From a first person perspective, a self-aware observer can inherently identify herself from other individuals. However, from a third person perspective this identity through introspection does not apply. On the other hand, because an observer’s own existence is a prerequisite for her reasoning she would always conclude she exists from a first person perspective. This means observers have to take a third person perspective to meaningfully contemplate her chance of not coming into existence. Combining the above points suggests arguments which utilize identity through introspection and information about one’s chance of coming into existence fails by not keeping a consistent perspective. This helps explaining questions such as doomsday argument and sleeping beauty problem. Furthermore, it highlights the problem with anthropic reasonings such as self-sampling assumption and self-indication assumption.
Any observer capable of introspection is able to recognize herself as a separate entity from the rest of the world. Therefore a person can inherently identify herself from other people. However, due to the first-person nature of introspection it cannot be used to identify anybody else. This means from a third-person perspective each individual has to be identified by other means. For ordinary problems this difference between first- and third-person reasoning bears no significance so we can arbitrarily switch perspectives without affecting the conclusion. However this is not always the case.
One notable difference between the perspectives is about the possibility of not existing. Because one’s existence is a prerequisite for her thinking, from a first person perspective an observer would always conclude she exists (cogito ergo sum). It is impossible to imagine what your experiences would be like if you don’t exist because it is self-contradictory. Therefore to envisage scenarios which oneself does not come into existence an observer must take a third person perspective. Consequently any information about her chances of coming into existence is only relevant from a third-person perspective.
Now with the above points in mind let’s consider the following problem as a model for the doomsday argument (taken from Katja Grace’s Anthropic Reasoning in the Great Filter):
God’s Coin Toss
Suppose God tosses a fair coin. If it lands on heads, he creates 10 people, each in their own room. If it lands on tails he creates 1000 people each in their own room. The people cannot see or communicate with the other rooms. Now suppose you wake up in a room and was told of the setup. How should you reason the coin fell? Should your reason change if you discover that you are in one of the first ten rooms?
The correct answer to this question is still disputed to this day. One position is that upon waking up you have learned nothing. Therefore you can only be 50% sure the coin landed on heads. After learning you are one of the first ten persons you ought to update to 99% sure the coin landed on heads. Because you would certainly be one of the first ten person if the coin landed on heads and only have 1% chance if tails. This approach follows the self-sampling assumption (SSA).
This answer initially reasons from a first-person perspective. Since from a first-person perspective finding yourself exist is a guaranteed observation it offers no information. You can only say the coin landed with an even chance at awakening. The mistake happens when it updates the probability after learning you are one of the first ten persons. Belonging to a group which would always be created means your chance of existence is one. As discussed above this new information is only relevant to third-person reasoning. It cannot be used to update the probability from first-person perspective. From a first person perspective since you are in one of the first ten rooms and know nothing outside this room you have no evidence about the total number of people. This means you still have to reason the coin landed with even chances.
Another approach to the question is that you should be 99% sure that the coin landed on tails upon waking up, since you have a much higher chance of being created if more people were created. And once learning you are in one of the first ten rooms you should only be 50% sure that the coin landed on heads. This approach follows the self-indication assumption (SIA).
This answer treats your creation as new information, which implies your existence is not guaranteed but by chance. That means it is reasoning from a third-person perspective. However your own identity is not inherent from this perspective. Therefore it is incorrect to say a particular individual or “I” was created, it is only possible to say an unidentified individual or “someone” was created. Again after learning you are one of the first ten people it is only possible to say “someone” from the first ten rooms was created. Since neither of these are new information the probability of heads should remains at 50%.
It doesn’t matter if one choose to think from first- or third-person perspective, if done correctly the conclusions are the same: the probability of coin toss remains at 50% after waking up and after learning you are in one of the first ten rooms. This is summarized in Figure 1.
Figure 1. Summary of Perspective Reasonings for God’s Coin Toss
The two traditional views wrongfully used both inherent self-identification as well as information about chances of existence. This means they switched perspective somewhere while answering the question. For the self-sampling assumption (SSA) view, the switch happened upon learning you are one of the first ten people. For the self-indication assumption (SIA) view, the switch happened after your self-identification immediately following the wake up. Due to these changes of perspective both methods require to defining oneself from a third-person perspective. Since your identity is in fact undefined from third-person perspective, both assumptions had to make up a generic process. As a result SSA states an observer shall reason as if she is randomly selected among all existent observers while SIA states an observer shall reason as if she is randomly selected from all potential observers. These methods are arbitrary and unimaginative. Neither selections is real and even if one actually took place it seems incredibly egocentric to assume you would be the chosen one. However they are necessary compromises for the traditional views.
One related question worth mentioning is after waking up one might ask “what is the probability that I am one of the first ten people?”. As before the answer is still up to debate since SIA and SSA gives different numbers. However, base on perspective reasoning, this probability is actually undefined. In that question “I” – an inherently self identified observer, is defined from the first-person perspective, whereas “one of the first ten people” – a group based on people’s chance of existence is only relevant from the third-person perspective. Due to this switch of perspective in the question it is unanswerable. To make the question meaningful either change the group to something relevant from first-person perspective or change the individual to someone identifiable from third-person perspective. Traditional approaches such as SSA and SIA did the latter by defining “I” in the third person. As mentioned before, this definition is entirely arbitrary. Effectively SSA and SIA are trying to solve two different modified versions of the question. While both calculations are correct under their assumptions, none of them gives the answer to the original question.
A counter argument would be an observer can identify herself in third-person by using some details irrelevant to the coin toss. For example, after waking up in the room you might find you have brown eyes, the room is a bit cold, dust in the air has certain pattern etc. You can define yourself by these characteristics. Then it can be said, from a third-person perspective, it is more likely for a person with such characteristics to exist if they are more persons created. This approach is following full non-indexical conditioning (FNC), first formulated by Professor Radford M.Neal in 2006. In my opinion the most perspicuous use of the idea is by Michael Titelbaum’s technicolor beauty example. Using this example he argued for a third position in the sleeping beauty problem.Therefore I will provide my counter argument while discussing the sleeping beauty problem.
The Sleeping Beauty Problem
You are going to take part in the following experiment. A scientist is going to put you to sleep. During the experiment you are going to be briefly woke up either once or twice depending the result of a random coin toss. If the coin landed on heads you would be woken up once, if tails twice. After each awakening your memory of the awakening would be erased. Now supposed you are awakened in the experiment, how confident should you be that the coin landed on heads? How should you change your mind after learning this is the first awakening?
The sleeping beauty problem has been a vigorously debated topic since 2000 when Adam Elga brought it to attention. Following self-indication assumption (SIA) one camp thinks the probability of heads should be 1/3 at wake up and 1/2 after learning it is the first awakening. On the other hand supporters of self-sampling assumption (SSA) think the probability of heads should be 1/2 at wake up and 2/3 after learning it is the first awakening.
Astute readers might already see the parallel between sleeping beauty problem and God’s coin toss problem. Indeed the cause of debate is exactly the same. If we apply perspective reasoning we get the same result – your probability should be 1/2 after waking up and remain at 1/2 after learning it is the first awakening. In first-person perspective you can inherently identify the current awakening from the (possible) other but cannot contemplate what happens if this awakening doesn’t exist. Whereas from third-person perspective you can imagine what happens if you are not awake but cannot justifiably identify this awakening. Therefore no matter from which perspective you chose to reason, the results are the same, aka double halfers are correct.
However, Titelbaum (2008) used the technicolor beauty example arguing for a thirder’s position. Suppose there are two pieces of paper one blue the other red. Before your first awakening the researcher randomly choose one of them and stick it on the wall. You would be able to see the paper’s color when awoke. After you fall back to sleep he would switch the paper so if you wakes up again you would see the opposite color. Now suppose after waking up you saw a piece of blue paper on the wall. You shall reason “there exist a blue awakening” which is more likely to happen if the coin landed tails. A bayesian update base on this information would give us the probability of head to be 1/3. If after waking up you see a piece of red paper you would reach the same conclusion due to symmetry. Since it is absurd to purpose technicolor beauty is fundamentally different from sleeping beauty problem they must have the same answer, aka thirders are correct.
Technicolor beauty is effectively identifying your current awakening from a third-person perspective by using a piece of information irrelevant to the coin toss. I purpose the use of irrelevant information is only justified if it affects the learning of relevant information. In most cases this means the identification must be done before an observation is made. The color of the paper, or any details you experienced after waking up does not satisfy this requirement thus cannot be used. This is best illustrated by an example.
Imagine you are visiting an island with a strange custom. Every family writes their number of children on the door. All children stays at home after sunset. Furthermore only boys are allowed to answer the door after dark. One night you knock on the door of a family with two children . Suppose a boy answered. What is the probability that both children of the family are boys? After talking to the boy you learnt he was born a Thursday. Should you change the probability?
A family with two children is equally likely to have two boys, two girls, a boy and a girl or a girl and a boy. Seeing a boy eliminates the possibility of two girls. Therefore among the other cases both boys has a probability of 1/3. If you knock on the doors of 1000 families with two children about 750 would have a boy answering, out of which about 250 families would have two boys, consistent with the 1/3 answer. Applying the same logic as technicolor beauty, after talking to the boy you shall identify him specifically as “a boy born on Thursday” and reason “the family has a boy born on Thursday”. This statement is more likely to be true if both the children are boys. Without getting into the details of calculation, a bayesian update on this information would give the probability of two boys to be 13/27. Furthermore, it doesn’t matter which day he is actually born on. If the boy is born on, say, a Monday, we get the same answer by symmetry.
This reasoning is obviously wrong and answer should remain at 1/3. This can be checked by repeating the experiment by visiting many families with two children. Due to its length the calculations are omitted here. Interested readers are encouraged to check. 13/27 would be correct if the island’s custom is “only boys born on Thursday can answer the door”. In that case being born on a Thursday is a characteristic specified before your observation. It actually affects the chance of you learning relevant information about whether a boy exists. Only then you can justifiably identify whoever answering the door as “a boy born on Thursday”and reason “the family has a boy born on Thursday”. Since seeing the blue piece of paper happens after you waking up does not affect your chance of awakening it cannot be used to identify you in a third-person perspective. Just as being born on Thursday cannot be used to identify the boy in the initial case.
On a related note, for the same reason using irrelevant information to identify you in the third-person perspective is justified in conventional probability problems. Because the identification happens before observation and the information learned varies depends one which person is specified. That’s why in general we can arbitrarily switch perspectives without changing the answer.