The Existential Coin Toss Thought Experiment: Self-Sampling v Self-Indication and the Presumptuous Philosopher Problem

A Version of this article is published in my book, ‘TWISTED LOGIC: PUZZLES, PARADOXES, AND BIG QUESTIONS’. CRC Press/Chapman & Hall, 2024. https://www.amazon.co.uk/Twisted-Logic-Puzzles-Paradoxes-Questions/dp/1032513349

The Existential Coin Toss is a thought experiment where the existence of two different worlds depends on a coin flip. World A (Heads) has one black-bearded individual, while World B (Tails) has two individuals, one with a black and another with a brown beard. Waking up in one of these worlds without prior knowledge of your world or beard colour, what probability would you assign to being in World B?

The Set-Up

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Heads creates World A, which is inhabited by one black-bearded individual.

Tails brings forth World B, which is populated by two individuals, one with a black beard and the other with a brown beard.

Awakening in the darkness, unaware of your world and beard colour, but aware of the rules of your existence, what probability would you assign to the coin having landed on Tails, placing you in World B?

The answer hinges on your basic assumptions about existence.

The Self-Sampling Assumption (SSA)

This approach encourages us to think of ourselves as a random selection from all entities that could have been us—our ‘reference class’. Consequently, we are a randomly selected bearded individual, with an equal chance of living in World A (Heads) or World B (Tails). If in World B, there’s a 50-50 probability of sporting either a black or brown beard.

But what happens when the light comes on and you see a black beard? Now, the probability of being in World A, where the sole inhabitant has a black beard, increases. Given the choice between World A (100% black beard chance) and World B (50% black beard chance), the likelihood of residing in World A is twice as much, making it a 2/3 chance the coin landed Heads.

The Self-Indication Assumption (SIA)

This alternative perspective suggests that you are twice as likely to be in a world where two observers exist than in a world with just one observer. Thus, you might lean towards World B, in which there are two observers, giving it twice the likelihood (2/3 chance) as World A (1/3 chance). But, once the lights are on and your beard is revealed to be black, the probability of living in World B reduces to 1/2, the same as the probability of living in World A, since your existence is confirmed in a scenario where both worlds have equal chances of your specific condition.

Implications

The contrasting perspectives of SSA and SIA in the Existential Coin toss thought experiment illustrate the complexities in assigning probabilities to our own existence. The assumptions we choose significantly influence our conclusions about our likelihood of existing in one world rather than another. As such, this not only sheds light on philosophical debates surrounding conditional probability but also challenges our understanding of existence and identity in uncertain contexts.

Unravelling the Sleeping Beauty Problem

The Sleeping Beauty Problem puts the Self-Sampling Assumption (SSA) and Self-Indication Assumption (SIA) to the test.

In this Problem, Sleeping Beauty volunteers for an experiment where she goes to sleep. A fair coin will be tossed to determine the next steps:

•       If the coin lands on Heads, she will be awakened once (on Monday).

•       If it lands on Tails, she will be awakened twice (on Monday and Tuesday).

On both awakenings, she has no memory of any previous awakenings, and thus can’t tell which day it is or how many times she’s been awoken. When she wakes, she’s asked: ‘What chance do you assign to the proposition that the coin landed Heads?’

Adopting the Self-Sampling Assumption (SSA)

Under this assumption, Sleeping Beauty would argue that there’s a 50-50 chance the coin landed on Heads or Tails, as those are the only two possible outcomes from a fair coin toss. This perspective doesn’t change upon waking. Only if she’s told that it’s her second awakening (which means it’s Tuesday and the coin must have landed Tails) will she change her belief to 100% Tails and 0% Heads.

Adopting the Self-Indication Assumption (SIA)

Under this assumption, Sleeping Beauty considers the number of observer-moments—points at which she is awake and observing. There are two such points if the coin lands Tails (one on Monday and one on Tuesday), but only one if the coin lands Heads (on Monday).

From this viewpoint, Sleeping Beauty would reason there’s a 1/3 chance the coin landed Heads and a 2/3 chance it landed Tails. This is because there are three observer-moments in total (Monday on Heads, Monday on Tails, and Tuesday on Tails), and each one is equally likely. So the coin landing Tails (which creates two observer-moments) is twice as likely as the coin having landed Heads (which creates only one observer-moment).

In summary, the Sleeping Beauty Problem involves SSA and SIA to determine probabilities based on the number of awakenings. Here, the SSA leads to a 50% chance of heads, while the SIA suggests a 1/3 probability, due to more observer-moments under tails.

Thus, the SSA and SIA lead to different conclusions in the Sleeping Beauty Problem, just as in the God’s Coin Toss problem. The correct approach remains a topic of debate among philosophers and statisticians, reflecting broader inquiries into how we interpret probability and make decisions under uncertainty.

Dilemma of the Presumptuous Philosopher

The Presumptuous Philosopher Problem introduces a critical examination of the Self-Indication Assumption (SIA) by presenting a scenario where SIA seems to lead to counterintuitive or problematic conclusions.

Consider a situation where scientists are evaluating two theories, each equally supported by prior evidence. Theory 1 predicts a universe with a million times more observers than Theory 2, but new evidence from a particle accelerator now strongly supports Theory 2. Despite the empirical evidence supporting Theory 2, philosophers using the SIA can insist that Theory 1 is much more likely to be correct. Look at it this way. You exist, and Theory 1 makes your existence a million times more likely than Theory 2, because there are a million times more observers that exist if Theory 1 is true than if Theory 2 is true. To put it another way, given the fact that you exist, a case can be made for supporting a theory that proposes that a very large number of observers exist over a theory that proposes a much smaller number, even if empirical evidence strongly contradicts it.

But should the sheer number of potential observers really sway our belief in a theory, especially when faced with concrete evidence to the contrary? Critics argue that this perspective cam lead to presumptuous conclusions, hence the name of the problem. In this way, the Presumptuous Philosopher Problem highlights the tension beytwee this approach and traditional evidence-based reasoning.

Conclusion: Choosing Our Assumptions

These thought experiments illustrate the complexity of probability and existence. They challenge us to ponder: Which assumption aligns with our intuition, and how reliable is our intuition in such abstract scenarios?