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The sleeping beauty paradox is a fallacy

The sleeping beauty paradox is a problem that has puzzled mathematicians and many other scholars for decades.

Here is a description of the problem:

Participants volunteer to undergo the following experiment and are told all of the following details: On Sunday she will be put to sleep. Depending on a coin toss (see below) during the experiment, she will be either awakened, interviewed just once, or after the first interview put back to sleep with a special drug that makes her forget that awakening, and is then awakened, interviewed the second time.

A fair coin will be tossed to determine which experimental procedure to undertake:

  • If the coin comes up heads, the participant will be awakened and interviewed just once (e.g., on Monday only).
  • If the coin comes up tails, she will be awakened and interviewed twice (e.g., on Monday and then Tuesday).

A key condition: Any time the participant is awakened and interviewed she will not be able to tell which day it is or whether she has been awakened before, because her memory has been erased.

During the interview participant is asked the following question:

“What is your estimate of the probability that the coin landed heads in your case?”

Here’s the puzzlement: depending on one’s viewpoint, you may conclude that the probability is either 1/2, or 1/3, and both conclusions would seem to be logical and flawless.

For example, suppose you conduct the same experiment 100 times, there would objectively be 50 heads, 50 tails. Therefore, if you objectively observe as an outsider, the probability for heads is 1/2 (50/100), and the fact that you are asking the participant to give an estimate shouldn’t change the probability.

But suppose you do not have the above outsider’s view of things, but are purely thinking from the participant viewpoint, things suddenly seem different. Considering that the survey question is asked 150 times, and 50 of which belong to heads, but 100 belong to tails, one may also logically conclude that the probability for heads is only 1/3 (50/150).

The puzzlement is understandable because it seems that depending on one’s viewpoint, the probability of the same thing can have two different values, and both seem to be logical and flawless.

As a result, some claim that sleeping beauty paradox proves that probability is subjective, not objective, or even it is an illusion, not real.

The fallacy

But the whole thing is a fallacy.

First, if the participant were a nonlife object, such as a piece of stone, there would be only one viewpoint, which is the outsider’s objective viewpoint. From this viewpoint, the dead stone on Tuesday is the same stone on Monday, and therefore everything on the tail side is identical to the head side. Just because you examine the tail side twice (Monday and Tuesday) rather than just once as you do with the head side (Monday only) does not make any difference. Thus the 1/2 answer becomes the only logical answer.

Therefore, the duality confusion arises from the fact that the participant is alive and has a mind, and the question is addressed to her subjective mind. There is of course nothing wrong with using a live person to conduct an experiment of this kind, but the contradiction arises from the fact that the participant’s memory is erased.

Why is erasing memory a problem?

Because it causes a loss of information, which creates a self-contradiction, and the missing information or lack of information leads to erroneous conclusion.

Specifically, had the participant on the tail side when tested on Tuesday had memory, she would have recognized immediately with 100% certainty that she is on the tail side (because the participant on the head side would never be examined on Tuesday). The certainty essentially eliminates the Tuesday sample from the probability estimate, because it is no longer an independent event, but rather an event 100% dependent on a preceding event, namely the tail side participant’s test on Monday. This leaves only two independent and valid samples, one from each side and both happens on Monday, and therefore the 1/2 answer (50% probability) from the participant’s viewpoint is completely consistent with the objective outsider’s viewpoint.

From an information science’s viewpoint, an event that has 100% certainty (or predictability) carries no new information. This fact may be actually more meaningful than the fallacious sleeping beauty problem itself.

This is because, it is interesting that a participant’s having memory would make the event non-informational, as if one type of information annihilates another type of information. Perhaps there is something that can be called ‘anti-information’? Or is it just an artifact of Shannon’s quirky definition of information?

In more practical terms however, it is just the availability of correct information cancels incorrect information.

One thing seems certain from what’s implicated in this example: information in the mind of an observer has a unique effect of eliminating states (samples in this case) and thus reduces entropy and improves the order of the system that is being observed. This touches upon a fundamental aspect of mind-universe relationship. But that is a different subject.

Therefore, the seeming paradox is artifacts of a basic self-contradiction: on the one hand, the participant’s memory is erased, effectively rendering her a nonliving object as far as information for estimating probabilities is concerned; but on the other hand, the participant is still hypothetically tested (inquired) as if she was still a qualified observer when she is not.

In other words, an object of test (participant in this case) cannot be both alive and dead at the same time, considered from an information point of view. If a logic premise includes such a condition, it would be a self-contradiction and a logical fallacy.

The fallacy can be exposed even more clearly with another thought experiment: just imagine you increase the number of tests (sleep-wake-inquire cycles) on the tail side from 2 to many, to an arbitrary number, or even to infinity. In the mind of the participant, because the number of possibilities that she is on the tail side and being tested approaches infinity, she is most certainly on the tail side and thus the probability that she is on the head side approaches zero. That would be an absurd conclusion. Even without the absurdity of infinity, just the fact that the probability would be arbitrarily altered by the number of tests conducted on the tail side already reviews the fallacy.

Probability and information; objective truth and subjective truth

The probability is information dependent. That is, the estimate of the probability of a certain thing depends on the information that is available to the estimator. See for example: Bayesian probability and the Monty Hall problem.

This is a universal truth, but the sleeping beauty problem is an excellent illustration.

However, some people like to use this fake paradox to jump into an ideological or philosophical conclusion.

Among the worst is the claim that probability is subjective, not objective, or that it is an illusion, not real.

“It is all in your head,” as those who have a problem with objective truth take pleasure to proclaim.

But that is absolutely wrong.

In fact, as illustrated above, the sleeping beauty paradox itself is a fallacy, because there’s no paradox at all.

The fallacious conclusions tend to come from an ambiguity which I call ‘the ambiguity of observer’s position’. People tend to unconsciously switch position between one that is of an objective outside observer and another that is of a subjective participant, depending on how people think and/or how the question being asked is phrased and interpreted. Because these two positions have different information, it is natural that they may have different estimates of the events including probabilities.

But regardless, there’s only one single framework of the objective truth, which is the outside observer.

It is an objective of the subjective participants to approximate the objective truth, rather than fallaciously boasting its ambiguity and subjectivity.

There can be a quite simple illustration using the sleeping beauty problem.

The seemingly unique question ‘”What is your estimate of the probability that the coin landed heads?” is not unique, but ambiguous, because there are really two different questions depending on who the observer is.

That is, the ambiguity is not that there are two different answers to the same question, but that there are two different questions in the ambiguous question that appears to be just one question.

Accordingly, the 1/2 answer is the correct answer to the following question: ‘judging from an outside observer, when a fair coin is tossed, what is the probability of you the participant is on the head side?” This question is a general and universal question even though it is addressed to the participant, and has nothing to do with specific experiment experienced by the participant. Therefore the answer to this question can only be 1/2.

But the 1/3 answer is the ‘correct’ answer to the following different, but more abstract, question: ‘judging from the participant’s viewpoint, among all the incidence of the survey samples which are divided into two groups, one group with heads, but another with tails, what is probability that you are in the first group?’ But this answer is only correct because it is logical based on the information the participant has in her head. It really is wrong because the participant is deprived of the correct information and given self-contradictory wrong information (specifically, she is considered both alive and dead at the same time).

You see, the confusion arises from ambiguity of definition or identity. A participant who falls in the tail group is one single physical identity (a specific person in this case), but at the same time represents two different abstract identities as a survey object (because she will be surveyed twice, each representing a survey object). The way the whole experiment is designed creates two different statistical ensembles. The first ensemble is the real universe in which the information reflects the reality. The second ensemble is an imaginary universe in which the information is distorted to reflect a substitute reality.

But as said above, a single framework of the objective truth always exists regardless of the relativity of our subjective experiences. The objective of the subjective participants must be to approximate the objective truth as close as possible, rather than fallaciously boasting its ambiguity and subjectivity.

A lesson of metaphysics

But the above imaginary universe in which the information is distorted can seem to be perfectly logical within the perimeter of its own information. It therefore reaches a wrong conclusion that seems to be perfectly logical.

This ought to serve as a warning to any human-centered thinking or philosophies. Without the objective truth, we the humans may all be the misguided sleeping beauties reaching conclusions that may seem to be perfectly logical to us, but in fact wrong.

We would not realize that the objective truth can be, and can only be, revealed from an outside observer (the Creator of the universe).

That humanity does not in fact live in complete falsehood, is only because in reality we do receive revelations from the outside observer. We connect with the physical world through our physical body and senses; we connect with our self and others through our soul; and we connect with God who is an outside Creator through our spirit. We are not sleeping beauties, if we choose to connect properly. The Creator has always communicated with us to provide proper information about our existence, the whole creation, and the Creator’s plan and purpose. We just need to humbly tune our ears to the voice of revelation and tune out from the noisy background.