Imprecise Epistemic Values and Imprecise Credences

IMPRECISE EPISTEMIC VALUES AND IMPRECISE CREDENCES Abstract. A number of recent arguments purport to show that imprecise credences are incompatible with accuracy-first epistemology. If correct, this conclusion suggests a conflict between evidential and alethic epistemic norms. In the first part of the paper, I claim that these arguments fail if we understand imprecise credences as indeterminate credences. In the second part, I explore why agents with entirely alethic epistemic values may end up in an indeterminate credal state. Following William James, I argue that there are many distinct alethic values a rational agent can have. Furthermore, such an agent is rationally permitted not to have settled on one fully precise value function. This indeterminacy in value will sometimes result in indeterminacy in epistemic behavior—i.e., because the agent’s values aren’t settled, what she believes may not be either.

0. Introduction Here are two brief but hard questions in epistemology: (1) What is the relationship between alethic and evidential norms? (2) How should an agent’s epistemic values affect her epistemic behavior? The first question is about the relationship between evidence and the truth. Presumably, following your evidence is generally a good guide to the truth, but truth and evidence can and often do come apart. Can the goal of truth ever come into conflict with the goal of following one’s evidence? The second question is about the relationship between an what an epistemic agent cares about on the one hand and what she ends up believing on the other. In the practical case, rational agents who like vanilla ice cream will behave differently from agents who prefer chocolate. Epistemic agents also have different epistemic values—some care about truth, while others care about explanation or justification. Should such differences lead to differences in what they think even when they have the same evidence? I don’t pretend to have full or complete answers to these questions. However, there has recently been an alleged conflict between two major programmes in formal epistemology whose resolution will, I think, at least shed some light on both. The first is the Imprecise Credences programme, which is motivated primarily by evidential considerations. According to orthodox philosophical bayesianism, rational agents ought to assign a precise level of confidence—standardly represented as a real number between 0 and 1—to each proposition under consideration. This requirement, simple and elegant as it is, seems like an undue restriction on rational doxastic states. Consider the claim that the person sitting next to you has at least three cans of garbanzo beans in her cupboard, or that Greece will leave the Euro by 2030, or that Homer was a woman. At first glance, it’s absurd to think that rational agents are required to assign exact numerical credences to any of these propositions. 1

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A more reasonable model allows for imprecise credences. Instead of using a single probability function to represent a doxastic state, we allow for a set of them. On this approach, an agent’s level of confidence in some proposition can be represented by, say, the interval [.2, .3]. Here, she is at least 20% confident and at most 30% confident, but there’s no number x such that she’s exactly x% confident. Imprecise bayesianism looks like a significant improvement over its orthodox rival in evidential terms. The evidence we have for certain claims simply doesn’t single out a single credence over the others. At the very least we ought to permit agents to have set-valued credences, or so it seems. Imprecise credences, however, seem incompatible with a second programme, called accuracy-first epistemology, which is motivated by alethic considerations. From this point of view, imprecise bayesianism looks like dead weight at best. Accuracy-firsters think that the only thing of final epistemic value is accuracy— proximity between doxastic attitude and truth-value. On the precise model, it’s relatively easy to cash out what this means: if p is true (false), the closer your credence is to 1 (0), the more accurate it is, and the better off you are all epistemic things considered. In general, following evidential norms is good, but not because you care per se about obeying the evidence. Instead, obeying evidential norms is a good means for attaining the end of accuracy. It’s far less clear how to understand what accuracy could amount to when we move to an imprecise model. Is it more accurate to have a confidence level of [.2, .3] in a falsehood than a confidence level of [.21, .29]? Is either of those more accurate than a precise credence of .26? It’s hard to say. As we’ll explore in a bit more detail below, a number of philosophers have argued that any attempt to measure the accuracy of imprecise credences will inevitably render them idle at best from an alethic point of view. So now we’re in a bit of a quandary. If, as I do, you favour an accuracy-first epistemology, then it appears you have no way of rationalising an imprecisecredence model. Given the intuitive attractiveness of imprecise credences from an evidential point of view, this seems like a cost. Conversely, if you favour imprecise credences, then it appears you have to deny that they have anything to do with pursuit of the truth. The evidential norms you favour are, by your own lights, at least partially disconnected from the aim of fitting your doxastic state to the world. The first goal of this paper is to propose one attractive way to resolve the tension. The second goal is to explore how purely alethic epistemic value can affect epistemic behaviour. The basic picture goes as follows. Even if all a rational epistemic agent should care about is accuracy, there are different reasonable ways of precisifying and pursuing accuracy. Rationality alone does not force agents to choose any precise notion of accuracy nor any precise strategy for achieving accuracy. Rational agents are therefore permitted to have indeterminate yet entirely alethic epistemic values. Indeterminate values in turn lead to indeterminate credences, which we can naturally identify with imprecise credences. On this interpretation, when we say an agent has imprecise credence [.2, .3] toward some proposition X, we don’t mean that her credence is literally the interval [.2, .3]. Instead, we mean that there’s no fact of the matter whether her credence in X is really .22, .29,

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or any other element of [.2, .3].1 This understanding of imprecise credences as indeterminate credences not only escapes the incompatibility arguments but also provides a positive account of how they fit with accuracy-first epistemology. In a slogan: Indeterminate values generate indeterminate credences. To be clear, the aim is not to propose a knock-down argument for imprecise credences nor for interpreting them as indeterminate. Indeed, there will be a number of optional choice points along the way. Instead, the goal is to present a plausible and well-motivated picture of epistemic value that renders accuracy-first epistemology and imprecise credences compatible and that allows for differences in epistemic value to affect epistemic behaviour. Here’s the plan. §1 presents imprecise credences, accuracy-first epistemology, and the basic argument for their incompatibility. §2 motivates the indeterminate interpretation of imprecise credences and shows why such an interpretation evades the incompatibility arguments. §3 explores the nature and variety of alethic epistemic value, demonstrates that indeterminate values lead to indeterminate credences, and discusses how we ought to make sense of imprecise credences from an accuracy-first perspective. §4 contrasts our account of imprecise credences with some alternatives. First we show that on the accuracy-first view, unlike on more orthodox accounts, agents with imprecise credences need not update by pointwise conditionalisation. Second, we argue that the imprecise view we’ve developed has some advantages over permissive bayesianism, according to which agents must select a single precise credence of their choosing from a set of maximally rational alternatives. §5 wraps up. 1. Imprecise Bayesianism and Accuracy-First Epistemology 1.1. Motivating Imprecise Credences. We often have evidence that is incomplete and non-specific. Consider, for instance, the following example from (Joyce 2010, 283): Black/Grey: An urn contains a large number of coins which have been painted black on one side and grey on the other. The coins were made at a factory that can produce coins of any bias β : (1 − β) where β, the objective chance of the coin coming up black, might have any value in the interval 0 < β < 1. You have no information about the proportions with which coins of various biases appear in the urn. If a coin is drawn at random from the urn, how confident should you be that it will come up black when tossed? Precise bayesians differ on exactly what you should think about the claim that the coin will come up black, which we’ll refer to as B. Some are subjectivists and think that any number of opinions consistent with the case are permissible, e.g., any level of confidence between 0 and 1.2 Others are objectivists, who in this case, at least, will usually advocate that you adopt a uniform prior over values of β and end up assigning credence 1/2 to B.3 1

This account of imprecise credences is developed in (Rinard 2015).

2 3

The most prominent example is (de Finetti 1964).

Although objective bayesians tend to agree that you ought to have credence 1/2 in this case, they disagree in general about how to choose a credence function. For a variety of different flavours of objective bayesianism, see (Carnap 1950; Jaynes 2003; Rosenkrantz 1981; Solomonoff 1964a,b; Williamson 2010).

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However, both of these responses require the agent to take a more definite stand on matters than the evidence alone seems to justify. Imprecise bayesians argue that, at the least, the nature of the evidence permits the agent not to have a completely precise attitude toward B. Her evidence is unspecific and incomplete, and her opinion can be too. Joyce (2010), for instance, argues: [A uniform prior] commits you to thinking that in a hundred independent tosses of the black/grey coin the chances of black coming up fewer than 17 times is exactly 17/101, just a smidgen ( = 1/606) more probable than rolling an ace with a fair die. Do you really think that your evidence justifies such a specific probability assignment? […] Or, to take another example, are you comfortable with the idea that upon seeing black on the first toss you should expect a black on the second toss with a credence of exactly 2/3, or, more generally, that seeing s blacks and N − s greys should lead you to expect a black on the next toss with a probability of precisely s + 1/N + 2? […] Again, the evidence you have about the coin’s bias (viz., nada!) is insufficient to justify such a specific inductive policy. Of course, any sharp credence function will have similar problems. Precise credences, whether the result of purely subjective judgments or “objective” rules […] always commit a believer to extremely definite beliefs about repeated events and very specific inductive policies, even when the evidence comes nowhere close to warranting such beliefs and policies. (pp. 283-4) One more general objection to precise credences that goes to the core of Joyce’s point is that they require agents to have an opinion about the comparative likelihood of each proposition under consideration. You must have your mind made up about whether X is more likely than Y , Y more likely than X, or that X and Y are equally likely. This requirement for totality sometimes goes beyond your evidence—you may have no basis for having a firm opinion about their relative likelihoods. Precise credences are thus in some evidential circumstances at best optional and at worst irrational according to imprecise bayesians. For what follows, we’ll look at the former option, i.e., making sense of the permissibility of imprecise credences. That is, the thesis we’ll try to make sense of from an accuracy-first perspective is: Imprecise: In the face of certain kinds of evidence, it is sometimes rationally permissible to adopt imprecise credences. This permissive version of Imprecise seems to me well-motivated from an evidentialist perspective. That is, imprecise credences seem like a perfectly rationally appropriate response to some kinds of evidence. However, as we’ll see over the next few pages, it’s far harder to make sense of them from an alethic perspective. 1.2. Accuracy-First Epistemology. There are lots of reasons a particular credal state might be epistemically good. It might, for instance, be justified by the evidence, or informative, or coherent, or explanatory. Crucially, it could also turn

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out to be highly accurate. The last of these properties, according to accuracyfirst epistemology, is what really matters. Accuracy is the sole source of epistemic value. The higher your credence in truths, and the lower your credence in falsehoods, the better off you are all epistemic things considered.4 Many norms of interest in epistemology—probabilism, conditionalisation, the principal principle, and so on—don’t tell us directly to be accurate. So, accuracyfirsters must justify them through their connection with the rational pursuit of accuracy. Either violating them renders agents unnecessarily inaccurate, or following them is somehow a good means toward the end of accuracy. For AFE to be successful, we then need an account of accuracy and explicit principles of rational choice. That is, we need to be able to show how an epistemic norm is part and parcel of the rational pursuit of epistemic value. We’ll first look at how we might carry this idea out in the context of precise bayesianism and then see why apparent problems arise when we extend it to imprecise credences. 1.3. The Epistemic Utility Programme. Because accuracy functions as a measure of a credence’s epistemic value at a world, it’s natural to treat accuracy as epistemic utility akin to practical utility. We can then turn to decision theory to discover candidate principles of rational choice to derive epistemic norms of interest. This idea is best illustrated with an example. 1.3.1. Probabilism. Joyce (1998, 2009) argues that a (precise) agent’s credence function should obey the axioms of probability as follows: Suppose Bob has credence function b, which doesn’t obey the axioms of probability. Then on any legitimate measure of accuracy, there’s some probability function b0 that is strictly more accurate at every world than b is. That is, on every legitimate measure, every non-probabilistic credence function b is strictly accuracy-dominated by some coherent function. Furthermore, no probability function is even weakly accuracy-dominated.5 Joyce then argues that only non-dominated options are rational.6 Therefore, only probability functions are candidates for being rationally permissible credence functions. More explicitly, Joyce’s argument works as follows: (I) The epistemic value of a credence function b at a world w is given by u(b, w) for some u ∈ U, where U is the set of legitimate accuracy measures.7 (II.a) For any non-probabilistically coherent b, there’s a b0 such that b0 is a probability function and b0 u-dominates b. (II.b) There is no probability b0 that is u-dominated by any function. (III) If a credence function b is u-dominated by an alternative credence function b0 that is itself undominated, then b is irrational. 4

We’ll be interested in the most recent wave of accuracy-first epistemology as it applies to partial belief. However, since it cares only about truth, it is a species of veritism. See (Goldman 1986) for a classic presentation of veritism in traditional epistemology. 5

I.e., if c is a probability function, then there’s no c 0 that’s at least as accurate as c at every world and strictly more accurate at some world. 6 More precisely: Choosing a dominated option is irrational assuming there are some options that aren’t dominated. 7

For now, we leave open whether there’s a single correct measure.

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(C) Therefore, all non-probabilistic credence functions are irrational. (I) pins down potentially legitimate measures of total epistemic value. (II) is a mathematical theorem. (III) is a principle of rational choice that connects epistemic behaviour and epistemic value. There are two clear ways to derive additional epistemic norms. The first is to change the principle of rational choice. Greaves and Wallace (2006) argue for conditionalisation, for example, by appealing to expected utility maximisation instead of dominance.8 That is, on the same measures of epistemic utility that Joyce appeals to, the updating policy that leads to the most expected epistemic accuracy is conditionalisation. The second is to change the class of legitimate epistemic utility functions, i.e., measures of accuracy. As we’ll see in a moment, however, epistemic utility theorists tend to agree on certain constraints that any good measure must satisfy. 1.3.2. Measuring Inaccuracy. Before seeing why imprecise credences and accuracy-first epistemology are supposed to be incompatible, it will be useful to get a bit clearer on what exactly accuracy-firsters mean by accuracy. For our purposes, we need not take issue with the particular constraints on legitimate measures they endorse, so instead we here just survey some general principles along with some brief motivation. First, a quick and non-substantive simplification. For technical convenience, it’ll be easier to use measures of inaccuracy or negative accuracy. That is, we seek constraints on acceptable measures divergence from truth-value, instead of proximity to truth-value. According to AFE, rational agents seek to minimise this divergence. Now back to the main question. If there’s one non-negotiable principle of a good measure, it’s that credences closer to truth-values are less inaccurate. A credence of .8 in a truth, for example, is less inaccurate than a credence of .7 in the same proposition. Likewise, an entire credence function that’s uniformly closer to the truth than another is overall less inaccurate. We now cash this out formally using standard possible world semantics. Let W be a set of worlds, and F be a set of propositions over W . We let w(X) = 1 (w(X) = 0) if X is true (false) at w. bel(F ) is the set of belief functions over F , where a belief function assigns some number x in [0,1] to each proposition in F . If a belief function obeys the Kolmogorov axioms, then it’s a probability function. A measure of inaccuracy (also known as a scoring rule) I is a function from bel(F ) × W → R≥0 that is intended to measure how close a belief function is to the truth at a given world. The fundamental constraint on legitimate inaccuracy measures is then the following: Truth-Directedness: If |b(X) − w(X)| ≤ |c(X) − w(X)| for all X, and |b(Y ) − w(Y )| < |c(Y ) − w(Y )| for some Y , then I(b, w) < I(c, w). 8

There are also structurally similar accuracy-firster arguments for a number of other epistemic norms, including but not limited to: the Principal Principle (Pettigrew 2013), the Principle of Indifference (Pettigrew 2014), the Principles of Reflection and Conglomerability (Easwaran 2013), and norms governing disagreement (Moss 2011; Levinstein 2015).

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Unfortunately, Truth-Directedness alone isn’t sufficient for the arguments of epistemic utility theory to work. To see why not, consider the absolute-value measure: X abs(b, w) = |b(X) − w(X)| X∈F

abs is clearly truth-directed since it simply sums up the absolute differences between credences and truth-values. However, under abs, some probability functions are dominated by non-probability functions.9 There is a long and ongoing discussion in the literature about which constraints in addition to Truth-Directedness are philosophically justified.10 However, we can at least mathematically characterise which measures will work. The most important additional constraint is one that requires a certain kind of immodesty. The idea is that if you have a credence of x in some proposition, you ought to expect x to be the least inaccurate of the alternatives. Otherwise, you could never rationally hold x as a credence, since it would automatically come out as dis-preferred to some alternative. In other words, you’d hold credence x toward a proposition while simultaneously thinking some alternative credence x 0 was less inaccurate.11 Likewise, every probability function should expect itself to be the least inaccurate. More formally: Propriety: If I is a legitimate measure of inaccuracy and b is a probability function, then for all distinct credence functions c, we have: Eb I(b) < Eb I(c) where Eb denotes the expected value function according to b. So, Propriety says that each probability function assigns itself lowest expected inaccuracy according to any legitimate measure. For our dialectical purposes, we can just accept this requirement, since we’ll be arguing below that imprecise bayesians should claim that there are multiple distinct measures of accuracy that are legitimate, and Propriety narrows rather than expands the space of legitimate measures.12 1.3.3. An Example. Later on, we’ll explore various proper measures in more detail. For now, let’s look at a single example for the sake of concreteness. One natural way to score an agent’s credal state at a world is to identify inaccuracy with mean squared error. I.e., 1 X Brier Score: BS(b, w) = (w(X) − b(X))2 |F | X∈F The Brier score simply sums up the square of the difference between the agent’s credence in each proposition and then takes the average. 9

Imagine, for instance, an urn had a Red ball, a Green ball, and a Blue ball, one of which will be drawn at random (i.e., each with a 1/3 chance). An agent with credence 0 in each of the three propositions Red, Green, and Blue is guaranteed to be less inaccurate under abs than an agent with a credence of 1/3 in each. The agent with a credence of 0 in all three will receive a total score of 1, whereas an agent with credence 1/3 in all three will receive a score of 4/3 > 1. 10 See, for example, (Joyce 1998, 2009; Leitgeb and Pettigrew 2010; Levinstein 2012; Predd et al. 2009; Pettigrew 2016; Selten 1998). 11

Compare: ‘I believe X, but I think a belief in ¬X is more accurate.’

12

For a direct defense of Propriety see (Joyce 2009).

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Sometimes, it will also be natural to score individual credences instead of entire credal states. We can likewise evaluate a particular credence simply by looking at its squared error. I.e., Local Brier Score: BS(x, i) = (i − x)2 Here, x is the agent’s credence, and i is the truth-value of the proposition in question. Context should make it clear below whether the local or global version is intended. The Brier Score (along with infinitely many other strictly proper measures of inaccuracy) will vindicate Joyce’s argument for probabilism along with the other accomplishments of epistemic utility theory. That is, if we use the Brier Score as a measure of epistemic disutility, then all and only non-probability functions are dominated, conditionalization is the policy that minimises expected epistemic disutility, and so on. 1.4. The Incompatibility Argument. Let’s now turn to the primary challenge of reconciling accuracy-first epistemology with imprecise credences. Recently, a number of authors have published a variety of impossibility results that aim to show that, in fact, accuracy-first epistemology and imprecise credences are incompatible (Seidenfeld et al. 2012; Mayo-Wilson and Wheeler 2015; Schoenfield 2015). More specifically, any way of measuring the inaccuracy of imprecise credal states is sure to yield an unattractive result. The formal arguments themselves can be rather nuanced, but I’ll here provide a simplified version without any bells and whistles. The aim isn’t to present anything water-tight (or nearly as water-tight as those found in the papers cited) but instead to give the reader a basic grasp of why reconciling AFE with imprecise credences looks especially challenging.13 Let’s compare the following cases: Mystery Coin: The only evidence you have that is relevant to whether Heads is that the objective chance of Heads is between 0.05 and 0.95. Fair Coin: You know the chance of Heads is exactly .5. Suppose in Mystery Coin, you adopt imprecise credal state [.05, .95] toward Heads, whereas in Fair Coin you adopt a precise state of .5. In each case, there are only two outcomes: one in which Heads is true, and one in which Heads is false. Inaccuracy is a function just of your credal state and how the world is. In particular, your actual level of inaccuracy doesn’t depend at all on the background evidence you have. Now, let’s make the reasonable assumption that on any good measure of inaccuracy, I(.5, 1) = I(.5, 0). That is, if you have credence .5, your level of inaccuracy is fixed.14 For instance, on the Brier Score, your inaccuracy is .25 regardless of whether Heads or Tails. Similarly, it seems, I([.05, .95], 1) should be the same as I([.05, .95], 0). Intuitively, you’re not any better off, from an alethic perspective, if Heads or Tails. The interval [.05, .95] doesn’t seem to favour one conclusion over the other. Let’s suppose, then, that I([.05, .95], 1) = I([.05, .95], 0) = m and that I(.5, 1) = I(.5, 0) = s. 13

The argument I give most closely follows that found in (Schoenfield 2015).

14

More sophisticated variations of the argument can do away with this assumption.

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Now, we might not want m to be a single number. After all, we might not want imprecise states to get precise numerical scores. So, we’ll just assume that one of the following holds: (1) m is a better score than s, (2) m is a worse score than s, (3) m is neither better nor worse than s If m is a better score than s, we’re in trouble. In that case, it must always be irrational to have credence .5 in any proposition at all. After all, since accuracy just depends on your credal state and how the world turns out, you’d do better in Fair Coin by adopting a credal state of [.05, .95] instead of .5. In other words, if m is better than s, .5 is accuracy-dominated and therefore always less epistemically valuable than a credence of [.05, .95]. Surely, however, it’s sometimes rational to have a credence of .5 in some propositions (e.g., if you know a coin is fair). So, m can’t be better than s. By analogous reasoning, if s is better than m, then it’s irrational to have an imprecise credence of [.05, .95] no matter what. A generalised version of this argument then rules out imprecise credences as rationally permissible. This option gives up the game. Those hoping to reconciling imprecise credences with AFE can’t accept that m is worse than s either. If s is neither better nor worse than m, then imprecise credences seem to do no alethic work. That is, an accuracy-seeking agent would never have reason to prefer an imprecise credence to some precise one. However, the situation is in fact worse than that from an accuracy perspective, as Schoenfield (2015) points out. In Fair Coin, having credence [.05, .95] toward Heads violates the Principal Principle, which is supposed to be a rational requirement. However, if s is neither better nor worse than m, then it’s never determinately better to be in credal state [.05, .95] than in credal state .5 toward some proposition. So, this option allows for violations of the Principal Principle. If we assume that the Principal Principle is better established than either accuracy-first epistemology or imprecise credences, one of the latter two should go. To respond to this argument, we should first look at two subtly different ways of understanding what imprecise credal states amount to.

2. Indeterminacy and Imprecise Bayesianism 2.1. The Formal Representation. As mentioned above, when we’re interested in an imprecise agent’s attitude toward a single proposition, we can represent it with a set or interval. For instance, in Mystery Coin, we might represent her with the interval [.05, .95]. We can also use sets of probability functions to represent her entire doxastic state over more than one proposition. Suppose, for instance, Alice has an imprecise credence [.2, .3] in X, imprecise credence [.3, .4] in Y , and precise credence .8 toward Z. How, formally, should we capture exactly what she thinks? Assuming she has no other precise views, Alice’s doxastic state R (called her Representor) is {c ∈ Prob : c(X) ∈ [.2, .3], c(Y ) ∈ [.3, .4], c(Z) = .8}

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On the orthodox picture, facts about Alice’s opinion correspond to facts that are true of each element of R.15 For instance, since every element of R assigns a lower precise probability to X than to Z, Alice thinks X is less likely than Z. However, since some credence function assigns .3 to both X and Y , Alice has no precise opinion as to whether X or Y is strictly more likely. 2.2. Interpretation. There are two importantly distinct ways to interpret how R represents Alice’s doxastic state: Determinate: Alice determinately identifies with the set of probability functions R. Indeterminate: It’s determinate that Alice’s credence function is a member of R, but it’s indeterminate which member of R it is. On the first option, there’s no indeterminacy at all. Alice’s doxastic state simply is the set R. On the second, there’s no fact of the matter what her doxastic state is exactly. The orthodox view above fits naturally with the Indeterminate interpretation. After all there is a clear analogy to supervaluationist semantics. On the supervaluationist approach, vague terms have admissible and inadmissible precisifications—roughly, reasonable and unreasonable ways of completely disambiguating the term. If, on every admissible precisification, a proposition comes out true, then it is determinately true. If, on every admissible precisification, it comes out false, then it is determinately false. If it comes out true on some but not other precisifications, then it is indeterminate whether it is true or false. As Rinard (2015) points out, imprecise credences behave very much like vague concepts under supervaluationism: We can apply [the] supervaluationist strategy to doxastic imprecision by seeing each function in your set as one admissible precisification of your doxastic state. Functions excluded from your set are inadmissible precisifications. Whatever is true according to all functions in your representor is determinately true; if something is true on some, but not all functions in your representor, then it’s indeterminate whether it’s true. For example, if all functions in your set have b(A) > b(B), then it’s determinate that you’re more confident of A than B. If different functions in your set assign different values to some proposition P , then for each such value, it’s indeterminate whether that value is your credence in P . (p. 2, minor changes) In other words, the orthodox interpretation is at least structurally supervaluationist, as it treats each function in Alice’s representor the same way supervaluationism treats precisifications in vagueness. So, the Indeterminate reading is at least a natural one and worth exploring further in the context of accuracy-first epistemology.16 15

See, for instance, (Hájek 2003; Levi 1985; Joyce 2005, 2010; van Fraassen 1990; Walley 1991).

16

Aside from (Rinard 2015), however, not much has been written explicitly on whether to endorse Determinate or Indeterminate. That is, few proponents of ICs have said directly whether Alice’s doxastic state is really her representor itself, or whether instead the representor is merely the set containing admissible precisifications.

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2.3. The Incompatibility Arguments Revisited. Let’s now return to the incompatibility arguments presented above. They show that if we assign a score of m to an imprecise credence of [.05, .95] and of s to .5 in Mystery Coin we apparently have no good options. Now, this seems like a big problem on the Determinate view. If Alice’s doxastic state in Mystery Coin is a set, then we need some way to score that set as a whole. That is, we need some way of saying when an accuracy-seeking agent ought to prefer being in the set-valued doxastic state to being in a precise credal state. So, given the truth of Determinate, we’re in trouble.17 However, on the Indeterminate view, Alice’s attitude toward Heads in Mystery Coin isn’t really the interval [.05, .95]. More precisely: her doxastic state isn’t some set of functions that assign credence between .05 and .95 to Heads. Instead, it’s indeterminate what her credal state is. So, assigning a score to the whole interval is, on this view, simply a category mistake. Instead, there’s no fact of the matter how inaccurate Alice is, since it’s indeterminate which credence function is hers. Indeterminate thus escapes the incompatibility arguments. It denies the presupposition that there’s some way or other to score a representor as a whole. However, by leaving an agent’s inaccuracy score indeterminate, it doesn’t tell us why imprecise credences might be at all desirable to an accuracy-firster. The answer I’ll develop below is based on the claim that Alice should be permitted to have indeterminate alethic values. Although all she ought to care about, epistemically speaking, is accuracy, there isn’t any single way she cares about accuracy, nor any precise notion of accuracy she places above all others. If her values are thus indeterminate, she can end up with indeterminate credences as a result. Let’s explore this idea further now. 3. Imprecise Epistemic Values At first blush, it seems that accuracy-first epistemology has already settled the question of epistemic value. All an agent should care about is having her doxastic state come close to matching actual truth-values. Epistemology thereby becomes a matter of determining what sorts of epistemic actions and policies are or are thought to be most truth-conducive. Quine (1986), for instance, once took a view of this sort: Normative epistemology is a branch of engineering. […] It is a matter of efficacy for an ulterior end, truth. […] The normative here, as elsewhere in engineering, becomes descriptive when the terminal parameter is expressed. (pp. 664-5) 17 Konek (2015) provides one way of scoring set-valued credences. On his approach, an agent with representor R receives a numerical score which is a weighted average of the least and most inaccurate members of R. E.g., for α ∈ [0, 1], we have:

Konek-Brierα : KBS([a, b], i) = α · minx∈[a,b] BS(x, i) + (1 − α) · max x∈[a,b] BS(x, i) Depending on the value of α, precise credences can be guaranteed to do worse than some imprecise credences. For instance, setting α = 3/4 will make the set-valued credence [.05, .95] dominate .5. For Konek’s approach to work, then, the weights have to change depending on the evidential situation, otherwise some agents would end up prohibited from ever adopting a precise credence regardless of the background evidence.

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However, as we’ll see shortly, merely caring about truth as the final end of epistemic action is not sufficient to make epistemology merely a problem of engineering. Instead, as William James famously argued, much remains unsettled: There are two ways of looking at our duty in the matter of opinion.[…] We must know the truth; and we must avoid error, — these are our first and great commandments as would be knowers; but they are not two ways of stating an identical commandment […] Believe truth! Shun error! — these, we see, are two materially different laws; and by choosing between them we may end by coloring differently our whole intellectual life. We may regard the chase for truth as paramount, and the avoidance of error as secondary; or we may, on the other hand, treat the avoidance of error as more imperative, and let truth take its chance. (1896, §VII) One way to bring out the distinction between the two great commandments in the context of full belief is to notice that each commandment is individually easy to satisfy. An agent can believe all truths simply by believing all propositions, yet she is thereby sure to violate the commandment to shun error. Likewise, an agent can suspend belief about each proposition and avoid error, yet she gives up the chance to believe truths. A similar lesson applies to precise credences. An agent who’s opinionated— with credences close to 0 or 1—has the chance to be extremely accurate, but she also risks great inaccuracy. In turn, an agent with credences closer to the middle of the spectrum protects herself from alethic disasters, but she also has no chance for very low inaccuracy. Deciding on a credence requires an agent to strike a balance between these two great alethic commandments. We’ll use a broadly Jamesian theme to develop an accuracy-first approach to imprecise credences. We’ll first look at two ways in which alethic values can plausibly rationally differ: just as with practical decision theory, an agent’s choices can be affected either by the precise nature of her utility function or by the method she uses to choose among her available options. If it’s rationally permissible not to have fully determinate epistemic values, then it’s rationally permissible to have imprecise credences. Thus, indeterminate values generate indeterminate credences. 3.1. Scoring Rules and Epistemic Value. In this section, we examine precise measures of alethic epistemic value. As mentioned above in §1.3.2, there are an infinite variety of measures of inaccuracy, known as proper scoring rules. In accuracy-first epistemology, proper scoring rules play the role of epistemic disutility functions, which reflect an agent’s alethic values (Joyce 2009; Moss 2011; Pettigrew 2016; Konek and Levinstein 2017). Scoring rules are purely alethic, in the sense that they are truth-directed and simply measure divergence between credence and truth-value. Nonetheless, they encapsulate different notions of value. First, we’ll see that different scoring rules disagree about the rank-order of epistemic options. That is, they disagree about which credence functions an agent should prefer at which worlds. Second, they importantly disagree about the cardinal level of epistemic risk involved in epistemic decisions. Third, no scoring rule in particular seems to give an answer

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that’s clearly better than the others. The upshot is that if no one scoring rule is privileged, the notion of accuracy that rational agents should care about is itself imprecise. 3.2. Examples. Although infinitely many different scoring rules satisfy the needed constraints, we’ll focus on three popular measures that will do the job accuracy-firsters want.18 That is, these measures will, combined with the right decision-theoretic principles, underwrite arguments for probabilism, conditionalisation, the Principal Principle, and so on. Brier Score: BS(x, i) = (i − x)2 Log Score: Log(x, i) = − ln(|(1 − i) − x|) 1 Spherical Score: Sph(x, i) = 1 − |1 − i − x|/(x 2 + (1 − x)2 ) /2 As before, x is the agent’s credence and i = 1, 0 depending on whether the proposition in question is true or false. As we saw earlier with the Brier Score, we can easily generate global versions of each score simply by averaging each of the local scores. Let’s now see how these scoring rules encode different alethic values. 3.3. Ordinal Differences. We begin with ordinal differences. Consider the following case: Lottery: An urn contains four balls: A, B, C, and D. As a matter of fact, A is chosen. Suppose Alice, Bob, and Carol have the credences shown in Table 1 over which ball was selected:

Alice Bob Carol

A .005 .033 .033

B .275 .127 .474

C .230 .137 .088

D .490 .703 .405

Table 1. Credences in Lottery

Given that A was actually chosen, who among our three characters is more accurate than who? Or, put differently, who is—from a purely alethic perspective— better off epistemically? It’s hard to say. On the one hand, Alice has the lowest credence in the true proposition A, so in one respect she’s doing worst. However, Bob has a very high credence (.703) in the false proposition D, whereas Alice’s highest credence in a false proposition is just .49. So, it’s not clear whether Alice is more or less inaccurate than Bob. What about Alice versus Carol? Again, it’s not clear. Carol is definitively more accurate than Alice on A, C, and D, but is much less accurate on B. No one ordering leaps out as the one all rational agents must agree upon. As you may suspect, our quantitative measures of inaccuracy also disagree about the ordering. According to the Brier Score, Carol is least inaccurate, followed by Alice and then Bob. According to the Log Score, Carol is again least inaccurate, followed by Bob, and then Alice. And according to the Spherical Score, Alice is least inaccurate, Carol is second, and Bob’s in last. 18

For further discussion, see (Joyce 2009).

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Each of these rules is truth-directed and proper. Nonetheless, they disagree about the accuracy rank-order in this case. So, agents with different scoring rules will disagree about which credence function is preferable to which. More abstractly: agents who differ in their alethic values can disagree about their epistemic preferences even when the state of the world is known. Now, if we agree that different orderings are reasonable, there’s a further question of whether rational agents are nonetheless obligated to make up their minds as to which among Alice, Bob, and Carol is more accurate than which other. More generally: Totality: Given a world w and two credence functions b1 and b2 , a rational agent would either prefer to have b1 to b2 as her credence function, prefer b2 to b1 , or be indifferent between b1 and b2 . According to this principle, agents must have, in effect, a single epistemic disutility function that completely ranks each credence function at each world. However, it’s not clear why Totality is a rational requirement even for agents who just care about seeking truth. After all, seeking accuracy seems fundamentally to mean that you prefer higher credences in truths to lower credences in falsehoods. That may well be rationally required. But that preference leaves a lot left to be determined, and there doesn’t, as yet, seem to be compelling epistemic reason to force agents to form a total preference ranking. Without specific reasons to the contrary, we can assume that some rational agents may not have any definitive views about whether Alice, Bob, or Carol is best off in this situation.19 3.4. Cardinal Comparisons. Scoring rules can also differ greatly on the amount of epistemic risk involved in epistemic decisions. For simplicity, let’s focus on how the Brier Score and the Log Score evaluate a single credence in a proposition H. Suppose Alice has credence .01 in H, and Bob has credence .001 in H.

Alice Bob

x .01 .001

BS(x, 1) .98 .998

BS(x, 0) 10−3 10−6

Log(x, 1) 4.6 6.9

Log(x, 0) 10−2 10−3

Table 2. Approximate Brier and Log Scores

Every truth-directed scoring rule agrees what the rank-order is at each world in this case. However, the Brier Score and Log Score disagree about how risky each credence is. To see this, we can look at the ratio of how much Alice stands to gain or lose in inaccuracy if she were to switch to Bob’s credence. In the case of the Brier Score, if she adopted Bob’s credence and H turned out false, her score would improve by 10−3 − 10−6 . If, however, H turned out true, she’d increase (i.e., worsen) her score by .998 − .98. The ratio of possible gain of inaccuracy to loss is approximately 18:1. That’s risky, to be sure, but not nearly as risky as the same change in credence is on the Log Score: around 256:1. So, in comparison, the Log Score is much more sensitive to small changes in 19

As we’ll see in §4.2, denying Totality comes with some alethic and evidential advantages.

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Figure 1. The Local Brier, Log, and Spherical Scores. The ascending curves represent I(x, 0) and the descending curves represent I(x, 1) for the respective scoring rules. Note that the Brier and Spherical Scores are bounded by 1, but the Log Score is unbounded. credence around 0. If H turns out true, on the Log Score, Bob is much, much worse off than Alice. Depending on what one cares about, either of these risk profiles can seem reasonable. On the one hand, both Alice and Bob are very confident that ¬H in absolute terms. Since .01 − .001 ≈ 0, this way of looking at their credences pushes us to assign them relatively similar scores. The Brier Score recognises them as similar credences. On the other hand, Alice is ten times more confident in H than Bob is. Looked at this way, it seems Alice should count as a lot more accurate than Bob when H turns out true.20 This point about the difference in risk-profile generalises: If an agent moves her credence in proposition X from x to x ± , how much she stands to gain or lose if X is true will depend on the value of x and the scoring rule.21 One can get some sense of the different risk profiles of the various scoring rules we’ve considered by examining Figure 1. Rationality seems to give us some leeway. Seeking the truth and avoiding error are compatible with different ways of balancing the two goals. Rational agents, it seems, need not make up their minds entirely about the exact details. However, as we’ll see, we have more work to do before such imprecise values can lead to imprecise credences. 3.5. An Oddity for Precise Bayesianism. We’ve now shown how important scoring rules are to the epistemic utility program and how they reflect different 20

One evocative but illicit way to build these two intuitions is to consider different types of claims. A forecast of a 1% chance of rain seems roughly as accurate as a forecast of a 0% chance. However, a forecast of a 1% chance of death is much different from a forecast of a 0% chance. 21

Each scoring rule is generated by an underlying measure on the unit interval. This measure represents, roughly, how important it is to have one’s credence on the correct side of a given point in the interval. Some rules care more about the middle of the probability spectrum (Spherical), some care more about the ends (Log), and some don’t privilege any particular part of the spectrum (Brier). For an extended discussion, see (Levinstein 2017).

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kinds of alethic values. Supposing there is some rational leeway for agents to choose between these different values, it seems such values would naturally influence epistemic behaviour. For instance, it seems that agents who use the Log Score would naturally be a bit more skittish about lowering a credence from .01 to .001 than would agents who use the Brier Score. However, the story is a bit more complicated than that. The problem here is that epistemic utility theorists require legitimate inaccuracy rules to obey Propriety. That is, on every acceptable rule, each probability function expects itself to be the least inaccurate. If a scoring rule isn’t strictly proper, then it won’t underwrite the main achievements of epistemic utility theory. For instance, probabilistically coherent credence functions won’t dominate incoherent ones. The requirement that scoring rules be strictly proper has severe effects on the role scoring rules can play in the heart of precise bayesian epistemology.22 Let’s see why. Suppose Bob has credence function b and at first uses the Brier Score to measure his epistemic disutility. Bob then has an epistemic change of heart and decides the Spherical Score really captures his alethic values better. Now that he’s converted to the spherical rule, how does his epistemic behaviour change? Answer: not at all! By Bob’s lights, b is still the function that he expects to minimise inaccuracy. So, despite the change in value, Bob sticks with the credence function he already had. This apparent epiphenomenalism of epistemic value is more surprising, perhaps, when we consider the intuitive effects of epistemic values on learning. Suppose Carol begins life as an epistemic risk-taker. Although Carol only updates by conditionalisation, her credence function ‘learns’ quickly in the sense that, without all that much evidence, she tends to arrive at credences close to 1 or 0. Carol is aware of the Jamesian tradeoff at play between the exhortations to Believe truth! and to Shun error!—although she can quickly become accurate, she also risks massive inaccuracy. Naturally, Carol uses a scoring rule that reflects these values, such as the Brier Score. As Carol gets older, she grows more conservative. She just doesn’t have the same tolerance for error she did when she was young. Although the epistemic highs of low inaccuracy are great when she gets things right, now that she’s aged, getting things massively wrong has become more punishing. Carol switches over to a more conservative scoring rule—the logarithmic function. What happens? Again, nothing. By Carol’s lights, planning to update her current credence function by conditionalisation minimises expected inaccuracy. So the change in value doesn’t manifest itself in any change in epistemic behaviour. So, since bayesianism at its core says to be probabilistically coherent and update by conditionalisation, how could your alethic values in any way influence your alethic behaviour? The answer, I think, is that they can affect how you choose a credence to have in the first place.23 When looking at your evidence, you don’t always have any credence—precise or imprecise—at all. You need to look at the available options and pick one that’s attractive. How you select your

22

See (Horowitz 2015) for an argument that different weightings of Jamesian goals can’t be made sense of in the context of epistemic utility theory for precise bayesianism. 23

More carefully: values affect which credences it’s rational to start with in a given epistemic situation. I do not mean to suggest any commitment to doxastic voluntarism.

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doxastic state in the first place can depend on your values. We now look at how this might work. 3.6. Two-Tier Conception of Evidence. To see how agents may end up with a particular doxastic state, it’s natural to adopt a common two-tier conception of evidence. To see how this works, let’s return to our Mystery Coin example from earlier. Let Heads be the proposition that the coin Marisol is holding will land heads on the next flip, and let E be the proposition that the chance of Heads is between .05 and .95. Epistemologists may disagree about what Marisol should think about Heads. Perhaps she’s obligated to have credence exactly .5. Perhaps she can rationally adopt any credence between .05 and .95. Or perhaps she should end up in some imprecise credal state. The important point now is just that, regardless of what she ultimately ought to do, we can think of Marisol’s selection of a doxastic state in the following way: First, E eliminates from contention all probability functions that assign credence less than .05 or greater than .95 to H. Second, Marisol selects her credal state (precise or imprecise) from the functions remaining.24 It may seem that this thesis requires us to deny unique, objective bayesianism. That is, it seems to fit naturally with: Non-Uniqueness: For some bodies of evidence E, there is sometimes no precise credence function that uniquely responds in the objectively most epistemically rational way to E. However, even the most austere objective bayesians have adopted a two-tier conception of evidence (Jaynes 1973; Williamson 2010). On their view, the first stage may leave a variety of options on the table, but the second stage always narrows the set of rational choices to exactly one. So, although the two-tier conception may not be compatible with every view of how evidence works, it’s a big tent. 3.7. Selection Rules. Assuming the two-tier conception is right, after the first stage, evidence can leave us with a bunch of probability functions still in contention. On our example above, Marisol has to select from the probability functions that assign anything between .05 and .95 to Heads. We now examine the process by which she chooses. Let’s begin with two temporary simplifying assumptions. First, we’ll assume Marisol will end up in a precise credal state. Second, we’ll assume that Marisol uses the Brier Score to measure inaccuracy. It may seem that if we’re forced into a precise view, Marisol ought to end up with credence .5 in heads. After all, she doesn’t have any evidence that favours or disfavours Heads over ¬Heads. But that move is a little too quick. Supposing Marisol adopts a credence of .5 in Heads, she’s sure to have a Brier-inaccuracy of .25: if Heads is true, she gets a score of (1 − .5)2 , and if Heads is false, she gets a score of .52 . On the other hand, if she adopts a credence of .4, she has the potential for a better score. If H is false, she’d end up with a score of .42 = .16. Of course, she 24

The first stage of this process may not impose any constraints. If an agent simply has no evidence whatever concerning some proposition, she can select her credence from the entire interval.

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risks a worse score of (1 − .4)2 = .36. So, while adopting a credence of .5 plays it safe and minimises the loss in the worst-case scenario, it also maximises the loss in the best-case scenario. Her ultimate credence therefore depends on her appetite for epistemic risk. This appetite is partly reflected in the selection rule she uses. That is, it’s reflected in the policy she uses to choose a credence from those allowed by the evidence. Although there are infinitely many potential selection rules, let’s take a moment to review three.25 Let O be the set of available credences after the first stage of the evidential process. Marisol can pick a credence in O via: MiniMax: Select the credence that has the best worst-case outcome. I.e., arg min max BS(x, i) x∈O

i=0,1

MiniMin: Select a credence that has the best best-case outcome: I.e., arg min min BS(x, i) x∈O

i=0,1

Hurwiczλ : Select the credence that has the best weighted average of the best- and worst-case outcomes, with weights given by λ ∈ [0, 1]. I.e., ! arg min λ max BS(x, i) + (1 − λ) min BS(x, i) x∈O

i=0,1

i=0,1

Each of these rules reflects a different epistemic risk-management policy. MiniMax heeds the commandment to Shun error! It looks at the options and chooses a credence based only on the maximum possible inaccuracy.26 In the case of Mystery Coin, it recommends credence of .5 under the Brier Score. MiniMin goes the opposite direction and zealously seeks to Believe truth! In this case, it recommends either a credence of .05 or .95. Hurwiczλ seeks a balance between the two great commandments.27 Depending on the value of λ, Hurwiczλ can end up recommending any credence within the interval [.05, .95]. Now, although MiniMin is to extreme to be a rational policy,various versions of Hurwiczλ are arguably rational. The risk-aversion of MiniMax leads to potentially undue scepticism, since it always requires the most agnostic credence that remains among any set of options. Given our ultimate goals of finding a plausible place for imprecise credences in accuracy-first epistemology, I’ll assume that no single selection rule is mandatory. However, even if a single rule is privileged, the same selection rule can lead to different credences depending on the scoring rule. As we saw above, different scoring rules disagree about which credence function is more accurate than which others at various worlds. Therefore, rules that appeal to best- and worsecase outcomes will yield different results depending on the utility function. 3.8. Epistemic Value and Imprecise Credence. It should now be clear how imprecise epistemic values—even purely alethic ones—can lead to imprecise credences. Suppose an agent only cares about having accurate credences. However, at least one of the following two claims is true of her: 25

For simplicity, we only look at these selection rules as applied to individual credences, not entire credence functions. 26

See (Pettigrew 2014) for a discussion of this rule in epistemology.

27

See (Konek 2015; Pettigrew 2015) for further discussion.

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(1) Her values don’t single out a single selection rule. For instance, her values may not decide between various versions of Hurwiczλ for λ in, say, [.5, .8]. (2) Her values don’t single out a single scoring rule. Although she strictly prefers higher credences in truths and lower credences in falsehoods, facts about her preferences don’t single out one scoring rule determinately. If either holds, there won’t always be a fact of the matter about which credence function is hers. All credence functions compatible with her epistemic values have equal claim. Note that our supervaluationist interpretation of imprecise credences is key here. Because the agent’s values are not fully determinate, there is no way to pin down a precise scoring rule and a precise selection rule that combine to reflect her values in a uniquely best way. So, various scoring rules paired with various selection rules can be equally reasonable precisifications of her epistemic values. In turn, various credence functions can be equally good precisifications of her doxastic state. 3.8.1. Scoring Imprecise Credences. We began with a puzzle that suggested that, from an accuracy-first perspective, imprecise credences were at best otiose. In our simplified version of the argument, a credal state of [.05, .95] would have ended up no more or less accurate than some particular precise credence, most likely .5. So, there could be no reason to prefer the imprecise credal state to the precise one, or vice versa. We’ve now seen that there is instead a more quietistic response available that’s natural on our supervaluationist picture. Suppose Alice’s representor assigns [.05, .95] to some proposition X. On our view, it’s not the case that Alice thinks that [.05, .95] is less inaccurate than .5. In fact, she simply has no determinate opinion about how the interval stacks up against .5 at all. Comparing the two is simply a category mistake. Instead, her views about the inaccuracy of any credence x in the interval [.05, .95] and .5 are indeterminate. On some precisifications of her credal state, she expects .23 to be more accurate than .5. On others, she expects it to be less inaccurate. Each x in [.05, .95] is, on some precisification of her doxastic state, one that minimises expected inaccuracy by her lights. Since her credal state is indeterminate—there’s simply no fact of the matter which element of [.05, .95] is really Alice’s credence toward x—her inaccuracy is likewise indeterminate. Put slightly differently: because it’s indeterminate whether .23 or .5 is the credence that she thinks does best at optimising her epistemic values and goals, it’s indeterminate whether .23 or .5 really is her credence. Now, one might object that if it’s indeterminate which x in [.05, .95] is really Alice’s credence, it’s also indeterminate whether she prefers .05 −  to .5 for small enough . After all, according to the precisification of her doxastic state that assigns .05 to X, .0499 is expected to be less inaccurate than .5. That’s true, but it misses the point. On no precisification of her epistemic values is .0499 a maximally good credence to have. We were also challenged to explain why imprecise credences didn’t lead to permitted violations of the Principal Principle. The answer is straightforward on

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our interpretation of what imprecise credences amount to. If Alice learns that the chance of Heads is x, it’s determinate that obeying the Principal Principle maximises objective expected epistemic utility according to every strictly proper scoring rule (Pettigrew 2013). So, if Alice is rational, then according to every precisification of her values, she ends up obeying the Principal Principle. That is, even though her epistemic standards and values will sometimes disagree with one another in certain evidential situations, they should all agree in cases where the chances are known. One might still raise a worry along the following lines. Suppose that Alice and Bob both know the chance of Heads is .5. Alice adopts credence .5, while Bob ends up in an imprecise credal state of [.05, .95]. Carol learns the chance of Heads as well, and she is deciding whether she should end up like Alice or Bob. Even though Bob’s credal state is indeterminate, Carol can—let us suppose— take a brain scan of Bob and of Alice and decide whether to switch her brain state to match either of theirs. Since the brain is just a physical object, the brain state is in fact determinate. So, in this way, Carol can decide which brain state she’d prefer to be in. Since there’s no fact of the matter whether Alice or Bob is more accurate, why should Carol prefer to be like Alice instead of like Bob? The answer is that it’s not agents or brains that are directly evaluated for accuracy, strictly speaking. Instead, accuracy is what gives value to doxastic states. Bob, although in a determinate brain state, has no determinate credal state. From a purely epistemic point of view, Carol shouldn’t judge directly whether she’d prefer to be Alice or Bob. Instead, she can judge whether various precisifications of Bob’s credal state are in accord with her epistemic values. If she’s rational, she’ll balk at all those that don’t assign credence .5 to Heads. It’s the mental states themselves—not the agents—that bear epistemic value. 4. Comparison to Alternatives We’ve so far seen that we can generate imprecise credences from imprecise values, and we’ve seen how imprecise credences are in fact compatible with accuracy-first epistemology. Before concluding, it’s worth briefly comparing the view we’ve developed with some alternatives. 4.1. Departure from Orthodoxy. We earlier noted that our Indeterminate interpretation fit with the orthodox view that facts about an imprecise agent’s doxastic state are those that each element of her representor agrees about. For instance, if for every c in Alice’s representor, c(X) > c(Y ), then Alice is more confident in X than in Y . However, there’s an important way in which we now depart from the orthodoxy. On the standard view, if Alice’s representor at t0 is R and she receives new evidence E, then she should update by pointwise conditionalisation.28 That is, her new imprecise state at t1 should be represented by RE = {c(·|E) : c ∈ R}.29 What’s the justification for this updating rule? We saw above, briefly, that in the case of precise bayesianism, conditionalisation can be justified through appeal to the standard decision-theoretic norm of expected utility maximisation. 28

To be less contentious, we might say she should plan to update by conditionalisation, since some might object to any diachronic updating norms. This wrinkle need not concern us here. 29

For another departure from the orthodox view, see (Weatherson 2007).

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Greaves and Wallace prove, roughly, that on any strictly proper scoring rule updating by conditionalisation minimises expected inaccuracy. The same cannot be said, however, for point-wise conditionalisation in the imprecise case. On our Indeterminate view, Alice is not, strictly speaking, in credal state R or RE at any point. So, there’s no way RE itself could be a state that minimises expected inaccuracy. Instead, we have to be a bit more subtle. According to every c in Alice’s representor R, the credence function that minimises expected inaccuracy is c(·|E). However, no function in particular need determinately minimise expected inaccuracy. If c 0 is also in R, then c 0 thinks c 0 (·|E) is best, which of course may not be equal to c(·|E). So, it’s indeterminate whether Alice thinks c(·|E) would be best. Now suppose that, after learning E, Alice’s representor at t1 is R0 ≠ RE . Did she do anything wrong? Not necessarily. Suppose c is in R, but c(·|E) isn’t in R0 . Bad move, according to c. However, so long as there’s some c 0 ∈ R such that c 0 (·|E) is in R0 , it’s not determinately true that Alice failed to minimise expected inaccuracy. Therefore, it’s not determinately true that she did anything irrational. In other words, there’s some precisification of her doxastic state at t0 and at t1 that makes Alice an expected inaccuracy-minimiser. One might try to object that Alice should determinately be an expected inaccuracy minimiser. After all, minimisation of expected disutility is the determinately rational thing to do. However, if Alice starts in R and ends up in RE , then it’s indeterminate whether she’s in c or c 0 at t0 and indeterminate whether she’s in c(·|E) or c 0 (·|E) at t1 . In turn, it’s indeterminate whether she really minimised expected inaccuracy. I take this heretical view—that Alice need not update by pointwise conditionalisation—to be a welcome departure from the orthodoxy. It means that if Alice begins epistemic life with some set R, she isn’t stuck with all the descendants of R forever after. If Alice changes or precisifies her epistemic values, her epistemic behaviour varies in a natural way. As we saw above, in precise bayesianism, once you have a credence function, you’re more or less stuck with updating it via conditionalisation forevermore. There’s no opportunity for values to affect how you learn. On this style of imprecise bayesianism, Alice has the option to let a change in values influence her learning behaviour without determinately violating the norm to maximise expected utility. 4.2. Imprecise Credences and Permissive Bayesianism. One popular alternative to imprecise bayesianism is permissivism. Permissivists agree with defenders of imprecise credences that sometimes evidence doesn’t single out a unique precise credence function as the maximally rational option. That is, Permissivists and ICers agree with Non-Uniqueness above. Permissivists and ICers disagree, however, about what’s rational to do in those situations. Unlike ICers, permissivists—or at least the species of them currently under discussion—think an agent is required pick a single precise credence function from the set of rational options. Is there a reason to favour imprecise credences over permissive ones? I think so, for two reasons. First, permissivists require agents to end up with opinions that, by everyone’s lights, must go beyond what the evidence objectively supports. They require agents to decide on a single credence function to adopt even

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though the evidence itself doesn’t privilege that credence function above the others. Perhaps such privileging is rationally licit, but it should not be mandatory. Imprecise credences rightly allow agents to remain undecided between alternative credence functions. Second, once agents adopt a single precise credence function, they are committed to a variety of firm opinions. In situations in which multiple credence functions are rationally on a par, some of these opinions are not supported by the evidence. For instance, suppose Alice and Bob share evidence that makes any credence in [.2, .3] toward Heads maximally rational. Alice ends up with credence .24, and Bob ends up with credence .29 based on their respective epistemic values. Both Alice and Bob recognise that they chose their credences from a set of options that were rationally on a par. After they end up in a determinate credal state, they must nonetheless think that that their own credence is uniquely best on every measure of epistemic value. Since whatever scoring rule they use is strictly proper, Alice thinks that .24 maximises expected epistemic utility, whereas Bob thinks .29 does. That wouldn’t be so bad if Alice and Bob could recognise that this apparent disagreement was due to different utility functions. If I like chocolate ice cream and you like kale, we can recognise that we’re pursuing optimal snackvalue given the difference in our tastes. In particular, I can say that given your preferences, you should expect to do best if you eat kale.30 However, Alice expects her credence function to be more accurate than Bob’s on every strictly proper rule. That is, she expects that on any measure acceptable to the epistemic utility theorist, her credence will come out less inaccurate. At the same time, she recognises that the evidence itself provides no reason to form this opinion over Bob’s view that his credence function is more accurate. So, (i) she realises that had she had a different epistemic utility function before she chose .24, she would have ended up with .29, (ii) she currently thinks that even on that alternative epistemic utility function, .24 is better than .29, and (iii) the evidence itself doesn’t support the claim that .24 is more accurate than .29 over the claim that .29 is more accurate than .24. Again, that may be all right if Alice were merely permitted to form opinions that went beyond the evidence—i.e., to adopt opinions that the evidence does not uniquely support. The problem is that permissivism mandates she form such unsupported opinions. So long as she ends up in some precise doxastic state or other, she must expect that her own credence function is the least inaccurate on every strictly proper measure. Imprecise credences, on our picture, allow for a bit more modesty. Suppose RA is Alice’s representor. She doesn’t expect any particular credence to be the least inaccurate. If Alice has both .2 and .25 in her representor, then there’s no fact of the matter which she expects to do better. Since the evidence, by stipulation, doesn’t objectively support any credence in the interval over any other, this seems like a superior response. She is not required to form opinions that exceed the evidence.

30

See (Horowitz 2015) for more on this issue.

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5. Conclusion We noted at the outset that imprecise credences can look attractive from an evidential perspective, but they also appear incompatible with accuracy-first epistemology. Appearances are deceiving, however. Different ways of valuing the truth—e.g., different scoring and selection rules—lead to different credences when evidence is unspecific. If agents have indeterminate values, they’ll in turn have indeterminate credences. Imprecise credences thus do not conflict with accuracy-first epistemology but instead naturally emerge from it. References Carnap, R. (1950). Logical Foundations of Probability. Chicago: University of Chicago Press. de Finetti, B. (1964). Foresight: Its Logical Laws, Its Subjective Sources. Wiley. Easwaran, K. (2013). Expected accuracy supports conditionalization – and congomerability and reflection. Philosophy of Science 80(1), 119–142. Goldman, A. (1986). Epistemology and Cognition. Harvard Univeristy Press. Greaves, H. and D. Wallace (2006). Justifying conditionalization: Conditionalization maximizes expected epistemic utility. Mind 115(632), 607–632. Hájek, A. (2003). What conditional probability could not be. Synthese 137 (3), 273–323. Horowitz, S. (2015). Epistemic value and the ‘Jamesian goals’. Unpublished manuscript. James, W. (1896). The Will to Believe and Other Essays in Popular Philosophy. Longmans, Green & Company. Jaynes, E. (1973). The well-posed problem. Foundations of Physics 3, 477–493. Jaynes, E. (2003). Probability Theory: The Logic of Science. Cambridge: Cambridge University Press. Joyce, J. M. (1998). A nonpragmatic vindication of probabilism. Philosophy of Science 65, 575–603. Joyce, J. M. (2005). How probabilities reflect evidence. Philosophical Perspectives 19, 153–178. Joyce, J. M. (2009). Accuracy and coherence: Prospects for an alethic epistemology of partial belief. In F. Huber and C. Schmidt-Petri (Eds.), Degrees of Belief, Volume 342, pp. 263–297. Springer. Joyce, J. M. (2010). A defense of imprecise credences in inference and decision making. Philosophical Perspectives 24, 281–323. Konek, J. (2015). Epistemic conservativity and imprecise credence. Philosophy and Phenomenological Research. Konek, J. and B. A. Levinstein (2017). The foundations of epistemic decision theory. Mind. Forthcoming. Leitgeb, H. and R. Pettigrew (2010). An objective justification of bayesianism I: Measuring inaccuracy. Philosophy of Science 77, 201–235. Levi, I. (1985). Imprecision and indeterminacy in probability judgment. Philosophy of Science 52(3), 390–409. Levinstein, B. A. (2012). Leitgeb and Pettigrew on accuracy and updating. Philosophy of Science 79(3), 413–424. Levinstein, B. A. (2015, March). With all due respect: The macro-epistemology of disagreement. Philosohers’ Imprint 15(13), 1–20.

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