This paper aims to investigate the so-called paradox of deterministic probabilities: in a deterministic world, all probabilities should be subjective; however, they also seem to play important explanatory and predictive roles which suggest they are objective. The problem is then to understand what these deterministic probabilities are. Recent proposed solutions of this paradox are the Mentaculus vision, the range account of probability, and a version of frequentism based on typicality. All these approaches aim at defining deterministic objective (...) probabilities as to make them compatible with determinism. In this paper I argue that one can think of the equivalent of subjective and objective deterministic probabilities in terms of typicality. Also, I show that only what I identify with objective probabilities play the necessary explanatory and predictive roles, while the subjective components essentially guide beliefs. In this way, the paradox is solved. (shrink)

The so-called Preface Paradox seems to show that one can rationally believe two logically incompatible propositions. We address this puzzle, relying on the notions of truthlikeness and approximate truth as studied within the post-Popperian research programme on verisimilitude. In particular, we show that adequately combining probability, approximate truth, and truthlikeness leads to an explanation of how rational belief is possible in the face of the Preface Paradox. We argue that our account is superior to other solutions of the paradox, including (...) a recent one advanced by Hannes Leitgeb (Analysis 74.1). (shrink)

Cumulative Prospect Theory (CPT) does not explain the St. Petersburg Paradox. We show that the solutions related to probability weighting proposed to solve this paradox, (Blavatskyy, Management Science 51:677–678, 2005; Rieger and Wang, Economic Theory 28:665–679, 2006) have to cope with limitations. In that framework, CPT fails to accommodate both gambling and insurance behavior. We suggest replacing the weighting functions generally proposed in the literature by another specification which respects the following properties: (1) to solve the paradox, the slope at (...) zero has to be finite. (2) to account for the fourfold pattern of risk attitudes, the probability weighting has to be strong enough. (shrink)

A festschrift for Dorothy Edgington, containing contributions from Cleo Condoravdi, Dorothy Edgington, Kit Fine, Alan Hájek, John Hawthorne, Sabine Iatridou, Nick Jones, Rosanna Keefe, Angelika Kratzer, David Over, Daniel Rothschild, Robert Stalnaker, Scott Sturgeon, and Timothy Williamson.

We begin by showing that every theory of the value of uncertain prospects must have one of three unpalatable properties. _Reckless_ theories recommend giving up a sure thing, no matter how good, for an arbitrarily tiny chance of enormous gain; _timid_ theories permit passing up an arbitrarily large potential gain to prevent a tiny increase in risk; _non-transitive_ theories deny the principle that, if A is better than B and B is better than C, then A must be better than (...) C. Having set up this trilemma, we study its horns. Non-transitivity has been much discussed; we focus on drawing out the costs and benefits of recklessness and timidity when it comes to axiology, decision theory, and normative uncertainty. (shrink)

For a finite universe of discourse, if Φ → and ~(Ψ → Φ) , then P(Ψ) > P(Φ), i.e., there is always a loss of information, there is an increase in probability, in a non reversible implication. But consider the two propositions, "All ravens are black", (i.e., "(x)(Rx ⊃ Bx)"), and "Some ravens are black" (i.e., "(∃x)(Rx & Bx)"). In a world of one individual, called "a", these two propositions are equivalent to "~Ra ∨ Ba" and "Ra & Ba" respectively. (...) However, (Ra & Ba) → (~Ra ∨ Ba) and ~[(~Ra ∨ Ba) → (Ra & Ba)]. Consequently, in a world of one individual it is more probable that all ravens are black than that some ravens are black! (shrink)

Consider the regularity thesis that each possible event has non-zero probability. Hájek challenges this in two ways: there can be nonmeasurable events that have no probability at all and on a large enough sample space, some probabilities will have to be zero. But arguments for the existence of nonmeasurable events depend on the axiom of choice. We shall show that the existence of anything like regular probabilities is by itself enough to imply a weak version of AC sufficient (...) to prove the Banach–Tarski Paradox on the decomposition of a ball into two equally sized balls, and hence to show the existence of nonmeasurable events. This provides a powerful argument against unrestricted orthodox Bayesianism that works even without AC. A corollary of our formal result is that if every partial order extends to a total preorder while maintaining strict comparisons, then the Banach–Tarski Paradox holds. This yields an argument that incommensurability cannot be avoided in ambitious versions of decision theory. (shrink)

Nebel argues for the Repugnant Conclusion via the “Intrapersonal Repugnant Conclusion,” on which certainty of a mediocre life is better for individuals than a sufficiently small chance of an excellent life. In this article, I deny that acceptance of the Intrapersonal Repugnant Conclusion leads us to the Repugnant Conclusion. I point out that on many views which avoid the Repugnant Conclusion we should discount very small probabilities down to zero. If we do, then Nebel’s crucial premise of Ex Ante (...) Pareto fails, because discounting at the individual level can fail to match up with discounting at the population level. (shrink)

Negative probabilities were several times proposed in the literature as a way to reconcile violation of Bell-type inequalities with the premise of local realism. It is argued that instead of using negative probabilities that have no physical meaning one can use for this purpose fuzzy probabilities that have sound and unambiguous interpretation.

Classic work by one of the most brilliant figures in post-war analytic philosophy.

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 – amongst them a relatively strong structural assumption to the effect that the underlying probability space is both finite and uniform. In (...) this paper, 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)

Exhumation and study of the 1945 paradox of confirmation brings out the defect of its formulation. In the context of quantifier conditional-probability logic it is shown that a repair can be accomplished if the truth-functional conditional used in the statement of the paradox is replaced with a connective that is appropriate to the probabilistic context. Description of the quantifier probability logic involved in the resolution of the paradox is presented in stages. Careful distinction is maintained between a formal logic language (...) and its semantics, as the same language may be outfitted with different semantics. An acquaintance with sections 1? 5 of Hailperin (2006) covering the sentential aspects of probability logic is assumed as background information for quantifier probability logic. (shrink)

Synthetic biology is a cutting-edge area of research that holds the promise of unprecedented health benefits. However, in tandem with these large prospective benefits, synthetic biology projects entail a risk of catastrophic consequences whose severity may exceed that of most ordinary human undertakings. This is due to the peculiar nature of synthetic biology as a ‘threshold technology’ which opens doors to opportunities and applications that are essentially unpredictable. Fears about these potentially unstoppable consequences have led to declarations from civil society (...) groups calling for the use of a precautionary principle to regulate the field. Moreover, the principle is prevalent in law and international agreements. Despite widespread political recognition of a need for caution, the precautionary principle has been extensively criticized as a guide for regulatory policy. We examine a central objection to the principle: that its application entails crippling inaction and incoherence, since whatever action one takes there is always a chance that some highly improbable cataclysm will occur. In response to this difficulty, which we call the ‘precautionary paradox,’ we outline a deliberative means for arriving at threshold of probability below which potential dangers can be disregarded. In addition, we describe a Bayesian mechanism with which to assign probabilities to harmful outcomes. We argue that these steps resolve the paradox. The rehabilitated PP can thus provide a viable policy option to confront the uncharted waters of synthetic biology research. (shrink)

We call something a paradox if it strikes us as peculiar in a certain way, if it strikes us as something that is not simply nonsense, and yet it poses some difficulty in seeing how it could make sense. When we examine paradoxes more closely, we find that for some the peculiarity is relieved and for others it intensifies. Some are peculiar because they jar with how we expect things to go, but the jarring is to do with imprecision and (...) misunderstandings in our thought, failures to appreciate the breadth of possibility consistent with our beliefs. Other paradoxes, however, pose deep problems. Closer examination does not explain them away. Instead, they challenge the coherence of certain conceptual resources and hence challenge the significance of beliefs which deploy those resources. I shall call the former kind weak paradoxes and the latter, strong paradoxes. Whether a particular paradox is weak or strong is sometimes a matter of controversy—sometimes it has been realised that what was thought strong is in fact weak, and vice versa,— but the distinction between the two kinds is generally thought to be worth drawing. In this Cchapter, I shall cover both weak and strong probabilistic paradoxes. (shrink)

루만에게 사회적 체계들은 커뮤니케이션의 체계들이다. 매체/형식의 구별은 체계의 폐쇄성과 개방성이라는 역설을 커뮤니케이션의 관점에서 나타낸다. 매체는 요소들의 느슨한 결합에 의한 비결정적 상태이므로 그 자체는 커뮤니케이션의 비개연성을 나타낼 뿐이다. 따라서 커뮤니케이션의 비개연성을 개연성으로 전환하기 위해서는 느슨한 결합을 단단한 결합으로 엮어주는 형식이 불가피하다. 형식은 체계와 마찬가지로 내부(체계)와 외부(환경)를 구별하는 엄격한 질서다. 그러나 커뮤니케이션의 체계가 지속적으로 유지되기 위해서는 형식의 변화 혹은 새로운 형식의 출현이 불가피하다. 이렇듯 매체는 형식을 통하여 커뮤니케이션의 개연성의 확립에 기여하지만, 여전히 제거되지 않은 비개연성을 통해서 형식의 변화를 촉진한다. 비개연적 개연성이라는 역설적 상황은 (...) 그가 들고 있는 세 가지 매체, 즉 언어, 확산매체, 성공매체에 모두 적용된다. (shrink)

The Borel-Kolmogorov paradox is often presented as an obscure problem that certain mathematical accounts of conditional probability must face. In this article, we point out that the paradox arises in the physical sciences, for physical probability or chance. By carefully formulating the paradox in this setting, we show that it is a puzzle for everyone, regardless of one’s preferred probability formalism. We propose a treatment that is inspired by the approach that scientists took when confronted with these cases.

We provide a 'verisimilitudinarian' analysis of the well-known Linda paradox or conjunction fallacy, i.e., the fact that most people judge the probability of the conjunctive statement "Linda is a bank teller and is active in the feminist movement" (B & F) as more probable than the isolated statement "Linda is a bank teller" (B), contrary to an uncontroversial principle of probability theory. The basic idea is that experimental participants may judge B & F a better hypothesis about Linda as compared (...) to B because they evaluate B & F as more verisimilar than B. In fact, the hypothesis "feminist bank teller", while less likely to be true than "bank teller", may well be a better approximation to the truth about Linda. (shrink)

Resolving Bertrand’s probability paradox.

Dorothy Edgington’s work has been at the centre of a range of ongoing debates in philosophical logic, philosophy of mind and language, metaphysics, and epistemology. This work has focused, although by no means exclusively, on the overlapping areas of conditionals, probability, and paradox. In what follows, I briefly sketch some themes from these three areas relevant to Dorothy’s work, highlighting how some of Dorothy’s work and some of the contributions of this volume fit in to these debates.

_Paradoxes from A to Z, Third edition_ is the essential guide to paradoxes, and takes the reader on a lively tour of puzzles that have taxed thinkers from Zeno to Galileo, and Lewis Carroll to Bertrand Russell. Michael Clark uncovers an array of conundrums, such as Achilles and the Tortoise, Theseus’ Ship, and the Prisoner’s Dilemma, taking in subjects as diverse as knowledge, science, art and politics. Clark discusses each paradox in non-technical terms, considering its significance and looking at likely (...) solutions. This third edition is revised throughout, and adds nine new paradoxes that have important bearings in areas such as law, logic, ethics and probability. Paradoxes from A to Z, Third edition is an ideal starting point for those interested not just in philosophical puzzles and conundrums, but anyone seeking to hone their thinking skills. (shrink)

Given any proposition, is it possible to have rationally acceptable attitudes towards it? Absent reasons to the contrary, one would probably think that this should be possible. In this paper I provide a reason to the contrary. There is a proposition such that, if one has any opinions about it at all, one will have a rationally unacceptable set of propositional attitudes—or if one doesn’t, one will end up being cognitively imperfect in some other manner. The proposition I am concerned (...) with is a self-referential propositional attitude ascription involving the propositional attitude of rejection. Given a basic assumption about what constitutes irrationality, and a few assumptions about the nature of cognitively ideal agents, a paradox results. This paradox is superficially like the Liar, but it is importantly different in that no alethic notions are involved at all. As such, it stands independent of the Liar and is not a ‘revenge’ version of it. After setting out the paradox I discuss possible responses. After considering several I argue that one is best off simply accepting that the paradox shows us something surprising and interesting about rationality: that some cognitive shortfall is unavoidable even for ideal agents. I argue that nothing disastrous follows from accepting this conclusion. (shrink)

In this paper I argue that a basic problem in philosophical discussions of culture is what I call the “integration problem”: the need to provide an account of how distinctive considerations of culture can be integrated within practical deliberation in general. I then show how the failure to resolve this problem generates three paradoxes, which I call the “cosmopolitan paradox,” the “inclusion paradox,” and the “representation paradox.” I argue that these paradoxes arise from a common source, the attempt to separate (...) out determinations of worth from demands of recognition, and both from socially contested deliberative practices. I conclude by suggesting that resolving these paradoxes probably requires not a theoretical solution but the achievement of a fully inclusive, cosmopolitan culture. (shrink)

Bayes Rule specifies how probabilities over parameters should be updated given any kind of information. But in some cases, the kind of information provided by both simulation and physical experiments is information on how certain output parameters may change when other input parameters are changed. There are three different approaches to this problem, one of which leads to the Garbage-In/garbage-out Paradox, the second of which violates the Borel Paradox, and the third of which is a supra-Bayesian heuristic. This paper (...) shows how to derive a fully Bayesian formula which avoids the Garbage-In/garbage-out and Borel Paradoxes. We also compare a Laplacian approximation of this formula with Bayesian synthesis and Bayesian melding and find that the Bayesian formula sometimes coincides with the Bayesian melding solution. (shrink)

A probability function is non-conglomerable just in case there is some proposition E and partition \ of the space of possible outcomes such that the probability of E conditional on any member of \ is bounded by two values yet the unconditional probability of E is not bounded by those values. The paradox of non-conglomerability is the counterintuitive—and controversial—claim that a rational agent’s subjective probability function can be non-conglomerable. In this paper, I present a qualitative analogue of the paradox. I (...) show that, under antecedently plausible assumptions, an analogue of the paradox arises for rational comparative confidence. As I show, the qualitative paradox raises its own distinctive set of philosophical issues. (shrink)

What is a paradox? -- Knowing and believing -- Vagueness and identity -- Logic and truth -- Mathematical paradoxes -- Probability paradoxes -- Space and time -- Impossibilities -- Deciding and acting -- Index of philosophers.

The book was planned and written as a single, sustained argument. But earlier versions of a few parts of it have appeared separately. The object of this book is both to establish the existence of the paradoxes, and also to describe a non-Pascalian concept of probability in terms of which one can analyse the structure of forensic proof without giving rise to such typical signs of theoretical misfit. Neither the complementational principle for negation nor the multiplicative principle for conjunction applies (...) to the central core of any forensic proof in the Anglo-American legal system. There are four parts included in this book. Accordingly, these parts have been written in such a way that they may be read in different orders by different kinds of reader. (shrink)

In their development of causal decision theory, Allan Gibbard and William Harper advocate a particular method for calculating the expected utility of an action, a method based upon the probabilities of certain counterfactuals. Gibbard and Harper then employ their method to support a two-box solution to Newcomb’s paradox. This paper argues against some of Gibbard and Harper’s key claims concerning the truth-values and probabilities of counterfactuals involved in expected utility calculations, thereby disputing their analysis of Newcomb’s Paradox. If (...) we are right, then Gibbard and Harper’s method of calculating expected utility does not adequately represent rational choice. (shrink)

I present a puzzle which seems simple, but is found to have interesting implications for confirmation. Its dissolution also helps us to throw light on the relationship between first- and second-order probabilities construed as rational degrees of belief.

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)

Hempel's high probability requirement asserts that any rationally acceptable answer to the question 'Why did event X occur?' must offer information which shows that X was to be expected at least with reasonable probability. Salmon rejected this requirement in his S-R model. This led to a series of paradoxical consequences, such as the assertion that an explanation of an event can both lower its probability and make it arbitrarily low, and the assertion that the explanation of an outcome would (...) have qualified as an explanation of its non-occurrence as well. We argue that if inductive explanations are to be seen as generalizations of the causal-deterministic model, or if they are to be seen as satisfying the requirement--fulfilled by the D-N model--that explanations ought to identify certain features of the universe that are nomically responsible for the explanadum event, then the high probability requirement seems to be unacceptable. If this is so, a realistically inspired theory of inductive explanation will be committed to the paradoxes that follow from Salmon's model. (shrink)

Many philosophers argue that Keynes’s concept of the “weight of arguments” is an important aspect of argument appraisal. The weight of an argument is the quantity of relevant evidence cited in the premises. However, this dimension of argumentation does not have a received method for formalisation. Kyburg has suggested a measure of weight that uses the degree of imprecision in his system of “Evidential Probability” to quantify weight. I develop and defend this approach to measuring weight. I illustrate the usefulness (...) of this measure by employing it to develop an answer to Popper’s Paradox of Ideal Evidence. (shrink)