Immortal Beauty: Does Existence Confirm Infinite Reincarnation? Jens Jaeger jj2674@nyu.edu 03-2020 1 Introduction Are we reborn after death? Do we continue existence in a different body, after our current body has died? One way to tackle this question is as a puzzle about personal identity. The puzzle involves the question of whether one is the same person as certain (past or future) individuals. Some people who believe in reincarnation derive their conviction from religious or spiritual beliefs. But as a question about personal identity, reincarnation is also open to be investigated through secular argument. Michael Huemer (forthcoming) recently proposed such an argument, employing Bayesian confirmation theory. According to him, our existence at any time is a probability-zero event unless we are infinitely reincarnated. Hence, upon observing our existence at the present time we should become confident of our own immortality. Part of this essay argues against Huemer's argument. More specifically, it argues that, while sound, Huemer's argument doesn't achieve what it sets out to show. It does not show that we should update in favor of infinite reincarnation upon observing our existence at the present time. The second part of the essay proposes a new argument for immortality, but also outlines its limitations. Here's the plan in more detail: I start by reconstructing Huemer's original argument (Section 2). I then say why I think it's unsuccessful: It fails to account for the total evidence we gain from observing our existence at the present time. In particular, it neglects certain essentially de se evidence (Section 3). I then present the new argument for reincarnation (Section 4), which takes its cue from the well-known Sleeping Beauty puzzle. It argues that, if the popular "thirder" solution to the puzzle is true, then 1 in certain cases one should become confident of reincarnation upon learning of one's own existence. Finally, I discuss some of the argument's limitations (Section 5). I argue that whether our evidence confirms immortality depends on what can be assumed about the qualitative makeup of the universe; in particular, about how many lives are lived in total, conditional on any given incarnation hypothesis. 2 Huemer's Argument Reconstructed I'll start by reconstructing Huemer's argument. Let SI (singular incarnation) be the conjunction of two theses: first, that the universe is temporally infinite in both directions; and second, that you are incarnated (born) exactly once. (Huemer calls the same hypothesis R, for 'restricted view of personal identity'.) SI excludes the hypothesis of infinite reincarnation, II. Hypothesis II also has two parts: first, that the universe is temporally infinite in both directions; second, that you exist but that there is no time at which you are first incarnated, nor a time at which you are last incarnated. On II, you are infinitely reincarnated throughout the universe's history. For simplicity, Huemer considers a partitioning of the time axis into countably infinitely many centuries. This yields a prior probability function that assigns values to any given century t expressing the prior probability of your being born in t. One way to understand the prior probability function is as encoding epistemic norms in the absence of any evidence at all. In particular, such a probability function permits non-trivial updating on hypotheses which the agent has already known (so-called 'old evidence'). Examples of old evidence may include facts such as whether or when the agent is alive.1 Huemer simplifies further by assuming that the life of any of your incarnations is with certainty fully confined to a single cell of the partition. This way, the prior probability of your being born in t equals the prior probability of your being alive in t. 1 A probability assignment prior to any evidence at all is also often called hypothetical prior. For concrete cases motivating the existence of hypothetical priors, see Manley ("On Being a Random Sample", pp.15-7). 2 Suppose your evidence entails that you exist at a particular century. Consider then the prior probability of this evidence, conditional on each of the two incarnation hypotheses. If one finds that the evidence is better supported by II than by SI, then one may conclude, from the Bayesian conception of evidence, that one's evidence confirms II over SI. This is Huemer's basic argumentative strategy. More explicitly, the argument is this. (A) Indifference: The prior probability P of the time of your (unique) incarnation, conditional on SI, is uniform over the centuries.2 From (A) and the probability axioms, it follows:3 For all t, P (I am alive at t|SI) = 0. (1) Huemer requires a crucial second premise for his argument (which he does not explicate): (B) Probable Existence Given Reincarnation: For any century t, the probability of your existence at t conditional on II is non-zero: For all t, P (I am alive at t|II) > 0. (2) Further, Huemer assumes (C) Non-Zero Priors: The prior probabilities of SI and II are 2Strictly speaking, Huemer merely requires that the prior probability is monotonically non-increasing over the centuries: Conditional on SI, one's incarnation "is initially no more likely to occur at any given time than at any earlier time" (Huemer, forthcoming, p.8). But given the universe's infinite temporal past, this is equivalent to the prior's uniformity, given SI. The framing in terms of uniformity is more convenient for our purposes. 3Given countable additivity, this would entail probability 0 that you are ever born, given SI. Let us here assume merely finite additivity. This permits a non-zero probability that you are ever born, despite vanishing probability at each century. 3 each non-zero: P (SI) > 0 and P (II) > 0. (3) Finally, he requires (D) Bayesian Conception of Evidence: If H and H ′ are hypotheses with P (H) > 0 and P (H ′) > 0, then evidence E confirms H overH ′ iff P (E|H) > P (E|H ′). (For anyH with P (H) > 0, the conditional probability is defined as usual: P (E|H) = P (E∧H) P (H) .) From (A), (B), (C), and (D), one can conclude (E) "Existence Is Evidence of Immortality": My evidence (for some t) that I am alive at t confirms II over SI. The argument is valid. Indeed, as I said in the introduction, I even grant that it is sound. Why believe Premise (A)? It follows from an application of the Principle of Indifference:4 (PI) If there is no reason to favor (epistemically) one possibility over another, then the two possibilities have the same probability. It seems that, a priori, there is no reason to favor one century as the time of one's incarnation over any other century. According to PI, each century should thus have equal probability of containing me. Conditional on SI- unique incarnation-PI then implies uniformly zero probability for each century. In this essay, I wish to grant the application of PI to the question of my temporal location. And so I accept this part of the argument: Conditional on SI, the prior probability of any century containing me is zero (Premise (A)). 4PI isn't Huemer's only argument for Premise (1). Additionally, he considers a coin flip analogy (cp. his Fn. 18). But throughout the essay, I will grant Huemer's application of PI, and thus the Indifference premise. So, to save space, I won't explicate potential alternative arguments. 4 Further, I even grant the remaining premises, (B)-(D).5 Still, I object. The argument, even if sound, does not show what Huemer claims it proves: that, upon observing our existence at the present time, we should become confident of our own immortality. The reason is that the proposition that I am alive at t does not exhaust the evidence gained from the observation. In addition, I learn also that t is now. Once this additional evidence is accounted for, observing our existence at the present time does not serve as evidence for immortality. 3 Total Evidence Considered 3.1 The Argument Suppose an agent observes her existence at the present time and then contemplates infinite reincarnation. I claim that the agent's epistemic situation is analogous to the following case. (Lottery) You are presented with a 100-ticket lottery. Of the 100 tickets an unknown number are winning tickets. More specifically, you are told that either the lottery contains only a single winning ticket, or ten winning tickets. The game master, who knows which tickets are winning tickets, reveals to you the number of some winning ticket. You are told that he does this by selecting a ticket at random from among the subset of winning tickets. The ticket number that's revealed is number 34. How should learning that number 34 is revealed affect your credence that there are many winning tickets? You might think that your credence shouldn't 5 Note that the truth of Premise (B) isn't entirely obvious. Conditional on infinite reincarnation, there is an uncountable set of (infinite) candidate sets of centuries specifying when I'm alive. For Premise (B) to hold, the subset of this uncountable set consisting of all and only sets which contain t has to have non-zero measure. This requires an argument. One way may be to establish that, conditional on infinite reincarnation, there's a non-zero probability that I'm alive in every century. This could be the implication of a multiverse theory together with a revisionary account of personal identity, according to which upon my local death I immediately continue existence in a counterpart of mine in another part of the multiverse. But other accounts may be feasible as well. 5 change. Whether the revealed ticket is number 34, or some other number, intuitively doesn't tell you much about how many winning tickets there are. But there is an analogue of Huemer's argument to the conclusion that you should become more confident that there are many winning tickets. After all, that number 34 is a winning ticket is more likely if there are many winning tickets. But in this case the Huemer-style argument fails, for it ignores the procedure by which the game master selects ticket number 34. This is easy to see formally. There are only two possibilities: either one out of the 100 tickets is a winning ticket, or ten are. Call these hypotheses OneWinner and TenWinners respectively. It's true that, conditional TenWinners, the probability of 34 being a winning ticket is higher-ten times higher-than conditional on OneWinner: P (#34 wins|TenWinners) = 1/10, P (#34 wins|OneWinner) = 1/100. However, our evidence is not merely that 34 is a winning ticket. We also know that 34 has been revealed specifically by a random selection over the subset of winning tickets. Given that 34 is a winning ticket, its being thus revealed is less likely-ten times less likely-conditional on TenWinners than conditional on OneWinner: P (#34 revealed|#34 wins ∧ TenWinners) = 1/10, whereas P (#34 revealed|#34 wins∧OneWinner) = 1. For, conditional on 34 winning and there being a total of ten winning tickets, there are in any case nine other tickets the game master could have revealed. Meanwhile, if 34 is the only winning ticket, the game master is guaranteed to reveal it. Overall, we recover the intuitive result that 34's being revealed doesn't tell us anything 6 about the total number of winning tickets in the lottery: P (#34 revealed|OneWinner) =P (#34 revealed ∧#34 wins|OneWinner) =P (#34 revealed|#34 wins ∧OneWinner) * P (#34 wins|OneWinner) =1 * 1/100 =1/100, and P (#34 revealed|TenWinners) =P (#34 revealed ∧#34 wins|TenWinners) =P (#34 revealed|#34 wins ∧ TenWinners) * P (#34 wins|TenWinners) =1/10 * 1/10 =1/100, (The first equality follows since'#34 revealed' entails '#34 wins'.) The analogy with the incarnation setup arises since prior to observing our temporal location we have irreducibly indexical uncertainty. Given Huemerstyle indifference assumptions, this indexical uncertainty plays an analogous role to the game master's random selection of a ticket over the subset of winning tickets. Including this indexical information renders our total evidence neutral on the question of infinite reincarnation. The parallel between the setups is best illustrated by a finite incarnation setup: (Finite Universe) The universe has a total lifespan of only 100 centuries (and you know this). You entertain two rival hypotheses: first, that you are alive in precisely one of the 100 centuries (SIf ); second, that you are alive in precisely ten of the 100 centuries (IIf ). In each case you are a priori ignorant about which century or centuries you are alive. Given this setup, upon being incarnated in century t, you are initially uncer7 tain about what century it is. When you learn that it is t, you acquire new information. Now, does observing my temporal location in (Finite Universe) provide evidence for IIf over SIf? Note that both hypotheses entail that you are alive sometime. Provided that SIf and IIf have non-zero prior probability, we can write this as P (I am alive sometime|SIf ) = P (I am alive sometime|IIf ) = 1. (4) Further, I'll initially make the following assumption: that learning the purely de se hypothesis that one is alive now should not change one's confidence in SIf or IIf . Let's call this assumption No Update From Pure Indexical: P (I am alive now |SIf ) = P (I am alive now |IIf ). For simplicity, I will thus regard P as already conditionalized on 'I am alive now'. Given No Update From Pure Indexical, this presupposition won't distort our assessment of the degrees of confirmation afforded to the incarnation hypotheses by our total evidence. (At this point I should explicate a linguistic difference between Huemer and me: At times, Huemer describes the evidence one acquires upon observing one's temporal location as 'I am alive now '. But he means by this not a rather trivial piece of de se information. Rather, in Huemer's usage, it is supposed to express, for some t and some person p, the proposition that p is alive at t.) By a finite analogue of Huemer's argument, we can show-correctly-that IIf makes it more likely that I am alive at t: A priori, I have no reason to favor any one ten-element subset of centuries over any other ten-element set as the set of centuries where I am alive. And so PI implies equal probability for each such subset. Since exactly 1/10 of those subsets have me exist at t, it follows that P (I am alive at t|IIf ) = 1/10. (5) 8 Analogous reasoning for SIf yields P (I am alive at t|SIf ) = 1/100. (6) If my total evidence consisted solely in 'I am alive at t', an update in favor of IIf upon observing my temporal location would be warranted. But my evidence is more specific. Upon observing my temporal location, I also acquire the de se information that t is now. Now, if I'm singularly incarnated, then my being alive at century t means I'm alive only at century t. Given that I am alive now (which we assume doesn't influence my probability in SIf or IIf ), the fact that I am alive only at t entails with certainty that it's now t. That is, P (t is now|I am alive at t ∧ SIf ) = 1. (7) By contrast, the analogous inference for multiple reincarnation IIf doesn't follow with certainty. Given IIf , the information that I am alive at t means that t is merely one out of a total of ten centuries where I am alive. If I know that I am alive now and that t is among ten centuries where I am alive, then PI suggests a mere 1/10 probability that t is now. This is because, for any given ten-element subset of centuries of which one is t, I have no particular reason to favor any member of the subset as being the current century. Thus, by PI, P (t is now|I am alive at t ∧ IIf ) = 1/10. (8) Putting Eqs. (5), (6), (7), (8) together, we get P (E|SIf ) = P (E|IIf ) = 1/100, (9) where E is my total evidence, 't is now ∧ I am alive at t'.6 And so our total 6Given the information that I am alive now, 't is now' entails 'I am alive at t'. Thus 't is now' suffices to capture our total evidence here. So we can also write P (t is now |SIf ) = P (t is now|IIf ) = 1/100. 9 evidence no more supports multiple reincarnation than singular incarnation. An update in favor of multiple reincarnation is unwarranted.7 This is not to say that there aren't situations in which learning about one's temporal location does warrant an update in favor of multiple reincarnation. This is typically the case if the information about one's temporal location(s) is learned without the stronger indexical proposition. Suppose a time-traveling alien grants you access to a Cosmic Demoscope: a large database, listing for every century all the persons alive in that century. Suppose you pick some century c at random, and look up its entry in the Demoscope. To your amazement, you find that you are alive at c. In this case, you are warranted to update in favor of multiple reincarnation. This is because in this case you don't learn any relevant indexical information. You only learn the proposition that you are alive at c. You don't learn that it's currently c. Similarly, if in the lottery case you simply pointed to ticket number 34 and demanded that it be revealed to you-irrespective of whether it is a winning ticket or not-then, upon learning that it is indeed a winning ticket, you should become more confident that there are many winning tickets. It remains to turn to the case of infinite reincarnation. The calculations remain essentially the same. The probability that I'm alive at t vanishes, by PI, conditional on SI. The probability that it's currently t, conditional on my being alive at t and SI, is 1. It's a bit harder to ascertain the probability that I'm alive at t, conditional on infinite reincarnation (as mentioned in Fn. 5). But grant that it is non-zero. It won't matter in the end, for the probability that it's currently t, conditional on my being alive at t and the truth of infinite reincarnation, simply vanishes. For infinite reincarnation induces infinite indexical uncertainty. Putting all this together, we have P (E|SI) = P (E|II) = 0, (10) 7I am relying here implicitly on a plausible principle known as the 'Total Evidence Requirement'. One possible formulation of the Requirement is that credence updates should be justified by one's total evidence, rather than some proper part of it. Huemer himself accepts the requirement, for familiar reasons (cp. Huemer (forthcoming, p.7)). 10 where E is 't is now ∧ I am alive at t', as above. Once the total evidence is in, you shouldn't update in favor of immortality.8 3.2 Interlude: Irreducibly Indexical Evidence At this point one might raise one of two objections. One might doubt that we do acquire irreducibly indexical evidence when observing our temporal location. Or one might grant that we acquire such evidence, but that the evidence shouldn't be treated in the way I suggested. I shall briefly address both worries. Start with the first. By 'irreducibly indexical' I mean that the evidence can't be satisfactorily translated into any de dicto proposition.9 In claiming this, I am leaning on influential arguments by (among others) Lewis (1979) and Perry (1979). Lewis famously imagines a scenario involving two gods, omniscient about all de dicto facts in the world, but nonetheless ignorant about certain de se facts: "[Two gods] inhabit a certain possible world, and they know exactly which world it is. Therefore they know every proposition that is true at their world. ... Still I can imagine them to suffer ignorance: neither one knows which of the two he is. ... One lives on top of the tallest mountain and throws down manna; the other lives on top of the coldest mountain and throws down 8Indexical evidence also explains why certain problematic generalizations of Huemer's argument fail. For example, it's straightforward to construct a spatial analogue of Huemer's original argument. The spatial analogue would entail that, upon observing your location in space, you should conclude that you are infinitely spatially extended (e.g. that you have infinitely many body parts scattered arbitrarily far away). It's a serious challenge for Huemer's original argument to explain how the spatial analogue fails, while the temporal argument remains sound. But once we recognize that our total evidence includes irreducibly indexical evidence, we can dissolve the spatial analogue. Our evidence isn't merely that we are at some spatial location p, but also that p is here. The conjunction of these two bits of evidence isn't any more likely given infinite spatial extent than given finite and small spatial extent. 9By 'de dicto proposition' I mean simply a Lewisian or Russellian proposition. I'm adding the qualifier 'de dicto' to distinguish these standard accounts of propositional content from e.g. Stalnaker's (2010) account. Stalnaker's account is (arguably) still a propositional account but adds additional propositional structure to accommodate indexicality. Cp. Fn. 12. 11 thunderbolts. Neither one knows whether he lives on the tallest mountain or on the coldest mountain; nor whether he throws manna or thunderbolts." (Lewis, 1979, pp.520-1) Each god has indexical uncertainty about which god he is. But ex hypothesi the two gods know all de dicto facts. Thus their indexical uncertainty can't be uncertainty de dicto. Analogous examples can be constructed involving the Cosmic Demoscope. We can imagine the Demoscope to also grant its user exhaustive de dicto knowledge about her world. Suppose the agent consulted this extended Demoscope, and found out that she is alive at century t. She would still be ignorant about what time it is now. This suggests that her temporal-indexical evidence is not reducible to any de dicto proposition. Another well-known argument for the irreducibility of temporal de se evidence is the 'argument from cognitive significance'. John Perry (1979) famously makes this argument, via examples such as the following: "[A] professor, who desires to attend the department meeting on time, and believes correctly that it begins at noon, sits motionless in his office at that time. Suddenly he begins to move. What explains his action? A change in belief. He believed all along that the department meeting starts at noon; he came to believe ... that it starts now." (ibid., p.4) Perry argues that no de dicto translation of the professor's de se evidence can satisfactorily explain his action. Notably, at noon the professor has two distinct pieces of evidence: that the department meeting starts at noon, and that noon is now. The evidence we receive when observing our temporal location is closely analogous: We learn that we exist at t, and that t is now. One can easily construct cases analogous to the tardy professor, which bring out the distinct cognitive significance of the two pieces of evidence. All of this strongly indicates that we do acquire irreducible temporal de se evidence when observing our temporal location. This concludes my discussion of the first worry. 12 The second worry is that the irreducibly indexical evidence doesn't warrant the updates I claim it does. To answer this worry, I spell out last section's argument more concretely, in terms of two popular accounts of indexical thought. 3.2.1 Property Theory Lewis (1979) defends a particularly popular account of indexical thought. According to Lewis, an agent's having de se thought is the agent's self-ascribing a certain property. (Properties are understood here in an undemanding sense. Any set of possible objects is a property.) More specifically, Lewis thinks that the thinkers of temporal de se thought are the agent's individual time-slices. A time-slice wondering whether it's Monday or Tuesday ascribes to herself the set of all possible objects located at a Monday or Tuesday. An equivalent formulation of Lewis' account represents belief content as sets of centered worlds-triples of a Lewisian world, an agent in that world, and a time in that world. (This is perhaps the most popular formulation among formal epistemologists.) The argument from the previous section is easily translated into Lewis' theory. For simplicity, I attend again to the finite case, (Finite Universe). 'I am alive at t' denotes the set of centered worlds with the center located at an agent who is alive at t (note that the center itself need not be located at t).Conditionalizing on ten-times reincarnation restricts this set of centered worlds to those where the agent at the center lives ten times. For any such world, and any such agent in that world, there are ten centers representing the ten times when the agent is alive. The singleton of a pair consisting of the world w and one of the ten agent-time centers, say the center (a, s), represents the belief that one is located in w and that one is agent a and that it is currently time s. Given any such world, and any ten-times reincarnated agent in that world, the epistemic probability distributes uniformly over the ten centers (by PI). The overall result is that, given that I am alive at t and ten-times reincarnation, the probability of 't is now' is 1/10. This is Eq. 8. Conditionalizing instead on one's singular incarnation restricts the set 13 of centered worlds to those where the center is located at an agent who is incarnated once. Given any such world, and a singularly incarnated agent in that world, the epistemic probability distributes over the single center consisting of the agent and her unique time of existence. This means that, given that I am alive at t and singular incarnation, the probability of 't is now' is 1. This is Eq. 7. Eqs. 5 and 6-Huemer's original conclusions-can also be explicitly derived in Lewis' framework: Of those centers located at a ten-times reincarnated agent, 1/10 are such that the agent is alive at t. Of those centered on a singularly incarnated agent, 1/100 are such that the agent is alive at t. Applying PI then yields Eqs. 5 and 6, respectively. 3.2.2 Guise-Russellianism Another popular variant, defended e.g. by Soames (1987), Salmon (1989), and Braun (2016), holds that the content of an agent's belief is a Russellian proposition (rather than a property, as per Lewis). Roughly, a Russellian proposition consists of all and only those objects and properties the proposition is about. This creates a prima facie tension with the irreducibility of indexical belief. One might worry that a Russellian proposition can't play the role of irreducibly de se content. If furthermore an agent's belief is fully determined by its content, then the Russellian can't account for irreducibly indexical belief. But most Russellians agree with the Lewisian and Perryian arguments for the irreducibility of de se belief. Thus Russellians widely acknowledge the need to further elaborate their account. A popular elaboration is to deny that an agent's belief is fully determined by its content. Following Salmon (1989), Braun (2016) considers, besides the binary relation believes, a ternary relation BEL. This relation relates an agent, a proposition, and a propositional guise. An agent then believes a proposition p iff she BELs p under some guise. Importantly, an agent's rationality is a matter of the BEL relations she exhibits.10 10More precisely: On Salmon's and Braun's accounts guises have rich logical structure. For example, for every propositional guise, there is precisely one guise that is its 'guisenegation'. Given these logical relations, Braun states rationality rules in terms of BEL. For 14 The sentences "t is now" and "t is t" represent two distinct guises of the same Russellian proposition, (t, t, is identical to). Further, none of the sentences represents a guise of the proposition that I am alive at t. The argument of the previous section can then be translated to this: Given that I am alive at t and ten-times reincarnation, I could come to learn any one guise of the form "s is now" for ten distinct times s (only one of which is t).11 That is, for any specification of ten centuries where I am alive, I could have come to BEL any one out of ten Russellian propositions of the form (s, s, is identical to) under the respective "s is now" guise. And so my previous argument goes through unproblematically. Conditional on the proposition-guise pair representing my knowledge that I am alive at t and the proposition-guise pair representing a specification of ten centuries where I am alive, the proposition-guise pair ((s, s, is identical to), "s is now") has probability 1/10, for any s at which I am alive according to the ten-times reincarnation thesis. The corresponding probability for ((t, t, is identical to), "t is now"), conditional on my being alive at t and singular incarnation, is 1. These are Eqs. 8 and 7 respectively. Eqs. 5 and 6 can be similarly translated. We conclude that my previous argument can be translated into the guise-Russellian account of indexical thought.12 example: No rational agent can BEL a proposition P under guise x while also BELing P under x's guise-negation. Note that on this account it is generally rational to believe both a proposition p and its negation ¬p simultaneously, as long as the negation isn't believed under the guise-negation of p's guise. 11Note that, while the proposition picked out by the guise is tautological, rational agents don't automatically know a tautology under every guise. 12 What about a propositional framework on which indexical thought isn't captured by guises? Some accounts employ a different entity as the third relatum for the beliefrelation; e.g. Perry's (1979) notion of belief states. Any such ternary account will enable my argument just as well as guise-Russellianism. What about accounts on which indexical thought isn't captured by a third relatum at all? On such accounts indexicality would need to be treated on the level of propositional content, rather than guises. Stalnaker (2010) seems to be an example of such an account. According to Stalnaker, the content of indexical thought is a set of (uncentered) worlds. Weatherson (2011) reconstructs Stalnaker's notion of world as follows: "In each world, each center, in Lewis's sense, has a haecceity. A world [in Stalnaker's sense] is the Cartesian product of a Lewisian world, i.e. a world without haecceities, and a function from each contextually salient haecceity to a location." (ibid., p.448) An analogue of my account will also go through on Stalnaker's account. The belief that you exist at t is represented by a set of Stalnakerian uncentered worlds, in each of which t exists and is inhabited. Suppose 15 4 Immortal Thirders-A New Argument For Immortality Once we account for the total evidence received from observing our existence at the present time, we don't have reason to update in favor of multiple reincarnation. But the story doesn't end here. Last section's treatment doesn't exhaust the relevant considerations about indexical uncertainty. As we'll see, an exhaustive treatment of indexical evidence in fact enables a different argument for immortality. The crux lies with an earlier assumption, No Update From Pure Indexical (Sec. 3); the new argument calls this assumption into question. One way to get started is by noting a striking parallel between the immortality case and the well-known Sleeping Beauty puzzle. Sleeping Beauty: Beauty goes to sleep on Sunday, planning on sleeping for two full days. On Sunday night she is told that, in the night from Sunday to Monday, a team of scientists will flip a coin. If the coin comes up Heads, Beauty is woken on Monday, and shortly after put back to sleep. If the coin comes up Tails, Beauty is woken on Monday and on Tuesday. After being put back to sleep on Monday, her memory of the Monday awakening is erased. Now consider the following variation: Immortal Beauty: God creates a universe temporally infinite toward the past and the future. She then flips a coin. If the coin further that you are at center c. Then each such world is equipped with a function mapping c's haecceity to a person-time pair (p, t) in that world such that p exists at t. Suppose for simplicity that you know that you are person p∗. Consider further a specific ten-times reincarnation thesis which says that you, p∗, exist at times s1, ..., s10. Conditional on that thesis and your existing at t, there are ten distinct functions, mapping c into one of each pair (p∗, si), i = 1, ..., 10. When observing your temporal location, besides learning the Lewisian proposition that p∗ exists at t, you also select precisely one of the ten distinct functions: the one which maps c to the pair (p∗, t). Again, the prior probability distributes uniformly over the ten world-function pairs. (For simplicity I consider here a very coarsegrained notion of world, such that the proposition that p∗ exists at t corresponds to exactly one world.) Hence, given a specific ten-times reincarnation thesis, and your existing at t, the prior probability of t's being now-i.e., of c's being mapped to (p∗, t)-is 1/10. This establishes Eq. 8. The other equations follow analogously. 16 comes up Heads, She creates Beauty in precisely one century (singular incarnation). If Tails, She incarnates Beauty in a countably infinite subset of centuries (infinite reincarnation), with no earliest or latest century. Prior to the coin flip, God had fixed plans for when to create Beauty conditional on either outcome of the coin flip. Upon creating Beauty, God informs her of the setup, though not about the outcome of the coin flip, nor about what her plans said. Thirding is a popular view about the original Sleeping Beauty puzzle: It's the view that, upon waking on Monday, Beauty should assign credence 1/3 to the hypothesis that the coin came up Heads (and thus 2/3 to Tails). Roughly, thirders think that, since there are more awakenings conditional on Tails, Beauty should update in favor of Tails upon awakening. More specifically, starting out with a 1/2 credence in Tails on Sunday, Beauty should update to a credence of 2/3 upon waking on Monday.13 The new argument is this: If thirding is correct in Sleeping Beauty, then 0-ing ('zero-ing' or 'nulling') is correct in Immortal Beauty. That is, after being informed that she is in an Immortal Beauty experiment, Beauty should assign credence 0 to Heads (singular incarnation) and credence 1 to Tails (infinite reincarnation). I'll make this argument by going through a series of four intermediate cases, demonstrating the parallel between Sleeping Beauty and Immortal Beauty. Consider first: Modified Sleeping Beauty: Beauty goes to sleep on Sunday, planning on sleeping for two full days. On Sunday night she is told that, in the night from Sunday to Monday, a team of scientists will flip a coin. If the coin comes up Heads, Beauty is woken precisely once during the next two days, either on Monday or on Tuesday. The scientists have received prior instructions specifying on what day Beauty should be woken if the coin comes up Heads. If the 13Prominent defenders of thirding include Elga (2000), Dorr (2002), Horgan (2004) and Titelbaum (2008). 17 coin comes up Tails, Beauty is woken both on Monday and on Tuesday, with her memory of the Monday awakening erased in the night from Monday to Tuesday. Next consider: Created Sleeping Beauty: After several significant medical breakthroughs, the scientists are now able to create adult humans in vitro, capable of rational thought and with an understanding of the English language. As in Modified Sleeping Beauty, the scientists flip a coin in the night from Sunday to Monday. If Heads, the scientists will create Beauty in vitro on either Monday or Tuesday. Which day she is created is again specified by previously received instructions. Upon creation, Beauty is initially asleep. After waking and briefly interacting with the scientists, Beauty is put pack to sleep, sleeping through until Wednesday. If Tails, the scientists create Beauty on Monday, again initially asleep. After interacting with the scientists, she is put back to sleep, and any memory she made post creation is erased. The scientists wake her again on Tuesday, briefly interact with her, before putting her back to sleep until Wednesday. Next up: Reincarnated Beauty: Pleased with the team's scientific progress, God decides to grant the scientists some of Her divine powers: The scientists gain the ability to reincarnate the deceased. They subsequently set up a case like Created Sleeping Beauty, with a few (morbid) differences. The Heads protocol remains unchanged. On Tails, however, rather than putting Beauty back to sleep after her Monday awakening, she is given a deadly potion. The scientists then create a new body in the night from Monday to Tuesday, indistinguishable from Beauty's when first created (in particular, the body will carry no memories of any previous day). With their newfound powers, the scientists then reincarnate Beauty into this new body on Tuesday. 18 Next: Multiply Reincarnated Beauty: The scientists are eager to test their newfound abilities more extensively. They set up a case like Reincarnated Beauty, with the following differences: Rather than just over two days, the scientists let the experiment run over n days. If the coin comes up Heads, Beauty is incarnated and woken precisely once during these next n days (according to previously received instructions). If the coin comes up Tails, Beauty is incarnated, and shortly after given the deadly potion, on each of m of the next n days (again, according to previously received instructions). The argument from thirding in Sleeping Beauty to 0-ing in Immortal Beauty goes as follows: Immortal Thirders: (1) If thirding is correct in Sleeping Beauty, thirding is correct in Modified Sleeping Beauty. (2) If thirding is correct in Modified Sleeping Beauty, thirding is correct in Created Sleeping Beauty. (3) If thirding is correct in Created Sleeping Beauty, thirding is correct in Reincarnated Beauty. (4) If thirding is correct in Reincarnated Beauty, 1/(m+ 1)-ing is correct in Multiply Reincarnated Beauty. (5) If 1/(m + 1)-ing is correct in Multiply Reincarnated Beauty, 0-ing is correct in Immortal Beauty. (C) If thirding is correct in Sleeping Beauty, 0-ing is correct in Immortal Beauty. 0-ing in Immortal Beauty means assigning posterior probability 0 to Heads, and 1 to Tails. Given the setup, Tails is equivalent to the hypothesis that 19 Beauty is immortal. Immortal Thirders thus shows the following: Given thirding, an agent, upon being informed that they are in an Immortal Beauty setup, should become confident of her own immortality. I'll say more about the argument's further implications later. First, I'll defend each of the five premises. Premise (1) has strong prima facie appeal. The only difference between Sleeping Beauty and Modified Sleeping Beauty is that, in the latter, Beauty doesn't know on which day she awakes conditional on Heads. Whether the Heads-awakening occurs on Monday or Tuesday intuitively doesn't seem significant for one's credences upon waking. This intuition can be made more rigorous. Let P be Beauty's credence function upon wakening.14 In Modified Sleeping Beauty she distributes her credence between two hypotheses: H-Mon : The scientists' instructions say to wake her on Monday if the coin comes up Heads. H-Tue : The scientists' instructions say to wake her on Tuesday if the coin comes up Heads. The two hypotheses are mutually exclusive and jointly exhaustive; they partition Beauty's space of epistemic possibilities. Thus we have P (Heads) = P (Heads|H-Mon) * P (H-Mon)+ (11) + P (Heads|H-Tue) * P (H-Tue). (12) Conditional on H-Mon the experiment is just the original Sleeping Beauty experiment. And so Beauty's credence in Heads conditional on H-Mon should just be Beauty's credence in Heads in Sleeping Beauty. And so, if thirding is correct in the original case, then P (Heads|H-Mon) = 1/3. Similarly, if H-Tue is true, then the current experiment is just like the original Sleeping Beauty case but with the Heads-awakening occurring on Tuesday rather than Monday. But it's hard to see what should motivate a different 14This way of speaking presupposes that Beauty's credence function is the same on each waking. This assumption is common enough (Pust, 2013, Fn.1), and motivated by the fact that Beauty's subjective evidential state is the same on each waking. 20 credence assignment to Heads in this modified case. The time of the single Heads-awakening relative to the Tails-awakenings plausibly shouldn't influence Beauty's credences upon waking. Several concrete accounts of thirding confirm this intuition.15 Regarding Premise (2), a salient difference between Modified Sleeping Beauty and Created Sleeping Beauty is that, in Created Sleeping Beauty, Beauty doesn't yet exist on Sunday. But in Modified Sleeping Beauty, Beauty forms an initial credence about Heads on Sunday. Certain accounts of thirding derive Beauty's intra-experiment credence by updating her Sunday credence upon the new indexical evidence Beauty receives upon waking.16 If no temporal prior is available, these diachronic schemes will be silent on what's rational for the agent to believe upon waking/creation. Further, even if there is such a prior, diachronic schemes may be subject to an instance of the problem of old evidence. This is because, in Created Sleeping Beauty, Beauty receives information about the setup only after she awakes, and so starts entertaining the hypotheses Heads/Tails only then. But 15 Among them is the Self-Indication Assumption (SIA) (cp. Bostrom (2010) and Manley ("On Being a Random Sample")). On this account, the probability of Heads only depends upon the ratio between the number of observer moments sharing Beauty's evidental state, given Heads, and the number of observer moments sharing Beauty's evidential state, given Tails. In the original Sleeping Beauty experiment, this ratio is 1 : 2. The relative position of the Heads-awakening doesn't change this ratio. Elga's (2000, p.143) first argument for thirding also supports thirding in Modified Sleeping Beauty: If Modified Sleeping Beauty were repeated many times over, in the long run about 1/3 of the wakings would be Heads-wakings. If long-run frequencies reflect rational single-case credences (in the absence of overriding evidence), then Beauty should assign credence 1/3 to Heads upon waking. As a third example, Horgan (2004) and Horgan (2008) also support thirding in Modified Sleeping Beauty. Horgan assumes that upon waking Beauty synchronically updates a 'preliminary probability function'. The preliminary probability initially merely assumes that it's either Monday or Tuesday; it doesn't assume that Beauty is being woken today. The update thus rules out one of the four possibilities: 'Heads and it's Monday', 'Tails and it's Monday', 'Heads and it's Tuesday', or 'Tails and it's Tuesday'. Conditional on H-Mon, it would rule out 'Heads and it's Tuesday'; conditional on H-Tue, it would rule out 'Heads and it's Monday'. Horgan argues that, prior to the update, probability is uniformly distributed over the four possibilities. If so, we'll get probability 1/3 for Heads after the update, both for H-Mon and H-Tue. Thus the relative position of the Heads-awakening doesn't influence the result of Horgan's account. 16E.g. Titelbaum (2008). 21 the fact that she is awake is evidentially relevant for Heads/Tails. Diachronic schemes may have trouble capturing this evidential connection. Synchronic updating schemes typically won't suffer from either of the two problems. On typical synchronic schemes, Beauty's probability at any given time is given by a constant hypothetical prior, conditionalized on her total evidence at that time. These schemes won't be silent in the absence of a temporally prior probability; for Beauty's probabilities at any given time depend only on the hypothetical prior and her current evidence, not on any temporally prior probability function. Nor will they suffer from the problem of old evidence; for the hypothetical prior is newly updated at any given time on Beauty's current total evidence. The two problems seem like defects of the diachronic schemes, rather than ideal features. Silence seems like a defect, since surely there are some constraints on what's rational for Beauty to believe upon waking in Created Sleeping Beauty. Further, the problem of old evidence is widely recognized as a serious problem, demanding a solution. Thus, the fact that some diachronic schemes are sensitive to the lack of a Sunday period isn't evidence that Beauty's credence function should be so sensitive. Synchronic updating schemes plausibly predict the same post-waking credence function for Created Sleeping Beauty as for Modified Sleeping Beauty. For the relevant total evidence Beauty has in Modified Sleeping Beauty after waking is the same as her relevant total evidence after being informed of the setup in Created Sleeping Beauty. This supports Premise (2). On to Premise (3). The only salient difference in Reincarnated Beauty is that, given Tails, rather than sleeping from Monday to Tuesday, Beauty is killed and reincarnated. This fact may make an emotional or ethical difference. But it shouldn't affect Beauty's credence in Heads. The argument for Premise (4) is similar to the argument for Premise (1). In Multiply Reincarnated Beauty, the scientist's instructions form a partition of Beauty's epistemic possibility space. In particular, the scientists' Tailsinstructions now too form a (non-trivial) partition. Conditionalizing on a cell of this partition, Beauty knows which m of the n days would be her Tailsawakenings. Again, the correct credence function is plausibly invariant under 22 switching the relative position of the Heads-awakening.17 Conditionalizing on any particular set of Tails-awakenings yields a case like Reincarnated Beauty, but with m rather than two Tails-awakenings. This suggests that if thirding is true in Reincarnated Beauty, then Beauty's credence in Heads upon incarnation and being informed of the setup should be 1/(m+ 1).18 On to the final Premise (5). Immortal Beauty is essentially an infinite limit of Multiply Reincarnated Beauty, with days swapped for centuries. The substitution of days with centuries shouldn't affect Beauty's probabilities. What about the infinite limit? Note that any one of God's possible plans for (re)incarnation can be viewed as the limit of an infinite sequence of scientists' instructions (where each set of instructions in the sequence specifies an earlier first incarnation, and a later last incarnation, than the previous set of instructions). For any finite m, a set of instructions specifying that Beauty exists at m centuries given Tails yields 1/(m + 1)-ing. And there is no particular reason to expect that Beauty's credences should be discontinuous in the infinite limit. So if she should have credence 1/(m+1) in Heads for any finite m, she should have credence 0 in Heads in Immortal Beauty. This supports Premise (5), and thus conclusion (C): Upon finding oneself in a setup like Immortal Beauty, thirders should become confident of their own immortality. It's helpful to formally represent the update in favor of Tails in the foregoing cases. Here's one way. Consider Multiply Reincarnated Beauty. If 1/(m + 1)-ing in Multiply Reincarnated Beauty is true, then the posteriors P ′(Heads) and P ′(Tails) (i.e., the probabilities after being informed of the setup), are related to the respective priors as follows: P ′(Heads) P (Heads) = 1 m * P ′(Tails) P (Tails) . (13) Suppose that Beauty achieves her posteriors in the original Sleeping Beauty by updating on new evidence.19 The same will then be plausible in Multiply 17It's a straightforward matter to confirm that the accounts previously examined in Fn. 15-SIA, Elga (2000) and Horgan (2008)-agree. 18Again, the previously examined accounts agree. 19Many thirders agree with this, cp. Horgan (2008) and Manley ("On Being a Random 23 Reincarnated Beauty. Let then I∗ be the evidence by which Beauty achieves her 1/(m + 1)-posterior in Multiply Reincarnated Beauty. Eq. (13) implies that P (I∗|Heads) = 1/m * P (I∗|Tails). (14) If furthermore P (I∗) > 0, we can express the conclusion that Tails is infinitely confirmed over Heads as the limit of Eq. 14 for infinitely many centuries: P (I∗|Heads) = 0 * P (I∗|Tails). (15) This makes precise the sense in which Beauty's credence in Immortal Beauty is the infinite limit of her credence in Multiply Reincarnated Beauty. Now, the result of Immortal Thirders can be generalized to further setups. Immortal Beauty features a coin-flipping deity. The coin flip serves to fix one's pre-observation probabilities of singular incarnation and immortality to 1/2 each, e.g. via the Principal Principle. (By 'pre-observation probabilities' I mean probabilities prior to finding oneself in the relevant setup. In the following, I will sometimes speak loosely and call these probabilities 'priors', even though these probabilities aren't the pure hypothetical priors. They are conditionalized on the setup, but not on finding oneself in the setup.) This, in turn, enables a clean analysis of the problem in terms of unique posterior probabilities. But we can make do with less; a coin-flipping deity isn't required for strong pro-immortality conclusions. Consider the following more general setup: Lone Immortal: Some time after being born, you learn that the universe is temporally infinite in both directions. You also learn that, tragically, you are (and will be and forever have been) completely alone in the universe. Thrown deep into an existential crisis, you start contemplating the question of reincarnation, Sample"). One account who doesn't is Elga (2000). Those who don't agree that Beauty gains new evidence upon waking have different arguments for why she should adopt the thirding posterior. My aim here is to exhibit one way to understand the update; I do not claim that it is the only way. 24 entertaining the two hypotheses that you only live once, for a finitely long time, or live infinitely often. Despite the generality of this setup, the epistemic state of the agent in Lone Immortal is still similar to the epistemic state of Beauty in Immortal Beauty. The salient difference is that Lone Immortal doesn't include a coin flip to fix the priors of singular incarnation and immortality to 1/2 each. How do different choices of priors affect the posterior of immortality? A realistic choice of priors has P (II) > 0 and P (SI) > 0. In Lone Immortal, you know that you actually exist sometime. (This piece of evidence doesn't favor one incarnation hypothesis over the other, as expressed in Eq. 4.) For another, you presumably have some grasp on the plausibility of various theories of personal identity. This plausibly warrants a non-zero probability for each of SI and II.20 Learning that you're currently in an Immortal Beauty setup infinitely confirms Tails over Heads: It sends the 1/2-priors in each hypothesis to posteriors 1 in Tails and 0 in Heads. The same thing would happen if the coin was biased, provided that the probabilities for Head and Tails each remained greater than 0. Suppose the agent in Lone Immortal entertains II and SI as the only two alternatives with credences P (SI) and P (II) (such that P (SI) + P (II) = 1). An Immortal Beauty setup employing a biased coin with corresponding chances, P (SI) and P (II), would then remove the disanalogy with Lone Immortal stemming from the priors. Thus we obtain 0-ing for Lone Immortal: Starting with non-zero probabilities P (SI) > 0 and P (II) > 0, summing to unity, an agent who finds herself in a setup like Lone Immortal should become confident of her own immortality. Now, the agent in Lone Immortal may entertain additional reincarnation hypotheses, besides SI and II. Indeed, a temporally infinite universe allows for infinitely many different incarnation hypotheses: For every natural number n there is the hypothesis that you are n times incarnated (in addition to the hypothesis of infinite reincarnation). Still, II will be infinitely confirmed 20In the case of II the combination of a non-zero credence in the conjunction of a multiverse theory and a revisionary theory of personal identity might do the trick, cp. Fn. 5. Huemer (forthcoming) argues for similar points. 25 over any finite reincarnation hypothesis.21 Given countable additivity, that entails that II is infinitely confirmed also over the (infinite) disjunction of all finite reincarnation hypotheses. Hence, even when the agent entertains other reincarnation hypotheses, she should still become confident in immortality after finding herself in a setup like Lone Immortal.22 What if the prior for II vanishes? In this case, we won't quite achieve confidence in immortality, but something reasonably close. For it's then plausible that, upon finding oneself in a scenario like Lone Immortal, there's a range of permissible posterior probabilities of II, which includes non-zero values. This still vindicates a weaker version of the desired conclusion: that it is rationally permissible to substantially increase one's credence in immortality, upon finding oneself in Lone Immortal. To summarize: In cases like Immortal Beauty and Lone Immortal, there is a sense in which existence is evidence of immortality: If thirding is correct, merely registering that one is in a scenario like that should convince one of one's own immortality. Further, there's a way to understand the setups which yields an even tighter sense of 'existence is evidence of immortality'. As noted before, most thirders think that Beauty achieves her posteriors by updating on new evidence. In Sleeping Beauty, this new evidence is essentially indexical. Let H1, T1, and T2, abbreviate 'Heads and it's Monday', 'Tails and it's Monday', and 'Tails and it's Tuesday', respectively. The new evidence is then that H1 ∨ T1 ∨ T2. The analogue of this evidence for Multiply Reincarnated Beauty is a disjunction of disjunctions: Each subordinate disjunction is associated with a specific set of scientists' instructions; it is of the form Hi ∨ Tj1 ∨ Tj2 ∨ ... ∨ Tjm , where the i and jk specify the days of waking conditional on Heads and Tails, according to the instruction. As before, let's call this evidence I∗. 21This can be formalized again as before. 22What if we deny countably additivity here? Then it's possible that II doesn't end up with a posterior close to 1, despite being infinitely confirmed over any finite reincarnation thesis. But finite additivity still yields a notable result: For any natural number n, you should have posterior 1 that you are incarnated at least n times. This means that, upon finding yourself in a case like Lone Immortal, you should at least become confident that you are reincarnated arbitrarily many times. 26 Crucially, once Beauty is informed of the setup, she can infer I∗ from another piece of essentially indexical information: that she is alive now. Indeed, given the background knowledge of the setup, the two pieces of information are equivalent. Hence, from Eq. 14, P (I am alive now|Heads) = 1/m * P (I am alive now|Tails). (16) On this view, then, thirders deny No Update From Pure Indexical (Sec. 3). To recall, this assumption says that conditionalizing on 'I am alive now ' shouldn't change your relative confidence in singular incarnation (Heads) or multiple/infinite reincarnation (Tails). But, given the setup of Multiply Reincarnated Beauty, Beauty can infer I∗ from the fact that she is alive now. And so she should become more confident in multiple reincarnation (Tails) once she learns that she is alive now. This expresses a good sense in which 'existence is evidence of immortality'. Note that it is to be distinguished from Huemer's original claim that observing your existence at the present time (de re) is evidence of immortality. As I have argued in Section 3, observing one's existence at the present time (de re) affords both the information that one is alive at t and the temporal de se information that t is now. Once the latter piece of information is accounted for, observing one's existence at the present time shouldn't alter one's confidence in reincarnation. Rather, on the account expounded here, you should be confident in immortality already prior to observing the present time.23 The cases discussed so far still make certain presuppositions (for example, Lone Immortal assumes that the agent knows that she is alone in the universe). As I explain in the next section, these limitations aren't easily eliminable. 23Since the argument is a conditional argument from thirding, people who are certain of halfing won't be moved much, of course. Rather, any halfer will likely accept my analysis of immortality in Section 3. But I take it that most people, if not thirders, at least lend thirding considerable credence, even after hearing the argument for immortality. Such people should still update in favor of immortality, in proportion to their credence in thirding. 27 5 Limitations On The Conclusion Most literature on Sleeping Beauty makes a certain tacit assumption: that there are no additional observer moments in the history of the universe with the same evidential state as Beauty intra-experiment. In a small universe, this assumption seems reasonable enough. Beauty's evidential state includes memories of her recent past, including Sunday. It also includes the knowledge that she is currently in a Sleeping Beauty experiment. In a universe with few observers, these information usually serve to restrict her self-locating uncertainty to a unique Sleeping Beauty experiment. In a universe with many observers, however, the assumption becomes more tenuous. (Potential candidates for evidentially equivalent observer moments in such a universe include Boltzmann brains, or repeated patterns of particles in a spatially infinite universe, or repeated patterns of particles in a temporally recurrent universe.) This matters if we try to extend Immortal Thirders to other scenarios besides Immortal Beauty and Lone Immortal. Consider One Immortal Or Many Mortals: Some time after being born you learn that the universe is temporally infinite in both directions. Moreover, you learn that one of two things is true: Either you are completely alone in the universe and are infinitely reincarnated ('Tails'). Or you are one out of infinitely many distinct people, each of whom lives alone in exactly one century, and is never reincarnated ('Heads'). As in previous setups, the 'Heads' scenario entails your own mortality. But now 'Heads' includes many additional and evidentially similar observer moments. This changes its posterior probability: The previous rationale in favor of immortality no longer obviously applies. Most accounts of thirding are sensitive to the ratio of the number of observer moments (lives) with the relevant evidential states given Heads, to the number of such observer moments given Tails. In One Immortal Or Many Mortals, the possibilities do not differ in the number of these observer moments. Thus, most thirders will think that Beauty's credence in immortality shouldn't change upon 28 finding herself in One Immortal Or Many Mortals. (This is the case e.g. on synchronic accounts of thirding, like Horgan (2008) or SIA; both accounts will treat the two possibilities symmetrically.)24 The lesson is that, whether existence is evidence for immortality, can depend on the number of lives that are lived, given each incarnation hypothesis. If infinite reincarnation infinitely multiplies the number of lives, 24One might think that there are views on which the epistemic situation between the many subjects on 'Heads' isn't relevantly similar; and thus that the original thirding rationale does apply. Consider specifically a view on which each subject is directly acquainted with herself (in the Russellian sense, cp. Russell (1910).) On such a view, one's own personal identity may figure in one's evidence. For example, Jane may have evidence of the form 'Jane exists at t', where 'Jane' denotes some access-restricted entity, perhaps a personal haecceity. (The entity has to be access-restricted since the personal identity of others plausibly isn't directly observable.) At first glance it might seem that this view effectively reduces One Immortal Or Many Mortals to a case of Immortal Beauty (with a biased coin): 'Heads' means Jane is singularly incarnated, 'Tails' means Jane is infinitely reincarnated; the existence of other agents on 'Tails' is epistemically irrelevant, since Jane's subjective evidential state differs from theirs. But this reasoning is dubious, for two reasons: First, it's simply not very plausible that Jane's subjective evidential state differs from the rest. More plausibly, acquaintance theory is simply another account of irreducibly indexical thought. If 'Jane' denotes some special access-restricted haecceity, it's plausible that the thought (only thinkable by Jane) that Jane is in the library just is Jane's thinking 'I am in the library'. Second, even if acquaintance theory doesn't reduce to an account of de se belief, it fails to reduce One Immortal Or Many Mortals to Immortal Beauty. For Jane's evidence now includes a new piece of information which will favor Heads over Tails. The new piece of information is that Jane-a certain special personal haecceity-exists. Given the setup of One Immortal Or Many Mortals, one can infer that there are at least as many possible personal haecceities as there are centuries in the universe (in this case: countably infinitely many). For these are required by the possibility of 'Heads'. An application of PI then suggests that Jane should take her own existence (qua haecceity) as evidence against 'Tails'. Indeed, she should discount 'Tails' relative to 'Heads' exactly by the number of centuries in the universe. This can be seen as follows: If M is the number of possible haecceities, then the probability of Jane's existing conditional on 'Tails' is merely 1/M . For there are M possible distributions of personal haecceities over centuries compatible with 'Tails'- namely those which assign a single hacceity to all centuries. Of those, precisely one has Jane exist sometime (namely the distribution which assigns Jane to every century). By contrast, conditional on 'Heads', Jane's existing has probability T/M , where T is the number of actual centuries. This is because a fraction of T/M of all personal haecceity distributions compatible with 'Heads' have Jane exist sometime. Hence PI implies that, upon observing her haecceity, Jane should update in favor of Heads. Only once she has done so, the case effectively becomes like Immortal Beauty: Given 'Heads', Jane lives exactly once; given 'Tails', Jane lives infinitely many time. This results in a probability boost for 'Tails'. However, the resulting boost exactly cancels out the previous update in favor of 'Heads'. The end result is that Jane shouldn't update in favor of immortality. This is precisely what the previous accounts said about One Immortal Or Many Mortals. 29 compared to singular incarnation, then Immortal Thirders generally applies: Thirders should become confident that they are immortal. But the overall number of lives in the universe might also be fixed, regardless of the incarnation hypothesis. In this case, singular incarnation and infinite reincarnation would merely amount to different distributions of personal identities over those lives: Singular incarnation would distribute many distinct identities, and each identity would occupy only one life. Infinite reincarnation would distribute fewer distinct identities, such that each identity can occupy many lives. If the overall number of lives lived is fixed in this way, many thirders may refrain from concluding their own immortality. 6 Conclusion I have argued that Huemer's argument for immortality is unsuccessful, since it ignores vital de se evidence gained from observing our existence at the present time. Yet, an exhaustive treatment of de se evidence also reveals a different, limited, argument for immortality. In certain cases, there's reason to think that registering the pure indexical 'I am alive now' should make you confident of your own immortality, even prior to observing the present time (de re). More generally, I have argued that, in these cases, contemplating the probability of immortality is analogous to contemplating the probability of Tails in Sleeping Beauty. Those who advocate thirding in the traditional Sleeping Beauty setup should update in favor of their own immortality in these cases. Finally, we also saw situations in which thirding doesn't warrant such an update. What do these results entail for our actual, real-life credences in immortality? This depends on our actual priors. Suppose we're mostly sure about who exists, but not about how often each person exists. Then we'll distribute most of our prior credence between a singular incarnation hypothesis and an infinite reincarnation hypothesis according to which all the same people exist. In this case, Immortal Thirders should make us quite confident in infinite reincarnation (it warrants an update in proportion to our credence in thirding). But we might also be quite uncertain about who exists; after all, the 30 identity relations between persons at different points in time and space aren't easily surveyable. If we are (moreover) mostly sure about the total number of lives lived, we'll distribute most of our prior credence between a singular incarnation hypothesis and an infinite reincarnation hypothesis that agree on the number of lives lived. In this case, Immortal Thirders doesn't imply an update in favor of infinite reincarnation. But most likely, we'll be quite uncertain both about who exists and how many lives are lived. In this case, Immortal Thirders implies that we should increase our confidence in infinite reincarnation relative to certain singular incarnation theses, but not relative to others. In particular, those singular incarnation theses that agree with infinite reincarnation on the total number of lives lived may maintain their prior confidence level relative to infinite reincarnation.25 References [1] Nick Bostrom. Anthropic Bias: Observation Selection Effects in Science and Philosophy. Routledge, 2010. [2] David Braun. "The Objects of Belief and Credence". In: Mind 125.498 (2016), pp. 469–497. doi: 10.1093/mind/fzv137. [3] Cian Dorr. "Sleeping Beauty: In Defence of Elga". In: Analysis 62.4 (2002), pp. 292–296. doi: 10.1111/1467-8284.00371. [4] Adam Elga. 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