Resolving Bertrand’s probability paradox.

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)

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)

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)

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)

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)

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)

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)

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

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 (...) class='Hi'>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)

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)

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)

_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)

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. (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.

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)

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.

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 mathematical prowess of pure mathematician J. E. Littlewood (1885-1977), and of his elder cousin the mathematical educator Philippa Fawcett (1868-1948), is illustrated in the context of the Mathematical Tripos examination at Cambridge. Littlewood’s brilliant though highly condensed treatment in his splendid Miscellany (1953) of a perplexing problem from an old Tripos paper – familiar to some as “The Paradox of the Second Ace” – is then expanded with reference to Coxeter’s treatment of it in his revision of Rouse (...) Ball’s Mathematical Recreations (1939) and ultimately explained in the light of Wittgenstein’s later philosophy. The discussion touches on the nature of mathematical supposition. (shrink)

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.

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)

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)

Many groups make decisions over multiple interconnected propositions. The “doctrinal paradox” or “discursive dilemma” shows that propositionwise majority voting can generate inconsistent collective sets of judgments, even when individual sets of judgments are all consistent. I develop a simple model for determining the probability of the paradox, given various assumptions about the probability distribution of individual sets of judgments, including impartial culture and impartial anonymous culture assumptions. I prove several convergence results, identifying when the probability (...) of the paradox converges to 1, and when it converges to 0, as the number of individuals increases. Drawing on the Condorcet jury theorem and work by Bovens and Rabinowicz , I use the model to assess the “truth-tracking” performance of two decision procedures, the premise- and conclusion-based procedures. I compare the present results with existing results on the probability of Condorcet’s paradox. I suggest that the doctrinal paradox is likely to occur under plausible conditions. (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)

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)

It is tempting to think that a process of choosing a point at random from the surface of a sphere can be probabilistically symmetric, in the sense that any two regions of the sphere which differ by a rotation are equally likely to include the chosen point. Isaacs, Hájek, and Hawthorne (2022) argue from such symmetry principles and the mathematical paradoxes of measure to the existence of imprecise chances and the rationality of imprecise credences. Williamson (2007) has argued from a (...) related symmetry principle to the failure of probabilistic regularity. We contend that these arguments fail, because they rely on auxiliary assumptions about probability which are inconsistent with symmetry to begin with. We argue, moreover, that symmetry should be rejected in light of this inconsistency, and because it has implausible decision- theoretic implications. -/- The weaker principle of probabilistic invariance says that the probabilistic comparison of any two regions is unchanged by rotations of the sphere. This principle supports a more compelling argument for imprecise probability. We show, however, that invariance is incompatible with mundane judgments about what is probable. Ultimately, we find reason to be suspicious of the application of principles like symmetry and invariance to nonmeasurable regions. (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)

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)

Ellsberg's (1961) famous paradox shows that decision-makers give events with ‘known’ probabilities a higher weight in their outcome evaluation. In the same article, Ellsberg suggests a preference representation which has intuitive appeal but lacks an axiomatic foundation. Schmeidler (1989) and Gilboa (1987) provide an axiomatisation for expected utility with non-additive probabilities. This paper introduces E-capacities as a representation of beliefs which incorporates objective information about the probability of events. It can be shown that the Choquet integral of an (...) E-capacity is the Ellsberg representation. The paper further explores properties of this representation of beliefs and provides an axiomatisation for them. (shrink)

Time travel is metaphysically possible. Nikk Effingham contends that arguments for the impossibility of time travel are not sound. Focusing mainly on the Grandfather Paradox, Effingham explores the ramifications of taking this view, discusses issues in probability and decision theory, and considers the potential dangers of travelling in time.

This paper engages with a specific problem concerning the relationship between descriptive and normative claims. Namely, if we understand that descriptive claims frequently contain normative assertions, and vice versa, how then do we interpret the traditionally rigid distinction that is made between the two, as ’Hume’s law’ or Moore’s ’naturalistic fallacy’ argument offered. In particular, Kripke’s interpretation of Wittgenstein’s ’rule-following paradox’ is specially focused upon in order to re-consider the rigid distinction. As such, the paper argues that if descriptive (...) and normative claims are not mutually exclusive, then we need a new framework with which to understand this relationship. In this regard, the paper borrows from concerns with vagueness, particularly using a degree-theoretic approach in terms of subjective probability, in an attempt to graphically figure out these differences. Consequently, the paper tentatively proposes the hyperbola model in which degrees of normativity and degrees of descriptivity could be expressed and measured. It is hoped, as a result, that this tentative proposal will contribute to deepening the debate on vagueness in general. (shrink)