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  1. Sorin Bangu (2010). On Bertrand's Paradox. Analysis 70 (1):30-35.
    The Principle of Indifference is a central element of the ‘classical’ conception of probability, but, for all its strong intuitive appeal, it is widely believed that it faces a devastating objection: the so-called (by Poincare´) ‘Bertrand paradoxes’ (in essence, cases in which the same probability question receives different answers). The puzzle has fascinated many since its discovery, and a series of clever solutions (followed promptly by equally clever rebuttals) have been proposed. However, despite the long-standing interest in this problem, an (...)
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  2. Paul Bartha (2004). Countable Additivity and the de Finetti Lottery. British Journal for the Philosophy of Science 55 (2):301-321.
    De Finetti would claim that we can make sense of a draw in which each positive integer has equal probability of winning. This requires a uniform probability distribution over the natural numbers, violating countable additivity. Countable additivity thus appears not to be a fundamental constraint on subjective probability. It does, however, seem mandated by Dutch Book arguments similar to those that support the other axioms of the probability calculus as compulsory for subjective interpretations. These two lines of reasoning can be (...)
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  3. Paul Bartha & Richard Johns (2001). Probability and Symmetry. Proceedings of the Philosophy of Science Association 2001 (3):S109-.
    The Principle of Indifference, which dictates that we ought to assign two outcomes equal probability in the absence of known reasons to do otherwise, is vulnerable to well-known objections. Nevertheless, the appeal of the principle, and of symmetry-based assignments of equal probability, persists. We show that, relative to a given class of symmetries satisfying certain properties, we are justified in calling certain outcomes equally probable, and more generally, in defining what we call relative probabilities. Relative probabilities are useful in providing (...)
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  4. Nick Bostrom (2008). The Doomsday Argument. Think 6 (17-18):23-28.
    A recent paper by Korb and Oliver in this journal attempts to refute the Carter-Leslie Doomsday argument. I organize their remarks into five objections and show that they all fail. Further efforts are thus called upon to find out what, if anything, is wrong with Carter and Leslie’s disturbing reasoning. While ultimately unsuccessful, Korb and Oliver’s objections do however in some instances force us to become clearer about what the Doomsday argument does and doesn’t imply.
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  5. Darren Bradley (2011). Confirmation in a Branching World: The Everett Interpretation and Sleeping Beauty. British Journal for the Philosophy of Science 62 (2):323-342.
    Sometimes we learn what the world is like, and sometimes we learn where in the world we are. Are there any interesting differences between the two kinds of cases? The main aim of this article is to argue that learning where we are in the world brings into view the same kind of observation selection effects that operate when sampling from a population. I will first explain what observation selection effects are ( Section 1 ) and how they are relevant (...)
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  6. Rudolf Carnap (1952). The Continuum of Inductive Methods. [Chicago]University of Chicago Press.
  7. Paul Castell (1998). A Consistent Restriction of the Principle of Indifference. British Journal for the Philosophy of Science 49 (3):387-395.
    I argue that a particular restricted version of the Principle of Indiference is a consistent, indispensible tool for guiding our probabilistic judgements.
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  8. Shoutir Kishore Chatterjee (2003). Statistical Thought: A Perspective and History. OUP Oxford.
    In this unique monograph, based on years of extensive work, Chatterjee presents the historical evolution of statistical thought from the perspective of various approaches to statistical induction. Developments in statistical concepts and theories are discussed alongside philosophical ideas on the ways we learn from experience. -/- Suitable for researchers, lecturers and students in statistics and the history of science this book is aimed at those who have had some exposure to statistical theory. It is also useful to logicians and philosophers (...)
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  9. Rodolfo De Cristofaro (2008). A New Formulation of the Principle of Indifference. Synthese 163 (3):329 - 339.
    The idea of a probabilistic logic of inductive inference based on some form of the principle of indifference has always retained a powerful appeal. However, up to now all modifications of the principle failed. In this paper, a new formulation of such a principle is provided that avoids generating paradoxes and inconsistencies. Because of these results, the thesis that probabilities cannot be logical quantities, determined in an objective way through some form of the principle of indifference, is no longer supportable. (...)
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  10. John F. Cyranski (1978). Analysis of the Maximum Entropy Principle “Debate”. Foundations of Physics 8 (5-6):493-506.
    Jaynes's maximum entropy principle (MEP) is analyzed by considering in detail a recent controversy. Emphasis is placed on the inductive logical interpretation of “probability” and the concept of “total knowledge.” The relation of the MEP to relative frequencies is discussed, and a possible realm of its fruitful application is noted.
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  11. Rodolfo de Cristofaro (2008). A New Formulation of the Principle of Indifference. Synthese 163 (3):329-339.
    The idea of a probabilistic logic of inductive inference based on some form of the principle of indifference has always retained a powerful appeal. However, up to now all modifications of the principle failed. In this paper, a new formulation of such a principle is provided that avoids generating paradoxes and inconsistencies. Because of these results, the thesis that probabilities cannot be logical quantities, determined in an objective way through some form of the principle of indifference, is no longer supportable. (...)
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  12. Cian Dorr (2010). The Eternal Coin: A Puzzle About Self-Locating Conditional Credence. Philosophical Perspectives 24 (1):189-205.
    The Eternal Coin is a fair coin has existed forever, and will exist forever, in a region causally isolated from you. It is tossed every day. How confident should you be that the Coin lands heads today, conditional on (i) the hypothesis that it has landed Heads on every past day, or (ii) the hypothesis that it will land Heads on every future day? I argue for the extremely counterintuitive claim that the correct answer to both questions is 1.
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  13. Adam Elga (2004). Defeating Dr. Evil with Self-Locating Belief. Philosophy and Phenomenological Research 69 (2):383–396.
    Dr. Evil learns that a duplicate of Dr. Evil has been created. Upon learning this, how seriously should he take the hypothesis that he himself is that duplicate? I answer: very seriously. I defend a principle of indifference for self-locating belief which entails that after Dr. Evil learns that a duplicate has been created, he ought to have exactly the same degree of belief that he is Dr. Evil as that he is the duplicate. More generally, the principle shows that (...)
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  14. Branden Fitelson (2006). Logical Foundations of Evidential Support. Philosophy of Science 73 (5):500-512.
    Carnap’s inductive logic (or confirmation) project is revisited from an “increase in firmness” (or probabilistic relevance) point of view. It is argued that Carnap’s main desiderata can be satisfied in this setting, without the need for a theory of “logical probability”. The emphasis here will be on explaining how Carnap’s epistemological desiderata for inductive logic will need to be modified in this new setting. The key move is to abandon Carnap’s goal of bridging confirmation and credence, in favor of bridging (...)
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  15. 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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  16. Ian Hacking (1971). Equipossibility Theories of Probability. British Journal for the Philosophy of Science 22 (4):339-355.
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  17. Alan Hájek & Michael Smithson (2012). Rationality and Indeterminate Probabilities. Synthese 187 (1):33-48.
    We argue that indeterminate probabilities are not only rationally permissible for a Bayesian agent, but they may even be rationally required . Our first argument begins by assuming a version of interpretivism: your mental state is the set of probability and utility functions that rationalize your behavioral dispositions as well as possible. This set may consist of multiple probability functions. Then according to interpretivism, this makes it the case that your credal state is indeterminate. Our second argument begins with our (...)
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  18. H. Haken (1993). Application of the Maximum Entropy Principle to Nonlinear Systems Far From Equilibrium. In E. T. Jaynes, Walter T. Grandy & Peter W. Milonni (eds.), Physics and Probability: Essays in Honor of Edwin T. Jaynes. Cambridge University Press. 239.
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  19. M. J. Hill, J. B. Paris & G. M. Wilmers (2002). Some Observations on Induction in Predicate Probabilistic Reasoning. Journal of Philosophical Logic 31 (1):43-75.
    We consider the desirability, or otherwise, of various forms of induction in the light of certain principles and inductive methods within predicate uncertain reasoning. Our general conclusion is that there remain conflicts within the area whose resolution will require a deeper understanding of the fundamental relationship between individuals and properties.
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  20. Colin Howson & Peter Urbach (1993). Scientific Reasoning: The Bayesian Approach. Open Court.
  21. Michael Huemer (2009). Explanationist Aid for the Theory of Inductive Logic. British Journal for the Philosophy of Science 60 (2):345-375.
    A central problem facing a probabilistic approach to the problem of induction is the difficulty of sufficiently constraining prior probabilities so as to yield the conclusion that induction is cogent. The Principle of Indifference, according to which alternatives are equiprobable when one has no grounds for preferring one over another, represents one way of addressing this problem; however, the Principle faces the well-known problem that multiple interpretations of it are possible, leading to incompatible conclusions. I propose a partial solution to (...)
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  22. Edwin T. Jaynes (1973). The Well-Posed Problem. Foundations of Physics 3 (4):477-493.
    Many statistical problems, including some of the most important for physical applications, have long been regarded as underdetermined from the standpoint of a strict frequency definition of probability; yet they may appear wellposed or even overdetermined by the principles of maximum entropy and transformation groups. Furthermore, the distributions found by these methods turn out to have a definite frequency correspondence; the distribution obtained by invariance under a transformation group is by far the most likely to be observed experimentally, in the (...)
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  23. Harold Jeffreys (1973). Scientific Inference. Cambridge [Eng.]Cambridge University Press.
    Thats logic. LEWIS CARROLL, Through the Looking Glass 1-1. The fundamental problem of this work is the question of the nature of scientific inference.
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  24. Richard Johns (2001). Probability and Symmetry. Philosophy of Science 68 (3):S109 - S122.
    The Principle of Indifference, which dictates that we ought to assign two outcomes equal probability in the absence of known reasons to do otherwise, is vulnerable to well-known objections. Nevertheless, the appeal of the principle, and of symmetry-based assignments of equal probability, persists. We show that, relative to a given class of symmetries satisfying certain properties, we are justified in calling certain outcomes equally probable, and more generally, in defining what we call relative probabilities. Relative probabilities are useful in providing (...)
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  25. 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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  26. 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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  27. Theo A. F. Kuipers (1984). Inductive Analogy in Carnapian Spirit. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1984:157 - 167.
    In this paper it is shown that there is a natural way of dealing with analogy by similarity in inductive systems by extending intuitive ways of introduction of systems without analogy. This procedure leads to Carnap-like systems, with zero probability for contingent generalizations, satisfying a general principle of so-called virtual analogy. This new principle is different from, but compatible with, Carnap's principle. It will be shown that the latter principle is satisfied, and should only be satisfied, if the underlying distance (...)
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  28. Jürgen Landes & Jon Williamson, Objective Bayesianism and the Maximum Entropy Principle.
    Objective Bayesian epistemology invokes three norms: the strengths of our beliefs should be probabilities, they should be calibrated to our evidence of physical probabilities, and they should otherwise equivocate sufficiently between the basic propositions that we can express. The three norms are sometimes explicated by appealing to the maximum entropy principle, which says that a belief function should be a probability function, from all those that are calibrated to evidence, that has maximum entropy. However, the three norms of objective Bayesianism (...)
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  29. 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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  30. Patrick Maher (2006). The Concept of Inductive Probability. Erkenntnis 65 (2):185 - 206.
    The word ‘probability’ in ordinary language has two different senses, here called inductive and physical probability. This paper examines the concept of inductive probability. Attempts to express this concept in other words are shown to be either incorrect or else trivial. In particular, inductive probability is not the same as degree of belief. It is argued that inductive probabilities exist; subjectivist arguments to the contrary are rebutted. Finally, it is argued that inductive probability is an important concept and that it (...)
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  31. Louis Marinoff (1994). A Resolution of Bertrand's Paradox. Philosophy of Science 61 (1):1-24.
    Bertrand's random-chord paradox purports to illustrate the inconsistency of the principle of indifference when applied to problems in which the number of possible cases is infinite. This paper shows that Bertrand's original problem is vaguely posed, but demonstrates that clearly stated variations lead to different, but theoretically and empirically self-consistent solutions. The resolution of the paradox lies in appreciating how different geometric entities, represented by uniformly distributed random variables, give rise to respectively different nonuniform distributions of random chords, and hence (...)
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  32. Elinor Mason (2004). Consequentialism and the Principle of Indifference. Utilitas 16 (3):316-321.
    James Lenman argues that consequentialism fails as a moral theory because it is impossible to predict the long-term consequences of our actions. I agree that it is impossible to predict the long-term consequences of actions, but argue that this does not count as a strike against consequentialism. I focus on the principle of indifference, which tells us to treat unforeseeable consequences as cancelling each other out, and hence value-neutral. I argue that though we cannot defend this principle independently, we cannot (...)
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  33. Ernan McMullin (1993). Indifference Principle and Anthropic Principle in Cosmology. Studies in History and Philosophy of Science Part A 24 (3):359-389.
    The successes scored by the big bang model of cosmic evolution in the 1960’s led to an intensive application of quantum theory to the problem of how the expansion might have begun and what its likely first stages were. It seemed as though an incredibly precise setting of the initial conditions would have been needed in order that a long-lived galactic universe containing heavy elements might develop. One response was to suppose that the fine-tuning could somehow be explained by the (...)
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  34. Christopher J. G. Meacham (2008). Sleeping Beauty and the Dynamics of de Se Beliefs. Philosophical Studies 138 (2):245-269.
    This paper examines three accounts of the sleeping beauty case: an account proposed by Adam Elga, an account proposed by David Lewis, and a third account defended in this paper. It provides two reasons for preferring the third account. First, this account does a good job of capturing the temporal continuity of our beliefs, while the accounts favored by Elga and Lewis do not. Second, Elga’s and Lewis’ treatments of the sleeping beauty case lead to highly counterintuitive consequences. The proposed (...)
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  35. Jeffrey M. Mikkelson (2004). Dissolving the Wine/Water Paradox. British Journal for the Philosophy of Science 55 (1):137-145.
    water paradox has long served as an argument against the Principle of Indifference. A solution to the paradox is proposed, with a view toward resolving general difficulties in applying the principle.
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  36. Amos Nathan (1984). The Fallacy of Intrinsic Distributions. Philosophy of Science 51 (4):677-684.
    Jaynes contends that in many statistical problems a seemingly indeterminate probability distribution is made unique by the transformation group of necessarily implied invariance properties, thereby justifying the principle of indifference. To illustrate and substantiate his claims he considers Bertrand's Paradox. These assertions are here refuted and the traditional attitude is vindicated.
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  37. 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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  38. John Norton (2008). Ignorance and Indifference. Philosophy of Science 75 (1):45-68.
    The epistemic state of complete ignorance is not a probability distribution. In it, we assign the same, unique, ignorance degree of belief to any contingent outcome and each of its contingent, disjunctive parts. That this is the appropriate way to represent complete ignorance is established by two instruments, each individually strong enough to identify this state. They are the principle of indifference (PI) and the notion that ignorance is invariant under certain redescriptions of the outcome space, here developed into the (...)
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  39. Greg Novack (2010). A Defense of the Principle of Indifference. Journal of Philosophical Logic 39 (6):655-678.
    The principle of indifference (hereafter ‘Poi’) says that if one has no more reason to believe A than B (and vice versa ), then one ought not to believe A more than B (nor vice versa ). Many think it’s demonstrably false despite its intuitive plausibility, because of a particular style of thought experiment that generates counterexamples. Roger White ( 2008 ) defends Poi by arguing that its antecedent is false in these thought experiments. Like White I believe Poi, but (...)
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  40. J. Paris & A. Vencovská (2011). Symmetry's End? Erkenntnis 74 (1):53-67.
    We examine the idea that similar problems should have similar solutions (to paraphrase van Fraassen’s slogan ‘Problems which are essentially the same must receive essentially the same solution’, see van Fraassen in Laws and symmetry, Oxford Univesity Press, Oxford, 1989, p. 236) in the context of symmetries of sentence algebras within Inductive Logic and conclude that by itself this is too generous a notion upon which to found the rational assignment of probabilities. We also argue that within our formulation of (...)
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  41. Matthew W. Parker, More Trouble for Regular Probabilitites.
    In standard probability theory, probability zero is not the same as impossibility. But many have suggested that only impossible events should have probability zero. This can be arranged if we allow infinitesimal probabilities, but infinitesimals do not solve all of the problems. We will see that regular probabilities are not invariant over rigid transformations, even for simple, bounded, countable, constructive, and disjoint sets. Hence, regular chances cannot be determined by space-time invariant physical laws, and regular credences cannot satisfy seemingly reasonable (...)
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  42. Tomasz Placek (1998). Łukasiewicz's Logical Probability and a Puzzle About Conditionalization. In. In Katarzyna Kijania-Placek & Jan Woleński (eds.), The Lvov-Warsaw School and Contemporary Philosophy. Kluwer Academic Publishers. 337--340.
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  43. Raghav Ramachandran, Arthur Ramer & Abhaya C. Nayak (2012). Probabilistic Belief Contraction. Minds and Machines 22 (4):325-351.
    Probabilistic belief contraction has been a much neglected topic in the field of probabilistic reasoning. This is due to the difficulty in establishing a reasonable reversal of the effect of Bayesian conditionalization on a probabilistic distribution. We show that indifferent contraction, a solution proposed by Ramer to this problem through a judicious use of the principle of maximum entropy, is a probabilistic version of a full meet contraction. We then propose variations of indifferent contraction, using both the Shannon entropy measure (...)
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  44. R. D. Rosenkrantz (1973). Probability Magic Unmasked. Philosophy of Science 40 (2):227-233.
    It has been alleged that Bayesian usage of prior probabilities allows one to obtain empirical statements on the basis of no evidence whatever. We examine this charge with reference to several examples from the literature, arguing, first, that the difference between probabilities based on weighty evidence and those based on little evidence can be drawn in terms of the variance of a distribution. Moreover, qua summaries of vague prior knowledge, prior distributions only transmit the empirical information therein contained and, therefore, (...)
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  45. Roger D. Rosenkrantz (1981). Foundations and Applications of Inductive Probability. Ridgeview Press.
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  46. 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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  47. Darrell P. Rowbottom & Nicholas Shackel (2010). Bangu's Random Thoughts on Bertrand's Paradox. Analysis 70 (4):689-692.
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  48. Teddy Seidenfeld (1979). Why I Am Not an Objective Bayesian; Some Reflections Prompted by Rosenkrantz. Theory and Decision 11 (4):413-440.
  49. Nicholas Shackel (2007). Bertrand's Paradox and the Principle of Indifference. Philosophy of Science 74 (2):150-175.
    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 (...)
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  50. Abner Shimony (1985). The Status of the Principle of Maximum Entropy. Synthese 63 (1):35 - 53.
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