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)
The lottery and preface paradoxes pose puzzles in epistemology concerning how to think about the norms of reasonable or permissible belief. Contextualists in epistemology have focused on knowledge ascriptions, attempting to capture a set of judgments about knowledge ascriptions and denials in a variety of contexts (including those involving lottery beliefs and the principles of closure). This article surveys some contextualist approaches to handling issues raised by the lottery and preface, while also considering some of the difficulties encountered by those (...) approaches. (shrink)
This 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)
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)
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: highly probable propositions are justifiably believable, justified believability is closed under conjunction introduction, known contradictions are not justifiably believable. This paper argues that a satisfactory solution to the lottery paradox must reject as versions of the paradox can be generated without appeal to either or and proposes a new solution to the paradox in terms of a novel account of justified believability.
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)
The safety analysis of knowledge, due to Duncan Pritchard, has it that for all contingent propositions, p, S knows that p iff S believes that p, p is true, and in most nearby worlds in which S forms his belief in the same way as in the actual world, S believes that p only if p is true. Among the other virtues claimed by Pritchard for this view is its supposed ability to solve a version of the lottery puzzle. In (...) this paper, I argue that the safety analysis of knowledge in fact fails to solve the lottery puzzle. I also argue that a revised version of the safety principle recently put forward by Pritchard fares no better. (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)
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)
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)
Duncan Pritchard's version of the safety analysis of knowledge has it that for all contingent propositions, p, S knows that p iff S believes that p, p is true, and (the “safety principle”) in most nearby worlds in which S forms his belief in the same way as in the actual world, S believes that p only if p is true. Among the other virtues claimed by Pritchard for this view is its supposed ability to solve a version of the (...) lottery puzzle. In this paper, I argue that the safety analysis of knowledge in fact fails to solve the lottery puzzle. I also argue that a revised version of the safety principle recently put forward by Pritchard fares no better. (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)
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)
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)
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.
Is it unethical to advertise lotteries? Many citizens think that states should not be actively promoting and encouraging the public tospend hard-earned dollars on a bet that they are virtually guaranteed to lose. Perhaps more importantly, business ethicists are concerned that lottery advertising may be targeting the most vulnerable markets: households with the lowest income and education levels. If this were true, then it would increase the already disproportionately large burden of lottery taxes on the poor. Fortunately, our research (...) finds no evidence to support the contention that advertising is responsible for high rates of lottery participation and expenditures by lower income groups or that low-income groups are more affected by advertising than high-income groups. On the contrary, awareness of lottery advertising seems to be associated with a higher probability to play Lotto only for the middle income group. This means that lottery advertising may actually reduce the regressivity of lottery taxes. (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)
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 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)
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)
A necessary criterion of Duncan Pritchard’s Anti-luck Virtue Epistemology is his safety condition. A believer cannot know p unless her belief is safe. Her belief is safe only if p could not have easily been false. But “easily” is not to be understood probabilistically. The chance that p is false might be extremely low and yet p remains unsafe. This is what happens, Pritchard argues, in lottery examples and explains why knowledge is not a function of the probabilistic strength of (...) one’s evidence. This paper argues that, contra Pritchard, modality holds no epistemic advantage over this type of “probabilistic evidentialism” that he criticizes. I begin with a review of Pritchard’s argument supporting modality over probability; second, I explain the problems with this argument, and third, I offer an alternative explanation of the lottery example. At the completion of the paper, modality and probability are on equal epistemic footing. (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)
We will present a new lottery-style paradox on counterfactuals and chance. The upshot will be: combining natural assumptions on the truth values of ordinary counterfactuals, the conditional chances of possible but non-actual events, the manner in which and relate to each other, and a fragment of the logic of counterfactuals leads to disaster. In contrast with the usual lottery-style paradoxes, logical closure under conjunction—that is, in this case, the rule of Agglomeration of counterfactuals—will not play a role in the derivation (...) and will not be entailed by our premises either. We will sketch four obvious but problematic ways out of the dilemma, and we will end up with a new resolution strategy that is non-obvious but less problematic: contextualism about what counts as a proposition. This proposal will not just save us from the paradox, it will also save each premise in at least some context, and it will be motivated by independent considerations from measure theory and probability theory. (shrink)
Safety accounts of knowledge claim, roughly, that knowledge that p requires that one's belief that p could not have easily been false. Such accounts have been very popular in recent epistemology. However, one serious problem safety accounts have to confront is to explain why certain lottery‐related beliefs are not knowledge, without excluding obvious instances of inductive knowledge. We argue that the significance of this objection has hitherto been underappreciated by proponents of safety. We discuss Duncan Pritchard's recent solution to the (...) problem and argue that it fails. More importantly, the problem reaches deeper and poses a threat to any current safety accounts that require a belief's modal stability in close possibilities (as well as safety accounts that appeal to ‘normality’). We end by arguing that ways out of the problem require substantial reconstruction for a safety‐based account of knowledge. (shrink)
This paper addresses an argument offered by John Hawthorne against 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)
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)
I argue that Schmidt et al, while correctly diagnosing the serious racial inequity in current ventilator rationing procedures, misidentify a corresponding racial inequity issue in alternative ‘unweighted lottery’ procedures. Unweighted lottery procedures do not ‘compound’ prior structural injustices. However, Schmidt et al do gesture towards a real problem with unweighted lotteries that previous advocates of lottery-based allocation procedures, myself included, have previously overlooked. On the basis that there are independent reasons to prefer lottery-based allocation of scarce lifesaving healthcare resources, (...) I develop this idea, arguing that unweighted lottery procedures fail to satisfy healthcare providers’ duty to prevent unjust population-level health outcomes, and thus that lotteries weighted in favour of Black individuals are to be preferred. (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)
For several centuries, economists, sociologists, and philosophers have been concerned with the magnitude and e¤ects of inequality. Economists have concentrated on inequality in income and wealth, and have linked this inequality to social welfare, aggregate savings and investment, economic development, and other issues. They have explained the observed degree of inequality by the e¤ect of random shocks, inherited position, and inequality..
This article investigates how subjects determine minimum selling prices for lotteries. We design an experiment where subjects have at every moment an incentive to state their minimum selling price and to adjust the price, if they believe that the price that they stated initially was not optimal. We observe frequent and sizeable price adjustments. We find that random pricing models cannot explain the observed price patterns. We show that earlier prices contain information about future price adjustments. We propose a (...) model of Stochastic Pricing that offers an intuitive explanation for these price adjustment patterns. (shrink)
The possibility to interpret expected and nonexpected utility theories in purely probabilistic terms has been recently investigated. Such interpretation proposes as guideline for the Decision Maker the comparison of random variables through their probability to outperform a stochastic benchmark. We apply this type of analysis to the model of Becker and Sarin, showing that their utility functional may be seen as the probability that an opportune random variable, depending on the one to be evaluated, does not outperform a non-random benchmark. (...) Further, the consequent choice criterion is equivalent to a sort of probability of ruin. Possible interpretations and financial examples are discussed. (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)
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. We solve the 'adding problems' that occur in these two contexts using a similar strategy, based on non-standard analysis.
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)
We present a compositional semantics for first-order logic with imperfect information that is equivalent to Sevenster and Sandu’s equilibrium semantics (under which the truth value of a sentence in a finite model is equal to the minimax value of its semantic game). Our semantics is a generalization of an earlier semantics developed by the first author that was based on behavioral strategies, rather than mixed strategies.