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The only ignorance you have left is about which of two locations you inhabit inside that world. In the case of Two Rooms, given that you inhabit either R1 or R2, it seems like the only non-arbitrary option is to be indifferent between R1 and R2. Similarly, given that you inhabit one of the locations described by c1 or c2, it seems like the only non-arbitrary option is to be indifferent between c1 and c2. An important point in defense of Center Indifference is that it avoids both of the major criticisms faced by Naive Indifference. First, unlike Naive Indifference, it does not lead to any contradictions.6 Naive Indifference doesn’t specify a unique way one should partition the space of possibilities that one is indifferent over, but Center Indifference specifies that one should be indifferent between maximally specific similar centered worlds. Second, the usual reasons for why one might favor one possibility over another don’t seem to be present in Center Indifference. Given that c1 and c2 agree on all the non-indexical facts, it’s hard to see how either hypothesis can plausibly be regarded as more theoretically virtuous than the other, for example by being more simple or explanatory. Even if one could somehow make sense of the idea that one of the locations described by c1 and c2 is (say) “simpler” than the other, you already know that someone has to occupy each of the locations described by c1 and c2. Call the individual that occupies the 4Although Lewis’ distinction between possible worlds and centered worlds is popular, it has its detractors. For example, Lewis’ main motivation for introducing centered worlds was to accommodate the content of self-locating belief, but Stalnaker (2008) has argued that the content of self-locating beliefs can be accommodated by ordinary possible worlds propositions. For a critique of Stalnaker’s approach to self-locating belief, see Weber (2015). Even if Lewis’ overall framework is rejected, the kind of principles that I will go on to discuss should be able to be formulated in other frameworks as well. 5 This principle corresponds to Weatherson’s (2005) principle of “C-INDIFFERENCE”, which is a slight strengthening of Elga’s (2004) original principle. Weatherson also formulates a stronger principle to CINDIFFERENCE, which he calls “P-INDIFFERENCE”, which allows the principle to be used in a much broader range of cases where it is unknown which possible world one is in. I will be focusing on C-INDIFFERENCE because the modifications that I will later be proposing to C-INDIFFERENCE straightforwardly generalize to Weatherson’s stronger P-INDIFFERENCE principle, and C-INDIFFERENCE is a more simple and intuitive principle to state. 6Later in this section, I address a potential worry with applying Center Indifference in infinite situations. 4 perspective of the centered world that you don’t occupy your “evidential twin”. Either you will occupy the simpler location or your evidential twin will. Why should the view that your evidential twin occupies the simpler location be an overall more complex theory than the view that you occupy the simpler location? Another way to support Center Indifference is by noting that violations of Center Indifference require a strange kind of forced epistemic disagreement. Suppose you deviated from Center Indifference in some way, say by being more confident in c1. Then, so long as you are self-aware, it will be implied by your evidence that you are more confident in c1. This implies that your evidential twin will also think that they are more likely to be located in c1. We can dramatize this disagreement by imagining that being located at c1 is tied up with something practically important: perhaps only the individual located at c2 will soon be tortured. While you think that you are most likely safe, your evidential twin will think that you will most likely be tortured (since they will think that they are more likely to be safe)! Insofar as it is natural to think of you and your evidential twin as epistemic peers with the same evidence, this forced disagreement about who is about to be tortured can seem strange. In other contexts, some have argued that disagreements of this kind can never be rational.7 An interesting benefit of Center Indifference is that these kinds of forced epistemic disagreements never arise when the centers occupied by c1 and c2 contain rational agents. Both you and your evidential twin will agree that you are both equally likely to be tortured. Before moving on, it is worth briefly addressing some important objections that Weatherson (2005) has raised to a related principle that Elga (2004) has defended. Elga (2004) formulates his own principle in terms of subjective indistinguishability: similar centered worlds that are centered on individuals undergoing indistinguishable subjective experiences should always be assigned the same credence. Weatherson criticizes Elga’s principle on the grounds that it presupposes a controversial internalist conception of evidence.8 For example, according to Elga’s principle, one should be indifferent between being a particular embodied human being and being a brain in a vat (BIV), so long as both the embodied being and the BIV occupy the same world and have indistinguishable experiences. However, on standard versions of externalism, it is part of your evidence that you have hands, so you shouldn’t assign any credence to being a BIV. My response to this objection is that the appeal to indistinguishable experiences was inessential to Elga’s principle. Center Indifference is entirely neutral on the correct conception of evidence. 7In particular, some have argued for the thesis of “Uniqueness” according to which any body of evidence rationalizes a unique doxastic attitude towards any proposition (e.g. White (2005a), Greco and Hedden (2016), Dogramaci and Horowitz (2016), and Horowitz (2019)). According to Uniqueness, if two agents share the very same evidence, then they cannot disagree over the likelihood of any proposition . The case here is related but somewhat different: it involves disagreement over de se matters rather than disagreement over an ordinary (uncentered) proposition. 8For defenses of an externalist conception of evidence, see Williamson (2000) and Srinivasan (2015). For defenses of an internalist conception of evidence, see Fumerton (2009), Schoenfield (2015), Duncan (2018), and Smithies (2019). 5 For example, given an externalist conception of evidence, Center Indifference does not require you to assign any credence to being a BIV, since being a BIV is not compatible with your evidence. A second criticism that Weatherson raises involves problems with infinity. Consider the following case: Infinite Rooms: There are a countable infinity of indistinguishable rooms R1, R2, …, each of which is occupied by a single agent. All of the agents know exactly which possible world they are in, but they don’t know which room they occupy, since they are all duplicates. How confident should any one of them be that they are in room R1? By Center Indifference, each agent is required to assign equal credence to being in any particular room. The only way to do this without violating the axiom of Finite Additivity is to assign a credence of 0 to being in any particular room. However, this violates the axiom of Countable Additivity.9 Weatherson recommends we should perform a modus tollens: since Countable Additivity should be upheld, Center Indifference should be rejected. It strikes me that the better thing to do is to perform the modus ponens, and conclude that there are violations of Countable Additivity. After all, there are already independent reasons to be skeptical of Countable Additivity, and it’s unclear what other credence one should assign in Infinite Rooms.10 However, rather than making the case for Center Indifference presuppose the falsity of Countable Additivity, one can also simply restrict Center Indifference to only apply in finitary situations: Finite Center Indifference: For any possible world w, if there are only finitely many similar centered worlds c1, c2, …, cn associated with w that are compatible with your evidence, then it is rationally required to set Cr(c1 | w) = 1/n. Finite Center Indifference is silent on Infinite Rooms and other infinitary cases, and so does not fall into trouble with Countable Additivity. For simplicity, I will mostly be focusing on Center Indifference in what follows, but it should be kept in mind that one can fall back to Finite Center Indifference to avoid getting entangled with puzzles about infinity.11 9The axiom of Finite Additivity states that, for two mutually exclusive hypotheses H1 and H2, it is a rational requirement that Cr(H1 or H2) = Cr(H1) + Cr(H2). The axiom of Countable Additivity states that, for any mutually exclusive hypotheses H1, H2, …, it is a rational requirement that Cr(H1 or H2 or …) = Cr(H1) + Cr(H2) + … . 10 For other cases that make trouble for Countable Additivity, see the treatment of the infinitary puzzles in Arntzenius et al. (2004), Builes (2020a), and Dorr et al. (2020). Easwaran (2013) presents some arguments in favor of Countable Additivity, but see Stewart and Nielsen (2021) for a critical response. 11 Singer (2014: 3170) raises another infinitary problem to a principle inspired by Elga (2004), which involves continuum-many different agents. While the case that Singer raises does pose a problem for the principle that he formulates, it does not pose any problem for Center Indifference, since Center Indifference is silent on what the right answer is to Singer’s infinitary puzzle. 6 Lastly, Weatherson also objects to the background assumption that one should assign a precise credence to cases like Two Rooms. Instead, Weatherson suggests that one should assign an imprecise credence to the relevant self-locating proposition, where an imprecise credence is a doxastic state that is representable by a set of multiple probability functions. For example, instead of having a credence of 0.7 in some proposition P, one could instead have an imprecise credence of [0.5, 0.7] in P, where this means that for every r ∈ [0.5, 0.7], one of the many probability functions that collectively represents you assigns a credence of r to P. The project of comparing the advantages and disadvantages of precise over imprecise credences is a large topic that I cannot fully address here. I will only provide a footnote to the sizeable literature of arguments against the rationality of imprecise credences (either that they are not rationally permissible or that they can never be rationally required).12 One simple point that can be made is that it is unclear how imprecise credences help with giving a non-arbitrary answer to Two Rooms. Surely you shouldn’t assign (say) a [0.317, 0.753] credence that you are in R1. Why that rather than [0.292, 0.819]? Perhaps the least arbitrary option is to be maximally imprecise and assign a credence of [0,1] to being in R1.13 Generalizing, we might formulate the following alternative to Center Indifference: Center Imprecision: For any two similar centered worlds c1 and c2, if both c1 and c2 are compatible with your evidence, then it is rationally required to set Cr (c1 | c1 or c2) = [0,1]. Beyond the general arguments against the rationality of imprecise credences, I only have two points to make regarding Center Imprecision. First, it seems that in Two Rooms one should be more confident that 2+2=4 than that one is in R1. However, on standard ways of interpreting imprecise credences, Center Imprecision contradicts this intuition.14 Second, all of the points that I will be making about Center Indifference in the next three sections equally apply to Center Imprecision. So, even if one is more sympathetic to Center Imprecision over Center Indifference, much of what I will say will still apply.15 12 See, for example, White (2009), Elga (2010), Carr (2020), and Builes et al. (2020). 13 In fact, this is what Weatherson (2005: 626) himself recommends about a related case. 14 I am assuming here that rational agents should be certain that 2+2=4. If one has a credence of 1 in P and an imprecise credence of [0,1] in Q, then one is not strictly more confident in P than Q because there is a probability function in one’s representor that is equally confident in P and Q. 15 A referee raises another worry to Center Indifference, namely that it seems to be in tension with the probabilities used in Everettian quantum mechanics given by the Born rule. I have two responses to this worry. First, Everettian quantum mechanics is highly controversial. According to the 2020 PhilPapers Survey conducted by Bourget and Chalmers (forthcoming), less than 20% of philosophers accept or lean towards such an interpretation. Second, it remains a matter of dispute how Center Indifference relates to the Everett Interpretation. For example, in their defense of the Everett interpretation, Carroll and Sebens (2018) write “We believe that the reasoning behind Elga’s principle, when properly applied to Everettian quantum mechanics, actually leads to the Born rule—not branchcounting” (27). They then go on to defend a more general epistemic principle that delivers the Born rule and “is compatible with indifference in standard cases of classical self-locating uncertainty” (41). 3. Two Ambiguities 7 Having given a preliminary defense of Center Indifference, I now wish to point out some cases where it’s unclear how Center Indifference applies, on the grounds that it is unclear how the operative notion of a “possible world” is supposed to apply.16 In the metaphysics of time, there is a debate between Presentists and Eternalists.17 Presentists hold that the whole of reality is three-dimensional: only present things exist. Nowhere in reality will you find any dinosaurs (although it is of course true that reality did once contain dinosaurs). Eternalists disagree. Eternalists hold that the whole of reality is four-dimensional: past, present, and future things all exist. In one portion of reality, there are dinosaurs, and in another portion of reality (perhaps) there are human outposts on Mars. For Eternalists, which time is the present time is merely an indexical fact, just as which place is “here” is merely an indexical fact. The present time is whichever time you happen to be located in. For Presentists, which time is present is not merely an indexical fact. The present time is the only time that exists! Now, suppose you were somehow convinced that Presentism were true, and suppose you were informed that throughout the history of the universe, there will be three individuals that share your evidence. When time t1 is present, one of those three individuals will exist. When a later time t2 is present, two of those three individuals will exist. According to Center Indifference, are you rationally required to assign a credence of 1/3 that you are any particular one of those individuals? It depends what “possible world” is meant to refer to. On one understanding of “possible world”, possible worlds are three-dimensional objects for the Presentist, because the whole of reality is a three-dimensional object. On this understanding, Center Indifference does not imply that you are required to assign a credence of 1/3 that you are any particular one of those individuals. Both individuals at t2 occupy the same possible world, so you are required to be indifferent between them, but all three individuals do not occupy the same possible world. After all, there is no way that reality could be in which all three exist. Alternatively, one could understand the notion of a “possible world” not to refer to everything that exists, but instead it might refer to everything that did exist, does exist, and will exist. In this second 16 The following material can be seen as an extending the discussion in Builes (2019, 2022b), where I also note the relevance of debates about the metaphysics of time to the epistemology of self-locating belief. However, in previous work, I did not connect the metaphysics of modality with the epistemology of self-locating belief, which is the purpose of this section. Furthermore, in previous work, I did not defend my own version of Elga’s (2004) indifference principle over other possible versions, which will be the focus of the next two sections. The current paper can be read as defending the kind of reasoning that I have employed in earlier work, while at the same time extending its coverage to a wide range of other debates, concerning the metaphysics of modality (sections 3-6), external world skepticism (section 7) and inductive skepticism (section 8). 17 For defenses of Eternalism, see Sider (2001: ch. 2), Skow (2015), and Turner (2019). For defenses of Presentism, see Markosian (2004), Bourne (2007), Ingram (2019), and Builes and Impagnatiello (forthcoming). 8 sense, the three individuals do occupy the same world, even if Presentism is true. So, in this second sense, Center Indifference requires one to be indifferent between all three individuals. Let us turn to a different case. Let a universe be a maximally interconnected spatiotemporal system. In other words, everything that lies inside a universe is spatiotemporally related to everything else in that universe, and everything that lies outside of a universe fails to be spatiotemporally related to anything inside that universe. Some philosophers have entertained the possibility that many universes exist. In fact, famously, Lewis (1986) argued that so many universes exist that there couldn’t be any more of them.18 Suppose you were convinced that many universes exist (perhaps because you were convinced that Lewis’ Modal Realism is true). Now consider three individuals that are located in two separate universes that share your same evidence, where one of them exists in universe u1 and two of them exist in universe u2. Conditional on your being one of these three individuals, are you required to assign a 1/3 credence that you are any particular one of them? Again, it depends what “possible world” is meant to refer to. On one natural reading, a “world” is supposed to refer to the entirety of reality, whatever that might be. On this understanding, it is entirely unproblematic for a single possible world to contain multiple universes, just as it is unproblematic for a single possible world to contain multiple planets. Given this reading, Center Indifference tells you to be indifferent between the three relevant individuals, since they all live in the same world. However, for reasons having to do with his overall understanding of modality, Lewis (1986) argued that possible worlds cannot contain multiple universes. Instead, Lewis’ notion of a “possible world” coincides exactly with the notion of a “possible universe”. On this understanding, Center Indifference does not imply that you need to be indifferent between the three individuals located in two different universes, because those individuals do not inhabit the same possible world after all. 4. Four Ways To Be Indifferent Towards Others Given these ambiguities, how should we make Center Indifference precise? In this section, I’ll distinguish a few natural ways this might be done. Let’s start by regimenting some terminology. I will be stipulatively using the notion of a “world” to refer to the whole of reality, unrestrictedly speaking. So, if Presentism is true, possible worlds are three-dimensional objects. If Modal Realism is true, then the plurality of all universes are contained in a single world (the actual world). I will also be stipulatively using the notion of a “universe” to refer to maximal spatiotemporal system, as before. However, my stipulated notion of a universe is meant to apply in a way that is metaphysically neutral between Presentism and 18 Also see Bricker (2020) for further defense of a distinctive kind of modal realism. See Builes (2022c) for an argument against modal realism. 9 Eternalism. So, for example, both Presentists and Eternalists should agree that Abraham Lincoln and George Washington are contained in the same universe. For the Presentist, a universe is not just a maximally interconnected spatial system, but it also includes everything that did exist and will exist throughout the evolution of a maximal spatial system. Let us say that two centered worlds are 3-D similar if their associated centers occupy the same time and universe. Two centered worlds are 4-D similar if their associated centers occupy the same universe. Any two centered worlds are 5-D similar. Lastly, two centered worlds are similar if their associated centers occupy the same possible world. Here is an example to illustrate these distinctions. Suppose I tell you that throughout the history of the universe there will be two individuals who share your evidence, and they will be located at different times. Presentists will think that those individuals do not occupy similar centered worlds, whereas Eternalists will think that those individuals do occupy similar centered worlds. However, whereas judgements of similarity are dependent on one’s prior metaphysical commitments, judgements of 3-D similarity, 4-D similarity, and 5-D similarity are metaphysically neutral. Regardless of your metaphysical views, both of those individuals live in 4-D similar and 5-D similar worlds without living in 3-D similar worlds. By using these stipulated distinctions, we can finally formulate different disambiguated variations of Center Indifference. Although I won’t do so here, it’s straightforward to note that exactly analogous variations can be formulated for Center Imprecision. (X-D) Center Indifference: For any two (X-D) similar centered worlds c1 and c2, if both c1 and c2 are compatible with your evidence, then it is rationally required to set Cr (c1 | c1 or c2) = 1/2. Roughly speaking, 3-D Center Indifference says to be indifferent across simultaneous individuals that share your evidence. 4-D Center Indifference says to be indifferent across spatiotemporally related individuals that share your evidence. 5-D Center Indifference says to be indifferent across any possible individuals that share your evidence. Center Indifference says to be indifferent across any individuals that live in the same world that share your evidence. 5. How To Be Indifferent Which of these four principles should we endorse? While I don’t have any decisive argument in favor of any one of these principles, I think we should endorse Center Indifference. First, both 3-D Center Indifference and 4-D Center Indifference seem to be restricted in an ad hoc way.