Two-Dimensional De Se Chance Deference

ABSTRACT

Standard principles of chance deference face two kinds of problems. In the first place, they face difficulties with a priori knowable contingencies. In the second place, they face difficulties in cases where you’ve lost track of the time. I provide a principle of chance deference which handles those problem cases. This principle has a surprising consequence for Adam Elga’s Sleeping Beauty puzzle.

KEYWORDS:

• Chance Deference
• De Se
• Two-Dimensionalism
• Sleeping Beauty

Acknowledgements

Thanks to Kevin Blackwell, Michael Caie, David Chalmers, Cian Dorr, Daniel Drucker, Jeremy Goodman, Jens Jäger, Barry Loewer, Anna Mahtani, Moritz Schulz, James Shaw, Juhani Yli-Vakkuri, and the referees and editor at this journal for helpful conversations, challenges, and feedback. Thanks also to audiences at the Rutgers Foundations of Probability Seminar, the Dianoia Institute of Philosophy, the University of Helsinki Formal Epistemology Workshop, the University of Toronto, and New York University.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Notes

1 Following philosophical tradition, I reserve the word ‘chance’ for objective probabilities. Throughout, my focus is on tychistic chance (like the chance of making certain observations in collapse interpretations of quantum mechanics), which I distinguish from deterministic chance (like the chance of a flipped coin landing heads in a Newtonian universe). See Gallow 2021.

2 Throughout, I will sloppily use regular quotation marks for quasi-quotation.

3 Similar cases are discussed in Schulz 2011, Nolan 2016, and Salmón 2019.

4 This isn’t Lewis’s Principal Principle, although it follows from the Principal Principle given the updating norm of conditionalization.

5 Throughout, whenever I write a schematic formula specifying what your conditional credences should be, I only mean to endorse substitution instances for which the ‘conditioning’ thought ‘$E$’ is epistemically possible. If ‘$E$’ is epistemically possible but is given a credence of zero, then the conditional credence $C(A|E)$ will only be defined relative to the additional parameter of a partition (Easwaran 2019). Lewis thought that no epistemically possible thought should be given a credence of zero, so he was not concerned with relativising conditional credences to partitions. (This required him to use infinitesimal credences—for more, see Williamson 2007, Easwaran 2014, and Hájek ms.) If we part ways with Lewis and allow that an epistemically possible thought may be given a credence of zero, then the natural partition to use in understanding Lewis’s principle is $\{\mathcal{C}{h}_t(A)=n\text{\%}\mid n\text{\%}\in [0,1]\}$.

6 Humeans about objective chance will have to revise Lewis’s principle for reasons unrelated to the problem cases I’m focused on here. See, for instance, the discussions by Thau (1994), Lewis (1994), Hall (1994), Ismael (2008), and Briggs (2009). Humeans should interpret ‘$\mathcal{C}{h}_t$’ as the definite description ‘the time $t$ objective chance function, conditioned on the proposition that it is the time $t$ objective chance function’. So understood, the principle will follow from the so-called ‘New Principle’, together with conditionalization.

7 ‘If the past contains seers with foreknowledge of what chance will bring, or time travellers who have witnessed the outcome of coin tosses to come, then patches of the past are enough tainted with futurity so that historical information about them may well seem inadmissible. That is why I qualified my claim that historical information is admissible, saying only that it is so “as a rule”’ (Lewis 1980: 274; see also Meacham 2010).

8 Lewis used the account of aboutness which he offered in his (1988). On this account, ‘$E$’ is entirely about times before $t$ (and so, as a rule, admissible) if, for any two worlds which have precisely the same history up until $t$, either ‘$E$’ is true at both or false at both.

9 See Frege 1892, Fodor 1975, Salmon 1986, Castañeda 1989, Moschvakis 1994, Chalmers 2011, Fitts 2014 and Braun 2016 among others.

10 In fact, the chance that a flipped coin lands heads is best understood as a deterministic chance, not a tychistic chance (which is my focus here). I’ll stick to coin flips in the interest of readability, but if we’re being ideally careful, we should think of the coin as a quantum system in the state $\sqrt{1/2}|\mathrm{heads}⟩+\sqrt{1/2}|\mathrm{tails}⟩$, and we should think of ‘flipping’ the coin as measuring whether it is in the state $|\mathrm{heads}⟩$ or $|\mathrm{tails}⟩$.

11 Admittedly, Lewis had adopted a much more liberal conception of inadmissibility by 2001, when he said that learning what time it is can provide you with inadmissible information about the future, ‘namely, that [you] are not now in it’ (Lewis 2001: 175). I won’t have anything to say about this view of admissibility beyond the following observation: if all it takes to have time $t$ inadmissible information is to know that it is now before $t$, then we would have inadmissible evidence about the outcome of a coin flip whenever we know that the coin flip will take place in the future. So, if we understand inadmissibility in this incredibly liberal sense, Lewis’s principle won’t constrain our credence in even this paradigm case.

12 This proposal appears in Wilson (2014: §6).

13 We may think that the doctor’s thought ‘I am sick’ is not the same as Beyoncé’s thought ‘I am sick’. However, we should not think that the doctor’s thought ‘Beyoncé is sick’ is the same as Beyoncé’s thought ‘I am sick’—for, presumably, if it were, the doctor’s thought ‘Beyoncé is sick’ would be the same as Beyoncé’s thought ‘Beyoncé is sick’. By the transitivity of sameness, Beyoncé’s thought ‘I am sick’ would be the same as her thought ‘Beyoncé is sick’. But if Beyoncé suffers from amnesia, her credence in these thoughts can differ, wherefore they must be different. So, regardless of whether or not the doctor’s thought ‘I am sick’ is the same as Beyoncé’s thought ‘I am sick’, the upshot is the same: in order for Beyoncé to defer to her doctor, she must find some way of associating her thoughts with the thoughts of her doctor.

14 Cf. Stalnaker (1978), Davies and Humberstone (1980), Kaplan (1989), and Chalmers (2004). To be clear: this is what Chalmers calls an epistemic two-dimensional array.

15 The notion of a de dicto locational surrogate therefore provides us with what Titelbaum (2008) calls ‘a context-insensitive claim’, which we can use as a surrogate for a ‘context-sensitive’ claim like ‘I am sick’.

16 To be clear: I am saying that you are at $\lambda$ iff ‘$\lambda$’ expresses a truth for you.

17 Conglomerability says that, for any thoughts, ‘$A$’ and ‘$E$’, and any collection of thoughts which are mutually exclusive and such that ‘$E$’ is a priori equivalent to $\underset{i}{\bigvee }{F}_i,$ $\mathrm{in}{f}_{i}C(A|F_i)\le C(A|E)\le \mathrm{su}{p}_{i}C(A|F_i).$When the number of possibilities is at most countably infinite, this principle follows from the assumption that your credences are a countably additive probability. I endorse conglomerability as a constraint on rational credences in general.

18 Any name or definite description which you know for sure to denote a unique time is an acceptable substituend for ‘$t$’. So, for instance, in the right circumstances, ‘5:55 Tuesday morning’ or ‘five minutes from now’ could be substituted for ‘$t$’. The same goes for the ‘$t$’ which appears in the principle Chance Deference below.

19 This follows if we assume that the objective chance function satisfies van Fraassen’s principle of Reflection (1984, 1995): for all (rigidly denoted) times $t$ and $t^{\ast }$ such that $t$ comes before $t^{\ast }$, any set of worlds $X$ and any number $n\text{\%}$, $\mathcal{C}{h}_t(X|\mathcal{C}{h}_{t^{\ast }}(X)=n\text{\%})=n\text{\%}$. For then, if $\mathcal{C}{h}_{\mathrm{mon}}$ were certain that the Tuesday chance of $[W]$ is 75%, $\mathcal{C}{h}_{\mathrm{mon}}$ would itself be 75% sure that $[W]$.

20 If you’re a Humean and this worries you, recall the discussion from footnote 6.

21 This follows from van Fraassen’s principle of Reflection, applied to the objective chances; see note 19.

22 See, for instance, Elga 2000, Dorr 2002, Arntzenius 2003, Hitchcock 2004, Horgan 2004, Weintraub 2004, and Titelbaum 2008.

23 See, for instance, Lewis 2001, Halpern 2004, Bostrom 2007, and Meacham 2008.

24 Of course, the thirder and halfer positions are not exhaustive. For one alternative, see the ‘imprecise’ suggestion discussed in Monton 2002 and defended in Singer 2014.

25 Horgan 2004 and Weintraub 2004 both observe that you learn ‘$\mathrm{Awake}$’ upon waking, and both suggest that this evidence breaks the usual connection between chance and credence.