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The puzzle of the unmarked clock and the new rational reflection principle

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Abstract

The “puzzle of the unmarked clock” derives from a conflict between the following: (1) a plausible principle of epistemic modesty, and (2) “Rational Reflection”, a principle saying how one’s beliefs about what it is rational to believe constrain the rest of one’s beliefs. An independently motivated improvement to Rational Reflection preserves its spirit while resolving the conflict.

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Notes

  1. Compare to Christensen (2007, 2010).

  2. Here and elsewhere I use talk of “beliefs” as shorthand for talk of degrees of belief.

  3. The clock example is due to Williamson (2007), as adapted by Christensen (2010, pp. 122–125), whose discussion I follow in this section. “Irritatingly austere” is from Williamson (2010, p. 13).

  4. Compare to Christensen (2010, p. 124): “it seems that [a subject looking at the clock] should think that .3 is probably too high a credence for her to have that [the clock reads a particular time], and certainly not too low.”

  5. This example is based on the case of Brayden from Christensen (2010, pp. 121–122).

  6. This constraint is a more cautious cousin of the “principle of non-akrasia” from Ross (2006, p. 277).

  7. RATIONAL REFLECTION is a variant of the principle “RatRef” from Christensen (2010, p. 122). A similar principle is introduced in Ross (2006, § 10.3) under the name “The epistemic principal principle”. Earlier work on related principles includes Goldstein (1983), Lewis (1986), van Fraassen (1984).

  8. To avoid complications, here and below I ignore self-locating beliefs, assume that every situation determines a unique ideally rational probability function, and assume that credences are countably additive and defined over a space of possibilities that is at most countably infinite.

  9. For readers familiar with the Reflection Principle from van Fraassen (1984), the Principal Principle from Lewis (1986), or RatRef from Christensen (2010), it may be helpful to note that RATIONAL REFLECTION entails

    RATREF P(H | I(H) = x) = x

    whenever P is the credence function of a possible rational subject S, H is a proposition, x is a real number, “I(H) = x” denotes the proposition that the ideal probability for S to have in H is x, and the conditional probability is well defined.

  10. Cf. Elga (2008), Christensen (2010).

  11. Williamson (2010, Appendix) proves related results, delivering conditions for the necessary truth of RatRef (a reflection principle closely related to RATIONAL REFLECTION—see note 7).

  12. As was pointed out to me by Michael Titelbaum, the proof in the text is analogous to the proof that the “Old” Principal Principle is in tension with Humeanism about laws of nature (Hall 1994; Lewis 1994).

  13. One might try to avoid the conflict by advocating not RATIONAL REFLECTION but the very similar principle RatRef (described in footnote 7). This does not work because it is unreasonable to accept both RatRef and MODESTY. That is because it is unreasonable to accept MODESTY without also accepting the following claim, which itself is incompatible with RatRef:

    • POSITIVE It can be rational to think something of the form “I’m not sure how confident I should be that P′ is ideal. Maybe I should be 20 % confident that it is, maybe 21 %, or maybe another value.” (Here 20 and 21 % are placeholders for any two positive values, and P′ can be any credence function.)

    Proof that POSITIVE is incompatible with RatRef: For brevity, use “V(H, v)” to denote the proposition that the ideal credence to have in H is v, and use “I(P′)” to denote the proposition that P′ is the ideally rational credence function to have. Now suppose POSITIVE is true. Then for some possible rational person who has credence function P, there are distinct positive values x and x′ such that P(V(I(P′), x)) and P(V(I(P′), x′)) are both greater than 0. But I(P′) is compatible with at most one of V(I(P′), x) and V(I(P′), x′), since it settles what the ideal probability to have in I(P′) is. So at least one of P(I(P′)|V(I(P′), x)) and P(I(P′)|V(I(P′), x′)) equals zero. But RatRef entails that these two terms equal x and x′ respectively, which are both positive. So POSITIVE contradicts RatRef.

  14. Compare to Christensen (2010, p. 124).

  15. Alternatively, we might reject the second sentence of MODESTY by saying that rationality requires one to be certain of the following falsehood: rationality requires one to be certain exactly what time the clock indicates. That seems just as desperate, since a rational viewer of the clock might well realize that she has an imperfect ability to discriminate nearby times. Thanks here to Kenny Easwaran.

  16. Williamson (2010) seems sympathetic to this approach.

  17. This is proposed as a fallback position in Christensen (2010, p. 136).

  18. It may be defensible, for example, if we are forced to accept a similar conclusion about independent cases, as is argued in Christensen (2007).

  19. Compare: “the ideal outcome of thinking about [the puzzle of the clock] would be our finding a way of accommodating the intuitions behind [a principle similar to RATIONAL REFLECTION] while avoiding the difficulties we’ve been examining” (Christensen 2010, p. 136).

  20. The line of reasoning, including the example involving the panel of experts, is due to Hall (1994, p. 510). Christensen (2010, p. 135) comes tantalizingly close to applying such reasoning to RATIONAL REFLECTION, but pulls back at the last moment.

  21. The constraint is named NEW RATIONAL REFLECTION because it stands to RATIONAL REFLECTION as the “New Principal Principle” stands to the “Principal Principle” (Lewis 1994; Hall 1994). (NEW RATIONAL REFLECTION also stands to RATIONAL REFLECTION as the “guru principle” from Elga (2007) stands to the “Reflection Principle” from van Fraassen (1995).) The derivation from footnote 23 of a special case of NEW RATIONAL REFLECTION can be adapted to yield a parallel derivation of a special case of the New Principal Principle.

  22. This second strategy adapts an idea from Ross (2006, pp. 277–299). A similar argument was independently suggested to me by Boris Kment.

  23. The special case: a rational agent has probability function P at time 0, realizes that rational agents update their beliefs by conditionalization, and realizes that she is about to conditionalize on the truth of what probability function is ideal-for-her-at-time-0. Together with CERTAIN, these assumptions entail that the agent satisfies the instance of NEW RATIONAL REFLECTION that applies to her.

    Proof: Under the above conditions, suppose that agent conditionalizes on the information that probability function P′ is ideal-for-her-at-time-0. As a result, at time 1 her new probability function is P(− |P′ is ideal-at-0). By the assumption that rational agents conditionalize, this new probability function is ideal-for-her-at-time-1.

    But at time 1, the agent is certain that P′ was ideal-for-her-at-time-0. And she is certain that at time 0 she remained ideally rational by conditionalizing on the truth about what function was ideal-for-her-at-time-0. So she is at time 1 certain that the following probability function is ideal-for-her-at-time-1: P′(− |P′ is ideal-at-0).

    Now CERTAIN applies to the agent at time 1, and entails that the agent’s probability function at time 1 is P′(− |P′ is ideal-at-0). We have now derived two expressions for the agent’s probability function at time 1. Equating them yields that for any proposition H:

    $$ P(H|P^{\prime}\,\hbox{is ideal-at-0}) = P^{\prime}(H|P^{\prime}\,\hbox{is ideal-at-0}), $$

    which is an instance of NEW RATIONAL REFLECTION.

  24. Nothing substantial would change if the 33/33/33 % distribution were changed to one that favored 12:17 over the other two times.

  25. For an independent and complementary diagnosis, see Horowitz and Sliwa (2011).

  26. It follows by the probability calculus that Pangloss’s credence in S is a weighted average of P G (S|99 %is ideal) and P G (S| 66 %is ideal). So this weighted average equals 99 %. But since each of these terms is no greater than 100 %, and since they are weighted roughly equally in the average, the only way for their average to be 99 % is for each of them to be close to 99 %.

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Acknowledgments

Thanks to David Christensen, Paulina Sliwa, Sophie Horowitz, Maria Lasonen-Aarnio, Michael Titelbaum, Jenann Ismael, participants in the 2011 Brown Epistemology workshop, the Corridor Group, the Princeton Formal Epistemology reading group, and the 2012 Bellingham Summer Philosophy Conference, an audience at Stanford University, and especially Joshua Schechter.

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    Adam Elga

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Elga, A. The puzzle of the unmarked clock and the new rational reflection principle. Philos Stud 164, 127–139 (2013). https://doi.org/10.1007/s11098-013-0091-0

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