It is possible that a fair coin tossed infinitely many times will always land heads. So the probability of such a sequence of outcomes should, intuitively, be positive, albeit miniscule: 0 probability ought to be reserved for impossible events. And, furthermore, since the tosses are independent and the probability of heads (and tails) on a single toss is half, all sequences are equiprobable. But Williamson has adduced an argument that purports to show that our intuitions notwithstanding, the probability of an (...) infinite sequence is 0. In this paper, I rebut his argument.No Abstract. (shrink)

A probability distribution is regular if no possible event is assigned probability zero. While some hold that probabilities should always be regular, three counter-arguments have been posed based on examples where, if regularity holds, then perfectly similar events must have different probabilities. Howson (2017) and Benci et al. (2016) have raised technical objections to these symmetry arguments, but we see here that their objections fail. Howson says that Williamson’s (2007) “isomorphic” events are not in fact isomorphic, but Howson is speaking (...) of set-theoretic representations of events in a probability model. While those sets are not isomorphic, Williamson’s physical events are, in the relevant sense. Benci et al. claim that all three arguments rest on a conflation of different models, but they do not. They are founded on the premise that similar events should have the same probability in the same model, or in one case, on the assumption that a single rotation-invariant distribution is possible. Having failed to refute the symmetry arguments on such technical grounds, one could deny their implicit premises, which is a heavy cost, or adopt varying degrees of instrumentalism or pluralism about regularity, but that would not serve the project of accurately modelling chances. (shrink)

We take it that we can exercise doxastic agency by reasoning and by making judgments. We take it, that is, that we can actively make up our minds by reasoning and judging. On what I call the ‘Standard View’ this is so because judgment can yield belief. It is typical to take it that judgments yield beliefs by causing them. But on the resultant understanding of the Standard View, I argue, it is unclear how judgment could play its role in (...) doxastic agency in the way we take it to. I therefore offer an alternative understanding of how judgment yields belief. Drawing on Ryle I argue that when one comes to believe by judging the event which is one's judging is token identical to the event which is one's coming to believe. This paves the way for version of the Standard View capable of explaining how we can actively make up our minds despite that we cannot believe or come to believe at will. (shrink)

Isn't probability 1 certainty? If the probability is objective, so is the certainty: whatever has chance 1 of occurring is certain to occur. Equivalently, whatever has chance 0 of occurring is certain not to occur. If the probability is subjective, so is the certainty: if you give credence 1 to an event, you are certain that it will occur. Equivalently, if you give credence 0 to an event, you are certain that it will not occur. And so on for other (...) kinds of probability, such as evidential probability. The formal analogue of this picture is the regularity constraint: a probability distribution over sets of possibilities is regular just in case it assigns probability 0 only to the null set, and therefore probability 1 only to the set of all possibilities. (shrink)

This book is an introduction to probability theory covering laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a comprehensive treatment concentrating on the results that are the most useful for applications. Its philosophy is that the best way to learn probability is to see it in action, so there are 200 examples and 450 problems.

Many philosophers have become worried about the use of standard real numbers for the probability function that represents an agent's credences. They point out that real numbers can't capture the distinction between certain extremely unlikely events and genuinely impossible ones—they are both represented by credence 0, which violates a principle known as “regularity.” Following Skyrms 1980 and Lewis 1980, they recommend that we should instead use a much richer set of numbers, called the “hyperreals.” This essay argues that this popular (...) view is the result of two mistakes. The first mistake, which this essay calls the “numerical fallacy,” is to assume that a distinction that isn't represented by different numbers isn't represented at all in a mathematical representation. In this case, the essay claims that although the real numbers do not make all relevant distinctions, the full mathematical structure of a probability function does. The second mistake is that the hyperreals make too many distinctions. They have a much more complex structure than credences in ordinary propositions can have, so they make distinctions that don't exist among credences. While they might be useful for generating certain mathematical models, they will not appear in a faithful mathematical representation of credences of ordinary propositions. (shrink)

Escapism is defined as the attempt to avoid awareness of aversive beliefs. Strategies, and a few examples, of escapism are discussed. It is argued that self-deception is one species of escapism and that entrenched escapism, escapism pursued with the intention of permanently avoiding any awareness of one's belief, no matter what happens, is theoretically irrational, except in the special case where it compensates for irrationality elsewhere, by guarding one from the formation of further irrational beliefs of more serious import than (...) the belief one wishes to avoid. The model of rationality employed in this argument is then extended to practical rationality, and it is argued that entrenched escapism is pragmatically irrational as well, unless it compensates for other irrationalities elsewhere in a person, as, for instance, when a person must avoid certain facts to avoid succumbing irrationally to despair, or unless it compensates for the effects of an environment in which it is otherwise impossible for an optimally functioning person to survive emotionally. The results for entrenched escapism would apply to self-deception as well, since it aims (in all but very odd cases) at the permanent removal of the offending belief. The function of escapism, then, is to compensate for irrational patterns of belief formation, and to maintain effectiveness (for the sake of continuance of the species), insofar as that is possible, in situation in which a rational person would succumb to despair and suicide. (shrink)

Classic analysis of the subject and the development of personal probability; one of the greatest controversies in modern statistcal thought.

000000001. Introduction Call a theory of the good—be it moral or prudential—aggregative just in case (1) it recognizes local (or location-relative) goodness, and (2) the goodness of states of affairs is based on some aggregation of local goodness. The locations for local goodness might be points or regions in time, space, or space-time; or they might be people, or states of nature.1 Any method of aggregation is allowed: totaling, averaging, measuring the equality of the distribution, measuring the minimum, etc.. Call (...) a theory of the good finitely additive just in case it is aggregative, and for any finite set of locations it aggregates by adding together the goodness at those locations. Standard versions of total utilitarianism typically invoke finitely additive value theories (with people as locations). A puzzle can arise when finitely additive value theories are applied to cases involving an infinite number of locations (people, times, etc.). Suppose, for example, that temporal locations are the locus of value, and that time is discrete, and has no beginning or end.2 How would a finitely additive theory (e.g., a temporal version of total utilitarianism) judge the following two worlds? Goodness at Locations (e.g. times) w1:..., 2, 2, 2, 2, 2, 2, 2, 2, 2, ..... w2:..., 1, 1, 1, 1, 1, 1, 1, 1, 1, ..... Example 1 At each time w1 contains 2 units of goodness and w2 contains only 1. Intuitively, we claim, if the locations are the same in each world, finitely additive theorists will want to claim that w1 is better than w2. But it's not clear how they could coherently hold this view. For using standard mathematics the sum of each is the same infinity, and so there seems to be no basis for claiming that one is better than the other.3 (Appealing to Cantorian infinities is of no help here, since for any Cantorian infinite N, 2xN=1xN.). (shrink)

What is it to deceive someone? And how is it possible to deceive oneself? Does self-deception require that people be taken in by a deceitful strategy that they know is deceitful? The literature is divided between those who argue that self-deception is intentional and those who argue that it is non-intentional. In this study, Annette Barnes offers a challenge to both the standard characterisation of other-deception and current characterizations of self-deception, examining the available explanations and exploring such questions as the (...) self-deceiver's false consciousness, bias, and the irrationality and objectionability of self-deception. She arrives at a non-intentional account of self-deception that is deeper and more complete than alternative non-intentional accounts and avoids the reduction of self-deceptive belief to wishful belief. (shrink)

Philosophical accounts of self-deception can be divided into two broad groups – the intentionalist and the anti-intentionalist. On intentionalist models what happens in the central cases of self-deception is parallel to what happens when one person intentionally deceives another, except that deceiver and deceived are the same person. This paper offers a positive argument for intentionalism about self-deception and defends the view against standard objections.

We address the question of how finitely additive moral value theories (such as utilitarianism) should rank worlds when there are an infinite number of locations of value (people, times, etc.). In the finite case, finitely additive theories satisfy both Weak Pareto and a strong anonymity condition. In the infinite case, however, these two conditions are incompatible, and thus a question arises as to which of these two conditions should be rejected. In a recent contribution, Hamkins and Montero (2000) have argued (...) in favor of an anonymity-like isomorphism principle and against Weak Pareto. After casting doubt on their criticism of Weak Pareto, we show how it, in combination with certain other plausible principles, generates a plausible and fairly strong principle for the infinite case. We further show that where locations are the same in all worlds, but have no natural order, this principle turns out to be equivalent to a strengthening of a principle defended by Vallentyne and Kagan (1997), and also to a weakened version of the catching-up criterion developed by Atsumi (1965) and by von Weizsäcker (1965). Footnotes1 For valuable comments, we would like to thank Marc Fleurbaey, Bart Capéau, Joel Hamkins, Barbara Montero, Tim Mulgan, and two anonymous referees for this journal. (shrink)

In Part I, I consider the normal contexts of assertions of belief and declarations of intentions, arguing that many action-guiding beliefs are accepted uncritically and even pre-consciously. I analyze the function of avowals as expressions of attempts at self-transformation. It is because assertions of beliefs are used to perform a wide range of speech acts besides that of speaking the truth, and because there is a large area of indeterminacy in such assertions, that self-deception is possible. In Part II, I (...) analyze the conditions of self-deception, and discuss the grounds on which it is regarded as irrational, even when particular instances may be beneficial. I consider some of the classical analyses of the motives for self-deception, and attempt to give an account of the occasions in which it is likely to occur. In the final section, I discuss the complex organization of the self that is presupposed by the phenomena of self-deception. (shrink)

The philosophical difficulties presented by self-deception are vexed and multifaceted. One such difficulty is what I call the ‘doxastic problem’ of self-deception. Solving the doxastic problem involves determining whether someone in a state of self-deception that ∼p both believes that p and believes that ∼p, simply holds one or the other belief, or, as I will argue, holds neither. This final option, which has been almost entirely overlooked to-date, is what I call ‘ nondoxasticism ’ about self-deception. In this article, (...) I present a negative case for nondoxasticism according to which, in the paradigm case of self-deception, there is no explanatory need to attribute the self-deceived person either their undesired belief that p, or their desired belief that ∼p. Folk psychology is replete with concepts other than belief, and if we bear this in mind, it becomes clear that the explanatory roles for which the self-deceived person's purported beliefs have traditionally been enlisted can be comfortably filled without recourse to belief. (shrink)

Kolmogorov''s axiomatization of probability includes the familiarratio formula for conditional probability: 0).$$ " align="middle" border="0">.

What if human joy went on endlessly? Suppose, for example, that each human generation were followed by another, or that the Western religions are right when they teach that each human being lives eternally after death. If any such possibility is true in the actual world, then an agent might sometimes be so situated that more than one course of action would produce an infinite amount of utility. Deciding whether to have a child born this year rather than next is (...) a situation wherein an agent may face several alternatives whose effects could well ramify endlessly on such suppositions, for the child born this year would be a different person—one who preferred different things, performed different actions, and had different descendants—from a child born next year. It has recently been suggested that traditional utilitarianism stumbles on such cases of infinite utility. Specifically, utilitarianism seems to require, for its application, that all experience of pleasure and pain cease at some time in the future or asymptotically approach zero.2 If neither of these conditions holds, then the utility produced by each of two alternative actions may turn out to be infinite, and utilitarianism thus loses its ability to discriminate morally between them. (shrink)

Is it possible for me to believe what I know not to be the case? It certainly does not seem possible for me, at the same time, to be aware of the fact that a given proposition is true and yet believe that the proposition is false. Models of self?deception which have the implication that this is possible are usually described as ?paradoxical?. However, many philosophers believe that there are genuine cases of self?deception which non?paradoxical models of self?deception mirror and (...) elucidate. In the present article the author considers what he takes to be the leading contenders among non?paradoxical (or seemingly non?paradoxical) models of self?deception. He concludes from his analysis that it is not clear that any of these models mirror actual cases of self?deception and that it is not even obvious that there are actual cases of self?deception to be mirrored. (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)