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An Ontological Solution to the Sleeping Beauty Problem

Franceschi Paul

Published Nov 1, 2008• 0 citations
OpenAlex

Abstract

ABSTRACT. I describe in this paper an ontological solution to the Sleeping Beauty problem. I begin with describing the Entanglement urn experiment. I restate first the Sleeping Beauty problem from a wider perspective than the usual opposition between halfers and thirders. I also argue that the Sleeping Beauty experiment is best modelled with the Entanglement urn. I draw then the consequences of considering that some balls in the Entanglement urn have ontologically different properties form normal ones. The upshot is that I endorse the halfer conclusion on the probability of Heads once beauty is awaken and the thirder conclusion on conditional probabilities, and that original conclusions ensue on the probability of waking on Monday. 1. The Entanglement urn Let us consider the following experiment. In front of you is a urn. The experimenter asks you to study very carefully the properties of the balls that are in the urn. You go up then to the urn and begin to examine carefully its content. You note first that the urn contains only red or green balls. By curiosity, you decide to take a sample in the urn of a red ball. Surprisingly, you notice that while you catch this red ball, another ball, but of green colour, also moves simultaneously. You decide then to replace the red ball in the urn and you notice that immediately, the latter green ball also springs back in the urn.

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A Two-Sided Ontological Solution to the Sleeping Beauty Problem

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I describe in this paper an ontological solution to the Sleeping Beauty problem. I begin with describing the Entanglement urn experiment. I restate first the Sleeping Beauty problem from a wider perspective than the usual opposition between halfers and thirders. I also argue that the Sleeping Beauty experiment is best modelled with the Entanglement urn. I draw then the consequences of considering that some balls in the Entanglement urn have ontologically different properties form normal ones. In this context, considering a Monday-waking (drawing a red ball) leads to two different situations that are assigned each a different probability. This leads to a two-sided account of the Sleeping Beauty problem. On the one hand, the first situation is handled by the argument for 1/3. On the other hand, the second situation corresponds to a reasoning that echoes the argument for 1/2 but that leads however, to different conclusions.

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A thirder and an Everettian: a reply to Lewis's 'Quantum Sleeping Beauty'

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Since the publication of Elga's seminal paper in 2000, the Sleeping Beauty paradox has been the source of much discussion, particularly in this journal. Over the past few decades the Everettian interpretation of quantum mechanics1 has also been much debated. There is an interesting connection between the way these two topics raise issues about subjective probability assignments. This connection is often alluded to, but as far as we know Peter J. Lewis's ‘Quantum Sleeping Beauty’ (2007a) is the first attempt to examine it explicitly. Lewis claims that the two debates are not independent: to be specific, he argues that accepting the Everettian interpretation of quantum mechanics requires you to be a ‘halfer’ about Sleeping Beauty, in opposition to the more widely accepted ‘thirder’ solution. This paper will argue that Lewis is wrong. Everettians do not have to be halfers. It is perfectly cogent to be both an Everettian and a thirder.

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Peter J. Lewis (2007) has argued that the Everettian interpretation of quantum mechanics implies the unpopular halfer position in the Sleeping Beauty debate. We retorted that it is perfectly coherent to be an Everettian and an ordinary thirder (Papineau and Durà-Vilà 2009). In a recent reply to our paper Lewis (2009) further clarifies the basis for his thinking. We think this brings out nicely where he goes wrong: he underestimates the importance of metaphysical considerations in determining rational credences. Lewis's original argument hinged on the epistemic parallels between (a) an Everettian scenario where the x-spin of an electron in an eigenstate of z-spin is measured, with the result hidden from an observer and (b) a simplified Sleeping Beauty scenario where Sleeping Beauty is told that she will be woken on Monday and on Tuesday, with memory erasure in between. Lewis (2007: 60) pointed out that in both cases the agent's subjective experience has the same branching structure, with the agent becoming uncertain about her location therein. He concluded that the agent should adopt the same credences in both cases. Given that Everettians are committed in the spin case to assigning pre-measurement credences of 1/2 to ‘up’ and ‘down’, Lewis infers that the Everettian interpretation implies that Sleeping Beauty should similarly assign pre-sleeping credences of 1/2 to waking on Monday and waking on Tuesday. But this would of course be very bad for the Everettian interpretation, since Sleeping Beauty's credences here should clearly be 1.

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I have argued (Lewis 2007) that anyone who adopts the Everettian approach to the foundations of quantum mechanics must also accept the (unpopular) ‘halfer’ solution to the Sleeping Beauty puzzle. Papineau and Durà-Vilà (2009) have responded with an argument that it is perfectly cogent both to be an Everettian and to accept the (popular) ‘thirder’ solution to Sleeping Beauty. Here I attempt to rebut their argument, and to clarify my original position. My argument depends on the claim that there are strong structural parallels between the Everettian treatment of a simple spin measurement and a simplified version of the Sleeping Beauty puzzle. The simple spin measurement is a measurement of the x-spin of a spin-1/2 particle whose state is an eigenstate of z-spin. The simplified Sleeping Beauty scenario is one in which Sleeping Beauty is told that she will (definitely) be woken on Monday and on Tuesday, with memory erasure in between. In each case, the agent's subjective experience has a branching structure, and she becomes uncertain about her location in this structure. On the basis of this analogy, I argue that the same subjective probability assignment is appropriate in each case to quantify the agent's uncertainty. But the accepted assignment in the Sleeping Beauty case does not allow for pre-measurement probabilities other than 1 when applied to the Everettian case, and the required assignment in the Everettian case entails the problematic halfer assignment when applied to the Sleeping Beauty case.

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ABSTRACT. I describe in this paper an ontological solution to the Sleeping Beauty problem. I begin with describing the Entanglement urn experiment. I restate first the Sleeping Beauty problem from a wider perspective than the usual opposition between halfers and thirders. I also argue that the Sleeping Beauty experiment is best modelled with the Entanglement urn. I draw then the consequences of considering that some balls in the Entanglement urn have ontologically different properties form normal ones. The upshot is that I endorse the halfer conclusion on the probability of Heads once beauty is awaken and the thirder conclusion on conditional probabilities, and that original conclusions ensue on the probability of waking on Monday. 1. The Entanglement urn Let us consider the following experiment. In front of you is a urn. The experimenter asks you to study very carefully the properties of the balls that are in the urn. You go up then to the urn and begin to examine carefully its content. You note first that the urn contains only red or green balls. By curiosity, you decide to take a sample in the urn of a red ball. Surprisingly, you notice that while you catch this red ball, another ball, but of green colour, also moves simultaneously. You decide then to replace the red ball in the urn and you notice that immediately, the latter green ball also springs back in the urn.
I describe in this paper an ontological solution to the Sleeping Beauty problem. I begin with describing the Entanglement urn experiment. I restate first the Sleeping Beauty problem from a wider perspective than the usual opposition between halfers and thirders. I also argue that the Sleeping Beauty experiment is best modelled with the Entanglement urn. I draw then the consequences of considering that some balls in the Entanglement urn have ontologically different properties form normal ones. In this context, considering a Monday-waking (drawing a red ball) leads to two different situations that are assigned each a different probability. This leads to a two-sided account of the Sleeping Beauty problem. On the one hand, the first situation is handled by the argument for 1/3. On the other hand, the second situation corresponds to a reasoning that echoes the argument for 1/2 but that leads however, to different conclusions.
Since the publication of Elga's seminal paper in 2000, the Sleeping Beauty paradox has been the source of much discussion, particularly in this journal. Over the past few decades the Everettian interpretation of quantum mechanics1 has also been much debated. There is an interesting connection between the way these two topics raise issues about subjective probability assignments. This connection is often alluded to, but as far as we know Peter J. Lewis's ‘Quantum Sleeping Beauty’ (2007a) is the first attempt to examine it explicitly. Lewis claims that the two debates are not independent: to be specific, he argues that accepting the Everettian interpretation of quantum mechanics requires you to be a ‘halfer’ about Sleeping Beauty, in opposition to the more widely accepted ‘thirder’ solution. This paper will argue that Lewis is wrong. Everettians do not have to be halfers. It is perfectly cogent to be both an Everettian and a thirder.
Peter J. Lewis (2007) has argued that the Everettian interpretation of quantum mechanics implies the unpopular halfer position in the Sleeping Beauty debate. We retorted that it is perfectly coherent to be an Everettian and an ordinary thirder (Papineau and Durà-Vilà 2009). In a recent reply to our paper Lewis (2009) further clarifies the basis for his thinking. We think this brings out nicely where he goes wrong: he underestimates the importance of metaphysical considerations in determining rational credences. Lewis's original argument hinged on the epistemic parallels between (a) an Everettian scenario where the x-spin of an electron in an eigenstate of z-spin is measured, with the result hidden from an observer and (b) a simplified Sleeping Beauty scenario where Sleeping Beauty is told that she will be woken on Monday and on Tuesday, with memory erasure in between. Lewis (2007: 60) pointed out that in both cases the agent's subjective experience has the same branching structure, with the agent becoming uncertain about her location therein. He concluded that the agent should adopt the same credences in both cases. Given that Everettians are committed in the spin case to assigning pre-measurement credences of 1/2 to ‘up’ and ‘down’, Lewis infers that the Everettian interpretation implies that Sleeping Beauty should similarly assign pre-sleeping credences of 1/2 to waking on Monday and waking on Tuesday. But this would of course be very bad for the Everettian interpretation, since Sleeping Beauty's credences here should clearly be 1.
I have argued (Lewis 2007) that anyone who adopts the Everettian approach to the foundations of quantum mechanics must also accept the (unpopular) ‘halfer’ solution to the Sleeping Beauty puzzle. Papineau and Durà-Vilà (2009) have responded with an argument that it is perfectly cogent both to be an Everettian and to accept the (popular) ‘thirder’ solution to Sleeping Beauty. Here I attempt to rebut their argument, and to clarify my original position. My argument depends on the claim that there are strong structural parallels between the Everettian treatment of a simple spin measurement and a simplified version of the Sleeping Beauty puzzle. The simple spin measurement is a measurement of the x-spin of a spin-1/2 particle whose state is an eigenstate of z-spin. The simplified Sleeping Beauty scenario is one in which Sleeping Beauty is told that she will (definitely) be woken on Monday and on Tuesday, with memory erasure in between. In each case, the agent's subjective experience has a branching structure, and she becomes uncertain about her location in this structure. On the basis of this analogy, I argue that the same subjective probability assignment is appropriate in each case to quantify the agent's uncertainty. But the accepted assignment in the Sleeping Beauty case does not allow for pre-measurement probabilities other than 1 when applied to the Everettian case, and the required assignment in the Everettian case entails the problematic halfer assignment when applied to the Sleeping Beauty case.
The Sleeping Beauty paradox in epistemology and the many-worlds interpretation of quantum mechanics both raise problems concerning subjective probability assignments. Furthermore, there are striking parallels between the two cases; in both cases personal experience has a branching structure, and in both cases the agent loses herself among the branches. However, the treatment of probability is very different in the two cases, for no good reason that I can see. Suppose, then, that we adopt the same treatment of probability in each case. Then the dominant ‘thirder’ solution to the Sleeping Beauty paradox becomes incompatible with the tenability of the many-worlds interpretation.
The Sleeping Beauty Problem (SBP) is a long-standing puzzle in classical probability theory and has been used to challenge the Many-Worlds Interpretation (MWI) of quantum mechanics, since both involve objective determinacy combined with subjective uncertainty about certain events. A common concern is that MWI yields a different answer to the quantum version of SBP than the widely supported Thirder position in the classical case. We argue that this concern is unwarranted. We show that in both the quantum and classical versions of SBP, the correct credence is given by the Halfer position. In the quantum (MWI) SBP, we show that if no unjustified renormalization is introduced, the correct credence is 1/2. We then extend this result to the classical SBP by refuting four major arguments for 1/3. First, we reject the Proportion Argument by distinguishing event weight from probability. Second, we rebut Elga's Variant Argument by extending an earlier critique that identifies the implicit introduction of additional information; we further clarify this point by constructing a new variant and explaining why the Principle of Indifference is inapplicable, drawing an analogy with a mistake by d'Alembert in the history of probability theory. Third, we identify a flaw in the Technicolor Beauty Variant Argument, which arises from treating overlapping events as disjoint. Finally, we argue that causal decision theory is inappropriate for SBP, rendering the Thirders vulnerable to a Dutch Book. Our results support the consistency of MWI under the challenge posed by SBP and suggest that the dominant position on SBP needs careful reconsideration.