The volume analyses and develops David Makinson’s efforts to make classical logic useful outside its most obvious application areas. The book contains chapters that analyse, appraise, or reshape Makinson’s work and chapters that develop themes emerging from his contributions. These are grouped into major areas to which Makinsons has made highly influential contributions and the volume in its entirety is divided into four sections, each devoted to a particular area of logic: belief change, uncertain reasoning, normative systems and the resources (...) of classical logic. Among the contributions included in the volume, one chapter focuses on the “inferential preferential method”, i.e. the combined use of classical logic and mechanisms of preference and choice and provides examples from Makinson’s work in non-monotonic and defeasible reasoning and belief revision. One chapter offers a short autobiography by Makinson which details his discovery of modern logic, his travels across continents and reveals his intellectual encounters and inspirations. The chapter also contains an unusually explicit statement on his views on the role of logic in philosophy. (shrink)

A theory of evidential probability is developed from two assumptions:(1) the evidential probability of a proposition is its probability conditional on the total evidence;(2) one's total evidence is one's total knowledge. Evidential probability is distinguished from both subjective and objective probability. Loss as well as gain of evidence is permitted. Evidential probability is embedded within epistemic logic by means of possible worlds semantics for modal logic; this allows a natural theory of higher-order probability to be developed. In particular, it is (...) emphasized that it is sometimes uncertain which propositions are part of one's total evidence; some surprising implications of this fact are drawn out. (shrink)

Sometimes appearances provide epistemic support that gets undercut later. In an earlier paper I argued that standard Bayesian update rules are at odds with this phenomenon because they are ‘rigid’. Here I generalize and bolster that argument. I first show that the update rules of Dempster–Shafer theory and ranking theory are rigid too, hence also at odds with the defeasibility of appearances. I then rebut three Bayesian attempts to solve the problem. I conclude that defeasible appearances pose a more difficult (...) and pervasive challenge for formal epistemology than is currently thought. 1 The Challenge for Bayesianism1.1 Updating and experience1.2 The problem1.3 Objections2 The Challenge for Dempster–Shafer Theory2.1 Background on Dempster–Shafer theory2.2 The problem for Dempster–Shafer theory3 The Challenge for Ranking Theory4 The Appeal to Metacognition5 The Appeal to Richer Inputs6 The Appeal to a Generic Underminer7 Conclusion. (shrink)

Conditionalization and Jeffrey Conditionalization cannot simultaneously satisfy two widely held desiderata on rules for empirical learning. The first desideratum is confirmational holism, which says that the evidential import of an experience is always sensitive to our background assumptions. The second desideratum is commutativity, which says that the order in which one acquires evidence shouldn't affect what conclusions one draws, provided the same total evidence is gathered in the end. (Jeffrey) Conditionalization cannot satisfy either of these desiderata without violating the other. (...) This is a surprising problem, and I offer a diagnosis of its source. I argue that (Jeffrey) Conditionalization is inherently anti-holistic in a way that is just exacerbated by the requirement of commutativity. The dilemma is thus a superficial manifestation of (Jeffrey) Conditionalization's fundamentally anti-holistic nature. (shrink)

The bootstrapping problem poses a general challenge, afflicting even strongly internalist theories. Even if one must always know that one’s source is reliable to gain knowledge from it, bootstrapping is still possible. I survey some solutions internalists might offer and defend the one I find most plausible: that bootstrapping involves an abuse of inductive reasoning akin to generalizing from a small or biased sample. I also argue that this solution is equally available to the reliabilist. The moral is that the (...) issues raised by bootstrapping are orthogonal to questions about internalism and basic knowledge, having more to do with the nature of good inductive reasoning. (shrink)

Richard Jeffrey’s “Conditioning, Kinematics, and Exchangeability” is one of the foundational documents of probability kinematics. However, the section entitled “Successive Updating” contains a subtle error regarding the applicability of updating by so-called relevance quotients in order to ensure the commutativity of successive probability kinematical revisions. Upon becoming aware of this error, Jeffrey formulated the appropriate remedy, but he never discussed the issue in print. To head off any confusion, it seems worthwhile to alert readers of Jeffrey’s “Conditioning, Kinematics, and Exchangeability” (...) to the aforementioned error and to document his remedy, placing it in the context of both earlier and subsequent work on commuting probability kinematical revisions.1. (shrink)

Given someone’s fully specified posterior probability distribution q and information about the revision method that they employed to produce q, what can you infer about their prior probabilistic commitments? This question provides an entrée into a thoroughgoing discussion of a class of parameterizations of Jeffrey conditioning in which the parameters furnish information above and beyond that incorporated in \. Our analysis highlights the ubiquity of Bayes factors in the study of probability revision.

The so-called "non-commutativity" of probability kinematics has caused much unjustified concern. When identical learning is properly represented, namely, by identical Bayes factors rather than identical posterior probabilities, then sequential probability-kinematical revisions behave just as they should. Our analysis is based on a variant of Field's reformulation of probability kinematics, divested of its (inessential) physicalist gloss.

Suppose that several individuals who have separately assessed prior probability distributions over a set of possible states of the world wish to pool their individual distributions into a single group distribution, while taking into account jointly perceived new evidence. They have the option of first updating their individual priors and then pooling the resulting posteriors or first pooling their priors and then updating the resulting group prior. If the pooling method that they employ is such that they arrive at the (...) same final distribution in both cases, the method is said to be externally Bayesian, a property first studied by Madansky . We show that a pooling method for discrete distributions is externally Bayesian if and only if it commutes with Jeffrey conditioning, parameterized in terms of certain ratios of new to old odds, as in Wagner , rather than in terms of the posterior probabilities of members of the disjoint family of events on which such conditioning originates. (shrink)

A simple rule of probability revision ensures that the final result ofa sequence of probability revisions is undisturbed by an alterationin the temporal order of the learning prompting those revisions.This Uniformity Rule dictates that identical learning be reflectedin identical ratios of certain new-to-old odds, and is grounded in the oldBayesian idea that such ratios represent what is learned from new experiencealone, with prior probabilities factored out. The main theorem of this paperincludes as special cases (i) Field's theorem on commuting probability-kinematical (...) revisions and (ii) the equivalence of two strategiesfor generalizing Jeffrey's solution to the old evidence problem tothe case of uncertain old evidence and probabilistic new explanation. (shrink)

This paper considers a special case of belief updating—when an agent learns testimonial data, or in other words, the beliefs of others on some issue. The interest in this case is twofold: (1) the linear averaging method for updating on testimony is somewhat popular in epistemology circles, and it is important to assess its normative acceptability, and (2) this facilitates a more general investigation of what it means/requires for an updating method to have a suitable Bayesian representation (taken here as (...) the normative standard). The paper initially defends linear averaging against Bayesian-compatibility concerns raised by Bradley (Soc Choice Welf 29:609-632, 2007), as well as problems associated with multiple testimony updates. The resolution of these issues, however, requires an extremely nuanced interpretation of the parameters of the linear averaging model—the so-called weights of respect. We go on to propose a role that the parameters of any 'shortcut' updating function should play, by way of minimal interpretation of these parameters. The class of updating functions that is consistent with this role, however, excludes linear averaging, at least in its standard form. (shrink)

We explore which types of probabilistic updating commute with convex IP pooling. Positive results are stated for Bayesian conditionalization, imaging, and a certain parameterization of Jeffrey conditioning. This last observation is obtained with the help of a slight generalization of a characterization of externally Bayesian pooling operators due to Wagner :336–345, 2009). These results strengthen the case that pooling should go by imprecise probabilities since no precise pooling method is as versatile.

Rational agents are supposed to maximize expected utility. But what are the options from which they choose? I outline some constraints on an adequate representation of an agent’s options. The options should, for example, contain no information of which the agent is unsure. But they should be sufficiently rich to distinguish all available acts from one another. These demands often come into conflict, so that there seems to be no adequate representation of the options at all. After reviewing existing proposals (...) for how to construe decision-theoretic options and finding them all wanting, I suggest that our model of rational agents should include a special domain of ‘virtual’ option propositions to serve as formal objects of deliberation and choice. (shrink)

How should rational beliefs change over time? The standard Bayesian answer is: by conditionalization (a.k.a. Bayes’ Rule). But conditionalization is not an adequate rule for updating beliefs in “centred” propositions whose truth-value may itself change over time. In response, some have suggested that the objects of belief must be uncentred; others have suggested that beliefs in centred propositions are not subject to diachronic norms. Iargue that these views do not offer a satisfactory account of self-locating beliefs and their dynamics. A (...) third response is to replace conditionalization by a new norm that can deal with centred propositions. I critically survey anumber of new norms that have been proposed, and defend one particular approach. (shrink)

How should a group with different opinions (but the same values) make decisions? In a Bayesian setting, the natural question is how to aggregate credences: how to use a single credence function to naturally represent a collection of different credence functions. An extension of the standard Dutch-book arguments that apply to individual decision-makers recommends that group credences should be updated by conditionalization. This imposes a constraint on what aggregation rules can be like. Taking conditionalization as a basic constraint, we gather (...) lessons from the established work on credence aggregation, and extend this work with two new impossibility results. We then explore contrasting features of two kinds of rules that satisfy the constraints we articulate: one kind uses fixed prior credences, and the other uses geometric averaging, as opposed to arithmetic averaging. We also prove a new characterisation result for geometric averaging. Finally we consider applications to neighboring philosophical issues, including the epistemology of disagreement. (shrink)

Famous results by David Lewis show that plausible-sounding constraints on the probabilities of conditionals or evaluative claims lead to unacceptable results, by standard probabilistic reasoning. Existing presentations of these results rely on stronger assumptions than they really need. When we strip these arguments down to a minimal core, we can see both how certain replies miss the mark, and also how to devise parallel arguments for other domains, including epistemic “might,” probability claims, claims about comparative value, and so on. A (...) popular reply to Lewis's results is to claim that conditional claims, or claims about subjective value, lack truth conditions. For this strategy to have a chance of success, it needs to give up basic structural principles about how epistemic states can be updated—in a way that is strikingly parallel to the commitments of the project of dynamic semantics. (shrink)

According to the recent Perceptual Confidence view, perceptual experiences possess not only a representational content, but also a degree of confidence in that content. The motivations for this view are partly phenomenological and partly epistemic. We discuss both the phenomenological and epistemic motivations for the view, and the resulting account of the interface between perceptual experiences and degrees of belief. We conclude that, in their present state of development, orthodox accounts of perceptual experience are still to be favoured over the (...) perceptual confidence view. (shrink)

Severity of Test (SoT) is an alternative to Popper's logical falsification that solves a number of problems of the logical view. It was presented by Popper himself in 1963. SoT is a less sophisticated probabilistic model of hypothesis testing than Oaksford & Chater's (O&C's) information gain model, but it has a number of striking similarities. Moreover, it captures the intuition of everyday hypothesis testing.

Traditional procedures for rational updating fail when it comes to self-locating opinions, such as your credences about where you are and what time it is. This paper develops an updating procedure for rational agents with self-locating beliefs. In short, I argue that rational updating can be factored into two steps. The first step uses information you recall from your previous self to form a hypothetical credence distribution, and the second step changes this hypothetical distribution to reflect information you have genuinely (...) learned as time has passed. While the second step resembles traditional procedures of updating by conditionalization, the first is best understood by analogy to traditional models of how agents transmit self-locating opinions through ordinary interpersonal communication. (shrink)

This paper discusses simultaneous belief updates. I argue here that modeling such belief updates using the Principle of Minimum Information can be regarded as applying Jeffrey conditionalization successively, and so that, contrary to what many probabilists have thought, the simultaneous belief updates can be successfully modeled by means of Jeffrey conditionalization.

In this article, I suggest an argument that seems to show a conflict between the reflection principle and conditionalization. In particular, I show that when the reflection principle is formulated in a standard way, the principle conflicts with Jeffrey conditionalization. And it is argued that the source of the conflict resides in an ambiguity of the standard formulation. Furthermore, I attempt to rescue the principle using Bayes factors. That is, I suggest a new formulation of the principle so as to (...) avoid the conflict. (shrink)

There are some candidates that have been thought to measure the degree to which evidence incrementally confirms a hypothesis. This paper provides an argument for one candidate—the log-likelihood ratio measure. For this purpose, I will suggest a plausible requirement that I call the Requirement of Collaboration. And then, it will be shown that, of various candidates, only the log-likelihood ratio measure \(l\) satisfies this requirement. Using this result, Jeffrey conditionalization will be reformulated so as to disclose explicitly what determines new (...) credences after experience. (shrink)

Human cognition requires coping with a complex and uncertain world. This suggests that dealing with uncertainty may be the central challenge for human reasoning. In Bayesian Rationality we argue that probability theory, the calculus of uncertainty, is the right framework in which to understand everyday reasoning. We also argue that probability theory explains behavior, even on experimental tasks that have been designed to probe people's logical reasoning abilities. Most commentators agree on the centrality of uncertainty; some suggest that there is (...) a residual role for logic in understanding reasoning; and others put forward alternative formalisms for uncertain reasoning, or raise specific technical, methodological, or empirical challenges. In responding to these points, we aim to clarify the scope and limits of probability and logic in cognitive science; explore the meaning of the explanation of cognition; and re-evaluate the empirical case for Bayesian rationality. (shrink)

Bayesians since Savage (1972) have appealed to asymptotic results to counter charges of excessive subjectivity. Their claim is that objectionable differences in prior probability judgments will vanish as agents learn from evidence, and individual agents will converge to the truth. Glymour (1980), Earman (1992) and others have voiced the complaint that the theorems used to support these claims tell us, not how probabilities updated on evidence will actually}behave in the limit, but merely how Bayesian agents believe they will behave, suggesting (...) that the theorems are too weak to underwrite notions of scientific objectivity and intersubjective agreement. I investigate, in a very general framework, the conditions under which updated probabilities actually converge to a settled opinion and the conditions under which the updated probabilities of two agents actually converge to the same settled opinion. I call this mode of convergence deterministic, and derive results that extend those found in Huttegger (2015b). The results here lead to a simple characterization of deterministic convergence for Bayesian learners and give rise to an interesting argument for what I call strong regularity, the view that probabilities of non-empty events should be bounded away from zero. (shrink)

This is a very selective survey of developments in epistemology, concentrating on work from the past twenty years that is of interest to philosophers of science. The selection is organized around interesting connections between distinct themes. I first connect issues about skepticism to issues about the reliability of belief-acquiring processes. Next I connect discussions of the defeasibility of reasons for belief to accounts of the theory-independence of evidence. Then I connect doubts about Bayesian epistemology to issues about the content of (...) perception. The last detailed connection is between considerations of the finiteness of cognition and epistemic virtues. To connect the connections I end by briefly discussing the pressure that consideration of social roles in the transmission of belief puts on the purposes of epistemology. (shrink)

Seeing a red hat can (i) increase my credence in the hat is red, and (ii) introduce a negative dependence between that proposition and po- tential undermining defeaters such as the light is red. The rigidity of Jeffrey Conditionalization makes this awkward, as rigidity preserves inde- pendence. The picture is less awkward given ‘Holistic Conditionalization’, or so it is claimed. I defend Jeffrey Conditionalization’s consistency with underminable perceptual learning and its superiority to Holistic Conditionalization, arguing that the latter is merely (...) a special case of the former, is itself rigid, and is committed to implausible accounts of perceptual con- firmation and of undermining defeat. (shrink)

At the heart of the Bayesianism is a rule, Conditionalization, which tells us how to update our beliefs. Typical formulations of this rule are underspecified. This paper considers how, exactly, this rule should be formulated. It focuses on three issues: when a subject’s evidence is received, whether the rule prescribes sequential or interval updates, and whether the rule is narrow or wide scope. After examining these issues, it argues that there are two distinct and equally viable versions of Conditionalization to (...) choose from. And which version we choose has interesting ramifications, bearing on issues such as whether Conditionalization can handle continuous evidence, and whether Jeffrey Conditionalization is really a generalization of Conditionalization. (shrink)

Jonathan Weisberg has argued that Jeffrey Conditioning is inherently “anti-holistic” By this he means, inter alia, that JC does not allow us to take proper account of after-the-fact defeaters for our beliefs. His central example concerns the discovery that the lighting in a room is red-tinted and the relationship of that discovery to the belief that a jelly bean in the room is red. Weisberg’s argument that the rigidity required for JC blocks the defeating role of the red-tinted light rests (...) on the strong assumption that all posteriors within the distribution in this example are rigid on a partition over the proposition that the jelly bean is actually red. But individual JC updates of propositions do not require such a broad rigidity assumption. Jeffrey conditionalizers should consider the advantages of a modest project of targeted updating focused on particular propositions rather than seeking to update the entire distribution using one obvious partition. Although Weisberg’s example fails to show JC to be irrelevant or useless, other problems he raises for JC (the commutativity and inputs problems) remain and actually become more pressing when we recognize the important role of background information. (shrink)

One of the fundamental problems of epistemology is to say when the evidence in an agent’s possession justifies the beliefs she holds. In this paper and its prequel, we defend the Bayesian solution to this problem by appealing to the following fundamental norm: Accuracy An epistemic agent ought to minimize the inaccuracy of her partial beliefs. In the prequel, we made this norm mathematically precise; in this paper, we derive its consequences. We show that the two core tenets of Bayesianism (...) follow from the norm, while the characteristic claim of the Objectivist Bayesian follows from the norm along with an extra assumption. Finally, we consider Richard Jeffrey’s proposed generalization of conditionalization. We show not only that his rule cannot be derived from the norm, unless the requirement of Rigidity is imposed from the start, but further that the norm reveals it to be illegitimate. We end by deriving an alternative updating rule for those cases in which Jeffrey’s is usually supposed to apply. (shrink)

In his book Prejudice, Endre Begby argues that people who hold paradigmatically prejudiced beliefs – for example, the belief that women are less adept at math than men – might be fully rational in holding those beliefs. In this article, I argue that Begby fails to provide compelling examples of beliefs that are both rational and prejudiced. On Begby’s account, whether a belief is prejudiced is determined by its content: it follows that any two token beliefs with the same content (...) will either both be prejudiced, or both unprejudiced, regardless of how they differ in other respects. I sketch an alternative account, on which whether a person’s belief counts as prejudiced might be sensitive to a greater range of factors. I then turn to Bebgy’s discussion of ‘evidential preemption,’ a phenomenon by which certain speech acts seem to inoculate themselves from having their contents disconfirmed or falsified by later counterevidence. I argue for skepticism about evidential preemption. To the extent that there is a genuine normative phenomenon in the neighborhood, it is the familiar one of testimonial defeat, in which testimony from one source is neutralized by conflicting testimony from another source that one has reason to think is more reliable. (shrink)

Rédei and Gyenis recently displayed strong constraints of Bayesian learning. However, they also presented a positive result for Bayesianism. Despite the limited significance of this positive result, I find it useful to discuss its two possible strengthenings to present new results and open new questions about the limits of Bayesianism. First, I will show that one cannot strengthen the positive result by restricting the evidence to so-called “certain evidence”. Secondly, strengthening the result by restricting the partitions—as parts of one’s evidence—to (...) Jeffrey-independent partitions requires additional constraints on one’s evidence to preserve its commutativity. So, my results provide additional grounds for caution and support for the limitations of Bayesian learning. (shrink)

– We offer a new motivation for imprecise probabilities. We argue that there are propositions to which precise probability cannot be assigned, but to which imprecise probability can be assigned. In such cases the alternative to imprecise probability is not precise probability, but no probability at all. And an imprecise probability is substantially better than no probability at all. Our argument is based on the mathematical phenomenon of non-measurable sets. Non-measurable propositions cannot receive precise probabilities, but there is a natural (...) way for them to receive imprecise probabilities. The mathematics of non-measurable sets is arcane, but its epistemological import is far-reaching; even apparently mundane propositions are liable to be affected by non-measurability. The phenomenon of non-measurability dramatically reshapes the dialectic between critics and proponents of imprecise credence. Non-measurability offers natural rejoinders to prominent critics of imprecise credence. Non-measurability even reverses some of the critics’ arguments—by the very lights that have been used to argue against imprecise credences, imprecise credences are better than precise credences. (shrink)

We explore the question of whether sustained rational disagreement is possible from a broadly Bayesian perspective. The setting is one where agents update on the same information, with special consideration being given to the case of uncertain information. The classical merging of opinions theorem of Blackwell and Dubins shows when updated beliefs come and stay closer for Bayesian conditioning. We extend this result to a type of Jeffrey conditioning where agents update on evidence that is uncertain but solid. However, merging (...) of beliefs does not generally hold for Jeffrey conditioning on evidence that is fluid. Several theorems on the asymptotic behavior of subjective probabilities are proven. Taken together they show that while a consensus nearly always emerges in important special cases, sustained rational disagreement can be expected in many other situations. (shrink)

This note is a sequel to Huber. It is shown that obeying a normative principle relating counterfactual conditionals and conditional beliefs, viz. the royal rule, is a necessary and sufficient means to attaining a cognitive end that relates true beliefs in purely factual, non-modal propositions and true beliefs in purely modal propositions. Along the way I will sketch my idealism about alethic or metaphysical modality.

Recent accounts of actual causation are stated in terms of extended causal models. These extended causal models contain two elements representing two seemingly distinct modalities. The first element are structural equations which represent the or mechanisms of the model, just as ordinary causal models do. The second element are ranking functions which represent normality or typicality. The aim of this paper is to show that these two modalities can be unified. I do so by formulating two constraints under which extended (...) causal models with their two modalities can be subsumed under so called which contain just one modality. These two constraints will be formally precise versions of Lewissystem of weights or priorities” governing overall similarity between possible worlds. (shrink)

The paper provides an argument for the thesis that an agent’s degrees of disbelief should obey the ranking calculus. This Consistency Argument is based on the Consistency Theorem. The latter says that an agent’s belief set is and will always be consistent and deductively closed iff her degrees of entrenchment satisfy the ranking axioms and are updated according to the ranktheoretic update rules.

Philosophers typically rely on intuitions when providing a semantics for counterfactual conditionals. However, intuitions regarding counterfactual conditionals are notoriously shaky. The aim of this paper is to provide a principled account of the semantics of counterfactual conditionals. This principled account is provided by what I dub the Royal Rule, a deterministic analogue of the Principal Principle relating chance and credence. The Royal Rule says that an ideal doxastic agent’s initial grade of disbelief in a proposition \(A\) , given that the (...) counterfactual distance in a given context to the closest \(A\) -worlds equals \(n\) , and no further information that is not admissible in this context, should equal \(n\) . Under the two assumptions that the presuppositions of a given context are admissible in this context, and that the theory of deterministic alethic or metaphysical modality is admissible in any context, it follows that the counterfactual distance distribution in a given context has the structure of a ranking function. The basic conditional logic V is shown to be sound and complete with respect to the resulting rank-theoretic semantics of counterfactuals. (shrink)

Belief revision theory studies how an ideal doxastic agent should revise her beliefs when she receives new information. In part I, I have first presented the AGM theory of belief revision. Then I have focused on the problem of iterated belief revisions. In part II, I will first present ranking theory (Spohn 1988). Then I will show how it solves the problem of iterated belief revisions. I will conclude by sketching two areas of future research.

Jeffrey updating is a natural extension of Bayesian updating to cases where the evidence is uncertain. But, the resulting degrees of belief appear to be sensitive to the order in which the uncertain evidence is acquired, a rather un-Bayesian looking effect. This order dependence results from the way in which basic Jeffrey updating is usually extended to sequences of updates. The usual extension seems very natural, but there are other plausible ways to extend Bayesian updating that maintain order-independence. I will (...) explore three models of sequential updating, the usual extension and two alternatives. I will show that the alternative updating schemes derive from extensions of the usual rigidity requirement, which is at the heart of Jeffrey updating. Finally, I will establish necessary and sufficient conditions for order-independent updating, and show that extended rigidity is closely related to these conditions. (shrink)

Conciliatory views of peer disagreement hold that when an agent encounters peer disagreement she should conciliate by adjusting her doxastic attitude towards that of her peer. In this paper I distinguish different ways conciliation can be understood and argue that the way conciliationism is typically understood violates the principle of commutativity of evidence. Commutativity of evidence holds that the order in which evidence is acquired should not influence what it is reasonable to believe based on that evidence. I argue that (...) when an agent encounters more than one peer, and applies the process of conciliation serially, the order she encounters the peers influences the resulting credence. I argue this is a problem for conciliatory views of disagreement, and suggest some responses available to advocates of conciliation. (shrink)

The externalist says that your evidence could fail to tell you what evidence you do or not do have. In that case, it could be rational for you to be uncertain about what your evidence is. This is a kind of uncertainty which orthodox Bayesian epistemology has difficulty modeling. For, if externalism is correct, then the orthodox Bayesian learning norms of conditionalization and reflection are inconsistent with each other. I recommend that an externalist Bayesian reject conditionalization. In its stead, I (...) provide a new theory of rational learning for the externalist. I defend this theory by arguing that its advice will be followed by anyone whose learning dispositions maximize expected accuracy. I then explore some of this theory’s consequences for the rationality of epistemic akrasia, peer disagreement, undercutting defeat, and uncertain evidence. (shrink)

An indifference principle says that your credences should be distributed uniformly over each of the possibilities you recognise. A chance deference principle says that your credences should be aligned with the chances. My thesis is that, if we are anti-Humeans about chance, then these two principles are incompatible. Anti-Humeans think that it is possible for the actual frequencies to depart from the chances. So long as you recognise possibilities like this, you cannot both spread your credences evenly and defer to (...) the chances. I discuss some weaker forms of indifference which will allow anti-Humeans to defer to the chances. (shrink)