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Sleeping Beauty and the Evidential Centered Principle

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Abstract

Since Elga published his “Self-locating belief and the Sleeping Beauty problem,” there has been an intense debate about which credence between 1/2 and 1/3 Beauty should assign to (H) the coin’s landing heads, when she is awakened on Monday. The Halfers claim that she ought to assign 1/2 to H at that moment. The Thirders argue that she ought to assign 1/3 to H then. Meanwhile, Pettigrew defended a new chance-credence coordination principle, called the “Evidential Temporal Principle” (ETP), in a recent edition of his book. In this paper, I provide a novel argument against the Halfer view and a partial defense of the Thirder view, respectively based on ETP. In addition, I generalize ETP to what I call the “Evidential Centered Principle,” which links one’s credences in centered propositions with their objective chances.

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Notes

  1. What actually happened was more complicated. In his (2012) work, Pettigrew presented a schematic accuracy-based argument for several chance-credence coordination principles, including Lewis’s (1980) Principal Principle. Then, Caie (2015) pointed out that, strictly speaking, Pettigrew’s argument fails to establish PP, but a modified principle, called “Temporal Principle,” would be defensible by a similar argument. Later, in a new edition of his book (2016), Pettigrew embraced Caie’s objection and suggested further generalizing TP, the result of which was ETP.

  2. See Titelbaum (2013) for a survey of the arguments for either position.

  3. As an anonymous reviewer points out, this implies that (IMP) a tensed proposition is true at t in w only if there is an individual located at t in w. This is problematic. Intuitively, for some moment t and world w, there is no individual located at t in w; for, we can imagine an empty world after everything is destroyed. So \(\llbracket \text {Nothing exists}\rrbracket\) is true at t in w. By IMP, there is some individual located at t in w. Contradiction. Seemingly, the above definition of a tensed proposition’s truth condition conflicts with our intuition. However, I am not convinced, because what is really intuitive is not that there might have been no individual whatsoever, but that there might have been no concrete individual. Perhaps we should count a temporal moment as a non-concrete individual while restricting the domain of ordinary quantifiers (e.g., “all,” “some,” and “nothing”) to concrete individuals. Then, for some moment t and some world w,  there is no concrete individual located at t in w (so that \(\llbracket \text {Nothing exists}\rrbracket\) is true at t in w), but there must be an individual located at t in w,  namely, t. This result is neither contradictory nor conflicting with IMP. Interestingly, Lewis’s Modal Realism (LMR) suffers from a similar problem: Intuitively, for some world w,  no individual exists in w but, according to LMR, there must be an individual in w because w is a maximal mereological sum of spatiotemporally connected individuals. Lewis’s solution is analogous to mine: “A world is not like a bottle that might hold no beer. The world is the totality of things it contains, so even if there’s no beer, there’s still the bottle” (1986, p. 73).

  4. See footnote 1.

  5. Caie uses “\(X^{t}\)” to mean the same thing, but I introduced a similar notion in an earlier paper (Kim, 2009) and, in the present paper, I wish to keep using my own notation.

  6. Notice that P,  S,  and R are proper tensed propositions.

  7. As Katherine knows at u that precipitation is already occurring, she also knows that \(tch_{n}\left( R|P\right) =tch_{n}\left( R\right) =0.4\). Similarly for \(tch_{d}\left( \cdot \right) .\)

  8. See Gaifman (1988) and Hall (1994) for general discussions of deferrence. See Kim (2009) for a discussion of a similar temporal mismatch between an agent’s prior and posterior credence functions.

  9. Currently, Pettigrew prefers a more decision-theoretic argument for ETP. See Pettigrew (2016, Chapter 6) for details.

  10. An omniscient function \(v_{w}:{\mathfrak {F}}^{G}\rightarrow \left\{ 0,1\right\}\) is the function \(v_{w}\) that assigns 1 to X if X is true in w and assigns 0 to X if X is false in w, for any \(X\in {\mathfrak {F}^G}\).

  11. If \({\mathcal {V}}_{{\mathfrak {F}}^{T}}\) is infinite, it will be necessary to substitute \(cl\left( {\mathcal {V}}_{{\mathfrak {F}}^{T}}^{+}\right)\), the smallest closed set containing \({\mathcal {V}}_{{\mathfrak {F}}^{T}}^{+}\), for \({\mathcal {V}}_{{\mathfrak {F}}^{T}}^{+}\) in this proof (Pettigrew, 2016, p. 91).

  12. See Bradley (2019) for a survey.

  13. Proof: Consider any \(c\in {\mathcal {V}}_{{\mathfrak {F}}^{T}}^{+}\). By definition, (i) \(c\left( \cdot \right) =\sum _{1\le j\le n}\lambda _{j}\cdot p^{j}\left( \cdot |E\right)\). By assumption, (ii) \(p^{i}\left( \left[ p^{i}\left( \cdot \right) =tch_{pres}\left( \cdot \right) \right] |E\right) =1\) for any \(i=1,\ldots ,n.\) Thus, (iii) \(p^{i}\left( X \& \left[ p^{i}\left( \cdot \right) =tch_{pres}\left( \cdot \right) \right] |E\right) =p^{i}\left( X|E\right)\) for any \(i=1,\ldots ,n\) and any \(X\in {\mathfrak {F}}^{T}.\) Whenever \(i\ne j\), \(\left[ p^{i}\left( \cdot \right) =tch_{pres}\left( \cdot \right) \right]\) and \(\left[ p^{j}\left( \cdot \right) =tch_{pres}\left( \cdot \right) \right]\) are mutually incompatible and, by (ii), \(p^{i}\left( \left[ p^{j}\left( \cdot \right) =tch_{pres}\left( \cdot \right) \right] |E\right) =0\). Thus, (iv) \(p^{j}\left( X \& \left[ p^{i}\left( \cdot \right) =tch_{pres}\left( \cdot \right) \right] |E\right) =0\) for any \(X\in {\mathfrak {F}}^{T}\) whenever \(i\ne j\). Hence, for any \(X\in {\mathfrak {F}}^{T}\),

    $$\begin{aligned} \begin{array}{cllcc} &{} c\left( X|\left[ p^{i}\left( \cdot \right) =tch_{pres}\left( \cdot \right) \right] \right) \\ = &{} \frac{\sum _{1\le j\le n}\lambda _{j}\cdot p^{j}\left( X \& \left[ p^{i}\left( \cdot \right) =tch_{pres}\left( \cdot \right) \right] |E\right) }{\sum _{1\le j\le n}\lambda _{j}\cdot p^{j}\left( p^{i}\left( \cdot \right) =tch_{pres}\left( \cdot \right) |E\right) } &{} \text { (by (i))}\\ = &{} \frac{p^{i}\left( X \& \left[ p^{i}\left( \cdot \right) =tch_{pres}\left( \cdot \right) \right] |E\right) }{p^{i}\left( \left[ p^{i}\left( \cdot \right) =tch_{pres}\left( \cdot \right) \right] |E\right) } &{} \text { (by (iv))}\\ = &{} p^{i}\left( X|E\right) . &{} \text { (by (ii) and (iii))} &{} &{} \text {Done.} \end{array} \end{aligned}$$

  14. Proof: It suffices to show that, when the domains of the chance and credence functions are restricted to tensed propositions, ECP and ETP are equivalent. Consider any \(X,E\in {\mathfrak {F}}^{T}\) and \(p\left( \cdot \right) \in \mathcal{PF}\mathcal{}\left( {\mathfrak {F}}^{T}\right)\). Clearly, for any \(Y\in {\mathfrak {F}}^{T},\) \(\left( {Y}\text { at }{\left\langle me,pres\right\rangle }\right)\) is equivalent to \(\left( {Y}\text { at }{pres}\right) .\) By definition, \(cch_{\left\langle me,pres\right\rangle }\upharpoonright {\mathfrak {F}}^{T}\) (i.e., the restriction of \(cch_{\left\langle me,pres\right\rangle }\left( \cdot \right)\) to tensed propositions) is identical to \(tch_{pres}\left( \cdot \right) .\) Hence, \(c_{t_{k}}\left( X|\left[ p\left( \cdot \right) =\left( cch_{\left\langle me,pres\right\rangle }\upharpoonright {\mathfrak {F}}^{T}\right) \right] \right) =c_{t_{k}}\left( X|\left[ p\left( \cdot \right) =tch_{pres}\left( \cdot \right) \right] \right)\). Therefore, \(c_{t_{k}}\left( X|\left[ p\left( \cdot \right) =\left( cch_{\left\langle me,pres\right\rangle }\upharpoonright {\mathfrak {F}}^{T}\right) \right] \right) =p\left( X|E\right)\) iff \(c_{t_{k}}\left( X|\left[ p\left( \cdot \right) =tch_{pres}\left( \cdot \right) \right] \right) =p\left( X|E\right) .\) Done.

  15. Several authors have discussed this variant of Sleeping Beauty. See Titelbaum (2013, endnote 31) for a reference list.

  16. ETP can impose an indirect constraint on the relation between one’s credal states at different moments. For example, ETP puts separate constraints on Beauty’s credal states on Sunday night and on Monday morning so that she is allowed to assign only \(\frac{1}{2}\) to H on Sunday night and only \(x<\frac{1}{2}\) on Monday morning. As a(n indirect) result, she ought to decrease her credence in H.

  17. However, see Kim (2009).

  18. For my purposes, it does not matter how the credence in a possible world w is split and assigned to the centered worlds associated with w (that is, centered worlds \(\left\langle \cdot ,\cdot ,w \right\rangle\)).

  19. As an anonymous reviewer points out, there are other updating models in tension with ETP. Following Conitzer (2015), let us introduce “\(\text {see }\phi\)” as a new notation. Judging from how he uses it, I take it that \((\text {see }\phi )=\llbracket \phi \text { is observed at some moment}\rrbracket .\) Assume that, for any possible world \(w_i\) in your old doxastic space \(\{w_1,\ldots ,w_n\}\) at \(t_k\), you previously assigned the same credence to every centered world \(\left\langle \cdot ,\cdot ,w_{i}\right\rangle .\) Now consider the following diachronic norm:

    (Halfer* Rule) \(c_{t_{k+1}}(G_i)= \frac{c_{t_k}(\text {see }E|G_i)c_{t_k}(G_i)}{c_{t_k}(\text {see }E|G_1)c_{t_k}(G_1)+\cdots +c_{t_k}(\text {see }E|G_n)c_{t_k}(G_n)},\)

    where E is your total evidence at \(t_{k+1}\) and, for any \(i\in \{1,\cdots ,n\},\) \(G_i=\{\left\langle \cdot ,\cdot ,w_i \right\rangle \in W\}.\) This supports the Halfer view because, on Sunday night, Beauty surely expected to receive WK as evidence at some moment whether the H-world or T-world is actual (formally, \(c_s(\text {see }WK|H)=c_s(\text {see }WK|T)=1\)). Thus, the Halfer* Rule is incompatible with ETP. Moreover, Conitzer (2015) convincingly argues that the Halfer Rule is vulnerable to a devastating counterexample but the Halfer* Rule is not. Thus, the Halfer* Rule appears to be a stronger challenger to ETP. Nonetheless, in this section, I discuss the Halfer Rule instead because it is much easier to understand. However, be warned: The reader should not simply conclude that, since there is a counterexample to the Halfer Rule, ETP has no serious problem despite the incompatibility between ETP and the Halfer Rule. The Halfer* Rule is also incompatible with ETP and not obviously vulnerable to any counterexample.

  20. See Curtis and Robson (2015) Chapters 4 and 5 for the introductory discussion and reference list on A-theory, the moving spot theory, and presentism.

  21. See Skow (2009) for a recent defense of the theory. See Curtis and Robson (2015) Section 4.2 for a general introduction.

  22. See Merlo (2016) for a similar view with an interesting twist.

References

  • Bradley, S. (2019). Imprecise probabilities. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy (Spring 2019 Edition). Retrieved August 26, 2022 from https://plato.stanford.edu/entries/imprecise-probabilities/

  • Caie, M. (2015). Credence in the image of chance. Philosophy of Science, 82(4), 626–648.

    Article  Google Scholar 

  • Conitzer, V. (2015). A devastating example for the Halfer Rule. Philosophical Studies, 172(8), 1985–1992.

    Article  Google Scholar 

  • Conitzer, V. (2020). The personalized A-theory of time and perspective. Dialetica, 74(1), 1–30.

    Google Scholar 

  • Curtis, B., & Robson, J. (2016). A critical introduction to the metaphysics of time. London: Bloomsbury Academic.

    Google Scholar 

  • Elga, A. (2000). Self-locating belief and the sleeping beauty problem. Analysis, 60, 143–147.

    Article  Google Scholar 

  • Gaifman, H. (1988). A theory of higher order probabilities. In B. Skyrms & W. L. Harper (Eds.), Causation, chance and credence (pp. 191–219). Dordrechet: Springer.

    Chapter  Google Scholar 

  • Hall, N. (1994). Correcting the guide to objective chance. Mind, 103(412), 505–518.

    Article  Google Scholar 

  • Hare, C. (2007). Self-bias, time-bias, and the metaphysics of self and time. Journal of Philosophy, 104(7), 350–373.

    Article  Google Scholar 

  • Kim, N. (2009). Sleeping Beauty and shifted Jeffrey condtionalization. Synthese, 168(2), 295–312.

    Article  Google Scholar 

  • Lewis, D. K. (1980). A subjectivist’s guide to objective chance. In R. C. Jeffrey (Ed.), Studies in inductive logic and probability (Vol. II, pp. 263–293). Berkeley: University of California Press.

    Chapter  Google Scholar 

  • Lewis, D. K. (1986). On the plurality of worlds. Oxford: Basil Blackwell.

    Google Scholar 

  • Lewis, D. K. (1994). Humean supervenience debugged. Mind, 103(412), 473–490.

    Article  Google Scholar 

  • Lewis, D. K. (2001). Sleeping beauty: Reply to Elga. Analysis, 61, 171–176.

    Article  Google Scholar 

  • Merlo, G. (2016). Subjectivism and the mental. Dialectica, 70(3), 311–342.

    Article  Google Scholar 

  • Pettigrew, R. (2012). Accuracy, chance, and the principal principle. The Philosophical Review, 121(2), 241–275.

    Article  Google Scholar 

  • Pettigrew, R. (2016). Accuracy and the laws of credence. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Skow, B. (2009). Relativity and the moving spotlight. Journal of Philosophy, 106(12), 666–678.

    Article  Google Scholar 

  • Titelbaum, M. G. (2013). Ten reasons to care about the sleeping beauty problem. Philosophy Compass, 8(11), 1003–17.

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the Gyeongsang National University Fund for Professors on Sabbatical Leave, 2021. I am grateful to Ilho Park and Darrel Rowbottom for their helpful comments on early drafts of this paper and to Misook Kim for helping with the editing of the final draft. In addition, I am grateful to the anonymous reviewers for their interesting and sometimes challenging comments. Finally, I would like to thank my dissertation supervisor, Phillip Bricker, who will retire this year. He not only taught me how to do good philosophy but also showed me how to become a better person. Thanks, and congratulations.

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    Namjoong Kim

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Kim, N. Sleeping Beauty and the Evidential Centered Principle. Erkenn (2022). https://doi.org/10.1007/s10670-022-00619-6

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