one’s credences should accord with one’s estimates of objective chances. As formulated by David Lewis, the Principal Principle came with a proviso: one’s credences are constrained by one’s beliefs about objective chances according to the principle only if one does not have relevant “inadmissible” information.3 Lewis’s reason for introducing the proviso was to bracket off cases involving oracles, time travel, and the like, where one might get information about the future that is not mediated by information about current chances. It is possible to maintain, however that some problems involving self-location also include inadmissible information and should hence be excepted from the Principal Principle’s domain of applicability. In light of the positive argument against the 1/3 view, which I will present below, we do in fact have strong grounds for regarding Beauty’s indexical information as inadmissible. Given Lewis’s own “best-system” analysis of chance, it would be unsurprising to find that indexical information can be sometimes inadmissible. According to the bestsystem analysis, chances are a kind of concise partial summary of patterns of local, nonmodal, occurrent facts. Since chances, defined in this way, do not even purport to summarize indexical information, there is no reason to suppose that all relevant information about future events is always implied by knowledge of the chances – not if we have reason for thinking that indexical information might also be relevant. (This is, in fact, the stance that Lewis adopted in his critique of Elga’s argument.4 ) Since our intuitions about what information is admissible in this type of case are no more secure than our direct intuitions about what credence Beauty should assign to HEADS, the Principal Principle fails to settle the Sleeping Beauty controversy. What counts as admissible information in the Sleeping Beauty problem must emerge from its solution – which needs to be independently justified – rather than assumed at the outset. Another way in which one may attempt to support the middle step in Elga’s argument is by invoking Bas van Fraassen’s Reflection Principle. 5 (The Reflection Principle, in its simplest form, postulates that your credence at a time t is constrained by your credence at a later time t’ according to Pt(X | Pt’ (X) = x) = x.) Here we encounter the same problem again: the applicability of the principle to Sleeping Beauty is at least as problematic as the 1/3 view itself. The Sleeping Beauty problem postulates a breakdown of rationality. Beauty faces the possibility of drug-induced amnesia. Even if we ask about Beauty’s credence on Monday, before the drug has been administered, she will not at that point know whether or not her memory has already been tampered with. It is independently known, from other cases, that the Reflection Principle should not be followed where forgetting takes place or is suspected. For example, if I knew that one year from now I will have credence 1/2 in the proposition that it rained today, that plainly does not imply that I should assign the same credence now – when I still vividly remember spending the day reading in the garden. The Reflection Principle, therefore, is of no more avail to the 1/3 view than is the Principal Principle. We are left with the unsupported appeal to intuition in the step of the argument that assumes that P(H1 | H1 ∨ T1) = 1/2. How are we to evaluate this intuition? One way is to examine what else we are led to accept if we adopt the 1/3 view. If the consequences 3 See (Lewis 1980; Lewis 1994). A precursor to the Principal Principle was formulated by Hugh Mellor (Mellor 1971). 4 See (Lewis 2001). A similar point was made, independently, in (Bostrom 2001). 5 See (van Fraassen 1984). 3 are unacceptable, we should revise our intuition. Let us therefore consider some variations of the original Sleeping Beauty to explore the wider ramifications of the 1/3 view. If we set P(H1) = P(T1) = P(T2), it follows that Beauty should, upon awakening, assign P(HEADS) = 1/3 and P(TAILS) = 2/3. The only relevant difference between HEADS and TAILS is that there would be more awakenings of Beauty on the latter hypothesis. Since the structure of the situation does not depend on the particular numbers involved, we can test our intuitions by considering a more extreme version of the problem. Extreme Sleeping Beauty This is like the original problem, except that here, if the coin falls tails, Beauty will be awakened on a million subsequent days. As before, she will be given an amnesia drug each time she is put to sleep that makes her forget any previous awakenings. When she awakes on Monday, what should be her credence in HEADS? By reasoning exactly parallel to that which Elga used to support the 1/3 view in the original version, we obtain P(H1) = P(T1) = P(T2) = P(T3) = … = P(T1,000,001). That is, upon awakening, Beauty should assign P(HEADS) = 1/1,000,002. The result can be generalized: following this line of reasoning, Beauty takes her observation “I am awake now” as evidence in favor of hypotheses that imply that there are many such awakenings of her. The degree of support is proportional to the number of awakenings postulated by the hypotheses. This consequence in Extreme Sleeping Beauty is counterintuitive. It seems like a rather excessive confidence in the proposition that a fair coin, yet to be tossed, will fall tails. We can bring out even worse implications if we consider cases in which more than one agent is involved. Suppose that a possible world contains several agent-parts {ai}i∈R that are subjectively indistinguishable, that is, these agent-parts are in such similar evidential situations that individual agent-parts cannot tell which one they are in. Elga has argued for an indifference principle stating that one should assign the same credence to each centered proposition of the form “My current agent-part is ai”, for i∈R, and this is supposed to hold whether or not the agent-parts in {ai}i∈R all belong to the same agent or to different agents.6 This highly restricted indifference principle follows as a special case from a somewhat stronger indifference principle that appears to be needed to make sense of many seemingly legitimate scientific inferences.7 The principle is thus well supported. But if we use it together with the reasoning in the 1/3 view, we obtain implausible results. Beauty and Doppelganger This is like the original Sleeping Beauty problem, except here Beauty is never woken up after being put to sleep on Monday. Instead, if the coin falls tails, 6 See (Elga 2004). 7 See (Bostrom 2002). 4 another person is created and awoken on Tuesday. This new person will spend her Tuesday wakening in a state that is subjectively indistinguishable from Beauty’s Monday state (she will have the same apparent memories and have experiences that feel just the same as Beauty’s). When Beauty awakes on Monday, what should be her credence in HEADS? By Elga’s weak indifference principle, Beauty should have the same conditional credence, given TAILS, in her being Beauty and Doppelganger. And by the same reasoning that the 1/3 view relies on in the original version of the problem, Beauty should have the same conditional credence, given MONDAY, in being Beauty as in being Doppelganger. From this we can derive, just as before, that the awakened Beauty should assign an equal credence in three centered propositions: P(“I am Beauty and HEADS”) = P(“I am Beauty and TAILS”) = P(“I am Doppelganger and TAILS”) Since the credence given to these three centered propositions sum to 1, it follows that upon awakening, Beauty should hold P(HEADS) = 1/3. Now consider the extreme version of Sleeping Beauty and Her Doppelganger, where on tails there will be a million different doppelgangers, each having an awakening on some subsequent day that will be subjectively indistinguishable from Beauty’s Monday awakening. It is easy to show, by the same steps as before, that Beauty should upon awaking have credence P(HEADS) = 1/1,000,002. The argument does not depend on whether the coin is tossed before the experiment starts or only after Beauty has been put back to sleep on Monday. It is, in fact, irrelevant what the objective chance of HEADS is when Beauty is making her assessment.8 The relevant factor is Beauty’s prior credence in HEADS (relative to her non-indexical background information). For example, Beauty’s posterior credence in HEADS would thus be unaffected if the existence of additional awakenings by her doppelgangers were made to depend, not on the outcome of a coin toss, but instead on whether the trillionth digit in the decimal expansion of π is even – provided only that Beauty’s prior credence in this proposition is 1/2. We can therefore generalize the foregoing thought experiment: Beauty and Doppelganger (generalized) Let ϕ be a (non-indexical) proposition, to which Beauty assigns a prior credence of 1/2. Beauty is never woken up again after being put to sleep on Monday. If ϕ is true then there will be a total of N > 0 awakenings of doppelgangers in states that are subjectively indistinguishable from Beauty’s Monday awakening. As before, we get P(HEADS) = 1/(N +2). Let us consider what this recommendation amounts too. Each (awake) agent-part, merely by taking into account the indexical fact that “I am currently an agent-part of this kind”, should give a greater 8 If the coin were tossed before she woke up, the chance of HEADS would not be 1/2, but either 1 or 0. 5 credence to hypotheses in proportion as they imply that there is a greater number of subjectively indistinguishable agent-parts of that kind. At this point, those familiar with the literature on observation selection theory may notice an uncanny similarity between the reasoning behind this position and the so-called “Self-Indication Assumption”. That assumption states that each observer should regard her own existence as evidence supporting hypotheses that imply the existence of a greater total population of observers in the world, the degree of support being proportional to the implied (expected) number of observers. The Self-Indication Assumption was originally introduced in discussions about the Doomsday argument, as an attempt to neutralize that argument. It turns out, however, that the assumption has implications of its own that are perhaps even more counterintuitive than those of the Doomsday argument. It seems that the following thought experiment, in particular, gives us fairly strong grounds for rejecting the SelfIndication Assumption. Presumptuous Philosopher It is the year 2100 and physicists have narrowed down the search for a theory of everything to only two remaining plausible candidate theories, T1 and T2 (using considerations from super-duper symmetry). According to T1 the world is very, very big but finite and there are a total of a trillion trillion observers in the cosmos. According to T2, the world is very, very, very big but finite and there are a trillion trillion trillion observers. The super-duper symmetry considerations are indifferent between these two theories. Physicists are preparing a simple experiment that will falsify one of the theories. Enter the presumptuous philosopher: “Hey guys, it is completely unnecessary for you to do the experiment, because I can already show to you that T2 is about a trillion times more likely to be true than T1!” (Whereupon the presumptuous philosopher explains the Self-Indication Assumption.)9 By modifying this example slightly, we can substitute the reasoning embodied in the 1/3-view for that of the presumptuous philosopher. To do this, suppose that the two theories that the physicists have come up with differ not only in regard to how many observers there are but also in regard to how many agent-parts there are that are subjectively indistinguishable from your own current one. Elga’s 1/3 view can now take the place of the Self-indication Assumption. It is worth noting that the situation described in this modified version of Presumptuous Philosopher is by no means a farfetched possibility. Contemporary cosmologists face essentially that predicament. They are trying to determine whether the universe is finite or infinite. Given the standard Big Bang model and the assumption that spacetime is singly connected, the universe is infinite if and only if it is either open or flat. Whether it is open or flat, or closed, depends on whether the cosmic energy density, Ω, exceeds a certain threshold value. Current measurements indicate that the actual density is very close to the critical value, Ω ≈ 1. It is an important open empirical question whether the actual value is above, below, or exactly at the critical level. 9 See (Bostrom 2002; Bostrom 2003).