Abstract
Philosophical interest in the role of self-locating information in the confirmation of hypotheses has intensified in virtue of the Sleeping Beauty problem. If the correct solution to that problem is 1/3, various attractive views on confirmation and probabilistic reasoning appear to be undermined; and some writers have used the problem as a basis for rejecting some of those views. My interest here is in two such views. One of them is the thesis that self-locating information cannot be evidentially relevant to a non-self-locating hypothesis. The other, a basic tenet of Bayesian confirmation theory, is the thesis that an ideally rational agent updates her credence in a non-self-locating hypothesis in response to new information only by conditionalization. I argue that we can disprove these two theses by way of cases that are much less puzzling than Sleeping Beauty. I present two such cases in this paper.
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Notes
Adam Elga (2000) introduced the Sleeping Beauty problem to Philosophy in a now well-known paper in Analysis.
In this paper I use the term “statement” to mean “object of credence.” I use “credence” and “degree of belief” interchangeably. Objects of credence can be true or false, but whether an object of credence actually is a statement in any ordinary sense of “statement” is a question that I do not address.
Suppose that one learns the truth of a self-locating statement without learning anything non-self-locating (e.g., I may know that some days are Mondays prior to learning that today is Monday). IT implies that in such a case the self-locating statement would be evidentially irrelevant to any non-self-locating hypothesis. Thus, IT entails what Michael Titelbaum (2008) calls the “Relevance-Limiting Thesis,” and formulates as follows: “It is never rational for an agent who learns only self-locating information to respond by altering a non-self-locating degree of belief.” (p. 556)
Notice that examples of rational changes in credence due to losing information (see Arntzenius 2003 for such an example) are not counterexamples to COT, for COT requires conditionalization for rational credence change only in cases where the change in credence is a response to gaining information. Cases of rationally changing one’s credence in a temporally self-locating statement in response to the passage of time (Arntzenius calls this “shifting”) also pose no threat to COT, for COT requires conditionalization for rational credence change only in cases where the object of the altered credence is non-self-locating.
Because the single halfer says that Beauty increases her credence in HEADS to 2/3 in response to learning “Today is Monday,” the single halfer is also committed to rejecting IT; for the non-self-locating implications of “Today is Monday” are all statements that Beauty knew prior to learning “Today is Monday.”
On Monday Beauty also knows “I am awake today,” but this statement also cannot be conditionalized upon to reach the conclusion that on Monday prob(HEADS) = 1/3. On the Pust and Draper view, this is because it has no probability at all (not even zero) on Sunday. On the more common view, conditionalization on “I am awake today” is impossible here because that statement already has a probability of one on Sunday and so its truth cannot be learned on Monday.
If it is defensible, then there is even greater motivation for my counterexamples to IT and COT. As we shall see, it is clear that in those cases the relevant change of credence is a response to gaining information.
Monton (2002) claims that on Monday Beauty alters her credence in HEADS because she forgets what day it is. The term “forgets” suggests a loss of information, but Monton’s proposal is that Beauty forgets what day it is in the sense that on Sunday she knows what day it is but on Monday she does not, and forgetting what day it is in that sense does not involve the loss of any information besides the usual shift from knowing (on Sunday) that it is Sunday to knowing (on Monday) that it is not Sunday. It also involves a failure on the part of Beauty to acquire on Monday precise information about what day it is, and that failure to acquire information is crucial to Beauty’s changing her credence in HEADS to 1/3 (assuming that she does). For a more thorough discussion of Monton’s position, see Draper (2008).
An exception here is Horgan (2004), who employs what he calls “synchronic Bayesian updating” to argue for 1/3. Although I do not address Horgan’s view here, I believe that it is worthy of serious consideration. I also believe that Joel Pust’s (2008) criticism of that view is worthy of serious consideration.
I am convinced, however, that he is correct.
Draper and Pust (2008) provide a Dutch book argument for 1/3 that avoids such difficulties, but it undermines only the single halfer’s position.
To be more precise, I should say that single and double halfers say that prob(HEADS/MON) = 2/3. The triple halfer says that prob(HEADS/MON) = 1/2. The triple halfer has his own problems, however, in virtue of his claim that when Beauty awakens on Monday, prob(MON) = 1. For if TAILS is true and so Beauty does awaken on Tuesday, then her epistemic situation on Tuesday is identical to her epistemic situation on Monday. Thus, the triple halfer must say that if Beauty awakens on Tuesday, then on that day no less than on Monday prob(MON) = 1. I take this to be an absurd consequence of triple halfism.
Although I have used simple, non-relational properties in the argument form above, direct inference arguments with the same basic structure can contain complex or relational properties. Furthermore, Pollock (1990; 111) points out that, typically, the premises in a direct inference argument will be more modest than (i) and (ii) above because they will affirm only that we are warranted in believing, say, that prob(Hx/Rx) = r and that Rc.
Pust does not use the expression “subset defeat,” preferring instead “Reichenbach’s principle.”
Paul Thorn (2011), one of Pollock’s co-authors, undertakes this task in a reply to Pust. I attack Thorn’s argument in an unpublished manuscript.
Notice that this sort of Dutch book cannot be used against you in case 1 even though you would increase your credence in HH from 1/4 to 1/3 should you learn on Monday that the coin lands heads at least once. The Dutch book works in case 3 because if you are committed to increasing your credence in HH to 1/3 on Monday (and on Tuesday) should you learn “The coin lands heads today,” then you are also committed to decreasing your credence in SAME to 1/3 on Monday regardless of whether you learn “The coin lands heads today” or “The coin lands tails today.” In case 1, on the other hand, your commitment to increase your credence in HH to 1/3 on Monday should you learn that the coin lands heads at least once does not mean that you are also committed to decrease your credence in SAME to 1/3 regardless of what you learn on that day. For it does not commit you to rejecting the obvious truth that if you learn on Monday that the coin does not land heads at least once (i.e., that TT is true), then your credence in SAME should increase to 1.
One might also try to avoid the conclusion that cases 3 and 4 show that IT is false by replacing the temporal indexical “today” in the relevant evidence statements with the demonstrative “this day,” for then the evidence statements (“The coin lands heads on this day” and “The red coin lands heads on this day”) might seem to be non-self-locating. (The same sort of suggestion might also be made in an attempt to show that neither the Sleeping Beauty problem nor Bradley’s example threatens IT.) Such an objection raises complex issues about the nature of both demonstratives and objects of credence, and so a thorough response here is not possible. Nevertheless, let me make two points. The first is that the argument against COT in the main text below is not threatened by the objection. This is because in cases 3 and 4 we can stipulate that you are certain that the relevant coin is not tossed on Sunday and so (using case 3 to make the point) on that day prob(The coin lands heads on this day) cannot possibly have a value greater than zero. The second point is that it is difficult to see how the revised evidence statements can fail to be self-locating, for on Monday there is no way for you to pick out a unique day as “this day” except by reference to its being the day of your current temporal location. Direct reference theorists might object that a proposition like “The coin lands heads on this day” cannot be self-locating because the content of “this day” is the day itself. If that is correct, however, it shows only that the proposition expressed by “The coin lands heads on this day” is not the object of credence expressed by that sentence. For on standard direct reference theories, on Monday “The coin lands heads on this day” and “The coin lands heads on Monday” express the same proposition. They do not, however, express the same object of credence, for on Monday your credence in “The coin lands heads on this day” increases to one even though your credence in “The coin lands heads on Monday” does not increase to one.
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Acknowledgments
I would like to thank Joel Pust and an anonymous referee for their contributions to this article.
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Draper, K. The evidential relevance of self-locating information. Philos Stud 166, 185–202 (2013). https://doi.org/10.1007/s11098-012-0033-2
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DOI: https://doi.org/10.1007/s11098-012-0033-2