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Time-Slice Rationality and Self-Locating Belief

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Abstract

The epistemology of self-locating belief concerns itself with how rational agents ought to respond to certain kinds of indexical information. I argue that those who endorse the thesis of Time-Slice Rationality ought to endorse a particular view about the epistemology of self-locating belief, according to which ‘essentially indexical’ information is never evidentially relevant to non-indexical matters. I close by offering some independent motivations for endorsing Time-Slice Rationality in the context of the epistemology of self-locating belief.

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Notes

  1. For some more recent criticisms and defenses of Time-Slice Rationality, see Podgorski (2016), Döring and Eker (2017), Snedegar (2017), and Hedden (2016, 2017).

  2. Authors who have defended the Thirder solution include Arntzenius (2003), Dorr (2002), Draper and Pust (2008), Elga (2000), Hitchcock (2004), Horgan (2004), Monton (2002), Stalnaker (2008), Titelbaum (2008), and Weintraub (2004). For a useful survey of the arguments for the Thirder position, see Ross (2010).

  3. For a book length defense of an accuracy-first approach to epistemology, see Pettigrew (2016).

  4. The following discussion will largely follow Kierland and Monton (2005).

  5. For much more on different inaccuracy measures and their justifications see Leitgeb and Pettigrew (2010).

  6. We can think of a world w as a function from uncentered propositions to the truth value of that proposition at w.

  7. The Halfer Rule is also equivalent to the updating procedures defended by Meacham (2008) and Halpern (2006).

  8. Kierland and Monton’s position is that one ought to be a Halfer in Duplicating Beauty, but either Thirding or Halfing is permissible in Sleeping Beauty.

  9. If her credence in the coin landing Heads is h, her total inaccuracy will be (1 − h)2 if the coin lands Heads, and her total inaccuracy will be (0 − h)2 if the coin lands Tails (because there will only be one center that is her). So, her expected total inaccuracy is (1/2)*(1 − h)2 + (1/2)*(0 − h)2 = h2 − h + 1/2. The value that minimizes this quantity is h = 1/2.

  10. It is worth noting that the expected inaccuracy of your current time-slice should actually be the weighted average of the inaccuracies of all the time-slices you may be (weighted by your credence in each of the corresponding centers). However, if one appeals to Elga’s (2000) ‘highly restricted principle of indifference’, which says that one should assign equal credence to all the epistemically possible centers within an uncentered word, one will be required to assign equal weights to all those time-slices within the same uncentered world. While Elga’s principle has proven very popular, for some pushback see Weatherson (2005). For a brief response to Weatherson (2005), see Bradley (2011a: 338–339).

  11. Here I am assuming Elga’s highly restricted principle of indifference (see footnote 11).

  12. To my knowledge, the only Thirder who has argued that the presence of objective chance makes a crucial difference in cases of self-locating uncertainty is Wilson (2014). Wilson also argues against the case of Quantum Measurement below on the grounds that the correct interpretation of Quantum Mechanics is not decided by a chancy process. For a response to this suggestion, see Bradley (2015: 689–692).

  13. Interestingly, while Titelbaum (2013a, b) Thirder framework does have counterintuitive consequences in Eternal Recurrence and Many Brains, Titelbaum has argued that it avoids the charge of easy confirmation in Quantum Measurement (pp. 273–282). However, Titelbaum only avoids the charge of easy confirmation given a certain theory about personal identity. If the experimenter before the experiment is not identical to either person after the experiment, or if the experimenter before the experiment is identical to both persons after the experiment, Titelbaum’s framework does not generate the desired probabilities. However, given an account of personal identity defended by Lewis (1976) and supplemented by Saunders and Wallace (2008), Titelbaum’s account does give the desired probabilities. On this account, there really are two observers before the experiment, each of which is identical to one of the observers after the experiment. So, there really isn’t an increase in the number of observers before and after the experiment! While this interpretation of the personal identity facts can be questioned on independent grounds (e.g. see Tappenden 2008), the Time-Slicer will also be wary of the fact that this framework rejects No Reference, since it gives different verdicts depending on the correct theory of personal identity.

  14. This assumes that in cases of fission (like Quantum Measurement), neither of the post-fission agents is identical to the pre-fission agent.

  15. Some philosophers who have questioned this diachronic constraint include Bostrom (2007), Meacham (2008), Briggs (2010), and Cozic (2011).

  16. Hedden himself endorses Uniqueness, which, in the context of Bayesian epistemology, is the claim that there is a uniquely rational prior probability function. While Uniqueness would guarantee that Beauty-is rationally required to have the same prior on Monday morning and Monday evening, for our purposes we need not rely on it.

  17. For suppose that at t1 your total evidence is E1, and at t2 you gain evidence E2 (which makes your total evidence E1 ^ E2). Then, if you have prior credence function C at both times, by Synchronic Conditionalization, your credences at t1 should be C1(−) = C(− | E1), and your credences at t2 should be C2(−) = C(− | E1 ^ E2). C2 is just the probability function that results from taking C1 and conditionalizing on E2.

  18. One important synchronic argument against Double Halfing is given in Titelbaum (2012). If one supplements Sleeping Beauty with the claim that a fair coin will also be flipped on Tuesday (which will have no effect on the rest of the experiment), then it turns out that Double Halfers must assign a credence greater than 1/2 in the proposition that ‘today’s coin flip will land Heads’ on Monday morning. In response, I grant that this a strong objection to Double Halfers who are primarily motivated by aligning their credences with the objective chances. However, this is not the relevant motivation for the Time-Slicer. The Time-Slicer is primarily motivated by having an impersonal epistemology, which is encapsulated in principles like No Reference, Synchronicity, and Impartiality. Given that the proposition ‘today’s coin will land Heads’ is a merely indexical proposition, the fact that the Double Halfer gives an unintuitive verdict on this indexical proposition shouldn’t be much of an embarrassment for the Time-Slicer.

  19. If you do not believe that Beauty ≠ Tuesday Beauty in Sorites Beauty100, then pick some other analogous soritical case where Beauty = Tuesday Beauty in Sorites Beauty0, and Beauty ≠ Tuesday Beauty in Sorites Beauty100.

  20. Of course, there are familiar externalist reasons for thinking that no epistemic theory can always give followable advice having to do with Williamson’s (2000) Luminosity argument. For a response to the Luminosity argument, see Berker (2008). For motivations to be an internalist that do not appeal to Luminosity, see Schoenfield (2015).

  21. If you have determinate intuitions for the case of X = 50, pick some other value of X for which you lack a determinate intuition.

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Acknowledgements

Thanks to Miriam Schoenfield, Jack Spencer, Roger White, and an anonymous referee for helpful feedback.

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    David Builes

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Builes, D. Time-Slice Rationality and Self-Locating Belief. Philos Stud 177, 3033–3049 (2020). https://doi.org/10.1007/s11098-019-01358-1

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