Abstract
How should we update de dicto beliefs in the face of de se evidence? The Sleeping Beauty problem divides philosophers into two camps, halfers and thirders. But there is some disagreement among halfers about how their position should generalize to other examples. A full generalization is not always given; one notable exception is the Halfer Rule, under which the agent updates her uncentered beliefs based on only the uncentered part of her evidence. In this brief article, I provide a simple example for which the Halfer Rule prescribes credences that, I argue, cannot be reasonably held by anyone. In particular, these credences constitute an egregious violation of the Reflection Principle. I then discuss the consequences for halfing in general.
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Notes
For my purposes, it is not necesssary to specify how credences in centered worlds are determined, i.e., how the total credence in a possible world is divided across its centers. This is because I will only consider credences in uncentered events in what follows. Titelbaum (2012) gives an example where halfers obtain an implausible credence in a centered event, if a certain condition on how the halfer distributes credence across centers holds.
One may wonder whether, similarly, we could set up a Dutch book against the thirder based on her alleged violation of the Reflection Principle. But this would involve her being offered bets on Monday awakenings, without being told that it is Monday, but not on Tuesday awakenings, and it has been argued that this does not constitute a fair Dutch book because the bookie is exploiting information that Beauty does not have (Hitchcock 2004). (Also, from being offered the bet Beauty might infer that it is Monday and thereby change her credences and decline the bet.)
Pittard nevertheless defends these credences, arguing that it may be reasonable to consider this a robustly perspectival context, one in which two disputants should end up having different beliefs in spite of them having the same evidence, being able to communicate without restriction, etc. This may be reminiscent of the perspectival realism described by Hare (2010) [see also Hare (2007, 2009)]. Hare (2009) goes into some detail discussing what conclusion two interlocutors, each of whom takes herself to be “the one with present experiences,” should reach. If indeed they should not be able to reach complete agreement, as seems likely, then this would appear to be a robustly perspectival context. However, in this case it does not seem possible to turn the situation into a money pump, because it does not seem possible to settle any bets made in a satisfactory way; we cannot adjudicate from a neutral perspective. Indeed, Hare concludes that the interlocutors should agree that the other is correct from the other’s point of view. In contrast, bets made by the participants in Pittard’s experiment could easily be settled from a neutral perspective.
Incidentally, applying the Thirder Rule does give the right answer: of all Heads awakenings, two are in the HH world, in which the coins come up the same, and the remaining two are in the HT and TH worlds, in which the coins do not come up the same. So if we use the Thirder Rule, the resulting credence in the event that both coins came up the same is \(2/4 = 1/2\). (I apologize for any confusion caused by the unfortunate coincidence that the Halfer Rule prescribes \(1/3\) in this context, and the Thirder Rule \(1/2\).)
The Thirder Rule still gives \(1/2\) as well, because there are only two possible centered worlds remaining, namely Monday in HH and Monday in HT.
On the face of it, the same happens in the Shangri La example given by Arntzenius (2003). (I thank an anonymous reviewer for Philosophical Studies for calling my attention to this.) In this example, someone experiences A or B according to the outcome of a coin toss. He knows, though, that at a certain point in time after the experience, any memories of B will be replaced by false memories of A, while any memories of A will be left intact, so that he will not be able to tell the two cases apart. Then, while experiencing A, he has credence \(1\) in Heads, in spite of knowing full well that he will later have credence \(1/2\) in Heads, without his memory being compromised in this particular case. Of course, this is entirely due to the fact that in a parallel case, his memory would be compromised to be indistinguishable from what he currently knows will be his (true) memory of A. Thereby, he will lose a piece of information that he currently has. But nothing similar happens in the enriched two-coins example. At the point in time when Beauty is told what day it is, her memory is never compromised, and she never loses information.
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Conitzer, V. A devastating example for the Halfer Rule. Philos Stud 172, 1985–1992 (2015). https://doi.org/10.1007/s11098-014-0384-y
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DOI: https://doi.org/10.1007/s11098-014-0384-y