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date: 20 January 2018

# Formal Epistemology

## Abstract and Keywords

Formal epistemology is a young but vibrant field of research in analytic philosophy characterized by both its matter and its method. The subject matter of this field is epistemology, the theory of knowledge. The method for investigating the subject matter of epistemology involves the use of formal, logicomathematical devices. This chapter highlights the major achievements of formal epistemology so far and gives a sense of what can be accomplished by addressing problems from mainstream epistemology with the use of logic, probability theory, computer simulations, and other formal tools. The historical roots of the field are also described, and there is a discussion of new questions that have been raised by formal epistemology that should also be of interest to mainstream epistemologists. We also pay attention to the currently emerging subfield of formal social epistemology.

Formal epistemology is a flourishing subfield of analytic philosophy characterized by both its matter and its method. The subject matter of this field is epistemology, the theory of knowledge. The method for investigating the subject matter of epistemology involves the use of formal, logicomathematical devices. Formal epistemologists attempt to break new ground on traditional epistemological questions, using an ever-expanding and -improving set of such devices. The philosophical application of various formal devices has itself given rise to a host of new, hotly debated epistemological questions. In this entry, we begin by discussing the discipline of formal epistemology, its historical background, and its foundations (section 1). Then, we summarize some recent work in formal epistemology, as it pertains to both traditional epistemological puzzles (section 2) and puzzles arising from the development of formal epistemology (section 3).

# 1. Historical Background and Foundations

As an identified and self-contained subfield of philosophy, formal epistemology is a relative newcomer in analytic philosophy. We are not aware of any references to formal epistemology in the literature before 1990. But in a short time, the field has firmly established itself. There are now many highly active formal epistemologists, philosophy departments advertise jobs explicitly in formal epistemology, and numerous workshops and conferences every year are exclusively devoted to topics in formal epistemology. Like most academic disciplines, philosophy has its fads and fashions. But formal epistemology, we can now say with some confidence, is here to stay.

While a new branch, formal epistemology grew on a tree with deep roots. First and foremost, this is true because many of the problems it deals with come from traditional epistemology, dating back to the ancient Greek philosophers. But even its hallmark methodological approach has origins in the history of philosophical and mathematical thinking—witness Aristotle’s development of formal logic and his application of this device in the characterization of scientific knowledge (epistêmê).

Formal epistemology combines tools taken from logic and probability theory. The logic referred to here is the still relatively modern type of logic initiated by Gottlob Frege, Bertrand Russell, and other mathematicians working at the dawn of the twentieth century. This logic saw important developments during much of the last century, certainly until the 1970s. Probability theory dates back further, with Christiaan Huygens and the Port Royal logicians (in collaboration with Blaise Pascal) laying the groundwork in the mid-seventeenth century. Jacob Bernoulli, Thomas Bayes, and Pierre-Simon Laplace made important contributions in the eighteenth century, and with Bruno de Finetti (1937/1964) and Andrey Kolmogorov (1950), probability theory received its contemporary form.

Like analytic philosophers generally, epistemologists have always relied on logic for clarifying or checking their arguments. But it was only after a proper semantics had been developed for modal logic (mainly in the work of Saul Kripke) that they started using logic to analyze epistemological concepts. Following Jaakko Hintikka’s pioneering work in Knowledge and Belief (1962) came the rise of “modal epistemology,” which seeks to analyze knowledge, justification, and related notions in terms of what goes on, not just in the actual world, but also in various nonactual worlds (typically worlds that are, in some sense, “close” to the actual one).

Mostly, however, the formal part of the work stopped with those definitions, and much of what goes under the name of “modal epistemology” is best classified as belonging to mainstream, rather than formal, epistemology. Some philosophers did go on to develop formal models of knowledge, justification, and belief using various modal logics. These formal models often make highly idealizing assumptions about the epistemic notion(s) they aim to represent—such as assuming “epistemic closure,” according to which knowledge is closed under logical entailment (which is validated by Hintikka’s epistemic logic), or assuming that everything that is believed is also believed to be believed (the “positive introspection principle,” as validated by KD45 and kindred logics of belief; see Meyer and van der Hoek [1995]). On the other hand, we know from the sciences that models that make idealizing assumptions can still be predictively accurate or valuable in other ways.

Nevertheless, epistemic logics have never gained as much traction in formal epistemology as probabilistic approaches. This is arguably because probability theory offers a modeling tool that is more versatile and flexible than epistemic logic. Moreover, in contrast to the latter, probability theory is built on the insight that, to understand how humans cognitively relate to the world, we must attend to the fact that our reasoning and thinking generally involve uncertainty along with more or less strong degrees of confidence or belief.

The insight that a full understanding of human thinking and rationality requires taking seriously a graded notion of belief received much of its impetus from work in psychology starting in the 1980s. Until then, it had been the received view among psychologists that good reasoning is a matter of obeying the laws of logic. But logic was developed to facilitate mathematical reasoning, in which we go from axioms to theorems via inferential steps that are necessarily truth-preserving. Psychologists have noticed that much of our nonmathematical reasoning can be good, despite being uncertain and defeasible. Accordingly, they claim that the standards of rationality for such reasoning are not provided by a monotonic logic but must be sought elsewhere. In work that marked the beginning of what is now generally known as the “New Paradigm” (Over [2009], Elqayam and Over [2013]), psychologists discovered that people did, overall, quite well at probabilistic reasoning, despite the fact that they were also prone to commit certain fallacies, as had in fact already been reported in earlier work (e.g., Kahneman, Slovic, and Tversky [1982]).

Because of its centrality to formal epistemology, it is worth being clear about what probability theory is. At bottom, the theory is quite simple: all there is to it are a couple of easy-to-grasp axioms plus a definition. Here is a simple presentation of the axioms:

1. A1. $0=Pr(φ∧¬φ) ⩽ Pr(φ) ⩽Pr(φ∨¬φ)=1;$

2. A2. $Pr(φ∨ψ)=Pr(φ)+Pr(ψ)−Pr(φ∧ψ).$

In words, these axioms state that all probabilities are between 0 and 1, with logical falsehoods receiving a probability of 0 and logical truths receiving a probability of 1; and that the probability of a disjunction is the sum of the probabilities of the disjuncts minus the probability of their conjunction. The definition concerns the notion of conditional probability, the probability of one proposition given, or on supposition of, the truth of another proposition. This definition states that the probability of φ given ψ, Pr(φ | ψ), equals the probability of the conjunction of φ and ψ divided by the probability of ψ, so $Pr(φ∧ψ) / Pr(ψ)$. Someone whose graded beliefs are representable by a probability function—a function Pr(·) satisfying A1 and A2—is said to be statically coherent.1 (This notion of probabilistic coherence is unrelated to the notion of coherence that is more commonly used in mainstream epistemology, which will be discussed below.)

Probability theory is silent on how one’s graded beliefs ought to change over time as new information comes in. By far the most formal epistemologists (though not all, as will be seen) have committed themselves to a further principle of dynamic coherence, according to which we rationally revise our graded beliefs after learning with certainty new information φ precisely if, for every proposition ψ, our new unconditional graded belief in ψ equals the degree to which we believed ψ conditional on φ before we learned about the truth of φ. Formally, where Pr(·) and Prφ(·) denote the probability functions representing our graded beliefs before and after the receipt of φ respectively, dynamic coherence requires that Prφ(ψ) = Pr(ψ | φ), for any ψ. For instance, suppose that in the morning you believe to a degree of .5 that it will rain in the evening, and you also believe to a degree of .8 that it will rain in the evening on the condition that the afternoon will be cloudy. Then later, when you see that the afternoon is cloudy (assuming you have learned no other relevant information besides this), you must, on pain of being dynamically incoherent, change your graded belief for rain in the evening to .8. If you do, then you are said to update via Bayes’ rule, also known as the rule of (strict) conditionalization.2

# 2. Formal Approaches to Mainstream and Traditional Epistemology

In this section, we briefly summarize a few of the ways that formal epistemologists have confronted important traditional and mainstream epistemological questions.

## Internalism and Externalism

What does it take for an agent to have an epistemically justified belief? Mainstream epistemologists famously divide on this question into two general camps, the internalist and the externalist. On the one hand, internalists emphasize the first-person perspective of an epistemic agent. Ask yourself: What does it take for me to be epistemically justified in believing something? The most compelling and common answers assert that you have a justified belief when this belief rests upon your internally having sufficient evidence or reason for that belief. Further questions immediately arise. In what sense must I have the requisite evidence or reason? When can my beliefs be said to “rest upon” specific grounds? How much evidence suffices for epistemic justification? And so on.

By contrast with the internalist’s “egocentric” notion of justification, externalists motivate their “naturalized” accounts by emphasizing a third-person perspective. Additionally, externalists typically place their focus on the concept of knowledge rather than justification. The most salient question for the externalist is: Under what conditions would we allow that an epistemic agent has knowledge? The most compelling and common answers assert that knowledge comes by way of there being the right sort of natural relation between the agent’s belief state and the world. Justification, when mentioned at all by the externalist, is often taken generically to stand for whatever external relation is involved in turning true belief into knowledge. Again, questions immediately arise. Most notably, what precisely constitutes the right sort of natural relation?

Formal epistemologists investigate all of the above questions. For example, some basic work in epistemic logic helps focus the internalism/externalism debate by articulating precise principles that bear on having justification and knowledge. In epistemic logic, one adopts a standard Kripkean modal logic but reinterprets the modal operators epistemically. The salient notion of necessity is meant to refer to the state of knowing—or what must be the case, given what we know. The corresponding notion of possibility is meant to refer to the state of not knowing not—or what might be the case, for all we know. To mark these distinct interpretations, we replace the standard modal operator □ with the more suggestive K.

One’s choice of epistemic logic is directed by reinterpreting and evaluating standard modal axioms. Most notably for present purposes, modal logic’s S4 axiom is restated as Kφ → KKφ and reinterpreted as requiring that one knows φ only if one knows that one knows φ. This controversial axiom has come to be known as the KK principle.3

The KK principle is sometimes taken as a formally exact criterion dividing internalists and externalists. Anyone who accepts this thesis will require that knowledge has a “luminosity” (Williamson [2000]) about it such that knowers of φ not only believe themselves to know φ, but they always believe this correctly in whatever way it takes for that true belief to be a case of knowing that they know φ. Various forms of internalism naturally motivate the idea that knowledge is so luminous. For example, perhaps knowledge of φ requires true belief in φ plus some justification-involving “warrant” condition; and perhaps epistemic agents have internal, reflective access to facts about whether their beliefs are warranted. If an epistemic agent knows any φ in this sense, this internal access naturally (always?) provides her with a warranted new meta-belief: I am warranted in believing φ; since I believe φ itself, I will also then know (i.e., truly believe, with the requisite warrant) that I know φ. Even granting the above form of internalism, the argument to the KK principle is far from airtight; more details would be needed to fully motivate the principle. Nonetheless, while internalism does not imply the KK principle, it does seem like the sort of philosophical framework needed to make this principle at all appealing.

On the other hand, it seems clear that externalists will want to reject the KK principle. Whatever the nature of the external relation between belief state and world that must be satisfied to convert true beliefs into knowledge, it is (qua externalist) not a relation that agents need be internally aware of satisfying. Indeed, it is tempting to characterize externalism about knowledge in such a way that it straightforwardly entails the denial of the KK principle: externalist accounts of knowledge allow that an agent can know some φ without believing (let alone knowing) that they satisfy the externalist condition for having this knowledge—and thus without believing (let alone knowing) that they have this knowledge. Although the question of how the KK principle bears on the internalism/externalism debate is still being explored (e.g., see Okasha [2013]), externalism is arguably much closer to implying the falsity of the KK principle than internalism is to implying its truth (Bird and Pettigrew [2016] uncover a precise sense in which this is true).

This is one example of how epistemic logic can be used to clarify precise principles over which internalists and externalists clash. Marking such exact points of disagreement helps to pinpoint what is at stake when internalists and externalists differ over whether justification (or the grounds of knowledge) must be internally had by a knower. In this way, a formal approach focuses the debate over the nature of justification.

In addition to epistemic logic, formal epistemologists have used probability and statistics to investigate issues bearing on internalism and externalism. This is a natural move given that concepts like justification and evidential support are thought of by contemporary epistemologists as defeasible, gradational, and fallible.4

Prima facie, an internalist theory of justification is nicely explicated with a Bayesian logic. For Bayesians, probabilities are inherently subjective at least in the sense that they are interpreted as a particular epistemic subject’s “degrees of (rational) belief.” Bayesians require that an agent’s degrees of belief be statically and dynamically coherent (see section 1). The epistemic agent has a stock of “background knowledge,” and that agent’s degrees of belief, to be (statically and dynamically) coherent, must be fixed by these known propositions in such a way that they satisfy the axioms of probability A1 and A2. Bayes’ rule (along with generalizations thereof) provides the Bayesian with a constraint on how an agent’s knowledge (new evidence or reasons) fixes rational degrees of belief.

This all does have an internalist ring to it. An agent’s own credences are the locus of attention on this account, and these seem plainly internal to the agent.5 Any rational change to degrees of belief is affected by the agent learning new evidence; adding such evidence to the agent’s background knowledge is what activates Bayes’ rule. This change too seems internal to the agent’s cognition. But where exactly in the formal framework should the Bayesian locate the notion of internalist justification? The answer seems to depend on what more the Bayesian may or may not want to say about how rational degrees of belief are constrained.

The straightforward answer seems to be that internalist justification is a matter of degree, measured by the Bayesian probabilities themselves; Bayesian degrees of rational belief are just degrees of justification. But this tempting idea at best only makes sense on an “objective Bayesian” account, according to which all rational credences are fixed at precise real values by background knowledge and logical principles.6 The majority of Bayesians take a “subjective” stance to some extent, allowing that rational degrees of belief are not entirely fixed in all cases. For example, the most subjective of Bayesians assert that rationality requires no more than that one’s degrees of belief satisfy axioms A1 and A2 and that one change one’s degrees of belief via Bayes’ rule. But two hypothetical epistemic agents sharing all of the same background knowledge could differ wildly in all of their probability assignments while satisfying the aforementioned requirements. Similarly, two agents could have identical probability assignments in all propositions, but differ greatly in how much background knowledge they have supporting these credences.

An example makes the problem clear. Two epistemic agents are confronted with a coin. Bill is a completely naive agent who has no information about the behavior of this coin, while Hazel has experimented tirelessly flipping this coin. Let us say that Hazel has observed exactly 750,000 heads and 250,000 tails in 1,000,000 flips of the coin. Bill knows that the coin will be flipped again and knows that it has to land heads or tails, but he has no reason whatever to think it will land one way over the other (Bill has not discussed the matter with Hazel, experimented with or even examined the coin, etc.). Nonetheless, Bill and Hazel have equal degrees of belief with respect to this coin landing heads (H) versus tails (T) on the next flip: Pr(H) = .75; Pr(T) = .25. Both agents are statically coherent, and we may stipulate that they are both dynamically coherent as well. Shall we allow that they are both equally justified (to degree .75) in believing that the coin will land heads on the next flip? Of course not; Hazel has far more evidence for believing this and so is far better justified.

This motivates another possible answer to the original question for the subjective Bayesian. The degree to which an agent is justified in having some degree of belief is not measured by the degree of belief itself but by the amount of background evidence serving to fix this credence. The subjective Bayesian can (and should) distinguish an agent’s credence from the “weight of evidence” grounding that credence. This weight of evidence is what plausibly explicates degree of justification for the subjective Bayesian—though how to formalize weight of evidence is a matter of ongoing investigation (Joyce [2005]).

With a Bayesian explication of justification in hand, internalists may gain a new foothold on resolving some venerable problems. Most famously, given the internalist’s notion of justification, we might doubt with Hume (1748/1912) whether beliefs about the world (“matters of fact”) can ever really be “founded on reasoning, or any process of the understanding.” But some formal epistemologists argue that probability theory provides us with new tools allowing us to respond to Hume’s problem of induction (Earman and Salmon [1992], Howson [2000], McGrew [2001]).

Even so, general complications remain for a Bayesian explication of internalist justification. First, note that weight of evidence really at best explicates the degree to which a credence is justified rather than any proposition or categorical belief. If a vast amount of evidence leads an agent’s credence to converge on Pr(φ) = .5, the weight of evidence might be very great indeed. But this does not mean that the agent is strongly justified in believing φ. In fact, the agent seems just as justified in believing ¬φ, which is just as strongly favored by all of the evidence in hand (since Pr(¬φ) = 1−Pr(φ) = .5). The agent is, however, strongly justified in having this credence. So, it is the justification of credences rather than beliefs or propositions that is being explicated by weight of evidence; the explicandum has shifted. Second, the idea that Bayesianism may nonetheless provide a formal internalist epistemology of degrees of belief has not gone unchallenged. Recently, Alvin Goldman (2009) and Christopher Meacham (2010) have independently (and all too briefly) argued that Bayesianism cannot explicate a pure form of internalism, since it requires external constraints on rational degrees of belief.

Can probability and statistics serve to illuminate and develop externalist accounts? This question is receiving increasing attention by formal epistemologists in recent years. To follow one example, Sherrilyn Roush (2005) uses probability theory to reexamine and develop Robert Nozick’s (1981) original counterfactual, tracking theory of knowledge. This theory analyzes the concept of epistemic agent $A$ knowing proposition φ as $A$ having a true belief in φ while satisfying two externalist conditions: the Sensitivity condition, according to which $A$ would not believe φ if φ were not true; and the Adherence condition, according to which $A$ would believe φ if φ were true.

Roush replaces these counterfactual conditions with the following probabilistic conditions:

• Probabilistic Sensitivity: The probability that a subject does not believe φ on the supposition that φ is false, Pr(¬Bφ | ¬φ), is greater than threshold t, where .95 < t < 1.

• Probabilistic Adherence: The probability that a subject believes φ on the supposition that φ is true, Pr(Bφ | φ), is greater than threshold t, where .95 < t < 1.

The effect is that Roush’s development of the tracking theory sidesteps a host of counterexamples to Nozick’s theory. For example (Roush [2005:98–100]), Sam sees Judy and believes P: that the girl he just saw is Judy. However, he is unaware of the fact that Judy’s identical twin Trudy was nearby, and that it was only by a twist of luck that he saw Judy herself and not Trudy—in which case Sam would have believed P falsely. In the nearest ¬P-worlds, Sam may not see Judy or Trudy, in which case his belief satisfies Sensitivity (and Adherence). But contrary to the original tracking account, few epistemologists would allow that Sam’s true belief is knowledge. Roush’s account gets this right. Probabilistic Sensitivity is not satisfied, since there is a significant chance that Sam sees Trudy and comes to believe P falsely; Pr(BP | ¬P) is not low, and so Pr(¬BP | ¬P) is not sufficiently high.

Several formal epistemologists offer alternative developments of externalists’ accounts, using distinct formal devices. Regarding the tracking theory, Horacio Arló-Costa and Rohit Parikh (2006) reject Roush’s approach and instead develop an account using doxastic logic. Kevin Kelly (2014) proposes a very different computational account of knowledge inspired by tracking accounts. Conor Mayo-Wilson (2016) proposes a modal-probabilistic formalization of Sensitivity and Adherence, but then goes further and tests his formal account in interesting ways against scientific practice. Interestingly, formal epistemologists working on tracking do not simply disagree over the best explication of Sensitivity and Adherence, but more fundamentally on what formal tool is best suited for explicating these conditions.

What can formal epistemology offer the mainstream epistemologist with respect to the internalism/externalism debate? While the answer is ultimately yet to be decided, the above discussion allows us already to say the following. Formal approaches serve to clarify this debate by uncovering exact points of agreement and disagreement between the sides. And the internalist and externalist positions themselves are being clarified; exactly what forms of internalism and externalism there are on offer is made clearer by formal epistemologists striving to get precise about what various accounts of justification and knowledge are claiming. Moreover, we have seen that particular problems and counterexamples for mainstream accounts are finding potential resolution in formally subtler developments of these accounts. To the extent that formal methods provide a fruitful means for developing both internalism and externalism, it may even turn out that formal epistemology does not so much advance this longstanding debate as show that there was something to each side’s position all along. Justification, after all, is plausibly polymorphous, and there may well be interesting philosophical aspects of internalist and externalist notions of justification (Staley and Cobb [2011]). All in all, in a variety of ways, formal epistemology seems to be offering promising new approaches to investigating these mainstream issues.

## The Structure of Justification

Granting that we can have epistemically justified beliefs, what structure does this justification take? Epistemologists often clarify this question through a famous and ancient puzzle known as the “Regress Problem” (e.g., BonJour [1999:118 f], Weisberg [2015, sec. 3]). Say that belief b is epistemically justified because it is connected in the relevant way to b1. This reason must itself be justified if it is to play the requisite role, and so we may press for the justification of b1. If b2 is the reason that suffices for justifying b1, then we may continue our inquiry and find that it is b3 that justifies b2, b4, which justifies b3, and so on. A regress of reasons threatens, and the question is how it terminates. Three alternate answers suggest themselves:

• Infinitism: The regress never stops; an infinite chain of ever-new reasons describes the structure of justification.

• Coherentism: The regress really moves between members of a large network of interconnected propositions; it is the coherence of this network that bestows justification on any of the propositions that form its nodes.

• Foundationalism: The regress eventually terminates in reasons that are justified, but not on the basis of other reasons; all beliefs are ultimately justified by being grounded in these “properly basic” beliefs.

The Regress Problem has often been cited in support of foundationalism (McGrew and McGrew [2008]). A simple version of such an argument takes the form of a reductio. Since legitimate epistemic justification could not possibly come by way of infinite chains of reasons, or entirely by way of (possibly circular) interconnections between propositions in a network—so the claim goes—rational regresses must eventually terminate in a foundational layer of properly basic beliefs. But formal epistemologists have breathed new life into the infinitist and coherentist alternatives by showing that, when explored more carefully, these alternatives to foundationalism are not obviously untenable.

Jeanne Peijnenburg (2007) and Peijnenburg and David Atkinson (2008) offer a probabilistic investigation into infinitism. Construing justification as a matter of degree explicated by probabilistic support, Peijnenburg and Atkinson (2008:333) ask the following question: “Consider [a] chain S0, S1, S2, …, etc., where each Sn+1 probabilistically justifies Sn. Are we able to justify S0 in the sense that we can compute Pr(S0)?” The thought is that, since the sequence ⟨S0, S1, S2, …⟩ is countably infinite, infinitism’s opponents would want to say that Pr(S0) either must be zero or just undefined. Peijnenburg and Atkinson (2008) demonstrate that, to the contrary, this probability can be calculated, at least under certain conditions.

The demonstration in Peijnenburg (2007) runs as follows: Without loss of generality, suppose that Pr(Sn | Sn+1) = α and Pr(Sn | ¬Sn+1) = β for all n. Additionally, given that Sn+1 is the probabilistic justifier of Sn (for each n), assume that α > β. Then, by the rule of total probability,

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Thus, for example,

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And so,

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Generalizing this calculation to any n, Peijnenburg derives the following equation:

(1)

$Display mathematics$

As Peijnenburg points out, as n increases without bound, the term (αβ)n+1 occurring in equation (1) becomes smaller and smaller. The result is that, in the limit with n going to infinity, we need not know the value of Pr(Sn+1) in order to derive Pr(S0), since the latter is only a function of known values α and β:

$Display mathematics$

Here is a case then in which we finite beings can calculate Pr(S0), and in which Pr(S0) need not be 0, despite S0 resting on an infinite sequence of reasons.

However, even allowing that it speaks appropriately to the notion of infinitist justification, one should be careful not to read too much into this formal result. According to Peter Klein, the foremost defender of infinitism today, the “essential claim of infinitism” is that “[t]he reasons that justify a belief are members of a chain (perhaps branching) that is infinitely long and non-repeating” (Klein [1998:919]). Infinitism makes a claim on the general structure of justification, and Peijnenburg and Atkinson’s formal work manifestly does not establish this general theory (nor do they argue that it does). Still, the result supports infinitism by dismantling the strong intuition shared by many that an infinite string of reasons cannot ultimately provide any reason at all. It thus takes away any quick argument from the impossibility of infinitist justification to foundationalism (or coherentism).

The structure of justification proposed by the infinitist is straightforwardly explicated, even if it remains doubtful that infinitism describes the general structure of justification. However, when one turns to consider coherentism, it is not even clear what the positive proposal is. Epistemologists speak of coherence vaguely as the property beliefs have to the extent that they “hang together,” “agree with each other,” “dovetail,” or “support one another.” But coherentists and their opponents alike bemoan this theory’s lack of substance given that there is no clearer explication of coherence; for instance, Laurence Bonjour (1985:94) famously writes, “[T]he main work of giving [a general account] which will provide some relatively clear basis for comparative assessments of coherence, has scarcely been begun, despite the long history of the concept.” Prior to investigating whether coherentism describes the structure of justification, one must first get a clear grasp on what coherence amounts to.

The most popular formal approach to this problem is again probabilistic.7 Tomoji Shogenji (1999) initiated a flurry of work developing and evaluating probabilistic measures of coherence (Olsson [2002], Bovens and Hartmann [2003], Fitelson [2003], Glass [2005], Douven and Meijs [2007], Schupbach [2011a], etc.). Shogenji’s original proposal is that the coherence of an agent’s set of believed propositions S = {B1, B2, …, Bn} be explicated by comparing the joint probability of the members of S, Pr(Bl ˄ B2 ˄ ··· ˄ Bn), with the value that this joint probability would take were these members statistically independent of one another Pr(B1) × Pr(B2) × ··· × Pr(Bn):

$Display mathematics$

The function CohS thereby provides a measure of the degree to which the members of this information set are probabilistically dependent on, or relevant to, one another. Such a measure thus intuitively captures the idea that coherence is a matter of how well beliefs “dovetail” or “support one another.” For simplicity, consider the case where we are evaluating the coherence of a pair of beliefs, S = {B1, B2}. Shogenji’s measure can be rewritten as follows:

$Display mathematics$

These last two ratios straightforwardly measure the degree to which either of the propositions supports the other (as the extent to which the truth of either proposition would make the other more likely).

Despite its intuitive appeal and simplicity, formal epistemologists have put forward various arguments against Shogenji’s account and developed alternative measures for evaluation. As one example of how this dialectic goes, Branden Fitelson (2003) argues that Shogenji’s general measure focuses exclusively on the coherence of all the propositions in set S at once and thus ignores support relations among various S’s proper subsets. Jonah Schupbach (2011a) develops an example showing that this “depth problem” does indeed lead Shogenji’s explication into counterintuitive results. Both Igor Douven and Wouter Meijs (2007) and Schupbach offer “subset-sensitive” developments of Shogenji’s original measure that do not fall prey to the depth problem, but other criticisms of these developments have now been published (e.g., Siebel and Schippers [2015]).

An interesting example of an early alternative explication of coherence is put forward by both Erik Olsson (2002) and David Glass (2005):

$Display mathematics$

While Shogenji’s measure (and measures inspired by it) nicely captures informal descriptions of coherence along the lines of “mutual support,” the Olsson-Glass measure (and measures inspired by it, such as Meijs’ [2006] measure) rather target the intuitive idea of coherence as “agreement.” Note that Cohog takes its maximal value 1 when Pr(B1B2 ∧ ⋯ ∧ Bn) = Pr(B1B2 ∨ ⋯ ∨ Bn). This occurs just when the various Bi all agree maximally with one another, being logically equivalent. Similarly, Cohog is minimal when the Bi stand contrary to one another.8

The debate over which, if any, proposed measure best explicates the epistemological notion of coherence continues today, but the examples above suggest the possibility that we need not automatically feel compelled to choose between alternatives. The concept of coherence finds precise explication through the work of formal epistemologists. But the formal approach also has the potential to disentangle distinct concepts otherwise conflated under the heading “coherence” (see Schippers [2014]).

What do such accounts say to the original question of epistemological interest, that is, whether coherentist structures can adequately describe the structure of justification? Of course, ultimately the answer will hinge on which of the precise formal characterizations of coherence one accepts. As with infinitists, coherentists first rephrase the question more precisely: Are more coherent sets of beliefs more probably true? Phrased in this way, the answer is clearly negative. In fact, using any proposed probabilistic measure of coherence, one can show that a more coherent set of beliefs may even be less probable. Using Shogenji’s account, the set of beliefs S = {B1, B2, …, Bn} may be more coherent than another set $S′={B′1, B′2, …, B′n}$,

$Display mathematics$

while simultaneously $Pr(B1∧B2∧⋯∧Bn). These formal accounts go beyond this negative conclusion, however. They also illuminate how and why this can be. In this case, the more coherent belief set can be less probable if $Pr(B1)×Pr(B2)×⋯×Pr(Bn)≪Pr(B′1)×Pr(B′2)×⋯×Pr(B′n)$; in words, the fact that S = {B1, B2, …, Bn} is more coherent than may be outweighed by the fact that the individual beliefs making up S are on average much less probable than those making up S′ (see Klein and Warfield [1994]).

Formal epistemology, in its current state, thus casts doubt on coherence theories of justification. This is, of course, not to say that coherence cannot be thought of as justification-conducive (or an epistemic virtue) in some weaker sense. In fact, another controversy in the contemporary literature on probabilistic coherentism concerns whether more coherent bodies of belief are demonstrably more probably true, ceteris paribus—where this “all else being equal” clause may arguably require formal concepts like Pr(B1) × Pr(B2) × ⋯ × Pr(Bn) (what Shogenji calls “total individual strength”) to be held constant between appropriately comparable sets of beliefs (Bovens and Hartmann [2003], Meijs and Douven [2007], Schupbach [2008], Huemer [2011]). Nonetheless, work in formal epistemology puts into doubt the thesis that coherence alone can describe the structure of justification generally—at least insofar as justification is thought of probabilistically. It seems rather that coherence is at best one epistemically relevant factor among potentially many that determines the extent to which some set of beliefs is justified.

Finally, probability theory has also been used to shed new light on various issues of foundationalism. We have already noted that formal work on infinitism might be taken to challenge the foundationalist idea that a terminal layer of properly basic beliefs is necessary if we are to have epistemic justification at all. A similar argument can be made against a “pure” form of foundationalism from the apparent need for circles in the architecture of justification. Susan Haack (1993:86) (see also Dancy [2003]) puts forward such an argument, asserting that a pure foundationalist theory cannot account for reasoning so mundane as that used for crossword puzzles, which is characterized by mutually supporting “circles” of justification.

In response to such arguments, Lydia McGrew and Timothy McGrew (2008:57) take on the foundationalist’s onus of “model[ing] the phenomenon of mutual support without violating the principle that circular reasoning is non-justificatory.” They proceed by constructing a foundationalist model of scenarios in which two beliefs, H1 and H2, support one another by filling in underlying foundational justifications for each of these separate beliefs. For their example, they stipulate that H1 is supported by foundational beliefs F1 and FA, while H2 is justified by foundational beliefs F2 and FB. With justification and support construed probabilistically, the mutual support between H1 and H2 amounts to noting that Pr(H1 | H2) > Pr(H1) if Pr(H2 | H1) > Pr(H2).

At this stage, the appearance of circular support remains. After all, F1 and FA jointly support H1, which supports H2, which supports H1 again! Correspondingly, F2 and FB jointly support H2, which supports H1, which supports H2 again! To dispel this appearance, McGrew and McGrew must show that the support which either belief receives from the other is somehow separable from the support it bestows on the other; for instance, the support H1 receives via H2 is distinguishable from the support H1 bestows upon H2. They do so using two formal maneuvers. First, either belief “screens off” its foundational justifiers from the other belief. Put formally:

$Display mathematics$

This means that all of the support that H1 receives from F2 and FB comes by way of H2; in McGrew and McGrew’s terminology, “H2 is a conduit through which F2 and FB support H1” (similarly, H1 serves as a conduit of support from F1 and FA to H2).

The second formal maneuver allows McGrew and McGrew to model the idea that the only support either belief (H1 or H2) provides for the other is in its role as such a conduit. They formalize this support using Jeffrey’s rule (see n, 2) on the intermediate, conduit beliefs. Continuing with the above example, imagine that FA and FB are already background knowledge for an agent, and then the agent comes to learn F1. Upon learning F1, the agent updates the probability of H1—using Bayes’ rule—to get Prnew(H1) = Prold(H1 | F1). The probability of H2 is then updated using Jeffrey’s rule:

$Display mathematics$

Given that the screening-off relation holds, this equation can be reduced as follows:

$Display mathematics$

The upshot is that the support that H2 receives from H1 just amounts to that which it receives from foundational belief F1; more generally in the above example, similar demonstrations show that the support H1 provides for H2 entirely comes by way of foundational beliefs F1 and FA, while the support that H2 provides for H1 is grounded entirely in F2 and FB. On this framework, then, the appearance of circular support is illusory, all justification ultimately deriving from foundational beliefs. Note that this work is not meant to be an argument for foundationalism, but rather aims to show that the “pure foundationalist” can make sense of mutual support—without allowing circular reasoning to be justificatory.

## Social Epistemology

Traditionally, epistemologists have written as though individuals seek knowledge entirely on their own, having to learn about the world in complete isolation of others. Mainstream epistemologists have only recently focused on the question of what difference it might make that, in reality, we socially interact with others who largely have the same epistemic goals as we. In groundbreaking work, Goldman (1999) showed that an exclusive focus on the isolated epistemic agent was deeply mistaken. There are important aspects of our epistemic lives that can only be understood by considering our interactions with fellow epistemic agents and by studying whole collectives of agents pursuing truth in a concerted effort. By pretending that our being members of such collectives is irrelevant to the study of epistemology, we are missing important pathways to justified belief and knowledge: much of what we justifiably believe or know is due to our interacting with others.

This insight led to increased attention investigating the epistemic significance of such topics as testimony, expertise, judgment aggregation, and disagreement. So far, much of the work undertaken on these and related topics is still nonformal and best classified as mainstream epistemology. But here we can briefly mention recent work suggesting the emergence of a subfield within formal epistemology, to wit, formal social epistemology.

We sometimes ascribe knowledge or belief to groups. We might say that the World Health Organization (WHO) believes climate change to call for more drastic measures, that the present government knows that researchers are underpaid, or that the court found the defendant guilty. And we may make such claims even if not every member of the WHO agrees that a more drastic response to climate change is necessary, if some members of the government actually disagree that researchers are underpaid, or if one of the judges dissented with the verdict. So, the consensus is that φ is seemingly not required to ascribe knowledge of or belief in φ to a collective. On the other hand, we presumably also would not want to ascribe knowledge or belief to the collective if at least a majority of its members knew or believed that φ.9 Interestingly, Lewis Kornhauser and Lawrence Sager (1986) have shown that aggregating what individuals know or believe on the basis of a simple majority rule—the collective knows/believes any proposition that most of its members know/believe—can give rise to inconsistencies at the aggregated level even if no individual is inconsistent.

The following is a classical demonstration of the problem: Suppose a committee of three has to decide about whether a certain applicant will be made a job offer, and define P ≔ as “The applicant has a strong publication record”; Q ≔ “The applicant has sufficient teaching experience”; R ≔ “The applicant should receive an offer.” All three committee members agree that the applicant should receive an offer if she has a strong publication record and sufficient teaching experience, but they do not quite agree on the atomic propositions. Specifically, member 1 believes P, Q, and R; member 2 believes P but believes both Q and R to be false; and member 3 believes Q but believes P and Q to be false. One readily verifies that all three members hold consistent beliefs. However, if group belief goes by majority, then the group as a whole is inconsistent. After all, a majority of its members (members 1 and 2) believe P; a majority (members 1 and 3) believe Q; all believe the biconditional (PQ) ↔ R; but while this biconditional in conjunction with P and Q entails R, a majority (members 2 and 3) believe that proposition to be false.

This discovery motivates various questions: Should aggregation procedures preserve consistency? Which candidate procedures preserve consistency (and under what conditions)? What other desiderata might we have for a satisfactory method of aggregation? Formal epistemologists debate these questions and more in a growing body of research studying the aggregation of epistemic attitudes. This research has, for example, provided a number of formal possibility and impossibility results, uncovering which aggregation procedures guarantee consistent attitudes (belief and knowledge, but also other propositional attitudes) at the group level under which precise conditions—see List and Puppe (2009), Dietrich and List (2010), and references given there.

The central question of these debates is how we should construct group attitudes on the basis of typically diverging—or at any rate partly diverging—individual attitudes. Whether such divergences in individual attitudes are themselves epistemically significant is the focus of a separate debate in social epistemology, the so-called peer disagreement debate. In particular, this debate has revolved around the question of whether the discovery that one holds a different view on a given matter than one or more of one’s peers—people with roughly equal intellectual capacities and access to basically the same evidence—should give one reason to revise that view. According to some participants in this debate, it is perfectly okay to stick to one’s own view in the face of peer disagreement, while others hold that such a case calls for revision of one’s view, with many believing that the revision should consist of a kind of compromise between one’s view and that of one’s disagreeing peer or peers. Much of the literature on peer disagreement does not belong to formal epistemology. But both the question of whether peer disagreement is a reason to compromise and the question of how to compromise (supposing that is found to be the right response to peer disagreement) have drawn much attention from formal epistemologists. In fact, proposed compromising models have drawn much inspiration from, and even heavily used, the earlier-cited literature on judgment aggregation. For representative formal work on peer disagreement, see for instance Fitelson and Jehle (2009), Lam (2011), Brössel and Eder (2014), Cevolani (2014), and Levinstein (2015).

If we want to study formal models of compromising in communities of agents, it will generally be difficult to obtain interesting analytic results even if the communities are only moderately large. To study epistemic interactions in such communities, we have been greatly helped by the development of computational environments that allow us to simulate large numbers of interacting agents while keeping track of the relevant epistemic features of those agents. A simulation model that has proven to be particularly helpful in this respect is one developed in Hegselmann and Krause (2002, 2006). The basic model is very simple and is populated by simulated agents in pursuit of some truth who change their views on what the truth is both on the basis of evidence they receive and on the basis of exchanges they have with some of their fellow agents. Already this simple model was shown to yield interesting results about the conditions under which the views of the agents in the community converge, the conditions under which these views diverge, and much more. Better still, the model is very flexible, making it easy to tweak and extend its basic machinery, and models building on the original model developed by Rainer Hegselmann and Ulrich Krause have been used to study normative questions pertinent to social epistemology (see, e.g., Olsson [2008] and Douven [2010]).

A model for studying epistemically interacting agents that is similar to, but not strictly an extension of, the model developed by Hegselmann and Krause is presented in Olsson [2011], Olsson and Vallinder [2013], and Vallinder and Olsson [2014]. The first paper uses the model for studying, and vindicating, certain theses implied by Goldman’s (1986) reliabilist epistemology. The second paper compares in computer simulations some prominent norms of assertion. And the third paper presents probabilistic models of trust among epistemic agents and shows that, under plausible assumptions, we are often better off putting greater trust in the reliability of our own inquiries than in those of others. For related work, see Sprenger, Martini, and Hartmann (2009) and Hartmann, Pigozzi, and Sprenger (2010).

# 3. New Questions Born Out of Formal Epistemology

In this section, we discuss a couple of the epistemological issues that have arisen out of the development of formal epistemology itself.

## Connecting Categorical and Graded Belief

With so much of formal epistemology focused on the notion of degrees of belief, questions arise regarding the status of the traditional notion of belief simpliciter. With a formally precise, probabilistic concept of credence in hand, some philosophers maintain that any talk of categorical belief is to be shunned as unscientific (most famously Jeffrey [2004]). People have degrees of belief, and probability is crucial to defining how such degrees of belief are rational and change in rational ways. But whenever we talk unqualifiedly about what we or others believe, we are talking loosely, and loose talk should not be the subject of serious philosophy. Needless to say, these philosophers take traditional epistemology to be a deeply misguided enterprise.

The majority of formal epistemologists, however, take both notions of belief (categorical and graded) seriously. There is an epistemology of belief and an epistemology of degrees of belief—as Richard Foley (1992) puts it—and in the view of these formal epistemologists neither is to be dismissed as unscientific or second-rate. Accordingly, many formal epistemologists today wrestle with the question of how degrees of belief and categorical beliefs bear on one another.

Surely graded and categorical beliefs do not just coexist in our heads, with graded beliefs being completely unconstrained by what we believe categorically and vice versa. It is thus reasonable to suppose that there must be some connection—some bridge principle or principles, if one likes—between the epistemology of belief and the epistemology of degrees of belief. One answer has been that the rationality of our categorical beliefs supervenes on our rationally held graded beliefs, in the sense that there cannot be a change in the former without there being some change in the latter, and where it is assumed that rationally held graded beliefs are degrees of belief that are representable by a probability function. Indeed, some have suggested the following straightforward connection between graded and categorical beliefs, which is sometimes called “the Lockean Thesis”:

• Lockean Thesis (LT): It is rational to believe φ (categorically) if and only if it is rational to believe φ to a degree above a certain threshold value t,

where t is then typically assumed to be close, but unequal, to 1.

However, (LT) is known to lead to trouble when combined with the following principles concerning categorical belief that, at least prima facie, appear difficult to deny:

• Conjunction Principle (CP): It is rational to believe the conjunction of any two propositions that are individually rational to believe.

• No Contradictions Principle (NCP): It is never rational to believe an explicit contradiction.

Specifically, these principles are known to give rise to the so-called lottery and preface paradoxes.

The lottery paradox was first presented in Henry Kyburg’s (1961). Consider a fair n-ticket lottery $L$ with exactly one winner, and with 1–1/n > t. Suppose you have been informed about the conditions of $L$ and consequently believe rationally to a degree exceeding t that ticket number i in $L$ is a loser, for all i: 1 ⩽ in. By (LT), it is rational for you to believe (categorically) that ticket number i is a loser, for all i. At the same time, you know, and hence rationally believe, that one of the tickets will be the winner. Now, the conjunction of all those propositions—that ticket number 1 will lose, that ticket number 2 will lose, that ticket number n will lose, and the proposition that one of tickets numbers 1 through n will be the winner—forms a contradiction relative to your background knowledge. As a result, you can rationally believe this conjunction, by repeated application of (CP), but then again you cannot, by (NCP).

David Makinson (1965) was the first to present the preface paradox. This paradox imagines that you have just finished writing a book. You have checked over and over again each of the n claims the book makes, and so you can rationally believe each of those claims to a degree above t. Thus, you can rationally believe them (categorically, that is), by (LT). However, you are also aware that colleagues spotted errors in your previous books after their publication, despite the fact that you had checked the claims in those books as thoroughly as you checked the claims in your new book, and despite the fact that you were as confident about the correctness of each of the claims in your previous books as you are about the correctness of the claims in the new book. On the basis of this evidence, it would seem rational for you to believe to a degree above t that the new book will not be entirely free from errors either. Hence, again by (LT), it would seem rational for you to believe categorically that at least one of the claims in the new book is incorrect. But then, by repeated application of (CP), it is rational for you to believe the conjunction of all the claims contained in the book and the proposition that at least one of them is false—which is an explicit contradiction, and according to (NC) you cannot rationally believe that.

It is a matter of ongoing controversy which of (LT), (CP), and (NCP) is to be abandoned in view of these paradoxes. Here, we can only provide some pointers to the relevant literature: Kyburg (1961), Klein (1985), Foley (1992), Christensen (2004), and Kroedel (2012) all favor rejecting (CP). Others, including Pollock (1990), Maher (1993), Nelkin (2000), Douven (2002), Wenmackers (2013), and Kelp (2016), believe that (LT) has to go, or at least needs qualification.10 Priest (1998) has criticized (NCP) on general grounds, and some recent proposals aim to salvage (CP), (NCP), and at least the gist of (LT) by making rational belief a contextual matter, so that it depends on which other propositions one considers as to whether it is rational to believe of a given ticket that it will lose; see Lin and Kelly (2012), Leitgeb (2014), and Easwaran (2016).11

## Formal Approaches to Explanatory Reasoning

There are typically many possibilities (possible worlds, if you like) compatible with our knowledge, some of which appear more likely than others in light of that knowledge. What we do when we receive new information about the world is to redistribute our credences across the possibilities. Bayes’ rule, as described in section 1, offers a systematic procedure for such redistributions. As also mentioned above, most formal epistemologists subscribe to Bayes’ rule. But in principle there are indefinitely many credence-redistribution procedures.

With the development of such mathematical models comes a set of interesting new questions pertaining to the proper place of less technical theories of inference and rationality. For example, consider the so-called Inference to the Best Explanation (IBE), which in the not-so-distant past enjoyed widespread popularity both among mainstream epistemologists (e.g., Harman [1965], Vogel [1990]) and among philosophers of science (Boyd [1984], Lipton [2004]). In its crudest formulation, IBE states that we ought to infer the hypothesis that best explains the available evidence. The reference to unqualified inference suggests that IBE belongs to the epistemology of categorical belief. Should IBE be dropped in favor of a formal credence-redistribution rule? Can IBE and various credence-redistribution rules be part of a larger, consistent account of epistemic rationality? Or perhaps can credence-redistribution procedures be developed (or reinterpreted) to validate IBE’s central insight that explanatory judgments carry legitimate normative weight? A few philosophers have argued that IBE should effectively be dropped in favor of Bayes’ rule or some other allegedly more fundamental model (Fumerton [1980], Salmon [2001]). However, an increasing number of formal epistemologists today are seeking more irenic accounts of how credence-redistribution principles and IBE can share in a full account of rational belief change.

“Bayesian explanationist” approaches attempt to show that Bayes’ rule and IBE are somehow compatible (or even mutually supportive) pieces of a general theory of rational inference and belief change. In one version of this approach, formal epistemologists argue that explanatory judgments are to be used as heuristics for assigning values necessary to running the Bayesian machinery (e.g., Huemer [2009], Lipton [2004, chap. 7], McGrew [2003], Poston [2014, chap. 7], Weisberg [2009]). This proposal is not unproblematic. For one thing, the explanationist is not likely to be attracted to a theory that reduces explanatory reasoning’s normative import merely to that which it gleans parasitically from the Bayesian framework (see Douven [2011, sec. 4] and Henderson [2014] for more extensive criticism of the heuristic view).

There are other options, contrasting with the heuristic picture, for how one might develop the general Bayesian explanationist view. For example, one might rather think of IBE and Bayes’ rule as situated at different levels of idealization in a unified hierarchy of logics. The idea here would be that by following either Bayes’ rule or IBE, epistemic agents (in certain contexts perhaps) tend to reason in accord with the other. For example, by following the less idealized, explanationist logic of IBE, reasoners tend to reason in accord with the more idealized Bayesian logic—under certain conditions, they may reason slightly less reliably and in others more reliably (Schupbach [2016]). Leah Henderson’s (2014) “emergent compatibilism” and the account defended in Jonah Schupbach’s (2011b, 2016) are both along these lines.

Another recent approach is to develop candidate credence-redistribution procedures distinct from Bayes’ rule, which give special attention to explanatory relationships between the various possibilities under consideration and the newly obtained information. Here, for example, is a “probabilistic version of IBE” studied in Igor Douven’s (2013) and Douven and Sylvia Wenmackers’ (2016): Let {ψi}in be a set of self-consistent, mutually exclusive, and jointly exhaustive hypotheses, and let Pr(·) represent one’s current graded beliefs. Then one probabilistically infers to the best explanation upon learning φ (and nothing else) if, for all i,

$Display mathematics$

with $E$ assigning a bonus to the hypothesis that explains the evidence best, and with Pr(·) one’s new graded belief function. One verifies that this probabilistic version of IBE agrees with Bayes’ rule if $E$ is set to be the constant function 0, meaning that no bonus points for explanatory bestness are ever attributed.

One reason to be interested in this and similar alternatives to Bayes’ rule is that psychologists have shown that, while people in some situations react to the receipt of new information in a way that suggests they are following Bayes’ rule, in other situations they appear to violate this rule; see, for instance, Robinson and Hastie (1985), Baratgin and Politzer (2007), Zhao et al. (2012), and Douven and Schupbach (2015a, 2015b). Those violations could have been entirely unsystematic, of course, but Douven and Schupbach (2015a, 2015b) found that, in some contexts, they arise systematically because in accommodating new information people take into account explanatory considerations. This finding was not entirely surprising, given that explanatory considerations were already known to play various roles in cognition; see Lombrozo (2006, 2012), Legare, Wellman, and Gelman (2009), Legare (2012), Williams and Lombrozo (2013), and Legare and Lombrozo (2014), among others. However, none of the previous studies had looked at the relationship between explanation and belief change.

That many formal epistemologists hesitate to consider such explanatory models of credence-redistribution is likely due to arguments asserting that any form of belief change at variance with Bayes’ rule betokens irrationality. For anyone buying into such arguments, probabilistic versions of IBE may be of interest to psychology (where they may be studied alongside other types of irrational behavior), but they should hold no appeal to philosophers who focus on normative issues. Even with respect to the normative claim, however, the bad reputation of probabilistic versions of IBE may well be undeserved, as the motivating arguments themselves appear doubtful.

For instance, Bas van Fraassen (1989) has made much of the fact that IBE in general will fail whenever the truth is not among the hypotheses under consideration. As explained in Schupbach (2014), however, that has no more significance than the fact that conjunction introduction will lead us to infer a false conclusion whenever one of the premises is false. Van Fraassen (1989) has also leveled David Lewis’s (1999) so-called dynamic Dutch book argument specifically against probabilistic versions of IBE. That argument purports to show that anyone whose belief changes are guided by such a version of IBE can be made to engage in a series of bets, the net payoff of which is necessarily negative. The argument is contentious, however (Douven [1999]), and even if it were sound, it is now generally recognized to concern practical rationality rather than epistemic rationality, which is the type of rationality at issue in a debate about how we ought to change our graded beliefs. Finally, it has been argued that changing our graded beliefs by any other rule than Bayes’ makes those graded beliefs less accurate than they would otherwise be (Rosenkrantz [1992]), but that turns out to be true only on one specific understanding of accuracy, where this understanding also appears to be of lesser epistemic importance (Douven [2013], [2016]).

# 4. Whither Formal Epistemology?

Formal epistemology is a young, vibrant field of research in analytic philosophy. In the foregoing discussion, we have highlighted some of its major achievements, offering a sense of what, in our opinion, can be accomplished if we address problems from mainstream epistemology by using logic, probability theory, computer simulations, and other formal tools. These tools have allowed researchers to state a number of long-known epistemological puzzles and long-held epistemological positions with unprecedented precision. For example, many of the metaphors that were, and still are, used so abundantly in mainstream work on the structure of justification can now be replaced by precise formal definitions, which offer the best prospects for progress in this area. Moreover, we saw that the rise of formal epistemology has raised new questions that should also be of interest to mainstream epistemologists. Graded beliefs have traditionally not been given much attention by epistemologists, but now that psychologists and formal epistemologists have pointed out how central graded beliefs are to an understanding of our cognitive functioning, any epistemologist will have to account for both their role and their relation to categorical beliefs.

Formal epistemology is here to stay, as we said earlier, and we offer all of the foregoing as reason to hold that it should stay. But this is not necessarily to say that formal epistemology will or should stay as a separate subfield of epistemology. Rather, our prediction is that it will gradually fade in, so to speak, as it will become more and more common to train philosophers in other formal methods besides logic, notably in probability theory; and philosophy departments may start offering courses in which students can learn how to write computer code for the purpose of running the kind of simulations discussed in section 2.12 It took analytic philosophers in the early twentieth century some time to fully familiarize themselves with the (then) new logic of Frege and Russell, but eventually this became a tool that all of them learnt to use more or less routinely. We believe, and hope, that the same will prove to be true for epistemologists and the use of probability theory and other formal methods discussed in this article.13

## References

Arló-Costa, H., and R. Parikh. 2006. “Tracking Truth: Knowledge and Conditionals in the Context of Branching Time.” Unpublished manuscript.Find this resource:

Baratgin, J., and G. Politzer. 2007. “The Psychology of Dynamic Probability Judgment: Order Effect, Normative Theories, and Experimental Methodology.” Mind and Society 6: 53–66.Find this resource:

Bird, A. 2010. “Social Knowing: The Social Sense of ‘Scientific Knowledge.’” Philosophical Perspectives 24: 23–56.Find this resource:

Bird, A., and R. Pettigrew. 2016. “Internalism, Externalism, and the KK Principle.” Unpublished manuscript.Find this resource:

BonJour, L. 1985. The Structure of Empirical Knowledge. Cambridge, MA: Harvard University Press.Find this resource:

BonJour, L. 1999. “The Dialectic of Foundationalism and Coherentism.” In J. Greco and E. Sosa, eds., The Blackwell Guide to Epistemology. Oxford: Blackwell, 117–142.Find this resource:

Bovens, L., and S. Hartmann. 2003. Bayesian Epistemology. Oxford: Oxford University Press.Find this resource:

Boyd, R. 1984. “The Current Status of Scientific Realism.” In J. Leplin, ed., Scientific Realism. Berkeley: University of California Press, 195–222.Find this resource:

Bradley, R. 2001. “Ramsey and the Measurement of Belief.” In D. Corfield and J. Williamson, eds., Foundations of Bayesianism. London: Kluwer Academic Publishers, 263–290.Find this resource:

Briggs, R., F. Cariani, K. Easwaran, and B. Fitelson. 2014. “Individual Coherence and Group Coherence.” in J. Lackey, ed., Essays in Collective Epistemology.; Oxford: Oxford University Press, 215–239.Find this resource:

Brössel, P., and A.-M. A. Eder. 2014. “How to Resolve Doxastic Disagreement.” Synthese 191: 2359–2381.Find this resource:

Cevolani, G. 2014. “Truth Approximation, Belief Merging, and Peer Disagreement.” Synthese 191: 2383–2401.Find this resource:

Chandler, J. 2010. “The Lottery Paradox Generalized?” British Journal for the Philosophy of Science 61: 667–679.Find this resource:

Chandler, J. 2013. “Acceptance, Aggregation and Scoring Rules.” Erkenntnis 78: 201–217.Find this resource:

Christensen, D. 2004. Putting Logic in its Place. Oxford: Oxford University Press.Find this resource:

Dancy, J. 2003. “A Defense of Coherentism.” In L. Pojman, ed., The Theory of Knowledge: Classical and Contemporary Readings. Belmont CA: Wadsworth, 206–215.Find this resource:

Decock, L. B., I. Douven, and M. Sznajder. 2016. “A Geometric Principle of Indifference.” Journal of Applied Logic, in press.Find this resource:

de Finetti, B. 1937/1964. “Foresight: Its Logical Laws, Its Subjective Sources.” In H. E. Kyburg and H. E. Smokler, eds., Studies in Subjective Probability. New York: Wiley, 55–118.Find this resource:

Dietrich, F., and C. List. 2010. “The Aggregation of Propositional Attitudes: Towards a General Theory.” Oxford Studies in Epistemology 3: 215–234.Find this resource:

Douven, I. 1999. “Inference to the Best Explanation Made Coherent.” Philosophy of Science 66: S424–S435.Find this resource:

Douven, I. 2002. “A New Solution to the Paradoxes of Rational Acceptability.” British Journal for the Philosophy of Science 53: 391–410.Find this resource:

Douven, I. 2010. “Simulating Peer Disagreements.” Studies in History and Philosophy of Science 41: 148–157.Find this resource:

Douven, I. 2011. “Abduction,” in E. N. Zalta (ed.) Stanford Encyclopedia of Philosophy (available at http://plato.stanford.edu/entries/abduction/).Find this resource:

Douven, I. 2013. “Inference to the Best Explanation, Dutch Books, and Inaccuracy Minimisation.” Philosophical Quarterly 69: 428–444.Find this resource:

Douven, I. 2016. “Inference to the Best Explanation: What Is It? And Why Should We Care?” In T. Poston and K. McCain, eds., Best Explanations: New Essays on Inference to the Best Explanation. Oxford: Oxford University Press, in press.Find this resource:

Douven, I., and W. Meijs. 2007. “Measuring Coherence.” Synthese 156: 405–425.Find this resource:

Douven, I., and J. N. Schupbach. 2015a. “The Role of Explanatory Considerations in Updating.” Cognition 142: 299–311.Find this resource:

Douven, I., and J. N. Schupbach. 2015b. “Probabilistic Alternatives to Bayesianism: The Case of Explanationism.” Frontiers in Psychology 6: 459.Find this resource:

Douven, I., and J. Uffink. 2003. “The Preface Paradox Revisited.” Erkenntnis 59: 389–420.Find this resource:

Douven, I., and S. Wenmackers. 2016. “Inference to the Best Explanation versus Bayes’ Rule in a Social Setting.” British Journal for the Philosophy of Science, in press.Find this resource:

Douven, I., and T. Williamson. 2006. “Generalizing the Lottery Paradox.” British Journal for the Philosophy of Science 57: 755–779.Find this resource:

Earman, J., and W. C. Salmon. 1992. “The Confirmation of Scientific Hypotheses.” In M. H. Salmon, J. Earman, C. Glymour, and J. Lennox, eds., Introduction to the Philosophy of Science. New York: Prentice-Hall, 42–103.Find this resource:

Easwaran, K. 2016. “Dr. Truthlove or: How I Learned to Stop Worrying and Love Bayesian Probabilities.” Noûs, in press.Find this resource:

Elqayam, S., and D. E. Over. 2013. “New Paradigm Psychology of Reasoning.” Thinking and Reasoning 19: 249–265.Find this resource:

Fitelson, B. 2003. “A Probabilistic Theory of Coherence.” Analysis 63: 194–199.Find this resource:

Fitelson, B., and D. Jehle. 2009. “What Is the ‘Equal Weight View’?” Episteme 6: 280–293.Find this resource:

Foley, R. 1992. “The Epistemology of Belief and the Epistemology of Degrees of Belief.” American Philosophical Quarterly 29: 111–124.Find this resource:

Fricker, M. 2012. “Group Testimony? The Making of a Collective Good Informant.” Philosophy and Phenomenological Research 84: 249–276.Find this resource:

Fumerton, R. 1980. “Induction and Reasoning to the Best Explanation.” Philosophy of Science 47: 589–600.Find this resource:

Fumerton, R. 1995. Metaepistemology and Skepticism. Boston: Rowman and Littlefield.Find this resource:

Gilbert, M. 1989. On Social Facts. London: Routledge.Find this resource:

Glass, D. H. 2005. “Problems with Priors in Probabilistic Measures of Coherence.” Erkenntnis 63: 375–385.Find this resource:

Goldman, A. I. 1986. Epistemology and Cognition. Cambridge MA: Harvard University Press.Find this resource:

Goldman, A. I. 1999. Knowledge in a Social World. Oxford: Oxford University Press.Find this resource:

Goldman, A. I. 2009. “Internalism, Externalism, and the Architecture of Justification.” Journal of Philosophy 106: 309–338.Find this resource:

Haack, S. 1993. Evidence and Inquiry. Oxford: Blackwell.Find this resource:

Harman, G. 1965. “The Inference to the Best Explanation.” Philosophical Review 74: 88–95.Find this resource:

Hartmann, S., G. Pigozzi, and J. Sprenger. 2010. “Reliable Methods of Judgment Aggregation.” Journal for Logic and Computation 20: 603–617.Find this resource:

Hawthorne, J., J. Landes, C. Wallmann, and J. Williamson. 2016. “The Principal Principle Implies the Principle of Indifference.” British Journal for the Philosophy of Science, in press.Find this resource:

Hegselmann, R., and U. Krause. 2002. “Opinion Dynamics and Bounded Confidence: Models, Analysis, and Simulations.” Journal of Artificial Societies and Social Simulation 5 (available at http://jasss.soc.surrey.ac.uk/5/3/2.html).

Hegselmann, R., and U. Krause. 2006. “Truth and Cognitive Division of Labor: First Steps towards a Computer Aided Social Epistemology.” Journal of Artificial Societies and Social Simulation 9 (available at http://jasss.soc.surrey.ac.uk/9/3/10.html).

Henderson, L. 2014. “Bayesianism and Inference to the Best Explanation.” British Journal for the Philosophy of Science 65: 687–715.Find this resource:

Hintikka, J. 1962. Knowledge and Belief. Ithaca, NY: Cornell University Press.Find this resource:

Howson, C. 2000. Hume’s Problem: Induction and the Justification of Belief. Oxford: Clarendon Press.Find this resource:

Huemer, M. 2009. “Explanationist Aid for the Theory of Inductive Logic.” British Journal for the Philosophy of Science 60: 345–375.Find this resource:

Huemer, M. 2011. “Does Probability Theory Refute Coherentism?” Journal of Philosophy 108: 35–54.Find this resource:

Hume, D. 1748/1912. An Enquiry Concerning Human Understanding. Chicago: Open Court Publishing.Find this resource:

Jeffrey, R. C. 1965. The Logic of Decision. New York: McGraw-Hill.Find this resource:

Jeffrey, R. 2004. Subjective Probability: The Real Thing. Cambridge: Cambridge University Press.Find this resource:

Joyce, J. M. 2005. “How Probabilities Reflect Evidence,” Philosophical Perspectives 19: 153–178.Find this resource:

Kahneman, D., P. Slovic, and A. Tversky. 1982. Judgment under Uncertainty: Heuristics and Biases. Cambridge: Cambridge University Press.Find this resource:

Kelly, K. T. 2014. “A Computational Learning Semantics for Inductive Empirical Knowledge.” In A. Baltag and S. Smets, eds., Johan van Benthem on Logic and Information Dynamics. Dordrecht: Springer, 289–337.Find this resource:

Kelp, C. 2016. “Lotteries and Justification.” Synthese, in press.Find this resource:

Keynes, J. M. 1921. A Treatise of Probability. London: Macmillan.Find this resource:

Klein, P. 1985. “The Virtues of Inconsistency.” Monist 68: 105–135.Find this resource:

Klein, P. 1998. “Foundationalism and the Infinite Regress of Reasons.” Philosophy and Phenomenological Research 58: 919–925.Find this resource:

Klein, P., and T. A. Warfield. 1994. “What Price Coherence?” Analysis 54: 129–132.Find this resource:

Kolmogorov, A. N. 1950. Foundations of the Theory of Probability. New York: Chelsea Publishing.Find this resource:

Kornhauser, L. A., and L. G. Sager. 1986. “Unpacking the Court.” Yale Law Journal 96: 82–117.Find this resource:

Koscholke, J., and M. Jekel. 2016. “Probabilistic Coherence Measures: A Psychological Study of Coherence Assessment.” Synthese, in press.Find this resource:

Kroedel, T. 2012. “The Lottery Paradox, Epistemic Justification and Permissibility.” Analysis 72: 57–60.Find this resource:

Kyburg, H. 1961. Probability and the Logic of Rational Belief. Middletown, CT: Wesleyan University Press.Find this resource:

Lackey, J. 2014. “Socially Extended Knowledge.” Philosophical Issues 24: 282–298.Find this resource:

Lam, B. 2011. “On the Rationality of Belief-Invariance in Light of Peer Disagreement.” Philosophical Review 120: 207–245.Find this resource:

Legare, C. H. 2012. “Exploring Explanation: Explaining Inconsistent Information Guides Hypothesis-testing Behavior in Young Children.” Child Development 83: 173–185.Find this resource:

Legare, C. H., and T. Lombrozo. 2014. “Selective Effects of Explanation on Learning during Early Childhood,” Journal of Experimental Child Psychology 126: 198–212.Find this resource:

Legare, C. H., H. M. Wellman, and S. A. Gelman. 2009. “Evidence for an Explanation Advantage in Naïve Biological Reasoning.” Cognitive Psychology 58: 177–194.Find this resource:

Leitgeb, H. 2014. “The Stability Theory of Belief.” Philosophical Review 123: 131–171.Find this resource:

Lewis, D. K. 1980. “A Subjectivist’s Guide to Objective Chance.” In R. C. Jeffrey, ed., Studies in Inductive Logic and Probability. Vol. 2. Berkeley: University of California Press, 83–132.Find this resource:

Lewis, D. K. 1999. Papers in Metaphysics and Epistemology. Cambridge: Cambridge University Press.Find this resource:

Levinstein, B. A. 2015. “With All Due Respect: The Macro-epistemology of Disagreement.” Philosophers’ Imprint 15 (available at http://hdl.handle.net/2027/spo.3521354.0015.013).

Lin, H., and K. T. Kelly. 2012. A Geo-logical Solution to the Lottery Paradox, with Applications to Conditional Logic.” Synthese 186: 531–575.Find this resource:

Lipton, P. 2004. Inference to the Best Explanation. 2nd ed. London: Routledge.Find this resource:

List, C., and C. Puppe. 2009. “Judgment Aggregation: A Survey.” In C. List and C. Puppe, eds., Handbook of Rational and Social Choice. Oxford: Oxford University Press.Find this resource:

Lombrozo, T. 2006. “The Structure and Function of Explanations.” Trends in Cognitive Sciences 10: 464–470.Find this resource:

Lombrozo, T. 2012. “Explanation and Abductive Inference.” In K. J. Holyoak and R. G. Morrison, eds., Oxford Handbook of Thinking and Reasoning. Oxford: Oxford University Press, 260–276.Find this resource:

Maher, P. 1993. Betting on Theories. Cambridge: Cambridge University Press.Find this resource:

Makinson, D. 1965. “The Paradox of the Preface.” Analysis 25: 205–207.Find this resource:

Mayo-Wilson, C. 2016. “(How) Is Scientific Knowledge Possible?” Manuscript.Find this resource:

McGrew, L., and T. McGrew. 2008. “Foundationalism, Probability, and Mutual Support.” Erkenntnis 68: 55–77.Find this resource:

McGrew, T. 2001. “Direct Inference and the Problem of Induction.” The Monist 84: 153–178.Find this resource:

McGrew, T. 2003. “Confirmation, Heuristics, and Explanatory Reasoning.” British Journal for the Philosophy of Science 54: 553–567.Find this resource:

Meacham, C. J. G. 2010. “Unravelling the Tangled Web: Continuity, Internalism, Non-uniqueness and Self-locating Beliefs.” Oxford Studies in Epistemology 3: 86–125.Find this resource:

Meijs, W. 2006. “Coherence as Generalized Logical Equivalence,” Erkenntnis 64: 231–252.Find this resource:

Meijs, W., and I. Douven. 2007. “On the Alleged Impossibility of Coherence.” Synthese 157: 347–360.Find this resource:

Meyer, J.-J., and W. van der Hoek. 1995. Epistemic Logic for AI and Computer Science. Cambridge: Cambridge University Press.Find this resource:

Nelkin, D. 2000. “The Lottery Paradox, Knowledge, and Rationality.” Philosophical Review 109: 373–409.Find this resource:

Nozick, R. 1981. Philosophical Explanations. Cambridge, MA: Harvard University Press.Find this resource:

Okasha, S. 2013. “On a Flawed Argument against the KK Principle,” Analysis 73: 80–86.Find this resource:

Olsson, E. J. 2002. “What Is the Problem of Coherence and Truth?” Journal of Philosophy 94: 246–272.Find this resource:

Olsson, E. J. 2008. “Knowledge, Truth, and Bullshit: Reflections on Frankfurt.” Midwest Studies in Philosophy 32: 94–110.Find this resource:

Olsson, E. J. 2011. “A Simulation Approach to Veritistic Social Epistemology.” Episteme 8: 127–143.Find this resource:

Olsson, E. J., and A. Vallinder. 2013. “Norms of Assertion and Communication in Social Networks.” Synthese 190: 2557–2571.Find this resource:

Over, D. E. 2009. “New Paradigm Psychology of Reasoning.” Thinking and Reasoning 15: 431–438.Find this resource:

Papineau, D. 2012. Philosophical Devices. Oxford: Oxford University Press.Find this resource:

Peijnenburg, J. 2007. “Infinitism Regained,” Mind 116:597–602.Find this resource:

Peijnenburg, J., and D. Atkinson. 2008. “Probabilistic Justification and the Regress Problem.” Studia Logica 89: 333–341.Find this resource:

Pettigrew, R. 2015. “What Chance–Credence Norms Should Not Be.” Noûs 49: 177–196.Find this resource:

Pollock, J. 1990. Nomic Probability and the Foundations of Induction. Oxford: Oxford University Press.Find this resource:

Poston, T. 2014. Reason and Explanation. Basingstoke: Palgrave Macmillan.Find this resource:

Priest, G. 1998. “What Is so Bad about Contradictions?” Journal of Philosophy 95: 410–426.Find this resource:

Ramsey, F. P. 1926/1990. “Truth and Probability.” In D. H. Mellor, ed., Philosophical Papers. Cambridge: Cambridge University Press, 52–94.Find this resource:

Robinson, L. B., and R. Hastie. 1985. “Revision of Beliefs hen a Hypothesis Is Eliminated from Consideration.” Journal of Experimental Psychology: Human Perception and Performance 11(4): 443–456.Find this resource:

Roche, W. 2013. “Coherence and Probability: A Probabilistic Account of Coherence.” In M. Araszkiewicz and J. Savelka, eds., Coherence: Insights from Philosophy, Jurisprudence and Artificial Intelligence. Dordrecht: Springer, 59–91.Find this resource:

Rosenkrantz, R. D. 1992. “The Justification of Induction.” Philosophy of Science 59: 527–539.Find this resource:

Roush, S. 2005. Tracking Truth: Knowledge, Evidence, and Science. Oxford: Oxford University Press.Find this resource:

Salmon, W. C. 2001. “Explanation and Confirmation: A Bayesian Critique of Inference to the Best Explanation.” In G. Hon and S. S. Rakover, eds., Explanation: Theoretical Approaches and Applications. Dordrecht: Kluwer, 61–91.Find this resource:

Schippers, M. 2014. “Probabilistic Measures of Coherence: From Adequacy Constraints toward Pluralism.” Synthese 191: 3821–3845.Find this resource:

Schupbach, J. N. 2008. “On the Alleged Impossibility of Bayesian Coherentism.” Philosophical Studies 141: 323–331.Find this resource:

Schupbach, J. N. 2011a. “New Hope for Shogenji’s Coherence Measure.” British Journal for the Philosophy of Science 62: 125–142.Find this resource:

Schupbach, J. N. 2011b. “Studies in the Logic of Explanatory Power”. PhD thesis, University of Pittsburgh.Find this resource:

Schupbach, J. N. 2014. “Is the Bad Lot Objection Just Misguided?” Erkenntnis 79: 55–64.Find this resource:

Schupbach, J. N. 2016. “Inference to the Best Explanation, Cleaned Up and Made Respectable.” In T. Poston and K. McCain, eds., Best Explanations: New Essays on Inference to the Best Explanation. Oxford: Oxford University Press, in press.Find this resource:

Shogenji, T. 1999. “Is Coherence Truth-conducive?” Analysis 59(4): 338–345.Find this resource:

Siebel, M., and M. Schippers. 2015. “Inconsistency as a Touchstone for Coherence Measures.” Theoria 30: 11–41.Find this resource:

Siebel, M., and W. Wolff. 2008. “Equivalent Testimonies as a Touchstone of Coherence Measures.” Synthese 161: 167–182.Find this resource:

Smith, M. 2010. “A Generalised Lottery Paradox for Infinite Probability Spaces.” British Journal for the Philosophy of Science 61: 821–831.Find this resource:

Sprenger, J., C. Martini, and S. Hartmann. 2009. “Consensual Decision-making among Epistemic Peers.” Episteme 6: 110–129.Find this resource:

Staley, K., and A. Cobb. 2011. “Internalist and Externalist Aspects of Justification in Scientific Inquiry.” Synthese 182: 475–492.Find this resource:

Thagard, P. 1989. “Explanatory Coherence.” Behavioral and Brain Sciences 12: 435–502.Find this resource:

Thagard, P. 2000. Coherence in Thought and Action. Cambridge, MA: MIT Press.Find this resource:

Vallinder, A., and E. J. Olsson. 2014. “Trust and the Value of Overconfidence: A Bayesian Perspective on Social Network Communication.” Synthese 191: 1991–2007.Find this resource:

van Fraassen, B. C. 1989. Laws and Symmetry. Oxford: Clarendon Press.Find this resource:

Vogel, J. 1990. “Cartesian Skepticism and Inference to the Best Explanation.” Journal of Philosophy 87: 658–666.Find this resource:

Weisberg, J. 2009. “Locating IBE in the Bayesian Framework.” Synthese 167: 125–143.Find this resource:

Weisberg, J. 2015. “Formal Epistemology.” In E. N. Zalta, ed., Stanford Encyclopedia of Philosophy (available at http://plato.stanford.edu/entries/formal-epistemology/).Find this resource:

Wenmackers, S. 2013. “Ultralarge Lotteries: Analyzing the Lottery Paradox Using Non-standard Analysis.” Journal of Applied Logic 11: 452–467.Find this resource:

Williams, J. J., and T. Lombrozo. 2013. “Explanation and Prior Knowledge Interact to Guide Learning.” Cognitive Psychology 66: 55–84.Find this resource:

Williamson, T. 2000. Knowledge and Its Limits. Oxford: Oxford University Press.Find this resource:

Zhao, J., V. Crupi, K. Tentori, B. Fitelson, and D. Osherson. 2012. “Updating: Learning versus Supposing.” Cognition 124: 373–378.Find this resource:

## Notes:

(1) Some philosophers and psychologists hold that static coherence requires more than obedience to the probability axioms. For instance, some believe that it also requires obedience to some principle that connects graded belief to objective probability, like Lewis’s (1980) Principal Principle (see Pettigrew [2015] for more recent discussion of the graded belief–objective probability link). Some researchers also hold that rational reasoners must obey a version of Keynes’ (1921) Principle of Indifference, according to which one ought to be equally confident in each of a set of mutually exclusive and jointly exhaustive hypotheses, absent reasons to the contrary. See for recent discussion Decock, Douven, and Sznajder (2016) and Hawthorne et al. (2016) (the latter paper discusses an interesting possible connection between the Principal Principle and the Principle of Indifference).

(2) Advocates of Bayes’ rule acknowledge that not all learning consists of coming to know with certainty the truth of some proposition. Sometimes we just become more certain of a proposition, without becoming entirely certain of it. Jeffrey (1965) proposed a rule for the accommodation of this type of learning event. The rule—now commonly referred to as “Jeffrey’s rule”—states that it ought to hold for all φ and ψ that , with Prold(⋅) and Prnew(⋅) representing one’s graded beliefs before and, respectively, after the learning event that shifts one’s confidence in φ. (Note that Bayes’ rule falls out of Jeffrey’s rule as the special case in which we become certain of ψ.)

(3) Relatedly, the S5 axiom becomes a “principle of negative introspection” ¬Kφ → K¬Kφ, requiring that one knows about all the cases in which they fail to have knowledge. Regardless of whether one is an internalist or externalist, this principle is dubious. There are many ways one might fail to have knowledge of some φ. Least controversially, φ may be believed in all the right ways but false. This principle would require one to know about all such cases.

(4) Observing that these features fit well with a notion of probabilistic support, mainstream epistemologists today often refer to justification as “probabilistic.” For example, Fumerton (1995:36) writes, “To be justified in believing one proposition P on the basis of another proposition E, one must be (1) justified in believing that E and (2) justified in believing that E makes probable P”; Klein (1998:923) also refers to justification as probabilistic when he weakens condition (2), requiring only “that it be true that E makes probable P.”

(5) Even if their accessibility may not be as plain. In fact, in what sense (if any), and to what extent, degrees of belief are accessible is a longstanding source of major controversy in formal epistemology (Ramsey [1926/1990]; Bradley [2001]).

(6) Even in the complete absence of background knowledge, objective Bayesians hold that there is exactly one unique, rational degree of belief that everyone ought to have—on pain of irrationality—in any particular proposition. In such a case, logical principles alone fix rational degrees of belief. See n. 1.

(7) For a different formal approach, the reader is directed to Thagard’s computational account of coherence as “constraint satisfaction.” In any “coherence problem,” we hunt for a subset of a set of elements E = {e1, e2, …, en}, which best satisfies a given set of constraints. These constraints may link elements in a positive way (e.g., if ej implies ek, a positive constraint may require that we accept both), or in a negative way (e.g., if ej is inconsistent with ek, a negative constraint may require that we accept only one). Each particular constraint is assigned a weight. Thagard (2000) explicates coherence as W, the sum of the weights corresponding to the constraints that get satisfied by a particular partition of E; thus, maximizing coherence amounts to partitioning E into two sets (accepted and rejected) in a way that maximizes W. Thagard (1989) develops a program (“ECHO”) that computes approximate solutions to explanatory coherence problems using a connectionist (neural network) algorithm.

(8) For still further probabilistic measures of coherence, see Siebel and Wolff (2008) and Roche (2013). Koscholke and Jekel (2016) report the results of experiments testing the descriptive accuracy of the currently best-known measures of coherence.

(9) Though see Gilbert (1989), Bird (2010), Fricker (2012), and Lackey (2014) for dissent on this point.

(10) Especially with regard to proposed qualifications, it is to be emphasized that the lottery and preface paradoxes have nothing specifically to do with lotteries or prefaces, so that limiting applications of (LT) to propositions not about lotteries, or propositions that do not assert anything about the correctness of claims made in a book, is not going to be of any help; see Douven and Uffink (2003), Douven and Williamson (2006), Chandler (2010), and Smith (2010).

(11) Chandler (2013) and Briggs et al. (2014) point out various interesting parallels between the lottery paradox and the earlier-cited literature on judgment aggregation.

(12) For instance, we expect to see more textbooks like Papineau (2012), which introduces students to logic (including modal logic and some metalogic) but also contains a part exclusively devoted to probability theory and its philosophical applications.

(13) We are grateful to an anonymous referee for helpful comments on a previous version of this chapter.