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  1. Horacio Arló-Costa & Richmond H. Thomason (2001). Iterative Probability Kinematics. Journal of Philosophical Logic 30 (5):479-524.
    Following the pioneer work of Bruno De Finetti [12], conditional probability spaces (allowing for conditioning with events of measure zero) have been studied since (at least) the 1950's. Perhaps the most salient axiomatizations are Karl Popper's in [31], and Alfred Renyi's in [33]. Nonstandard probability spaces [34] are a well know alternative to this approach. Vann McGee proposed in [30] a result relating both approaches by showing that the standard values of infinitesimal probability functions are representable as Popper functions, and (...)
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  2. John C. Bigelow (1977). Semantics of Probability. Synthese 36 (4):459--72.
  3. John C. Bigelow (1976). Possible Worlds Foundations for Probability. Journal of Philosophical Logic 5 (3):299--320.
  4. Michael E. Cuffaro (2010). Wittgenstein on Prior Probabilities. Proceedings of the Canadian Society for History and Philosophy of Mathematics 23:85-98.
    Wittgenstein did not write very much on the topic of probability. The little we have comes from a few short pages of the Tractatus, some 'remarks' from the 1930s, and the informal conversations which went on during that decade with the Vienna Circle. Nevertheless, Wittgenstein's views were highly influential in the later development of the logical theory of probability. This paper will attempt to clarify and defend Wittgenstein's conception of probability against some oft-cited criticisms that stem from a misunderstanding of (...)
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  5. Georg J. W. Dorn (2002). Induktion und Wahrscheinlichkeit. Ein Gedankenaustausch mit Karl Popper. In Edgar Morscher (ed.), Was wir Karl R. Popper und seiner Philosophie verdanken. Zu seinem 100. Geburtstag. Academia Verlag
    Zwischen 1987 und 1994 sandte ich 20 Briefe an Karl Popper. Die meisten betrafen Fragen bezüglich seiner Antiinduktionsbeweise und seiner Wahrscheinlichkeitstheorie, einige die organisatorische und inhaltliche Vorbereitung eines Fachgesprächs mit ihm in Kenly am 22. März 1989 (worauf hier nicht eingegangen werden soll), einige schließlich ganz oder in Teilen nicht-fachliche Angelegenheiten (die im vorliegenden Bericht ebenfalls unberücksichtigt bleiben). Von Karl Popper erhielt ich in diesem Zeitraum 10 Briefe. Der bedeutendste ist sein siebter, bestehend aus drei Teilen, geschrieben am 21., 22. (...)
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  6. Georg J. W. Dorn (1992/93). Popper’s Laws of the Excess of the Probability of the Conditional Over the Conditional Probability. Conceptus: Zeitschrift Fur Philosophie 26:3–61.
    Karl Popper discovered in 1938 that the unconditional probability of a conditional of the form ‘If A, then B’ normally exceeds the conditional probability of B given A, provided that ‘If A, then B’ is taken to mean the same as ‘Not (A and not B)’. So it was clear (but presumably only to him at that time) that the conditional probability of B given A cannot be reduced to the unconditional probability of the material conditional ‘If A, then B’. (...)
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  7. Kenny Easwaran (2011). Varieties of Conditional Probability. In Prasanta Bandyopadhyay & Malcolm Forster (eds.), Handbook for 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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  8. F. Y. Edgeworth (1922). The Philosophy of Chance. Mind 31 (123):257-283.
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  9. F. Y. Edgeworth (1884). The Philosophy of Chance. Mind 9 (34):223-235.
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  10. J. Franklin (2001). Resurrecting Logical Probability. Erkenntnis 55 (2):277-305.
    The logical interpretation of probability, or ``objective Bayesianism''''– the theory that (some) probabilitiesare strictly logical degrees of partial implication – is defended.The main argument against it is that it requires the assignment ofprior probabilities, and that any attempt to determine them by symmetryvia a ``principle of insufficient reason'''' inevitably leads to paradox.Three replies are advanced: that priors are imprecise or of little weight, sothat disagreement about them does not matter, within limits; thatit is possible to distinguish reasonable from unreasonable priorson (...)
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  11. Joseph S. Fulda (1988). Estimating Semantic Content: An A Priori Approach. International Journal of Intelligent Systems 3 (1):35-43.
    Gives a general method as well as some results (inspired by Asimov, 1951; since discovered to be in Bar-Hillel and Carnap [several versions; Charles Parsons referred me to /Language and Information/]) to recover meaning (eventually automatically) from logical form/logical probability, which are mirror images. (Sets are taken as extensions of predicates, and knowledge of the sizes is needed; to that extent the method is a posteriori).
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  12. Joseph S. Fulda & Kevin De Fontes (1989). The A Priori Meaningfulness Measure and Resolution Theorem Proving. Journal of Experimental and Theoretical Artificial Intelligence 1 (3):227-230.
    Demonstrates the validity of the measure presented in "Estimating Semantic Content" on textbook examples using (binary) resolution [a generalization of disjunctive syllogism] theorem proving; the measure is based on logical probability and is the mirror image of logical form; it dates to Popper.
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  13. Rolf Haenni, Jan-Willem Romeijn, Gregory Wheeler & Jon Williamson (2011). Probabilistic Logics and Probabilistic Networks. Synthese Library.
    Additionally, the text shows how to develop computationally feasible methods to mesh with this framework.
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  14. James Hawthorne (2005). Degree-of-Belief and Degree-of-Support: Why Bayesians Need Both Notions. Mind 114 (454):277-320.
    I argue that Bayesians need two distinct notions of probability. We need the usual degree-of-belief notion that is central to the Bayesian account of rational decision. But Bayesians also need a separate notion of probability that represents the degree to which evidence supports hypotheses. Although degree-of-belief is well suited to the theory of rational decision, Bayesians have tried to apply it to the realm of hypothesis confirmation as well. This double duty leads to the problem of old evidence, a problem (...)
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  15. C. Howson (1973). Must the Logical Probability of Laws Be Zero? British Journal for the Philosophy of Science 24 (2):153-163.
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  16. Richard Johns, An Epistemic Theory of Objective Chance.
    A theory of objective, single-case chances is presented and defended. The theory states that the chance of an event E is its epistemic probability, given maximal knowledge of the possible causes of E. This theory is uniquely successful in entailing all the known properties of chance, but involves heavy metaphysical commitment. It requires an objective rationality that determines proper degrees of belief in some contexts.
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  17. Michael Harry Kelley (1969). Methodological Problems of Logical Probability. Dissertation, The University of Wisconsin - Madison
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  18. John Maynard Keynes (1921/2004). A Treatise on Probability. Dover Publications.
    With this treatise, an insightful exploration of the probabilistic connection between philosophy and the history of science, the famous economist breathed new life into studies of both disciplines. Originally published in 1921, this important mathematical work represented a significant contribution to the theory regarding the logical probability of propositions. Keynes effectively dismantled the classical theory of probability, launching what has since been termed the “logical-relationist” theory. In so doing, he explored the logical relationships between classifying a proposition as “highly probable” (...)
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  19. N. V. Khovanov (1969). Probability as a Measure of Necessity. Soviet Studies in Philosophy 11 (2):141--51.
    One of the characteristic features of the dynamic development of science and technology in recent decades is the constantly rising significance of probabilistic, statistical and information-theory methods in research, both theoretical and applied. Nor is the mathematical theory of probability standing still. The internal logic of its development is leading steadily to enrichment of the traditional study of probability with new axioms and constructive formal calculi.
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  20. D. Klyve (2013). In Defense of Bertrand: The Non-Restrictiveness of Reasoning by Example. Philosophia Mathematica 21 (3):365-370.
    This note has three goals. First, we discuss a presentation of Bertrand's paradox in a recent issue of Philosophia Mathematica, which we believe to be a subtle but important misinterpretation of the problem. We compare claims made about Bertrand with his 1889 Calcul des Probabilités. Second, we use this source to understand Bertrand's true intention in describing what we now call his paradox, comparing it both to another problem he describes in the same section and to a modern treatment. Finally, (...)
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  21. Isaac Levi (2010). Probability Logic, Logical Probability, and Inductive Support. Synthese 172 (1):97 - 118.
    This paper seeks to defend the following conclusions: The program advanced by Carnap and other necessarians for probability logic has little to recommend it except for one important point. Credal probability judgments ought to be adapted to changes in evidence or states of full belief in a principled manner in conformity with the inquirer’s confirmational commitments—except when the inquirer has good reason to modify his or her confirmational commitment. Probability logic ought to spell out the constraints on rationally coherent confirmational (...)
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  22. Stefan Lukits (2013). The Principle of Maximum Entropy and a Problem in Probability Kinematics. Synthese 191 (7):1-23.
    Sometimes we receive evidence in a form that standard conditioning (or Jeffrey conditioning) cannot accommodate. The principle of maximum entropy (MAXENT) provides a unique solution for the posterior probability distribution based on the intuition that the information gain consistent with assumptions and evidence should be minimal. Opponents of objective methods to determine these probabilities prominently cite van Fraassen’s Judy Benjamin case to undermine the generality of maxent. This article shows that an intuitive approach to Judy Benjamin’s case supports maxent. This (...)
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  23. Halina Mortimer (1973). A Rule of Acceptance Based on Logical Probability. Synthese 26 (2):259 - 263.
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  24. Jill North (2010). An Empirical Approach to Symmetry and Probability. Studies in History and Philosophy of Science Part B 41 (1):27-40.
    We often use symmetries to infer outcomes’ probabilities, as when we infer that each side of a fair coin is equally likely to come up on a given toss. Why are these inferences successful? I argue against answering this with an a priori indifference principle. Reasons to reject that principle are familiar, yet instructive. They point to a new, empirical explanation for the success of our probabilistic predictions. This has implications for indifference reasoning in general. I argue that a priori (...)
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  25. Daniel Osherson, Inductive Inference Based on Probability and Similarity.
    We advance a theory of inductive inference designed to predict the conditional probability that certain natural categories satisfy a given predicate given that others do (or do not). A key component of the theory is the similarity of the categories to one another. We measure such similarities in terms of the overlap of metabolic activity in voxels of various posterior regions of the brain in response to viewing instances of the category. The theory and similarity measure are tested against averaged (...)
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  26. John F. Phillips (2005). A Theory of Objective Chance. Pacific Philosophical Quarterly 86 (2):267–283.
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  27. Darrell P. Rowbottom (2013). Bertrand's Paradox Revisited: Why Bertrand's 'Solutions' Are All Inapplicable. Philosophia Mathematica 21 (1):110-114.
    This paper shows that Bertrand's proposed 'solutions' to his own question, which generates his chord paradox, are inapplicable. It uses a simple analogy with cake cutting. The problem is that none of Bertrand's solutions considers all possible cuts. This is no solace for the defenders of the principle of indifference, however, because it emerges that the paradox is harder to solve than previously anticipated.
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  28. Darrell P. Rowbottom (2013). Popper's Measure of Corroboration and P(H|B). British Journal for the Philosophy of Science 64 (4):axs029.
    This article shows that Popper’s measure of corroboration is inapplicable if, as Popper argued, the logical probability of synthetic universal statements is zero relative to any evidence that we might possess. It goes on to show that Popper’s definition of degree of testability, in terms of degree of logical content, suffers from a similar problem. 1 The Corroboration Function and P(h|b) 2 Degrees of Testability and P(h|b).
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  29. Darrell P. Rowbottom (2008). On the Proximity of the Logical and 'Objective Bayesian' Interpretations of Probability. Erkenntnis 69 (3):335-349.
    In his Bayesian Nets and Causality, Jon Williamson presents an ‘Objective Bayesian’ interpretation of probability, which he endeavours to distance from the logical interpretation yet associate with the subjective interpretation. In doing so, he suggests that the logical interpretation suffers from severe epistemological problems that do not affect his alternative. In this paper, I present a challenge to his analysis. First, I closely examine the relationship between the logical and ‘Objective Bayesian’ views, and show how, and why, they are highly (...)
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  30. Darrell P. Rowbottom & Nicholas Shackel (2010). Bangu's Random Thoughts on Bertrand's Paradox. Analysis 70 (4):689-692.
    Bangu (2010) claims that Bertrand’s paradox rests on a hitherto unrecognized assumption, which assumption is sufficiently dubious to throw the burden of proof back onto ‘objectors to [the principle of indifference]’ (2010: 31). We show that Bangu’s objection to the assumption is ill-founded and that the assumption is provably true.
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  31. D. Stove (1970). Deductivism. Australasian Journal of Philosophy 48 (1):76 – 98.
  32. D. C. Stove (2010). Is the Theory of Logical Probability Groundless? In Antony Eagle (ed.), Philosophy of Probability: Contemporary Readings. Routledge
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  33. D. C. Stove (1986). The Rationality of Induction. Oxford University Press.
    Writing on the justification of certain inductive inferences, the author proposes that sometimes induction is justified and that arguments to prove otherwise are not cogent. In the first part he examines the problem of justifying induction, looks at some attempts to prove that it is justified, and responds to criticisms of these proofs. In the second part he deals with such topics as formal logic, deductive logic, the theory of logical probability, and probability and truth.
  34. D. C. Stove (1973). Probability and Hume's Inductive Scepticism. Oxford,Clarendon Press.
    This book aims to discuss probability and David Hume's inductive scepticism. For the sceptical view which he took of inductive inference, Hume only ever gave one argument. That argument is the sole subject-matter of this book. The book is divided into three parts. Part one presents some remarks on probability. Part two identifies Hume's argument for inductive scepticism. Finally, the third part evaluates Hume's argument for inductive scepticism.
  35. Ludwig van den Hauwe (2011). John Maynard Keynes and Ludwig von Mises on Probability. Journal of Libertarian Studies 22 (1):471-507.
    The economic paradigms of Ludwig von Mises on the one hand and of John Maynard Keynes on the other have been correctly recognized as antithetical at the theoretical level, and as antagonistic with respect to their practical and public policy implications. Characteristically they have also been vindicated by opposing sides of the political spectrum. Nevertheless the respective views of these authors with respect to the meaning and interpretation of probability exhibit a closer conceptual affinity than has been acknowledged in the (...)
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  36. Bas C. van Fraassen (2010). Indifference : The Symmetries of Probability. In Antony Eagle (ed.), Philosophy of Probability: Contemporary Readings. Routledge
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  37. Lukas M. Verburgt (2014). Remarks on the Idealist and Empiricist Interpretation of Frequentism: Robert Leslie Ellis Versus John Venn. BSHM Bulletin: Journal of the British Society for the History of Mathematics 29 (3):184-195.
    The goal of this paper is to correct a widespread misconception about the work of Robert Leslie Ellis and John Venn, namely that it can be considered as the ‘British empiricist’ reaction against the traditional theory of probability. It is argued, instead, that there was no unified ‘British school’ of frequentism during the nineteenth century. Where Ellis arrived at frequentism from a metaphysical idealist transformation of probability theory’s mathematical calculations, Venn did so on the basis of an empiricist critique of (...)
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  38. Hermann Vetter (1969). Logical Probability, Mathematical Statistics, and the Problem of Induction. Synthese 20 (1):56 - 71.
    In this paper I want to discuss some basic problems of inductive logic, i.e. of the attempt to solve the problem of induction by means of a calculus of logical probability. I shall try to throw some light upon these problems by contrasting inductive logic, based on logical probability, and working with undefined samples of observations, with mathematical statistics, based on statistical probability, and working with representative random samples.
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  39. John M. Vickers (1988). Chance and Structure: An Essay on the Logical Foundations of Probability. Oxford University Press.
    Discussing the relations between logic and probability, this book compares classical 17th- and 18th-century theories of probability with contemporary theories, explores recent logical theories of probability, and offers a new account of probability as a part of logic.
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  40. Gregory Wheeler & Jon Williamson (2011). Evidential Probability and Objective Bayesian Epistemology. In Prasanta S. Bandyopadhyay & Malcolm Forster (eds.), Handbook of the Philosophy of Statistics. Elsevier
    In this chapter we draw connections between two seemingly opposing approaches to probability and statistics: evidential probability on the one hand and objective Bayesian epistemology on the other.
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  41. Jon Williamson, Philosophies of Probability: Objective Bayesianism and its Challenges.
    This chapter presents an overview of the major interpretations of probability followed by an outline of the objective Bayesian interpretation and a discussion of the key challenges it faces. I discuss the ramifications of interpretations of probability and objective Bayesianism for the philosophy of mathematics in general.
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