to this calculation is a principle of indifference among all brains, which is valid if reasoned as an impartial observer. Yet it also specifically considers the first-person center I which contradict the indifference. As a result the probability it trying to calculate is logically inconsistent to begin with. There is no answer to it. Instead of using the indexical I the brain in question shall be specified from impartial observers’ perspective. E.g. A randomly selected one among all brains would almost 100% being a Boltzmann brain. This calculation would be correct. But also way less interesting. The same principle also refutes Nick Bostrom’s Simulation Hypothesis.
The non-existence of such probabilities can also be shown by the frequentist interpretation. Recall in the cloning example, I (participant) can re-experience the experiment as the first person. From my perspective after taking part in a large number of iterations the relative frequency of Heads or seeing my friend would both approach a certain value (1/2). However there is no reason for the relative frequency of me being the clone or the original of each experiment to converge toward any particular value. This again suggests such probabilities do not exist. Instead of using indexicals a participant must be specified from impartial observers’ perspective. Only then it is valid to ask the probability of this individual being the original or clone. E.g. the probability that the participant in the chosen room (if it exists) being the original is valid. A relative frequency can be calculated by an outsider without having to take a participant’s first-person perspective.
Sleeping Beauty Paradox and Conclusion
The Sleeping Beauty Paradox is without a doubt the most debated problem in anthropic reasoning. Nonetheless the same principle applies. The answer to it can be derived either from beauty’s first-person perspective or from impartial observers’ perspective. From the first-person perspective I have gained no new information. I did find myself awake today specifically. Yet that is just a logical truth in first-person reasoning. So even before falling asleep on Sunday it is already known that I would wake up in the experiment and identify that day as today. The probability of Heads remains at 1/2. From impartial observers’ perspective there is no new information either. While beauty being awake on a specific day is not guaranteed from this perspective, it could not use beauty’s perspective center to specify today. So all that is known is there is an unspecific awakening, i.e. there is at least one awakening. The probability of Heads should remain at 1/2 as well.
More importantly “the probability of today being Monday”, or “the probability of this awakening being the first” do not exist. Because they use indexicals in some default reference class (actual awakenings or potential awakenings) which is inconsistent. No Bayesian updating shall be performed after learning “Today is Monday”. The probability of Heads is 1/2 at awakening and remains at 1/2 after beauty finds out it is Monday.
In conclusion, perspectives play a significant role in anthropic related problems. Different perspectives could potentially give completely different answers. Most notably the special interest to the perspective center of the first-person and the general indifference of impartial observers are not compatible. Reasoning from these two perspectives must be kept separate to avoid paradoxes.