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Belief, Credence, and Norms by Lara Buchak.
UC Berkeley ABSTRACT: There are currently two robust traditions in philosophy dealing with doxastic attitudes: the tradition that is concerned primarily with all-or-nothing belief, and the tradition that is concerned primarily with degree of belief or credence. This paper concerns the relationship between belief and credence for a rational agent, and is directed at those who may have hoped that the notion of belief can either be reduced to credence or eliminated altogether when characterizing the norms governing ideally rational agents. It presents a puzzle which lends support to two theses. First, that there is no formal reduction of a rational agent’s beliefs to her credences, because belief and credence are each responsive to different features of a body of evidence. Second, that if our traditional understanding of our practices of holding each other responsible is correct, then belief has a distinctive role to play, even for ideally rational agents, that cannot be played by credence. The question of which avenues remain for the credence-only theorist is considered. 1. Introduction Full belief (hereafter, just “belief”) is a familiar attitude: it is the attitude that the folk talk about, and it has been a subject of epistemology since epistemology began. Partial or degreed belief (hereafter, “credence”), on the other hand, is a semi-technical notion that has come to the forefront of epistemology more recently. Although the idea that probability features into epistemology traces back to at least John Locke, Frank Ramsey was the first to formalize the idea that beliefs come in precise degrees that can be measured by betting behavior.1 Since then, credences have been closely associated with preferences about gambles. Some have proposed that disposing an agent to take certain bets is merely part of the functional role of credences, whereas others have proposed that the link is definitional: one’s credence in p is the amount of money one is willing to pay in ordinary circumstances for a bet that yields $1 if p obtains and $0 if not. A particular kind of belief will be important in the ensuing discussion: belief in propositions of the form there is a chance c that p. For example: “there is a 50% chance that the coin will land heads”; “there is a 99% chance that my lottery ticket will lose”; “there is a very low chance that this table will spontaneously combust.” These objective-chance propositions are not necessarily claims about what the chances are according to our best theory of physics. Rather, they are claims about the chance (frequency, propensity, etc.) of an event relative to an implied fixed background: the bias of the coin or the number of lottery tickets, but not a complete physical description of the workings of the coin-toss or ticket-picking 1Ramsey (1926). 2 mechanism. These propositions aren’t merely reports of credences: when I tell you that the coin has a 50% chance of landing heads, I am not reporting a fact about my mental state or my evidence but a (purported) fact about the coin. Full belief in a chance-c-that-p proposition will ordinarily be accompanied by credence cr(p) = c. However, we will see that the fact that an agent believes a chance-cthat-p proposition for a particular c (even a very high c) doesn’t necessarily mean that she believes p. There is another important kind of chance-belief: belief in an epistemic-chance propositions. For example, one might believe that there is an 80% chance that a particular broken bone will heal without surgery or that there’s only a small chance one’s co-worker will make it into work on time. This is not a belief about objective chance relative to some background: there are no “chance mechanisms” of the kind involved in the coin-flip operating here. Rather, it is a belief about the relationship between one’s evidence and the world.2 How do belief and credence each correspond to how an agent sees the world? When an agent believes p, she in some sense rules out worlds in which not-p holds. The truth of not-p is incompatible with the attitude she holds towards p (though it is not incompatible with her holding that attitude, since she may be mistaken). On the other hand, having a particular credence in p, at least if it is not 0 or 1, does not rule out either the p-worlds or the not-p-worlds. The truth of not-p is compatible with the attitude the agent holds towards p, even when she assigns a very high (not-1) credence to p. One way to see this is to notice that if two agents, one who believes p and the other who assigns a high credence to p, each learn not-p, then the former, but not the latter, takes himself to have been incorrect.3 Thus, belief that p involves an on-off commitment to p in a way that credence doesn’t. Believing an objective-chance-proposition amounts to representing the world as one in which the relevant event has the relevant objective chance. Like ordinary propositional beliefs, objective-chance propositional beliefs rule out worlds in which the chance-proposition is false. When I believe that p has an r chance of obtaining, what I believe is incompatible with p having a different chance of obtaining relative to the implied background. For example, consider the belief that a coin has an 80% chance of landing heads. The chance-proposition is true just in case 80% is the actual bias of the coin (the objective chance, relative to the implied background), and so believing it rules out worlds in which this is false. The coin in fact landing heads and in fact landing tails are each compatible with the chance-proposition, and so the possibilities that are left open are worlds in which there is an 80% chance of heads and the coin lands heads, and worlds in which there is an 80% chance of heads and the coin lands tails. Thus, belief in 2Here is another example to illustrate the difference between objective and epistemic chance. One may believe that a coin has an objective chance of either 80% or 20% of landing heads, and have symmetric evidence with respect to each bias; as a result, one can believe that the coin has an epistemic chance of 50% of landing heads, but one will not believe that it has an objective chance of 50% of landing heads. 3See, e.g., Fantl and McGrath (2010: 141). 3 chance-propositions about p does not rule out worlds in which not-p obtains, but it does rule out some worlds. Similarly, beliefs in epistemic-chance propositions rule out the world being some way. Whereas the belief that there is an 80% chance of the coin landing heads rules out certain hypotheses about the coin, the belief that there is an 80% chance of a bone healing without surgery rules out certain hypotheses about the character of the agent’s evidence in relation to the world. For example, it rules out the hypothesis that agent’s total evidence (the x-ray appearing a certain way, the frequency of broken bones healing in the population) strongly indicates that the bone will not heal without surgery. Does credence rule out worlds in a similar manner? I won’t take a stand on this, but note that to the extent one thinks of credence as playing the same role as belief in an epistemic chance proposition, one will think that the attitude one takes by having cr(p) = c rules out, for example, worlds in which c is an inappropriate credence for one to have. But to the extent that one thinks of credence as a state formed unreflectively (as a state, for example, that some animals can be represented as having) or that doesn’t represent anything in the world, one need not think that the attitude rules out particular worlds. There are two different, but related, questions concerning the relationship between belief and credence: how the mental states of belief and credence relate to each other, and how the normative states of rational belief and rational credence relate to each other. I am concerned here with the latter question: for an agent doing what she ought, how to do the credences she has relate to what she believes? Naturally, the answer to this question bears on the question of how the mental states relate to each other, but I will not directly address this question here. Rationality is to be taken in the “reasonableness” sense of rationality rather than the “coherence” sense: rational agents are agents who form the beliefs and credences they ought to form given their evidence and, if it is relevant, the situation they find themselves in. I will as far as possible avoid relying on particular normative epistemic notions (justification, warrant, and the like) that each entail reasonableness, since my main argument will rely on data about what a reasonable person ought to believe in certain cases rather than why they ought to believe it. As for credences, whether there are requirements beyond coherence is a matter of debate among formal epistemologists. However, the only fact I will make use of in my argument is a relatively uncontroversial one: if it is part of one’s evidence that the frequency of truths in the reference class to which p belongs is x, and if there is no narrower or competing reference class for which one has evidence, then it is at least rationally permissible to set cr(p) = x. There is another debate in epistemology which I seek largely to cross-cut: the debate about whether belief-like or knowledge-like states underlie action.4 It may at first appear that anyone interested 4Two prominent theories that claim rational agents act only on what they know (rather than only on what they believe) include those of Fantl and McGrath (2002) and Hawthorne and Stanley (2011). See also the debate about 4 in discussing the role of credence and belief in mental life cannot avoid taking a stand here, since while full belief has its epistemic counterpart in knowledge, it appears there is no corresponding epistemic notion associated with credence. After all, belief can be true or false – a belief that p is true just in case p is true – but a credence cannot, and since knowledge is factive, only beliefs can constitute knowledge. However, there is a degreed sense in which credences can be assessed by their truth: a credence can be closer or farther from the truth. For example, if p is true, then cr(p) = 0.99 is closer to the truth than cr(p) = 0.1.5 Furthermore, Moss (2013) has argued that credences can constitute knowledge because they can satisfy the factivity criterion when it is properly understood. Therefore, I tentatively accept that in addition to the ordinary notion of knowledge associated with belief, which we can call belief-knowledge, there is a notion of knowledge associated with credence, which we can call credence-knowledge, that plays a role parallel to the role that belief-knowledge is supposed to play with respect to belief. Thus, while I will be concerned with arguing that notions in the “belief package” and notions in the “credence package” each play particular justificatory roles, I won’t take a stand on whether these roles are played by belief and credence or belief-knowledge and credence-knowledge. I will present a puzzle that, I will argue, lends support to two theses. The first is that (assuming the view known as the Certainty View is false) a rational agent’s having a particular credal state does not entail that she has a particular belief state, even within a given context and set of stakes: belief cannot be reduced to credence. The second is that the notion of belief is ineliminable from our moral practices of holding each other responsible: we cannot construct the norms associated with these practices using credences alone. Thus, unless there is a way of resisting the puzzle, we have either to revise these practices or accept two epistemic notions that don’t fit together well. (I will not in this paper consider a third route: trying to reduce credence to belief or to do without credence altogether.) 2. Assumptions and the State of Play Let me begin by outlining where the project of reducing belief to credence currently stands. One initial thought, of course, is that (at least for a rational agent) to believe p is to assign cr(p) = 1: to believe something is to be certain of it. This view, the Certainty View, is naturally suggested by the description of credences as “partial beliefs.” For the purposes of this paper, however, I will bracket this view and simply assume it is false. This assumption represents a mainstream view, though of course not everyone will be on board with it (I will later discuss whether accepting the Certainty View can solve the problem I norms of assertion, where Douven (2006) argues that the norm of assertion is not (as the consensus view holds) “assert only what you know” but rather “assert only what is rationally credible to you,” where what is rationally credible to one is what we can or could rationally believe. 5Scoring rule arguments for probabilism have made use of the idea that it is an epistemic virtue to have cr(p) closer to 1 in worlds in which p is true and closer to 0 in worlds in which p is false. See Pettigrew (2011) for an overview of these arguments. 5 raise for the eliminativist about belief).6 If belief cannot be reduced to credence 1, then there are two initially promising proposals. The first is the Threshold View: there is a threshold t such that a rational agent believes p if and only if cr(p) ≥ t. Of course, t may be somewhat vague. The second is the Modified Threshold View: that credence above a threshold, where the threshold is relative to the context or stakes involved, is necessary and sufficient for belief for a rational agent.7 The (unmodified) Threshold View has met with problems in the form of familiar paradoxes such as the Lottery Paradox and the Preface Paradox.8 I will concentrate on the former.9 Consider a candidate threshold t. Now consider a fair lottery with n tickets, where n > 1/(1 – t). For a rational agent, propositions of the form “Ticket m is not the winning ticket” are all rationally given credence above the threshold, as is the proposition “Some ticket will be the winning ticket.” On the Threshold View, this implies that it is rational to believe all of these propositions, even though they jointly contradict. Therefore, under the assumption that one ought to believe the conjunction of what one believes (the “conjunction principle”), a rational agent ought to believe a contradiction. Furthermore, Douven and Williamson (2006) show that “defeasible threshold views” – views that say that there is a presumptive threshold credence for belief but that belief can be defeated in the presence of some specified condition that is purely formal in nature – run into modified versions of the lottery paradox. However, not everyone is willing to accept that the Lottery Paradox defeats the Threshold View. The “Lockean,” for example, holds that the Threshold View is correct, and denies the conjunction principle: he thus allows that a rational agent can hold each of the “lottery beliefs” without holding their contradictory conjunction. In any case, we might worry about resting a claim about the relationship between rational belief and rational credence on a case which seems independently to present a puzzle for the belief package. I will argue that neither kind of threshold view can be correct, using pairs of cases that have the same stakes. The initial pair of cases is familiar to legal scholars, although these cases have also been discussed somewhat in the epistemology literature. The pairs of cases all have a structure in which it is clear what the rational credences are. Furthermore, they illustrate a point that goes beyond refuting the 6Authors generally focus on objecting to the claim that cr(p) = 1 is necessary for belief. Nonetheless, the claim that cr(p) = 1 is sufficient for belief has also met with challenges: see Maher (1993) and Hájek (ms.). For an alternative picture on which full beliefs have maximal credence, see van Fraassen (1995), who takes conditional credences to be basic and full beliefs to be derived from them. 7That justified belief requires credence over a threshold, which is relative to the stakes involved, is motivated in Fantl and McGrath (2002) by consideration of the phenomenon of “pragmatic encroachment.” One kind of Modified Threshold View is what Schroeder and Ross (2012) call Pragmatic Credal Reductivism, spelled out in Weatherson (2005), and (under one interpretation) Fantl and McGrath (2010). (See also Harsanyi (1985) for a view of this type.) This view also fits with the general spirit of Hawthorne (2004) and Stanley (2005), although they both formulate their views in terms of epistemic probability rather than subjective probability or credence. 8A more recent argument against Threshold Views that I won’t discuss, but that is worth examining, is Jane Friedman’s (forthcoming) argument from the rationality of suspending judgment on high-credence propositions. 9Kyburg (1961). 6 threshold views: that there can be no purely formal reduction of belief to credence. Finally, consideration of these cases will help uncover the domain in which belief plays an essential role. 3. No Formal Reduction We begin with a famous court case, the classic example of what is known as “the problem of naked statistical evidence” in legal scholarship.10 Here is the court case in broad outline. We will examine the hypothetical version that is usually presented in legal scholarship, the “Blue Bus Case,” which abstracts from the non-critical details of the actual case.11 As Fred Schauer presents it: “Suppose it is late at night…and an individual’s car is hit by a bus. This individual cannot identify the bus, but she can establish that it is a blue bus, and she can prove as well that 80 percent of the blue buses in the city are operated by the Blue Bus Company, that 20 percent are operated by the Red Bus Company, and that there are no buses in the vicinity except those operated by one of these two companies. Moreover, each of the other elements of the case – negligence, causation, and, especially, the fact and the extent of the injury – is either stipulated or established to a virtual certainty.”(81-82) In civil cases, the standard of proof is that the plaintiff must prove her case “by a preponderance of the evidence.” This is usually taken to mean “by a balance of the probabilities” (Schauer notes that that is the phrase used in English law), which we might think means cr(p) > 0.5, where p is the proposition the plaintiff is trying to establish.12 However, in the actual case, and “as the overwhelming majority of courts would conclude,” according to Schauer, the plaintiff cannot win the lawsuit, because the evidence that the plaintiff was hit by a Blue Bus is ‘merely statistical’. It is important to note that the statistical evidence is not inadmissible; rather, it is insufficient on its own.13 Given these facts, let us consider another hypothetical case, which we will call the “Green Bus Case”: Suppose it is late at night, and an individual’s car is hit by a green bus. The two bus companies in the area, the Green Bus Company and the Yellow Bus Company, each operate 50 percent of the green busses. There is an eyewitness, who identifies the bus as belonging to the Green Bus Company (the two bus companies operate busses with distinctive shapes). It is night-time, and so her vision is not ideal: let us say she makes mistakes 25% of the time. All of the other elements of the case remain the same. 10 Central discussions of this case and others involving naked statistical evidence appear Nesson (1985); Cohen (1977); Thomson (1986); Colyvan, Regan, and Ferson (2001); and Redmayne (2008). 11 Presentation based on Schauer, Chapter 3. See that chapter for further details of the actual case. 12 But see Cohen (1977) for arguments (in addition to the one considered here) against the thesis that evidential standards can be cashed out in terms of credences or other “Pascalian” notions of probability. 13 See Cohen (1977: 82). 7 Given the standard of preponderance of the evidence, we could speculate that in this case, the plaintiff would win a suit against the Green Bus Company.14 The situations appear to license the following credences as rational: cr(BB) = 0.8; cr(GB) = 0.75, where BB stands for the claim that a bus belonging to the Blue Bus Company hit the woman in the first case, and GB stands for the claim that a bus belonging to the Green Bus Company hit the woman in the second case. However, only in the second case – the one with the lower credence – could the court judge that the plaintiff has won the suit. Let us use the language “a verdict that p is (or is not) licensed” to mean that a court ought (or ought not) to conclude that p. Here we have a case with the same stakes and context, in which cr(GB) = 0.75 does license a verdict that GB, but cr(BB) = 0.8 fails to license a verdict that BB. This is to say: threshold views of the relationship between licensed court verdicts and rational credence are false. I don’t want to rest too much on the undoubtedly vexed relationship between it being licensed for a court to conclude that p on the basis of some evidence and it being rational for an epistemic agent to believe that p on the basis of that evidence. What is important about this example for our purposes is that the claims about belief analogous to those about licensed verdicts are intuitive in these cases. It seems clear that when we reflect on all the evidence available in the case, and reflect on what we ought to believe, we don’t have a clear (rational) belief about whether the Blue Bus hit the woman.15 But in the case of the Green Bus, we do. (If you are worried that 0.75 is never high enough for belief – that there is some necessary (possibly stakes-dependent) credence threshold that is higher – then vary the examples to increase both numbers above whatever threshold you think is high enough for the Green Bus Case, e.g. make 95% of the busses Blue Buses in the first case and make the eyewitness 90% reliable in the second.) So I want to tentatively conclude that rational beliefs about this case track the licensed verdicts. Here is another case, with the same form, that seems to prompt the same intuitions. You leave the seminar room to get a drink, and you come back to find that your iPhone has been stolen.16 There were only two people in the room, Jake and Barbara. You have no evidence about who stole the phone, and you don’t know either party very well, but you know (let’s say) that men are 10 times more likely to steal iPhones than women. I contend that this isn’t enough to make you rationally believe that Jake stole the phone. If you accused Jake, he could, it seems to me, rightly point out that you don’t have evidence that he in particular stole the phone. He could protest that you only know something about men in general 14 In any event, if “preponderance of the evidence” sometimes requires only that the claim is more probable than not, we could tweak the information given so that it would license the same credence as in this hypothetical case and also license a court verdict. Since the argument in this section only hinges on what we ought to believe in these cases, the complexities of the actual legal system are unimportant to the discussion here. 15 See also Thomson (1986), who argues that in the Blue Bus case, we don’t know whether the blue bus hit the woman. 16 I thank the students in Robert Audi’s graduate seminar at Notre Dame for suggesting this case. 8 or on average. But you should have a high credence that Jake stole the phone: if you had to place a bet with only monetary gain and loss at stake, it is clear that you should bet on Jake (given the statistics, you can expect to do better in general by betting on the man in in these kinds of cases: assuming there are an equal number of men and women in the population, then for every 11 cases of iPhone-stealing, 10 are perpetrated by men). On the other hand, if we modified the case so that you know that men and women are equally likely to steal, but a fairly but not perfectly reliable eyewitness (let’s say, 90% reliable) tells you she saw Jake take it, it seems that you can rationally form the belief that Jake took it, even though you would have a lower credence in this case. A similar point holds if Jake has a guilty look and if guilty looks indicate strongly but not perfectly that the individual has perpetrated the crime in question. Statistical evidence generally produces a rational belief in a chance-c-that-p proposition. It also presumably produces a rational credence of cr(p) = c. But what is interesting about statistical evidence is that it is often by itself not enough to produce a belief that p, even when c is very high. Admittedly, it will be hard to say what counts as merely statistical evidence, and I am leaving open whether statistical evidence can in some cases produce belief: I only claim that in many cases it cannot, even though it produces a higher credence than a rational agent will have in other cases in which she does believe. In at least some instances, belief is not fixed by credence, even in combination with stakes and context. That bare statistical evidence cannot produce belief is a common enough position in the literature. The Blue Bus case has been discussed extensively in the legal literature, and to a certain extent in the epistemology literature.17 Furthermore, in the epistemological literature, Thomson (1986), Kaplan (1996), and Nelkin (2000) have each proposed to solve the Lottery Paradox by claiming that purely statistical evidence should not produce belief.18 There is disagreement about why exactly these cases don’t give rise to a verdict or to rational belief.19 But most scholars seem to focus on the fact that beliefs formed on the basis of statistical evidence, if true, are correct as a matter of luck, and moreover, that the believer knows this (this makes them different from, say, Gettier cases). For example, as Thomson and Nelkin both point out, beliefs formed on the basis of statistical evidence are unsafe: crucially, they are not causally connected to the truth of the proposition. But the belief in the chance-c-of-p proposition can be safe – or, more generally, correct not as a matter of luck – and so need not run afoul of rationality. Furthermore, the relevant credences are not going to run afoul of rationality. If one’s credence in p is 17 For references to the legal scholarship, see footnote 10. Discussions that focus on both legal and epistemological issues include Thomson (1986) and, more recently, Enoch et al (ms.). These do not explicitly focus on credence. 18 These accounts have come under fire. See, for example, Douven’s (2000) reply to Nelkin. Douven’s reply is specifically aimed at Nelkin’s claim that the “One False Belief” accounts of Bonjour (1985) and Ryan (1996) cannot handle an additional case she proposes. The cases here, however, have a different structure than Nelkin’s cases. 19 For an outline of the disagreement about why they don’t give rise to a guilty verdict in the legal case, see Redmayne (2008). 9 based only on statistical evidence, then one’s credence exactly matches the frequency in the relevant class. What we’ve seen is that a certain kind of evidential basis can give rise to a justified high credence without giving rise to a justified belief, whereas other kinds of evidential bases can give rise to a justified lower (but still high) credence and yet also give rise to justified belief. What is important about the cases here, and has not historically been the focus of the literature (primarily because the literature on statistical evidence has focused on what makes a belief justified rather than on the relationship between belief and credence), is that (1) the statistical-evidence cases here can be paired with non-statistical evidence cases that have the same stakes and context; and (2) it is clear what the rational (or at least rationally permissible) credence is in these cases. Thus, the argument here against the Threshold View rests on few auxiliary assumptions about rational belief (it does not, for example, assume the conjunction principle) and contains fewer “escape routes” in the form of allowing the threshold to change in response to other facts about the agent’s situation. Again, I want to be clear that I don’t have a general thesis about the role of statistical evidence in belief-formation. Clearly, statistical evidence, when paired with other kind of evidence, can figure into rational belief-formation: for example, evidence that the fingerprints found at a crime scene are a statistical match with those of the defendant, in combination with some evidence suggesting that she had motive to commit the crime, can lead to both a verdict and a belief in her guilt, when motive alone would not. Furthermore, it is possible that there are some cases in which statistical evidence on its own can give rise to belief. I am making the rather modest point that in at least some cases of bare statistical evidence, the evidence fails to produce a rational belief but does produce a rational high credence: higher than the credence in analogous cases in which the evidence does give rise to belief. Why not try to build in the type of evidence into a reduction of belief to credence? The problem is that we aren’t going to be able to read off the type of evidence from purely formal features of one’s credal state. Granted, when the statistical evidence is about objective chance, one will have a high credence, if not credence 1, in a chance-c-of-p proposition. But consider again the iPhone theft cases, in which the statistical evidence is clearly not about objective chance. In the first case, you know that Jake is a man and that men are more likely to steal. In the second case, you know that Jake looks guilty and that people are more likely to look guilty after they’ve stolen. In both cases, you have a high conditional credence that Jake stole, given, alternately, that Jake is a man and that Jake looks guilty.20 But only in the 20 I’m leaving open how we want to represent the statistical evidence in the credal framework, as cr(p(Js) = 0.9 | Jm) ≈ 1, or as cr(Js | Jm) = 0.9. The latter seems more straightforward, but if we want to interpret statistical evidence as being evidence about the epistemic probabilities, we might want to employ the former. As for the suggestion that believing or having a high credence in an epistemic-chance proposition blocks outright belief, this won’t work because epistemic-chance propositions are not believed only in response to statistical evidence: presumably one also 10 second case do we think you ought to believe that Jake stole. A plausible explanation of this is that the counterfactual “if Jake hadn’t stolen, Jake wouldn’t look guilty” is true if Jake did in fact steal, but the counterfactual “if Jake hadn’t stolen, Jake wouldn’t be a man” is false regardless of whether Jake stole. Or, alternatively, that if Jake is guilty, then his guilty look is caused by his guilt but his being a man is not. And there need be no formal differences in credences between the cases. The crucial point is that one can’t in general read the difference between causation and correlation off of a probability function; one needs to intervene in the world in order to establish a causal relationship.21 Even though there won’t be a “local” difference in credence in the cases, one might wonder whether there will be a “global” or “holistic” difference, a difference in credences related to the target credences. For example, one might hypothesize that credences based on statistical evidence are less resilient than credences based on non-statistical evidence.22 One might think that in most cases in which you have an extremely high credence, most pieces of evidence that you might get will not lower your credence very much, but that in the lottery case, for example, the announcement of the winner has the potential to drastically change your credence. Similarly, one might think that a second eyewitness will make less of a difference to the Green Bus case than a first eyewitness would make to the Blue Bus case. Cashed out formally, one might hypothesize that there will be a difference in the probabilities of BB and GB conditional on other relevant evidence. The problem with this response is that there won’t be a difference between these conditional credences when the new evidence is independent of both the old eyewitness and the statistical evidence. Consider in each case the effect of an independent eyewitness, with reliability 0.75, who states that the bus belonged to the other company. In the Blue Bus case, the rational agent’s credence on the new evidence will be cr(BB) = 0.57, and in the Green Bus case, her credence on the new evidence will be cr(GB) = 0.5.23 And if it is true that the wrong causal direction is why the evidence that should produce a high credence should not produce a belief, then this point generalizes. The statistical indistinguishability of causation from correlation (in non-intervention believes that there is a high epistemic chance Jake stole in the “guilty look” case – that is just what it means to believe the guilty look is evidence of Jake’s guilt in this case. 21 See Spites, Glymour, Scheines (1993). There are a few exceptions to this general claim but they are not relevant to the present case. Perhaps an objector could claim there will be a difference in one’s credences in the relevant counterfactuals. But I doubt that an agent needs to formulate a credal opinion about counterfactuals in order to count as rational. Alternatively, one could try to add more to structure to credence functions. If one wants to take these escape route, it will be an interesting upshot of the argument here that rational agents need to have much more complex credences than is ordinarily supposed. 22 I thank Brian Weatherson and Roger White for raising this point. 23 In the Blue Bus case, where E is the new eyewitness’s testimony and S is the statistical evidence, cr(BB | E & S) = cr(E | BB)cr(BB | S)/[cr(E | BB & S)cr(BB | S) + cr(E | ~BB & S)cr(~BB | S)] = (0.25)(0.8)/[(0.25)(0.8) + (0.75)(0.2)] = 0.2/0.35 ≈ 0.57. In the Green Bus case, where E is the new eyewitness’s testimony and O is the old eyewitness’s testimony, cr(GB | E & O) = cr(E | GB & O)cr(GB | O)/[cr(E | GB & O)cr(GB | O) + cr(E | ~GB & O)cr(~GB | O)] = (0.25)(0.75)/[(0.25)(0.75) + (0.75)(0.25)] = 0.5. 11 settings) means that taking all the formal properties of a credence function into account – even the global ones – won’t be enough to distinguish between causation and correlation. What these cases bring out is that rational credence and rational belief are sensitive to different features of evidence. So while a given body of evidence will usually support a belief just in case it supports a high credence, there is no necessary connection between the two. The statistical cases show that credences don’t distinguish between certain facts about our evidence in the way that belief does. What this suggests is the following picture: at the “base level,” we have a body of evidence, which separately determines rational credence and rational belief. Since evidence that supports a high credence is often evidence that supports belief, there is generally a connection between the two. But the in-general connection is not intrinsic: it occurs because of the way both credence and belief are related to evidence, not because of the way they are related to each other. Consideration of the fact that two different evidential bases can be such that the one produces a higher credence in p and no belief that p, and the other a lower credence in p but belief that p, also allows us to question an initially plausible sounding tenet about the relationship between credence and belief: if one believes p, and one’s credence in p increases, then one continues to believe p. The statistical cases provide an easy example. Consider Kelly, a rational agent who is participating in a game show where she might win a prize. She has a very high credence (and belief) that the winner is determined by another contestant’s choice, and she has a very high credence (and belief) that the contestant hates her, so she has cr(WON’T WIN) = 0.95. Let’s say that she also believes she won’t win the prize. She then discovers that the winner is determined by a fair 100-ticket lottery. Her credence increases to cr(WON’T WIN) = 0.99, but she no longer believes that she won’t win; rather, she believes that she will almost certainly not win. If you are torn about this case, consider an analogous case involving judgment about a person’s guilt, e.g., you learn that Jake didn’t have a guilty look on his face (just a bad reaction to cold medicine) and simultaneously learn that men are more likely to steal. I submit that your credence in Jake’s guilt will increase, but you will lose your belief. The principle that belief is stable in response to an increase in credence (we might say, that belief is “monotonic” with respect to credence) is generally true. However, 12 the fact that it is sometimes false shows that its appeal might be explained not by a tight relationship between credence and belief, but by the fact that in most ordinary cases, evidence that leads to an increase in credence also preserves belief. Belief cannot be read off the purely formal properties of a credal state, even if we take into account stakes and context. However, as I will argue in the remainder of this paper, belief is ineliminable from our best theories about the norms associated with holding each other responsible. 4. Belief and Blame Given that belief is not reducible to credence, we might hope that we can do away with the notion of belief entirely by precisifying the principles in which it plays a role, or by relegating it role in the mental life of non-ideal agents, e.g., as a heuristic. However, as I will argue in this section, it turns out that we need belief, and its accompanying epistemology, precisely because there is a domain in which our norms involving belief are sensitive to the kinds of evidential connections that belief tracks but credence doesn’t.24 Let us consider the context in which the idea of credence was developed, and the norm in which it is well-suited to play a role: that of decision theory. Initially, decision theory was developed to characterize how one should bet in explicit betting contexts where the payoff of a bet depends on an objective-chance mechanism, such as the roll of a dice or the arrangement of a deck of cards. The norm of decision theory in its initial form, as developed by Pascal, was that one ought to choose, among the available actions, the action that maximizes expected monetary value, given the objective probabilities involved.25 That is, when facing a choice among lotteries of the form L = {$x1, p1; $x2, p2; …; $xn, pn}, where L yields $xi with probability pi, one ought to choose the one with the highest value of EV(L) = ∑ . 24 Theories that seek to eliminate belief altogether include Jeffrey (1970) and Christensen (2004). The latter argues that the notion of binary belief is useful, though “may not in the end capture any important aspect of rationality”(ix). Theories in which belief and credence play different roles in the same domain include the “reasoning disposition account” of Ross and Schroeder (2012). Theories in which credence and belief play the same role but occupy a different discourse include that of Frankish (2009). Sturgeon (2008) is a difficult theory to categorize, since he thinks that everyday evidence does not always rationalize sharp credence, and fuzzy confidence of a certain sort is identical with belief, but I tentatively place his theory in the category of theories in which credence and belief play a role in the same domain. Two theories that do recognize different primary roles for credence and belief are Mark Kaplan’s (1996) Assertion View and Patrick Maher’s (1993) notion of “acceptances.” Both Kaplan and Maher claim that our ordinary notion of belief is not coherent, and each propose to replace it by a notion that shares many of the features of belief and does much of the same work. (Therefore, there is a sense in which these theories are eliminativist.) These theories are not reductionist in the sense that they don’t reduce assertions or acceptances to credence, but they are reductionist in that they reduce the rationality of assertions or acceptances to facts about the agent’s credences plus something else: for example, according to Maher, one rationally accepts a proposition if doing so maximizes expected “cognitive” utility. I think these theories are on the right track in their recognition of two very different kinds of activity, one which involves credence and one which involves something else. 25 See Fermat and Pascal (1654). 13 Decision theory in its modern form is the result of several modifications to this norm.