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.

The lottery problem is often regarded as a successful counterexample to reliabilism. The process of forming your true belief that your ticket has lost solely on the basis of considering the odds is, from a purely probabilistic viewpoint, much more reliable than the process of forming a true belief that you have lost by reading the results in a normally reliable newspaper. Reliabilism thus seems forced, counterintuitively, to count the former process as knowledge if it so counts the (...) latter process. -/- I offer a theory of empirical knowledge which, while being recognizably reliabilist, restricts empirical knowledge to cases in which the fact that p and the belief that p are causally connected. I show that this form of reliabilism solves the lottery problem, avoids the problems that beset the causal theory of knowledge, and show how it handles a number of problematic cases in the recent literature. (shrink)

(1998). Contextualist solutions to epistemological problems: Scepticism, Gettier, and the lottery. Australasian Journal of Philosophy: Vol. 76, No. 2, pp. 289-306. doi: 10.1080/00048409812348411.

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)

This paper attempts to show that contextualism cannot adequately handle all versions of ‘The Lottery Paradox.” Although the application of contextualist rules is meant to vindicate the intuitive distinction between cases of knowledge and non-knowledge, it fails to do so when applied to certain versions of “The Lottery Paradox.” In making my argument, I first briefly explain why this issue should be of central importance for contextualism. I then review Lewis’ contextualism before offering my argument that the (...) class='Hi'>lottery paradox persists on all contextualist accounts. Although I argue that the contextualist does not fare well, hope nevertheless remains. For, on Lewis’ behalf, I offer what I take to be the best solution for the contextualist and argue that once this solution is adopted, contextualism will be in a better position to handle the lottery paradox than any other substantive epistemological theory. (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)

According to the safety principle, if one knows that p, one's belief that p could not easily have been false. One problem besetting this principle is the lottery problem – that of explaining why one does not seem to know that one will lose the lottery purely based on probabilistic considerations, prior to the announcement of the lottery result. As Greco points out, it is difficult for a safety theorist to solve this problem, without (...) paying a heavy price. In this paper, I first reject three existing safety-based solutions to the lottery problem, due to Pritchard, Sosa, and Broncano-Berrocal. Failure of these accounts reveals that there is something crucial missing in the safety principle. By way of remedying this, I propose to integrate a safety principle with a condition regarding one's own awareness of (nearby) error-possibilities. The resulting account, as I argue, enjoys a number of theoretical advantages, including its capacity to handle the lottery problem. (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)

The main aim of this paper is to analyse David Lewis’ version of contextualism and his solution to the Gettier problem and the lottery problem through the employment of his Rule of Relevance and Stewart Cohen’s response to these problems. Here I analyse whether Stewart Cohen’s response to David Lewis’ solutions to these problems is on the right track or not. Hence, I try to analyse some concept in David Lewis and Stewart Cohen which has remained unanalysed. (...) Cohen tries to show that when we try to solve some variation of the lottery problem and the Gettier problem by applying Lewis’s Rule of Relevance, then it generates some counterintuitive result. So Cohen gives Lewis some alternatives (which are explained in “The Strategy of Biting the Bullet and the Problem of Interference” and “Biting the Bullet Strategy as a Natural Extension of Contextualist Resolution to the Sceptical Problem and the Pity Poor Bill Variation of the Lottery Problem” sections) to avoid this counterintuitive result; this attempt, however, affects some other presuppositions of the contextual theory of David Lewis. My aim in this paper is to show how without taking these alternatives suggested by Cohen, Lewis can apply his Rules of Relevance to solve the lottery problem and the Gettier problem without any counterintuitive result. (shrink)

The lottery problem is the problem of explaining why mere reflection on the long odds that one will lose the lottery does not yield knowledge that one will lose. More generally, it is the problem of explaining why true beliefs merely formed on the basis of statistical evidence do not amount to knowledge. Some have thought that the lottery problem can be solved by appeal to a violation of the safety principle for knowledge, (...) i.e., the principle that if S knows that p, not easily would S have believed that p without p being the case. Against the standard safety‐based solution, I argue that understanding safe belief as belief that directly covaries with the truth of what is believed in a suitably defined set of possible worlds forces safety theorists to make a series of theoretical choices that ultimately prevent a satisfactory solution to the problem. In this way, I analyze several safety principles that result from such choices—the paper thus gives valuable insights into the nature of safety—and explain why none solves the lottery problem, including their inability to explain away Gettierized lottery cases. On a more positive note, I show that there is a viable solution in terms of safety if we get rid of the unquestioned assumption that safe beliefs directly track the truth. The alternative is a conception of safe belief according to which what safe beliefs directly track is the appropriateness of the circumstances and, indirectly, the truth. The resulting safety principle, I argue, explains why mere statistical evidence is not a safe source of knowledge. (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)

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)

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)

I argue that Schmidtet 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’ (in the relevant sense) prior structural injustices. However, Schmidtet aldo 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 (and others who experience serious health injustice) are to be preferred. (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)

In this article I criticize the non-consequentialist Weighted Lottery (WL) solution to the choice between saving a smaller or a larger group of people. WL aims to avoid what non-consequentialists see as consequentialism's unfair aggregation by giving equal consideration to each individual's claim to be rescued. In so doing, I argue, WL runs into another common objection to consequentialism: it is excessively demanding. WL links the right action with the outcome of a fairly weighted lottery, which means that (...) an agent can only act rightly if s/he has actually run the lottery. In many actual cases, this involves epistemic demands that can be almost impossible to meet. I argue that plausible moral principles cannot make such extreme epistemic demands. (shrink)

Several philosophers have claimed that S knows p only if S’ s belief is safe, where S's belief is safe iff (roughly) in nearby possible worlds in which S believes p, p is true. One widely held intuition many people have is that one cannot know that one's lottery ticket will lose a fair lottery prior to an announcement of the winner, regardless of how probable it is that it will lose. Duncan Pritchard has claimed that a chief (...) advantage of safety theory is that it can explain the lottery intuition without succumbing to skepticism. I argue that Pritchard is wrong. If a version of safety theory can explain the lottery intuition, it will also lead to skepticism. Content Type Journal Article Category Original Article Pages 1-26 DOI 10.1007/s10670-011-9305-z Authors Dylan Dodd, Department of Philosophy, Northern Institute of Philosophy, University of Aberdeen, Aberdeen, UK Journal Erkenntnis Online ISSN 1572-8420 Print ISSN 0165-0106. (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)

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)

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)

In this imaginative and provocative book, Barbara Goodwin explores the question of how lottery systems can achieve egalitarian social justice in societies with seemingly ineradicable inequalities. She begins with the utopian fable of Aleatoria, a country not unlike our own in the not-too-distant-future, where most goods are distributed by lottery--even the right to have children. She then analyzes the philosophical arguments for and against lottery distribution and a comparison of "justice by lottery" with other contemporary theories (...) of justice. Goodwin also applies her theory to practical problems in the real world which could be--or have been--justly resolved by the use of lotteries, such as military drafts, jury duty, and immigration eligibility. She demonstrates that in many areas, including that of political power, a regular and random reallocation of goods would be a fairer and more democratic method than the distributive systems found in liberal democracies today. (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)

It is a platitude in epistemology to say that knowledge excludes luck. Indeed, if one can show that an epistemological theory allows ‘lucky’ knowledge, then that usually suffices to warrant one in straightforwardly rejecting the view. Even despite the prevalence of this intuition, however, very few commentators have explored what it means to say that knowledge is incompatible with luck. In particular, no commentator, so far as I am aware, has offered an account of what luck is and on this (...) basis identified what it means for a true belief to be non-lucky. It is just such a view that I propose, however, and I hope to give a flavour of what this strategy involves here. In particular, I have two goals in this paper. The first is to outline the general contours of the position and show how such a view can account for the attraction of adducing a safety condition on knowledge, with all the epistemic benefits that this principle holds. Relatedly, I will also explain how an anti-luck epistemology can assist us in determining the best formulation of this principle. The second goal of the paper is to show anti-luck epistemology in action by highlighting how such a view can deal with the various problems posed by lottery-style examples. (shrink)

This article discusses how the concept of a fair finite lottery can best be extended to denumerably infinite lotteries. Techniques and ideas from non-standard analysis are brought to bear on the problem.

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

This article discusses Keith DeRose’s treatment of the lottery problem in Chapter 5 of his recent The Appearance of Ignorance. I agree with a lot of it but also raise some critical points and questions and make some friendly proposals. I discuss different ways to set up the problem, go into the difference between knowing and ending inquiry, propose to distinguish between two different kinds of lotteries, add to the defense of the idea that one can know (...) lottery propositions, give a critical discussion of DeRose’s contextualist solution to the problem, and support his defense against an absurdity objection with additional arguments. (shrink)

At present, peer review is the most common method used by funding agencies to make decisions about resource allocation. But how reliable, efficient, and fair is it in practice? The ex ante evaluation of scientific novelty is a fundamentally uncertain endeavor; bias and chance are embedded in the final outcome. In the current study, I will examine some of the most central problems of peer review and highlight the possible benefits of using a lottery as an alternative decision-making mechanism. (...) Lotteries are driven by chance, not reason. The argument made in the study is that the epistemic landscape could benefit in several respects by using a lottery, thus avoiding all types of bias, disagreement, and other limitations associated with the peer review process. Funding agencies could form a pool of funding applicants who have minimal qualification levels and then select randomly within that pool. The benefits of a lottery would not only be that it saves time and resources, but also that it contributes to a more dynamic selection process and increases the epistemic diversity, fairness, and impartiality within academia. (shrink)

The three most common responses to Taurek’s ‘numbers problem’ are saving the greater number, equal chance lotteries and weighted lotteries. Weighted lotteries have perhaps received the least support, having been criticized by Scanlon What We Owe to Each Other ( 1998 ) and Hirose ‘Fairness in Life and Death Cases’ ( 2007 ). This article considers these objections in turn, and argues that they do not succeed in refuting the fairness of a weighted lottery, which remains a potential (...) solution to cases of conflict. Moreover, it shows how these responses actually lead to a new argument for weighted lotteries, appealing to fairness and Pareto-optimality. (shrink)

Adversaries of Moral Luck (AMLs) are at pains to explain why wrongdoers are liable to bear burdens (punishment, compensation etc.) which are related to the harm they cause, because the consequences of what we do are a matter of luck. One attempt to solve this problem suggests that wrongdoers who cause more harm are liable to bear a greater burden not because they are more blameworthy but rather because they get the short straw in a liability lottery (represented (...) by the apparently indeterminate causal process). In this paper I argue that this attempt fails on several grounds. Apart from the fact that it is hard to see how the implementation of liability lotteries can be motivated and the fact that such scheme presupposes a political order (whereas the notion of liability does not seem to presuppose one), detaching liability from the outcomes of a culpable action undermines whichever justifications there were for imposing liability in the first place. Moreover, relying on the determination of the causal process as a good indication of the wrongdoer’s degree of culpability is mistaken, because the luck brought about through the causal process is not necessarily the only element involved in cases of harmful conduct which lies beyond the wrongdoers’ control. (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.

Despite worries about the fairness of lotteries or the sources of the human psyche’s strong attraction to them, Americans have made lotteries a part of their civic lives. The popularity of gaming does not, however, gainsay the unease many Americans feel about state sponsorship of lotteries. The debates that surrounded the introduction of lotteries remain to this day, but the arguments are tired and the camps deadlocked. One camp argues that a lottery is simply a properly randomized drawing that (...) determines who among a freely chosen group of participants shall be awarded all or some of the monetary contributions of the group. These proponents suggest that the randomness of the drawing and the autonomy of the participants render the lottery fair and sponsorship by the state unobjectionable. Opponents of state-supported gambling argue, by contrast, that states market lotteries by making inappropriate emotional appeals and by supplying information of dubious veracity. Consequently, so this group argues, lotteries must be judged as unfair gaming devices and state support viewed as improper. I shall show that both camps have fundamentally misunderstood the problem. Evaluating whether state lotteries are sales or swindles relies neither on an analysis of subjective attitudes nor on an examination of purely procedural aspects of play. Correct analysis depends on a determination of what lotteries are. That is, there is a difference between claiming what a lottery does and what it claims to be, between how it works and what it is. If a lottery is claimed to be something that it is not, then regardless of what one gets for one’s money, one has been swindled. I will show that performing an ontological examination of the state-supported lottery reveals it to be a swindle. I conclude by suggesting that some of the confusion regarding the legitimacy of the state-sponsored lottery stems from misunderstandings of several tenets of liberalism. It is these misunderstandings that at times are employed to justify lotteries. (shrink)

There are many conceivable circumstances in which some people have to be sacrificed in order to give others a chance to survive. The fair and rational method of selection is a lottery with equal chances. But why should losers comply, when they have nothing to lose in a war of all against all? A novel solution to this Compliance Problem is proposed. The lottery must be made self-enforcing by making the lots themselves the means of enforcement of (...) the outcome. This way no external authority is needed to make the losers’ compliance rational both ex ante and ex post. A fairly realistic concrete scenario is sketched to show that the solution could be made to work in practice, particularly since making it work is in everyone’s enlightened self-interest in the circumstances. (shrink)

This work is a complete mathematical guide to lottery games, covering all of the problems related to probability, combinatorics, and all parameters describing the lottery matrices, as well as the various playing systems. The mathematics sections describe the mathematical model of the lottery, which is in fact the essence of the lotto game. The applications of this model provide players with all the mathematical data regarding the parameters attached to the gaming events and personal playing systems. By (...) applying these data, one can find all the winning probabilities for the play with one line, and how these probabilities change with playing the various types of systems containing several lines, depending on their structure. Also, each playing system has a formula attached that provides the number of possible multiple prizes in various circumstances. Other mathematical parameters of the playing systems and the correlations between them are also presented. The generality of the mathematical model and of the obtained formulas allows their application for any existent lottery and any playing system. Each formula is followed by numerical results covering the most frequent lottery matrices worldwide and by multiple examples predominantly belonging to the 6/49 lottery. The listing of the numerical results in dozens of well-organized tables, along with instructions and examples of using them, makes possible the direct usage of this guide by players without a mathematical background. The author also discusses from a mathematical point of view the strategies of choosing involved in the lotto game. The book does not offer so-called winning strategies, but helps players to better organize their own playing systems and to confront their own convictions with the incontestable reality offered by the direct applications of the mathematical model of the lotto game. As a must-have handbook for any lottery player, this book offers essential information about the game itself and can provide the basis for gaming decisions of any kind. (shrink)

BackgroundThe Health Research Council of New Zealand is the first major government funding agency to use a lottery to allocate research funding for their Explorer Grant scheme. This is a somewhat controversial approach because, despite the documented problems of peer review, many researchers believe that funding should be allocated solely using peer review, and peer review is used almost ubiquitously by funding agencies around the world. Given the rarity of alternative funding schemes, there is interest in hearing from the (...) first cohort of researchers to ever experience a lottery. Additionally, the Health Research Council of New Zealand wanted to hear from applicants about the acceptability of the randomisation process and anonymity of applicants.MethodsThis paper presents the results of a survey of Health Research Council applicants from 2013 to 2019. The survey asked about the acceptability of using a lottery and if the lottery meant researchers took a different approach to their application.ResultsThe overall response rate was 39%, with 30% from applicants in the years 2013 to 2018, and 68% for those in the year 2019 who were not aware of the funding result. There was agreement that randomisation is an acceptable method for allocating Explorer Grant funds with 63% in favour and 25% against. There was less support for allocating funds randomly for other grant types with only 40% in favour and 37% against. Support for a lottery was higher amongst those that had won funding. Multiple respondents stated that they supported a lottery when ineligible applications had been excluded and outstanding applications funded, so that the remaining applications were truly equal. Most applicants reported that the lottery did not change the time they spent preparing their application.ConclusionsThe Health Research Council’s experience through the Explorer Grant scheme supports further uptake of a modified lottery. (shrink)

In most penal systems, success is punished more than failure. For example, murder is punished more severely than attempted murder. But success or failure is often determined by luck. It thus appears that punishment is allotted on the basis of arbitrary factors. The problem of criminal attempts is the question of how to best resolve this apparent tension. One particularly sophisticated attempt at resolution, first developed by David Lewis, holds that such differential punishment is not unjust when understood as (...) a natural penal lottery. What is most interesting about this view is that it does not appear to involve a commitment to resultant moral luck. I argue that the natural penal lottery fails to deliver justice. Upon analysis, it carries the same implication that it sought to avoid—namely, a commitment to resultant moral luck. I then argue that there can be, in principle, no penal lottery that delivers justice, natural or otherwise. (shrink)

After a discussion of the evolution and criticisms of state run lotteries, this article examines the ethics of lottery advertising. A discussion of the appeals used by lottery advertisers is followed by evidence concerning the impact of expected value information on lottery purchase intentions. Findings point toward less emphasis on the lottery as a solution to financial and job problems and more emphasis on information about the actual value of a lottery bet. Using accepted standards (...) from the marketing literature, lottery advertising is found to be deceptive. (shrink)

The central issue of this dissertation is known in bioethics as the problem of fair chances versus best outcomes. The decision-making context is patient selection for scarce, transplantable organs. This problem poses two options for patient selection: either select by a procedure which affords fair chances to all medically suitable transplant candidates or select those whose prognoses indicate the highest levels of prospective medical benefit. The fair chances/best outcomes problem is essentially a problem of choosing between (...) lives. An egalitarian approach to patient selection favours fair chances. A utilitarian approach favours best outcomes. My project is a sustained defence of an egalitarian approach. It targets first a rule utilitarian argument for maximizing medical benefit. It targets second an argument that fairness in patient selection requires a preference for saving younger lives. I argue that patients should have prognoses at or above a threshold level of medical benefit to be medically suitable candidates. I propose random selection by lottery as a means of final selection that equally values lives. I argue that equality of opportunity best reflects an egalitarian commitment to equal concern and respect. (shrink)

In this paper, I argue that there is a kind of evil, namely, the unequal distribution of natural endowments, or natural inequality, which presents theists with a new evidential problem of evil. The problem of natural inequality is a new evidential problem of evil not only because, to the best of my knowledge, it has not yet been discussed in the literature, but also because available theodicies, such the free will defense and the soul-making defense, are not (...) adequate responses in the face of this particular evil, or so I argue. (shrink)

A critical debate has blossomed within the field of research policy, science and technology studies, and philosophy of science regarding the possible benefits and limitations of allocating extramural grants using a lottery system. The most common view among those supporting the lottery idea is that some form of modified lottery is acceptable, if properly combined with peer review. This means that partial randomization can be applied only after experts have screened the pursuit-worthiness of all submitted proposals and (...) sorted out those of lowest quality. In the present paper, I will argue against the use of partial lotteries or partial randomization and instead promote use of a _pure lottery_ in combination with a radical increase in block funding. The main reason for holding this position is that a partial lottery cannot solve the problems inherent in the current funding system, which is based on grant competitions and peer review. A partial lottery cannot decrease the enormous time-waste, reduce the uneven distribution of time between researchers, neutralize expert biases or mitigate academic power asymmetries. Instead, we need a stronger focus on improving general time management in academia by implementing a more holistic model for organizing research opportunities in the future. (shrink)

Over the past two decades, gamblers have begun taking mathematics into account more seriously than ever before. While probability theory is the only rigorous theory modeling the uncertainty, even though in idealized conditions, numerical probabilities are viewed not only as mere mathematical information, but also as a decision-making criterion, especially in gambling. This book presents the mathematics underlying the major games of chance and provides a precise account of the odds associated with all gaming events. It begins by explaining in (...) simple terms the meaning of the concept of probability for the layman and goes on to become an enlightening journey through the mathematics of chance, randomness and risk. It then continues with the basics of discrete probability, combinatorics and counting arguments for those interested in the supporting mathematics. These mathematic sections may be skipped by readers who do not have a minimal background in mathematics; these readers can skip directly to the Guide to Numerical Results to pick the odds and recommendations they need for the desired gaming situation. Doing so is possible due to the organization of that chapter, in which the results are listed at the end of each section, mostly in the form of tables. The chapter titled The Mathematics of Games of Chance presents these games not only as a good application field for probability theory, but also in terms of human actions where probability-based strategies can be tried to achieve favorable results. Through suggestive examples, the reader can see what are the experiments, events and probability fields in games of chance and how probability calculus works there. The main portion of this work is a collection of probability results for each type of game. Each game s section is packed with formulas and tables. Each section also contains a description of the game, a classification of the gaming events and the applicable probability calculations. The primary goal of this work is to allow the reader to quickly find the odds for a specific gaming situation, in order to improve his or her betting/gaming decisions. Every type of gaming event is tabulated in a logical, consistent and comprehensive manner. The complete methodology and complete or partial calculations are shown to teach players how to calculate probability for any situation, for every stage of the game for any game. Here, readers can find the real odds, returned by precise mathematical formulas and not by partial simulations that most software uses. Collections of odds are presented, as well as strategic recommendations based on those odds, where necessary, for each type of gaming situation. The book contains much new and original material that has not been published previously and provides great coverage of probabilities for the following games of chance: Dice, Slots, Roulette, Baccarat, Blackjack, Texas Hold em Poker, Lottery and Sport Bets. Most of games of chance are predisposed to probability-based decisions. This is why the approach is not an exclusively statistical one, but analytical: every gaming event is taken as an individual applied probability problem to solve. A special chapter defines the probability-based strategy and mathematically shows why such strategy is theoretically optimal.". (shrink)

Rationalism, my target, says that in order to have perceptual knowledge, such as that your hand is making a fist, you must “antecedently” (or “independently”) know that skeptical scenarios don’t obtain, such as the skeptical scenario that you are in the Matrix. I motivate the specific form of Rationalism shared by, among others, White (Philos Stud 131:525–557, 2006) and Wright (Proc Aristot Soc Suppl Vol 78:167–212, 2004), which credits us with warrant to believe (or “accept”, in Wright’s terms) that our (...) senses are reliably veridical, where that warrant is one we enjoy by default, that is, without relying on any evidence or engaging in any positive argument. The problem with this form of Rationalism is that, even if you have default knowledge that your senses are reliable, this is not adequate to rule out every kind of skeptical scenario. The problem is created by one-off skeptical scenarios, scenarios that involve a highly reliable perceiver who, by a pure fluke, has a one-off, non-veridical experience. I claim you cannot infer that your present perceptual experience is veridical just on the basis of knowledge of your general reliability. More generally, if you infer that the present F is G, just on the basis of your knowledge that most Fs are Gs, this is what I call statistical inference, and, as I argue, statistical inference by itself does not generate knowledge. I defend this view of statistical inference against objections, including the objection that radical skepticism about our ordinary inductive knowledge will follow unless statistical inference generates knowledge. (shrink)

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?

One of the most important problems of Muslim societies and humanity is the conflicts and troubles among spouses and between parents and their children in the family. problems. Research is carried out and answers are sought for the solution of these problems, but the dissolution in families cannot be prevented and the divorce rate increases day by day. Besides being a prophet, the Messenger of Allah (pbuh) is a servant of Allah and a human being. It is seen that there (...) were problems in the homes of the Prophet (pbuh), both between him and his wives, and among his wives. Since the creation of human beings has not changed, the Prophet's (pbuh) attitudes and solutions to these problems are important for the solution of family problems happening today. It is seen that the Messenger of Allah (pbuh) reacted to these problems and discussions far above his age and he treated his wives and solved the problems as modern researchers and scientists recommend in the recent world. The purpose of sending the Messenger of Allah (pbuh) is to convey the religion of Allah to people. For this reason, he was given orders regarding the most private matters of his private life when appropriate. Human's happiness in this world and in the hereafter can only be achieved by recognizing this perfect example. This can be achieved by learning about his personality as an individual and his family life as a family. The conflict of values among spouses in the family, life perspective, personality and priority differences are the basis of the problems. In addition, changes in personality especially after marriage, loss of love and respect for each other, and jealousy are the main situations that harm marriage. Many marriages end because of miscommunication. There are many people who try to solve problems by being cold towards their spouse and taking a stand towards him/her. If a man is married to many women, as in the example of the Prophet (pbuh), the main problem he encounters is jealousy and the resulting unrest. In a family, there are joyful and happy moments as well as unhappy moments and times. It is very natural for family members to quarrel with each other and to have disagreements among them. The Prophet's family is not far from it. The wives of the Messenger of Allah (pbuh) are not ordinary people, but women who are either at the forefront of their tribe with their ancestry, intelligence and beauty, or who are one step ahead with their conversion to Islam and their services. According to the laws of God, the same poles repel each other, while the opposite poles attract each other. They are all very precious and it is perfectly natural for a repulsion force to develop between them. It is necessary to think of the Prophet's (pbuh) houses as a school. The state of our mothers when they entered that house is not the same as the degree they reached when the Prophet (pbuh) passed away. Differences of opinion and conflicts between members of the family, deterioration of the balance in relations, and weakening of solidarity are among the events seen in every family. The Prophet (pbuh) was able to have controversies with his wives, even if it was a rarely example. However, in these discussions, it is seen that he does not allow violence and tries to solve the issues patiently and by talking to each other, and sometimes asks for help from his close circle or relatives of his spouses. ʿĀʾIs̲h̲a is the most prominent among the wives of the Messenger of Allah, except for Khadija. She himself knew this and was voicing this claim against the other wives of the Messenger of Allah (pbuh). When the Prophet (pbuh) married Umm Salama, people talked about his beauty. ʿĀʾIs̲h̲a, who heard these praises, was very upset about the new rival who came to her. With Zaynab bt. ḎJ̲aḥs̲h̲ stepping into the house of the prophet, the balances changed and a fierce rivalry began between her and ʿĀʾIs̲h̲a. Just as ʿĀʾIs̲h̲a was proud of her superiority, Zaynab began to brag about her superiority. ʿĀʾIs̲h̲a had a special place among the wives of the Prophet. This was evident from the fact that the Messenger of Allah (pbuh) said at every opportunity that he loved him very much. The Messenger of Allah (pbuh) did not hesitate to express this either. When the Prophet (pbuh) went on an expedition, he would treat his wives fairly and draw lots so that he would not cause hatred or envy among them. When the lottery went out to another wife besides ʿĀʾIs̲h̲a, the dissatisfaction on his face could be felt. Among his other wives, the Prophet tried to protect and watch over his wife, Safiyya, at every opportunity, which he felt was strange because he was later included among them and was of Jewish origin. Although Mariye was a slave, she became one of the envied women among the wives of the Prophet after she gave birth to her son Ibrahim to the Messenger of Allah (pbuh). Undoubtedly, it was ʿĀʾIs̲h̲a who envied him the most. (shrink)

Introduction -- Value theory : the nature of the good life -- Epicurus letter to Menoeceus -- John Stuart Mill, Hedonism -- Aldous Huxley, Brave new world -- Robert Nozick, The experience machine -- Richard Taylor, The meaning of life -- Jean Kazez, Necessities -- Normative ethics : theories of right conduct -- J.J.C. Smart, Eextreme and restricted utilitarianism -- Immanuel Kant the good will & the categorical imperative -- Thomas Hobbes, Leviathan -- Philippa Foot, Natural goodness -- Aristotle, Nicomachean (...) ethics -- W.D. Ross, What makes right acts right? -- Hilde Lindemann, What is feminist ethics? -- Metaethics : the status of morality -- David Hume, Moral distinctions not derived from reason -- J.L. Mackie, The subjectivity of values -- Gilbert Harman, Ethics and observation -- Mary Midgley, Trying out one's new sword -- Michael Smith, Rrealism -- Renford Bambrough, Pproof -- Moral problems -- Peter Singe, The Singer solution to world poverty -- Heidi Malm, Paid surrogacy: arguments and responses -- Ronald Dworkin, Playing God : genes, clones, and luck -- James Rachels, The morality of euthanasia -- John Harris, The survival lottery -- Peter Singer, Unsanctifying human life -- William F. Baxter, People or penguins : the case for optimal pollution -- Judith Jarvis, Tthomson a defense of abortion -- Don Marquis, Why abortion is immoral -- Jonathan Bennett, The conscience of Huckleberry Finn -- Michael Walzer, Terrorism : a critique of excuses -- David Luban, Liberalism, torture, and the ticking bomb -- Martin Luther King, Jr., Letter from Birmingham City Jail -- Igor Primoratz, Justifying legal punishment -- Stephen Nathanson, An eye for an eye -- Michael Huemer, America's unjust drug war -- John Corvino, Why shouldn't Tommy and Jimmy have sex? : a defense of homosexuality -- Bonnie Steinbock, Adultery -- Hugh Lafollette, Licensing parents -- Jane English, What do grown children owe their parents? (shrink)

In this paper we survey some main arguments for and against epistemological contextualism. We distinguish and discuss various kinds of contextualism, such as attributer contextualism (the most influential version of which is semantic, conversational, or radical contextualism); indexicalism; proto-contextualism; Wittgensteinian contextualism; subject, inferential, or issue contextualism; epistemic contextualism; and virtue contextualism. Starting with a sketch of Dretske's Relevant Alternatives Theory and Nozick's Tracking Account of Knowledge, we reconstruct the history of various forms of contextualism and the ways contextualists try to (...) handle some notorious epistemological quandaries, especially skepticism and the lottery paradox. Then we outline the most important problems that contextualist theories face, and give overviews of their criticisms and defenses as developed in this issue. (shrink)

The problems that arise from the presence of self-attacking ar- guments and odd-length cycles of attack within argumentation frameworks are widely recognized in the literature on defeasible argumentation. This paper introduces two simple semantics to capture different intuitions about what kinds of arguments should become justified in such scenarios. These semantics are modeled upon two extensions of argumentation frameworks, which we call sustainable and tolerant. Each one is constructed on the common ground of the powerful concept of admissibility introduced by (...) Dung in [10]. The novelty of this approach consists in viewing the admissibility of a subset of arguments as relative to potentially challenging subsets of arguments. Both sustainable and tolerant semantics are more credulous than preferred semantics (i.e. they justify at least the same arguments, and possibly more). Given certain sufficient conditions they coincide among them as well as with other semantics introduced by Dung. (shrink)

In a recent paper, Sylvia Wenmackers and Leon Horsten discuss how the concept of a fair infinite lottery can best be extended to denumerably infinite lotteries. Their paper uses and builds on the alpha-theory of Vieri Benci and Mauro Di Nasso. The purpose of this paper is to demonstrate a potential problem for alpha-theory.

You must either save a group of m people or a group of n people. If there are no morally relevant diff erences among the people, which group should you save? is problem is known as the number problem. e recent discussion has focussed on three proposals: (i) Save the greatest number of people, (ii) Toss a fair coin, or (iii) Set up a weighted lottery, in which the probability of saving m people is m / m (...) + n , and the probability of saving n people is n / m + n . is contribution examines a fourth alternative, the mixed solution, according to which both fairness and the total number of people saved count. It is shown that the mixed solution can be defended without assuming the possibility of interpersonal comparisons of value. (shrink)

The aim of this paper is to present and analyse Bruno de Finetti's view that the axiom of countable additivity of the probability calculus cannot be justified in terms of the subjective interpretation of probability. After presenting the core of the subjective theory of probability and the main de Finetti's argument against the axiom of countable additivity (the so called de Finetti's infinite lottery) I argue against de Finetti's view. In particular, I claim that de Finetti does not prove (...) the impossibility of using Dutch Book argument for the axiom of countable additivity. Consequently, we can use Dutch Book argument for the justification of the axiom of countable additivity and regard de Finetti's lottery as a special case when the axiom does not hold, or we can justify countable additivity by Dutch Book argument and reject de Finetti's lottery as irrational. The second strategy, represented especially by Jon Williamson, is much more compatible with the idea of subjective interpretation of probability. (shrink)