Au xviiie siècle, la figure insistante de la «femme philosophe» s'articule à un imaginaire ambivalent de la différence des sexes, entre hantise d'une confusion délétère et quête d'un modèle d'harmonie. La femme travestit-elle la philosophie? Les Lumières ont-elles un genre?

The lottery paradox can be solved if epistemic justification is assumed to be a species of permissibility. Given this assumption, the starting point of the paradox can be formulated as the claim that, for each lottery ticket, I am permitted to believe that it will lose. This claim is ambiguous between two readings, depending on the scope of ‘permitted’. On one reading, the claim is false; on another, it is true, but, owing to the general failure of permissibility (...) to agglomerate, does not generate the paradox. The solution generalizes to formulations of the paradox in terms of rational acceptability and doxastic rationality. (shrink)

The lottery paradox plays an important role in arguments for various norms of assertion. Why is it that, prior to information on the results of a draw, assertions such as, “My ticket lost,” seem inappropriate? This paper is composed of two projects. First, I articulate a number of problems arising from Timothy Williamson’s analysis of the lottery paradox. Second, I propose a relevant alternatives theory, which I call the Non-Destabilizing Alternatives Theory , that better explains the pathology of (...) asserting lottery propositions, while permitting assertions of what I call fallible propositions such as, “My car is in the driveway.” Le paradoxe de la loterie joue un rôle important dans l’argumentation visant à défendre diverses normes de l’assertion. Comment se fait-il que, avant que les résultats d’un tirage soient connus, des assertions comme «Mon billet a perdu» semblent inappropriées? Cet article se compose de deux projets. Premièrement, je relève certains problèmes issus de l’analyse du paradoxe de la loterie par Timothy Williamson. Deuxièmement, je propose une théorie des alternatives pertinentes que j’appelle la «théorie des alternatives non-déstabilisantes» , et qui explique d’une meilleure façon la pathologie de l’assertion de propositions concernant la loterie, tout en permettant des assertions faillibles, telles que «Ma voiture est dans l’entrée». (shrink)

The lottery paradox shows seemingly plausible principles of rational acceptance to be incompatible. It has been argued that we shouldn’t be concerned by this clash, since the concept of (categorical) belief is otiose, to be supplanted by a quantitative notion of partial belief, in terms of which the paradox cannot even be formulated. I reject this eliminativist view of belief, arguing that the ordinary concept of (categorical) belief has a useful function which the quantitative notion does not serve. I (...) then propose a solution to the paradox it engenders. (shrink)

The lottery paradox shows that the following three individually highly plausible theses are jointly incompatible: (i) highly probable propositions are justifiably believable, (ii) justified believability is closed under conjunction introduction, (iii) known contradictions are not justifiably believable. This paper argues that a satisfactory solution to the lottery paradox must reject (i) as versions of the paradox can be generated without appeal to either (ii) or (iii) and proposes a new solution to the paradox in terms of a novel (...) account of justified believability. (shrink)

Safety accounts of knowledge intend to explain why certain true and intuitively justified beliefs fail to be knowledge in terms of such beliefs falling prey to a modal veritic type of luck. In particular, they explain why true and intuitively justified beliefs in “lottery propositions” (highly likely propositions reporting that a particular statistical outcome obtains) are not knowledge. In this paper, I argue that there is a type of case involving lottery propositions that inevitably lies beyond the scope (...) of any reasonable safety account of epistemic luck. I offer counterexamples to accounts of epistemic luck in terms of safety conditions that involve both “locally” and “globally” reliable ways of forming beliefs in nearby worlds. All such counterexamples present a lottery case illustrating the next possibility: the process of selecting the lottery winner might be such that any world in which it delivers a different outcome is extremely far away from the actual world. In addition to being a case of safe ignorance, this type of lottery case shows that, ultimately, either veritic epistemic luck is not unsafe true belief or beliefs in lottery propositions are not epistemically luckily true. (shrink)

How should we make distributive decisions when there is not enough of the good to go around, or at least not enough of it right now? What does fairness require in such cases? In what follows, we distinguish between cases of scarcity and bottleneck cases, and we argue that both arguments for lotteries and arguments for queues have merit, albeit for different distributive scenarios. When dealing with scarcity not everyone can get the good. A secondary good that can be distributed (...) fairly is the chances of obtaining the good. In cases of scarcity, lotteries are the best way of allocating chances of obtaining the good fairly. When dealing with bottlenecks, the secondary good that can and ought to be distributed fairly is waiting time. Queues are best suited to distribute the good of waiting time fairly. (shrink)

In a recent article, Douven and Williamson offer both (i) a rebuttal of various recent suggested sufficient conditions for rational acceptability and (ii) an alleged ‘generalization’ of this rebuttal, which, they claim, tells against a much broader class of potential suggestions. However, not only is the result mentioned in (ii) not a generalization of the findings referred to in (i), but in contrast to the latter, it fails to have the probative force advertised. Their paper does however, if unwittingly, bring (...) us a step closer to a precise characterization of an important class of rationally unacceptable propositions—the class of lottery propositions for equiprobable lotteries. This helps pave the way to the construction of a genuinely lottery-paradox-proof alternative to the suggestions criticized in (i). (shrink)

Suppose that I hold a ticket in a fair lottery and that I believe that my ticket will lose [L] on the basis of its extremely high probability of losing. What is the appropriate epistemic appraisal of me and my belief that L? Am I justified in believing that L? Do I know that L? While there is disagreement among epistemologists over whether or not I am justified in believing that L, there is widespread agreement that I do not (...) know that L. I defend the two-pronged view that I am justified in believing that my ticket will lose and that I know that it will lose. Along the way, I discuss four different but related versions of the lottery paradox—The Paradox for Rationality, The Paradox for Knowledge, The Paradox for Fallibilism, and The Paradox for Epistemic Closure—and offer a unified resolution of each of them. (shrink)

In this paper, we present the results of two surveys that investigate subjects’ judgments about what can be known or justifiably believed about lottery outcomes on the basis of statistical evidence, testimonial evidence, and “mixed” evidence, while considering possible anchoring and priming effects. We discuss these results in light of seven distinct hypotheses that capture various claims made by philosophers about lay people’s lottery judgments. We conclude by summarizing the main findings, pointing to future research, and comparing our (...) findings to recent studies by Turri and Friedman. (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)

The allocation of vaccines and therapeutics for Covid‐19 obviously raises ethical questions, and physicians and ethicists have begun to address them. Writers have identified various criteria that should guide allocation decisions, but the criteria often conflict and need to be balanced against one another. This article proposes a model for thinking about how different considerations that are relevant to the distribution of vaccines and scarce treatments for Covid‐19 could be integrated into an allocation procedure. The model employs the construct of (...) a weighted lottery, which is a construct that has been employed in other contexts that involve the distribution of scarce resources. The article highlights the advantages of applying a weighted lottery to the Covid‐19 context and offers an illustration for how it might work in practice. The primary aim of the article is to articulate the structural features of a weighted lottery for this context and to bring out its advantages over other methods for allocating Covid‐19 medications. (shrink)

Survival lotteries play a considerable role in philosophers’ thinking on the ethics of killing. They have also entered popular culture. The lotteries are not widely accepted, yet they seem to be reasonable, if only we can set some of our taboos to one side. In this chapter, deontology, the moral rights theory, and utilitarianism are confronted with famous ‘survival lotteries’. Here this chapter examines cases such as John Harris’s organ lottery, John Taurek’s rescue lottery, and Gilbert Harman’s ‘organ’ (...) thought experiments. In the final analysis, it is argued that utilitarianism can best explain our considered intuitions in relation to these lotteries. It is possible to debunk intuitions commonly held to the opposite effect, once they are submitted to cognitive psychotherapy. (shrink)

Jim buys a ticket in a million-ticket lottery. He knows it is a fair lottery, but, given the odds, he believes he will lose. When the winning ticket is chosen, it is not his. Did he know his ticket would lose? It seems that he did not. After all, if he knew his ticket would lose, why would he have bought it? Further, if he knew his ticket would lose, then, given that his ticket is no different in (...) its chances of winning from any other ticket, it seems that by parity of reasoning he should also know that every other ticket would lose. But of course he doesn’t know that; in fact, he knows that not every ticket will lose. (shrink)

The lottery preparation, a new general kind of Laver preparation, works uniformly with supercompact cardinals, strongly compact cardinals, strong cardinals, measurable cardinals, or what have you. And like the Laver preparation, the lottery preparation makes these cardinals indestructible by various kinds of further forcing. A supercompact cardinal κ, for example, becomes fully indestructible by <κ-directed closed forcing; a strong cardinal κ becomes indestructible by κ-strategically closed forcing; and a strongly compact cardinal κ becomes indestructible by, among others, the (...) forcing to add a Cohen subset to κ, the forcing to shoot a club Cκ avoiding the measurable cardinals and the forcing to add various long Prikry sequences. The lottery preparation works best when performed after fast function forcing, which adds a new completely general kind of Laver function for any large cardinal, thereby freeing the Laver function concept from the supercompact cardinal context. (shrink)

A popular way to relate probabilistic information to binary rational beliefs is the Lockean Thesis, which is usually formalized in terms of thresholds. This approach seems far from satisfactory: the value of the thresholds is not well-specified and the Lottery Paradox shows that the model violates the Conjunction Principle. We argue that the Lottery Paradox is a symptom of a more fundamental and general problem, shared by all threshold-models that attempt to put an exact border on something that (...) is intrinsically vague. 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’ and satisfies a moderately weakened form of the Conjunction Principle. We also propose an adaptation of the model that is able to deal with beliefs that are less firm than ‘almost certainty’. The adapted version is also of interest for the epistemicist account of vagueness. (shrink)

We talk and think about our beliefs both in a categorical and in a graded way. How do the two kinds of belief hang together? The most straightforward answer is that we believe something categorically if we believe it to a high enough degree. But this seemingly obvious, near-platitudinous claim is known to give rise to a paradox commonly known as the 'lottery paradox' – at least when it is coupled with some further seeming near-platitudes about belief. How to (...) resolve that paradox has been a matter of intense philosophical debate for over fifty years. This volume offers a collection of newly commissioned essays on the subject, all of which provide compelling reasons for rethinking many of the fundamentals of the debate. (shrink)

Intuition suggests that a fair lottery is the appropriate way to allocate a scarce good when two or more people have equally strong claims to it. This article lays out three conditions that any conception of justice compatible with this intuition must satisfy — efficiency of outcomes, fairness of outcomes, and fairness of treatment. The third, unlike the first two, manifests itself only in the intentions of the allocative agent, not in the final allocation itself. For this reason, while (...) justice generally requires publicity — requires, that is, that the justice of public practices be as visible as possible — for fairness of treatment publicity is indispensable. This fact has implications for defining a fair lottery. On most accounts, fair lotteries must be equiprobable. But while some theories of probability facilitate the connection between the equiprobability of fair lotteries and the contribution they can make to justice, others do not. (shrink)

One result of this note is about the nonconstructivity of countably infinite lotteries: even if we impose very weak conditions on the assignment of probabilities to subsets of natural numbers we cannot prove the existence of such assignments constructively, i.e., without something such as the axiom of choice. This is a corollary to a more general theorem about large-small filters, a concept that extends the concept of free ultrafilters. The main theorem is that proving the existence of large-small filters requires (...) a nonconstructive axiom like AC. (shrink)

This paper defends the permissibility solution to the lottery paradox against an objection by Anna-Maria Asunta Eder. Eder argues that the permissibility solution should also be applicable to the preface paradox, but conflicts with a plausible principle about epistemic permissions when so applied. This paper replies by first criticizing Eder’s considerations in defense of her principle; in particular, it argues that the plausibility of her principle is to a large extent parasitic on the spurious plausibility of the principle of (...) factual detachment. The paper then presents a direct argument against Eder’s principle, which shows that her principle conflicts with a Pareto-style condition on permissible belief. (shrink)

There are many ordinary propositions we think we know. Almost every ordinary proposition entails some "lottery proposition" which we think we do not know but to which we assign a high probability of being true (for instance: “I will never be a multi-millionaire” entails “I will not win this lottery”). How is this possible - given that some closure principle is true? This problem, also known as “the Lottery puzzle”, has recently provoked a lot of discussion. In (...) this paper I discuss one of the most promising answers to the problem: Stewart Cohen’s contextualist solution which is based on ideas about the salience of chances of error. After presenting some objections to it I sketch an alternative solution which is still contextualist in spirit. (shrink)

According to the safety account of knowledge, one knows that p only if one’s belief could not easily have been false. The account is open to two different readings which, in turn, give rise to a weak version and a strong version of the safety condition. In this paper, it is argued that, if we opt for the weak version, then we are not able to account for why one’s belief in a lottery proposition based on statistical evidence is (...) true as a matter of luck. But, if we opt for the strong version, then we are not able to accommodate some cases of perceptual knowledge. Therefore, the safety account of knowledge is undermined jointly by the lottery case and perceptual knowledge. (shrink)

Lin and Kelly :957–981, 2012) and Leitgeb :1338–1389, 2013, Philos Rev 123:131–171, 2014), offer similar solutions to the Lottery Paradox, defining acceptance rules which determine a rational agent’s beliefs in terms of broader features of her credal state than just her isolated credences in individual propositions. I express each proposal as a method for obtaining an ordering over a partition from a credence function, and then a belief set from the ordering. Although these proposals avoid the original Lottery (...) Paradox, I raise a diachronic case which illustrates that neither satisfies both Lin and Kelly’s constraint that the update on orderings track the update on credence functions, and the intuitive constraint that credence of at least 0.5 is necessary for rational belief. I conclude by suggesting that we reformulate these proposals in terms of orderings over entire algebras based on partitions rather than orderings just over the partitions themselves. Reformulating both rules in this way yields acceptance rules which avoid the Lottery Paradox while satisfying both the tracking and likeliness constraints. (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)

In order to accommodate empirically observed violations of the independence axiom of expected utility theory Becker and Sarin (1987) proposed their model of lottery dependent utility in which the utility of an outcome may depend on the lottery being evaluated. Although this dependence is intuitively very appealing and provides a simple functional form of the resulting decision criterion, lottery dependent utility has been nearly completely neglected in the recent literature on decision making under risk. The goal of (...) this paper is to revive the lottery dependent utility model. Therefore, we derive first a sound axiomatic foundation of lottery dependent utility. Secondly, we develop a discontinuous variant of the model which can accommodate boundary effects and may lead to a lexicographic non-expected utility model. Both analyses are accompanied by considering some functional specifications which are in accordance with recent experimental results and may have significant applications in business and management science. (shrink)

We analyze the results from three different risk attitude elicitation methods. First, the broadly used test by Holt and Laury, HL, second, the lottery-panel task by Sabater-Grande and Georgantzis, SG, and third, responses to a survey question on self-assessment of general attitude towards risk. The first and the second task are implemented with real monetary incentives, while the third concerns all domains in life in general. Like in previous studies, the correlation of decisions across tasks is low and usually (...) statistically non-significant. However, when we consider only subjects whose behavior across the panels of the SG task is compatible with constant relative risk aversion, the correlation between HL and self-assessed risk attitude becomes significant. Furthermore, the correlation between HL and SG also increases for CRRA-compatible subjects, although it remains statistically non-significant. (shrink)

There is a wide consensus among epistemologists that we fail to know lottery propositions (that is, highly likely propositions solely supported by statistical evidence), but there is no agreed-upon explanation of why we fail to know them. Yet, paradoxes surrounding lotteries continue pressing on the need to identify the correct explanation. This dissertation evaluates various explanations of why we do not know lottery propositions, which normally take one of two forms. The first argues that lottery beliefs are (...) true by mere epistemic luck (roughly, luck, chance, or accident in possessing a true belief in a proposition p, which prevents us from knowing p), and provides an account of epistemic luck that entails that lottery beliefs are epistemically luckily true. Here I evaluate and reject an explanation that relies on the largely dominant account of epistemically lucky beliefs as “unsafe true beliefs”. The second argues that lottery beliefs are unjustified (where being justified in believing p is necessary for knowing p), and provides an account of justification with a condition that lottery beliefs or propositions do not satisfy. I divide conditions on justification into “probabilistic” and “non-probabilistic” and argue, contrary to Douven and Williamson (2016), that justification can have a probabilistic condition that lottery beliefs fail to satisfy, thus restoring such accounts as a theoretical possibility to solve the “lottery paradox”. I also argue that among the dominant candidates of non-probabilistic conditions, Smith’s (2016) normic support condition on justification is the only suitable one to explain why we do not know lottery propositions. Alternatively, if lottery propositions are justified, there is nonetheless a solution to the lottery paradox compatible with their possessing such epistemic status, which consists in rejecting the “aggregativity of justification”. I argue that this principle is incorrect, and that preserving it obstructs solving what I identify as the most basic form of the lottery paradox. Finally, I address the legal correlates of lottery propositions –i.e., litigated claims supported by bare statistical evidence. I divide explanations of why such claims do not meet a given standard of proof into two types: externalist and internalist. I present independent problems for two externalist explanations –Pritchard’s (2018, 2022) modal explanation and Blome-Tillmann’s (2017) knowledge-first explanation, and argue that they face such problems in virtue of being externalist. This finding motivates the need to evaluate externalist accounts qua externalist to determine if an externalist account is a viable solution to the “proof paradox”. (shrink)

One of the problems that Bayesian regularity, the thesis that all contingent propositions should be given probabilities strictly between zero and one, faces is the possibility of random processes that randomly and uniformly choose a number between zero and one. According to classical probability theory, the probability that such a process picks a particular number in the range is zero, but of course any number in the range can indeed be picked. There is a solution to this particular problem on (...) the books: a measure that assigns the same infinitesimal probability to each number between zero and one. I will show that such a measure, while mathematically interesting, is pathological for use in confirmation theory, for the same reason that a measure that assigns an infinitesimal probability to each possible outcome in a countably infinite lottery is pathological. The pathology is that one can force someone to assign a probability within an infinitesimal of one to an unlikely event. (shrink)

This paper addresses an argument offered by John Hawthorne gainst the propriety of an agent’s using propositions she does not know as premises in practical reasoning. I will argue that there are a number of potential structural confounds in Hawthorne’s use of his main example, a case of practical reasoning about a lottery. By drawing these confounds out more explicitly, we can get a better sense of how to make appropriate use of such examples in theorizing about norms, knowledge, (...) and practical reasoning. I will conclude by suggesting a prescription for properly using lottery propositions to do the sort of work that Hawthorne wants from them. (shrink)

Many have the intuition that the right response to the Lottery Paradox is to deny that one can justifiably believe of even a single lottery ticket that it will lose. The paper shows that from any theory of justification that solves the paradox in accordance with this intuition, a theory not of that kind can be derived that also solves the paradox but is more conducive to our epistemic goal than the former. It is argued that currently there (...) is no valid reason not to give preference to the derived accounts over the accounts from which they come. (shrink)

ABSTRACT In 1975 John Harris envisaged a survival lottery to redistribute organs from one to a greater number in order to reduce number of deaths as a consequence of organ failure. In this paper I reach a conclusion about when running a survival lottery is permissible by looking at the reason prospective participants have for allowing the procedure from a contractual perspective. I identify three versions of the survival lottery. In a National Lottery, everyone within a (...) jurisdiction is a candidate for being a donor for everyone else, disregarding all differences between individuals' eventual possibility of needing an organ. In a Group Specific Lottery, it is a question of running a lottery among members of a specific group who share the same probability of getting organ failure. In a Local Lottery one randomises among individuals who are already in need of a new organ but who happen to be compatible and in need of different organs. While the first is vulnerable to considerations of fairness, it is difficult to perceive a feasible way to implement the second option that does not come with a host of unwelcome consequences. I argue, however, that it is permissible to run Local Lotteries. (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)

Thomas Kroedel argues that we can solve a version of the lottery paradox if we identify justified beliefs with permissible beliefs. Since permissions do not agglomerate, we might grant that someone could justifiably believe any ticket in a large and fair lottery is a loser without being permitted to believe that all the tickets will lose. I shall argue that Kroedel’s solution fails. While permissions do not agglomerate, we would have too many permissions if we characterized justified belief (...) as sufficiently probable belief. If we reject the idea that justified beliefs can be characterized as sufficiently probably beliefs, Kroedel’s solution is otiose because the paradox can be dissolved at the outset. (shrink)

It can often be heard in the hallways, and occasionally read in print, that reliabilism runs into special trouble regarding lottery cases. My main aim in this paper is to argue that this is not so. Nevertheless, lottery cases do force us to pay close attention to the relation between justification and probability.

Faced with a choice between saving one stranger and saving a group of strangers, some people endorse weighted lotteries, which give a strictly greater chance of being saved to the group of strangers than the single stranger. In this paper I attempt to criticize this view. I first consider a particular version of the weighted lotteries, Frances Kamm's procedure of proportional chances, and point out two implausible implications of her proposal. Then, I consider weighted lotteries in general, and claim (1) (...) that the correct thing to distribute is not the chance of being saved but the good of being saved, (2) that assigning some chance to the single stranger is not the only way to give a positive (and equal) respect to the people concerned, and (3) that the weighted lottery appears to be deceptive since it would show the respect to the single stranger in a negligible way. (shrink)

The thesis that high probability suffices for rational belief, while initially plausible, is known to face the Lottery Paradox. The present paper proposes an amended version of that thesis which escapes the Lottery Paradox. The amendment is argued to be plausible on independent grounds.

The consensus is that a normal person, regarding a normal fair lottery, under ordinary, realistic conditions, can’t know ahead of time that a given ticket is a loser. I disagree, sort of. Suppose you learn that a lottery involves extreme probabilities. You also hear, perhaps via the lottery’s website, that someone who bought just one ticket got the winning number. In response to the acquisition of those two pieces of knowledge, you come to think that either the (...) announcement is false (no one who bought a single ticket won), or it's true but either something superhuman was going on or the lottery was fixed. I say: if your belief is true, then it amounts to knowledge. That is, you can know that the lottery’s outcome won’t be an ordinary case of someone with one ticket winning a fair lottery, with no superhuman assistance. -/- . (shrink)

Weighted lottery proposals give guidance in rescue dilemma situations by balancing the demands of comparative and absolute fairness. While they do not advocate for saving the greater number outright, they are responsive to absolute fairness insofar as they show a certain sensitivity to the numbers involved. In this paper we investigate what criterion of absolute fairness we should demand weighted lotteries to fulfill. We do so by way of critically examining what is probably the most sophisticated weighted lottery (...) on the market: Gerard Vong's Exclusive Composition-Sensitive (EXCS) lottery. We find that both the standard that seems most common in the debate, and a different standard Vong uses to criticize Jens Timmermann's Individualist Lottery are in contradiction to basic demands placed upon weighted lotteries and are therefore unsuitable as necessary conditions for absolute fairness. We instead propose a purely gradual understanding of absolute fairness. (shrink)

As an application of his Material Theory of Induction, Norton (2018; manuscript) argues that the correct inductive logic for a fair infinite lottery, and also for evaluating eternal inflation multiverse models, is radically different from standard probability theory. This is due to a requirement of label independence. It follows, Norton argues, that finite additivity fails, and any two sets of outcomes with the same cardinality and co-cardinality have the same chance. This makes the logic useless for evaluating multiverse models (...) based on self-locating chances, so Norton claims that we should despair of such attempts. However, his negative results depend on a certain reification of chance, consisting in the treatment of inductive support as the value of a function, a value not itself affected by relabeling. Here we define a purely comparative infinite lottery logic, where there are no primitive chances but only a relation of ‘at most as likely’ and its derivatives. This logic satisfies both label independence and a comparative version of additivity as well as several other desirable properties, and it draws finer distinctions between events than Norton's. Consequently, it yields better advice about choosing between sets of lottery tickets than Norton's, but it does not appear to be any more helpful for evaluating multiverse models. Hence, the limitations of Norton's logic are not entirely due to the failure of additivity, nor to the fact that all infinite, co-infinite sets of outcomes have the same chance, but to a more fundamental problem: We have no well-motivated way of comparing disjoint countably infinite sets. (shrink)

We reflect on lessons that the lottery and preface paradoxes provide for the logic of uncertain inference. One of these lessons is the unreliability of the rule of conjunction of conclusions in such contexts, whether the inferences are probabilistic or qualitative; this leads us to an examination of consequence relations without that rule, the study of other rules that may nevertheless be satisfied in its absence, and a partial rehabilitation of conjunction as a ‘lossy’ rule. A second lesson is (...) the possibility of rational inconsistent belief; this leads us to formulate criteria for deciding when an inconsistent set of beliefs may reasonably be retained. (shrink)

As John Rawls makes clear in A Theory of Justice, there is a popular and influential strand of political thought for which brute luck – that is, being lucky in the so-called “lottery of life” – ought to have no place in a theory of distributive justice. Yet the debate about luck, desert, and fairness in contemporary political philosophy has recently been rekindled by a handful of philosophers who claim that desert should play a bigger role in theories of (...) distributive justice. In the present paper, we present the results of our attempts to fill in some of the missing empirical details of this debate. Our findings provide some preliminary evidence that, contrary to what most contemporary political philosophers have assumed, people are not as worried by natural luck as previously thought. Instead, people’s worries seem to be focused exclusively on inequalities generated by social luck. (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)

Lottery puzzles involve an ordinary piece of knowledge which seems to imply knowledge of a so-called “lottery proposition,” which itself seems unknown: I might be said to know that I won’t be going on safari next year. But if I were to win the lottery, I would go, and I don’t know that I won’t win the lottery. Examples can be multiplied. Thus we seem left either with the paradoxical position of knowing certain ordinary propositions, but (...) failing to know the lottery propositions they imply, or else conceding to the skeptic. I present a version of reliabilism according to which empirical knowledge is true belief produced by a reliable process causally connecting belief and fact. According to this theory, if my ordinary belief and my belief in the lottery proposition are suitably connected to the facts that render them true, both count as knowledge. In cases where my ordinary belief and my belief in the lottery proposition are not suitably connected to the relevant facts, neither count as knowledge. Thus the paradoxical air of lottery puzzles is removed, and skepticism is avoided. (shrink)

In this article, I discuss three distinct but related puzzles involving lotteries: Kyburg’s lottery paradox, the statistical evidence problem, and the Harman-Vogel paradox. Kyburg’s lottery paradox is the following well-known problem: if we identify rational outright belief with a rational credence above a threshold, we seem to be forced to admit either that one can have inconsistent rational beliefs, or that one cannot rationally believe anything one is not certain of. The statistical evidence problem arises from the observation (...) that people seem to resist forming outright beliefs whenever the available evidence for the claim under consideration is purely statistical. We need explanations of whether it is in fact irrational to form such beliefs, and of whether a clear distinction can be drawn between statistical and non-statistical evidence. The Harman-Vogel paradox is usually presented as a paradox about knowledge: we tend to assume that we can know so-called ordinary propositions, such as the claim that I will be in Barcelona next spring. Yet, we hesitate to make knowledge claims regarding so-called lottery propositions, such as the claim that I won’t die in a car crash in the next few months, even if these lottery propositions are obviously entailed by the ordinary propositions we claim to know. Depending on one’s view about the relationship between rational belief and knowledge, the Harman-Vogel paradox has ramifications for a theory of rational outright belief. Formal theories of the relationship between rational credence and rational belief, such as Leitgeb’s stability theory, tend to focus mostly on handling Kyburg’s lottery paradox, but not the other two puzzles I mention. My aim in this article is to draw out relationships and differences between the puzzles, and to examine to what extent existing formal solutions to Kyburg’s lottery paradox help with answering the statistical evidence problem and the Harman-Vogel paradox. (shrink)

Knowledge and Lotteries is organized around an epistemological puzzle: in many cases, we seem consistently inclined to deny that we know certain propositions, while crediting ourselves with knowledge of propositions that imply them. In its starkest form, the puzzle is this: we do not think we know that a given lottery ticket will be a loser, yet we normally count ourselves as knowing all sorts of things which entail that its holder will not suddenly acquire a large fortune. After (...) providing a number of specific and general characterizations of the puzzle, Hawthorne carefully examines the competing merits of candidate solutions, addressing along the way a range of central questions concerning the nature and importance of knowledge. (shrink)

Many goods are distributed by processes that involve randomness. In lotteries, randomness is used to promote fairness. When taking social risks, randomness is a feature of the process. The losers of such decisions ought to be given a reason why they should accept the outcome. Surprisingly, good reasons demand more than merely equalex antechances. What is also required is a true statement of the form: ‘the result could easily have gone the other way and you could have been the winner’. (...) This rules in standard lotteries but rules out many lotteries based on merely epistemic probability. (shrink)

I seem to know that I won't experience spaceflight but also that if I win the lottery, then I will take a flight into space. Suppose I competently deduce from these propositions that I won't win the lottery. Competent deduction from known premises seems to yield knowledge of the deduced conclusion. So it seems that I know that I won't win the lottery; but it also seems clear that I don't know this, despite the minuscule probability of (...) my winning (if I have a lottery ticket). So we have a puzzle. It seems to generalize, for analogues of the lottery-proposition threaten almost all ordinary knowledge attributions. For example, my apparent knowledge that my bike is parked outside seems threatened by the possibility that it's been stolen since I parked it, a proposition with a low but non-zero probability; and it seems that I don't know this proposition to be false. Familiar solutions to this family of puzzles incur unacceptable costs—either by rejecting deductive closure for knowledge, or by yielding untenable consequences for ordinary attributions of knowledge or of ignorance. After canvassing and criticizing these solutions, I offer a new solution free of these costs. Knowledge that p requires an explanatory link between the fact that p and the belief that p. This necessary but insufficient condition on knowledge distinguishes actual lottery cases from typical, apparently analogous ‘quasi-lottery’ cases. It does yield scepticism about my not winning the lottery and not experiencing spaceflight, but the scepticism doesn't generalize to quasi-lottery cases such as that involving my bike. (shrink)

Shortages of new therapeutics to treat coronavirus disease (COVID-19) have forced clinicians, public health officials, and health systems to grapple with difficult questions about how to fairly allocate potentially life-saving treatments when there are not enough for all patients in need (1). Shortages have occurred with remdesivir, tocilizumab, monoclonal antibodies, and the oral antiviral Paxlovid (2) -/- Ensuring equitable allocation is especially important in light of the disproportionate burden experienced during the COVID-19 pandemic by disadvantaged groups, including Black, Hispanic/Latino and (...) Indigenous communities, individuals with certain disabilities, and low-income persons. However, many health systems have resorted to first-come, first-served approaches to allocation, which tend to disadvantage individuals with barriers in access to care (3). There is mounting evidence of racial, ethnic, and socioeconomic disparities in access to medications for COVID-19 (4, 5). -/- One potential method to promote equitable allocation is to use a weighted lottery, which is an allocation strategy that gives all eligible patients a chance to receive the scarce treatment while also allowing the assignment of higher or lower chances according to other ethical considerations (6). We sought to assess the feasibility of implementing a weighted lottery to allocate scarce COVID-19 medications in a large U.S. health system and to determine whether the weighted lottery promotes equitable allocation. (shrink)

Recently, several funding agencies have introduced the distribution of funds by a lottery system; however, the effects of this system on the productivity of the research community are unclear. Simulation studies in philosophy of science have argued that a combination of peer review and lottery is an optimal method. However, these models overlook several important aspects of research activities, such as baseline funding through block grants. In this article, I present a general theoretical model that incorporates these aspects (...) and argue that the conventional combination of peer review and baseline funding outperforms the combination of peer review and lottery in many situations. (shrink)