Two-Dimensional Deference J. Dmitri Gallow ∗ Abstract Principles of expert deference say that you should align your credences with those of an expert. This expert could be your doctor, your future, better informed self, or the objective chances. These kinds of principles face difficulties in cases in which you are uncertain of the truth-conditions of the thoughts in which you invest credence, as well as cases in which the thoughts have different truth-conditions for you and the expert. For instance, you shouldn't defer to your doctor by aligning your credence in the de se thought 'I am sick' with the doctor's credence in that same de se thought. Nor should you defer to the objective chances by setting your credence in the thought 'The actual winner wins' equal to the objective chance that the actual winner wins. Here, I generalize principles of expert deference to handle these kinds of problem cases. [Note: this paper is long, but much of it may be skipped by the casual reader. I've colored passages which may be ignored in gray.] 1 Introduction I have opinions about how likely various things are. I think that it's unlikely to snow inMelbourne, that a flipped coin is just as likely to land heads as tails, and that global leadership is unlikely to take serious steps to address climate change. Others have these kinds of opinions, too. For instance, weather reporters have opinions about how likely it is to snow in Melbourne; the objective chances have opinions about how likely it is that a flipped coin will land heads;1 andmy future, better informed, selves have opinions about how likely global leadership is to take serious steps to address climate change. Principles of expert deference say that I should treat the opinions of these experts-the weather reporters, the objective chances, or my future, better informed self-as a particularly strong Draft of September 4, 2020. Please do not cite without permission. Comments appreciated. B: dmitri.gallow@gmail.com ∗ Thanks to James Shaw and Daniel Drucker for helpful conversations on this material. 1. I speak about the objective chances as though they were people with probabilistic opinions. This is a satisfying metaphor, though I hope it's clear that I do not take the metaphor at all seriously. 1 Two-Dimensional Deference kind of evidence. Roughly, given that one of these experts thinks that 'p' is n% likely, I too should think that 'p' is n% likely.2 Principles like these are plausible, but their plausibility depends upon an implicit restriction on the thoughts which we are permitted to substitute for 'p'. It is plausible that, when it comes to matters of my health, I should defer to my doctor. Let 'p' be 'I am sick'. Presumably, 'p' concerns my health. But, given that my doctor thinks 'I am sick' is n% likely, I shouldn't think that 'I am sick' is n% likely. After all, when my doctor thinks 'I am sick', she thinks that she is sick, not that I am sick. Or let 'p' be 'It is now Monday'. On Sunday, I can know that one of my future, better informed selves will think that 'It is now Monday' is nearly 100% likely. But it doesn't follow that, on Sunday, I should think that 'It is now Monday' is nearly 100% likely. These substituends for 'p' are about matters de se et nunc. They contain concepts like 'I' and 'now', and therefore concern who I am, and where I am located in space and time. But similar concerns arise when 'p' has nothing to do with self-location. Consider the following example, from Hawthorne & Lasonen-Aarnio (2009): tomorrow, we will draw one of 100 names from an urn, and the person whose name is drawn will win a prize. Before the draw takes place, we introduce the name 'Lucky' for the person whose name is actually drawn. We don't yet know the truth-conditional content of 'Lucky wins'. If Sabeen actually wins, then the truth-conditional content of 'Lucky wins' is that Sabeenwins. If Evin actually wins, then the truth-conditional content of 'Lucky wins' is that Evin wins. Even though we don't know what the truth-conditions of 'Lucky wins' are, we know for sure that those truth-conditions have a 1% chance of being satisfied. Whoever Lucky is, they have a 1% chance of winning the prize-same as everyone else. So it appears that objective chance thinks 'Lucky wins' is 1% likely. But surely we should think 'Lucky wins' is nearly 100% likely. It is, after all, a priori knowable that, if anybody wins, then Lucky wins.3 What thoughts like 'I am sick' and 'Lucky wins' have in common is that their truth-conditions depend upon who entertains them, when and where they entertain them, or what the world is like when they entertain them. If 'I am sick' is entertained by me, then it has the truth-conditional content that Dmitri is sick. If it is entertained bymy doctor, then it has the truth-conditional 2. I enclose a thought in single quotation marks to form a name for the thought. When using a schematic variable, p, I will use "p" for the result of writing ' ' ', followed by the substituend of 'p', followed by ' ' '. (That is: I'll sloppily use single quotation marks for quasi-quotation.) 3. Similar cases are discussed in Titelbaum (2015a), Nolan (2016), and Salmón (2019). Youmay think that introducing a name like 'Lucky' has provided uswith some 'inadmissible' information. I'll address this in §3.1 below. 2 §1 Introduction content that she is sick. If 'Luckywins' is entertained in aworld inwhich Sabeen wins, then it has the truth-conditional content that Sabeen wins. If it is entertained in a world in which Evin wins, then it has the truth-conditional content that Evin wins. Several authors have modeled thoughts like 'I am sick' and 'Lucky wins' with a two-dimesional semantics.4 As I'll be understanding it here, the first dimension models your a priori ignorance of who you are, when and where you are, and what the world is like. The second dimension models how the truth-conditional content of your thoughts depends upon who you are, when and where you are, and what the world is like. If we confine our attention to thoughts like 'Somebody is sick onAugust 16, 2020', this second dimensionwill be uninteresting. These thoughts all have the same truth-conditions, nomatter who, when, or where you are, and no matter what the world is like. Principles of expert deference work well when we only concern ourselves with thoughts like these. They face difficulties when we begin to entertain thoughts like 'I am sick' and 'Lucky wins'-thoughts whose truth-conditions vary depending upon who, when, and where you are, or what the world is like. Here, I will introduce and explore an emendation of principles of expert deferencewhich allows them to deal with thesemore interesting thoughts. This emendation comes in two parts. Standard principles of expert deference say that you should align your opinions about 'p' with the expert's opinions about 'p'. However, when it comes to thoughts like 'I am sick', I should not align my opinion with my doctor's opinions about 'I am sick', but instead with her opinion about some other thought-a surrogate for 'p'. The first part of the emendation concerns this surrogate. The second part of the emendation will be required to deal with expert deference in cases in which uncertainty about who, when, or where you are has led to uncertainty about the truth-conditions of your thoughts. Roughly, I'll deal with these cases by suggesting that you should align your opinions with those of the expert, conditional on the expert's opinions, conditional on who you both are, and conditional on when andwhere you both are located in space and time. This emendation will reduce to the familiar principles of expert deference when you only have opinions about boring thoughts like 'Somebody is sick on August 16, 2020'. But, once you have some opinions about interesting thoughts like 'I am sick', the emendation will disagree with the familiar principles, even when it comes to the boring thoughts. I'll illustrate this with a discussion of Elga (2000)'s Sleeping Beauty puzzle. Lewis (2001) took his principle of 4. See, in particular, Kaplan (1978, 1989), Stalnaker (1978), Evans (1979), Davies & Humberstone (1980), Jackson (1998), and Chalmers (2006a,b). 3 Two-Dimensional Deference chance deference tomilitate against Elga's 'thirder' solution to that puzzle. However, once I've emended his principle of chance deference in the ways adumbrated above, it will be perfectly consistent with Elga's 'thirder' solution. However, it will be inconsistent with Lewis's own 'halfer' solution. 2 Credences andThoughts Theprinciples of expert deference I'm concernedwith here focus on the kind of opinion you express when you say that Fermat probably didn't have a proof of Fermat's last theorem.5 There are multiple ways of understanding claims about what's likely or probable. We could understand them as claims about what's objectively likely, or as claims about what's subjectively likely. I'll reserve the word 'chances' for objective probabilities. Subjective judgments of likelihood, on the other hand, I'll call 'credences'. Understood as a claim about credence, when you say that Fermat probably didn't have a proof of Fermat's last theorem, you aren't saying that the objective chance of him having a proof is low. Instead, you are simply expressing your own confidence that he didn't have a proof. I'll represent your credenceswith a function,C, from thoughts to real numbers between 0 and 1. When I say that you're quite confident that Fermat didn't have a proof of Fermat's last theorem, I mean that you have a high credence in the thought you would express with the sentence 'Fermat didn't have a proof of Fermat's last theorem'. I'll call the real number C(p) 'your credence in the thought 'p", or 'your credence that p'. Of course, strictly speaking, your credence in a thought is a psychological state, and not the abstract number we use to represent it. Likewise, your credences are not strictly speaking a function; instead, they are represented with a function. But I'll adopt this way of speaking as a convenient shorthand. In the foregoing, I amusing 'thoughts' as a technical term for the arguments of your credence function; they are those things to which you assign degrees of confidence. I'll suppose that, in many cases, at least, one and the same thought may be entertained by different people and at different times, that thoughts can be true or false, and that it makes sense to talk about the negation of a thought, p, which I'll write '∼p', as well as the disjunction and conjunction of two thoughts, p and q, which I'll write 'p ∨ q' and 'p ∧ q', respectively. I will also assume that a thought, together with the information of who entertains the thought, when and where, and in what possible world, determines the truthconditions of that thought (that is: the set of possible worlds, or locations in possible worlds, in which the thought is true)-more on this below. Beyond 5. For a principle of expert deference which concerns 'full' or 'outright' belief, see Zagzebski (2012). 4 §2 Credences andThoughts these assumptions, I hope to remain officially neutral on what a thought is.6 But, just to orient the reader, let me say something brief about the menu of options. (The uninterested reader may skip ahead to §3.) Some terminology: as I'll use the term here, a proposition is the referent of a 'that'-clause in an attitude ascription. So the referent of 'that Fermat didn't have a proof ' in an attitude ascription like 'John suspects that Fermat didn't have a proof ' is a proposition. Some hold that propositions are fine-grained, in the following sense: the referent of 'that Twain is gifted humorist' is a different proposition than the referent of 'that Clemens is a gifted humorist'.7 Others want to identify these two propositions. Call the first group fine-grainers, and the second, coarse-grainers. Coarse-grainers say that, if you believe that Twain is a gifted humorist and yet disbelieve that Clemens is a gifted humorist, then you both believe and disbelieve one and the same proposition.8 Nonetheless, coarse-grainers will want to allow that you may still be rational (after all, you may not know that 'Twain' and 'Clemens' co-refer). They will want to distinguish your rational belief state from an irrational state of believing that Twain both was and was not a gifted humorist. To do this, they will appeal to the notion of a guise. A guise is a way of being acquainted with a proposition. For coarse-grainers, when you bear an attitude to a proposition, you do so under some guise or other. The reason you can rationally believe and disbelieve one and the same proposition is that the guise under which you believe it is distinct from the guise under which you disbelieve it. A coarse-grainer will distinguish thoughts from propositions. For I have supposed that your credences are a function from thoughts to real numbers- or, equivalently, a one-one relation between thoughts and real numbers. It follows that you cannot give one and the same thought two different credences: if C(p) , C(q), then p , q. But, according to the coarse-grainer, you can give one and the same proposition two different credences. For instance, you can be confident that Twain is a gifted humorist but not very confident that Clemens is. So, if you are a coarse-grainer, you should individuate thoughts by something other-or something more-than their propositional content. I see two natural suggestions for the coarse-grainer: thoughts could be guises, or they could be guise-proposition pairs (where the paired proposi6. Notice, in particular, that nothing I'll say here requires thoughts to be narrow-that is, nothing I'll say here requiresme to suppose that which thought you entertain strongly locally supervenes upon your qualitative intrinsic properties (see Yli-Vakkuri & Hawthorne 2018). 7. This is the view of Frege (1892). The most prolific contemporary defender is Chalmers (2004, 2006a,b). 8. When I say that you disbelieve that p, I mean you believe that ∼p. 5 Two-Dimensional Deference tion is the one you entertain via that guise).9 Braun (2016) opts for the second option, though from my perspective, the first is more attractive. On a coarse-grained view of propositions, rational credence has everything to do with the guises under which propositions are entertained, and nothing to do with the propositions thereby entertained. Just to illustrate the point, consider a coarse-grained view on which propositions are individuated by their truth-conditions-that is, a proposition is individuated by the set of metaphysically possible worlds in which the proposition is true. Take any thought, 'p', and let 'P ' be the proposition which your thought 'p' expresses. So long as 'p' is true, there is a guise under which you may be rationally certain that P . To appreciate this, first note that there is a guise under which you can be rationally certain in the maximally strong true truth-conditional content-the truth-conditional content which is true at the actual world, and false at every other possible world. For it is a priori that a thought is true iff it is actually true: ∀q(q↔ @q) (here, I am quantifying over thoughts, and writing '@q' for 'actually, q'). Since '∀q(q ↔ @q)' is a priori, it is rational to be certain that ∀q(q ↔ @q). But '∀q(q ↔ @q)' is true at the actual world and false at every other possible world.10 Now, so long as the thought 'p' is true, 'p' and 'p ∨ ∀q(q↔@q)' have the very same truth-conditions. Since it's rational to be certain of '∀q(q↔ @q)', and rational certainty is preserved by a priori knowable entailment, it's also rational to be certain of 'p ∨ ∀q(q ↔ @q)'. So you can be rationally certain of a thought with the same truth-conditions as 'p', for any true 'p'. (I owe this observation to Gibbard, 2012, appendix 1; in the same appendix, Gibbard shows that similar results hold for other kinds of coarsegrainers.) So, from my perspective, when it comes to rational credence, it's most natural for a coarse-grainer to think that propositions are an idle wheel, and so to identify thoughtswith guises, not guise-proposition pairs. (Of course, propositions have other important roles to play for the coarse-grainer. They are needed; they're just not needed in the domain of your credence function.) Whether a coarse-grainer identifies thoughtswith guises or guise-proposition pairs maymake a difference to the book-keeping in what follows. Consider the guise associated with my thought 'I am now sick'. Suppose, for the sake of illustration, that both my future self and my doctor are also capable of entertaining a proposition under this guise-though that guise will determine different propositions for me, my future self, and my doctor. For me, the guise determines the proposition that Dmitri is sick on August 16, 2020; whereas, for my future self, it determines the proposition that Dmitri is sick onAugust 17, 2020, 9. See the proposals discussed in Chalmers (2011), Braun (2016), and Fitts (2014). 10. I am supposing a standard semantics for 'actually' ('@'), on which '@q' is true at a worldw iff 'q' is true at the actual world. 6 §3 Experts and Deference and, for my doctor, the guise determines the proposition that she is sick on August 16, 2020. If the thought 'I am sick now' is just the guise, then both me, my future self, and my doctor can have a credence in this one thought. On the other hand, if the thought 'I am sick now' is a pair of a guise and a proposition, then my doctor is not capable of entertaining my thought 'I am sick'. I doubt that there is any substantive issue here. Suppose that thoughts are identified with guise-proposition pairs. Then, we may say that two thoughts are equivalent iff they have a guise-component in common. Then, even if my doctor cannot entertain my thought 'I am sick', she can entertain a thought which is equivalent to it. And this will be enough for my purposes. In what follows, I'll opt for the first form of book-keeping, supposing that me and my doctor can both have credences in the thought I'd express with 'I am sick', but the reader should feel free to keep their own books differently, supposing instead that me and my doctor's thoughts are merely equivalent, and not identical. So far as I can see, nothing substantive will change. Insofar as fine-grainers are happy to say that the proposition I express when I say 'I am sick' is the same as the proposition you expresses when you say 'I am sick', theymay identify thoughts with propositions. If they distinguish between these propositions, then their propositions are finer than my thoughts. Even if you and I believe different propositions when we each believe the propositions we'd express with 'I am sick', there is nonetheless something that our belief states have in common. Perhaps both of our beliefs are mediated by the same sentence in the language of thought, perhaps they are mediated by the same guise, or belief state. Then, if you are this kind of fine-grainer, you may understand my talk about thoughts as talk about equivalence classes of belief states which have that feature in common. 3 Experts and Deference In addition to thoughts about whether Fermat had a proof of Fermat's last theorem, whether you are sick, whether a flipped coin landed heads, and so on, you may also have opinions about others' opinions about these matters. If you take another's opinions to bemore reliable than your own, then I'll say that you take them to be an expert. For example, you may take your doctor, the objective chances, or your future selves to be experts. In that case, you may wish to show deference to the expert's opinions-that is, you may wish to align your credences with theirs. If you're going to attempt to align your credences with those of the expert, then you must have some views about what the expert's opinions are. So I'll suppose that you have credences in thoughts of the form 'E = E', where the script 'E ' stands for the definite description 'the expert's credence function', 7 Two-Dimensional Deference and 'E' is a particular credence function. When I'm speaking about a generic expert, I'll use 'E '. When I'm talking about the time t chances, I'll use 'Cht '. And when I'm talking about your time t self, I'll use 'Ct '. Thus, 'Cht = Cht ' says that the time t chance function is given by Cht , and 'Ct = Ct ' says that, at time t, your credence function is Ct . Given thoughts of the form 'E = E', we can construct thoughts of the form 'E(p) = n%'. To get this thought, we take every potential expert function E which is such that E(p) = n% and disjoin them all. That is: the thought E(p) = n% is the disjunction ∨ E :E(p)=n%E = E. How should you show deference to an expert? Several authors11 suggest the following general recipe: if you are certain that the expert thinks that 'p' is n% likely, then you too should think that 'p' is n% likely. If, however, you are uncertain about what the expert thinks, then, conditional on the expert thinking that 'p' is n% likely, you too should think that 'p' is n% likely. That is, your credences should satisfy D1, for every p and every n% such thatC(E(p) = n%) > 0, (D1) C(p | E(p) = n%) = n% (By the way, 'C(p | q)' is your credence that p on the indicative supposition that q. I will suppose throughout that these conditional credences satisfy the product rule, C(p ∧ q) = C(p | q) * C(q). This means that, if C(q) > 0, then C(p | q) must be the ratio C(p ∧ q)/C(q). In the remainder, I'll frequently write things like 'C(p | q) = n%' without explicitly mentioning the proviso that C(q) > 0, but this is just in the interests of brevity. I only ever intend to make claims about conditional credences when the thought on the right-hand-side of the ' | ' has positive credence.) Other authors suggest the following: if you are uncertain about what the expert thinks, then, conditional on E being the expert's probability function, your credence in 'p' should be E(p). That is, your credences should satisfy D2, for every p and every E. (D2) C(p | E = E) = E(p) D2 is nearly equivalent toD1. If you satisfy D2, then youwill satisfy D1, as well; and, if you satisfy D1-and you are not in an incredibly singular and special case-then you will satisfy D2, as well. The two constraints are not precisely equivalent, but they are equivalent for all philosophical purposes.12 So I will 11. See, in particular, Skyrms (1980) and Gaifman (1988). 12. Gaifman (1988) provides a case in which you satisfy D1 but not D2. In Gallow (ms), I show that Gaifman's case is, in a good sense, the only kind of case in which D1 and D2 come apart. 8 §3 Experts and Deference allow myself to go back and forth between principles of the form D1 and D2. (See Gallow (ms) for more discussion.) Finally, some authors suggest that, if you are uncertain about what the expert thinks, then your credence in 'p' should be your expectation of the expert's credence in 'p'.13 That is, your credences should satisfy D3, for every 'p', (D3) C(p) = ∑ E E(p) *C(E = E) D3 is strictly weaker than D1 and D2; both D1 and D2 entail D3, and D3 does not entail either D1 or D2. Unlike the differences between D1 and D2, the difference between D3 and the other two principles is not negligible. There are other forms which principles of expert deference take, but those other forms are introduced to deal with complications orthogonal to my purposes here. For instance, D2 entails that the expert knows for sure what its own probabilities are. In some applications, this seems implausible, and several authors have suggested replacing the right-hand-side of D2 with 'E(p | E = E)'.14 For the substituends for 'p' I'll be looking at here, it is safe to suppose that whatever uncertainty the expert may have about their own probabilities won't make a significant difference to their probability in 'p', so I'll ignore these additional complications here.15 3.1 Chance Deference For an example of a principle of expert deference of the formD1: Lewis (1980) contends that your credences should satisfy LCD (forLewis's principle of chance deference), for every future time t, every proposition P , and every number n%, (LCD) C(P | Cht(P ) = n%) = n% That is: conditional on the time t chance of P being n%, your credence in P should ben%. More carefully, Lewis thinks that you should satisfy LCD as long as you lack evidence which is inadmissible for the time t chances.16 Roughly, 13. See, for instance, Ismael (2008, 2015). 14. See, e.g., Hall (1994), Lewis (1994), and Elga (2013). 15. When the expert is the credences which are rational for you, Dorst (forthcoming) defends a very different principle of expert deference which he calls 'Trust'. I think you should defer to this expert in the way recommended by Elga (2013)'s 'New Rational Reflection' principle (see Gallow, forthcoming), but I believe Dorst's Trust could be generalized in ways analogous to the generalization of Reflection I will offer in §5 below. 16. This isn't the way Lewis explicitly formulates his chance deference principle, but it follows from his formulation given the update norm of conditionalization (which he also accepts-see Lewis, 9 Two-Dimensional Deference evidence is time t inadmissiblewhen it is about matters at or after time t. Lewis thought that ordinary humans left to their own devices will only have admissible evidence. The kinds of situations in which you might have inadmissible evidence are situations in which you're dealing with oracles, crystal balls, time travelers, or some other exotic source of information about the outcome of future chancy processes.17 In this subsection, I will raise two problems for LCD. The first problem is that the principle gives bad advice when it comes to a priori knowable contingent thoughts like 'Lucky wins' (from the introduction). The second problem is that the principle gives bad advice in cases where you've lost track of the time. 3.1.1. A Priori Knowable Contingencies. Suppose that we are soon to flip a coin; before we do, I introduce the name 'Beatrice' with the following speech: 'Let's call whichever side of the coin actually lands up 'Beatrice". Because you are certain that the coin is fair, you are certain that the current chance that the coin lands Beatrice up is 50%. (Either the coin actually lands heads, or it actually lands tails. If it actually lands heads, then 'Beatrice' names heads, and the chance that the coin lands Beatrice up is 50%. If it actually lands tails, then 'Beatrice' names tails, and the chance that the coin lands Beatrice up is 50%. So, eitherway, the chance that the coin lands Beatrice up is 50%.) So you are certain that chance is 50% confident that the coin lands Beatrice up. If you are certain that chance thinks that P is n% likely, then LCD says that you must think that P is n% likely. So LCD says that youmust think it 50% likely that the coin lands Beatrice up. But, given the way that the name 'Beatrice' was introduced, it is a priori that the coin will land Beatrice up. So your credence that the coin lands 1999). Lewis's explicit formulation was this: if t is a time, 'C0' is a reasonable initial credence function, P is any proposition, n% is any number between 0% and 100%, and A is any time t admissible proposition compatible with Cht(P ) = n%, then C0(P | Cht(P ) = n% ∧ A) = n%. Call this principle 'PP'. Conditionalization tells us that, if your total evidence is E, then your current credence functionC should beC0(− | E). So, if your total evidence is time t admissible, then PP implies that C(P | Cht(P ) = n%) should be n%. And this is the principle LCD in the body. 17. Lewis supplies the following "almost sufficient" condition on admissibility: "If a proposition is entirely about matters of particular fact at times no later than t, then as a rule that proposition is admissible at t" (Lewis, 1980, p. 272). Lewis thinks that this holds as a rule, and not across the board; it is an almost sufficient condition, rather than a sufficient condition full stop. He explains his reasons for this restriction by noting that "if the past contains seers with foreknowledge of what chancewill bring, or time travelers who havewitnessed 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, p. 274, emphasis added). See Meacham (2010) for further discussion of Lewis's reasons for qualifying his sufficient condition for admissibility. 10 §3 Experts and Deference Beatrice up should bemuch higher than 50%. This looks like a counterexample to LCD. One reaction to this kind of case is to suggest that my naming ceremony has provided youwith some kind of inadmissible information. This kind of flatfooted appeal to inadmissibility often shows up in conversation about cases like these, and more careful versions of the reaction show up in Schwarz (2014) and Spencer (2020). In the remainder of this sub-section, I'll address this reaction. I'll ultimately conclude that, while the flatfooted version of the response faces some serious problems, the more careful approach of Schwarz and Spencer is able to successfully deal with this problem. However, even the principles advocated by Schwarz and Spencer face the second difficulty I'll raise in §3.1.2 below. That is: these principles give bad advice in cases in which you've lost track of the time. The uninterested reader should feel free to skip ahead to §3.1.2. Let me make three points about the flat-footed version of this response (I'll address Schwarz's and Spencer's more careful versions below). Firstly, if my dubbing ceremony provides youwith inadmissible evidence, then inadmissible evidence is much easier to come by than Lewis thought. As I mentioned above, Lewis thought that ordinary humans left to their own devices would only have admissible evidence; but we are perfectly capable of engaging in naming ceremonies like the one I used to introduce 'Beatrice' without the assistance of crystal balls, oracles, or time travelers.18 Secondly-and more importantly- we can generate this problem for LCD without any dubbing ceremony or the introduction of any name at all. All we need is the rigidified definite description 'the side of the coin which actually lands up'. For we can substitute 'The side of the coin which actually lands up lands up' in for 'P ' in LCD. You should be certain, or nearly certain, that the side of the coin which actually lands up lands up, but you are also certain, or nearly certain, that the chance of this happening is 50%. We can even create this kind of trouble for LCD with just demonstratives like 'this coin'. So we have a similar counterexample to LCD without the putatively inadmissible evidence of a dubbing ceremony. Thirdly, if a dubbing ceremony like this is all that's required to escape the rational requirement of LCD, then the principle is all too easily defanged. If introducing the name 'Beatrice' means that LCD doesn't constrain your credence in 'the coin lands Beatrice up', then it alsomeans that LCDwon't constrain your credence in 'the coin lands heads up'. The reason is that, when Lewis explicitly 18. At least, I will suppose that we are capable of introducing names in this way, though some have disagreed. I won't be engaging with that position here, since it's increasingly unpopular, and it doesn't ultimately get us out of our puzzle, as we can raise the same troubles with definite descriptions and demonstratives (see the discussion in the body above). 11 Two-Dimensional Deference formulates his principle of chance deference, he assumes that the objects of credence and the objects of chance are both propositions-which are, for Lewis, truth-conditional contents, or sets of metaphysically possible worlds. (It's for this reason, by the way, that I used the uppercase 'P ', rather than the lowercase 'p', in my statement of LCD.) Suppose that, actually, the coin lands heads up. Then, 'the coin lands Beatrice up' and 'the coin lands heads up' have the same truth-conditional content. For this reason, if dubbing ceremonies give you inadmissible evidence, you are always in a position to easily escape the rational requirement imposed by LCD. Suppose you know that the chance of Mudskipper winning the race is 5%, but you wish nonetheless to be very confident than Mudskipper will win. Then, you can simply introduce the verbs 'flerm' and 'glurg' with the following speech: 'If Mudskipper actually wins, then 'flerm' means win and 'glurg' means 'lose', and if Mudskipper actually loses, then 'flerm' means 'lose' and 'glurg' means 'win'.' Now, you should not defer to chance about either 'Mudskipper flerms' or 'Mudskipper glurgs', and you know that 'Mudskipper wins' has the same truth-conditional content as one of those thoughts. If this has given you inadmissible evidence, then you are now free to adopt a high credence that Mudskipper wins without violating LCD. The bounds of rationality are not so easily slipped. So it's unsurprising that, when Schwarz (2014, §3) presents an 'inadmissibility' response to puzzles like these, he does not assume that the objects of credence are truth-conditional contents. Instead, he assumes that they are thoughts.19 He offers the following emendation of LCD: for any thought 'p', any number n%, and any future time t, (SCD) C(p | Cht(p) = n%) = n% 19. More carefully, he assumes (with Lewis, 1979) that the objects of credence are sets of centered possible worlds, and that, when we attribute a property to an individual, we do so with what Schwarz calls an identifier: a relation we bear to the individual, and by which we pick the individual out. For instance, we can identify the heads side of the coin in two ways: either as the side with George Washington's face on it, or as the side which actually lands up. These two different identifiers make for two different objects of credence, so they make for two different thoughts. Inmy broad and ecumenical terminology from §2, an 'identifier' like this is associated with a guise-it gives us a way of being related to a proposition (i.e., a truth-conditional content). Schwarz, then, takes the objects of credence to be guise-proposition pairs, and not guises on their own. Since nothing I have to say in this section depends upon the identity conditions for your thoughts, this complication won't make any difference to my discussion here. One final caveat: Schwarz intends his principle to apply to deterministic chances as well as tychistic chances; for this reason, his explicit presentation of the principle has some additional bells and whistles which mine lacks. In the present context, I'm only concerned with tychistic chance (for this reason, the examples involving coin flips should really involve quantum measurement instead, but I've gone with coin flips for ease of exposition). 12 §3 Experts and Deference so long as you don't have any inadmissible information, and so long as the thought 'p' is admissible at t. In this principle, 'Cht(p) = n%' is also a thought. Just as 'the coin lands Beatrice up' has the same truth-conditional content as 'the coin lands heads up', 'the chance that the coin lands Beatrice up is 50%' has the same truth-conditional content as 'the chance that the coin lands heads up is 50%'. As I'll understand it here, Schwarz's principle SCD requires the embedded thought 'p' on the right-hand-side of the ' | ' to be the same as the unembedded 'p' on the left-hand-side of the ' | '.20 So long as the thought 'the coin lands Beatrice up' is inadmissible, SCD won't fall prey to the counterexample which beset LCD. What SCD is capable of telling us depends uponhowmany thoughts are admissible; and, before it tells us anything at all, wemust have an account ofwhich thoughts are admissible and which are not. A natural first suggestion is this: a thought is inadmissible whenever the truth-conditional content of that thought depends upon matters which are chancy at t. That is: 'p' is inadmissible at t iff, for some P , there's a positive chance at t that P is the truth-conditional content of 'p', and there is a positive chance at t that P is not the truth-conditional content of 'p'. This first pass suggestion could use further Chisholming, but it will do as a rough-and-ready characterization of which thoughts are admissible. Spencer (2020, fn 20) floats a superficially different way of responding to these kinds of worries. Where 'p' and 'q' are any thoughts, n% is any number, and 'Ex('p',q)' says that 'p' expresses the truth-conditional content that q, Spencer suggests modifying LCD to this: C(p | Cht(q) = n%∧Ex('p',q)) = n% so long as you don't have any time t inadmissible information, and so long as 'Ex('p',q)' is only about matters before t.21 If 'p' is 'the coin lands Beatrice up', and 'q' says that the coin lands heads up, then 'Ex('p',q)' will be partly about matters after t. So Spencer's proposed principle won't fall prey to the counterexample which beset LCD. Whatever truth-conditional content 'p' expresses, this content could be entertained via the thought p. That is, if 'p' in fact expresses the truth-conditional content that q, then one way of entertaining the truth-conditional content of 20. A very similar principle is also suggested by Chalmers (2011, p. 630), though Chalmers doesn't include Schwarz's restriction to admissible thoughts. 21. Spencer formulates his principle in terms of an initial credence function; but, given conditionalization, his principle will entail the one in the body, as I explain in footnote 16. Additionally, instead of using the thought 'Ex('p',q)', he uses 'Of(g,q)', where g is the guise of the thought 'p', and 'Of(g,q)' says that g is a guise of the truth-conditional content that q. I've used 'Ex('p',q)' instead in an attempt to translate Spencer's principle into into my more ecumenical idiom. 13 Two-Dimensional Deference 'Ex('p',q)' ("p' expresses the truth-conditional content that q') iswith the thought 'Ex('p',p)' ("p' expresses the truth-conditional content thatp'). And this thought is knowable a priori. Thus, it is certain to be true. If a thought is certain to be true, then we can ignore it whenever it shows up on the right-hand-side of the ' | '. So if we set 'q' to 'p' in Spencer's proposed principle, then it reduces to the following: C(p | Cht(p) = n%) = n% so long as 'Ex('p',p)' is only about matters before t. And this is just SCD with a more explicit specification of which thoughts are admissible at t. According to Spencer's principle, a thought is admissible at t iff 'Ex('p',p)' is about matters prior to t. This is in the same ballpark as the rough-and-ready characterization of admissibility I offered Schwarz above. There is some additional strength in Spencer's principle, due to the fact that we don't have to co-ordinate the thought 'q' embedded in 'Cht(q) = n%' on the right-hand-side of the ' | ' with the thought 'p' on the right-hand-side. However, this additional strength won't be relevant to my discussion below. In §3.1.2 below, I'll focus on Schwarz's principle, but everything I have to say about it there applies to Spencer's principle as well. 3.1.2. Losing Track of the Time. Principles like SCD are able to deal with a priori knowable contingencies like 'the coin will land Beatrice up'. However, they face difficulties with cases where you've lost track of the time. For a case like this, suppose that you're not sure whether today is Tuesday or Wednesday, but you know that today the chance of 'Mudskipper wins' ('m') is 75%, and yesterday the chance of 'm' was 25%. Then, the Tuesday chance of 'm' might be 25%, and it might be 75%. The truth-conditions of the thought 'm' are not chancy on Tuesday-nor is the thought "m' expresses the truth-conditional content that m' about the future. Moreover, you don't have any information which is inadmissible on Tuesday. Just to make this crystal clear, we can suppose that today is in fact Tuesday. There are no crystal balls, oracles, or time travelers to speak of-so it's hard to see how you could have come by any information about times after Tuesday. If that's correct, then SCD says that, conditional on Chtues(m) = 25%, your credence in 'm' must be 25%, and conditional on Chtues(m) = 75%, your credence in 'm' must be 75%. Suppose you're 50% sure that today is Tuesday and 50% sure that today is Wednesday. Then, you are 50% sure that the Tuesday chance of 'm' is 75%, and 50% sure that the Tuesday chance of 'm' is 25%. It follows from the law of total probability that your credence in 'm' must be 50%.22 22. The law of total probability says that C(m) = C(m | Chtues(m) = 25%) *C(Chtues(m) = 25%)+ 14 §3 Experts and Deference This seems to be clearly the incorrect result. You know for sure that the current chance ofMudskipper winning is 75%-so surely you should be 75% confident that Mudskipper wins. Losing track of the time doesn't give you any reason to be less confident that Mudskipper will win. But perhaps I was too quick when I said that you don't have any information which is inadmissible for the Tuesday chances. Today may be Tuesday, but you don't currently know that today is Tuesday. For all you're in a position to know, it is Wednesday. And if it is Wednesday, then your knowledge that today's chance of 'm' is 75% is Tuesday-inadmissible information. This information won't count as admissible according to Lewis's criterion, since that information is not in fact about times after after Tuesday. But perhaps we should revise Lewis's criterion. Perhaps we should say that information is time t inadmissible iff, for all you know, it is about times after t. Then, we could say that losing track of the time has given you the Tuesday-inadmissible information that the Wednesday chance of 'm' might be 75%. Let me make three observations about this reply. Firstly, while it's true that 'the Wednesday chance of 'm' is 75%' is inadmissible information for the Tuesday chances, we should be cautious about the inference from "e' is inadmissible' to "it might be that e' is inadmissible'. After all, 'Mudskipper wins' itself is inadmissible, and you know that Mudskipper might win. But surely knowing that Mudskipper might win isn't a sufficient reason to stop deferring to the known chance that Mudskipper wins. Secondly, suppose that I have been keeping track of the day, and I inform you that today is Tuesday. At that point, you know for sure that it is Tuesday and that the Tuesday chance of 'm' is 75%. Once I've told you the day, there is no difference between your epistemic situation with respect to 'm' and my own. Since I don't have any information which is inadmissible for the Tuesday chances, you shouldn't have any inadmissible information, either. But, in learning that it is Tuesday, you didn't lose any information. You only gained information. Since you don't end up with any inadmissible information, you must not have had any inadmissible information to begin with. I can foresee some denying this by claiming that, whenever you gain the information that p, you will thereby lose the information that it might be that ∼p. So gaining the information that it is Tuesday means losing the information that the Wednesday chance of 'm' might be 75%. It seems to me that, if you learn something new and you haven't forgotten anything, then you shouldn't count as losing information in any interesting sense of the word 'information'. But I don't have much to say to anyone who disagrees with me on this point. C(m | Chtues(m) = 75%) *C(Chtues(m) = 75%) = 25% * 50%+75% * 50% = 50%. 15 Two-Dimensional Deference Thirdly: suppose that there is a countable infinity of times, t1, t2, t3, . . . , which might, for all you know, be the current time. As before, while you don't know the time, you do know that the current chance of Mudskipper winning is 75%. In this kind of case, a principle of chance deference should tell you to set your credence in 'm' to 75%. However, if all that it takes for you to have information which is inadmissible for the time ti chances is for you to know something about what the ti+1 chances might be, then you will have information which is inadmissible for every time ti . And SCD won't tell you to defer to the chances at any time. So I don't think it's plausible to suggest that losing track of the time gives you some inadmissible information about the future. However, let me emphasize that I think that there is something deeply right about this response. In particular, I think that you have sufficient reason to have a credence of 75% in 'm', in spite of the fact that your expectation of the Tuesday chance of 'm' is 50%. Moreover, I think that this is true precisely because you think that it might be Wednesday. (To foreshadow, this will follow from the principle of chance deference I will endorse in §5 below.) However, I don't see how to understand this in terms of your possessing inadmissible information about the future. And if you don't have any inadmissible evidence for the Tuesday chances, SCD will require your credence in 'm' to be 50%. This seems bad enough, but it seems that SCD will also require your credence in 'm' to be 75%. After all, your expectation of the Wednesday chance of 'm' is 75%.23 So, if you don't have any inadmissible information, then SCD will require your credence in 'm' to be both 50% and 75%. Since I don't think that loosing track of the time means that you have inadmissible information about the current chances, I am inclined to reject SCD. 3.2 Deference to Your Future Self For another principle of expert deference: van Fraassen (1984, 1995) proposes the principle that your credences should satisfy the following constraint: for any future time t, any thought 'p', and any credence function Ct , C(p | Ct = Ct) = Ct(p) That is: conditional on your time t credence function being Ct , your credence that p should be Ct(p). From this principle, it follows that your current cre23. If today is Wednesday, then you know for sure that the Wednesday chance of 'm' is 75%, and if today is Tuesday, then you know for sure that the Tuesday chance of 'm's is 75%, and so you'll expect the Wednesday chance of 'm' to be 75% as well. By the law of iterated expectations, your current expectation of the Wednesday chance of 'm' is 75%. 16 §3 Experts and Deference dence that p should equal your expectation of your time t credence that p, for any thought 'p' and any future time t. That is, for any 'p' and any future t: C(p) = ∑ Ct Ct(p) *C(Ct = Ct) There are a host of counterexamples to van Fraassen (1984)'s principle. Many of these counterexamples are cases in which you expect your future self to be irrational, or to have strictly less information that you have now. For instance, Talbott (1991) notes that, in ten years' time, you will have little to no idea what you ate for lunch today; but this is no reason to drop your current credence that you had spaghetti for lunch today. Likewise, suppose you are about to be brainwashed to believe that Elmo from Sesame Street is the Antichrist. That's unfortunate, but it's no reason for you to now think that Elmo is the Antichrist.24 Dwelling on these kinds of counterexamples encouarages thinking of van Fraassen (1984)'s principle as a constraint on your current credences. But the principle is more plausible, andmore defensible, if we think of it as a constraint on your learning dispositions. Suppose that you are about to learn exactly one of the thoughts in E = {e1, e2, . . . , eN }. Let 'D' be the definite description 'the credence function you are disposed to adopt after learning', and let 'De' be the credence function which you are disposed to adopt if you learn 'e'. Then, the version of van Fraassen (1984)'s principle which I will focus on says that the credence in 'p' which you are disposed to adopt after learning shouldn't be expected to be any higher or lower than your current credence in 'p'. That is, for any 'p', your learning dispositions should satisfy the following constraint, which I'll call 'Reflection'. (Reflection) ∑ e∈E De(p) *C(D =De) = C(p) As I understand Reflection, it imposes the constraint that your learning dispositions not be biased. If, after learning, your credence that p is expected to be higher than your current credence that p, then your learning dispositions are biased in favor of 'p'. On ther other hand, if your credence that p is expected to be lower than C(p) after learning, then your learning dispositions are biased against 'p'. So, if your learning dispositions are not biased in favor or of against 'p', then they will satisfy Reflection.25 In a slogan, Reflection says that, 24. For more on these kinds of objections to van Fraassen's principle, see Hitchcock & Green (1994), Elga (2007), and Briggs (2009). 25. See Salow (2018) and Gallow (forthcoming) for more discussion. 17 Two-Dimensional Deference upon learning, you should be disposed to become somebody worthy of being deferred to.26 Talbott's counterexamples involve irrationality and forgetting. Take forgetting first. I contend that, at least in mundane cases, if you're currently disposed to forget something you now know upon learning, that is a rational defect in your learning dispositions. These kinds of rational defects may be widespread and forgivable, but they are nonetheless departures from ideal epistemic rationality. Likewise, in cases where you are disposed to adopt irrational credences after learning, this is a rational defect in your learning dispositions. As an aside: in some cases of foreseen irrationality or information loss, we may want to insist that, even though you will forget or adopt irrational credences, you are not currently disposed to forget or adopt irrational credences. Perhaps time, alcohol, brainwashing, belief-inducing pills, or what-have-you will mask or otherwise interfere with your current learning dispositions. If so, then your current learning dispositions could be perfectly rational, and the trouble is just that those rational dispositions will not manifest themselves. Note that, even if your learning dispositions are irrational, Reflection does not put any pressure on you to adjust your current credences. This may be a way of getting your learning dispositions to satisfy Reflection, but it is not a way of meeting the demands of rationality. Compare: a legal goal requires getting the ball between the goal posts; but it doesn't follow that, if the ball is off-course, moving the goal posts is a way of getting a legal goal. Moving the goal posts is itself illegal. Likewise, adjusting your current credences to forget something you now know is irrational. Arntzenius (2003) presents some more pressing counterexamples to Reflection. These counterexamples all involve thoughts de se et nunc-thoughts about when and where you are located in time and space. Thoughts like these can give rise to some straightforward counterexamples to Reflection. For instance: on Sunday before going to sleep, I am nearly certain to awake, and I am disposed to be nearly certain that it is Monday upon awaking. However, I am now nearly certain that it is not Monday. So my expectation of my credence in the thought 'It is Monday' after learning is much greater than my current credence in the thought 'It is Monday', in violation of Reflection. One reaction to counterexamples like these is to limit Reflection to apply only to certain kinds of thoughts: thoughts which aren't about where and when you are located in space and time. Arntzenius's counterexamples 26. More specifically, you should be disposed to become somebody worthy of being deferred to in expectation. For reasons I don't have the space to discuss here, I don't think that you should be disposed to become somebody worthy of being deferred to directly-that is, I don't think that your learning dispositions should satisfy C(p | D = De) = De(p), for every e. See Gallow (forthcoming) for more discussion. 18 §3 Experts and Deference show that Reflection is not defensible, even after it is restricted to thoughts like these. Let me introduce two of these counterexamples. Both involve your credence that a fair coin flip landed heads. This isn't a thought about where and when you are located in space and time. First counterexample: you know that it is now 8:30, and you know that, if the coin landed heads, then the lights in your room will turn off at 12:00. However, you don't currently have a clock. As you sit in your room, you will learn that time has passed, although you know that youwon't keep perfect track of the time. So, by the time 11:30 rolls around, you'll have learnt that it's close to midnight, but the most you'll know for sure is that between two and four hours have passed, so that it's between 10:30 and 12:30. Let's suppose that, at 11:30, your credence that it's earlier than 11:00 will be 25%, your credence that it's later than 12:00 will be 25%, and your credence that it's between 11:00 and 12:00 will be 50%. But at 11:30, the lights in your room will certainly be on. So at 11:30, after learning that somewhere between two and four hours have passed, you'll think it's 75% likely that the lights don't tell you anything about whether the coin landed heads, and you'll think it's 25% likely that the lights tell you that the coin landed tails. So your credence that the coin landed heads will be 37.5%, or 3/8. So at 11:30, after learning that between two and four hours has passed, you expect your credence that the coin landed heads to be lower than it currently is, in violation of Reflection. But it doesn't appear that these dispositions to update your credences after learning that between two and four hours have passed are irrational. The second counterexample is due originally to Elga (2000), and is known as Sleeping Beauty: on Sunday evening, you will be put to sleep with a powerful sedative and awakened on Monday morning. On Monday evening, after putting you back to sleep, I will flip a fair coin. If the coin lands heads, then I will not awaken you until Wednesday. If, however, the coin lands tails, then I will erase your memories of Monday, and awaken you again on Tuesday. If the coin lands tails and you're awoken on Tuesday, this awakening will be indistinguishable from the Monday awakening. On Monday evening, before putting you back to sleep, I will let you know that it is Monday evening. You are told all of this on Sunday. (And, just by the way: you're beautiful.) There are three prominent schools of thought on Sleeping Beauty, which we can categorize by their answers to the following two questions. First question: how confident should you be that Monday's coin flip lands heads when you awaken on Monday? Second question: how confident should you be that Monday's coin flip lands heads on Monday evening, after you're told that it is Monday? Thirders answer the first question with 'one third', and the sec19 Two-Dimensional Deference ond with 'one half '.27 Lewisian Halfers answer the first question with 'one half ' and the second with 'two thirds'.28 Double Halfers answer both questions with 'one half '.29 These answers are of course not exhaustive, but I'll focus on these three positions here.30 All three of these positions violate Reflection in Sleeping Beauty. Start with the Thirder. Before going to sleep on Sunday, the Thirder is disposed to lower their credence in 'Monday's flip lands heads' ('h') from1/2 down to 1/3 upon learning that they've awoken. On Sunday evening, before going to sleep, they know for sure that they will learn this upon awaking Monday morning. So they know for sure that after awaking on Monday morning, their credence in 'h' will be lower than their current credence in 'h', in violation of Reflection. Similarly, before going to sleep on Sunday, the Lewisian Halfer is disposed to raise their credence in 'h' to 2/3 upon learning that they've awoken and that it is Monday on Monday evening. So, on Sunday evening, they know for sure that Monday evening, their credence in 'h' will be higher than their current credence in 'h', in violation of Reflection. Finally, consider the Double Halfer on Monday morning, before learning what day it is. They know that they will either learn that it is Monday or that it is Tuesday. If they learn that it is Monday, then they are disposed to keep their credence in 'h' fixed at 1/2. If they learn that it is Tuesday, they are disposed to lower their credence in 'h' to zero. They think that it's 75% likely to be Monday,31 so their expectation of their credence in 'h' after learning is 3/8,32 which is less than their current credence in 'h', in violation of Reflection. (Looking ahead: the Thirder will satisfy the revision of Reflection I will propose in §5; though neither the Lewisian Halfer nor the Double Halfer will.) Incidentally, Sleeping Beauty also raises issues for principles of chance deference. It's not clear whether the Thirder, the Lewisian Halfer, or the Double 27. See, for instance, Elga (2000, 2004), Dorr (2002), Arntzenius (2003), Hitchcock (2004), Horgan (2004), and Weintraub (2004) 28. See Lewis (2001) 29. See, for instance, Halpern (2004), Bostrom (2007), and Meacham (2008). 30. For one alternative, see the 'imprecise' suggestion discussed in Monton (2002) and defended in Singer (2014) 31. I am supposing that the Double Halfer thinks that, conditional on the coin landing tails, it's just as likely to be Monday as Tuesday. This doesn't ultimately matter, however. So long as their credence that it's Tuesday is greater than zero, their expectation of their credence in 'h' after learning will be lower than 1/2. 32. Their expectation of their updated credence in heads is given by Dmonday(h) * C(monday) + Dtuesday(h) * C(tuesday) = 1/2 * 3/4 + 0 * 1/4 = 3/8, where 'Dmonday' and 'Dtuesday' are the credence functions they're disposed to adopt upon learning that it is Monday or Tuesday, respectively. 20 §3 Experts and Deference Halfer will abide the principle of chance deference SCD in Sleeping Beauty. Start with the Thirder. On Monday morning, the Thirder's credence in 'Monday's flip lands heads' is 1/3, even though they know for sure that the Monday evening chance of the coin landing heads is 1/2. Moreover, the thought 'Monday's flip lands heads' is admissible, since the truth-conditional content of this thought is not chancy on Monday evening. As I argued in §3.1 above, I don't think that losing track of the time provides you with any inadmissible information, so I don't think that you have any information which is inadmissible for the Monday evening chances. So I don't think that either the Thirder has inadmissible evidence when they violate SCD. Or consider the Lewisian Halfer. After learning that it is Monday, the Lewisian Halfer has a credence of 2/3 in 'Monday's flip lands heads', though, again, they know for sure that the chance of the coin landing heads is 1/2, and the thought 'Monday's flip lands heads' is admissible. Lewis (2001) insists that, when you are told that it is Monday, you are getting some inadmissible evidence. As Lewis admits, this is a "novel and surprising" kind of inadmissible evidence, for it is neither "from a prophet [nor] by way of backwards causation".33 When you learn that it is Monday, Lewis claims that you are "getting evidence...about the future: namely that [you] are not now in it."34 To be blunt, I find Lewis's position here baffling. The de se information that you are not currently in the future of a chance event is not information about the future of a chance event in any usual sense of the word 'about'-nor is it about the future of a chance event on Lewis's own technical understanding of the word 'about'.35 Even setting aside the question of whether this information is about the future, consider how persuasive you find the following: "Yes, I admit that Mudskipper has a 5% chance of winning the race, but I'm still quite confident that Mudskipper wins, for I've got some privileged information: that race isn't until tomorrow." If the information that you are not in the future of a chance event is inadmissible, then all it takes to escape the rational imperative to defer to the chances is a wristwatch. The bounds of rationality are not so easily slipped. Finally, consider the Double Halfer. The Double Halfer will violate SCD in a slight variant of Sleeping Beauty, due to Titelbaum (2015a). In this version of the case, on Monday, before I tell you what day it is, I put the coin in your hand, and let you flip it. As before, if the flip lands heads, then youwill be not be awoken until Wednesday. And if it lands tails, then your memories of Monday 33. Lewis (2001, p. 175). 34. Lewis (2001, p. 175) 35. See Lewis (1980, p. 272). 21 Two-Dimensional Deference will be erased and youwill be awoken again onTuesday. Tuesday evening, I will also give you a coin to flip; although the outcome of Tuesday's flip won't make any difference. You'll be awoken on Wednesday with your memories intact eitherway. You are told all of this on Sunday evening. In this version of the case, on Monday, before they flip the coin, the Double Halfer will have a credence of 5/8 in the thought 'this flip lands heads' (where 'this' demonstrates the flip they are about to perform). For the Double Halfer will think that it is 3/4 likely to be Monday and 1/4 likely to be Tueday. Conditional on it being Monday, the Double Halfer thinks 'this flip lands heads' is 2/3 likely; and conditional on it being Tuesday, the Double Halfer thinks 'this flip lands heads' is 1/2 likely. So their credence in 'this flip lands heads' is 2/3 * 3/4 + 1/2 * 1/4 = 5/8. But the truth-conditional content of 'this flip lands heads' is not chancy on Monday; so this thought should count as admissible. And it's incredibly hard to see how you could have any inadmissible information about whether this flip-the one you're about to perform-lands heads. You know for sure that you are in the past of this flip, and there are no crystal balls, oracles, or time travelers in sight. In summary: principles of chance deference face difficulties with thoughts like 'the coin will land Beatrice up'. We can attempt to deal with these difficulties by restricting the kinds of thoughts to which the principles apply, but this leads to some implausible consequences in cases in which you don't knowwhat time it is. Similarly, the principle of reflection faces difficulties with thoughts like 'today is Monday'. Moreover, both principles of chance deference and the principle of Reflection face difficulties with mundane thoughts like 'the coin flipped on Monday landed heads', once you start entertaining thoughts like 'today is Monday'. In §4 below, I'll introduce a two-dimensional framework formodeling how the truth-conditional contents of thoughts like 'the coin will land Beatrice up', 'today isMonday', and 'I am sick' vary depending uponwho you are, where and when you are located in space and time, and what the world is like. Then, in §5, I will use this framework to develop a general theory of how to defer to experts about thoughts like these. This theory of deference will solve the problem cases we encountered in this section. 4 A Two-Dimensional Framework In §2, I surveyed several options for what a thought may be. Importantly, no matter what we say a thought is, it will turn out that you can be uncertain about the truth-conditions of your thoughts. When you think 'I am sick', you may not know whether you think a thought which has the same truth-conditions as 'Beyoncé is sick'. For, if you suffer from amnesia, you may not know whether 22 §4 A Two-Dimensional Framework you are Beyoncé; and you know that 'I am sick' has the same truth-conditions as 'Beyoncé is sick' iff you are Beyoncé. So your uncertainty about whether you are Beyoncé translates into uncertainty about the truth-conditions of your thought 'I am sick'. And this will be so whether the thought 'I am sick' is a Fregean, fine-grained proposition, a guise, or a pair of a guise and a coarsegrained proposition. To model the ways that the truth-conditions of your thoughts can vary depending upon what the world is like, who you are, and where you are located, I will introduce a two-dimensional framework. This framework will be similar to the two-dimensional frameworks offered by authors like Kaplan (1989), Stalnaker (1978), Evans (1979), Davies & Humberstone (1980), Jackson (1998), and Chalmers (2006a,b), though there will be some formal and interpretational differences. As an aside, the framework I'll develop her is nuanced, and not wholly necessary for understanding the basic outline of the theory I'll present in §5. Whenever I appeal to the formal details of the framework in §5, I will accompany those details with an informal gloss. If the reader is willing to trust that I'm not misrepresenting things too badly with this informal gloss, they should feel free to skip ahead to §5. 4.1 Indices The framework begins with a domain of indices, generated from an underlying set of thoughts.36 An index, as I will be understanding it, is a maximally consistent set of thoughts. A consistent set of thoughts is maximal just in case no proper superset of it is also consistent. Let me offer three clarificatory comments about 'consistency'. Firstly, consistency is not metaphysical compossibility. The thoughts in the set {'Twain is gifted', 'Clemens is not gifted'} are not metaphysically compossible; but they are consistent, as I use the term here. Secondly, even if a set of thoughts can be a priori known to contain a falsehood, it can nonetheless be consistent. For instance, any set containing both the thoughts 'q' and '∼@q' can be a priori known to contain a falsehood, as can any set containing the thought 'I am not here now'.37 Nonetheless, these sets may still count as consistent. Although it is a priori knowable that their members do not describe actuality, it is not a priori knowable that their members do not describe a possibility. Consistency concerns the latter notion, not 36. I'll assume that this set of thoughts is closed under negation, disjunction, and conjunction. 37. Some may disagree on the grounds that the question of whether you exist is a posteriori. I don't think that's the right way to think about a priority, but it doesn't matter. If you think 'I am not here now' cannot be known a priori to be false for this reason, feel free to substitute 'I exist and I am not here now'. 23 Two-Dimensional Deference the former. Thirdly, if you have thoughts de se et nunc-thoughts about who you are and where and when you are located in space and time-then an index will include this kind of information. For instance, one indexmight include the thought 'I am Beyoncé'. Another indexmight include the thought 'I am not Beyoncé'. In some sense, these two thoughts are consistent. Beyoncé could truly think the first while Kelly truly thinks the second. However, they are not consistent in the sense that I'm using the term here. For, when you entertain these two thoughts, here and now, they explicitly contradict each other. Summing up, we may think of consistency like this: can you a priori determine that it is not possible for each thought in the set to be true (given the truth-conditional content they have for you, here and now)? (As the examples of {'q', '∼@q'} and {'I am not here now'} demonstrate, determining that it's not possible for each thought in a set to be true is more demanding than determining that it's not actually the case that each thought in the set is true.) If the answer is 'yes', then the set is inconsistent; if the answer is'no', then the set is consistent. Exactly one index will contain all and only the true thoughts (as actually entertained by you, here and now). Call that 'the actual index'. Call the other indices 'non-actual'. Some indices may be known, a priori, to be non-actual. For other indices, their non-actuality will not be a priori knowable. Call the latter kinds of indices epistemically possible. We can lift your credence distribution over thoughts to a credence function over sets of epistemically possible indices, as follows. For every thought, there will be the corresponding set of epistemically possible indices at which that thought is true. Set your credence in that set equal to your credence in the thought. If your credences over thoughts were probabilities, then this new credence function over sets of epistemically possible indices will be a probability function, as well.38 4.2 Metaphysical Possibility Consider an index at which both 'Twain is clever' and 'Clemens is not clever' are true. This index is not actual-nor is it even possible, given the metaphys38. More carefully, I mean this: suppose that your credences satisfy the following constraints: 1) your credence in each thought is no lower than zero; 2) if it is a priori knowable that t is true, then your credence in t is 100%; and 3) if it is a priori knowable that at least one of p and q is false, then your credence in the disjunction p∨ q is equal to the sum of your credence in p and your credence in q. Then, when we lift your credences to a function over sets of epistemically possible indices, the lifted function will satisfy the following constraints: 1∗) your credence in every set is non-negative; 2∗) your credence in the set of all epistemically possible indices is 100%; and 3∗) if P andQ are disjoint sets, then your credence in their union is equal to the sum of your credence in P and your credence in Q. 24 §4 A Two-Dimensional Framework ical necessity of identity (which I will assume here).39 Nonetheless, it is not a priori knowable that it is metaphysically impossible. If the actual index is one at which Twain , Clemens, then this index is metaphysically possible. Similarly, if the actual index is one at which I am not Beyoncé and today is Monday, then any index at which I am Beyoncé or today is Tuesday will be impossible. So which indices are metaphysically possible varies depending upon which index is actual. To model this, let me introduce a binary accessibility relation, R, defined over the set of indices. R will model relative metaphysical accessibility between indices. If Rij , then I'll say that j is metaphysically possible from i. I'll work my way up to the relation R by first defining a smaller relation, R−, which I'll then extend to R. Say that an epistemically possible index, i, bears the relation R− to another index, j , R−ij , iff, if i is actual, then j is metaphysically possible. The relation R− tells us which indices are metaphysically possible from each epistemically possible index. I assume that the correct logic of metaphysical possibility is S5; so, every index should bemetaphysically possible from itself (i.e., R should be reflexive), and, if both j and k are metaphysically accessible from i, then k should be metaphysically possible from j (i.e., R should be Euclidean). When we restrict attention to the epistemically possible indices, R− will already be reflexive and Euclidean; but this won't hold for the epistemically impossible indices. So we should extend R− by taking its reflexive and Euclidean closure. R is the relation which results. For example, suppose you introduce 'Beatrice' as a name for the side of the coin which actually lands face-up, and let us limit our attention to the following two thoughts: 'The coin lands heads up'-which we can dub 'h'-and 'The coin lands Beatrice up'-which we can dub 'b'. Then, there are four indices, which we can name 'ihb', 'ihb', 'ihb', and 'ihb'. At ihb, the coin lands heads up, and it lands Beatrice up. At ihb, the coin lands heads up, but not Beatrice up. At ihb, the coin lands tails up, and Beatrice up. And at ihb, the coin lands tails up, but not Beatrice up. The second and fourth of these indices, ihb and ihb, are not epistemically possible. (Given the way 'Beatrice' was introduced, it is a priori knowable that the coin lands Beatrice up. So it is a priori knowable that ihb and ihb contain at least one falsehood.) All four, however, are consistent. Take ihb. If ihb is the actual index, then 'Beatrice' refers to heads. Since it's possible that the coin not land heads up, it is possible that the coin not land Beatrice up. So ihb is metaphysically possible from ihb. And, if ihb is metaphysically possible from ihb, then ihb must be metaphysically possible from ihb, and ihb must be metaphysically possible from itself. Figure 1 shows which indices are metaphysically possible from which other indices. 39. The metaphysical necessity of identity says that, if x = y, then it is metaphysically necessary that x = y. 25 Two-Dimensional Deference  R row col ihb ihb ihb ihb ihb X × × X ihb × X X × ihb × X X × ihb X × × X  Figure 1: Thematrix on the left shows the relation of relative metaphysical possibility, R. (A 'X' means that the row index bears R to the column index; an '×' means that it does not.) The same relation is displayed graphically on the right. ihb and ihb are epistemically possible, and ihb and ihb are not epistemically possible. Notice that, when we ask about whether an index j is metaphysically possible from i, we give the thoughts in j the truth-conditions they have when entertained at i, and not the truth-conditions they have when entertained at j . Were the thought '∼b' entertained at ihb, it would have the truth-conditional content that the coin didn't land tails up. It is because these truth-conditions conflict with 'the coin didn't land heads up' that ihb is epistemically impossible.40 However, at ihb, '∼b' is true iff the coin didn't land heads up. And this clearly is consistent with '∼h'. That's why, from ihb, the index ihb is metaphysically possible (though it is known a priori to be non-actual). This highlights an important lesson which we will need in the next subsection: an index represents the world as being a certain way-in particular, it represents the world as meeting the truth-conditions of the thoughts it contains. However, it represents the world as meeting the actual truth-conditions of those thoughts, and not the truth-conditions those thoughts would have, were they entertained in a possibility described by that index. The index ihb describes a possibility at which the coin doesn't land Beatrice up. However, it also describes a possibility at which the thought 'the coin lands Beatrice up' would express a truth. Suppose the coin actually lands heads, and consider a possibility whichwewould accurately describe with the thought '∼h∧∼b'. The inhabitants of this possibility would not accurately describe their world with the thought '∼h ∧ ∼b'. Instead, they would accurately represent their world with '∼h∧ b'. That's because, even though our thought 'b' is true iff the coin lands heads up, their thought 'b' is true iff the coin lands tails up. 40. I'm treating 'the coin landed tails up' as the negation of 'the coin landed heads up'. So the coin must either land heads or tails up-if the coin lands on its edge or doesn't land, that counts as a tails landing. 26 §4 A Two-Dimensional Framework  ~hrow, col ihb ihb ihb ihb ihb T ∗ ∗ F ihb ∗ F T ∗ ihb ∗ F T ∗ ihb T ∗ ∗ F  (a)  ~brow, col ihb ihb ihb ihb ihb T ∗ ∗ F ihb ∗ T F ∗ ihb ∗ F T ∗ ihb F ∗ ∗ T  (b) Figure 2: Two-dimesional valuations for the thoughts h = 'the coin lands heads up' (in figure 2a) and b = 'the coin lands Beatrice up' (in figure 2b). In both matrices, the first index comes from the row, and the second index comes from the column. 4.3 Two-Dimensional Valuations With this lesson appreciated, we may introduce a two-dimensional valuation function, ~. Hand this valuation function a thought, 'p', and it hands you back a two-place function, ~pi,j , where i and j are indices. The output of the function ~pi,j depends upon, firstly, whether j is metaphysically possible from i, and, secondly, whether j satisfies the truth-conditions which the thought 'p' has when it is entertained at i. If j is not metaphysically possible from i, then ~pi,j will be undefined. In that case, I'll write '~pi,j = ∗'. If j is metaphysically possible from i, then, if j satisfies the truth-conditions which 'p' has when entertained at i, then ~pi,j = T . If, on the other hand, j is metaphysically possible from i, but j doesn't satisfy the truth-conditions 'p' has when entertained at i, then ~pi,j = F. ('T ' is the truth-value true, and 'F' is the truth-value false.) For instance, figure 2 shows the outputs of the two-place functions ~h (in figure 2a) and ~b (in figure 2b). In these matrices, the truth-values along each row correspond to the truthconditional contents which 'h' and 'b' express, if that row's index is actual. Look at the first and fourth rows in figure 2a. If ihb is actual, then there are two metaphysically possible indices: ihb and ihb. We know a priori that we are not at the index ihb; but, even so, we can askwhich truth-conditions 'h' would have, when entertained at ihb, and when entertained at ihb. And the answer is: its truth-conditions are exactly the same, nomatter which of these metaphysically possible indices it is entertained at: the thought is true at ihb and false at ihb no matter which index it is entertained at. Likewise, if ihb is actual, then there are two metaphysically possible indices: ihb and ihb. And, again, the thought 'h' has the same truth-conditions, whether it is entertained at ihb or ihb. No matter which index it is entertained at, 'h' is false at ihb and true at ihb. In general, if a thought has the same truth-conditions, no matter which index it is entertained at, say that it is a boring thought. (Or, more carefully, a thought 'p' is boring iff, for every epistemically possible index i, and every index j metaphysically possible from i, the truth-conditions 'p' has when en27 Two-Dimensional Deference tertained at i is the same as the truth-conditions 'p' has when entertained at j .) Figure 2b demonstrates that 'b' is not boring. Start with the first and fourth rows. If 'b' is entertained at ihb, it expresses a truth-conditional content which is true at ihb and false at ihb. On the other hand, if 'b' is entertained at ihb, then 'b' expresses a truth-conditional content which is true at ihb and false at ihb. Similarly, if 'b' is entertained at ihb, it expresses a truth-conditional content which is true at ihb and false at ihb; whereas, if 'b' is entertained at ihb, it expresses a content which is false at ihb and true at ihb. If a thought isn't boring, say that it's interesting. There are two important asymmetries between the boring 'h' and the interesting 'b'. The first asymmetry is that, while the actual truth-value of 'h' determines the truth-conditional content of 'b', the actual truth-value of 'b' underdetermines the truth-conditional content of 'h'. Note that we could coarse-grain by ignoring the thought 'b', partitioning our indices into cells by grouping together any indices which agree on thoughts not involving 'b'. Let 'H ' be the cell in which 'h' is true, {ihb, ihb}, and let '¬H ' be the cell in which 'h' is false, {ihb, ihb}. Then, we may show how the truth-conditional content of 'b' depends upon the actual truth of 'h' with a two-dimensional valuation function like the one shown below. [~brow, col H ¬H H T F ¬H F T ] That is: the truth-conditional content of 'b' only depends upon whether the coin landed heads or tails. The same cannot be said in reverse. If we coarsegrain by ignoring the thought 'h', and grouping together any indices which agree on thoughts not involving 'h', then we will not have enough information to determine the truth-conditional content of 'b'. Following Chalmers (2012), we may call {H,¬H} a scrutability base for 'b', since 'b"s truth-conditional content is a priori implied by either cell of this partition. Chalmers (2006a,b, 2012) argues that there is an appropriate scrutability base which allows us to provide a two-dimensional array like this for every thought not in the base. I'm sympathetic to Chalmers's position, but I won't require this additional thesis for my purposes here. 41 41. I won't require any of this additional apparatus myself, but just to dot my 'i's and cross my 't's: I've extended the two-dimensional valuation function so that it takes as inputs sets of indices, and not just single indices. Given any two sets of indices, I and J , the output of the function ~pI,J will depend upon, firstly, whether any j ∈ J is metaphysically possible from any i ∈ I , and secondly, whether 'p' is true or false at every j ∈ J which is metaphysically possible from some i ∈ I . If no j ∈ J is metaphysically possible from any i ∈ I , then ~pI,J is undefined. If 'p' is true at some j ∈ J which is metaphysically possible from some i ∈ I , and false at some other j ∈ J which is metaphysically possible from some i ∈ I , then ~pI,J ismultiply defined. If, 28 §4 A Two-Dimensional Framework Thesecond asymmetry concerns the diagonal entries in thematrices shown in figures 2a and 2b. (As I use the term here, the diagonal entries are the ones where the row index is the same as the column index, and the row index is epistemically possible.) When we set both the first and second indices to ihb, the thought 'h' is false, ~hihb ,ihb = F. In contrast, the thought 'b' is true when we set both the first and second indices to ihb, ~b ihb ,ihb = T . We may define a diagonal valuation function as follows. Given a thought, 'p', and an epistemically possible index, i, let [p]i = ~pi,i . Since every epistemically possible index is metaphysically possible from itself, [p]i will always be defined. Call the function [p]i the diagonal content of the thought 'p'. In these terms, the second asymmetry is that the diagonal content of 'h' is contingent, whereas the diagonal content of 'b' is necessary.42 By construction, a thought's diagonal content will be necessary iff the thought is a priori knowable, and its diagonal content will be contingent iff it is only knowable a posteriori.43 An important point for my purposes is this: while truth-conditional contents live along the rows-i.e., while the truth-conditional content of a thought is given by the row of the actual index-rational credence lives along the diagonal. Earlier, I lifted your credence distribution over thoughts to a credence distribution over sets of epistemically possible indices by setting your credence in the set of epistemically possible indices containing 'p' to your credence that p. But the epistemically possible indices containing 'p' are precisely those indices i such that [p]i = T . So we should sharply distinguish a rational credence in 'p' from a rational credence in 'p"s truth-conditional content. We can illustrate the distinction with the example of 'the coin lands Beatrice up', or 'b'. Suppose that is rational to have a credence of 50% that ihb is the actual index, and a credence of 50% that ihb is the actual index. And suppose the coin actually lands heads up. Then, the truth-conditional content of 'b' is: true at ihb and false at ihb. Since your credence in ihb is 50% and your credence in ihb is 0%, your credence in the truth-conditional content of 'b' is 50%. In contrast, the diagonal content of 'b' is true at both ihb and ihb. So your credence in the diagonal content of 'b' is 100%. So rational credence goes with diagonal contents, and not with truth-conditional contents. however, there's just one truth-value which 'p' takes on at every j ∈ J which is metaphysically possible from any i ∈ I , then ~pI,J is defined to be that truth-value. 42. I say that 'p"s diagonal content is necessary iff [p]i = T for every epistemically possible index i; and 'p"s diagonal content is contingent iff [p]i = T for some epistemically possible index i, and [p]j = F for some epistemically possible index j . 43. This is a consequence of our stipulation that an epistemically possible index is one whose nonactualilty is not a priori knowable. If 'p' is a priori knowable, then it will belong to every index whose non-actuality is not a priori knowable. So 'p' will be true at every index along the diagonal. 29 Two-Dimensional Deference 4.4 Worlds and Locations Some of the thoughts in an index only concern what the world is like, and other thoughts concern who you are, or when and where you are located in the world. Following Lewis (1979), I'll call the former kinds of thoughts de dicto, and the latter de se. Note that a de se thought may tell youmore than just when, where, or who you are; it may also tell you something about what the world is like. For instance, the thought 'the wedding is in ten minutes' tells you both that the wedding takes place, and that you are located ten minutes prior to the wedding. If a thought tells you only when, where, or who you are, then I'll call that thought purely de se. An index will settle exactly what the world is like-or rather, it will settle the truth-value of any of your de dicto thoughts. So it will completely describe the world in as rich a detail as your de dicto thoughts permit. Similarly, it will settle exactly when, where, and who you are-or, at least, it will settle the truthvalue of any of your purely de se thoughts. So it will completely describe your location in as rich a detail as your purely de se thoughts permit. In §5, I'll want to pull apart these two kinds of information. Here, I'll explain how to do that. If there's no de dicto thought included in i whose negation is included in j , then say that j is i's world-mate. World-mates do not disagree about any de dicto matters, though they may disagree about de se matters. Beingworld-mates is an equivalence relation,44 so it partitions the indices. I'll call a cell of this partition a 'world'. And I'll write 'Wi ' for the cell of this partition to which the index i belongs. If Wi =Wj , then i and j are world-mates. Similarly, if there's no purely de se thought included in i whose negation is included in j , then say that j is i's location-mate. Location-mates do not disagree about any purely de sematters, though they may disagree about other de sematters, as well as de dictomatters. Being location-mates is an equivalence relation, so it partitions the indices. Take a cell of this partition, Λ, and let 'λ' be a thought equivalent to the conjunction of every thought contained in the intersection of the indices inΛ. Then, 'λ' is a thought which specifies who you are, as well as when and where you are, in as rich a detail as your purely de se thoughts will permit. I'll call 'λ' a location. For illustration, consider the following three thoughts: 'I am sick' ('s'), 'Beyoncé is sick' ('b'), and 'I am Beyoncé' ('β'). There are six indices, which we can call 'iβbs', 'iβbs', 'iβbs', 'iβbs', 'iβbs', and 'iβbs'. At iβbs, you are Beyoncé, Beyoncé 44. The reflexivity of being world-mates is guaranteed by indices' consistency. To see that the relation is Euclidean, suppose (for reductio) that both j and k are world-mates of i, but that j is not a world-mate of k. Then, there is a de dicto thought included in j whose negation is included in k-call it 'p'. By i's maximality, either 'p' or '∼p' is included in i. If 'p' ∈ i, then k is not a world-mate of i; if '∼p' ∈ i, then j is not a world-mate of i. Either way, we have a contradiction. 30 §4 A Two-Dimensional Framework  R row col iβbs iβbs iβbs iβbs iβbs iβbs iβbs X X × × × × iβbs X X × × × × iβbs × × X X X X iβbs × × X X X X iβbs × × X X X X iβbs × × X X X X  Figure 3: The matrix shows the relation of relative metaphysical possibility, R. (A 'X' means that the row index bears R to the column index; an '×' means that it does not.) is sick, and (therefore) you are sick. At iβbs, you are not Beyoncé, Beyoncé is sick, but you are not. And so on for the other indices, in the obvious way. The relation of relative metaphysical possibility for these indices is shown in figure 3. If you are Beyoncé, then it's not possible that you're not Beyoncé; likewise, if you're not Beyoncé, then it's not possible that you are Beyoncé. The thought 'β' is purely de se. 'b' is de dicto. 's' is neither purely de se nor de dicto. So the indices iβbs, iβbs, and iβbs are world-mates-they are all indices belonging to the world in which Beyoncé is sick. Likewise, iβbs, iβbs, and iβbs are world-mates-they are all indices belonging to the world in which Beyoncé is not sick. The indices iβbs and iβbs are location-mates. They are both indices inwhich you are Beyoncé. Similarly, iβbs, iβbs, iβbs, and iβbs are location-mates. They are all indices in which you are not Beyconcé. Thus, with this collection of thoughts, there are two different locations: β and ∼β. There is of course more to be said about who, when, and where you are than the location ∼β tells us, but the thoughts β,b, and s are not expressive enough to allow you to say those things. Notice also that the thoughts β,b, and s are not expressive enough for us to say what the truth-conditions of 's' would be, were it entertained at a possibility described by the index iβbs. Since the index iβbs doesn't say who you are, it doesn't tell us what truth-conditional content 'I am sick' would express, were it to be entertained at a possibility described by that index. If we are going to define a two-dimensional valuation function like the one I introduced in §4.3, then we will have to begin with a set of thoughts expressive enough that each index can settle the truth-conditions of every thought in the set. In the following, I'll assume that the set of thoughts we start with is at least this expressive. (Though, in the interests of simplicity, I will continue to allow myself to focus on expressively impoverished subsets of thoughts like {'β','b','s'}.) Distinguishing worlds from locations allows us to define a kind of thought which will be important in §5. Take any thought 'p', and any location 'λ'. With these, I'll want to find a new de dicto surrogate for 'p' which is true anywhere in 31 Two-Dimensional Deference a world so long as the thought 'p' is true when entertained at the location λ in that world. I'll call this the de dicto λ-surrogate of 'p', and I'll write it 'pλ'. For instance, take the thought 'I am sick' and the location 'I am Beyoncé'. Then, the de dicto Beyoncé-surrogate of 'I am sick' is the thought 'Beyoncé is sick'. This surrogate thought is true at any world iff 'I am sick' is true when entertained at the location 'I am Beyoncé'. In general, in terms of the diagonal valuation function from §4.3, we may say that 'pλ' is true at an epistemically possible index i iff there is some index j which is a world-mate of i (Wj = Wi), which is at the location λ ([λ]i = T ), and at which 'p' is true ([p]i = T ). That is,45 (pλ) [pλ]i = { T if ∃j : Wj =Wi ∧ [p]j = [λ]j = T F else The other kind of thought which will be important in §5 is the thought that some location is occupied. Suppose, for instance, that 'I am Twain and I am Clemens' is a location. This location will not be occupied at every world. For instance, it will not be occupied at any world at which Twain is not Clemens; nor will it be occupied at any world at which Twain does not exist. Knowing that Twain is Clemens just is knowing that this location is occupied. In general, if 'λ' is a location, then the thought that 'λ' is occupied is just the de dicto λsurrogate of 'λ'. That is, 'λλ' is the thought that 'λ' is occupied. For 'λλ' is true at an index i iff there is some index, j , which is a world-mate of i's, and at which both 'λ' and 'λ' is true-that is to say, iff there is some index, j , which is a world-mate of i's, and at which 'λ' is true. In the next section, I'll use these two kinds of thoughts to provide a twodimensional principle of expert deference. This principlewill allowus to escape the problem cases we encountered in §3. 5 Two-Dimensional Deference In this section, I'm going to suggest that, once you or the expert have opinions in interesting thoughts like 'the coin flip will land Beatrice up', 'I am sick', and 'Today is Monday', we must modify traditional principles which take the form 45. The constraint (pλ) singles out a class of a priori equivalent thoughts, any of which could serve as the de dicto λ-surrogate of 'p'. For instance, either 'Beyoncé is sick' or 'Both Beyoncé is sick and Beyoncé is sick' will be true at any location in any world at which 'I am sick' is true at the location 'I am Beyoncé'. So either could be the de dicto Beyoncé-surrogate of 'I am sick'. Since any probability function will assign the same probability to any two a priori equivalent thoughts, it won't matter which of these a priori equivalent thoughts we take to be the de dicto β-surrogate of 'p'. 32 §5 Two-Dimensional Deference of D1 or D2 (from §3). C(p | E(p) = n%) = n%(D1) C(p | E = E) = E(p)(D2) In the first place, your thoughts 'I am sick' and 'Today is Monday' could be true without the expert's thoughts 'I am sick' and 'Today is Monday' being true. More generally, the truth-conditions of your thought 'p' could differ from the truth-conditions of the expert's thought 'p'. So you shouldn't want to align your credence in 'p' with the expert's credence in that same thought, as the traditional principles of expert deference assume. To deal with this problem, I will suggest that, instead of deferring to the expert by aligning your credence in 'p' with the expert's credence in 'p', you should defer by aligning your credence in 'p' with the expert's credence in a surrogate of 'p'. Roughly, we want to find a thought which, when entertained by the expert, will have the same truthconditions as 'p', when entertained by you. Whatmakes the interesting thoughts interesting is that their truth-conditions depend upon matters about which you may be uncertain. For instance, you may not know who you are, in which case you may not know which truthconditional content your thought 'I am sick' expresses. This will complicate our search for the expert's surrogate for your thought 'p'. For instance, if you are Beyoncé, then your doctor's surrogate for 'I am sick' should presumably be a thought like 'Beyoncé is sick'; and, if you are Kelly, then your doctor's surrogate for 'I am sick' should be 'Kelly is sick'. But what if you're not certain whether you are Beyoncé or Kelly? In that case, it doesn't look like we will be able to find any one surrogate for your thought 'I am sick'. Instead, I'll suggest that, conditional on you being Beyoncé, you should defer to your doctor's opinion about 'Beyoncé is sick'; and, conditional on you being Kelly, you should defer to your doctor's opinion about 'Kelly is sick'. In general, our principles of expert deference must control for any uncertainty you or the expert may have about your location. I'll also suggest that these principles must control for any uncertainty either you or the expert may have about their location. Then, in rough outline, the principle I'll propose says that you should defer to the expert by setting your credence in 'p' to the expert's credence in an appropriate surrogate of 'p', when both your and the expert's credence functions have been conditioned on both your and the expert's locations. (And not just your actual locations, but any locations which you and the expert may occupy.) That's the rough outline. More carefully, here's themodification ofD2 I'll be proposing: for any thought, 'p', any locations λ and ε, and any potential expert function E, your credence in 'p', given that the expert's credence function is E, 33 Two-Dimensional Deference you are located atλ, and ε is occupied, should be equal toE's credence in the de dicto λ-surrogate, 'pλ', given that λ is occupied, and given that they are located at ε. (D4) C(p | E = E ∧λ∧ εε) = E(pλ | λλ ∧ ε) A similar modification could replace D1; I'll leave the details in this footnote.46 In §5.1 below, I'll explain the principle D4 by walking through some examples. I'll also explain how to generalize principles like D3, which say that you should align your credences with your expectation of the expert's credences. Then, in §5.2, I'll apply the lessons of §5.1 to the case of chance. I'll defend a principle of chance deference and use it to treat the problematic cases from §3.1, as well as the case of Sleeping Beauty, discussed in §3.2. In §5.3, I'll apply the lessons of §5.1 to the case of your future self, defend a generalization of van Fraassen's principle of Reflection, and use it to treat the problem cases from §3.2. 5.1 The Principle Explained Suppose you know for sure that you are Beyoncé, and you wish to set your credence in 'I am sick' by deferring to the opinion of your doctor. To do so, you should not set your credence in 'I am sick' equal to your doctor's credence in that same thought. Instead, you must find some surrogate for your thought 'I am sick'. This should be a thought which will be true when entertained by the doctor iff 'I am sick' is entertained by you. In §4.4, I introduced the notions of a location and aworld. A location specifies exactly who you are, as well as when andwhere you are in time and space, in as much detail as your thoughts permit. A world specifies exactly what things are like, apart from the question of who, when, and where you are located. Following Lewis (1979), I called thoughts which are at least partly about your location in the world de se. And thoughts which are only about what the world is like, and aren't at all about your location within the world, I called de dicto. You and your doctor may faultlessly disagree about the de se-for instance, you may truly think 'I am sick' while she truly thinks 'I am not sick'. But you and your doctor may not faultlessly disagree about the de dicto. For this reason, coordination of opinion between you and your doctor should go by way of the de dicto. If your doctor's opinions are going to constrain your credence in the thought 'I am sick', then we must find a de dicto 46. For any thought 'p', any locations λ and ε, and any number n%, your credence in 'p', given that you are located atλ, ε is occupied, and given that the expert's credence in 'pλ' is n%, conditional onλ being occupied and thembeing at ε, should ben%: C(p | λ∧εε∧E(p | λλ∧ε) = n%) = n%. 34 §5 Two-Dimensional Deference surrogate for this thought. Of course, in the present case, a suitable de dicto surrogate is clear: 'Beyoncé is sick' will fit the bill. But this particular solution won't offer any guidance in other cases. It would be preferable to have a general solution, one which will allow us to find a suitable de dicto surrogate for any thought of yours.47 The question to ask yourself is this: 'how confident is the doctor that my thought 'I am sick' is true?' That is: 'how confident is she that, when I entertain the thought 'I am sick', it expresses a truth?' However confident she is of that is how confident you should be in 'I am sick'. In §4.4, I defined the thought 'pλ', which is true at any location in a world so long as 'p' is true at location λ within that world. For this reason, 'pλ' says that the thought 'p', entertained at λ, expresses a truth. It is a general surrogate for your thought that p, given that you are at location λ. Return to the simple example from §4.4. Recall that 'β' is the location 'I am Beyoncé', 's' is the de se thought 'I am sick', and 'b' is the de dicto thought 'Beyoncé is sick'. Then, it is a priori knowable that 'sβ ' is true if and only if b is true. If 'D' is the definite description 'the doctor's credence function', then, for every potential credence functionD , you should defer to the doctor's opinion by setting your credence in s, given that the doctor's credence function is D , equal to D 's credence in the surrogate sβ , which will just be D 's credence in b: C(s | D = D) = D(sβ) = D(b). And, in general, your credence in any thought 'p', given that the expert E 's probability function is E, should be E's credence in pλ: C(p | E = E) = E(pλ). This works well in the present case, but in constructing the surrogate 'sβ ', we took for granted that you were Beyoncé. This is something about which you may uncertain. Suppose that you don't know whether you're Beyoncé or Kelly, and you know that your doctor thinks Beyoncé is very likely sick and Kelly is very likely not sick. In that case, you shouldn't be very confident in 'I am sick'. Instead, you should proportion your confidence in 'I am sick' to your confidence that you are Beyoncé. That is, for each potential credence function D , and each locationλ, you should defer to the doctor's opinion by setting your credence in s, given thatD =D and you are at locationλ, equal toD 's credence in the surrogate 'sλ': C(s | D = D ∧ λ) = D(sλ). If 'λ' is the location 'I am Beyoncé', then 'sλ' will be a priori equivalent to the de dicto thought 'Beyoncé is sick'. And if 'λ' is the location 'I am Kelly', then 'sλ' will be a priori equivalent to the de dicto thought 'Kelly is sick'. In general, your credence in 'p', given that the expert E 's probability function is E and given that you are at location λ, should be E's credence in pλ: C(p | E = E ∧λ) = E(pλ). Just as you can be uncertain about which locations you occupy, your doc47. There is overlap between our task of finding a surrogate for your thought 'p' and the literature on how to update your de se beliefs. Formore, see Titelbaum (2016) and the references therein. 35 Two-Dimensional Deference tor could be uncertain about which locations are occupied. Suppose you know for sure that you are Jekyll, and you are 50% confident that you are also Hyde. However, your doctor is very confident that Jekyll and Hyde are two different people. She has one medical record filed under 'Jekyll' and another filed under 'Hyde'. There are positive test results in the 'Hyde' file, and no test results in the 'Jekyll' file. The test is very reliable and the disease is very rare, so she thinks 'Hyde is sick' is very likely, and she thinks 'Jekyll is sick' is very unlikely. However, were she to learn that Jekyll and Hyde are one and the same person, she would think it's very likely that that person is sick.48 Now, you don't want to treat your doctor as an expert about whether or not you are Hyde. You have more and better evidence about this than she does. So you shouldn't become very confident that you are not Hyde on the basis of her opinion about whether Jekyll is Hyde. However, you do want to treat her as an expert about whether or not you are sick. Let 'η' be the location 'I am Jekyll and I am Hyde'. Then, if we follow the suggestion from the previous paragraph, we'll say that your credence in 'I am sick' ('s'), given that you are Jekyll and Hyde ('η'), should be the doctor's credence in the de dicto thought 'sη '. But the thought 'sη ' is only true in worlds at which the location η is occupied. At any world at which Jekyll and Hyde are different people, 'sη ' will be false.49 Since your doctor is very confident in ∼η, she will think that 'sη ' is very unlikely. So the principleC(p | E = E∧λ) = E(pλ)would tell you to be very confident that you're not sick. This is the wrong verdict. Given that you think you're 50% sure that you're Hyde, and given that the doctor is very confident that Hyde is sick, you should be somewhere around 50% confident that you are sick. The trouble with the principleC(p | E = E∧λ) = E(pλ) is that, on the lefthand-side, you have conditioned on some information-the information that you occupy location λ-which the expert may lack. If the expert doesn't have this information, then before deferring to them, you should first bring them up to speed by conditioning the function E on this information. Of course, the location 'λ' is a purely de se thought; so you don't want to bring the function E up to speed by conditioning it on this information. Instead, you want to bring it up to speed by conditioning it on an appropriate de dicto surrogate of this information. That is, you want to condition it on the de dicto information that location λ is occupied: 'λλ'.50 This thought, 'λλ' is true in any world at which 48. Cf. Chalmers (2011). 49. Recall from §4.4 that 'sη ' is true at an index i iff there is some index j which is a world-mate of i, and at which both 's' and 'η' is true. If Jekyll and Hyde are different people at an index i, then at any index which is a world-mate of i, 'η' will be false. So 'sη ' will be false at any index at which Jekyll and Hyde are different people. 50. By the way, on the left-hand-side, you've also conditioned on the information that the expert's 36 §5 Two-Dimensional Deference 'λ' is true somewhere. In the case of the location 'η', 'ηη ' is the de dicto thought which says that Jekyll andHyde are the same person. Thus, your credence in 's', given that your doctor's credence function is D , and given that you are Hyde ('η'), should be equal to D 's credence in the de dicto η-surrogate 'sη ', given that η is occupied: C(s | D = D ∧ η) = D(sη | ηη). Since your doctor is very confident that Jekyll/Hyde is sick ('sη '), conditional on Jekyll and Hyde being the sameperson ('ηη '), this principlewill tell you, correctly, to be very confident that you are sick, given that you are Hyde. In general, your credence in a thought 'p', given that the expert E 's probability function is E and you are at location λ, should be E's credence in the de dicto λ-surrogate 'pλ', given that location λ is occupied, 'λλ': C(p | E = E ∧λ) = E(pλ | λλ) Notice that, if 'p' is a boring thought, you are certain that you are in location 'λ', and the expert's credence in 'p' is independent of whether λ is occupied, then this principle reduces to the more familiar principle of expert deference, D2. For, if 'p' is a boring thought, then 'pλ' is a priori equivalent to 'p'; and if E's credence in 'p' is independent of whether λλ, then the right-hand-side of the equationmust equal E(p). And, if you are certain of 'λ', then this conjunct may be ignored on the right-hand-side of the ' | '. So, in these special circumstances, the equation reduces to C(p | E = E) = E(p). But wait-what if the expert is uncertain about their location? In some cases, thismaynotmatter. If her location is irrelevant to the question ofwhether you are sick, then you may defer to your doctor about whether you are sick without either of you knowing her location. However, in some cases, her location may be relevant to whether you are sick. Suppose, for instance, that both you and your doctor suffer from amnesia. However, you both know the following four things: 1) you are either Alfred or Byron, and she is either Colleen or Dinah; 2) Alfred and Colleen are fraternal twins, as are Byron and Dinah; 3) the disease is congenital, so if one fraternal twin is sick, then the other one is sick, too; and 4) the doctor is very confident that she is sick. Again, suppose you don't wish to treat your doctor as an expert about who you are or who she is. Suppose that you have more and better evidence about that than she does. However, you do want to treat her as an expert about whether or not you are sick. She's the one who's run all the tests, and she's in a much better position to appraise those tests and apprise you of their improbability function is E; if the expert lacks this information, then you should likewise bring it up to speed by conditioning it on E = E. (See Hall, 1994.) In all the examples I'm considering here, I'm taking it for granted that the expert knows what its own probability function is, so I'm ignoring this complication here. 37 Two-Dimensional Deference port. In this kind of situation, you should only defer to your doctor conditional on a hypothesis about where you and she are located. Let 'α', 'γ ', and 'δ' be the locations 'I am Alfred and the patient', 'I am Colleen and the doctor', and 'I am Dinah and the doctor', respectively. Then, conditional on you being Alfred and the doctor being Colleen, your credence in 'I am sick' ('s') should be the doctor's credence in the de dicto α-surrogate 'sα ', given that you are Alfred and she is Colleen: C(s | D = D ∧ α ∧ γγ ) = D(sα | αα ∧ γ). Since the doctor's credence in sα is high, given that you are Alfred and she is Colleen, you should have a high credence in s, given that you are Alfred and she is Colleen. Likewise, conditional on you being Alfred and her being Dinah, your credence in 'I am sick' should be her credence in the surrogate 'sα ', given that you are Alfred and she is Dinah: C(s | D = D ∧α ∧ δδ) = D(sα | αα ∧ δ). Since the doctor's credence in sα is low, given that you are Alfred and she is Dinah, you should have a low credence in s, given that you are Alfred and she is Dinah. There are three general lessons to learn from the cases we've considered in this section. Firstly, when your thoughts are interesting, coordinating your opinions with those of the expert requires us to find appropriate surrogates for your thought. Secondly, a principle of expert deference doesn't strictly speaking implore you to coordinate your credences with those of the expert. Instead, it directs you to align your conditional credences with those of the expert. So, if your credences have been conditioned on some information, like that you occupy the location 'λ', then you should only align those credences with the expert function's after it has also been conditioned on this information (or a suitable surrogate). Thirdly, if there are some thoughts you do not wish to defer to the expert about, and the expert takes those thoughts to be relevant to the question of whether your thought 'p' is true, then you must bracket those thoughts by aligning your credence with the expert's only after conditioning both your and the expert's credence functions on these thoughts. Supposing that the expert has opinions about their location, and supposing that you don't wish to defer to the expert about your or their location, we can summarize these three lessons with the following principle: for any thought 'p', any locations λ and ε, and any potential expert function E, your credence in 'p', given that the expert function is E, you are at λ and the expert is at location ε, should be equal to E's credence in the de dicto λ-surrogate 'pλ', given that you are at λ and they are at ε: (D4) C(p | E = E ∧λ∧ εε) = E(pλ | λλ ∧ ε) For some experts, we may be able to ignore the expert's location, ε in D4. For instance, the objective chance function at t, Cht , does not have a spatial location. It's only location is temporal, and that location is a priori knowable. If the 38 §5 Two-Dimensional Deference objective chances at t were at t∗, they wouldn't be the objective chance at t.51 In some applications, we may be able to ignore 'λ', 'ε', 'λλ', or 'εε' because either you know your or the expert's location for sure, or else the expert knows their or your location for sure. In particular, notice that, if 'p' is a boring thought, and additionally, your and E 's credence in 'p' is independent of your and the expert's location, then this proposed principle entails the more familiar principle of expert deference D2. For, if your credence in 'p' is independent of your and the expert's location (conditional on E = E) then the left-hand-side of D4 equals C(p | E = E). And, if 'p' is a boring thought, then the right-hand-side of D4 will be equal to E(p | λλ ∧ ε). If E's credence in 'p' is independent of λλ and ε, then this is equal to E(p). So the principle reduces to C(p | E = E) = E(p). 5.2 Two-Dimensional Chance Deference As Imentioned above, when the relevant expert is the objective chances at some future time, t, we can ignore the expert's location in D4. In that case, we get the following principle of chance deference, which I'll call 'CD' (for chance deference): So long as you lack any time t inadmissible information, your credence in 'p', given that the time t objective chance function is Cht and you are located at λ, should be equal to the objective chance of 'pλ', given that you are located at λ. (CD) C(p | Cht = Cht ∧λ) = Cht(pλ | λλ) In this section, I'd like to take this principle out for a test drive by applying it to the problem cases we encountered in §3.1. Firstly, take the case in which, before we flip the coin, we name whichever side lands up 'Beatrice'. In this example, you need not suffer from any uncertainty about your location. Since the coin toss is in your future, the objective chance function will also be certain of your location. Let 'λ' be your known location. If you are certain of 'λ', then we can ignore 'λ' whenever it shows up on the right-hand-side of the ' | '. Likewise, when chance is certain of 'λλ', we can ignore it whenever it shows up on the right-hand-side of the ' | '. So in the present case, our principle tells us this: C(b | Cht = Cht) = Cht(bλ) (Recall, 'b' is the thought that the coin lands Beatrice up.) 51. Of course, you may wish to defer, not to the time t chances, but instead the current chances. Then, even though the current chances know their temporal location, you may not (if you've lost track of the time). In that application, the current chance's temporal location cannot be ignored. 39 Two-Dimensional Deference Recall from §4.3 that the thought 'b"s diagonal content is necessary. It will express a truth no matter how the coin lands. By definition, the thought 'bλ' is true at a world so long as the diagonal content of 'b' is true at the location λ in that world. Since the diagonal content of 'b' is true at every location in every world with positive chance, the thought 'bλ' will be true at every world with positive chance. So you know for sure that the chance of 'bλ' will be 100%. So CD tells you to have a credence of 100% that the coin will land Beatrice up. Letme remind the reader ofwhy I defined 'bλ' in terms of 'b"s diagonal content, rather than, for instance, 'b"s truth-conditional content: as we saw at the end of §4.3, your credences are invested in thoughts, and not truth-conditional contents. In terms of the matrices shown in figures 1, 2, and 3, they do not live along the rows; they instead live along the diagonals. Of course, that's not the situation with objective chance. Objective chances are something like brute propensities for the world to develop in certain ways; so the objects of objective chance are not thoughts, but rather truth-conditional contents. The chance of P is the world's propensity to develop in such a way as to make P true.52 So, when we are coordinating your credences with the objective chances, we need to find a truth-conditional content which acts as a surrogate for your thought. CD's choice of surrogate is the truth-conditional content that your thought, entertained at your location, expresses a truth. It thereby mirrors the solution which D4 applies to experts like your doctor. We don't ask how confident your doctor is in 'I am sick'. Instead, we ask how confident your doctor is that your thought 'I am sick' expresses a truth. Likewise, we don't ask how confident objective chance is in the thought 'the coin lands Beatrice up'-chances don't attach to thoughts, so this question is ill-posed. Instead, we ask how confident chance is in the truth-conditional content that your thought 'the coin lands Beatrice up' expresses a truth. Since chance is maximally confident of this, you should be maximally confident in 'the coin lands Beatrice up'. ContrastCD's prescriptions about 'b' with its prescriptions about the thought 'the coin lands heads up' ('h'). The diagonal content of 'h' is not necessary. The thought 'h' expresses a falsehood at possibilities where the coin lands tails. You know for sure that the objective chance that your thought 'h' expresses a truth is 50%. So CD says that your credence in 'h' should be 50%. Next, consider Sleeping Beauty, discussed in §3.2. When you awake on Monday morning, you will not know whether it is Monday morning or Tuesday morning. Let 'h' be the thought 'the coin flipped Monday evening lands heads', let 'μ' be the location 'it is Monday', and let 'Chm' be the (known) objec52. Or, at least, I will take it for granted that objective chances attach to truth-conditional contents. But see Nolan (2016), who takes the kind of puzzles we're discussing here as a reason to think that chances attach to entities more fine-grained than truth-conditional contents. 40 §5 Two-Dimensional Deference tive chance function on Monday. Then, CD tells us that your credence in 'h', conditional on 'μ', should be 50%. To see this, note that 'h' is a boring thought. It has the same truth-conditional content, no matter which day it is. So we can ignore the subscripts on chance's surrogate. Then, CD tells us: C(h | Chm = Chm ∧μ) = Chm(h | μμ) Since you know for sure what the Monday chances will be, we can ignore the thought 'Chm = Chm' on the right-hand-side of the ' | ' in your credences. Since chance knows for sure that you will be around on Monday morning, we can ignore 'μμ' on the right-hand-side of the ' | ' in the function Chm. Then, since Chm(h) = 50%, we have: (1) C(h | μ) = 50% The constraint given in (1) is powerful. So long as your credence that it is Monday is less than 100% and your credence in 'h', given that it is Tuesday, is 0%, (1) rules out the Halfer's position. For the Halfer says that your unconditional credence in 'h' should be 50%. But the laws of probability require your unconditional credence in 'h' to be a weighted average of your conditional credences C(h | μ) and C(h | τ) (where τ is the location 'It is Tuesday'), with weights given by your credences C(μ) and C(τ), respectively. Since C(h | τ) is uncontroversially 0%, C(h) = Ch(h | μ) *C(μ). So (1) tells us that, if C(μ) < 100%, then C(h) < 50%. In contrast, the constraint (1) is perfectly compatible with the Thirder's position. In fact, if we additionally take for granted that, conditional on the coin landing tails, it's just as likely to be Monday as Tuesday, (1) entails the Thirder's position. For there are three relevant possibilities: either the coin lands heads, 'h', (in which case it must be Monday), or the coin lands tails and it's Monday, '∼h ∧ μ', or the coin lands tails and it's Tuesday '∼h∧τ '. Equation 1 entails that your credence in the first possibility equals your credence in the second, C(h) = C(∼h∧μ). And, if you think it's just as likely to be Monday as Tuesday, given that the coin lands tails, then your credence in the second possibility equals your credence in the third, C(∼h∧μ) = C(∼h∧τ). If you give the same credence to each of the only three possibilities, then your credence in each must be 1/3.53 In my statement of CD, I restricted it to cases in which you lack inadmissible evidence. However, the principle will still constrain your credence, even when you have inadmissible evidence. Suppose you have the inadmissible evidence 'e'. Then, CD says that your credence in 'p', given 'Cht = Cht ' and 'λ', 53. This argument comes from Elga (2000). 41 Two-Dimensional Deference must be Cht(pλ | λλ ∧ eλ). Or at least, this is true if we take for granted the principle of conditionalization, according to which you current credence function C should be some prior credence function C0 conditioned on your total evidence, e. For then, C(p | Cht = Cht ∧λ) = C0(p | Cht = Cht ∧λ∧ e) = C0(p∧ e | Cht = Cht ∧λ) C0(e | Cht = Cht ∧λ) = Cht(pλ ∧ eλ | λλ) Cht(eλ | λλ) = Cht(pλ | λλ ∧ eλ) (The first line follows from conditionalization. The second line then follows from the product rule on the assumption that C(p ∧ e ∧ Cht = Cht ∧λ) > 0. The third line follows from two applications ofCD-once in the numerator, and once in the denominator.54 Finally, the fourth line follows from the product rule.)55 This observation allows us to provide a different account of which evidence is admissible. Rather than saying that evidence is time t admissible iff it is information entirely about the past at t, we may say that the information 'e' is time t admissible iff, for every potential chance function Cht and every potential location 'λ', Cht(eλ) = 100%.56,57 So long as 'e' is admissible, it won't make any difference to the chance of 'pλ', in the sense that Cht(pλ | λλ ∧ eλ) = Cht(pλ | λλ). (Recall that, if Cht(eλ) = 100%, then 'eλ' may be ignored on the right-hand-side of the ' | '.) One important difference between this criterion of admissibility and the criterion given in terms of aboutness concerns whether inadmissibility agglomerates. If 'e' is about times after t, then 'e∧ f ' will be about times after t, for any 'f '. So, with the aboutness criterion of admissibility, once you have some inadmissible information, gaining any additional information will still leave you with inadmissible information. However, this doesn't follow from the revision I am proposing here. To appreciate this point, notice that, in Sleeping Beauty, on Monday morning, before learning what day it is, the de se evidence that you are now awake 54. On the third line, I am also taking it for granted that (p ∧ e)λ = pλ ∧ eλ. Given the usual definition of '∧', this is a consequence of the definition (pλ) from §4.4. 55. A similar point is made about Lewis's principle LCD in Hall (1994). 56. The chance function Cht is a potential chance function iff C(Cht = Cht) > 0. And 'λ' is a potential location iff C(λ) > 0. 57. Similar definitions of 'admissibility' appear throughout the literature. See, e.g., Pettigrew (2012). 42 §5 Two-Dimensional Deference ('a') will count as inadmissible (for the Monday evening chances).58 After all, the Monday evening chances are not certain that you will be awake on Tuesday, Chm(aτ ) = 50%. Since you assign positive credence to being at the location 'τ ' ('It is Tuesday'), this means that 'a' is inadmissible.59 However, when you learn that it is Monday, the evidence a∧ μ will not be inadmissible. For, once you know for sure that it is Monday, 'τ ' is no longer a potential location. Once you learn that it is Monday, 'μ' is the only potential location, and Chm(aμ∧μμ) = 100%. So, while the evidence awas inadmissible, the evidence a∧μ is admissible.60 Finally, recall the example we discussed in §3.1 below: You have lost track of the day, and you aren't sure whether it is Tuesday or Wednesday. But you know for sure that the current chance of 'Mudskipper wins' ('m') is 75%, and that yesterday, the chance of 'm' was 25%. I argued that, in this case, you don't have any information which is about matters after Tuesday. And we saw that, if you don't have any evidence like this, then SCD implies, implausibly, that your credence in 'm' should be 50%. In contrast, CD implies that your credence in 'm' should be 75%. Let 'τ ' be the location 'It is Tuesday', and let 'ω' be the location 'It is Wednesday'. Then, you have the evidence that the current chance of 'm' is 75%-call that evidence 'Chnow(m) = 75%'. This evidence is inadmissible for the Tuesday chances. To see this, note first that the de dicto ω-surrogate of 'Chnow(m) = 75%' is 'Chω(m) = 75%'. And there there is a potential Tuesday chance function-the one which gives 'm' a chance of 25%, call it 'Ch25%'-according to which 'm' is not independent of 'Chω(m) = 75%'. You are certain that this is the Tuesday objective chance function iff today is Wednesday, so CD says that: C(m | Cht = Ch25%) = C(m | Cht = Ch25% ∧ω) = C(m | Cht = Ch25% ∧ω∧Chnow(m) = 75%) = Ch25%(m | Chω(m) = 75%) 58. Cf. Horgan (2004) and Weintraub (2004). 59. Notice that the de se evidence that you are nowawakewon'tmake any difference to our derivation of constraint (1) above, since C(aμ) = Chm(aμ) = 100%. So C(h | Chm = Chm ∧ a ∧ μ) = Chm(h | μμ ∧ aμ) = Chm(h | μμ) = Chm(h) = 50%. 60. Note that this criterion of admissibility could be accepted by somebody who accepts SCD, though the resulting view would tell us strictly less than CD does. For instance, CD implies that, in Sleeping Beauty, your credence in h, given that it is Tuesday, should be Cht(h | aτ ), whereas SCD would say that you have inadmissible evidence and fall silent. (Note that the derivation of the general constraintC(p | Ch = Cht∧λ) = Cht(pλ | λλ∧eλ) took for granted that 'eλ' was in the domain of the chance function. And while the de dicto Tuesday-surrogate of a self-locating thought like 'I am now awake' will be in the domain of the chance function, the de se thought 'I am now awake' will not be.) 43 Two-Dimensional Deference = 75% This first line is true because you know that Ch = Ch25% iff ω. The second line follows because you are certain that the current chance of 'm' is 75%. The third line follows from CD, together with the fact that 'Chω(m) = 75%' is the de dicto ω-surrogate of 'Chnow(m) = 75%'. And the final line follows on the assumption that the objective chance function defers to its future self. There is only one other potential Tuesday chance function. That's the one which gives 'm' a chance of 75%-call it 'Ch75%'. You are certain that this is the Tuesday objective chance function iff today is Tuesday. So CD says that: C(m | Cht = Ch75%) = C(m | Cht = Ch75% ∧ τ) = C(m | Cht = Ch75% ∧ τ ∧Chnow(m) = 75%) = Ch75%(m | Cht(m) = 75%) = Ch75%(m) = 75% So your credence in 'm' is 75%, conditional on both Ch = Ch25% and Ch = Ch75%. Since these are the only two potential Tuesday objective chance functions, your unconditional credence in 'm' must be 75%, by the law of total probability. 5.3 Two-Dimensional Deference to Your Future Self Let the relevant expert be the credences you are disposed to adopt after you learn, when you are guaranteed to learn exactly one of the thoughts in E = {e1, e2, . . . , eN }. Then, the principle D4 says that, for any thought 'p', any locations 'λ' and 'ε', and any potential updated credence functionDe (where 'De' is the credence function you're disposed to adopt if you learn e), your credence in 'p', given that you are at λ, your future credences are given by De, and your future self is at ε, should be equal toDe's credence in 'pλ', given that you are at λ and they are at ε. C(p | D =De ∧λ∧ εε) =De(pλ | λλ ∧ ε) From this principle, it follows that, conditional on you and your future self 's locations, your current credence that p should equal your expectation of your future credence that p.∑ e∈E De(pλ | λλ ∧ ε) *C(D =De | λ∧ εε) = C(p | λ∧ εε) 44 §5 Two-Dimensional Deference This principle says that, once any confusion about you and your future self 's locations have been cleared up, your learning dispositions should be unbiased. In all the cases I'll consider in this section, neither you nor your future self suffers from any uncertainty about your current location, and you are certain about which location your future selves will occupy, in which case the above equation entails: (Reflection∗) ∑ e∈E De(pλ | ε) *C(D =De) = C(p) In §3.2, we saw some counterexamples to the principle Reflection. For instance, take the following straightforward counterexample: on Sunday, before going to sleep, I know for sure that it is Sunday. However, I also know for sure that tomorrow morning I'll learn that I've awoken, and upon learning that I've awoken, I'm disposed to be not very confident at all in 'It is Sunday'. These learning dispositions violate Reflection, but they do not violate Reflection∗. For, supposing that yourMondaymorning self suffers fromno ignorance about their location, Reflection∗ tells us that your current credence in 'It is Sunday' should equal your expectation of your Monday morning credence, not in 'It is Sunday', but instead in its de dicto Sunday-surrogate of 'It is Sunday'. This surrogate says that the thought 'It is Sunday' expresses a truth when it is entertained on Sunday. Since this is a priori, your Monday morning self will know it for sure, and, while you will violate Reflection, you will not violate Reflection∗. Or consider Arntzenius's case in which you sit in a room without a clock, knowing that the lights will go off at 12:00 iff a fair coin landed heads. At 9:00, when you know the time, your credence that the coin lands heads is 50%, or 1/2. At 11:30, after learning that somewhere between 2 and 4 hours have passed, suppose your credence that the coin has landed heads will be 37.5%, or 3/8. In this case, after learning, you are uncertain about your location. Let 'ε' be the location 'It is 11:30'. Then, at 11:30, you are uncertain about whether ε. However, at 9:00 you know for sure that, at 11:30 you will be at 11:30-so you know 'εε' for sure. So your current credence that the coin landed heads ('h'), conditional on 'εε', will be the same as your current unconditional credence in 'h', which is 50%. And even though your 11:30 credence in 'h' is 37.5%, your 11:30 credence in 'h' conditional on it being 11:30, is 50%. So your learning dispositions do not violate Reflection∗ in this case. Or consider Sleeping Beauty. As we saw in §3.2, both the Thirder and the Halfer violates Reflection in this case. However, the Thirder will satisfy Reflection∗. Let 'μ' be the location 'It is now Monday morning'. You know for sure what your credences will be on Monday morning; for you know for sure that, on Monday morning, you will learn only that you've awoken, a, and nothing else. So, if Da is the credence function you're disposed to adopt upon 45 Two-Dimensional Deference learning that you've awoken, then Reflection∗ requires that Da(hσ | μ) = C(h) (where 'σ ' is the location 'It is Sunday'.) Since 'h' is a de dicto thought, 'hσ '= 'h', so we can ignore the subscripted 'σ '. If you are a Thirder, then you are disposed to adopt a credence function upon learning that a on Monday morning which is such thatDa(h | μ) = 1/2. And on Sunday, you think the coin is just as likely to land heads as tails, C(h) = 1/2. So, when it comes to your Monday morning credences, the Thirder will satisfy the principle of Reflection∗. Your Tuesday morning credences are more complicated, and more interesting. Before discussing them, however, let me consider a different case.61 Colonel Aureliano Buendía faces the firing squad, the ranks of which are filled with men who once served under his command. He does not know whether these men are still loyal or not. However, in the minute after the order is given, one of two thingswill be the case: either Colonel Buendíawill be dead, inwhich case he won't have any credences about whether his men are loyal, or he will survive (perhaps because themen do not fire, perhaps because they all miss). If he survives, he will be nearly certain that the men are loyal-and rationally so. Learning that all of the men have either refrained from firing or have missed is strong evidence that those men are loyal.62 This case might be seen as a problem for a principle like Reflection∗, since Buendía has only one potential future credence function, and it is muchmore confident in the de dicto thought 'The men are loyal' than Buendía is currently. However, I think that, properly undestood, Reflection∗ does not apply in this case. For Reflection∗ simply presupposes that, no matter what happens, at the relevant future time, your learning dispositions will manifest themselves with the adoption of some credence function or other. Against the backdrop of this presupposition, it goes on to assert that, in expectation, these learning dispositions shouldn't be biased for or against any thought. However, if we are taking seriously the possibility of you not having any credences at all at the relevant time, then this presupposition is false. In that case, there will be potential outcomes in which your future credence is undefined. And if a quantity is undefined in some possibilities with positive credence, then you cannot take an expectation of that quantity. Nonetheless, you can take a conditional expectation of the quantity-you can ask what value you expect it to take on, conditional on it taking on a value at 61. I borrow this case from Leslie (1989), who introduces it in his discussion of the 'fine-tuning' argument. My presentation of the case differs in superficial ways from Leslie's. 62. At least, I'm inclined to take this as a datum, but there's been some dispute about this in the literature. See, for instance, Sober (2005); but note that, in his 2009, Sober ends up changing his mind and agreeing that Buendía's survival is evidence of loyalty. 46 all. So, when it comes to principles like Reflection∗, we can ask about what credence in 'p' you expect to have, conditional on your credences being defined. And we can say that this conditional expectation should match your current credence in 'p', conditional on your credences being defined. Let 'D , ∗' say that your future credences are defined. Then, we should modify Reflection∗ so that it says that your expectation of your future credence that p, conditional on your future credences being defined, should equal your current credence that p, conditional on your future credences being defined. That is, if λ is your location and ε is your future self 's location, then for any thought 'p', (Reflection∗∗) ∑ e∈E De(pλ | ε) *C(D =De | D , ∗) = C(p | D , ∗) Return to your Tuesday morning credence in 'Monday's coin flip lands heads' ('h'). If the coin flip on Monday lands heads, then you will be asleep on Tuesday, and you will not be around to have any credences about the day or about whether the coin flip on Monday lands heads or tails. Let 'τ ' be the location 'It is Tuesday', and let 'D' be the definite description 'your Tuesday morning credences'. Then, you know for sure that, if D is defined, then you will have learnt that you are awake, a. So, ifD is defined, there is only one potential credence function, Da. And Reflection∗∗ says that it should satisfy the following equality: Da(h | τ) = C(h | D , ∗) Upon waking on Tuesday, your credence that the coin lands heads, given that it is Tuesday, will be 0%. And, on Sunday, your credence that the coin lands heads, given that you're around to have credences on Tuesday, will be 0%. So you will satisfy the extended principle Reflection∗∗. References Arntzenius, Frank. 2003. "Some Problems for Conditionalization and Reflection." Journal of Philosophy, vol. 100 (7): 356–370. [18], [20], [45] Bostrom, Nick. 2007. "Sleeping beauty and self-location: A hybrid model." Synthese, vol. 157: 59–78. [20] Braun, David. 2016. "The Objects of Belief and Credence." Mind, vol. 125 (498): 469–497. [6] Briggs, R. A. 2009. "Distorted Reflection." The Philosophical Review, vol. 118 (1): 59–85. [17] Chalmers, David J. 2004. "Epistemic Two-Dimensional Semantics." Philosophical Studies, vol. 118 (1/2): 153–226. [5] -. 2006a. "The Foundations of Two-Dimensional Semantics." In Two-Dimensional Semantics: Foundations and Applications, M. GarciaCarpintero & J. Macia, editors. Oxford University Press, Oxford. [3], [5], [23], [28] -. 2006b. "Two-Dimensional Semantics." In Oxford Handbook of the Philosophy of Language, E. Lepore & B. Smith, editors. Oxford University Press, Oxford. [3], [5], [23], [28] -. 2011. "Frege's Puzzle and the Objects of Credence." Mind, vol. 120 (479): 587–635. [6], [13], [36] -. 2012. Constructing the World. Oxford University Press, Oxford. [28] Davies, Martin & Lloyd Humberstone. 1980. "Two Notions of Necessity." Philosophical Studies, vol. 38 (1): 1–30. [3], [23] Dorr, Cian. 2002. "Sleeping Beauty: In Defense of Elga." Analysis, vol. 62 (276): 292–296. [20] Dorst, Kevin. forthcoming. "Evidence: A Guide for the Uncertain." Philosophy and Phenomenological Research. [9] Elga, Adam. 2000. "Self-locating belief and the Sleeping Beauty problem." Analysis, vol. 60 (2): 143–147. [3], [19], [20], [41] -. 2004. "Defeating Dr. Evil with Self-Locating Belief." Philosophy and Phenomenological Research, vol. 69 (2): 383–396. [20] -. 2007. "Reflection and Disagreement." Noûs, vol. 41 (3): 478–502. [17] -. 2013. "The puzzle of the unmarked clock and the new rational reflection principle." Philosophical Studies, vol. 164: 127–139. [9] Evans, Gareth. 1979. "Reference and Contingency." The Monist, vol. 62 (2): 161–189. [3], [23] Fitts, Jesse. 2014. "Chalmers on the Objects of Credence." Philosophical Studies, vol. 170 (2): 343–358. [6] Frege, Gottlob. 1892. "Über Sinn und Bedeutung." Zeitschrift für Philosophie und philosophische Kritik, vol. 100: 25–50. English translation in P. Geach and M. Black (eds. and trans.), 1980, Translations from the Philosophical Writings of Gottlob Frege, Oxford: Blackwell, third edition. [5] Gaifman,Haim. 1988. "ATheory ofHigherOrder Probabilities." InCausation, Chance, and Credence: Proceedings of the Irvine Conference on Probability and Causation, Brian Skyrms & William L. Harper, editors, vol. 1, 191– 220. Kluwer Academic Publishers, Dordrecht. [8] Gallow, J. Dmitri. forthcoming. "Updating for Externalists." Noûs. [9], [17], [18] -. ms. "Local and Global Deference." [8], [9] Gibbard, Allan. 2012. Meaning and Normativity. Oxford University Press, Oxford. [6] Hall, Ned. 1994. "Correcting the Guide to Objective Chance." Mind, vol. 103 (412): 505–517. [9], [37], [42] Halpern, Joseph Y. 2004. "Sleeping Beauty Reconsidered: Conditioning and reflection in asynchronous systems." In Proceedings of the Twentieth Conference on Uncertainty in AI, 226–234. [20] Hawthorne, John & Maria Lasonen-Aarnio. 2009. "Knowledge and Objective Chance." InWilliamson onKnowledge, PatrickGreenough&Duncan Pritchard, editors, 92–108. Oxford University Press, Oxford. [2] Hitchcock, Christopher. 2004. "Beauty and the Bets." Synthese, vol. 139: 405–420. [20] Hitchcock, Christopher & Mitchell S. Green. 1994. "Reflections on Reflection: van Fraassen on Belief." Synthese, vol. 98 (2): 297–324. [17] Horgan, Terrence. 2004. "Sleeping Beauty awakened: New odds at the dawn of the new day." Analysis, vol. 64: 10–24. [20], [43] Ismael, Jeann. 2008. "Raid! Dissolving the Big, Bad Bug." Noûs, vol. 42 (2): 292–307. [9] -. 2015. "In Defense of IP: A Response to Pettigrew." Noûs, vol. 49 (1): 197– 200. [9] Jackson, Frank. 1998. From Metaphysics to Ethics: A Defence of Conceptual Analysis. Oxford University Press, Oxford. [3], [23] Kaplan, David. 1978. "Dthat." In Syntax and Semantics, Peter Cole, editor, 221–243. Academic Press, New York. [3] -. 1989. "Demonstratives: An Essay on the Semantics, Logic, Metaphysics and Epistemology of Demonstratives and other Indexicals." InThemes From Kaplan, John Perry Joseph Almog & Howard Wettstein, editors, 481– 563. Oxford University Press, Oxford. [3], [23] Leslie, John. 1989. Universes. Routledge, London. [46] Lewis, David K. 1979. "Attitudes De Dicto and De Se." The Philosophical Review, vol. 88 (4): 513–543. [12], [30], [34] -. 1980. "A Subjectivist's Guide to Objective Chance." In Studies in Inductive Logic and Probability, Richard C. Jeffrey, editor, vol. II, 263–293. University of California Press, Berkeley. [9], [10], [11], [15], [21] -. 1994. "Humean Supervenience Debugged." Mind, vol. 103 (412): 473–490. [9] -. 1999. "Why Conditionalize?" In Papers in Metaphysics and Epistemology, vol. 2, chap. 23, 403–407. Cambridge University Press, Cambridge. [9] -. 2001. "Sleeping Beauty: reply to Elga." Analysis, vol. 61 (3): 171–176. [3], [20], [21] Meacham, Christopher J. G. 2008. "Sleeping Beauty and the Dynamics of De Se Belief." Philosophical Studies, vol. 138 (2): 245–269. [20] -. 2010. "Two Mistakes Regarding the Principal Principle." British Journal for the Philosophy of Science, vol. 61 (2): 407–431. [10] Monton, Bradley. 2002. "Sleeping Beauty and the Forgetful Bayesian." Analysis, vol. 62 (1): 47–53. [20] Nolan, Daniel. 2016. "Chance and Necessity." Philosophical Perspectives, vol. 30 (1): 294–308. [2], [40] Pettigrew, Richard. 2012. "Accuracy, Chance, and the Principal Principle." Philosophical Review, vol. 121 (2): 241–275. [42] Salmón, Nathan. 2019. "ImpossibleOdds." Philosophy and Phenomenological Research, vol. 99 (3): 644–662. [2] Salow, Bernhard. 2018. "The Externalist's Guide to Fishing for Compliments." Mind, vol. 127 (507): 691–728. [17] Schwarz, Wolfgang. 2014. "Proving the Principal Principle." In Chance and Temporal Asymmetry, Alistair Wilson, editor, 81–99. Oxford University Press. [11], [12], [14] Singer, Daniel Jeremy. 2014. "Sleeping beauty should be imprecise." Synthese, vol. 191 (14): 3159–3172. [20] Skyrms, Brian. 1980. "Higher Order Degrees of Belief." In Prospects for Pragmatism, D. H. Mellor, editor, chap. 6, 109–137. Cambridge University Press. [8] Sober, Elliot. 2005. "The Design Argument." In Blackwell Guide to the Philosophy of Religion, William E. Mann, editor, chap. 6, 117–147. Blackwell Publishing, Malden, MA. [46] -. 2009. "Absence of evidence and evidence of absence: evidential transitivity in connection with fossils, fishing, fine-tuning, and firing squads." Philosophical Studies, vol. 143 (1): 63–90. [46] Spencer, Jack. 2020. "No Crystal Balls." Noûs, vol. 54 (1): 105–125. [11], [13], [14] Stalnaker, Robert C. 1978. "Assertion." In Syntax and Semantics 9: Pragmatics, P. Cole, editor, 315–332. Academic Press, New York. [3], [23] Talbott, W. J. 1991. "Two Principles of Bayesian Epistemology." Philosophical Studies, vol. 62: 135–150. [17], [18] Titelbaum, Michael G. 2015a. "An Embarrassment for Double‐Halfers." Thought, vol. 1 (2): 146–151. [2], [21] -. 2015b. "Rationality's Fixed Point (Or: In Defense of Right Reason)." Oxford Studies in Epistemology, vol. 5: 253–294. -. 2016. "Self-LocatingCredences." InTheOxfordHandbook of Probability and Philosophy, Alan Hájek & Christopher R. Hitchcock, editors, chap. 31. Oxford University Press, Oxford. [35] van Fraassen, Bas C. 1984. "Belief and the Will." The Journal of Philosophy, vol. 81 (5): 235–256. [16], [17], [34] -. 1995. "Belief and the Problem of Ulysses and the Sirens." Philosophical Studies, vol. 77: 7–37. [16] Weintraub, Ruth. 2004. "Sleeping Beauty: the Simple Solution." Analysis, vol. 64 (1): 8–10. [20], [43] Yli-Vakkuri, Juhani & John Hawthorne. 2018. Narrow Content. Oxford University Press, Oxford. [5] Zagzebski, Linda Trinkaus. 2012. Epistemic Authority: A Theory of Trust, Authority, and Autonomy in Belief. Oxford University Press, Oxford. [4]