A ‘lottery belief’ is a belief that a particular ticket has lost a large, fair lottery, based on nothing more than the odds against it winning. The lottery paradox brings out a tension between the idea that lottery beliefs are justified and the idea that that one can always justifiably believe the deductive consequences of things that one justifiably believes – what is sometimes called the principle of closure. Many philosophers have treated the (...) class='Hi'>lottery paradox as an argument against the second idea – but I make a case here that it is the first idea that should be given up. As I shall show, there are a number of independent arguments for denying that lottery beliefs are justified. (shrink)

In the first chapter of his Knowledge and Lotteries, John Hawthorne argues that thinkers do not ordinarily know lottery propositions. His arguments depend on claims about the intimate connections between knowledge and assertion, epistemic possibility, practical reasoning, and theoretical reasoning. In this paper, we cast doubt on the proposed connections. We also put forward an alternative picture of belief and reasoning. In particular, we argue that assertion is governed by a Gricean constraint that makes no reference to knowledge, (...) and that practical reasoning has more to do with rational degrees of belief than with states of knowledge. (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)

Thomas Kroedel argues that the lottery paradox can be solved by identifying epistemic justification with epistemic permissibility rather than epistemic obligation. According to his permissibility solution, we are permitted to believe of each lottery ticket that it will lose, but since permissions do not agglomerate, it does not follow that we are permitted to have all of these beliefs together, and therefore it also does not follow that we are permitted to believe that all tickets will lose. I (...) present two objections to this solution. First, even if justification itself amounts to no more than epistemic permissibility, the lottery paradox recurs at the level of doxastic obligations unless one adopts an extremely permissive view about suspension of belief that is in tension with our practice of doxastic criticism. Second, even if there are no obligations to believe lottery propositions, the permissibility solution fails because epistemic permissions typically agglomerate, and the lottery case provides no exception to this rule. (shrink)

The lottery paradox involves a set of judgments that are individually easy, when we think intuitively, but ultimately hard to reconcile with each other, when we think reflectively. Empirical work on the natural representation of probability shows that a range of interestingly different intuitive and reflective processes are deployed when we think about possible outcomes in different contexts. Understanding the shifts in our natural ways of thinking can reduce the sense that the lottery paradox reveals something problematic about (...) our concept of knowledge. However, examining these shifts also raises interesting questions about how we ought to be thinking about possible outcomes in the first place. (shrink)

I explore how rational belief and rational credence relate to evidence. I begin by looking at three cases where rational belief and credence seem to respond differently to evidence: cases of naked statistical evidence, lotteries, and hedged assertions. I consider an explanation for these cases, namely, that one ought not form beliefs on the basis of statistical evidence alone, and raise worries for this view. Then, I suggest another view that explains how belief and credence relate to (...) evidence. My view focuses on the possibilities that the evidence makes salient. I argue that this makes better sense of the difference between rational credence and rational belief than other accounts. (shrink)

I review recent empirical findings on knowledge attributions in lottery cases and report a new experiment that advances our understanding of the topic. The main novel finding is that people deny knowledge in lottery cases because of an underlying qualitative difference in how they process probabilistic information. “Outside” information is generic and pertains to a base rate within a population. “Inside” information is specific and pertains to a particular item’s propensity. When an agent receives information that 99% of (...) all lottery tickets lose (outside information), people judge that she does not know that her ticket will lose. By contrast, when an agent receives information that her specific ticket is 99% likely to lose (inside information), people judge that she knows that her ticket will lose. Despite this difference in knowledge judgments, people rate the likelihood of her ticket losing the exact same in both cases (i.e. 99%). The results shed light on other factors affecting knowledge judgments in lottery cases, including formulaic expression and participants’ own estimation of whether it is true that the ticket will lose. The results also undermine previous hypotheses offered for knowledge denial in lottery cases, including the hypotheses that people deny knowledge because they either deny justification or acknowledge a chance for error. (shrink)

If someone disagrees with my moral views, or more generally if I’m in a group of n people who all disagree with each other, but I don’t have any special evidence or basis for my epistemic superiority, then it’s at best a 1-in-n chance that my views are correct. The skeptical threat from disagreement is thus a kind of moral lottery, to adapt a similar metaphor from Sharon Street. Her own genealogical debunking argument, as I discuss, relies on a (...) premise of such disagreement among evolutionary counterparts. -/- In this paper, I resist the threat from disagreement by showing that, on some of the most influential and most attractive theories of content determination, the premise of moral disagreement cannot serve any skeptical or revisionary purposes. I examine and criticize attempts, made by Gilbert Harman and Sharon Street, to argue from disagreement to relativism by relying on a theory of content determination that involves a principle that, within certain constraints, maximizes the attribution to us of true beliefs. And I examine and criticize Robert Williams’s attempt to show there is moral disagreement by relying on a theory of content determination that involves a principle that instead maximizes the attribution to us of rationality. My overall aim is to defend commonsense moral realism via a careful look at the theory of content and concepts. (shrink)

I argue that we should solve the Lottery Paradox by denying that rational belief is closed under classical logic. To reach this conclusion, I build on my previous result that (a slight variant of) McGee’s election scenario is a lottery scenario (see Lissia 2019). Indeed, this result implies that the sensible ways to deal with McGee’s scenario are the same as the sensible ways to deal with the lottery scenario: we should either reject the Lockean Thesis (...) or Belief Closure. After recalling my argument to this conclusion, I demonstrate that a McGee-like example (which is just, in fact, Carroll’s barbershop paradox) can be provided in which the Lockean Thesis plays no role: this proves that denying Belief Closure is the right way to deal with both McGee’s scenario and the Lottery Paradox. A straightforward consequence of my approach is that Carroll’s puzzle is solved, too. (shrink)

This paper elaborates a new solution to the lottery paradox, according to which the paradox arises only when we lump together two distinct states of being confident that p under one general label of ‘belief that p’. The two-state conjecture is defended on the basis of some recent work on gradable adjectives. The conjecture is supported by independent considerations from the impossibility of constructing the lottery paradox both for risk-tolerating states such as being afraid, hoping or hypothesizing, (...) and for risk-averse, certainty-like states. The new proposal is compared to views within the increasingly popular debate opposing dualists to reductionists with respect to the relation between belief and degrees of belief. (shrink)

According to the permissibility solution to the lottery paradox, the paradox can be solved if we conceive of epistemic justification as a species of permissibility. Clayton Littlejohn has objected that the permissibility solution draws on a sufficient condition for permissible belief that has implausible consequences and that the solution conflicts with our lack of knowledge that a given lottery ticket will lose. The paper defends the permissibility solution against Littlejohn's objections.

Sometimes epistemologists theorize about belief, a tripartite attitude on which one can believe, withhold belief, or disbelieve a proposition. In other cases, epistemologists theorize about credence, a fine-grained attitude that represents one’s subjective probability or confidence level toward a proposition. How do these two attitudes relate to each other? This article explores the relationship between belief and credence in two categories: descriptive and normative. It then explains the broader significance of the belief-credence connection and concludes with (...) general lessons from the debate thus far. (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.

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)

This essay discusses the difficulty to reconcile two paradigms about beliefs: the binary or categorical paradigm of yes/no beliefs and the probabilistic paradigm of degrees of belief. The possibility for someone to hold both types of belief simultaneously is challenged by the lottery paradox, and more recently by a general impossibility theorem by Dietrich and List (2018, 2021). The nature, relevance, and implications of the tension are explained and assessed.

A theory of rational belief should get the cases right. It should also reach its verdicts using the right theoretical assumptions. Leading theories seem to predict the wrong things. With only one exception, they don't accommodate principles that we should use to explain these verdicts. We offer a theory of rational belief that combines an attractive picture of epistemic desirability with plausible principles connecting desirability to rationality. On our view, it's rational to believe when it's sufficiently likely that (...) you'd know because believing when it's sufficiently likely that you'd know minimises expected objective epistemic undesirability. (shrink)

Agents are often assumed to have degrees of belief (“credences”) and also binary beliefs (“beliefs simpliciter”). How are these related to each other? A much-discussed answer asserts that it is rational to believe a proposition if and only if one has a high enough degree of belief in it. But this answer runs into the “lottery paradox”: the set of believed propositions may violate the key rationality conditions of consistency and deductive closure. In earlier work, we showed (...) that this problem generalizes: there exists no local function from degrees of belief to binary beliefs that satisfies some minimal conditions of rationality and non-triviality. “Locality” means that the binary belief in each proposition depends only on the degree of belief in that proposition, not on the degrees of belief in others. One might think that the impossibility can be avoided by dropping the assumption that binary beliefs are a function of degrees of belief. We prove that, even if we drop the “functionality” restriction, there still exists no local relation between degrees of belief and binary beliefs that satisfies some minimal conditions. Thus functionality is not the source of the impossibility; its source is the condition of locality. If there is any non-trivial relation between degrees of belief and binary beliefs at all, it must be a “holistic” one. We explore several concrete forms this “holistic” relation could take. (shrink)

We consider a basic logic with two primitive uni-modal operators: one for certainty and the other for plausibility. The former is assumed to be a normal operator, while the latter is merely a classical operator. We then define belief, interpreted as “maximally plausible possibility”, in terms of these two notions: the agent believes \ if she cannot rule out \ ), she judges \ to be plausible and she does not judge \ to be plausible. We consider four interaction (...) properties between certainty and plausibility and study how these properties translate into properties of belief. We then prove that all the logics considered are minimal logics for the highlighted theorems. We also consider a number of possible interpretations of plausibility, identify the corresponding logics and show that some notions considered in the literature are special cases of our framework. (shrink)

It is one thing to hold that merely statistical evidence is _sometimes_ insufficient for rational belief, as in typical lottery and profiling cases. It is another thing to hold that merely statistical evidence is _always_ insufficient for rational belief. Indeed, there are cases where statistical evidence plainly does justify belief. This project develops a dispositional account of the normativity of statistical evidence, where the dispositions that ground justifying statistical evidence are connected to the goals (= proper (...) function) of objects. There are strong intuitive motivations for doing this. For we can turn almost any case of _non-justifying_ merely statistical evidence into a case of _justifying_ merely statistical evidence by adding information about the dispositions and goals of the objects involved. The resulting view not only helps us understand when and why merely statistical evidence is normatively significant, but it also helps us understand how statistical evidence relates to more standard forms of evidence (perceptual, testimonial). The emerging view also has surprising applications, as it imposes limitations on the epistemic value of fine-tuning arguments for theism as well as undermines a standard class of case-based arguments for moral encroachment. (shrink)

In Knowledge and Lotteries, John Hawthorne offers a diagnosis of our unwillingness to believe, of a given lottery ticket, that it will lose a fair lottery – no matter how many tickets are involved. According to Hawthorne, it is natural to employ parity reasoning when thinking about lottery outcomes: Put roughly, to believe that a given ticket will lose, no matter how likely that is, is to make an arbitrary choice between alternatives that are perfectly balanced given (...) one’s evidence. It’s natural to think that parity reasoning is only applicable to situations involving lotteries dice, spinners etc. – in short, situations in which we are reasoning about the outcomes of a putatively random process. As I shall argue in this paper, however, there are reasons for thinking that parity reasoning can be applied to any proposition that is less than certain given one’s evidence. To see this, we need only remind ourselves of a kind of argument employed by John Pollock and Keith Lehrer in the 1980s. If this argument works, then believing any uncertain proposition, no matter how likely it is, involves a (covert) arbitrary or capricious choice – an idea that contains an obvious sceptical threat. (shrink)

The paper begins by reviewing two problems for fallibilism: the lottery problem, or the problem of explaining why fallible evidence, though otherwise excellent, is not enough to know that one will lose the lottery, and Gettier problems. It is then argued that both problems can be resolved if we note an important illocutionary force of knowledge attributions: namely, that when we attribute knowledge to someone we mean to give the person credit for getting things right. Alternatively, to say (...) that a person knows is to say that she got things right due to her own abilities, efforts and actions, rather than due to dumb luck, or blind chance, or something else. (shrink)

ABSTRACT Drawing inspiration from Fred Dretske, L. S. Carrier, John A. Barker, and Robert Nozick, we develop a tracking analysis of knowing according to which a true belief constitutes knowledge if and only if it is based on reasons that are sensitive to the fact that makes it true, that is, reasons that wouldn’t obtain if the belief weren’t true. We show that our sensitivity analysis handles numerous Gettier-type cases and lottery problems, blocks pathways leading to skepticism, (...) and validates the epistemic closure thesis that correct inferences from known premises yield knowledge of the conclusions. We discuss the plausible views of Ted Warfield and Branden Fitelson regarding cases of knowledge acquired via inference from false premises, and we show how our sensitivity analysis can account for such cases. We present arguments designed to discredit putative counterexamples to sensitivity analyses recently proffered by Tristan Haze, John Williams and Neil Sinhababu, which involve true statements made by untrustworthy informants and strange clocks that sometimes display the correct time while running backwards. Finally, we show that in virtue of employing the paradox-free subjunctive conditionals codified by Relevance Logic theorists instead of the paradox-laden subjunctive conditionals codified by Robert Stalnaker and David Lewis. (shrink)

An influential proposal is that knowledge involves safe belief. A belief is safe, in the relevant sense, just in case it is true in nearby metaphysically possible worlds. In this paper, I introduce a distinct but complementary notion of safety, understood in terms of epistemically possible worlds. The main aim, in doing so, is to add to the epistemologist’s tool-kit. To demonstrate the usefulness of the tool, I use it to advance and assess substantive proposals concerning knowledge and (...) justification. (shrink)

English abstract: This paper discusses the delicate relationship between traditional epistemology and the increasingly influential probabilistic (or ‘Bayesian’) approach to epistemology. The paper introduces some of the key ideas of probabilistic epistemology, including credences or degrees of belief, Bayes’ theorem, conditionalization, and the Dutch Book argument. The tension between traditional and probabilistic epistemology is brought out by considering the lottery and preface paradoxes as they relate to rational (binary) belief and credence respectively. It is then argued that (...) this tension can be alleviated by rejecting the requirement that rational (binary) beliefs must be consistent and closed under logical entailment. Instead, it is suggested that this logical requirement applies to a different type of binary propositional attitude, viz. acceptance. (shrink)

In recent work, Thomas Kroedel has proposed a novel solution to the lottery paradox. As he sees it, we are permitted/justified in believing some lottery propositions, but we are not permitted/justified in believing them all. I criticize this proposal on two fronts. First, I think that if we had the right to add some lottery beliefs to our belief set, we would not have any decisive reason to stop adding more. Suggestions to the contrary run into (...) the wrong kind of reason problem. Reflection on the preface paradox suggests as much. Second, while I agree with Kroedel that permissions do not agglomerate, I do not think that this fact can help us solve the lottery paradox. First, I do not think we have any good reason to think that we’re permitted to believe any lottery propositions. Second, I do not see any good reason to think that epistemic permissions do not agglomerate. (shrink)

I show that the Lottery Paradox is just a version of the Sorites, and argue that this should modify our way of looking at the Paradox itself. In particular, I focus on what I call “the Cut-off Point Problem” and contend that this problem, well known by Sorites scholars, ought to play a key role in the debate on Kyburg’s puzzle. Very briefly, I show that, in the Lottery Paradox, the premises “ticket n°1 will lose”, “ticket n°2 will (...) lose”… “ticket n°1000 will lose” are equivalent to soritical premises of the form “~(the winning ticket is in {…, (tn)}) ⊃ ~(the winning ticket is in {…, tn, (tn + 1)})” (where “⊃” is the material conditional, “~” is the negation symbol, “tn” and “tn + 1” are “ticket n°n” and “ticket n°n + 1” respectively, and “{}” identify the elements of the lottery tickets’ set. The brackets in “(tn)” and “(tn + 1)” are meant to point out that in the antecedent of the conditional we do not always have a “tn” (and, as a result, a “tn + 1” in the consequent): consider the conditional “~(the winning ticket is in {}) ⊃ ~(the winning ticket is in {t1})”). As a result, failing to believe, for some ticket, that it will lose comes down to introducing a cut-off point in a chain of soritical premises. In this paper I explore the consequences of the different ways of blocking the Lottery Paradox with respect to the Cut-off Point Problem. A heap variant of the Lottery Paradox is especially relevant for evaluating the different solutions. One important result is that the most popular way out of the puzzle, i.e., denying the Lockean Thesis, becomes less attractive. Moreover, I show that, along with the debate on whether rational belief is closed under classical logic, the debate on the validity of modus ponens should play an important role in discussions on the Lottery Paradox. -/- . (shrink)

There is much to like about the idea that justification should be understood in terms of normality or normic support (Smith 2016, Goodman and Salow 2018). The view does a nice job explaining why we should think that lottery beliefs differ in justificatory status from mundane perceptual or testimonial beliefs. And it seems to do that in a way that is friendly to a broadly internalist approach to justification. In spite of its attractions, we think that the normic support (...) view faces two serious challenges. The first is that it delivers the wrong result in preface cases. These cases suggest that the view is either too sceptical or too externalist. The second is that the view struggles with certain kinds of Moorean absurdities. It turns out that these problems can easily be avoided. If we think of normality as a condition on *knowledge*, we can characterise justification in terms of its connection to knowledge and thereby avoid the difficulties discussed here. The resulting view does an equally good job explaining why we should think that our perceptual and testimonial beliefs are justified when lottery beliefs cannot be. Thus, it seems that little could be lost and much could be gained by revising the proposal and adopting a view on which it is knowledge, not justification, that depends directly upon normality. (shrink)

In this paper I will compare two competing accounts of assertion: the knowledge account and the justified belief account. When it comes to the evidence that is typically used to assess accounts of assertion – including the evidence from lottery propositions, the evidence from Moore’s paradoxical propositions and the evidence from conversational patterns – I will argue that the justified belief account has at least as much explanatory power as its rival. I will argue, finally, that a (...) close look at the ways in which assertions can be challenged and retracted reveals a certain advantage for the justified belief account. The paper will touch upon a number of further topics along the way, including the logical interaction between knowledge and justified belief, the nature of defeat, and the hypothesis that knowledge and justified belief are normatively coincident goals. (shrink)

To understand something involves some sort of commitment to a set of propositions comprising an account of the understood phenomenon. Some take this commitment to be a species of belief; others, such as Elgin and I, take it to be a kind of cognitive policy. This paper takes a step back from debates about the nature of understanding and asks when this commitment involved in understanding is epistemically appropriate, or ‘acceptable’ in Elgin’s terminology. In particular, appealing to lessons from (...) the lottery and preface paradoxes, it is argued that this type of commitment is sometimes acceptable even when it would be rational to assign arbitrarily low probabilities to the relevant propositions. This strongly suggests that the relevant type of commitment is sometimes acceptable in the absence of epistemic justification for belief, which in turn implies that understanding does not require justification in the traditional sense. The paper goes on to develop a new probabilistic model of acceptability, based on the idea that the maximally informative accounts of the understood phenomenon should be optimally probable. Interestingly, this probabilistic model ends up being similar in important ways to Elgin’s proposal to analyze the acceptability of such commitments in terms of ‘reflective equilibrium’. (shrink)

As the ongoing literature on the paradoxes of the Lottery and the Preface reminds us, the nature of the relation between probability and rational acceptability remains far from settled. This article provides a novel perspective on the matter by exploiting a recently noted structural parallel with the problem of judgment aggregation. After offering a number of general desiderata on the relation between finite probability models and sets of accepted sentences in a Boolean sentential language, it is noted that a (...) number of these constraints will be satisfied if and only if acceptable sentences are true under all valuations in a distinguished non-empty set W. Drawing inspiration from distance-based aggregation procedures, various scoring rule based membership conditions for W are discussed and a possible point of contact with ranking theory is considered. The paper closes with various suggestions for further research. (shrink)

According to an increasingly popular view in epistemology and philosophy of mind, beliefs are sensitive to contextual factors such as practical factors and salient error possibilities. A prominent version of this view, called credal sensitivism, holds that the context-sensitivity of belief is due to the context-sensitivity of degrees of belief or credence. Credal sensitivism comes in two variants: while credence-one sensitivism (COS) holds that maximal confidence (credence one) is necessary for belief, threshold credal sensitivism (TCS) holds that (...) belief consists in having credence above some threshold, where this threshold doesn’t require maximal confidence. In this paper, I argue that COS has difficulties in accounting for three important features about belief: i) the compatibility between believing p and assigning non-zero credence to certain error possibilities that one takes to entail not-p, ii) the fact that outright beliefs can occur in different strengths, and iii) beliefs held by unconscious subjects. I also argue that TCS can easily avoid these problems. Finally, I consider an alleged advantage of COS over TCS in terms of explaining beliefs about lotteries. I argue that lottery cases are rather more problematic for COS than TCS. In conclusion, TCS is the most plausible version of credal sensitivitism. (shrink)

This dissertation is a contribution to formal and computational philosophy. -/- In the first part, we show that by exploiting the parallels between large, yet finite lotteries on the one hand and countably infinite lotteries on the other, we gain insights in the foundations of probability theory as well as in epistemology. Case 1: Infinite lotteries. We discuss how the concept of a fair finite lottery can best be extended to denumerably infinite lotteries. The solution boils down to the (...) introduction of infinitesimal probability values, which can be achieved using non-standard analysis. Our solution can be generalized to uncountable sample spaces, giving rise to a Non-Archimedean Probability (NAP) theory. Case 2: Large but finite lotteries. We propose application of the language of relative analysis (a type of non-standard analysis) to formulate a new model for rational belief, called Stratified Belief. This contextualist model seems well-suited to deal with a concept of beliefs based on probabilities ‘sufficiently close to unity’. -/- The second part presents a case study in social epistemology. We model a group of agents who update their opinions by averaging the opinions of other agents. Our main goal is to calculate the probability for an agent to end up in an inconsistent belief state due to updating. To that end, an analytical expression is given and evaluated numerically, both exactly and using statistical sampling. The probability of ending up in an inconsistent belief state turns out to be always smaller than 2%. (shrink)

According to the traditional analysis of propositional knowledge (which derives from Plato's account in the Meno and Theaetetus), knowledge is justified true belief. This chapter develops the traditional analysis, introduces the famous Gettier and lottery problems, and provides an overview of prospective solutions. In closing, I briefly comment on the value of conceptual analysis, note how it has shaped the field, and assess the state of post-Gettier epistemology.

This entry introduces the Lockean Thesis and sketches the ways in which the lottery paradox, the preface paradox, and the problem of merely statistical evidence can be used to put pressure on the Lockean Thesis.

Epistemologists have proposed various norms of assertion to explain when a speaker is in an epistemic position to assert a proposition. In this article I propose a distinct necessary condition on assertibility: that a speaker should assert only what she sensitively believes, where a subject's belief is sensitive just in case the subject would not hold it if it were false. I argue that the Sensitivity Rule underwrites simple explanations for three key features of assertibility that pose explanatory challenges (...) to other prominent proposals: the fact that assertibility is open under known entailment, the general impropriety of assertions that a lottery ticket has lost made purely on the basis of the speaker's knowledge of the odds, and the fact that assertibility varies widely with features of the conversational context. I close by considering three distinct roles the Sensitivity Rule might play in the overall theory of assertibility. (shrink)

This essay is an accessible introduction to the proof paradox in legal epistemology. -/- In 1902 the Supreme Judicial Court of Maine filed an influential legal verdict. The judge claimed that in order to find a defendant culpable, the plaintiff “must adduce evidence other than a majority of chances”. The judge thereby claimed that bare statistical evidence does not suffice for legal proof. -/- In this essay I first motivate the claim that bare statistical evidence does not suffice for legal (...) proof. I then introduce and motivate a knowledge-centred explanation of this fact. The knowledge-centred explanation rests on two premises. The first is that legal proof requires knowledge of culpability. The second is that one cannot attain knowledge that p from bare statistical evidence that p. To motivate the second premise, I suggest that beliefs based on bare statistical evidence fail to be safe—they could easily be wrong—and bare statistical evidence cannot eliminate relevant alternatives. -/- I then cast doubt on the first premise; I argue that legal proof does not require knowledge. I thereby dispute the knowledge-centred explanation of the inadequacy of bare statistical evidence for legal proof. Instead of appealing to the nature of knowledge, I suggest we should seek a more direct explanation by appealing to those more foundational epistemic properties, such as safety or eliminating relevant alternatives. (shrink)

The Preface Paradox is often discussed for its implications for rational belief. Much less discussed is a variant of the Preface Paradox for knowledge. In this paper, I argue that the most plausible closure-friendly resolution to the Preface Paradox for Knowledge is to say that in any given context, we do not know much. I call this view “Socraticism”. I argue that Socraticism is the most plausible view on two accounts—(1) this view is compatible with the claim that most (...) of our knowledge ascriptions are true, and (2) provided that (1) is true, the costs of accepting Socraticism are much less than the costs of accepting any other resolution to the Paradox. I argue for (1) in Section 2 by developing a question-sensitive contextualist model for knowledge that shows how Socraticism is compatible with the claim that most of our knowledge ascriptions are true. I also argue how this contextualist model can achieve this result where other contextualist models fail. I then consider other closure-friendly solutions to the paradox in Section 3 and show how accepting those solutions forces us to give up a number of plausible epistemic principles. (shrink)

Stalnaker's Thesis about indicative conditionals is, roughly, that the probability one ought to assign to an indicative conditional equals the probability that one ought to assign to its consequent conditional on its antecedent. The thesis seems right. If you draw a card from a standard 52-card deck, how confident are you that the card is a diamond if it's a red card? To answer this, you calculate the proportion of red cards that are diamonds -- that is, you calculate the (...) probability of drawing a diamond conditional on drawing a red card. Skyrms' Thesis about counterfactual conditionals is, roughly, that the probability that one ought to assign to a counterfactual equals one's rational expectation of the chance, at a relevant past time, of its consequent conditional on its antecedent. This thesis also seems right. If you decide not to enter a 100-ticket lottery, how confident are you that you would have won had you bought a ticket? To answer this, you calculate the prior chance--that is, the chance just before your decision not to buy a ticket---of winning conditional on entering the lottery. The central project of this article is to develop a new uniform theory of conditionals that allows us to derive a version of Skyrms' Thesis from a version of Stalnaker's Thesis, together with a chance-deference norm relating rational credence to beliefs about objective chance. (shrink)

How does evidence figure into the reasoning of an agent? This is an intricate philosophical problem but also one we all encounter in our daily lives. In this chapter, we identify the problem and outline a possible solution to it. The problem arises, because the fact that it is up to us whether we do something makes a difference to how we should think of the evidence concerning whether we will actually do it. Otherwise we regard something that is up (...) to us as if it were not: We regard something that is up to us as if it were the outcome of a lottery. Nonetheless, we would be wrong to ignore the evidence. Otherwise we could not make a rational decision. In this chapter, we first show that a decision-theoretic approach to this problem cannot succeed. This approach does not explain how we can take evidence into account in practical reasoning without making predictions concerning matters that are up to us. It also gives rise to incoherence between our practical and our theoretical conclusions. We then argue that what is required to solve the problem is recognizing that beliefs about matters that are up to us can be grounded in practical reasoning. We argue that such beliefs are not subject to an evidential norm, because they are not meant to reflect a reality that is independent of them but instead are meant to bring about the reality they represent. Finally, we argue that, even if we are fully rational agents, we will sometimes lack practical knowledge of what we will do. That is because when it is practically rational to do something that will require resolve, we may be in a position to rationally conclude that we will do it, even though we have evidence that there is a non-negligible chance that we won’t. In such cases, our evidence serves as a defeater for our practical knowledge. (shrink)

This paper is about the alethic aspect of epistemic rationality. The most common approaches to this aspect are either normative (what a reasoner ought to/may believe?) or evaluative (how rational is a reasoner?), where the evaluative approaches are usually comparative (one reasoner is assessed compared to another). These approaches often present problems with blindspots. For example, ought a reasoner to believe a currently true blindspot? Is she permitted to? Consequently, these approaches often fail in describing a situation of alethic maximality, (...) where a reasoner fulfills all the alethic norms and could be used as a standard of rationality (as they are, in fact, used in some of these approaches). I propose a function α, which accepts a set of beliefs as inputand returns a numeric alethic value. Then I use this function to define a notion of alethic maximality that is satisfiable by finite reasoners (reasoners with cognitive limitations) and does not present problems with blindspots. Function α may also be used in alethic norms and evaluation methods (comparative and non-comparative) that may be applied to finite reasoners and do not present problems with blindspots. A result of this investigation isthat the project of providing purely alethic norms is defective. The use of function α also sheds light on important epistemological issues, such as the lottery and the preface paradoxes, and the principles of clutter avoidance and reflection. (shrink)

A number of prominent epistemologists claim that the principle of sensitivity “play[s] a starring role in the solution to some important epistemological problems”. I argue that traditional sensitivity accounts fail to explain even the most basic data that are usually considered to constitute their primary motivation. To establish this result I develop Gettier and lottery cases involving necessary truths. Since beliefs in necessary truths are sensitive by default, the resulting cases give rise to a serious explanatory problem for the (...) defenders of sensitivity accounts. It is furthermore argued that attempts to modally strengthen traditional sensitivity accounts to avoid the problem must appeal to a notion of safety—the primary competitor of sensitivity in the literature. The paper concludes that the explanatory virtues of sensitivity accounts are largely illusory. In the framework of modal epistemology, it is safety rather than sensitivity that does the heavy explanatory lifting with respect to Gettier cases, lottery examples, and other pertinent cases. (shrink)

This paper defends a new norm of assertion: Assert that p only if you are in a position to know that p. We test the norm by judging its performance in explaining three phenomena that appear jointly inexplicable at first: Moorean paradoxes, lottery propositions, and selfless assertions. The norm succeeds by tethering unassertability to unknowability while untethering belief from assertion. The PtK‐norm foregrounds the public nature of assertion as a practice that can be other‐regarding, allowing asserters to act (...) in the best interests of their audience when psychological pressures would otherwise prevent them from communicating the knowable truth. (shrink)

Statistical evidence—say, that 95% of your co-workers badmouth each other—can never render resenting your colleague appropriate, in the way that other evidence (say, the testimony of a reliable friend) can. The problem of statistical resentment is to explain why. We put the problem of statistical resentment in several wider contexts: The context of the problem of statistical evidence in legal theory; the epistemological context—with problems like the lottery paradox for knowledge, epistemic impurism and doxastic wrongdoing; and the context of (...) a wider set of examples of responses and attitudes that seem not to be appropriately groundable in statistical evidence. Regrettably, we do not come up with a fully general, fully adequate, fully unified account of all the phenomena discussed. But we give reasons to believe that no such account is forthcoming, and we sketch a somewhat messier account that may be the best that can be had here. (shrink)

[Significantly updated in Chapter 6 of Awareness and the Substructure of Knowledge] In order for a reason to justify an action or attitude it must be one that is possessed by an agent. Knowledge-centric views of possession ground our possession of reasons, at least partially, either in our knowledge of them or in our being in a position to know them. On virtually all accounts, knowing P is some kind of non-accidental true belief that P. This entails that knowing (...) P is a kind of non-accidental true representation that P. I outline a novel theory of the epistemic requirement on possession in terms of this more general state of non-accidental true representation. It is just as well placed to explain the motivations behind knowledge-centric views of possession, and it is also better placed to explain the extent of the reasons we possess in certain cases of deductive belief-updates and cases involving environmental luck. I conclude with three reflections. First, I indicate how my arguments generate a dilemma for Errol Lord’s view that possessing reasons is just a matter of being in a position to manifest one’s knowledge how to use them. Second, I explain how my view can simultaneously manage cases of environmental luck without falling prey to lottery cases. Finally, I sketch the direction for a further range of counterexamples to knowledge-centric theories of possession. (shrink)

Socrates' attitude towards falsehood is quite puzzling in the Republic. Although Socrates is clearly committed to truth, at several points he discusses the benefits of falsehood. This occurs most notably in Book 3 with the "noble lie" (414d-415c) and most disturbingly in Book 5 with the "rigged sexual lottery" (459d-460c). This raises the question: What kinds of falsehoods does Socrates think are beneficial, and what kinds of falsehoods does he think are harmful? And more broadly: What can this tell (...) us about the relationship between ethics and epistemology? The key to answering these questions lies in an obscure and paradoxical passage in Book II; at 382a-d Socrates distinguishes between "true falsehoods" and "impure lies." True falsehoods are always bad, but impure lies are sometimes beneficial. Despite Socrates' insistence that he is not saying anything deep, his distinction is far from straightforward. Nevertheless, in order to determine why some falsehoods are beneficial and why some are always harmful, we must understand what exactly true falsehoods are and how they differ from impure lies. In this paper, I argue that true falsehoods are a restricted class of false beliefs about ethics; they are false beliefs about how one should live and what one should pursue. I refer to these beliefs as "normative commitments." False normative commitments are always pernicious because they create and sustain psychological disharmony. Unlike true falsehoods, impure lies can be about anything. Nevertheless, they are only beneficial when they help produce and sustain true normative commitments. I argue that the upshot of this is that practical concerns have a kind of primacy over theoretical concerns. (shrink)

When faced with multiple claims to a particular good, what does distributive justice require? To answer this question, we need a substantive moral theory that will enable us assign relative moral weights to the parties' claims. But this is not all we need. Once we have assessed the moral weight of each party's claim, we still need to decide what method of distribution to employ, for there are two methods open to us. We could take the winner-take-all approach, and award (...) the good to the party with the strongest claim. On the other hand, we could divide the good proportionally, according to the relative strength of each party's claim. Because the choice between these two methods of distribution can have a dramatic impact on the resulting pattern of distribution, the choice presents a question of justice. But this is a question of justice that is often overlooked. As a result, we currently employ the principle of proportionality far less often than justice actually requires. If we focus on the question of distributive method, however, we are not only better able to understand how certain reasons enter into our all-things-considered moral judgments, we are also able to explain some perplexing but common aspects of our moral beliefs: how rights can be said to have peremptory force, yet still be balanced against other important interests; how justice can sometimes require compromise, yet sometimes require victory; and how a moral theory can avoid being too demanding while still being demanding enough. Key Words: autonomy • equality • indivisibility • inviolability • weighted lotteries • rights. (shrink)

We examine two leading theories of rational belief, the Lockean view and the explanationist view. The first is appealing because it fits with some independently plausible claims about the ways that rational persons pursue their aims. The second is appealing because it seems to account for intuitions that cause trouble for the Lockean view. While fitting the intuitive data is desirable, we are troubled that the explanationist view seems to clash with our theoretical beliefs about what rationality must be (...) like. We think that upon further examination, the intuitive appeal of the explanationist view starts to diminish. We also think that these further intuitions that spell trouble for the explanationist spell trouble for any theory that is not expectationist. We propose a novel expectationist theory of rational belief that improves upon the Lockean and the explanationist views. We think that recent defences of the Lockean view contain an important insight. A substantive theory of rational response should be based on a suitable theory of prizes and a suitable theory of how we should pursue prizes in the face of uncertainty. Most theories of rational belief typically take for granted a truth-centred picture of epistemic prizes (e.g., that epistemic desirability and undesirability can be fully understood in terms of accuracy) and then differ in terms of how they recommend pursuing prizes so understood. We think the Lockeans embrace plausible principles of how prizes should be pursued. We trace the difficulties that this view faces to veritistic assumptions about prizes. We suggest that some prizes are epistemically loaded in that a complete description of the prize will itself make reference to our epistemic states or standards. We argue that knowledge matters to rational belief and choice because in the epistemic domain, knowledge is the prize. We see this in practical domains, too. In some choice settings, what's desired is desired, in part, because it involves a kind of connection to reality only knowledge provides. (shrink)

In this dissertation I present a new solution to the renowned Gettier problem. My solution, which in a sense represents a defense of a rather traditional epistemological approach, is based upon a distinction between primary and secondary beliefs. I argue that primary beliefs are known if justified and true, whereas secondary beliefs are known if they are believed on the basis of a known primary belief. Much emphasis is put upon defending this approach against potential objections, but I also (...) draw some epistemological and semantical consequences pertaining to such issues as the nature of epistemological justification, the Lottery Paradox, philosophical skepticism and the semantics of belief contexts. (shrink)

Abstract: Lotteries have long been a source of fascination and intrigue, offering the tantalizing prospect of unexpected fortunes. In this research paper, we delve into the world of lottery predictions, employing cutting-edge AI techniques to unlock the secrets of lottery outcomes. Our dataset, obtained from Kaggle, comprises historical lottery draws, and our goal is to develop predictive models that can anticipate future winning numbers. This study explores the use of deep learning and time series analysis to achieve (...) this elusive feat. Through rigorous experimentation and data-driven approaches, we seek to determine the viability of AI in the realm of lottery predictions. Our findings reveal both the promise and limitations of AI in this context, shedding light on the complexities of lottery data and the potential need for quantum computing as a last resort. (shrink)

To resolve the lottery paradox, the “no-justification account” proposes that one is not justified in believing that one's lottery ticket is a loser. The no-justification account commits to what I call “the Harman-style skepticism”. In reply, proponents of the no-justification account typically downplay the Harman-style skepticism. In this paper, I argue that the no-justification reply to the Harman-style skepticism is untenable. Moreover, I argue that the no-justification account is epistemically ad hoc. My arguments are based on a rather (...) surprising finding that the no-justification account implies that people living in Taiwan typically suffer from the Harman-style skepticism. (shrink)