Abstract
This article is an attempt to use the insights of objective probability theory to solve the Sleeping Beauty problem. The approach is to develop a partial theory of direct inference and then apply that partial theory to the problem. One of the crucial components of the partial theory is the thesis that expected indefinite probabilities provide a reliable basis for direct inference. The article relies heavily on recent work by Paul D. Thorn to defend that thesis. The article’s primary conclusion is that Beauty (the perfectly rational agent of the Sleeping Beauty story) can by way of a justifiable direct inference from a statement of expected indefinite probability reach the conclusion that the epistemic probability that the relevant coin toss lands heads is 1/3. The article also provides an account of why the self-locating information that Beauty acquires on Monday is evidentially relevant to the question of whether the coin toss lands heads or tails.
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Notes
Pollock (1990) was the first objectivist to treat direct inference as defeasible. Nevertheless, some objectivist theories that predate Pollock’s work on objective probability can, as he shows, be reformulated in terms of defeasible direct inference.
Where r and s are variables ranging over numbers, intervals can be closed (expressed as “[r, s]”), open (expressed as “(r, s)”), or half-open (expressed as “(r, s]” or as “[r, s)”). A specific number r can be expressed as the interval [r, r].
Kyburg (1961) was the first to notice that an exception to standard rules of direct inference must be made for certain gerrymandered reference classes. Isaac Levi’s theory (1982) avoids the need for the exception by requiring for justified direct inference that the object of interest is randomly selected from the reference class, but most objectivists regard that restriction on direct inference as too limiting. (A background assumption of this article is that most objectivists are right about that).
In defending the Oscar seminar’s argument against Pust’s objection, Thorn (2011) relies on a broader notion of “logically stronger”. But any property R1 that is logically stronger than a property R2 on my definition will also be logically stronger on his definition. Thus, Thorn would have no reason to reject, as opposed to merely expanding the scope of, my rule of subproperty defeat.
A reason to believe that P is directly rebutted by a reason to believe that ~ P. A reason to believe that P is indirectly rebutted by a reason to believe that Q if the relevant agent has a reason to believe that if Q then ~ P that is independent of any reason she might have to believe that ~ P.
Pollock would formulate the principle in different terms. He distinguishes levels of arguments and would say that the premises of an undercut level 0 argument P could rebut the premises of another level 0 argument Q. At any level n such that n > 0, however, P would no longer be “in contention” and so its premises would not rebut the premises of Q at that level (unless the defeater that undercuts the premises of P is itself defeated). The upshot is the same: at least ultimately, reasons that are undercut do not rebut (1990, pp. 89–92).
Some theories of direct inference are formulated in terms of sets (or classes) rather than properties and so, strictly speaking, do not include a rule of subproperty defeat. But if one reformulates the rule of subproperty defeat in terms of sets, the reformulated rule of “subset defeat” is a component of such theories. My argument for 1/3 can easily be reformulated in terms of sets rather than properties.
The details of Lewis’s position (2010, pp. 374–381) are worth exploring, but here I will only report that I do not find his Dutch book argument for 1/2 convincing. Beyond his Dutch book arguments, Lewis’s only positive argument in favor of rejecting assumption 2 is that one would otherwise be forced to give up the principle of conditionalization or the principal principle. He identifies his principle of conditionalization as the proposition that “for an uncentered proposition H, your new credence in H on acquiring evidence E should be… your conditional credence in H given E” (p. 370). That principle, unlike Assumption 2, is not obviously true, and I am happy to abandon it.
In hypothetical frequency theories, “the long run” should not be taken to imply an unlimited temporal sequence of trials as opposed to simply an unlimited number of trials all of which might be simultaneous.
From probb(Hτ) = 1/2, one can reach probb(Ut) = 1/4 by the following reasoning:
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(A) probb(Hτ) = 1/2.
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(B) probb(Hτ or ~ Hτ) = 1.
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(C) Thus, probb(~ Hτ) = 1/2. (from (A) and (B))
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(D) probb(Ut/Hτ) = 0.
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(E) probb(Ut/~ Hτ) = 1/2.
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(F) probb(Ut) = [probb(Hτ) x probb(Ut/Hτ)] + [probb(~ Hτ) × probb(Ut/~ Hτ)].
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(G) Therefore, probb(Ut) = 1/4. (from (A), (C), (D), (E), and (F)).
Thus, if Beauty has a prima facie reason to believe that probb(Hτ) = 1/2, then she also has a prima facie reason to believe that probb(Ut) = 1/4.
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As a basis for my objection to the Oscar Seminar’s argument, I (2017) identify a defeasible reason for Beauty to believe that prob(Hτ) = 1/2. However, that reason can also be undercut by incorporating the information that Beauty is awakened today into a reference property that is a subproperty of the one I use.
References
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Acknowledgements
I would like to thank Joel Pust for his substantial contribution to this paper.
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Draper, K. Direct inference and the sleeping beauty problem. Synthese 198, 2253–2271 (2021). https://doi.org/10.1007/s11229-019-02203-y
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DOI: https://doi.org/10.1007/s11229-019-02203-y