Sleeping Beauty in a grain of rice

Abstract

In the Sleeping Beauty problem, Beauty is woken once if a coin lands heads or twice if the coin lands tails but promptly forgets each waking on returning to sleep. Philosophers have divided over whether her waking credence in heads should be a half or a third. Beauty has centered beliefs about her world and about her location in that world. When given new information about her location she should update her worldly beliefs before updating her locative beliefs. When she conditionalizes in this way, her credence in heads is a half before and after being told it is Monday. In applications of Dutch Book arguments to the Sleeping Beauty problem, the probability of a particular outcome has often been confounded with consequences of that outcome. Heads and tails are equally likely but twice as much is at stake if the coin falls tails because Beauty is fated to make the same choice twice. As a consequence, the possibility of tails should be given twice the weight of the possibility of heads when deciding whether to bet on heads even though heads and tails are equally likely.

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Fig. 1

Notes

  1. 1.

    Locative beliefs are de se beliefs. Worldly beliefs are not otherworldly beliefs. They are Beauty’s centered beliefs about her actual world and what it might be. If her actual world is conceived as an object with properties, then worldly beliefs might be considered de re (but I am ill-educated on the philosophical nuances of these Latin phrases).

  2. 2.

    Lewis (1979) might have said that when Beauty is told it is Monday, she learns something about her location in ordinary space that changes her location in logical space. Her propositional attitude changes from ‘week in which heads or tails’ to ‘Monday in which heads or tails.’ ‘Waking on Tuesday’ is a property of the first propositional attitude that does not have a counterpart in the second propositional attitude.

  3. 3.

    This procedure appears similar to, perhaps is the same as, Meacham’s (2008, p. 249) compartmentalized conditionalization.

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Acknowledgments

Lucas Mix brought the Sleeping Beauty problem to my attention. Carl Veller patiently explained thirder reasoning and critically read the manuscript. Ned Hall and the anonymous reviewers provided valuable input.

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Correspondence to David Haig.

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Haig, D. Sleeping Beauty in a grain of rice. Biol Philos 31, 23–37 (2016). https://doi.org/10.1007/s10539-015-9503-1

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Keywords

  • Sleeping Beauty
  • Hamilton’s rule
  • Credence
  • Relatedness
  • Endosperm
  • Conditionalization
  • De se beliefs