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  1. Probability Theory with Superposition Events.David Ellerman - manuscript
    In finite probability theory, events are subsets S⊆U of the outcome set. Subsets can be represented by 1-dimensional column vectors. By extending the representation of events to two dimensional matrices, we can introduce "superposition events." Probabilities are introduced for classical events, superposition events, and their mixtures by using density matrices. Then probabilities for experiments or `measurements' of all these events can be determined in a manner exactly like in quantum mechanics (QM) using density matrices. Moreover the transformation of the density (...)
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  2. Logics of Imprecise Comparative Probability.Yifeng Ding, Wesley H. Holliday & Thomas F. Icard - 2021 - International Journal of Approximate Reasoning 132:154-180.
    This paper studies connections between two alternatives to the standard probability calculus for representing and reasoning about uncertainty: imprecise probability andcomparative probability. The goal is to identify complete logics for reasoning about uncertainty in a comparative probabilistic language whose semantics is given in terms of imprecise probability. Comparative probability operators are interpreted as quantifying over a set of probability measures. Modal and dynamic operators are added for reasoning about epistemic possibility and updating sets of probability measures.
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  3. The Borel Complexity of von Neumann Equivalence.Inessa Moroz & Asger Törnquist - 2021 - Annals of Pure and Applied Logic 172 (5):102913.
  4. The Differential of Probabilistic Entailment.Daniele Mundici - 2021 - Annals of Pure and Applied Logic 172 (6):102945.
  5. Counterexamples to Some Characterizations of Dilation.Michael Nielsen & Rush T. Stewart - 2021 - Erkenntnis 86 (5):1107-1118.
    We provide counterexamples to some purported characterizations of dilation due to Pedersen and Wheeler :1305–1342, 2014, ISIPTA ’15: Proceedings of the 9th international symposium on imprecise probability: theories and applications, 2015).
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  6. Moving Beyond Sets of Probabilities.Gregory Wheeler - 2021 - Statistical Science 36 (2):201--204.
    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 (...)
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  7. The Structure of Epistemic Probabilities.Nevin Climenhaga - 2020 - Philosophical Studies 177 (11):3213-3242.
    The epistemic probability of A given B is the degree to which B evidentially supports A, or makes A plausible. This paper is a first step in answering the question of what determines the values of epistemic probabilities. I break this question into two parts: the structural question and the substantive question. Just as an object’s weight is determined by its mass and gravitational acceleration, some probabilities are determined by other, more basic ones. The structural question asks what probabilities are (...)
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  8. Comparative Infinite Lottery Logic.Matthew W. Parker - 2020 - Studies in History and Philosophy of Science Part A 84:28-36.
    As an application of his Material Theory of Induction, Norton (2018; manuscript) argues that the correct inductive logic for a fair infinite lottery, and also for evaluating eternal inflation multiverse models, is radically different from standard probability theory. This is due to a requirement of label independence. It follows, Norton argues, that finite additivity fails, and any two sets of outcomes with the same cardinality and co-cardinality have the same chance. This makes the logic useless for evaluating multiverse models based (...)
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  9. More Than Impossible: Negative and Complex Probabilities and Their Philosophical Interpretation.Vasil Penchev - 2020 - Logic and Philosophy of Mathematics eJournal (Elsevier: SSRN) 12 (16):1-7.
    A historical review and philosophical look at the introduction of “negative probability” as well as “complex probability” is suggested. The generalization of “probability” is forced by mathematical models in physical or technical disciplines. Initially, they are involved only as an auxiliary tool to complement mathematical models to the completeness to corresponding operations. Rewards, they acquire ontological status, especially in quantum mechanics and its formulation as a natural information theory as “quantum information” after the experimental confirmation the phenomena of “entanglement”. Philosophical (...)
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  10. Should Mathematicians Play Dice?Don Berry - 2019 - Logique Et Analyse 246 (62):135-160.
    It is an established part of mathematical practice that mathematicians demand -/- deductive proof before accepting a new result as a theorem. However, a wide -/- variety of probabilistic methods of justification are also available. Though such -/- procedures may endorse a false conclusion even if carried out perfectly, their -/- robust structure may mean they are actually more reliable in practice once implementation -/- errors are taken into account. Can mathematicians be rational -/- in continuing to reject these probabilistic (...)
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  11. Dilation and Asymmetric Relevance.Arthur Paul Pedersen & Gregory Wheeler - 2019 - Proceedings of Machine Learning Research 103:324-26.
    A characterization result of dilation in terms of positive and negative association admits an extremal counterexample, which we present together with a minor repair of the result. Dilation may be asymmetric whereas covariation itself is symmetric. Dilation is still characterized in terms of positive and negative covariation, however, once the event to be dilated has been specified.
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  12. Inferring Probability Comparisons.Matthew Harrison-Trainor, Wesley H. Holliday & Thomas Icard - 2018 - Mathematical Social Sciences 91:62-70.
    The problem of inferring probability comparisons between events from an initial set of comparisons arises in several contexts, ranging from decision theory to artificial intelligence to formal semantics. In this paper, we treat the problem as follows: beginning with a binary relation ≥ on events that does not preclude a probabilistic interpretation, in the sense that ≥ has extensions that are probabilistically representable, we characterize the extension ≥+ of ≥ that is exactly the intersection of all probabilistically representable extensions of (...)
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  13. Another Approach to Consensus and Maximally Informed Opinions with Increasing Evidence.Rush T. Stewart & Michael Nielsen - 2018 - Philosophy of Science (2):236-254.
    Merging of opinions results underwrite Bayesian rejoinders to complaints about the subjective nature of personal probability. Such results establish that sufficiently similar priors achieve consensus in the long run when fed the same increasing stream of evidence. Initial subjectivity, the line goes, is of mere transient significance, giving way to intersubjective agreement eventually. Here, we establish a merging result for sets of probability measures that are updated by Jeffrey conditioning. This generalizes a number of different merging results in the literature. (...)
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  14. Quantum Mechanics Over Sets: A Pedagogical Model with Non-Commutative Finite Probability Theory as its Quantum Probability Calculus.David Ellerman - 2017 - Synthese (12):4863-4896.
    This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or toy model of quantum mechanics over sets (QM/sets). There have been several previous attempts to develop a quantum-like model with the base field of ℂ replaced by ℤ₂. Since there are no inner products on vector spaces over finite fields, the problem is to define the Dirac brackets and the probability calculus. The previous attempts (...)
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  15. Bayesian Decision Theory and Stochastic Independence.Philippe Mongin - 2017 - TARK 2017.
    Stochastic independence has a complex status in probability theory. It is not part of the definition of a probability measure, but it is nonetheless an essential property for the mathematical development of this theory. Bayesian decision theorists such as Savage can be criticized for being silent about stochastic independence. From their current preference axioms, they can derive no more than the definitional properties of a probability measure. In a new framework of twofold uncertainty, we introduce preference axioms that entail not (...)
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  16. Rewording the Rules on Disjunctive Probability.Ronald Cordero - 2016 - Metaphilosophy 47 (4-5):719-727.
    Logic is a central and highly useful part of philosophy. Its value is particularly evident when it comes to keeping our thinking about disjunctive probabilities clear. Because of the two meanings of “or”, logic can show how the likelihood of a disjunction being true can be determined quite easily. To gauge the chance that one of two or more exclusive alternatives is true, one need only sum up their respective likelihoods. And to know the chance that at least one of (...)
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  17. Probability and Randomness.Antony Eagle - 2016 - In Alan Hájek & Christopher Hitchcock (eds.), The Oxford Handbook of Probability and Philosophy. Oxford, U.K.: Oxford University Press. pp. 440-459.
    Early work on the frequency theory of probability made extensive use of the notion of randomness, conceived of as a property possessed by disorderly collections of outcomes. Growing out of this work, a rich mathematical literature on algorithmic randomness and Kolmogorov complexity developed through the twentieth century, but largely lost contact with the philosophical literature on physical probability. The present chapter begins with a clarification of the notions of randomness and probability, conceiving of the former as a property of a (...)
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  18. Scoring Imprecise Credences: A Mildly Immodest Proposal.Conor Mayo-Wilson & Gregory Wheeler - 2016 - Philosophy and Phenomenological Research 92 (1):55-78.
    Jim Joyce argues for two amendments to probabilism. The first is the doctrine that credences are rational, or not, in virtue of their accuracy or “closeness to the truth” (1998). The second is a shift from a numerically precise model of belief to an imprecise model represented by a set of probability functions (2010). We argue that both amendments cannot be satisfied simultaneously. To do so, we employ a (slightly-generalized) impossibility theorem of Seidenfeld, Schervish, and Kadane (2012), who show that (...)
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  19. Dilation, Disintegrations, and Delayed Decisions.Arthur Paul Pedersen & Gregory Wheeler - 2015 - In Thomas Augistin, Serena Dora, Enrique Miranda & Erik Quaeghebeur (eds.), Proceedings of the 9th International Symposium on Imprecise Probability: Theories and Applications (ISIPTA 2015). Aracne Editrice. pp. 227–236.
    Both dilation and non-conglomerability have been alleged to conflict with a fundamental principle of Bayesian methodology that we call \textit{Good's Principle}: one should always delay making a terminal decision between alternative courses of action if given the opportunity to first learn, at zero cost, the outcome of an experiment relevant to the decision. In particular, both dilation and non-conglomerability have been alleged to permit or even mandate choosing to make a terminal decision in deliberate ignorance of relevant, cost-free information. Although (...)
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  20. A Merton Model of Credit Risk with Jumps.Hoang Thi Phuong Thao & Quan-Hoang Vuong - 2015 - Journal of Statistics Applications and Probability Letters 2 (2):97-103.
    In this note, we consider a Merton model for default risk, where the firm’s value is driven by a Brownian motion and a compound Poisson process.
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  21. On the Search For Objective Truths of Reality and Human Society: Part 1.Dale C. Gillman - 2014 - Dissertation, Kennesaw State University
    Abstract Pioneers in the respective field of philosophy have made remarkable progress in the world. Thinkers starting with Plato and more contemporary thinkers such as Descartes and many others have tried pushing the limits of what humans can learn about the world around us. In my paper I am trying to establish probability as a recognized law of our physical reality. I also briefly discuss logic, reason and the need for modern philosophy. My hopes are also to show that though (...)
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  22. Infinitesimal Chances.Thomas Hofweber - 2014 - Philosophers' Imprint 14.
    It is natural to think that questions in the metaphysics of chance are independent of the mathematical representation of chance in probability theory. After all, chance is a feature of events that comes in degrees and the mathematical representation of chance concerns these degrees but leaves the nature of chance open. The mathematical representation of chance could thus, un-controversially, be taken to be what it is commonly taken to be: a probability measure satisfying Kolmogorov’s axioms. The metaphysical questions about chance (...)
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  23. Demystifying Dilation.Arthur Paul Pedersen & Gregory Wheeler - 2014 - Erkenntnis 79 (6):1305-1342.
    Dilation occurs when an interval probability estimate of some event E is properly included in the interval probability estimate of E conditional on every event F of some partition, which means that one’s initial estimate of E becomes less precise no matter how an experiment turns out. Critics maintain that dilation is a pathological feature of imprecise probability models, while others have thought the problem is with Bayesian updating. However, two points are often overlooked: (1) knowing that E is stochastically (...)
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  24. A Consistent Set of Infinite-Order Probabilities.David Atkinson & Jeanne Peijnenburg - 2013 - International Journal of Approximate Reasoning 54:1351-1360.
    Some philosophers have claimed that it is meaningless or paradoxical to consider the probability of a probability. Others have however argued that second-order probabilities do not pose any particular problem. We side with the latter group. On condition that the relevant distinctions are taken into account, second-order probabilities can be shown to be perfectly consistent. May the same be said of an infinite hierarchy of higher-order probabilities? Is it consistent to speak of a probability of a probability, and of a (...)
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  25. The Game of Probability: Literature and Calculation From Pascal to Kleist. [REVIEW]William Deringer - 2013 - Isis 104:841-842.
  26. Robert Leslie Ellis's Work on Philosophy of Science and the Foundations of Probability Theory.Lukas M. Verburgt - 2013 - Historia Mathematica 40 (4):423-454.
    The goal of this paper is to provide an extensive account of Robert Leslie Ellisʼs largely forgotten work on philosophy of science and probability theory. On the one hand, it is suggested that both his ‘idealist’ renovation of the Baconian theory of induction and a ‘realism’ vis-à-vis natural kinds were the result of a complex dialogue with the work of William Whewell. On the other hand, it is shown to what extent the combining of these two positions contributed to Ellisʼs (...)
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  27. Sets of Probability Distributions, Independence, and Convexity.Fabio G. Cozman - 2012 - Synthese 186 (2):577-600.
    This paper analyzes concepts of independence and assumptions of convexity in the theory of sets of probability distributions. The starting point is Kyburg and Pittarelli’s discussion of “convex Bayesianism” (in particular their proposals concerning E-admissibility, independence, and convexity). The paper offers an organized review of the literature on independence for sets of probability distributions; new results on graphoid properties and on the justification of “strong independence” (using exchangeability) are presented. Finally, the connection between Kyburg and Pittarelli’s results and recent developments (...)
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  28. Infinite Lotteries, Perfectly Thin Darts and Infinitesimals.Alexander R. Pruss - 2012 - Thought: A Journal of Philosophy 1 (2):81-89.
    One of the problems that Bayesian regularity, the thesis that all contingent propositions should be given probabilities strictly between zero and one, faces is the possibility of random processes that randomly and uniformly choose a number between zero and one. According to classical probability theory, the probability that such a process picks a particular number in the range is zero, but of course any number in the range can indeed be picked. There is a solution to this particular problem on (...)
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  29. B. De Finetti, Philosophical Lectures on Probability[REVIEW]Jon Williamson - 2010 - Philosophia Mathematica 18 (1):130-135.
  30. Do We Need Second-Order Probabilities?Sven Ove Hansson - 2008 - Dialectica 62 (4):525-533.
    Although it has often been claimed that all the information contained in second-order probabilities can be contained in first-order probabilities, no practical recipe for the elimination of second-order probabilities without loss of information seems to have been presented. Here, such an elimination method is introduced for repeatable events. However, its application comes at the price of losses in cognitive realism. In spite of their technical eliminability, second-order probabilities are useful because they can provide models of important features of the world (...)
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  31. 14. Experimenting with the Insights of Mathematicians and Scientists.William A. Mathews - 2005 - In Lonergan's Quest: A Study of Desire in the Authoring of Insight. University of Toronto Press. pp. 221-240.
  32. On the Probability of Absolute Truth for And/Or Formulas.Alan Woods - 2005 - Bulletin of Symbolic Logic 12 (3).
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  33. The Theory of Probability. [REVIEW]Osher Doctorow - 2003 - Isis 94:549-550.
  34. Probability, Logic, and Probability Logic.Alan Hájek - 2001 - In Lou Goble (ed.), The Blackwell Guide to Philosophical Logic. Blackwell. pp. 362--384.
  35. Logic, Probability, and Coherence.John M. Vickers - 2001 - Philosophy of Science 68 (1):95-110.
    How does deductive logic constrain probability? This question is difficult for subjectivistic approaches, according to which probability is just strength of (prudent) partial belief, for this presumes logical omniscience. This paper proposes that the way in which probability lies always between possibility and necessity can be made precise by exploiting a minor theorem of de Finetti: In any finite set of propositions the expected number of truths is the sum of the probabilities over the set. This is generalized to apply (...)
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  36. Observables and Statistical Maps.Stan Gudder - 1999 - Foundations of Physics 29 (6):877-897.
    This article begins with a review of the framework of fuzzy probability theory. The basic structure is given by the σ-effect algebra of effects (fuzzy events) $\mathcal{E}{\text{ }}\left( {\Omega ,\mathcal{A}} \right)$ and the set of probability measures $M_1^ + {\text{ }}\left( {\Omega ,\mathcal{A}} \right)$ on a measurable space $\left( {\Omega ,\mathcal{A}} \right)$ . An observable $X:\mathcal{B} \to {\text{ }}\mathcal{E}{\text{ }}\left( {\Omega ,\mathcal{A}} \right)$ is defined, where $\begin{gathered} X:\mathcal{B} \to {\text{ }}\mathcal{E}{\text{ }}\left( {\Omega ,\mathcal{A}} \right) \\ \left( {\Lambda ,{\text{ }}\mathcal{B}} \right) (...)
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  37. Randomness by Deborah J. Bennett. [REVIEW]Patti Hunter - 1999 - Isis 90:345-346.
  38. Proof of Kolmogorovian Censorship.Gergely Bana & Thomas Durt - 1997 - Foundations of Physics 27 (10):1355-1373.
    Many argued that Kolmogorov's axioms of classical probability theory are incompatible with quantum probabilities, and that this is the reason for the violation of Bell's inequalities. Szabó showed that, in fact, these inequalities are not violated by the experimentally observed frequencies if we consider the real, “effective” frequencies. We prove in this work a theorem which generalizes this results: “effective” frequencies associated to quantum events always admit a Kolmogorovian representation, when these events are collected through different experimental setups, the choice (...)
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  39. Book Reviews: JAN VON PLATO. Creating Modern Probability: Its Mathematics, Physics and Philosophy in Historical Perspective. Cambridge and New York: Cambridge University Press, 1994. [REVIEW]David Bellhouse - 1996 - Philosophia Mathematica 4 (3):290-291.
  40. The Logic of Probability.Bruno De Finetti & Brad Angell - 1995 - Philosophical Studies 77 (1):181 - 190.
  41. Zero-One Laws with Variable Probability.Joel Spencer - 1993 - Journal of Symbolic Logic 58 (1):1-14.
  42. Probability Functions: The Matter of Their Recursive Definability.Hugues Leblanc & Peter Roeper - 1992 - Philosophy of Science 59 (3):372-388.
    This paper studies the extent to which probability functions are recursively definable. It proves, in particular, that the (absolute) probability of a statement A is recursively definable from a certain point on, to wit: from the (absolute) probabilities of certain atomic components and conjunctions of atomic components of A on, but to no further extent. And it proves that, generally, the probability of a statement A relative to a statement B is recursively definable from a certain point on, to wit: (...)
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  43. Some Connections Between Epistemic Logic and the Theory of Nonadditive Probability.Philippe Mongin - 1992 - In Paul Humphreys (ed.), Patrick Suppes: Scientific Philosopher. Dordrecht: Kluwer. pp. 135-171.
    This paper is concerned with representations of belief by means of nonadditive probabilities of the Dempster-Shafer (DS) type. After surveying some foundational issues and results in the D.S. theory, including Suppes's related contributions, the paper proceeds to analyze the connection of the D.S. theory with some of the work currently pursued in epistemic logic. A preliminary investigation of the modal logic of belief functions à la Shafer is made. There it is shown that the Alchourrron-Gärdenfors-Makinson (A.G.M.) logic of belief change (...)
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  44. Maximum Likelihood Estimation on Generalized Sample Spaces: An Alternative Resolution of Simpson's Paradox. [REVIEW]Matthias P. Kläy & David J. Foulis - 1990 - Foundations of Physics 20 (7):777-799.
    We propose an alternative resolution of Simpson's paradox in multiple classification experiments, using a different maximum likelihood estimator. In the center of our analysis is a formal representation of free choice and randomization that is based on the notion of incompatible measurements.We first introduce a representation of incompatible measurements as a collection of sets of outcomes. This leads to a natural generalization of Kolmogoroff's axioms of probability. We then discuss the existence and uniqueness of the maximum likelihood estimator for a (...)
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  45. How to Solve Probability Teasers-Discussion.Maya Bar-Hillel - 1989 - Philosophy of Science 56 (2):348-358.
    Recently, Nathan criticized Bar-Hillel and Falk's analysis of some classical probability puzzles on the grounds that they wrongheadedly applied mathematics to the solving of problems suffering from “ambiguous informalities”. Nathan's prescription for solving such problems boils down to assuring in advance that they are uniquely and formally soluble—though he says little about how this is to be done. Unfortunately, in real life problems seldom show concern as to whether their naturally occurring formulation is or is not ambiguous, does or does (...)
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  46. The Autonomy of Probability Theory (Notes on Kolmogorov, Rényi, and Popper).Hugues Leblanc - 1989 - British Journal for the Philosophy of Science 40 (2):167-181.
    Kolmogorov's account in his [1933] of an absolute probability space presupposes given a Boolean algebra, and so does Rényi's account in his [1955] and [1964] of a relative probability space. Anxious to prove probability theory ‘autonomous’. Popper supplied in his [1955] and [1957] accounts of probability spaces of which Boolean algebras are not and [1957] accounts of probability spaces of which fields are not prerequisites but byproducts instead.1 I review the accounts in question, showing how Popper's issue from and how (...)
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  47. Probability and Statistics in Historical Perspective.Donald Mackenzie - 1989 - Isis 80:116-124.
  48. De Finetti's Earliest Works on the Foundations of Probability.Jan Plato - 1989 - Erkenntnis 31 (2-3):263-282.
    Bruno de Finetti's earliest works on the foundations of probability are reviewed. These include the notion of exchangeability and the theory of random processes with independent increments. The latter theory relates to de Finetti's ideas for a probabilistic science more generally. Different aspects of his work are united by his foundational programme for a theory of subjective probabilities.
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  49. The Philosophers of Gambling.N. Rescher - 1989 - Boston Studies in the Philosophy of Science 116:203-220.
    In An Intimate Relation: Studies in the History and Philosophy of Science Presented to Robert E. Butts on his 60th Birthday.
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  50. Conditions on Upper and Lower Probabilities to Imply Probabilities.Patrick Suppes & Mario Zanotti - 1989 - Erkenntnis 31 (2-3):323 - 345.
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