This article explores some open questions related to the problem of verification of theories in the context of empirical sciences by contrasting three epistemological frameworks. Each of these epistemological frameworks is based on a corresponding central metaphor, namely: (a) Neo-empiricism and the gambling metaphor; (b) Popperian falsificationism and the scientific tribunal metaphor; (c) Cognitive constructivism and the object as eigen-solution metaphor. Each of one of these epistemological frameworks has also historically co-evolved with a certain statistical theory and method for testing (...) scientific hypotheses, respectively: (a) Decision theoretic Bayesian statistics and Bayes factors; (b) Frequentist statistics and p-values; (c) Constructive Bayesian statistics and e-values. This article examines with special care the Zero Probability Paradox (ZPP), related to the verification of sharp or precise hypotheses. Finally, this article makes some remarks on Lakatos’ view of mathematics as a quasi-empirical science. (shrink)

The topic of this thesis is how we should treat tiny probabilities of vast value. This thesis consists of six independent papers. Chapter 1 discusses the idea that utilities are bounded. It shows that bounded decision theories prescribe prospects that are better for no one and worse for some if combined with an additive axiology. Chapter 2, in turn, points out that standard axiomatizations of Expected Utility Theory violate dominance in cases that involve possible states of zero probability. (...) Chapters 3–6 discuss the idea that we should ignore tiny probabilities in practical decision-making. Chapter 3 argues that discounting small probabilities solves the ‘Intrapersonal Addition Paradox’ and thus helps avoid the Repugnant Conclusion. Chapter 4 explores what the most plausible version of this view might look like and what problems the different versions have. Chapter 5 focuses on one type of problem, namely, money pumps. The Independence Money Pump, in particular, presents a difficult challenge for those who wish to discount small probabilities. Finally, Chapter 6 discusses the implications of discounting small probabilities for the value of the far future. (shrink)

The most well-known example of the impossibility of traveling in time is the grandfather paradox or self-infanticide argument: a person who travels in the past and kills his own grandfather, thus preventing the existence of one of his parents and thus his own existence. A philosophical response to this paradox would be the impossibility of changing the past, like Novikov self-consistency principle (if an event exists that would cause a paradox or any "change" to the past whatsoever, (...) then the probability of that event is zero, thus it would be impossible to create time paradoxes). DOI: 10.13140/RG.2.2.31279.79521. (shrink)

A thesis (“weak BCP”) nearly universally held among philosophers of probability connects the concepts of objective chance and metaphysical modality: Any prospect (outcome) that has a positive chance of obtaining is metaphysically possible—(nearly) equivalently, any metaphysically impossible prospect has zero chance. Particular counterexamples are provided utilizing the monotonicity of chance, one of them related to the four world paradox. Explanations are offered for the persistent feeling that there cannot be chancy metaphysical necessities or chancy metaphysical impossibilities. Chance (...) is objective but contrary to popular opinion it is also largely epistemic. Chancy necessities are analogous to necessary a posteriori truths. Philosophy and Phenomenological Research, Volume 99, Issue 3, Page 644-662, November 2019. (shrink)

In this paper, a metaphysics is proposed that includes everything that can be represented by a well-founded multiset. It is shown that this metaphysics, apart from being self-explanatory, is also benevolent. Paradoxically, it turns out that the probability that we were born in another life than our own is zero. More insights are gained by inducing properties from a metaphysics that is not self-explanatory. In particular, digital metaphysics is analyzed, which claims that only computable things exist. First of (...) all, it is shown that digital metaphysics contradicts itself by leading to the conclusion that the shortest computer program that computes the world is infinitely long. This means that the Church-Turing conjecture must be false. Secondly, the applicability of Occam’s razor is explained by evolution: in an evolving physics it can appear at each moment as if the world is caused by only finitely many things. Thirdly and most importantly, this metaphysics is benevolent in the sense that it organizes itself to fulfill the deepest wishes of its observers. Fourthly, universal computers with an infinite memory capacity cannot be built in the world. And finally, all the properties of the world, both good and bad, can be explained by evolutionary conservation. (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)

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)

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)

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)

In response to cases that involve tiny probabilities of huge payoffs, some argue that we ought to discount small probabilities down to zero. However, this paper shows that doing so violates Independence and Continuity, and as a result of these violations, those who discount small probabilities can be exploited by money pumps. Various possible ways of avoiding exploitation will be discussed. This paper concludes that the money pump for Independence undermines the plausibility of discounting small probabilities. Much of the (...) discussion on Independence generalizes to other views that also violate Independence. (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.

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)

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)

This paper presents a paradox based on the following assumptions: that emotions are warranted when you are justified in thinking that the emotion is fitting, that there are warranted cases of past-directed fear, that fear is fitting in the face of its formal object: dangerousness, and that this formal object consists in a probability of damage or harm to something of value. The paper then discusses three likely solutions: (1) denying that past-directed fear can be warranted, (2) using (...) an alternative formulation of fear's formal object, and (3) giving up warrant as justified as fittingness. Finally, it provides a case for the third solution and presents a warrant-based approach to fittingness and the appropriateness of emotions in general. (shrink)

Enjoying great popularity in decision theory, epistemology, and philosophy of science, Bayesianism as understood here is fundamentally concerned with epistemically ideal rationality. It assumes a tight connection between evidential probability and ideally rational credence, and usually interprets evidential probability in terms of such credence. Timothy Williamson challenges Bayesianism by arguing that evidential probabilities cannot be adequately interpreted as the credences of an ideal agent. From this and his assumption that evidential probabilities cannot be interpreted as the actual credences (...) of human agents either, he concludes that no interpretation of evidential probabilities in terms of credence is adequate. I argue to the contrary. My overarching aim is to show on behalf of Bayesians how one can still interpret evidential probabilities in terms of ideally rational credence and how one can maintain a tight connection between evidential probabilities and ideally rational credence even if the former cannot be interpreted in terms of the latter. By achieving this aim I illuminate the limits and prospects of Bayesianism. (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)

The standard treatment of conditional probability leaves conditional probability undefined when the conditioning proposition has zero probability. Nonetheless, some find the option of extending the scope of conditional probability to include zero-probability conditions attractive or even compelling. This article reviews some of the pitfalls associated with this move, and concludes that, for the most part, probabilities conditional on zero-probability propositions are more trouble than they are worth.

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)

Paradoxes and their Resolutions is a ‘thematic compilation’ by Avi Sion. It collects in one volume the essays that he has written in the past (over a period of some 27 years) on this subject. It comprises expositions and resolutions of many (though not all) ancient and modern paradoxes, including: the Protagoras-Euathlus paradox (Athens, 5th Cent. BCE), the Liar paradox and the Sorites paradox (both attributed to Eubulides of Miletus, 4th Cent. BCE), Russell’s paradox (UK, 1901) (...) and its derivatives the Barber paradox and the Master Catalogue paradox (also by Russell), Grelling’s paradox (Germany, 1908), Hempel's paradox of confirmation (USA, 1940s), and Goodman’s paradox of prediction (USA, 1955). This volume also presents and comments on some of the antinomic discourse found in some Buddhist texts (namely, in Nagarjuna, India, 2nd Cent. CE; and in the Diamond Sutra, date unknown, but probably in an early century CE). (shrink)

This book is devoted to the presentation of the new quantum mechanical formalism based on the probability representation of quantum states. In the 20s and 30s it became evident that some properties in quantum mechanics can be assigned only to the quantum mechanical system, but not necessarily to its constituents. This led Einstein, Podolsky and Rosen (EPR) to their remarkable 1935 paper where they concluded that quantum mechanics is not a complete theory of nature (EPR paradox). In order (...) to avoid the contradiction which arises from instantaneous action at a distance mentioned above we introduce an extension of the canonical relativity by using measure algebra of physical events in Minkowski space-time. The canonical QM formalism is extended by additional new postulate of EPRB nonlocality for continuous and discrete observables, chapter I. The postulate of EPRB nonlocality is supported by new quantum mechanical formalism based on the probability representation of quantum states. Chapter II is devoted to the new quantum mechanical formalism based on the probability representation of quantum states. Chapter III is devoted to the Einstein's 1927 gedanken experiment resolution. Chapter IV is devoted to the EPR paradox resolution. Chapter V is devoted to the EPR-B paradox resolution. Chapter VI is devoted to the Schrödinger's cat (measured spin) paradox resolution. Chapter VII is devoted to the Bell inequalities revisited. Remind that the canonical arguments which were presented by many authors, namely, that violations of Bell type inequalities signal us that the classical Kolmogorovian model of probability is inapplicable to quantum phenomena. We claimed that the canonical assumption, under which Bell type inequalities were derived, is not supported by real physical nature of the EPRB experiments. The fundamental physical nature violations of the canonical Bell type inequalities explained by Postulate of EPR-Nonlocality and Heisenberg noise-disturbance uncertainty relations. (shrink)

Events between which we have no epistemic reason to discriminate have equal epistemic probabilities. Bertrand’s chord paradox, however, appears to show this to be false, and thereby poses a general threat to probabilities for continuum sized state spaces. Articulating the nature of such spaces involves some deep mathematics and that is perhaps why the recent literature on Bertrand’s Paradox has been almost entirely from mathematicians and physicists, who have often deployed elegant mathematics of considerable sophistication. At the same (...) time, the philosophy of probability has been left out. In particular, left out entirely are the philosophical ground of the principle of indifference, the nature of the principle itself, the stringent constraint this places on the mathematical representation of the principle needed for its application to continuum sized event spaces, and what these entail for rigour in developing the paradox itself. This book puts the philosophy and its entailments back in and in so doing casts a new light on the paradox, giving original analyses of the paradox, its possible solutions, the source of the paradox, the philosophical errors we make in attempting to solve it and what the paradox proves for the philosophy of probability. The book finishes with the author’s proposed solution—a solution in the spirit of Bertrand’s, indeed—in which an epistemic principle more general than the principle of indifference offers a principled restriction of the domain of the principle of indifference. (shrink)

I defend the thesis that legal standards of proof are reducible to thresholds of probability. Many reject this thesis because it appears to permit finding defendants liable solely on the basis of statistical evidence. To the contrary, I argue – by combining Thomson's (1986) causal analysis of legal evidence with formal methods of causal inference – that legal standards of proof can be reduced to probabilities, but that deriving these probabilities involves more than just statistics.

There is a trade-off between specificity and accuracy in existing models of belief. Descriptions of agents in the tripartite model, which recognizes only three doxastic attitudes—belief, disbelief, and suspension of judgment—are typically accurate, but not sufficiently specific. The orthodox Bayesian model, which requires real-valued credences, is perfectly specific, but often inaccurate: we often lack precise credences. I argue, first, that a popular attempt to fix the Bayesian model by using sets of functions is also inaccurate, since it requires us to (...) have interval-valued credences with perfectly precise endpoints. We can see this problem as analogous to the problem of higher order vagueness. Ultimately, I argue, the only way to avoid these problems is to endorse Insurmountable Unclassifiability. This principle has some surprising and radical consequences. For example, it entails that the trade-off between accuracy and specificity is in-principle unavoidable: sometimes it is simply impossible to characterize an agent’s doxastic state in a way that is both fully accurate and maximally specific. What we can do, however, is improve on both the tripartite and existing Bayesian models. I construct a new model of belief—the minimal model—that allows us to characterize agents with much greater specificity than the tripartite model, and yet which remains, unlike existing Bayesian models, perfectly accurate. (shrink)

We have ignored for a century that the incompleteness of Quantum Theory (QT) is inseparable from the incompleteness of Special Relativity (RT). In this article, I claim that the latter has been gravely incomplete vis à vis the former from 1927 until today. But completing RT in the light of QT is not as simple as merely postulating nonlocality and stochasticity as “elements of reality” (which is de facto done by most physicists and pragmatic philosophers); otherwise, RT would not still (...) be in a “peaceful” conflict with QT after a century. Vice versa, I contend that QT is incomplete vis à vis RT, though not for the reasons claimed in the iconic EPR paper. We then show how to complete the Ontology, Foundation, and Structure of both RT and QT and merge them into an internally consistent embracive theory I call QR/TOPI. This theory offers a more cogent and simpler avenue to integrate RT with QT than positing exotic causal structures like ‘retrocausality’, ‘future-input dependence’, ‘superdeterminism’ – not to mention the extravagant ‘Many-Worlds’, ‘Many-Minds’, ‘Parallel-Lives’, and other interpretations of QT like Many-Histories, QBism, etc. -/- QR/TOPI provides the “radical conceptual renewal” wished by John Bell so as to integrate probability and nonlocality into an upgraded RT and, reciprocally, to integrate Frame-Invariance into QT while at the same time, as demanded by 2022 Nobel laureate Anton Zeilinger, providing basic physical meaning to the resulting encompassing theory. The old outcast notion of absolute simultaneity is resurrected without any conflict with Einstein’s relative simultaneity, while Frame-Invariance is preserved via our Quantumlike Transformation (QLT), which is an extension of the Lorentz Transformation (LT): QLT includes what LT excludes: nonlocality. -/- Section 1 examines the philosophical foundations of Space and Time, focusing on RT, its plethora of empirical validations, and the tenets which make it incompatible with QT. Section 2 incorporates stochasticity into RT. Sections 3 through 5 gradually introduce QR/TOPI for mono-, bi-, and tri-quanton systems, with full consideration of Bell Theorem, nonlocality, teleportation, and their implications. Section 6 attempts to review the current status quo. Section 7 makes the case for the incompleteness of RT and QT. Section 8 explains how to complete and integrate both theories so as to formally develop QR/TOPI. Finally, in Section 9, via multiple experimental setups, I zero in on Zeilinger’s basic question: “what does this really mean in a basic way?”. (shrink)

Brock and Glasgow recently introduced a new definition of paradox and argue that this conception of paradox itself leads to paradox, the so-called Paradox Paradox. I show that they beg the questions during the course of their argument, but, more importantly, do so in a philosophically interesting way: it reveals a counterexample to the equivalence between being a logical truth and having a probability of one. This has consequences regarding norms of rationality, undermining the (...) grounds for the Conjunction Fallacy. (shrink)

The theory of lower previsions is designed around the principles of coherence and sure-loss avoidance, thus steers clear of all the updating anomalies highlighted in Gong and Meng's "Judicious Judgment Meets Unsettling Updating: Dilation, Sure Loss, and Simpson's Paradox" except dilation. In fact, the traditional problem with the theory of imprecise probability is that coherent inference is too complicated rather than unsettling. Progress has been made simplifying coherent inference by demoting sets of probabilities from fundamental building blocks to (...) secondary representations that are derived or discarded as needed. (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)

The logical interpretation of probability, or "objective Bayesianism'' – the theory that (some) probabilities are strictly logical degrees of partial implication – is defended. The main argument against it is that it requires the assignment of prior probabilities, and that any attempt to determine them by symmetry via a "principle of insufficient reason" inevitably leads to paradox. Three replies are advanced: that priors are imprecise or of little weight, so that disagreement about them does not matter, within limits; (...) that it is possible to distinguish reasonable from unreasonable priors on logical grounds; and that in real cases disagreement about priors can usually be explained by differences in the background information. It is argued also that proponents of alternative conceptions of probability, such as frequentists, Bayesians and Popperians, are unable to avoid committing themselves to the basic principles of logical probability. (shrink)

According to Hempel's paradox, evidence (E) that an object is a nonblack nonraven confirms the hypothesis (H) that every raven is black. According to the standard Bayesian solution, E does confirm H but only to a minute degree. This solution relies on the almost never explicitly defended assumption that the probability of H should not be affected by evidence that an object is nonblack. I argue that this assumption is implausible, and I propose a way out for Bayesians. (...) Introduction Hempel's paradox, the standard Bayesian solution, and the disputed assumption Attempts to defend the disputed assumption Attempts to refute the disputed assumption A way out for Bayesians Conclusion. (shrink)

The lottery paradox involves a set of judgments that are individually easy, when we think intuitively, but ultimately hard to reconcile with each other, when we think reflectively. Empirical work on the natural representation of probability shows that a range of interestingly different intuitive and reflective processes are deployed when we think about possible outcomes in different contexts. Understanding the shifts in our natural ways of thinking can reduce the sense that the lottery paradox reveals something problematic (...) about our concept of knowledge. However, examining these shifts also raises interesting questions about how we ought to be thinking about possible outcomes in the first place. (shrink)

Harman’s lottery paradox, generalized by Vogel to a number of other cases, involves a curious pattern of intuitive knowledge ascriptions: certain propositions seem easier to know than various higher-probability propositions that are recognized to follow from them. For example, it seems easier to judge that someone knows his car is now on Avenue A, where he parked it an hour ago, than to judge that he knows that it is not the case that his car has been stolen (...) and driven away in the last hour. Contextualists have taken this pattern of intuitions as evidence that ‘knows’ does not always denote the same relationship; subject-sensitive invariantists have taken this pattern of intuitions as evidence that non-traditional factors such as practical interests figure in knowledge; still others have argued that the Harman Vogel pattern gives us a reason to abandon the principle that knowledge is closed under known entailment. This paper argues that there is a psychological explanation of the strange pattern of intuitions, grounded in the manner in which we shift between an automatic or heuristic mode of judgment and a controlled or systematic mode. Understanding the psychology behind the pattern of intuitions enables us to see that the pattern gives us no reason to abandon traditional intellectualist invariantism. The psychological account of the paradox also yields new resources for clarifying and defending the single premise closure principle for knowledge ascriptions. (shrink)

David Builes presents a paradox concerning how confident you should be that any given member of an infinite collection of fair coins landed heads, conditional on the information that they were all flipped and only finitely many of them landed heads. We argue that if you should have any conditional credence at all, it should be 1/2.

This thesis focuses on expressively rich languages that can formalise talk about probability. These languages have sentences that say something about probabilities of probabilities, but also sentences that say something about the probability of themselves. For example: (π): “The probability of the sentence labelled π is not greater than 1/2.” Such sentences lead to philosophical and technical challenges; but can be useful. For example they bear a close connection to situations where ones confidence in something can affect (...) whether it is the case or not. The motivating interpretation of probability as an agent's degrees of belief will be focused on throughout the thesis. -/- This thesis aims to answer two questions relevant to such frameworks, which correspond to the two parts of the thesis: “How can one develop a formal semantics for this framework?” and “What rational constraints are there on an agent once such expressive frameworks are considered?”. (shrink)

Brentano never published on Bertrand’s paradox but claimed to have a solution. Adrian Maître has recovered from the Franz Brentano Archive Brentano’s remarks on his solution. They do not give us a worked demonstration of his solution but only an incomplete and in places obscure justification of it. Here I attempt to identify his solution, to explain what seem to me the clearly discernible parts of his justification and to discuss the extent to which the justification succeeds in the (...) light of current work on Bertrand’s paradox. (shrink)

The principle of indifference is supposed to suffice for the rational assignation of probabilities to possibilities. Bertrand advances a probability problem, now known as his paradox, to which the principle is supposed to apply; yet, just because the problem is ill‐posed in a technical sense, applying it leads to a contradiction. Examining an ambiguity in the notion of an ill‐posed problem shows that there are precisely two strategies for resolving the paradox: the distinction strategy and the well‐posing (...) strategy. The main contenders for resolving the paradox, Marinoff and Jaynes, offer solutions which exemplify these two strategies. I show that Marinoff’s attempt at the distinction strategy fails, and I offer a general refutation of this strategy. The situation for the well‐posing strategy is more complex. Careful formulation of the paradox within measure theory shows that one of Bertrand’s original three options can be ruled out but also shows that piecemeal attempts at the well‐posing strategy will not succeed. What is required is an appeal to general principle. I show that Jaynes’s use of such a principle, the symmetry requirement, fails to resolve the paradox; that a notion of metaindifference also fails; and that, while the well‐posing strategy may not be conclusively refutable, there is no reason to think that it can succeed. So the current situation is this. The failure of Marinoff’s and Jaynes’s solutions means that the paradox remains unresolved, and of the only two strategies for resolution, one is refuted and we have no reason to think the other will succeed. Consequently, Bertrand’s paradox continues to stand in refutation of the principle of indifference. (shrink)

A classical probabilistics explanation for a typical quantum effect in Hardy's paradox is demonstrated.

This work deals with problems involving infinities and infinitesimals. It explores the ideas behind zero, its relationship to ontological nothingness, finititude (such as finite numbers and quantities), and the infinite. The idea of infinity and zero are closely related, despite what many perceive as an intuitive inverse relationship. The symbol 0 generally refers to nothingness, whereas the symbol infinity refers to ``so much'' that it cannot be quantified or captured. The notion of finititude rests somewhere between complete nothingness (...) and something having no end. -/- My concern is that many of the philosophers arguing for or against the ontology (or possibility) of an actual infinite set are unaware or unfamiliar with the mathematical literature attempting to clearly and rigorously define these terms. I believe it is a mistake to leave mathematicians out of this conversation, as analysts in particular have defined (and used) infinity in a way that is relevant to the ongoing debate between philosophers. -/- For this paper, I have selected examples from Principles of Analysis by Walter Rudin, which has become a standard text for Real Analysis classes taken by pure mathematicians and engineers. I will also restrict the scope of examples to Euclidean spaces for simplicity. Finally, I will be using addition, + and multiplication * in their usual way. (shrink)

Nicholas Rescher claims that rational decision theory “may leave us in the lurch”, because there are two apparently acceptable ways of applying “the standard machinery of expected-value analysis” to his Dr. Psycho paradox which recommend contradictory actions. He detects a similar contradiction in Newcomb’s problem. We consider his claims from the point of view of both Bayesian decision theory and causal decision theory. In Dr. Psycho and in Newcomb’s Problem, Rescher has used premisses about probabilities which he assumes to (...) be independent. From the former point of view, we show that the probability premisses are not independent but inconsistent, and their inconsistency is provable within probability theory alone. From the latter point of view, we show that their consistency can be saved, but then the contradictory recommendations evaporate. Consequently, whether one subscribes to evidential or causal decision theory, rational decision theory is not in any way vitiated by Rescher’s arguments. (shrink)

After long arguments between positivism and falsificationism, the verification of universal hypotheses was replaced with the confirmation of uncertain major premises. Unfortunately, Hemple proposed the Raven Paradox. Then, Carnap used the increment of logical probability as the confirmation measure. So far, many confirmation measures have been proposed. Measure F proposed by Kemeny and Oppenheim among them possesses symmetries and asymmetries proposed by Elles and Fitelson, monotonicity proposed by Greco et al., and normalizing property suggested by many researchers. Based (...) on the semantic information theory, a measure b* similar to F is derived from the medical test. Like the likelihood ratio, measures b* and F can only indicate the quality of channels or the testing means instead of the quality of probability predictions. Furthermore, it is still not easy to use b*, F, or another measure to clarify the Raven Paradox. For this reason, measure c* similar to the correct rate is derived. Measure c* supports the Nicod Criterion and undermines the Equivalence Condition, and hence, can be used to eliminate the Raven Paradox. An example indicates that measures F and b* are helpful for diagnosing the infection of Novel Coronavirus, whereas most popular confirmation measures are not. Another example reveals that all popular confirmation measures cannot be used to explain that a black raven can confirm “Ravens are black” more strongly than a piece of chalk. Measures F, b*, and c* indicate that the existence of fewer counterexamples is more important than more positive examples’ existence, and hence, are compatible with Popper’s falsification thought. (shrink)

Humility is a paradoxical virtue. This should come as no great surprise. It doesn’t take much explanation for one to realize that if someone is boasting about how humble he is, then he probably is not humble. In fact, as we shall see, the paradoxical nature of humility has a long history, going back to at least Thomas Aquinas in the thirteenth century. While it may not be a novel claim that there exists an apparent paradox of humility, I (...) will argue that there is more than one humility paradox, perhaps as many as five: the epistemic paradox, the paradox of self-attribution, the inculcation paradox, the agentic paradox, and the axiological paradox. All these paradoxes are distinct, yet seemingly intertwined. Upon examining them each in turn, we will be left with a Gordian knot of humility paradoxes. (shrink)

This note discusses three issues that Allen and Pardo believe to be especially problematic for a probabilistic interpretation of standards of proof: (1) the subjectivity of probability assignments; (2) the conjunction paradox; and (3) the non-comparative nature of probabilistic standards. I offer a reading of probabilistic standards that avoids these criticisms.

Skeptical hypotheses such as the brain-in-a-vat hypothesis provide extremely poor explanations for our sensory experiences. Because these scenarios accommodate virtually any possible set of evidence, the probability of any given set of evidence on the skeptical scenario is near zero; hence, on Bayesian grounds, the scenario is not well supported by the evidence. By contrast, serious theories make reasonably specific predictions about the evidence and are then well supported when these predictions are satisfied.

Using “brute reason” I will show why there can be only one valid interpretation of probability. The valid interpretation turns out to be a further refinement of Popper’s Propensity interpretation of probability. Via some famous probability puzzles and new thought experiments I will show how all other interpretations of probability fail, in particular the Bayesian interpretations, while these puzzles do not present any difficulties for the interpretation proposed here. In addition, the new interpretation casts doubt on (...) some concepts often taken as basic and unproblematic, like rationality, utility and expectation. This in turn has implications for decision theory, economic theory and the philosophy of physics. (shrink)

This article examines the prisoner's dilemma paradox and argues that confessing is the rational choice, despite this probably entailing a less-than-ideal outcome.

This paper (first published under the same title in Journal of Mathematical Economics, 29, 1998, p. 331-361) is a sequel to "Consistent Bayesian Aggregation", Journal of Economic Theory, 66, 1995, p. 313-351, by the same author. Both papers examine mathematically whether the the following assumptions are compatible: the individuals and the group both form their preferences according to Subjective Expected Utility (SEU) theory, and the preferences of the group satisfy the Pareto principle with respect to those of the individuals. While (...) the 1995 paper explored these assumptions in the axiomatic context of Savage's (1954-1972) SEU theory, the present paper explores them in the context of Anscombe and Aumann's (1963) alternative SEU theory. We first show that the problematic assumptions become compatible when the Anscombe-Aumann utility functions are state-dependent and no subjective probabilities are elicited. Then we show that the problematic assumptions become incompatible when the Anscombe-Aumann utility functions are state-dependent, like before, but subjective probabilities are elicited using a relevant technical scheme. This last result reinstates the impossibilities proved by the 1995 paper, and thus shows them to be robust with respect to the choice of the SEU axiomatic framework. The technical scheme used for the elicitation of subjective probabilities is that of Karni, Schmeidler and Vind (1983). (shrink)

According to the so-called Lockean thesis, a rational agent believes a proposition just in case its probability is sufficiently high, i.e., greater than some suitably fixed threshold. The Preface paradox is usually taken to show that the Lockean thesis is untenable, if one also assumes that rational agents should believe the conjunction of their own beliefs: high probability and rational belief are in a sense incompatible. In this paper, we show that this is not the case in (...) general. More precisely, we consider two methods of computing how probable must each of a series of propositions be in order to rationally believe their conjunction under the Lockean thesis. The price one has to pay for the proposed solutions to the paradox is what we call “quasi-dogmatism”: the view that a rational agent should believe only those propositions which are “nearly certain” in a suitably defined sense. (shrink)

This paper defends the heretical view that, at least in some cases, we ought to assign legal liability based on purely statistical evidence. The argument draws on prominent civil law litigation concerning pharmaceutical negligence and asbestos-poisoning. The overall aim is to illustrate moral pitfalls that result from supposing that it is never appropriate to rely on bare statistics when settling a legal dispute.

Unlike other classical arguments for the existence of God, Pascal’s Wager provides a pragmatic rationale for theistic belief. Its most popular version says that it is rationally mandatory to choose a way of life that seeks to cultivate belief in God because this is the option of maximum expected utility. Despite its initial attractiveness, this long-standing argument has been subject to various criticisms by many philosophers. What is less discussed, however, is the rationality of this choice in situations where the (...) decision-makers are confronted with greater uncertainty. In this paper, I examine the imprecise version of Pascal’s Wager: those scenarios where an agent’s credence that God exists is imprecise or vague rather than precise. After introducing some technical background on imprecise probabilities, I apply five different principles for decision-making to two cases of state uncertainty. In the final part of the paper, I argue that it is not rationally permitted to include zero as the lower probability of God’s existence. Although the conditions for what makes an act uniquely optimal vary significantly across those principles, I also show how the option of wagering for God can defeat any mixed strategy under two distinct interpretations of salvation. (shrink)

Logical Probability (LP) is strictly distinguished from Statistical Probability (SP). To measure semantic information or confirm hypotheses, we need to use sampling distribution (conditional SP function) to test or confirm fuzzy truth function (conditional LP function). The Semantic Information Measure (SIM) proposed is compatible with Shannon’s information theory and Fisher’s likelihood method. It can ensure that the less the LP of a predicate is and the larger the true value of the proposition is, the more information there is. (...) So the SIM can be used as Popper's information criterion for falsification or test. The SIM also allows us to optimize the true-value of counterexamples or degrees of disbelief in a hypothesis to get the optimized degree of belief, i. e. Degree of Confirmation (DOC). To explain confirmation, this paper 1) provides the calculation method of the DOC of universal hypotheses; 2) discusses how to resolve Raven Paradox with new DOC and its increment; 3) derives the DOC of rapid HIV tests: DOC of “+” =1-(1-specificity)/sensitivity, which is similar to Likelihood Ratio (=sensitivity/(1-specificity)) but has the upper limit 1; 4) discusses negative DOC for excessive affirmations, wrong hypotheses, or lies; and 5) discusses the DOC of general hypotheses with GPS as example. (shrink)

One of the Bell's assumptions in the original derivation of his inequalities was the hypothesis of locality, i.e., the absence of the in uence of two remote measuring instruments on one another. That is why violations of these inequalities observed in experiments are often interpreted as a manifestation of the nonlocal nature of quantum mechanics, or a refutation of a local realism. It is well known that the Bell's inequality was derived in its traditional form, without resorting to the hypothesis (...) of locality and without the introduction of hidden variables, the only assumption being that the probability distributions are nonnegative. This can therefore be regarded as a rigorous proof that the hypothesis of locality and the hypothesis of existence of the hidden variables not relevant to violations of Bell's inequalities. The physical meaning of the obtained results is examined. Physical nature of the violation of the Bell inequalities is explained under new EPR-B nonlocality postulate.We show that the correlations of the observables involved in the Bohm{Bell type experiments can be expressed as correlations of classical random variables. The revisited Bell type inequality in canonical notatons reads <AB>+<A′B>+<AB′>-<A′B′>≤6. (shrink)