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)
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.
Knowledge and Lotteries is organized around an epistemological puzzle: in many cases, we seem consistently inclined to deny that we know a certain class of 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 ordinary things that 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. In so doing, he explores a number of central questions concerning the nature and importance of knowledge, including the relationship of knowledge to assertion and practical reasoning, the status of epistemic closure principles, the merits of various brands of scepticism, the prospects for a contextualist account of knowledge, and the potential for other sorts of salience-sensitive accounts. Along the way, he offers a careful treatment of pertinent issues at the foundations of semantics. His book will be of interest to anyone working in the field of epistemology, as well as to philosophers of language. (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)
I. Introduction 1.1 George M avrodes seems to say that, as reasonable persons, on reading reports of winners of really big lotteries believe these reports, so must reasonable persons, on hearing or reading testimony for what in their view would be miracles, believe this testimony. That a randomly selected entrant should win a big lottery is immensely improbable, and yet a single report can reverse this improbability. He says it shows “that there is nothing incredible, or even unusual, in (...) the power of a single testimony to reverse an enormous initial improbability” (Mavrodes 2005, p. 317) Of miracles he says that “almost every aficionado” of them holds that they are improbable in the same way, and, he seems to say, to the same effect. (shrink)
THIS PAPER ANSWERS RECENT CRITICISMS OF HUME’S SKEPTICISM WITH REGARD TO MIRACLES BY THOSE WHO ARGUE THAT THERE ARE COUNTEREXAMPLES, ILLUSTRATED BY LOTTERIES, TO HUME’S ACCOUNT OF HOW THE TRUTH OF REPORTS ABOUT IMPROBABLE EVENTS MUST BE EVALUATED. THE AUTHOR FIRST SHOWS THAT THESE ARGUMENTS ARE ANALOGOUS TO BUTLER’S CRITICISM OF HUME’S PREDECESSORS IN THE DEBATE ABOUT MIRACLES. IT IS THEN ARGUED THAT EACH OF THESE CRITICISMS COLLAPSES THE DISTINCTION BETWEEN PROBABILITIES PERTAINING TO EVENTS QUA UNIQUE OCCURRENCES AND PROBABILITIES (...) PERTAINING TO EVENTS QUA TOKENS INSTANTIATING EVENT-TYPES. (shrink)
We conducted five experiments that reveal some main contours of the folk epistemology of lotteries. The folk tend to think that you don't know that your lottery ticket lost, based on the long odds ("statistical cases"); by contrast, the folk tend to think that you do know that your lottery ticket lost, based on a news report ("testimonial cases"). We evaluate three previous explanations for why people deny knowledge in statistical cases: the justification account, the chance account, and the (...) statistical account. None of them seems to work. We then propose a new explanation of our own, the formulaic account, according to which some people deny knowledge in statistical cases due to formulaic expression. (shrink)
In this paper 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 nonconsequentialists 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)
If payoffs are tickets for binary lotteries, which involve only two money prizes, then rationality requires expected value maximization in tickets. This payoff scheme was increasingly used to induce risk neutrality in experiments. The experiment presented here involved lottery choice and evaluation tasks. One subject group was paid in binary lottery tickets, another directly in money. Significantly greater deviations from risk neutral behavior are observed with binary lottery payoffs. This discrepancy increases when subjects have easy access to the alternatives' (...) expected values and mean absolute deviations. Behavioral regularities are observed at least as often as with direct money payoffs. (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)
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)
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)
Abstract 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)
John Hawthorne’s recent monograph Knowledge and Lotteries1 is centred on the following puzzle: Suppose you claim to know that you will not be able to afford to summer in the Hamptons next year. Aware of your modest means, we believe you. But suppose you also claim to know that a ticket you recently purchased in a multi-million dollar lottery is a loser. Most of us have the intuition that you do not know that your ticket is a loser. However, your (...) alleged knowledge of not being able to afford to summer in the Hamptons puts you in a position to know that your ticket is a loser. For the proposition that you will not be able to afford to summer in the Hamptons entails the proposition that you will lose the lottery. And the following principle, what Hawthorne calls ‘Single Premise Closure’ ( p. 34), is very plausible: If you know that p, p entails q, and you competently deduce q from p thereby coming to believe that q (all the while retaining your knowledge of p), then you come to know q. (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)
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 Cohens 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)
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)
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)
Hume’s main argument against rational belief in miracles might seem to rule out rational belief in other antecedently improbable occurrences as well--for example, a certain person’s having won the lottery. Dorothy Coleman has recently defended Hume against the lottery counterexample, invoking Hume’s distinction between probability of chances and probability of causes. I argue that Coleman’s defence fails.
John Hawthorne’s marvelous book contains a wealth of arguments and insights based on an impressive knowledge and understanding of contemporary discussion. We can address only a small aspect of the topic. In particular, we will offer our own answers to two questions about knowledge that he discusses.
This paper revisits a puzzle that arises for theories of knowledge according to which one can know on the basis of merely inductive grounds. No matter how strong such theories require inductive grounds to be if a belief based on them is to qualify as knowledge, there are certain beliefs (namely, about the outcome of fair lotteries) that are based on even stronger inductive grounds, while, intuitively, they do not qualify as knowledge. This paper discusses what is often regarded (...) as the most promising classical invariantist solution to the puzzle, namely, that beliefs about the outcomes of fair lotteries do not qualify as knowledge because they are too lucky to do so (or, relatedly, because they do not satisfy a safety condition on knowledge), while other beliefs based on potentially weaker inductive grounds are not too lucky (or, relatedly, because they are safe). A case is presented that shows that this solution to the puzzle is actually not viable. It is argued that there is no obvious alternative solution in sight and that therefore the puzzle still awaits a classical invariantist solution. (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..
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)
In light of the failure of attempts to analyse knowledge as a species of justified belief, a number of epistemologists have suggested that we should instead understand justification in terms of knowledge. This paper focuses on accounts of justification as a kind of knowledge. According to such accounts a belief is justified just in case any failure to know is due to uncooperative external circumstances. I argue against two recent accounts of this sort due to Alexander Bird and Martin Smith. (...) A further aim is to defend a more traditional conception, according to which justification is a matter of sufficiently high evidential likelihood. In particular, I suggest that this conception of justification offers a plausible account of lottery cases: cases in which one believes a true proposition on the basis of probabilistic evidence. (shrink)
In the first chapter of his Knowledge and Lotteries, John Hawthorne argues that thinkers do not ordinarily know lottery propositions. His arguments depend on claims about the intimate connections between knowledge and assertion, epistemic possibility, practical reasoning, and theoretical reasoning. In this paper, we cast doubt on the proposed connections. We also put forward an alternative picture of belief and reasoning. In particular, we argue that assertion is governed by a Gricean constraint that makes no reference to knowledge, and (...) that practical reasoning has more to do with rational degrees of belief than with states of knowledge. (shrink)
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)
In his recent article, ‘Lottery puzzles and Jesus’ return’, Donald Smith says that Christians should accept a very robust scepticism about the future because a Christian ought to think that the probability of Jesus’ return happening at any future moment is inscrutable to her. But I think that Smith’s argument lacks the power rationally to persuade Christians who are antecedently uncommitted as to whether or not we can or do have any substantive knowledge about the future. Moreover, I think that (...) Christians who are so antecedently uncommitted have available objections they can reasonably press against Smith’s arguments. In the article, I attempt to bring out these objections. (shrink)
This paper argues in favor of the epistemic properties of inclusiveness in the context of democratic deliberative assemblies and derives the implications of this argument in terms of the epistemically superior mode of selection of representatives. The paper makes the general case that, all other things being equal and under some reasonable assumptions, more is smarter. When applied to deliberative assemblies of representatives, where there is an upper limit to the number of people that can be included in the group, (...) the argument translates into a defense of a specific selection mode of participants: random selection. (shrink)
In “The possibility of morality,” Phil Brown considers whether moral error theory is best understood as a necessary or contingent thesis. Among other things, Brown contends that the argument from relativity, offered by John Mackie—error theory’s progenitor—supports a stronger modal reading of error theory. His argument is as follows: Mackie’s is an abductive argument that error theory is the best explanation for divergence in moral practices. Since error theory will likewise be the best explanation for similar divergences in possible worlds (...) similar to our own, we may conclude that error theory is true at all such worlds, just as it is in the actual world. I contend that Brown’s argument must fail, as abductive arguments cannot support the modal conclusions he suggests. I then consider why this is the case, concluding that Brown has stumbled upon new and interesting evidence that agglomerating one’s beliefs can be epistemically problematic—an issue associated most famously with Henry Kyburg’s lottery paradox. (shrink)
According to the permissibility solution to the lottery paradox, the paradox can be solved if we conceive of epistemic justification as a species of permissibility. Clayton Littlejohn has objected that the permissibility solution draws on a sufficient condition for permissible belief that has implausible consequences and that the solution conflicts with our lack of knowledge that a given lottery ticket will lose. The paper defends the permissibility solution against Littlejohn's objections.
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)
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)
This dissertation is a contribution to formal and computational philosophy. -/- In the first part, we show that by exploiting the parallels between large, yet finite lotteries on the one hand and countably infinite lotteries on the other, we gain insights in the foundations of probability theory as well as in epistemology. Case 1: Infinite lotteries. We discuss how the concept of a fair finite lottery can best be extended to denumerably infinite lotteries. The solution boils (...) down to the introduction of infinitesimal probability values, which can be achieved using non-standard analysis. Our solution can be generalized to uncountable sample spaces, giving rise to a Non-Archimedean Probability (NAP) theory. Case 2: Large but finite lotteries. We propose application of the language of relative analysis (a type of non-standard analysis) to formulate a new model for rational belief, called Stratified Belief. This contextualist model seems well-suited to deal with a concept of beliefs based on probabilities ‘sufficiently close to unity’. -/- The second part presents a case study in social epistemology. We model a group of agents who update their opinions by averaging the opinions of other agents. Our main goal is to calculate the probability for an agent to end up in an inconsistent belief state due to updating. To that end, an analytical expression is given and evaluated numerically, both exactly and using statistical sampling. The probability of ending up in an inconsistent belief state turns out to be always smaller than 2%. (shrink)
Many epistemologists have responded to the lottery paradox by proposing formal rules according to which high probability defeasibly warrants acceptance. Douven and Williamson () present an ingenious argument purporting to show that such rules invariably trivialise, in that they reduce to the claim that a probability of 1 warrants acceptance. Douven and Williamson’s argument does, however, rest upon significant assumptions—among them a relatively strong structural assumption to the effect that the underlying probability space is both finite and uniform . In (...) this article, I will show that something very like Douven and Williamson’s argument can in fact survive with much weaker structural assumptions—and, in particular, can apply to infinite probability spaces. (shrink)
This paper is concerned with formal solutions to the lottery paradox on which high probability defeasibly warrants acceptance. It considers some recently proposed solutions of this type and presents an argument showing that these solutions are trivial in that they boil down to the claim that perfect probability is sufficient for rational acceptability. The argument is then generalized, showing that a broad class of similar solutions faces the same problem. An argument against some formal solutions to the lottery paradox The (...) argument generalized Some variations Adding modalities Anticipated objections. (shrink)
John Locke proposed a straightforward relationship between qualitative and quantitative doxastic notions: belief corresponds to a sufficiently high degree of confidence. Richard Foley has further developed this Lockean thesis and applied it to an analysis of the preface and lottery paradoxes. Following Foley's lead, we exploit various versions of these paradoxes to chart a precise relationship between belief and probabilistic degrees of confidence. The resolutions of these paradoxes emphasize distinct but complementary features of coherent belief. These features suggest principles that (...) tie together qualitative and quantitative doxastic notions. We show how these principles may be employed to construct a quantitative model - in terms of degrees of confidence - of an agent's qualitative doxastic state. This analysis fleshes out the Lockean thesis and provides the foundation for a logic of belief that is responsive to the logic of degrees of confidence. (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)
De Finetti would claim that we can make sense of a draw in which each positive integer has equal probability of winning. This requires a uniform probability distribution over the natural numbers, violating countable additivity. Countable additivity thus appears not to be a fundamental constraint on subjective probability. It does, however, seem mandated by Dutch Book arguments similar to those that support the other axioms of the probability calculus as compulsory for subjective interpretations. These two lines of reasoning can be (...) reconciled through a slight generalization of the Dutch Book framework. Countable additivity may indeed be abandoned for de Finetti's lottery, but this poses no serious threat to its adoption in most applications of subjective probability. Introduction The de Finetti lottery Two objections to equiprobability 3.1 The ‘No random mechanism’ argument 3.2 The Dutch Book argument Equiprobability and relative betting quotients The re-labelling paradox 5.1 The paradox 5.2 Resolution: from symmetry to relative probability Beyond the de Finetti lottery. (shrink)
This paper is concerned with formal solutions to the lottery paradox on which high probability defeasibly warrants acceptance. It considers some recently proposed solutions of this type and presents an argument showing that these solu 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.
One version of the argument for design relies on the assumption that the apparent fine-tuning of the universe for the existence of life requires an explanation. I argue that the assumption is false. Philosophers who argue for the assumption usually appeal to analogies, such as the one in which a person was to draw a particular straw among a very large number of straws in order not to be killed. Philosophers on the other side appeal to analogies like the case (...) of winning a lottery. I analyze the two analogies and explain why the lottery analogy is the right one to use. In the light of such an analysis, we can see that although the cosmic feature of being life-permitting is rare, it does not allow life-permitting possible universes to stand out because there are other rare cosmic features that other possible universes have. (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)
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.
Henry Kyburg’s lottery paradox (1961, p. 197) arises from considering a fair 1000 ticket lottery that has exactly one winning ticket. If this much is known about the execution of the lottery it is therefore rational to accept that one ticket will win. Suppose that an event is very likely if the probability of its occurring is greater than 0.99. On these grounds it is presumed rational to accept the proposition that ticket 1 of the lottery will not win. Since (...) the lottery is fair, it is rational to accept that ticket 2 won’t win either—indeed, it is rational to accept for any individual ticket i of the lottery that ticket i will not win. However, accepting that ticket 1 won’t win, accepting that ticket 2 won’t win, . . . , and accepting that ticket 1000 won’t win entails that it is rational to accept that no ticket will win, which entails that it is rational to accept the contradictory proposition that one ticket wins and no ticket wins. (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)
In this article I show that the argument in John Harris's famous "Survival Lottery" paper cannot be right. Even if we grant Harris's assumptions—of the justifiability of such a lottery, the correctness of maximizing consequentialism, the indistinguishability between killing and letting die, the practical and political feasibility of such a scheme—the argument still will not yield the conclusion that Harris wants. On his own terms, the medically needy should be less favored (and more vulnerable to being killed), than Harris suggests.
We defend a set of acceptance rules that avoids the lottery paradox, that is closed under classical entailment, and that accepts uncertain propositions without ad hoc restrictions. We show that the rules we recommend provide a semantics that validates exactly Adamsâ€™ conditional logic and are exactly the rules that preserve a natural, logical structure over probabilistic credal states that we call probalogic . To motivate probalogic, we first expand classical logic to geo-logic , which fills the entire unit cube, and (...) then we project the upper surfaces of the geo-logical cube onto the plane of probabilistic credal states by means of standard, linear perspective, which may be interpreted as an extension of the classical principle of indifference. Finally, we apply the geometrical/logical methods developed in the paper to prove a series of trivialization theorems against question-invariance as a constraint on acceptance rules and against rational monotonicity as an axiom of conditional logic in situations of uncertainty. (shrink)
This article compares the ‘enfranchisement lottery’, a novel method for allocating the right to vote, with universal suffrage. The comparison is conducted exclusively on the basis of the expected consequences of the two systems. Each scheme seems to have a relative advantage. On the one hand, the enfranchisement lottery would create a better informed electorate and thus improve the quality of electoral outcomes. On the other hand, universal suffrage is more likely to ensure that elections are seen to be fair, (...) which is important for political stability. This article concludes that, on balance, universal suffrage is prima facie superior to the enfranchisement lottery. Yet the analysis shows that the instrumental case for the ‘one person, one vote’ principle is less conclusive than democratic theorists usually suppose. (shrink)
List and Pettit have stated an impossibility theorem about the aggregation of individual opinion states. Building on recent work on the lottery paradox, this paper offers a variation on that result. The present result places different constraints on the voting agenda and the domain of profiles, but it covers a larger class of voting rules, which need not satisfy the proposition-wise independence of votes.
The Lottery Paradox is generally thought to point at a conflict between two intuitive principles, to wit, that high probability is sufficient for rational acceptability, and that rational acceptability is closed under logical derivability. Gilbert Harman has offered a solution to the Lottery Paradox that allows one to stick to both of these principles. The solution requires the principle that acceptance licenses conditionalization. The present study shows that adopting this principle alongside the principle that high probability is sufficient for rational (...) acceptability gives rise to another paradox. (shrink)
Abstract I aim to explain why majority voting can be assumed to have an epistemic edge over lottery voting. This would provide support for majority voting as the appropriate decision mechanism for deliberative epistemic accounts of democracy. To argue my point, I first recall the usual arguments for majority voting: maximal decisiveness, fairness as anonymity, and minimal decisiveness. I then show how these arguments are over inclusive as they also support lottery voting. I then present a framework to measure accuracy (...) so as to compare the two decision mechanisms. I go over four arguments for lottery voting and three arguments for majority voting that support their respective accuracy. Lottery voting is then shown to have, compared to majority voting, a decreased probability of discrimination. That is, I argue that with lottery voting it is less probable under conditions of normal politics that if the procedure selects X, X is reasonable. I then provide two case scenarios for each voting mechanism that illustrate my point. Content Type Journal Article Pages 1-17 DOI 10.1007/s11158-011-9176-9 Authors Yann Allard-Tremblay, Departments of Philosophy, University of St Andrews, Edgecliffe, The Scores, St Andrews, Fife KY16 9AL, UK Journal Res Publica Online ISSN 1572-8692 Print ISSN 1356-4765. (shrink)
In a well-known paper entitled, ‘Survival Lottery’, published in a philosophical journal, John Harris proposed for discussion an interesting idea for saving the lives of certain kinds of patients who are at the point of death. Let us assume that there are two such patients, one that could be saved by a heart transplant and the other by the transplantation of a pair of lungs. However, no suitable organs are available for this purpose. Might it perhaps not be immoral to (...) select, by national lottery, a healthy person, who would be sacrificed, his organs used as transplants, and thus two lives be saved through the sacrifice of only one? This proposal is subjected first to a critical philosophical and ethical analysis, and then to a critical analysis from the point of view of Jewish Ethics as embodied in Halakhah. CiteULike Connotea Del.icio.us What's this? (shrink)
We will present a new lottery-style paradox on counterfactuals and chance. The upshot will be: combining natural assumptions on (i) the truth values of ordinary counterfactuals, (ii) the conditional chances of possible but non-actual events, (iii) the manner in which (i) and (ii) relate to each other, and (iv) 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 (consequents 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 (as we hope) 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)
The Lottery Paradox has been thought to provide a reductio argument against probabilistic accounts of inductive inference. As a result, much work in artificial intelligence has concentrated on qualitative methods of inference, including default logics, which are intended to model some varieties of inductive inference. It has recently been shown that the paradox can be generated within qualitative default logics. However, John Pollock's qualitative system of defeasible inference (named OSCAR), does avoid the Lottery Paradox by incorporating a rule designed (...) specifically for that purpose. I shall argue that Pollock's system instead succumbs to a worse disease: it fails to allow for induction at all (a disease sometimes known as "Conjunctivitis'). (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 (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)
Since utilities and probabilities jointly determine choices, event-dependent utilities complicate the elicitation of subjective event probabilities. However, for the usual purpose of obtaining the information embodied in agent beliefs, it is suﬃcient to elicit objective probabilities, i.e., probabilities obtained by updating a known common prior with that agent’s further information. Bayesians who play a Nash equilibrium of a certain insurance game before they obtain relevant information will afterward act regarding lottery ticket payments as if they had event-independent risk-neutral utility and (...) a known common prior. Proper scoring rules paid in lottery tickets can then elicit objective probabilities. (shrink)
Numerous instruments have been developed to elicit numerical values that represent the strength of preference for different health states. However, relatively few studies have attempted to analyse the reasoning processes that people employ when they are asked to answer questions based on these elicitation methods. The lottery equivalents method is a preference elicitation instrument that has recently received some attention in the literature. This study attempts a qualitative analysis of the use of this instrument on a group of 25 relatively (...) highly educated respondents. For each of three health states considered in the study, a substantial number of respondents refused to trade the chance of survival for a possible improvement in the health state. Therefore, many respondents violated an assumption that is necessary for the lottery equivalents instrument to generate cardinal health state values. These findings place a question mark against the usefulness of the lottery equivalents method, and add weight to the suspicion that ‘preferences’ are constructed according to how questions are framed. (Published Online July 31 2007). (shrink)
Any theory of knowledge that is fallibilist—i.e., that allows for one to have knowledge that could have been false or accidentally true—faces the lottery paradox. The paradox arises from the combination of two plausible claims: first, no one can know that one’s lottery ticket will lose prior to learning that it in fact has lost, and, second, the justification one has for the belief that one’s ticket will lose is just as good as the justification one has for paradigmatic instances (...) of knowledge. In thispaper, I offer a solution to the lottery paradox that is grounded in a thorough-going acceptance of fallibilism. (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)
The theory of justice pioneered by John Rawls explores a simple idea--that the concern of distributive justice is to compensate individuals for misfortune. Some people are blessed with good luck, some are cursed with bad luck, and it is the responsibility of society--all of us regarded collectively--to alter the distribution of goods and evils that arises from the jumble of lotteries that constitutes human life as we know it. Some are lucky to be born wealthy, or into a favorable (...) socializing environment, or with a tendency to be charming and intelligent and persevering and the like. These people are likely to be successful in the economic marketplace and to achieve success in other important ways over the course of their lives. On the other hand some people are, as we say, born to lose. Distributive justice stipulates that the lucky should transfer some or all of their gains due to luck to the unlucky. (shrink)
The oddities in lottery cases and Moore’s paradox appear to support the knowledge account of assertion, according to which one should assert only what one knows. This paper preserves an emphasis on epistemic norms but presents grounds for an alternative explanation. The alternative divides the explanandum, explaining the error in lottery and Moorean assertions with one move and their deeper incoherence with another. The error derives from a respect in which the assertions are uninformative: the speaker is not being appropriately (...) responsive to her addressee’s epistemic needs. And the incoherence derives from a deeper respect in which lottery and some (but not all) Moorean assertions are uninformative: it is difficult to see how the speaker’s assertion could express any judgment she has made or would relevantly make, since she transparently lacks epistemic authority to inform any conceivable interlocutor on the subject. This diagnosis suggests an epistemic approach not directly to assertion but to judgment. Without judging that p, how could a speaker be in the business of informing her addressee that p? If the speaker transparently lacks authority to inform anyone whether p – to give anyone her word that p – how could she without confusion count as judging that p? (shrink)
As the ongoing literature on the paradoxes of the Lottery and the Preface reminds us, the nature of the relation between probability and rational acceptability remains far from settled. This article provides a novel perspective on the matter by exploiting a recently noted structural parallel with the problem of judgment aggregation. After offering a number of general desiderata on the relation between finite probability models and sets of accepted sentences in a Boolean sentential language, it is noted that a number (...) of these constraints will be satisfied if and only if acceptable sentences are true under all valuations in a distinguished non-empty set W. Drawing inspiration from distance-based aggregation procedures, various scoring rule based membership conditions for W are discussed and a possible point of contact with ranking theory is considered. The paper closes with various suggestions for further research. (shrink)
My starting point is some widely accepted and intuitive ideas about justified, well-founded belief. By drawing on John Pollock’s work, I sketch a formal framework for making these ideas precise. Central to this framework is the notion of an inference graph. An inference graph represents everything that is relevant about a subject for determining which of her beliefs are justified, such as what the subject believes based on what. The strengths of the nodes of the graph represent the degrees of (...) justification of the corresponding beliefs. There are two ways in which degrees of justification can be computed within this framework. I argue that there is not any way of doing the calculations in a broadly probabilistic manner. The only alternative looks to be a thoroughly non-probabilistic way of thinking wedded to the thought that justification is closed under competent deduction. However, I argue that such a view is unable to capture the intuitive notion of justification, for it leads to an uncomfortable dilemma: either a widespread scepticism about justification, or drawing epistemically spurious distinctions between different types of lotteries. This should worry anyone interested in well-founded belief. (shrink)
The naïve see causal connections everywhere. Consider the fact that Evelyn Marie Adams won the New Jersey lottery twice. The naïve find it irresistible to think that this cannot be a coincidence. Maybe the lottery was rigged or perhaps some uncanny higher power placed its hand upon her brow. Sophisticates respond with an indulgent smile and ask the naïve to view Adams’ double win within a larger perspective. Given all the lotteries there have been, it isn’t at all surprising (...) that someone would win one of them twice. No need to invent conspiracy theories or invoke the paranormal – the double win was a mere coincidence. (shrink)
Fair lotteries offer familiar ways to pose a number of epistemological problems, prominently those of closure and of scepticism. Although these problems apply to many epistemological positions, in this paper I develop a variant of a lottery case to raise a difficulty with the reliabilist's fundamental claim that justification or knowledge is to be analyzed as a high truth-ratio (of the relevant belief-forming processes). In developing the difficulty broader issues are joined including fallibility and the relation of reliability to (...) understanding. (shrink)
Fairness is a central, but under-theorized, notion in moral and political philosophy. This paper makes two contributions. Firstly, it criticizes Broome’s seminal account of fairness in ( 1990–1991 ) Proc Aristotelian Soc 91:87–101, showing that there are problems with restricting fairness to a matter of relative satisfaction and holding that it does not itself require the satisfaction of the claims in question. Secondly, it considers the justification of lotteries to resolve cases of ties between competing claims, which Broome claims (...) as support for his theory, and contrasts random procedures to contests of skill, which may also be considered lotteries in a broader sense. I offer no alternative account of fairness of my own, but hope to point the way for future research on the nature of fairness. (shrink)