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  1. 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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  2. Weintraub’s Response to Williamson’s Coin Flip Argument.Matthew W. Parker - 2021 - European Journal for Philosophy of Science 11 (3):1-21.
    A probability distribution is regular if it does not assign probability zero to any possible event. Williamson argued that we should not require probabilities to be regular, for if we do, certain “isomorphic” physical events must have different probabilities, which is implausible. His remarks suggest an assumption that chances are determined by intrinsic, qualitative circumstances. Weintraub responds that Williamson’s coin flip events differ in their inclusion relations to each other, or the inclusion relations between their times, and this can account (...)
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  3. Countable Additivity, Idealization, and Conceptual Realism.Yang Liu - 2020 - Economics and Philosophy 36 (1):127-147.
    This paper addresses the issue of finite versus countable additivity in Bayesian probability and decision theory -- in particular, Savage's theory of subjective expected utility and personal probability. I show that Savage's reason for not requiring countable additivity in his theory is inconclusive. The assessment leads to an analysis of various highly idealised assumptions commonly adopted in Bayesian theory, where I argue that a healthy dose of, what I call, conceptual realism is often helpful in understanding the interpretational value of (...)
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  4. 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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  5. On Linear Aggregation of Infinitely Many Finitely Additive Probability Measures.Michael Nielsen - 2019 - Theory and Decision 86 (3-4):421-436.
    We discuss Herzberg’s :319–337, 2015) treatment of linear aggregation for profiles of infinitely many finitely additive probabilities and suggest a natural alternative to his definition of linear continuous aggregation functions. We then prove generalizations of well-known characterization results due to :410–414, 1981). We also characterize linear aggregation of probabilities in terms of a Pareto condition, de Finetti’s notion of coherence, and convexity.
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  6. 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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  7. Declarations of Independence.Branden Fitelson & Alan Hájek - 2017 - Synthese 194 (10):3979-3995.
    According to orthodox (Kolmogorovian) probability theory, conditional probabilities are by definition certain ratios of unconditional probabilities. As a result, orthodox conditional probabilities are undefined whenever their antecedents have zero unconditional probability. This has important ramifications for the notion of probabilistic independence. Traditionally, independence is defined in terms of unconditional probabilities (the factorization of the relevant joint unconditional probabilities). Various “equivalent” formulations of independence can be given using conditional probabilities. But these “equivalences” break down if conditional probabilities are permitted to have (...)
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  8. 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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  9. A Note on Cancellation Axioms for Comparative Probability.Matthew Harrison-Trainor, Wesley H. Holliday & Thomas F. Icard - 2016 - Theory and Decision 80 (1):159-166.
    We prove that the generalized cancellation axiom for incomplete comparative probability relations introduced by Rios Insua and Alon and Lehrer is stronger than the standard cancellation axiom for complete comparative probability relations introduced by Scott, relative to their other axioms for comparative probability in both the finite and infinite cases. This result has been suggested but not proved in the previous literature.
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  10. Non-Archimedean Probability.Vieri Benci, Leon Horsten & Sylvia Wenmackers - 2013 - Milan Journal of Mathematics 81 (1):121-151.
    We propose an alternative approach to probability theory closely related to the framework of numerosity theory: non-Archimedean probability (NAP). In our approach, unlike in classical probability theory, all subsets of an infinite sample space are measurable and only the empty set gets assigned probability zero (in other words: the probability functions are regular). We use a non-Archimedean field as the range of the probability function. As a result, the property of countable additivity in Kolmogorov’s axiomatization of probability is replaced by (...)
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  11. Axioms for Non-Archimedean Probability (NAP).Vieri Benci, Leon Horsten & Sylvia Wenmackers - 2012 - In De Vuyst J. & Demey L. (eds.), Future Directions for Logic; Proceedings of PhDs in Logic III - Vol. 2 of IfColog Proceedings. College Publications.
    In this contribution, we focus on probabilistic problems with a denumerably or non-denumerably infinite number of possible outcomes. Kolmogorov (1933) provided an axiomatic basis for probability theory, presented as a part of measure theory, which is a branch of standard analysis or calculus. Since standard analysis does not allow for non-Archimedean quantities (i.e. infinitesimals), we may call Kolmogorov's approach "Archimedean probability theory". We show that allowing non-Archimedean probability values may have considerable epistemological advantages in the infinite case. The current paper (...)
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  12. A Note on Comparative Probability.Nick Haverkamp & Moritz Schulz - 2012 - Erkenntnis 76 (3):395-402.
    A possible event always seems to be more probable than an impossible event. Although this constraint, usually alluded to as regularity , is prima facie very attractive, it cannot hold for standard probabilities. Moreover, in a recent paper Timothy Williamson has challenged even the idea that regularity can be integrated into a comparative conception of probability by showing that the standard comparative axioms conflict with certain cases if regularity is assumed. In this note, we suggest that there is a natural (...)
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  13. Varieties of Conditional Probability.Kenny Easwaran - 2011 - In Prasanta Bandyopadhyay & Malcolm Forster (eds.), Handbook of the Philosophy of Science, Vol. 7: Philosophy of Statistics. North Holland.
    I consider the notions of logical probability, degree of belief, and objective chance, and argue that a different formalism for conditional probability is appropriate for each.
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  14. Probabilistic Substitutivity at a Reduced Price.David Miller - 2011 - Principia: An International Journal of Epistemology 15 (2):271-.
    One of the many intriguing features of the axiomatic systems of probability investigated in Popper (1959), appendices _iv, _v, is the different status of the two arguments of the probability functor with regard to the laws of replacement and commutation. The laws for the first argument, (rep1) and (comm1), follow from much simpler axioms, whilst (rep2) and (comm2) are independent of them, and have to be incorporated only when most of the important deductions have been accomplished. It is plain that, (...)
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  15. On Generalizing Kolmogorov.Richard Dietz - 2010 - Notre Dame Journal of Formal Logic 51 (3):323-335.
    In his "From classical to constructive probability," Weatherson offers a generalization of Kolmogorov's axioms of classical probability that is neutral regarding the logic for the object-language. Weatherson's generalized notion of probability can hardly be regarded as adequate, as the example of supervaluationist logic shows. At least, if we model credences as betting rates, the Dutch-Book argument strategy does not support Weatherson's notion of supervaluationist probability, but various alternatives. Depending on whether supervaluationist bets are specified as (a) conditional bets (Cantwell), (b) (...)
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  16. The Lockean Thesis and the Logic of Belief.James Hawthorne - 2009 - In Franz Huber & Christoph Schmidt-Petri (eds.), Degrees of Belief. Synthese Library: Springer. pp. 49--74.
    In a penetrating investigation of the relationship between belief and quantitative degrees of confidence (or degrees of belief) Richard Foley (1992) suggests the following thesis: ... it is epistemically rational for us to believe a proposition just in case it is epistemically rational for us to have a sufficiently high degree of confidence in it, sufficiently high to make our attitude towards it one of belief. Foley goes on to suggest that rational belief may be just rational degree of confidence (...)
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  17. Probabilistic Coherence and Proper Scoring Rules.Joel Predd, Robert Seiringer, Elliott Lieb, Daniel Osherson, H. Vincent Poor & Sanjeev Kulkarni - 2009 - IEEE Transactions on Information Theory 55 (10):4786-4792.
    We provide self-contained proof of a theorem relating probabilistic coherence of forecasts to their non-domination by rival forecasts with respect to any proper scoring rule. The theorem recapitulates insights achieved by other investigators, and clarifi es the connection of coherence and proper scoring rules to Bregman divergence.
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  18. Degrees of Belief.Franz Huber & Christoph Schmidt-Petri (eds.) - 2008 - Dordrecht and Heidelberg: Springer.
    Various theories try to give accounts of how measures of this confidence do or ought to behave, both as far as the internal mental consistency of the agent as ...
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  19. Probability Semantics for Quantifier Logic.Theodore Hailperin - 2000 - Journal of Philosophical Logic 29 (2):207-239.
    By supplying propositional calculus with a probability semantics we showed, in our 1996, that finite stochastic problems can be treated by logic-theoretic means equally as well as by the usual set-theoretic ones. In the present paper we continue the investigation to further the use of logical notions in probability theory. It is shown that quantifier logic, when supplied with a probability semantics, is capable of treating stochastic problems involving countably many trials.
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  20. 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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  21. Probabilité Conditionnelle Et Certitude.Bas C. Van Fraassen - 1997 - Dialogue 36 (1):69-.
    Personal probability is now a familiar subject in epistemology, together with such more venerable notions as knowledge and belief. But there are severe strains between probability and belief; if either is taken as the more basic, the other may suffer. After explaining the difficulties of attempts to accommodate both, I shall propose a unified account which takes conditional personal probability as basic. Full belief is therefore a defined, derivative notion. Yet we will still be able to picture opinion as follows: (...)
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  22. Getting the Constraints on Popper's Probability Functions Right.Hugues Leblanc & Peter Roeper - 1993 - Philosophy of Science 60 (1):151-157.
    Shown here is that a constraint used by Popper in The Logic of Scientific Discovery (1959) for calculating the absolute probability of a universal quantification, and one introduced by Stalnaker in "Probability and Conditionals" (1970, 70) for calculating the relative probability of a negation, are too weak for the job. The constraint wanted in the first case is in Bendall (1979) and that wanted in the second case is in Popper (1959).
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  23. On the Structure of the Quantum-Mechanical Probability Models.Nicola Cufaro-Petroni - 1992 - Foundations of Physics 22 (11):1379-1401.
    In this paper the role of the mathematical probability models in the classical and quantum physics is shortly analyzed. In particular the formal structure of the quantum probability spaces (QPS) is contrasted with the usual Kolmogorovian models of probability by putting in evidence the connections between this structure and the fundamental principles of the quantum mechanics. The fact that there is no unique Kolmogorovian model reproducing a QPS is recognized as one of the main reasons of the paradoxical behaviors pointed (...)
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  24. 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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  25. On Harold Jeffreys' Axioms.S. Noorbaloochi - 1988 - Philosophy of Science 55 (3):448-452.
    It is argued that models of H. Jeffreys' axioms of probability (Jeffreys [1939] 1967) are not monotone even with I. J. Good's proposed modification (Good 1950). Hence the additivity axiom seems essential to a theory of probability as it is with Kolmogorov's system (Kolmogorov 1950).
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  26. On the Impossibility of Events of Zero Probability.Asad Zaman - 1987 - Theory and Decision 23 (2):157-159.
  27. Pragmatic Probability.Newton C. A. Costa - 1986 - Erkenntnis 25 (2):141-162.
  28. Probability Functions and Their Assumption Sets — the Binary Case.Hugues Leblanc & Charles G. Morgan - 1984 - Synthese 60 (1):91 - 106.
  29. On Probability Theory and Probabilistic Physics—Axiomatics and Methodology.L. S. Mayants - 1973 - Foundations of Physics 3 (4):413-433.
    A new formulation involving fulfillment of all the Kolmogorov axioms is suggested for acomplete probability theory. This proves to be not a purely mathematical discipline. Probability theory deals with abstract objects—images of various classes of concrete objects—whereas experimental statistics deals with concrete objects alone. Both have to be taken into account. Quantum physics and classical statistical physics prove to be different aspects ofone probabilistic physics. The connection of quantum mechanics with classical statistical mechanics is examined and the origin of the (...)
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  30. Likelihood.Anthony William Fairbank Edwards - 1972 - Cambridge University Press.
    Dr Edwards' stimulating and provocative book advances the thesis that the appropriate axiomatic basis for inductive inference is not that of probability, with its addition axiom, but rather likelihood - the concept introduced by Fisher as a measure of relative support amongst different hypotheses. Starting from the simplest considerations and assuming no more than a modest acquaintance with probability theory, the author sets out to reconstruct nothing less than a consistent theory of statistical inference in science.
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  31. Richard T. Cox. Probability, Frequency and Reasonable Expectation. American Journal of Physics, Vol. 14 , Pp. 1–13. - Richard T. Cox. The Algebra of Probable Inference. The Johns Hopkins Press, Baltimore1961, X + 114 Pp. [REVIEW]David Miller - 1972 - Journal of Symbolic Logic 37 (2):398-399.
  32. A Set of Independent Axioms for Probability.Karl R. Popper - 1938 - Mind 47 (186):275-277.