## Sleeping Beauty and Fair Coin

Posted on February 12, 2023 — 4 Minutes Read

Originally formulated by Arnold Zuboff, in his paper, One Self: The Logic of Experience, the Sleeping Beauty problem has invited debates from philosophers and mathematicians in the last 30 years. The problem, like many other philosophical puzzles, is rather simple. In a slight deviation from the original formulation, Sleeping Beauty volunteers to participate in an experiment, in which she will be put to sleep on Sunday, and be awakened on Monday should a fair coin come up heads, or on Monday and then again on Tuesday should it come up tails instead. When she is awakened, she does not know which day it is or whether she has been awaken before, and she is to an answer one question: what is your degree of belief that the coin came up heads?

Simple may it seem on the surface, this problem has resulted in two distinct positions. The first of which is the thirder position which holds that the answer is 1/3. Adam Elga argued for this position in his paper, Self-locating Belief and the Sleeping Beauty Problem. It rests upon the notion that there are two possible worlds, and three predicaments within these worlds, namely:

- H1: Heads and Monday
- T1: Tails and Monday
- T2: Tails and Tuesday

Before the experiment, it is well agreed that the probability of the coin coming up heads is 1/2. Given the way the problem is set up however, being in T1 is identical to being in T2, and as such the principle of indifference requires the two of them being assigned the same credence, or degree of belief i.e. P(T1) = P(T2). Knowing also that a fair coin requires that P(H1) = P(T1), and that H1, T1 and T2 are the three exhaustive outcomes of this experiment, and therefore their probability must add to 1, without any information as to which day it is, upon waking up, Sleeping Beauty must update her answer from 1/2 to P(H1) = P(T1) = P(T2) = 1/3.

The other, halfer position, believes otherwise. David Lewis argued for this in his paper, Sleeping Beauty: Reply to Elga. In which he disagrees that upon waking up, Sleeping Beauty should update her answer from 1/2 to 1/3, because in the way the experiment is set up, she is told of the details of the experiment before it begins, and receives no additional information of any relevance upon waking up, therefore it does not hold that her degree of belief should be any different.

Debate on-going, personally I view the dispute at least in part due to the way the question is posed. If instead one approach the problem by way of transcendental deduction that Immanuel Kant, the 18th-century philosopher and the central figure in modern philosophy, devised to distinguish his transcendental philosophy from the dogmatic metaphysics of his time, or the by way of hermeneutical phenomenology that Martin Heidegger, one of the most influential philosophers of the 20th century, developed to discern his fundamental ontology from the traditional one; one will be inclined to ask a different set of questions instead i.e. in what conditions would one say that the answer is 1/2 and, what other conditions, 1/3. Two set of questions may emerge as a result:

- What is probability of the coin coming up heads, regardless of what is to happen before or after, and independent of the observer and of the consequences thereby imposed on the result of the outcome? Most I imagine would say that it is 1/2.
- What is the probability of Sleeping Beauty, waking up, as a result of the coin coming up heads? Likewise, I imagine most would agree that it is 1/3.

Better questions do seem to yield better answers.