Measurements are being conducted to obtain a better estimate of the cosmic energy density. If the universe is infinite then with probability one there are an infinite number of agent-moments in states subjectively indistinguishable to your current one.10 Therefore, if Elga’s 1/3 view is correct, we could conclude that we already have “infinitely strong” evidence that the universe is infinite. The consequence that it would be a waste of money to carry out the planned experiments because we can predict the outcome from our armchair (with probability 1), is extremely implausible. Somebody wishing to toe this line should be willing to bet at practically any odds on the outcome of these future experiments.11 We have seen that the original argument given for the 1/3 view is inconclusive, that neither the Principal Principle nor the Reflection Principle could be successfully invoked to buttress it, and that when we unfold the reasoning embedded in the 1/3 view, we find that highly counterintuitive consequences follow. We have good reason to reject the 1/3 view. We have not yet considered another argument in favor of the 1/3 view, one that is based on long-run frequency or betting considerations. We will discuss this argument in a later section. But first, let us turn our gaze to the 1/2 view. We shall argue that this view, too, should be rejected. The 1/2 view According to the 1/2 view presented by David Lewis, Beauty should upon awakening have credence 1/2 in HEADS, and her conditional credence in HEADS given MONDAY should be 2/3. P(H1) = 1/2 P(H1 | H1 ∨ T1) = 2/3 Suppose that Beauty is informed that it is Monday, and let P+ be her new credence function after she has obtained this information. Lewis claims that P+ should be obtained by conditionalizing P on MONDAY. Thus, P+(HEADS) = P(HEADS | MONDAY) = 2/3 Lewis’s argument for this claim is simple: Before the experiment, Beauty should assign credence 1/2 to the proposition that a fair coin to be tossed in the future will fall heads. She already knows that she will be awakened. Therefore, when she awakes, she obtains no new relevant information; so her credence in HEADS should remain 1/2. This argument starts to look peculiar when we compare it to Lewis’s explanation of why Beauty should increase her credence in HEADS upon being informed that it is Monday: 10 See (Bostrom 2002). 11 Even if during the next two hundred years we obtained overwhelming empirical evidence that the universe is finite, we should, on this view, continue to assign credence 1 to the universe being infinite. 7 Now when Beauty is told during her Monday awakening that it’s Monday, … she is getting evidence – centered evidence – about the future: namely that she is not now in it. That’s new evidence: before she was told that it is Monday, she did not yet have it…. This new evidence is relevant to HEADS, since it raises her credence in it by 1/6 [i.e. from 1/2 to 2/3].12 On this reasoning, it would seem, one could similarly argue that when Beauty awakes on Monday (but before she is informed that it is Monday) she likewise gets relevant evidence – centered evidence – about the future: namely that she is now in it. Since it makes no difference whether the coin is tossed before the experiment begins or on Monday evening (a point of agreement between Lewis and Elga), let us suppose the case where the coin is tossed just before Beauty awakens on Monday. If being in “the future” means being in the period after the coin has been tossed, Beauty now has new relevant information about her current location relative to this period (namely, that she is in it now). Lewis is thus committed to the view that one’s beliefs about a chance event such as a coin toss can be affected by obtaining evidence that is purely about one’s own current location. Yet he offers no argument for why only centered evidence that it is Monday, but not centered evidence that one is currently in the “experimental phase” (i.e. that it is either Monday or Tuesday, rather than, say, the preceding Sunday) can be relevant to HEADS. Absent such an argument, his claim that Beauty upon awakening should assign credence 1/2 to HEADS is a completely unsupported assumption, one which those who disagree with the 1/2 view should feel free to reject. Opponents of the 1/2 view can simply insist that Beauty does get centered relevant evidence when she finds herself awake in the experimental (Monday or Tuesday) phase. Lewis’s argument for the 1/2 view therefore fails. If we unpack the implications of accepting the 1/2 view, we find that it has implications no less counterintuitive than those of the 1/3 view. Let us begin by considering again the amplified version of the Sleeping Beauty 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? The adherent of the 1/2 view will maintain that Beauty, upon awakening, should retain her credence of 1/2 in HEADS, but also that, upon being informed that it is Monday, she should become extremely confident in HEADS: P+(HEADS) = 1,000,001/1,000,002 This consequence is itself quite implausible. It is, after all, rather gutsy to have credence 0.999999% in the proposition that an unobserved fair coin will fall heads. 12 (Lewis 2001), p. 175 8 We can extract an even more counterintuitive consequence by modifying the example slightly. Instead of using a single coin toss, with a prior probability of heads equal to 1/2, we could stipulate a sequence of 10 independent tosses of the same coin. The prior probability that all of these tosses will come up heads is 2-10, which is less than one in a thousand (≈ 0.00098%). Suppose that unless the coin comes up heads all ten times, Beauty will not be awakened again after the Monday awakening. If, however, the tossing does yield ten heads, then Beauty will be awakened on a million subsequent days. We can then ask what odds Beauty could reasonably accept if offered to bet on such a sequence of coin tosses. Beauty the High Roller Beauty is awakened on Monday and after having been awake for an hour she is offered a bet. She is told that a fair coin will be tossed ten times. If it lands heads all ten times then Beauty wins $1,000. If it lands tails at least once, then Beauty loses $100,000. But there is a twist: If Beauty wins, the experiment ends at that point. If Beauty loses, she will be put to sleep, given an amnesia drug that causes her to forget her awakening, and then awoken again the next day; and this procedure will be repeated for a total of one million days. (On each of these subsequent awakenings, Beauty will spend an hour in a state of ignorance about what day it is before she is put to sleep. No bet is offered after the initial Monday awakening.) Beauty awakes on Monday and prudently decides to reject the bet that she is offered. But just as she is about to declare her decision, David Lewis’s ghost appears in a puff of smoke. The ghost explains the 1/2-view reasoning and argues that Beauty’s credence in the proposition that all ten tosses will come up heads should be very close to unity. In fact, the ghost calculates that, even taking into account the low prior probability of this proposition, Beauty should nevertheless assign it a posterior credence of 99.8% after taking into account that she has just learnt that her current awakening is the initial Monday awakening.13 The expected value of the gamble to Beauty is therefore positive: EV ≈ 0.998 * $1,000 + 0.002 * (– $100,000) = $998 – $200 = $798 So according to the ghost’s reckoning, Beauty ought to take the bet. But surely it would be crazy for Beauty to follow the ghost’s advice.14 Hence we should reject the 1/2 view. 13 Let H* be the proposition that the coin falls heads all ten times. Let M be the centered proposition stating that Beauty’s is currently awake on the first Monday. We then have *)(*)|(*)(*)|( *)(*)|( )|*( HPHMPHPHMP HPHMP MHP ¬¬+ = . With P(H*) = 2-10, P(¬H*) = 1 - P(H*), P(M | H*) = 1, and P(M | ¬H*) = 2/1,000,002, it easy to check the ghost’s calculation. 14 We assume that Beauty is risk-neutral and that her utility function is linear in money. If she has a diminishing marginal utility of money, or is risk-averse, we can simply adjust the stakes or the number of awakenings that would occur so that the calculation still favors her taking the gamble, without affecting the basic point of the thought experiment. 9 A hybrid model? If the 1/3- and the 1/2-view both have unacceptable consequences, how can we build a better model for reasoning with indexical information? Consider again the 1/3 view. The problem with that view was that it led to a bias in favor of hypotheses entailing that there are many subjectively indistinguishable duplicates of one’s current agent-part. When all the hypotheses under consideration agree on the number of such duplicates, the bias does not manifest itself; but it causes trouble in cases like Sleeping Beauty. The obvious way to correct this “many-duplicates” bias is to divide one’s credence in being any one particular duplicate with the total number of duplicates. Consider the following two possible worlds, in which the only agent-parts are those in the Sleeping Beauty experiment. (We shall assume throughout that all “agentparts” are of equal duration.) w1: h1 [The “heads” world] w2: t1 t2 [The “tails” world] As before, H1, T1, and T2 are the centered propositions expressing that one is currently h1, t1, and t2, respectively; HEADS is the proposition that the actual world is w1; and TAILS the proposition that the actual world is w2. The proposal for removing the many-duplicate bias is that we set P(H1) = 1/2 P(T1) = P(T2) = 1/4 Of course, we still have P(H1 | HEADS) = 1 P(T1 | TAILS) = P(T2 | TAILS) = 1/2 It follows that P(HEADS) = P(TAILS) = 1/2. Thus, there is no general tendency, upon finding oneself awakened in the experiment, to favor hypotheses implying that there are many such awakenings. This means that we are immune from objections of the “Presumptuous Philosopher”-type. Our new de-biasing postulate implies that the conditional probability of HEADS given that it is MONDAY is greater than 50%: P(HEADS | H1 ∨ T1) = 2/3 Constraint P Now, if the credence function P+ that Beauty should have upon learning that it is MONDAY were obtained by conditionalizing P on (H1 ∨ T1), then we would fall into the trap illustrated in Beauty the High Roller. To avoid falling into this trap, it seems as though we require that P+(HEADS | H1 ∨ T1) = 1/2 Constraint P+