According to so-called epistemic theories of conditionals, the assertability/acceptability/acceptance of a conditional requires the existence of an epistemically significant relation between the conditional’s antecedent and its consequent. This paper points to some linguistic data that our current best theories of the foregoing type appear unable to explain. Further, it presents a new theory of the same type that does not have that shortcoming. The theory is then defended against some seemingly obvious objections.

List and Pettit have stated an impossibility theorem about the aggregation of individual opinion states. Building on recent work on the lottery paradox, this paper offers a variation on that result. The present result places different constraints on the voting agenda and the domain of profiles, but it covers a larger class of voting rules, which need not satisfy the proposition-wise independence of votes.

As part of an exceptionally lucid analysis of the Lottery Paradox, Dana Nelkin castigates the solutions to that paradox put forward by Laurence Bonjour and Sharon Ryan. According to her, these are “so finely tailored to lottery-like cases that they are limited in their ability to explain [what seem the intuitively right responses to such cases]”. She then offers a solution to the Lottery Paradox that allegedly has the virtue of being independently motivated by our intuitions regarding certain non-lottery-like cases. (...) This note argues that Nelkin fails to show that her solution applies to other than lottery-like cases or in any event that it is more general than Bonjour’s and Ryan’s solutions. (shrink)

This paper is concerned with formal solutions to the lottery paradox on which high probability defeasibly warrants acceptance. It considers some recently proposed solutions of this type and presents an argument showing that these solutions are trivial in that they boil down to the claim that perfect probability is sufficient for rational acceptability. The argument is then generalized, showing that a broad class of similar solutions faces the same problem. An argument against some formal solutions to the lottery paradox The (...) argument generalized Some variations Adding modalities Anticipated objections. (shrink)

External world skeptics are typically opposed to admitting as evidence anything that goes beyond the purely phenomenal, and equally typically, they disown the use of rules of inference that might enable one to move from premises about the phenomenal alone to a conclusion about the external world. This seems to bar any a posteriori resolution of the skepticism debate. This paper argues that the situation is not quite so hopeless, and that an a posteriori resolution of the debate becomes possible (...) once it is recognized that the skeptic holds overly defensive and ill-motivated positions vis-à-vis both evidence and inference, and that more reasonable ones are available. In stating these more reasonable positions, as well as in showing how they make possible an a posteriori resolution of the skepticism debate, the paper draws on the machinery of Bayesian epistemology. (shrink)

The Lottery Paradox and the Preface Paradox both involve the thesis that high probability is sufficient for rational acceptability. The standard solution to these paradoxes denies that rational acceptability is deductively closed. This solution has a number of untoward consequences. The present paper suggests that a better solution to the paradoxes is to replace the thesis that high probability suffices for rational acceptability with a somewhat stricter thesis. This avoids the untoward consequences of the standard solution. The new solution will (...) be defended against a seemingly obvious objection. 1 The paradoxes of rational acceptability 2 The standard solution 3 A new solution to the paradoxes 4 Basic assumptions 5 The new solution defended 6 Conclusion 7 Appendix. (shrink)

There is widespread agreement that we cannot know of a lottery ticket we own that it is a loser prior to the drawing of the lottery. At the same time we appear to have knowledge of events that will occur only if our ticket is a loser. Supposing any plausible closure principle for knowledge, the foregoing seems to yield a paradox. Appealing to some broadly Gricean insights, the present paper argues that this paradox is apparent only.

We’ve all been at parties where there's one cookie left on what was once a plate full of cookies, a cookie no one will eat simply because everyone is following a rule of etiquette, according to which you’re not supposed to eat the last cookie. Or at least we think everyone is following this rule, but maybe not. In this paper I present a new paradox, the Cookie Paradox, which is an argument that seems to prove that in any situation (...) in which everyone is truly following the rule, no one eats any cookies at all, no matter how many there are to be eaten. The ‘Cookie Argument’ resembles the more familiar argument that surprise exams are impossible, but it's not exactly the same. I argue that the biggest difference is that, unlike the surprise exam argument, the Cookie Argument is actually sound! I conclude the paper by explaining how it could be possible for a group of people to engage in behavior (eating cookies) that guarantees that at least one of the members of the group will violate a rule, even when it's common knowledge in the group that everyone is committed to following that very rule. (shrink)

I argue that believing that p implies having a credence of 1 in p. This is true because the belief that p involves representing p as being the case, representing p as being the case involves not allowing for the possibility of not-p, while having a credence that’s greater than 0 in not-p involves regarding not-p as a possibility.

What is the relationship between degrees of belief and binary beliefs? Can the latter be expressed as a function of the former—a so-called “belief-binarization rule”—without running into difficulties such as the lottery paradox? We show that this problem can be usefully analyzed from the perspective of judgment-aggregation theory. Although some formal similarities between belief binarization and judgment aggregation have been noted before, the connection between the two problems has not yet been studied in full generality. In this paper, we seek (...) to fill this gap. The paper is organized around a baseline impossibility theorem, which we use to map out the space of possible solutions to the belief-binarization problem. Our theorem shows that, except in limiting cases, there exists no belief-binarization rule satisfying four initially plausible desiderata. Surprisingly, this result is a direct corollary of the judgment-aggregation variant of Arrow’s classic impossibility theorem in social choice theory. (shrink)

The lottery and preface paradoxes pose puzzles in epistemology concerning how to think about the norms of reasonable or permissible belief. Contextualists in epistemology have focused on knowledge ascriptions, attempting to capture a set of judgments about knowledge ascriptions and denials in a variety of contexts (including those involving lottery beliefs and the principles of closure). This article surveys some contextualist approaches to handling issues raised by the lottery and preface, while also considering some of the difficulties encountered by those (...) approaches. (shrink)

This thesis modifies the wording of William Clifford’s 1877 evidence principle (that ‘it is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence’) to propose an explicitly moral principle, restricted to descriptive beliefs (about what is or is not the case) and excluding prescriptive beliefs (about what ought or ought not to be the case). It considers potential counter-examples, particularly William James’s 1896 defence of religious belief; and concludes that the modified principle survives unscathed. It then searches (...) for suitable criteria for ‘sufficient’ and ‘insufficient’ evidence, first within a Bayesian framework enriched by pragmatic considerations, and finally by returning to Clifford’s conception of our shared responsibilities to our shared epistemic asset. Combined with elements of Edward Craig’s (1990, 1999) ‘state-of-nature’ theory of knowledge this gets us to a minimum threshold to avoid insufficient evidence and an aspirational criterion of sufficient evidence. (shrink)

Epistemology is the study of knowledge and justified belief. Belief is thus central to epistemology. It comes in a qualitative form, as when Sophia believes that Vienna is the capital of Austria, and a quantitative form, as when Sophia's degree of belief that Vienna is the capital of Austria is at least twice her degree of belief that tomorrow it will be sunny in Vienna. Formal epistemology, as opposed to mainstream epistemology (Hendricks 2006), is epistemology done in a formal way, (...) that is, by employing tools from logic and mathematics. The goal of this entry is to give the reader an overview of the formal tools available to epistemologists for the representation of belief. A particular focus will be the relation between formal representations of qualitative belief and formal representations of quantitative degrees of belief. (shrink)

Paradoxes have played an important role both in philosophy and in mathematics and paradox resolution is an important topic in both fields. Paradox resolution is deeply important because if such resolution cannot be achieved, we are threatened with the charge of debilitating irrationality. This is supposed to be the case for the following reason. Paradoxes consist of jointly contradictory sets of statements that are individually plausible or believable. These facts about paradoxes then give rise to a deeply troubling epistemic problem. (...) Specifically, if one believes all of the constitutive propositions that make up a paradox, then one is apparently committed to belief in every proposition. This is the result of the principle of classical logical known as ex contradictione (sequitur) quodlibetthat anything and everything follows from a contradiction, and the plausible idea that belief is closed under logical or material implication (i.e. the epistemic closure principle). But, it is manifestly and profoundly irrational to believe every proposition and so the presence of even one contradiction in one’s doxa appears to result in what seems to be total irrationality. This problem is the problem of paradox-induced explosion. In this paper it will be argued that in many cases this problem can plausibly be avoided in a purely epistemic manner, without having either to resort to non-classical logics for belief (e.g. paraconsistent logics) or to the denial of the standard closure principle for beliefs. The manner in which this result can be achieved depends on drawing an important distinction between the propositional attitude of belief and the weaker attitude of acceptance such that paradox constituting propositions are accepted but not believed. Paradox-induced explosion is then avoided by noting that while belief may well be closed under material implication or even under logical implication, these sorts of weaker commitments are not subject to closure principles of those sorts. So, this possibility provides us with a less radical way to deal with the existence of paradoxes and it preserves the idea that intelligent agents can actually entertain paradoxes. (shrink)

This paper is about conjunctions and disjunctions in the scope of non-doxastic atti- tude verbs. These constructions generate a certain type of ignorance implicature. I argue that the best way to account for these implicatures is by appealing to a notion of contex- tual redundancy (Schlenker, 2008; Fox, 2008; Mayr and Romoli, 2016). This pragmatic approach to ignorance implicatures is contrasted with a semantic account of disjunctions under `wonder' that appeals to exhausti cation (Roelofsen and Uegaki, 2016). I argue that (...) exhausti cation-based theories cannot handle embedded conjunctions, so a pragmatic account of ignorance implicatures is superior. (shrink)

This paper presents statistical default logic, an expansion of classical (i.e., Reiter) default logic that allows us to model common inference patterns found in standard inferential statistics, including hypothesis testing and the estimation of a populations mean, variance and proportions. The logic replaces classical defaults with ordered pairs consisting of a Reiter default in the first coordinate and a real number within the unit interval in the second coordinate. This real number represents an upper-bound limit on the probability of accepting (...) the consequent of an applied default and that consequent being false. A method for constructing extensions is then defined that preserves this upper bound on the probability of error under a (skeptical) non-monotonic consequence relation. (shrink)

What is the epistemological structure of situations where many small risks amount to a large one? Lottery and preface paradoxes and puzzles about quantum-mechanical blips threaten the idea that competent deduction is a way of extending our knowledge. Seemingly, everyday knowledge involves small risks, and competently deducing the conjunction of many such truths from them yields a conclusion too risky to constitute knowledge. But the dilemma between scepticism and abandoning MPC is false. In extreme cases, objectively improbable truths are known. (...) Safety is modal, not probabilistic, in structure, with closure and factiveness conditions. It is modelled using closeness of worlds. Safety is analogous to knowledge. It suggests an interpretation of possible worlds semantics for epistemic logic. To avoid logical omniscience, a relation of epistemic counterparthood between formulas is introduced. This supports a safety conception of knowledge and formalizes how extending knowledge by deduction depends on logical competence. (shrink)

Bayesian Epistemology is a general framework for thinking about agents who have beliefs that come in degrees. Theories in this framework give accounts of rational belief and rational belief change, which share two key features: (i) rational belief states are represented with probability functions, and (ii) rational belief change results from the acquisition of evidence. This dissertation focuses specifically on the second feature. I pose the Evidence Question: What is it to have evidence? Before addressing this question we must have (...) an understanding of Bayesian Epistemology. The first chapter argues that we should understand Bayesian Epistemology as giving us theories that are evaluative and not action-guiding. I reach this verdict after considering the popular ‘ought’-implies-‘can’ objection to Bayesian Epistemology. The second chapter argues that it is important for theories in Bayesian Epistemology to answer the Evidence Question, and distinguishes between internalist and externalist answers. The third and fourth chapters present and defend a specific answer to the Evidence Question. The account is inspired by reliabilist accounts of justification, and attempts to understand what it is to have evidence by appealing solely to considerations of reliability. Chapter 3 explains how to understand reliability, and how the account fits with Bayesian Epistemology, in particular, the requirement that an agent’s evidence receive probability 1. Chapter 4 responds to objections, which maintain that the account gives the wrong verdict in a variety of situations including skeptical scenarios, lottery cases, scientific cases, and cases involving inference. After slight modifications, I argue that my account has the resources to answer the objections. The fifth chapter considers the possibility of losing evidence. I show how my account can model these cases. To do so, however, we require a modification to Conditionalization, the orthodox principle governing belief change. I present such a modification. The sixth and seventh chapters propose a new understanding of Dutch Book Arguments, historically important arguments for Bayesian principles. The proposal shows that the Dutch Book Arguments for implausible principles are defective, while the ones for plausible principles are not. The final chapter is a conclusion. (shrink)

In “Can it be rational to have faith?”, it was argued that to have faith in some proposition consists, roughly speaking, in stopping one’s search for evidence and committing to act on that proposition without further evidence. That paper also outlined when and why stopping the search for evidence and acting is rationally required. Because the framework of that paper was that of formal decision theory, it primarily considered the relationship between faith and degrees of belief, rather than between faith (...) and belief full stop. This paper explores the relationship between rational faith and justified belief, by considering four prominent proposals about the relationship between belief and degrees of belief, and by examining what follows about faith and belief according to each of these proposals. It is argued that we cannot reach consensus concerning the relationship between faith and belief at present because of the more general epistemological lack of consensus over how belief relates to rationality: in particular, over how belief relates to the degrees of belief it is rational to have given one’s evidence. (shrink)

Handbook of the History of Logic, vol. 10, eds. Dov Gabbay, Stephan Hartmann, and John Woods, forthcoming.

Modern belief revision theory is based to a large extent on partial meet contraction that was introduced in the seminal article by Carlos Alchourrón, Peter Gärdenfors, and David Makinson that appeared in 1985. In the same year, Alchourrón and Makinson published a significantly different approach to the same problem, called safe contraction. Since then, safe contraction has received much less attention than partial meet contraction. The present paper summarizes the current state of knowledge on safe contraction, provides some new results (...) and offers a list of unsolved problems that are in need of investigation. (shrink)

In legal proceedings, a fact-finder needs to decide whether a defendant is guilty or not based on probabilistic evidence. We defend the thesis that the defendant should be found guilty just in case it is rational for the fact-finder to believe that the defendant is guilty. We draw on Leitgeb’s stability theory for an appropriate notion of rational belief and show how our thesis solves the problem of statistical evidence. Finally, we defend our account of legal proof against challenges from (...) Staffel and compare it to a recent competitor put forth by Moss. (shrink)

According to the Lockean thesis, a proposition is believed just in case it is highly probable. While this thesis enjoys strong intuitive support, it is known to conflict with seemingly plausible logical constraints on our beliefs. One way out of this conflict is to make probability 1 a requirement for belief, but most have rejected this option for entailing what they see as an untenable skepticism. Recently, two new solutions to the conflict have been proposed that are alleged to be (...) non-skeptical. We compare these proposals with each other and with the Lockean thesis, in particular with regard to the question of how much we gain by adopting any one of them instead of the probability 1 requirement, that is, of how likely it is that one believes more than the things one is fully certain of. (shrink)

Everyday life reasoning and argumentation is defeasible and uncertain. I present a probability logic framework to rationally reconstruct everyday life reasoning and argumentation. Coherence in the sense of de Finetti is used as the basic rationality norm. I discuss two basic classes of approaches to construct measures of argument strength. The first class imposes a probabilistic relation between the premises and the conclusion. The second class imposes a deductive relation. I argue for the second class, as the first class is (...) problematic if the arguments involve conditionals. I present a measure of argument strength that allows for dealing explicitly with uncertain conditionals in the premise set. (shrink)

Why can testimony alone be enough for findings of liability? Why statistical evidence alone can’t? These questions underpin the “Proof Paradox” (Redmayne 2008, Enoch et al. 2012). Many epistemologists have attempted to explain this paradox from a purely epistemic perspective. I call it the “Epistemic Project”. In this paper, I take a step back from this recent trend. Stemming from considerations about the nature and role of standards of proof, I define three requirements that any successful account in line with (...) the Epistemic Project should meet. I then consider three recent epistemic accounts on which the standard is met when the evidence rules out modal risk (Pritchard 2018), normic risk (Ebert et al. 2020), or relevant alternatives (Gardiner 2019 2020). I argue that none of these accounts meets all the requirements. Finally, I offer reasons to be pessimistic about the prospects of having a successful epistemic explanation of the paradox. I suggest the discussion on the proof paradox would benefit from undergoing a ‘value-turn’. (shrink)

"Partial realism" is a common position in the contemporary philosophy of science literature. It states that the "essential" elements of empirically successful scientific theories accurately represent corresponding features the world. This thesis makes several novel contributions related to this position. Firstly, it offers a new definition of the concept of “empirical success”, representing a principled merger between the use-novelty and unification accounts. Secondly, it provides a comparative critical analysis of various accounts of which elements are "essential" to the success of (...) a theory, including structural realism and the divide et impera strategy. A novel account of essentialness, entitled the “empirically successful sub-theory "account", is defended. Thirdly, it is argued that the realism/anti-realism debate should put to the side metaphysical questions and focus instead on partial realism's commitment to the continuity of science. Because this commitment lacks metaphysical implications, it is referred to as "deflationary realism". Anti-realists cannot reject deflationary realism as a matter of a priori principle; its overall viability can therefore only be assessed by a careful examination of the history of science. Finally, another consequence of partial realism, named "partial rationalism", is defended. Partial rationalism states that, in cases where several competing theories have been suggested, scientists are rational just in case they accept the essential elements of each of the scientific theories on offer. This novel position subverts the existing literature on scientific "revolutions", as it sometimes demands that scientists devise a synthesis between competing scientific theories, instead of "choosing" only one. The philosophical points defended in this thesis are illustrated and supported by case studies from the history of science, including Fresnel’s wave theory of light, the Copernican revolution, the "neo-Darwinian synthesis" in evolutionary biology, the "prion revolution" in molecular biology, the miasma theory of disease, and the chemical revolution. (shrink)

Infallibilism is the claim that knowledge requires that one satisfies some infallibility condition. I spell out three distinct such conditions: epistemic, evidential and modal infallibility. Epistemic infallibility turns out to be simply a consequence of epistemic closure, and is not infallibilist in any relevant sense. Evidential infallibilism i s unwarranted but it is not an satisfactory characterization of the infallibilist intuition. Modal infallibility, by contrast, captures the core infallibilist intuition, and I argue that it is required to solve the Gettier (...) problem and account for lottery cases. Finally, I discuss whether modal infallibilism has sceptical consequences and argue that it is an open question whose answer depends on one’s account of alethic possibility. (shrink)

The terms “model” and “model-building” have been used to characterize the field of formal philosophy, to evaluate philosophy’s and philosophical logic’s progress and to define philosophical logic itself. A model is an idealization, in the sense of being a deliberate simplification of something relatively complex in which several important aspects are left aside, but also in the sense of being a view too perfect or excellent, not found in reality, of this thing. Paraconsistent logic is a branch of philosophical logic. (...) It is however not clear how paraconsistent logic can be seen as model-building. What exactly is modeled? In this paper I adopt the perspective of looking at a particular instance of paraconsistent logic—paranormal modal logic—which might be seen as a model of a specific kind of agent: inductive agents. After ntroducing what I call the highlevel and low-level models of inductive agents, I analyze the extent to which the above-mentioned idealizing features of model-building appear in paranormal modal logic and how they affect its philosophical significance. (shrink)

Belief is a familiar attitude: taking something to be the case or regarding it as true. But we are more confident in some of our beliefs than in others. For this reason, many epistemologists appeal to a second attitude, called credence, similar to a degree of confidence. This raises the question: how do belief and credence relate to each other? On a belief-first view, beliefs are more fundamental and credences are a species of beliefs, e.g. beliefs about probabilities. On a (...) credence-first view, credences are more fundamental and beliefs are a species of credence, e.g. credence above some threshold. In this thesis, I develop and defend a third view that I call belief-credence dualism. On this view, belief and credence are independent, equally fundamental attitudes, and neither reduces to the other. I begin by motivating the project: why should we care about the relationship between belief and credence? I argue it has broad implications for many debates in epistemology and beyond. Then, I defend dualism, arguing that it can explain features of our mental lives that a credence-first view and a belief-first view cannot. I also argue that dualism has attractive, interesting implications when applied to the pragmatic encroachment debate. Finally, I explore implications of dualism, both for the nature of evidence and how faith might go beyond the evidence but nonetheless be epistemically rational. I conclude that the human mind is, in some ways, complex, but we should be happy with this conclusion also long as each mental state we posit has a clear role to play. (shrink)

I am concerned with epistemic closure—the phenomenon in which some knowledge requires other knowledge. In particular, I defend a version of the closure principle in terms of analyticity; if an agent S knows that p is true, then S knows that all analytic parts of p are true as well. After targeting the relevant notion of analyticity, I argue that this principle accommodates intuitive cases and possesses the theoretical resources to avoid the preface paradox. I close by arguing that contextualists (...) who maintain that knowledge attributions are closed within—but not between—linguistic contexts are tacitly committed to this principle’s truth. (shrink)

Vann McGee has presented a putative counterexample to modus ponens. I show that (a slightly modified version of) McGee’s election scenario has the same structure as a famous lottery scenario by Kyburg. More specifically, McGee’s election story can be taken to show that, if the Lockean Thesis holds, rational belief is not closed under classical logic, including classical-logic modus ponens. This conclusion defies the existing accounts of McGee’s puzzle.

Probabilism says an agent is rational only if her credences are probabilistic. This paper is concerned with the so-called Accuracy Dominance Argument for Probabilism. This argument begins with the claim that the sole fundamental source of epistemic value for a credence is its accuracy. It then shows that, however we measure accuracy, any non-probabilistic credences are accuracy-dominated: that is, there are alternative credences that are guaranteed to be more accurate than them. It follows that non-probabilistic credences are irrational. In this (...) paper, I identify and explore a lacuna in this argument. I grant that, if the only doxastic attitudes are credal attitudes, the argument succeeds. But many philosophers say that, alongside credences, there are other doxastic attitudes, such as full beliefs. What's more, those philosophers typically claim, these other doxastic attitudes are closely connected to credences, either as a matter of necessity or normatively. Now, since full beliefs are also doxastic attitudes, it seems that, like credences, the sole source of epistemic value for them is their accuracy. Thus, if we wish to measure the epistemic value of an agent's total doxastic state, we must include not only the accuracy of her credences, but also the accuracy of her full beliefs. However, if this is the case, there is a problem for the Accuracy Dominance Argument for Probabilism. For all the argument says, there might be non-probabilistic credences such that there is no total doxastic state that accuracy-dominates the total doxastic state to which those credences belong. (shrink)

This paper argues for two theses: that degrees of belief are context sensitive; that outright belief is belief to degree 1. The latter thesis is rejected quickly in most discussions of the relationship between credence and belief, but the former thesis undermines the usual reasons for doing so. Furthermore, identifying belief with credence 1 allows nice solutions to a number of problems for the most widely-held view of the relationship between credence and belief, the threshold view. I provide a sketch (...) of a formal framework on which both theses are true. This is a modified Bayesian framework; I argue that despite making credences context-sensitive, the framework lets Bayesians hold on to their signature explanatory successes. The sort of context-sensitivity claimed for credences here mirrors the sort of context-sensitivity I have elsewhere claimed for outright belief: one's credences depend, in part, on the space of alternative possibilities one takes seriously in a context. (shrink)