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Titelbaum’s Theory of De Se Updating and Two Versions of Sleeping Beauty

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Abstract

In his “Relevance of Self-locating Belief” (2008), Titelbaum suggests a general theory about how to update one’s degrees of self-locating belief. He applies it to the Sleeping Beauty problem, more specifically, Lewis’s (Analysis 61(3):171–176, 2001) version of that problem. By doing so, he defends the Thirder solution to the puzzle. Unfortunately, if we modify the puzzle very slightly, and if we apply his general updating theory to the thus modified version, we get the Halfer view as a result. In this paper, we will argue that the difference between the two versions of Sleeping Beauty isn’t sufficient for justifying the different verdicts on them. Since this is a counter-intuitive result, we should reject Titelbaum’s theory of de se updating.

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Notes

  1. This problem has been pointed out by various authors (Arntzenius 2002; Halpern 2004; Hitchcock 2004; Meacham 2008; Kim 2009).

  2. The symbols, “\(\lceil \)” and “\(\rceil \),” are ours.

  3. Sometimes, we will write “\(\left\{ t_{n}\right\} _{n\in N^{\alpha }}\)”, attaching the superscript to the name of the index set rather than to that of the time set itself.

  4. This is an obvious fact because there are only four truth-functions with one sentential variable.

  5. Titelbaum himself does not use or mention the definitions or principle in this paragraph. However, this is entirely harmless, as we could rewrite every claim or argument in this paper using only Titelbaum’s own terminology and principles. In other words, they are just useful shortcuts.

  6. According to Titelbaum’s theory of updating, \(H^{*}\) is actually irrational. But, in this stage of discussion, we are trying to demonstrate the descriptive power of his modelling framework rather than to discuss any principle of rational updating.

  7. The sentence and time need to be from the domain and time set of the given model.

  8. Some philosophers accept conditonal probability as a primitive notion and define unconditional probability in terms of it (Hájek 2003). However, not all problems will be solved by this approach (Kim 2009).

  9. The former is Titelbaum’s (2008) original formulation, but we use the latter more frequently in this paper.

  10. Given this definition, we can rewrite C4 as follows: (C4′) every \(X\in L^{\alpha }\) has a doxastic equivalent \(Y\in L^{\beta }\) at any \(t_{n}\in T^{\alpha }\).

  11. This is (almost) the same claim as the right-to-left direction of Theorem E.6 in Titelbaum (2013). The only difference is the fact that the latter has “\(L^{\alpha }\supseteq L^{\alpha }\)” instead of “\(S^{\alpha }\supseteq S^{\alpha }\)” but this is ignorable thanks to ClosureSubset. See p. 324 in that book for a proof.

  12. Neither (2) nor (3) can be an analogue for \(M^{A}\) of any arithmetic statement for \(M^{A-}\), because they include “\(\lnot WAKEUP\)” and “\(\lnot MON\),” which are not included in \(L^{A-}\).

  13. Clearly, \(\left( S^{A}-S^{A-}\right) =\left\{ WAKEUP,MON\right\} \). By (2) and (3), \(P_{s}^{A}(WAKEUP\equiv {\mathbf{F}})=1.0=P_{m}^{A}(WAKEUP\equiv {\mathbf{T}})\) and \(P_{s}^{A}(MON\equiv {\mathbf{F}})=1.0=P_{m}^{A}(MON\equiv {\mathbf{T}})\). Since \({\mathbf{T}},{\mathbf{F}}\in L^{A-}, {\mathbf{T}}\) and \({\mathbf{F}}\) are the doxastic equivalents at \(t_{s}\) and \(t_{m}\) of \(WAKEUP\) and \(MON\).

  14. Titelbaum writes “Elga analyzes the Sleeping Beauty problem by adding a feature to the story. He imagines that, as a part of the experimental protocol, on each day that Beauty is awakened, the researchers chat with her for a bit and then reveal to her what day it is before putting her back to sleep...” (Titelbaum 2008, 582) However, we are not completely sure that Elga had this version of the Sleeping Beauty problem in mind, although there is some textual evidence supporting this interpretation. In this paper, we focus on Lewis’s version of the problem, remaining neutral about the exegetical issue.

  15. According to (8), \( P_{m}^{B-}\left( \lnot HEADS \& MON\right) ,P_{m}^{B-}\left( \lnot HEADS \& \lnot MON\right) >0\). It follows that \( P_{m}^{B-}\left( {\mathbf{T}} \& \lnot MON\right) ,P_{m}^{B-}\left( \lnot {\mathbf{F}} \& MON\right) >0\). Thus, \( P_{m}^{B-}\left( HEADS\not \equiv MON\right) =P_{m}^{B-}\left( \lnot HEADS \& MON\right) +P_{m}^{B-}\left( HEADS \& \lnot MON\right) >0\). Similarly, \(P_{M}^{B-}\left( \lnot HEADS\not \equiv MON\right) ,P_{M}^{B-}\left( {\mathbf{T}}\not \equiv MON\right) ,P_{M}^{B-}\left( {\mathbf{F}}\not \equiv MON\right) >0\). Therefore, \(P_{m}^{B-}\left( HEADS\equiv MON\right) ,P_{m}^{B-}\left( \lnot HEADS\equiv MON\right) ,P_{m}^{B-}\left( {\mathbf{T}}\equiv MON\right) ,P_{m}^{B-}\left( {\mathbf{F}}\equiv MON\right) <1.0\). In other words, none of \(HEADS, \lnot HEADS, {\mathbf{T}}\), and \({\mathbf{F}}\) are doxastic equivalents at \(t_{m}\) of \(MON\).

  16. By (10) and (11), \(P_{s}^{B1}(MON\equiv {\mathbf{F}})=1.0=P_{m+}^{B1}(MON\equiv {\mathbf{T}}), P_{s}^{B1}(WAKEUP\equiv {\mathbf{F}})=1.0=P_{m+}^{B1}(WAKEUP\equiv {\mathbf{T}})\), and \({\mathbf{T}},{\mathbf{F}}\in L^{B-}\).

  17. Here, we use “wake” in the atemporal sense. It will be more natural to use the past tense, but we think that this sentence expresses an intelligible proposition.

  18. Here, we use “wakes” in the atemporal sense and it does not imply that the event of waking up is occurring now.

  19. Indeed, if the coin lands tails, she will wake up three times, the first time on Monday, the second time on Tuesday, and the third on Thursday. Regarding this case, I will ignore her awakening on Monday, and call her awakening on Tuesday “the first awakening” and her awakening on Wednesday “the second awakening.” This is because her memory of Monday is erased in this case and so her experience on that day plays no role in Beauty’s updating from \(t_{s}\) to \(t_{w}\).

  20. For \(P_{s}^{C1}(WED\equiv {\mathbf{F}})=1.0=P_{w}^{C1}(WED\equiv {\mathbf{T}}), P_{s}^{C1}(WAKEUP_{BEFORE}^{1}\equiv {\mathbf{F}})=1.0=P_{w}^{C1}(WAKEUP_{BEFORE}^{1}\equiv WAKEUP_{WED}^{1})\) and \(P_{s}^{C1}(WAKEUP^{2}\equiv {\mathbf{F}})=1.0=P_{w}^{C1}(WAKEUP^{2}\equiv WAKEUP_{WED}^{2})\).

  21. It would be nicer if we proved this point in a more rigorous way, without appealing to any intuition. Unfortnately, we could not find a purely formal proof of \(C_{s}^{C1-}\subseteq C_{w}^{C1-}\). Still, the claim itself seems to be uncontroversial in the given setting. So this will do for my purpose, but if Titelbaum’s theory becomes standard, it will be desirable to have a general formal procedure for deciding whether the given model is LC-friendly.

  22. \(P_{m}^{C2}(WED\equiv {\mathbf{F}})=1.0=P_{w}^{C2}(WED\equiv {\mathbf{T}})\) and \(P_{m}^{C2}(WAKEUP^{2}\equiv {\mathbf{F}})=1.0=P_{w}^{C2}(WAKEUP^{2}\equiv WAKEUP_{WED}^{2})\).

  23. Moss (2012) provides a counterexample for Titelbaum’s theory. To solve this problem, Titelbaum modifies ProperExpansionPrinciple as follows: If \(M^{\beta }\) is a context-insensitive proper reduction of \(M^{\alpha }\), then the analogue for \(M^{\alpha }\) of any verdict of \(M^{\beta }\) is a verdict of \(M^{\alpha }\) (Titelbaum 2013, 191–200). Because of this modification, no counterintuitive verdict results from his theory regarding Moss’s example. However, he cannot deal with our example in the same manner because \(L^{C2-}\) includes only non-indexical sentences, which means that the new and old versions of ProperExpansionPrinciple apply equally well to the Sleeping BeautyC case. Of course, this is a bad news for Titelbaum because the absurd result shown in this section can be derived by using the new version, too.

  24. The same comments apply here as those we made in 21.

  25. By “think about,” we mean something like “have any credal opinion about.” In addition, you may regard UniversalInclusion as ambiguous between the “think” reading and the “may think” reading, but for my purpose, this does not matter. For my objection applies equally well to both readings.

  26. Suppose, for reductio, that \(M^{A1}\) has an LC-friendly proper reduction, say, \(M^{A1-}\). Then, its domain \(L^{A1-}\) will include none of \(MON, MON_{8AM}\), and \(WAKEUP\). To see why, focus on \(MON\) first. If \(MON\in L^{A1-}\), then \(\lnot MON\in C_{s}^{A1-}-C_{m}^{A1-}\). Because \(M^{A1-}\) is LC-friendly, \(MON\notin L^{A1-}\). By similar reasonings, \(MON_{8AM},WAKEUP\notin L^{A1-}\). By ClosureEquivalence, \(L^{A1-}=\left\{ {\mathbf{T}},{\mathbf{F}},HEADS,\lnot HEADS\right\} \). By (33), none of the four sentences can be a doxastic equivalent at \(t_{m}\) of \(MON_{8AM}\). This means that \(M^{A1-}\) is not a proper reduction of \(M^{A1}\), which contradicts the supposition. By reductio, done.

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Acknowledgments

This work was supported by the National Research Foundation of Korea (NRF) under the Grant funded by the Korean government (NRF-2012S1A5B5A01025359). I would like to thank David Etlin, Byeondoek Lee, Chris Meacham, Ilho Park, Wolfgang Schwarz, and Yeongseo Yeo for their comments and assistance. Especially, I owe many thanks to Michael Titelbaum, who kindly gave me a detailed reply to (an earlier draft of) this paper. Finally, I also want to say “Thank you” to my mother Jungsook Kim, who passed away last year.

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  1. Sunkyunkwan University, 50-7 Ogin 6gagil, Noosang-dong, APT #202, Jongro-gu, Seoul, 110-041, South Korea

    Namjoong Kim

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Kim, N. Titelbaum’s Theory of De Se Updating and Two Versions of Sleeping Beauty. Erkenn 80, 1217–1236 (2015). https://doi.org/10.1007/s10670-015-9721-6

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