theorem, the risk that she shall bear a child is less than one in a billion. Therefore, my dear friends, indulge your desires and worry not about the consequences!” Given the assumption that the same method of reasoning should be applied as in Incubator, and using some plausible prior probability of pregnancy given carnal embrace (say, 1/100), it is easy to verify that there is nothing wrong with the serpent’s mathematics. The question, of course, is whether the assumption should be granted. ≈ Let us review some of the differences between Incubator and Serpent’s Advice to see if any of them are relevant in the sense of providing a rational ground for treating the two cases differently. • In the Incubator experiment there was a point in time, stage (a), when the subject was actually ignorant about her position among the observers. By contrast, Eve presumably knew all along that she was the first woman. But it is not clear why that should matter. We can imagine that Eve and Adam were created on a remote island, and that they didn’t know whether there are other people on Earth, until one day they were informed that they are thus far the only ones. It is still counterintuitive to say that the couple needn’t worry about the possibility of Eve getting pregnant. • When the subject is making the inference in Dungeon, the coin has already been tossed. In the case of Eve, the relevant chance event has not yet taken place. This difference does not seem crucial either. We can modify Serpent’s Advice by supposing that the deciding chance event has already taken place. Let’s say the couple has just sinned and they are now brooding over the ramifications. Should the serpent’s argument completely reassure them that nothing bad will happen? It seems not. So the worry remains. 11 • At stage (b) in Dungeon, any observers resulting from the toss have already been created, whereas Eve’s potential progeny do not yet exist at the time when she is assessing the odds. We can consider a variant of Dungeon where each cell exists in a different century. That is, let us suppose that cell #1, along with its observer, are created in the first century, and destroyed after, say, 30 years. In each of the subsequent 99 centuries, a new cell is built, allowed to exist for 30 years, and is then destroyed. At some point in the first century a coin is tossed and, depending on how it lands, these subsequent cells will or will not contain observers. Stage (a) can now be defined to take place in the first century after the first prisoner has been created but before the coin has been tossed and before the prisoner has been allowed to come out of his cell to observe its number. At this stage (stage (a)) it seems that he should assign the same credence to tails and the same conditional credences of tails given that he is in a particular cell as he did in the original version—for precisely the same reasons. But then it follows, just as before, that his posterior credence of tails, after finding that he is in cell #1, should be much greater than the prior credence of tails. This version of Dungeon is analogous to Serpent’s Advice with respect to the non-existence of the later humans at the time when the odds are being assessed. • In Dungeon, the two hypotheses under consideration (heads and tails) have well-defined known prior probabilities (50%), whereas Eve and Adam must rely on vague subjective considerations to assess the risk of pregnancy. True, but would we want to say that if Eve’s getting pregnant were determined by some distinct microbiological process with a well-defined objective chance which Eve and Adam knew about, then they ought to accept the serpent’s advice? If anything, the knowledge of such an objective chance would make the consequence even weirder. 12 8. The mystery that we are facing here is that it seems clear that both the serpent and the presumptuous philosopher are wrong, yet it seems as if the only model that yields this double result (model 1) is incoherent. One may be tempted to blame the strength of SSA for these troubles and think that we should reject it. But that, it appears, would transfix us on another horn of the dilemma, for we would then have to reject the cogent argument about the Dungeon thought experiment presented above, and, perhaps even more seriously, we would have failed to account for a number of very well-founded scientific applications in cosmology and elsewhere (which I lack the space to fully explore in this article). There are a number of possible moves and objections that one can try at this point. But most of these maneuvers and objections rest on simple misunderstandings, or else they fail to provide a workable alternative to how to reason about the range of problems that need to be addressed. It is easy enough to come up with a method of reasoning that works in one particular case, but when one then tests it against other cases—philosophical thought experiments and legitimate scientific inferences—one usually soon discovers that it yields paradoxes or otherwise unacceptable results. Yet by seriously confronting this central conundrum of self-locating belief, we can glean important clues about what a general theory of observation selection effects must look like. 9. So where do we go from here? The full answer is complicated and difficult and cannot be fully explored in a relatively short paper like this one. But by helping myself to a fair amount of hand-waving, I can at least try to indicate the direction in which I think the solution is to be found. 13 One key to the solution is to realize that the problem with SSA is not that it is too strong but that it isn’t strong enough. SSA tells you to take into account a certain kind of indexical information—information about which observer you are. But you have more indexical information than that about who you are; you also know when you are. That is, you know which temporal segment—which “observer-moment”—of an observer that you are at the current time. We can formulate a ‘Strong Self-Sampling Assumption’ that takes this information into account: (SSSA) Each observer-moment should reason as if it were randomly sampled from its reference class. Arguments can be given for SSSA along lines parallel to those of the arguments for SSA provided above. For example, one can consider cases in which a person is unaware of what time it is and has to assign credence to different temporal possibilities. A second key to the solution is to see how the added analytical power of SSSA enables us to relativize the reference class. What this means is that different observer-moments of the same observer may use different reference classes without that observer being incoherent over time. To illustrate, let us again consider the Incubator thought experiment. Before, we rejected model 3 because it seemed to imply that the reasoner should be incoherent. But we can now construct a new model, model 4, which agrees with the answers that model 3 gave, that is, a credence of 1/2 of heads at both stage (a) and stage (b), but which modifies the reasoning that led to these answers in a such a way as to avoid incoherency. Suppose that just as before and for the same reasons, we assign, at stage (a), the credences: 2 1 Pr(tails) = Pr(I'm in cell #1| tails) = 1 14 100 1 Pr(I'm in cell #1| heads) = Now, if the only epistemic difference between stage (a) and stage (b) is that at the latter stage you have the additional piece of information that you are in cell #1, then Bayesian conditionalization of the above conditional credences entails (as in model 1) that your posterior credence must be: 101 100 Prposterior (tails) = Pr(tails | I'm in cell #1) = . However, when we take SSSA into account, we see that there are other epistemic differences between stages (a) and (b). In addition to gaining the information that you are in cell #1, you also lose information when you enter stage (b). At stage (a), you knew that you were currently an observer-moment who is ignorant about which cell you are in and who is pondering different possibilities. At stage (b), you no longer know this piece of indexical information, because it is no longer true of you that you currently are such an observer-moment. You do know that you are an observer who previously was at stage (a), but this is an indexically different piece of knowledge from knowing that you are currently at stage (a). Since your total information at stage (b) is not equal to the information you had at stage (a) conjoined with the proposition that you are in cell #1, there is therefore no requirement that your beliefs at stage (b) be obtained by conditionalizing your stage (a) credence function on the proposition that you are in cell #1.