Abstract
We present a new argument for the claim that in the Sleeping Beauty problem, the probability that the coin comes up heads is 1/3. Our argument depends on a principle for the updating of probabilities that we call ‘generalized conditionalization’, and on a species of generalized conditionalization we call ‘synchronic conditionalization on old information’. We set forth a rationale for the legitimacy of generalized conditionalization, and we explain why our new argument for thirdism is immune to two attacks that Pust (Synthese 160:97–101, 2008) has leveled at other arguments for thirdism.
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Notes
In another commonly discussed version, the coin is flipped on Monday evening while Sleeping Beauty is asleep, rather than Sunday evening. It is widely agreed among those who have discussed the problem, and we concur, that there is no important difference between the two versions. We focus on the Sunday-flip version here (at the suggestion of a referee) because doing so slightly simplifies exposition. But our discussion below is readily adaptable to the Monday-flip variant too.
Here and throughout, we will use the expression ‘epistemic probability’ rather than the term ‘credence’. The latter evokes subjectivist conceptions according to which epistemic probability is degree of belief, or rational degree of belief, or degree of confidence, or something of this sort. We think that epistemic probability is better construed as quantitative degree of evidential support, relative to a specific body of information. (Quantitative degree of evidential support should be distinguished from qualitative degree of evidential support, as expressible by locutions like ‘makes highly likely’, ‘makes highly unlikely’, ‘makes somewhat likely’, etc.) But for the most part, except where we indicate otherwise, our discussion in this paper will be neutral about the nature of epistemic probability.
The expression ‘candidate-possibility’ is intended to accommodate the thought that some of the items that fall under this rubric will count sometimes as epistemically possible and sometimes as epistemically impossible, depending on the specific body of background-information relative to which one attributes epistemic probabilities to various propositions.
Typically there are a number of ways one could bracket the information B. One way would be to excise not only B itself, but also any information that one actually possesses only by virtue of actually possessing B. We will call this strong bracketing of B. But another way, which we will call weak bracketing of B, would be to retain certain information that is logically weaker than B but which one actually possesses only by virtue of actually possessing B—e.g., certain information that is logically entailed by B, which does not logically entail B, and which one actually possesses only because it is logically entailed by B.
Not surprisingly, the idea of generalizing the method of conditionalization in this way has arisen before, and has been invoked approvingly in the literature on “the problem of old evidence” in philosophy of science. See, for instance, Barnes (1999), Glymour (1980, 1987–1991), Howson (1984, 1985, 1991), Jeffrey (1995), and Monton (2006).
I–B might be the proper portion of I that results from strongly bracketing B, or it might be some proper portion of I that results from weakly bracketing B in one way or another. Cf. note 4.
Thus, R1 is being bracketed weakly rather than strongly (cf. note 4 above).
In correspondence, Joel Pust has pointed out that if (as Horgan assumed in his original argument for thirdism) it were acceptable sometimes to assign a non-zero preliminary probability to a proposition entailing that one is not presently conscious, then Horgan’s original assignment of equal preliminary probabilities of 1/4 each to the four hypotheses TodayH,Mon,TodayH,Tues TodayT,Mon and TodayT,Tues could be justified this way: “by the conjunction of (a) the Elga-Lewis limited indifference principle, according to which indistinguishable centered worlds associated with the same possible world should get equal credence, and (b) a suitable version of the principal principle according to which possible worlds which differ only in the outcome of a fair coin toss should, in the absence of relevant evidence, be assigned the same credence” (quoting Pust). These two principles alone are not enough to justify the assignment of preliminary probabilities in Table 1; rather, one also must appeal to evidential indifference as grounds for assigning equal probabilities to the three subcases within each of the four cases 1.a, 1.b, 2.a, 2.b. But because the claim that the evidence is indifferent vis-à-vis the three subcases is so intuitively obvious here and so unproblematic-looking, this claim strikes us as beyond serious question—even though the task of formulating suitable general principles of probabilistic indifference remains difficult and elusive.
By the definition of conditional probability, P_[(TodayH,Mon & R1)|R1] = P_[(TodayH,Mon & R1)]/P_(R1). This is equivalent to P_[(TodayH,Mon & R1)]/(P_[(TodayH,Mon & R1)] + P_[(TodayT,Mon & R1)] + P_[(TodayT,Tues & R1)]) = (1/12)/((1/12) + (1/12) + (1/12)) = 1/3. One calculates P_[(TodayT,Mon & R1)|R1] and P_[(TodayT,Tues & R1)|R1] in a similar way.
The conditionalization that leads to these conclusions can be implemented in a non-calculation-intensive way, directly on the basis of Table 2 and without doing the calculations that generate Table 3. First eliminate, from the original twelve finest-grained candidate-possibilities in Table 2 (each of which has an unconditional preliminary probability of 1/12), the nine finest-grained candidate-possibilities that are incompatible with R1; then normalize one’s probability distribution over the remaining three finest-grained candidate-possibilities—viz., (TodayH,Mon & R1), (TodayT,Mon & R1), and (TodayT,Tues & R1)—in a way that preserves the pairwise ratios (all 1:1) of their preliminary probabilities. These three possibilities now have updated probabilities of 1/3 each, replacing their preliminary probabilities of 1/12 each.
A referee has suggested, with reference to our presentation of this argument in earlier drafts, that the argument would work equally well with a two-way disjunction of potential rules—e.g., (R1 or R2)—rather than the four-way disjunction we employ. This is not so, although our previous drafts failed to make the matter sufficiently clear. The trouble is that a two-way disjunction of rules would not generate a strongly symmetric hierarchical partition-structure of candidate-possibilities. Without strong symmetry, it becomes much more tendentious what counts as the correct way to assign preliminary probabilities to the various cells in the relevant partition-structure—which gives the halfers various opportunities for defensive resistance that are precluded when the partition-structure is strongly symmetric.
This objection was posed by a referee.
In fairness to him, we acknowledge that Horgan’s prior writings have not explicitly articulated either notion of conditionalization as clearly or explicitly as would be ideal.
This reply to Pust invokes the specific conception of epistemic probability described in note 2 above.
Pust has recently argued, in response to Horgan (2008), that “the most plausible account of quantitative degree of (evidential) support, when conjoined with any of the three major accounts of indexical thought in such a way as to rationally constrain rational credence, contradicts essential elements of Horgan’s argument” (Pust 2011, p. 20). We are inclined to think that if this conclusion is correct, then it points up the need for a more adequate account of indexical thought.
Let ‘R1 +’ abbreviate ‘R1 and I know that R1’. Presumably, R1 + is an item of information that is known by Sleeping Beauty in her actual epistemic situation. Does that mean that her synchronic conditionalization should be keyed to R1 +, rather than to R1? No. For, in the envisioned epistemic situation she considers when bracketing the information R1, the proposition R1 + has epistemic probability zero—which means that conditional probabilities that are conditional on R1 + are undefined in that envisioned epistemic situation. Conditionalization should be geared to one’s strongest bracketed conditionalizable-upon information that is relevant to epistemic probabilities. For Sleeping Beauty, that information is R1. In order to apply conditionalization, one must know (at the time one does so) the conditionalized-upon proposition—which arguably means that if one is sufficiently rational, one thereby knows that one knows that proposition. But it is not required that the knowledge-claim is itself a component of the conditionalized-upon proposition. (This note was prompted by correspondence with Pust.)
Arntzenius gives a closely related argument in Arntzenius (2003).
Thanks to Adam Elga, two anonymous referees, and especially Joel Pust for helpful comments on earlier drafts.
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Horgan, T., Mahtani, A. Generalized Conditionalization and the Sleeping Beauty Problem. Erkenn 78, 333–351 (2013). https://doi.org/10.1007/s10670-011-9316-9
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DOI: https://doi.org/10.1007/s10670-011-9316-9