Abstract
I will present five puzzles which show that rational people can update their degrees of belief in manners that violate Bayesian Conditionalization and van Fraassen’s Reflection Principle. I will then argue that these violations of Conditionalization and Reflection are due to the fact that there are two, as yet unrecognized, ways in which the degrees of belief of rational people can develop.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Thus, for instance, if the light is not switched off, there must be a moment (which could be before or after midnight) such that you have equal degree of belief in each of the 3 possibilities: heads and it is before midnight, tails and it is before midnight, tails and its after midnight.
- 2.
One might wonder why I inserted the phrase “almost certainly” in this sentence. The reason for this is that there is a subtlety as to whether you know at 6 pm that you will have an increased degree of belief in tails at 11.59 pm. There is an incoherence in assuming that at 6 pm you know with certainty what your degree of belief distribution over possible times will be at 11.59 pm. For if you knew that you could simply wait until your degree of belief distribution is exactly like that. (You can presumably establish by introspection what your degree of belief distribution is.) And when you reach that distribution, you would know that it has to be 11.59 pm. So when that happens you should then collapse your degree of belief distribution completely on it being 11.59 pm. But this is incoherent. Thus, the fact that you do not have a perfect internal clock also implies that you can not know in advance what your degree of belief distribution is going to look like after it has developed (guided only by your internal clock). Thus you can not in advance be certain how your degree of belief distribution over possible times will develop. Nonetheless you can be certain at 6 pm that your degree of belief in tails will not decrease prior to midnight, and that it is extremely likely to have increased by 11.59 pm. At 6 pm your expectation for you degree of belief in tails at 11.59 pm will be substantially greater than 0.5.
- 3.
Elga, A. (2000): “Self-locating belief and the Sleeping Beauty problem”, Analysis 60, pp 143–147.
- 4.
Cian Dorr has independently arrived at the idea of using commutativity in order to argue for the degrees of belief that Elga advocates in the Sleeping Beauty case. See Dorr, C.: “Sleeping Beauty: in defence of Elga”, forthcoming, Analysis.
- 5.
This scenario is similar to the “Dr Evil scenario” in Elga, A. (manuscript): “Defeating Dr. Evil with self-locating belief”.
- 6.
See van Fraassen (1995): “Belief and the problem of Ulysses and the sirens”, Philosophical Studies 77: 7–37.
- 7.
Strict conditionalization: when one learns proposition X at t, one’s new degrees of belief Dt equals one’s old degrees of belief D0 conditional upon X: Dt(Y) = D0(Y/X). One might also allow Jeffrey conditionalization. It matters not for our purposes.
- 8.
Bas van Fraassen has, in conversation with me, has suggested that in such situations Conditionalization indeed should be violated, but Reflection should not. In particular he suggested that the degrees of belief of the traveler should become completely vague, upon arrival in Shangri La. This does not strike me as plausible. Surely upon arrival in Shangri La our traveler is effectively in the same epistemic situation as someone who simply knows that a fair coin has been tossed. One can make this vivid by considering two travelers, A and B. Traveler A never looks out of the window of the car, and hence maintains degree of belief ½ in heads all the way. (The memory replacement device does not operate on travelers who never look out of the window.) Traveler A, even by van Fraassen’s lights, upon arrival in Shangri La, should still have degree of belief ½ in Heads. However, traveler B, does look out of the window during the trip. Upon arrival, by van Fraassen’s lights, B’s degrees of belief should become completely vague. But it seems odd to me that traveler B is epistemically penalized, i.e. is forced to acquire completely vague degrees of belief, just because he looked out of the window during the trip, when it seems clear that he ends up in exactly the same epistemic position as his companion, who did not look out of the window.
- 9.
It is obvious how to generalize this case to a case in which there are memory replacement devices at the end of both roads, where these memory replacement devices are indeterministic, i.e. when it is the case that for each possible path there are certain objective chances for certain memories upon arrival in Shangri La. For, given such chances (and the Principal Principle), one can easily calculate the degrees of belief that one should have (in heads and tails) given the memory state that one ends up with. And, generically, one will still violate Conditionalization and Reflection.
- 10.
Some people will balk at some of the degrees of belief that I have argued for in this paper, in particular in the self-locating cases. For instance, some people will insist that tomorrow one should still be certain that one is on Earth, even when one now knows (for sure) that a perfect duplicate of oneself will be created on Mars at midnight tonight. I beg to differ. However, even if in this case, and other cases, one disagrees with me as to which degrees of belief are rationally mandated, the main claim of this paper still stands. The main claim is that in such cases of possible experience duplication, it is at the very least rationally permissible that one’s degrees of belief become more spread out as time progresses, and hence rational people can violate Conditionalization and Reflection.
- 11.
One might model the prisoner here as having unique distinct experiences at each distinct, external clock, time, and as initially having precise degrees of belief over the possible ways in which those experiences could correlate to the actual, external clock, time. If one were to do so then the prisoner would merely be initially uncertain as to which world he was in (where worlds are distinguished by how his experiences line up with the actual, external clock, time), but for each such possible world would be always certain as to where he was located in it. And, if one were to do so, then the original prisoner case would be essentially the same case as Collins’s prisoner case: no uncertainty of location in any given world, merely an initial uncertainty as to which world one is in, and a subsequent shifting of the locally concentrated degrees of belief within each of the possible worlds. However, there is no need to represent the original prisoner case that way. Indeed it seems psychologically somewhat implausible to do so. More importantly, the arguments and conclusions of this paper do not depend on how one models this case.
Acknowledgements
I would like to thank John Collins, Adam Elga, John Hawthorne, Isaac Levi, Barry Loewer, and Tim Maudlin for extensive and crucial comments and discussions on earlier versions of this paper.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Arntzenius, F. (2016). Some Problems for Conditionalization and Reflection. In: Arló-Costa, H., Hendricks, V., van Benthem, J. (eds) Readings in Formal Epistemology. Springer Graduate Texts in Philosophy, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-20451-2_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-20451-2_10
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-20450-5
Online ISBN: 978-3-319-20451-2
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)