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  1. added 2020-02-05
    How Probable is an Infinite Sequence of Heads?Timothy Williamson - 2007 - Analysis 67 (3):173-180.
    Isn't probability 1 certainty? If the probability is objective, so is the certainty: whatever has chance 1 of occurring is certain to occur. Equivalently, whatever has chance 0 of occurring is certain not to occur. If the probability is subjective, so is the certainty: if you give credence 1 to an event, you are certain that it will occur. Equivalently, if you give credence 0 to an event, you are certain that it will not occur. And so on for other (...)
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  2. added 2020-01-29
    Utilitarianism with and Without Expected Utility.David McCarthy, Kalle Mikkola & Joaquin Teruji Thomas - 2020 - Journal of Mathematical Economics 87:77-113.
    We give two social aggregation theorems under conditions of risk, one for constant population cases, the other an extension to variable populations. Intra and interpersonal welfare comparisons are encoded in a single ‘individual preorder’. The theorems give axioms that uniquely determine a social preorder in terms of this individual preorder. The social preorders described by these theorems have features that may be considered characteristic of Harsanyi-style utilitarianism, such as indifference to ex ante and ex post equality. However, the theorems are (...)
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  3. added 2018-06-05
    Symmetry Arguments Against Regular Probability: A Reply to Recent Objections.Matthew W. Parker - 2018 - European Journal for Philosophy of Science 9 (1):8.
    A probability distribution is regular if no possible event is assigned probability zero. While some hold that probabilities should always be regular, three counter-arguments have been posed based on examples where, if regularity holds, then perfectly similar events must have different probabilities. Howson (2017) and Benci et al. (2016) have raised technical objections to these symmetry arguments, but we see here that their objections fail. Howson says that Williamson’s (2007) “isomorphic” events are not in fact isomorphic, but Howson is speaking (...)
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  4. added 2017-11-08
    Reasoning with Plenitude.Roger White - 2018 - In Matthew A. Benton, John Hawthorne & Dani Rabinowitz (eds.), Knowledge, Belief, and God: New Insights in Religious Epistemology. Oxford: Oxford University Press. pp. 169-179.
  5. added 2016-04-26
    Philosophical Perspectives on Infinity.Graham Oppy - 2006 - Cambridge University Press.
    This book is an exploration of philosophical questions about infinity. Graham Oppy examines how the infinite lurks everywhere, both in science and in our ordinary thoughts about the world. He also analyses the many puzzles and paradoxes that follow in the train of the infinite. Even simple notions, such as counting, adding and maximising present serious difficulties. Other topics examined include the nature of space and time, infinities in physical science, infinities in theories of probability and decision, the nature of (...)
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  6. added 2016-01-19
    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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  7. added 2015-03-20
    Indeterminacy of Fair Infinite Lotteries.Philip Kremer - 2014 - Synthese 191 (8):1757-1760.
    In ‘Fair Infinite Lotteries’ (FIL), Wenmackers and Horsten use non-standard analysis to construct a family of nicely-behaved hyperrational-valued probability measures on sets of natural numbers. Each probability measure in FIL is determined by a free ultrafilter on the natural numbers: distinct free ultrafilters determine distinct probability measures. The authors reply to a worry about a consequent ‘arbitrariness’ by remarking, “A different choice of free ultrafilter produces a different ... probability function with the same standard part but infinitesimal differences.” They illustrate (...)
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  8. added 2015-03-19
    Infinitesimals Are Too Small for Countably Infinite Fair Lotteries.Alexander R. Pruss - 2014 - Synthese 191 (6):1051-1057.
    We show that infinitesimal probabilities are much too small for modeling the individual outcome of a countably infinite fair lottery.
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  9. added 2014-04-02
    Regularity and Hyperreal Credences.Kenny Easwaran - 2014 - Philosophical Review 123 (1):1-41.
    Many philosophers have become worried about the use of standard real numbers for the probability function that represents an agent's credences. They point out that real numbers can't capture the distinction between certain extremely unlikely events and genuinely impossible ones—they are both represented by credence 0, which violates a principle known as “regularity.” Following Skyrms 1980 and Lewis 1980, they recommend that we should instead use a much richer set of numbers, called the “hyperreals.” This essay argues that this popular (...)
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  10. added 2013-06-25
    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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  11. added 2013-06-25
    Ultralarge and Infinite Lotteries.Sylvia Wenmackers - 2012 - In B. Van Kerkhove, T. Libert, G. Vanpaemel & P. Marage (eds.), Logic, Philosophy and History of Science in Belgium II (Proceedings of the Young Researchers Days 2010). Koninklijke Vlaamse Academie van België voor Wetenschappen en Kunsten.
    By exploiting the parallels between large, yet finite lotteries on the one hand and countably infinite lotteries on the other, we gain insights in the foundations of probability theory as well as in epistemology. We solve the 'adding problems' that occur in these two contexts using a similar strategy, based on non-standard analysis.
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  12. added 2013-03-13
    More Trouble for Regular Probabilitites.Matthew W. Parker - manuscript
    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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  13. added 2012-07-12
    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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  14. added 2011-08-30
    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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  15. added 2011-08-25
    Fair Infinite Lotteries.Sylvia Wenmackers & Leon Horsten - 2013 - Synthese 190 (1):37-61.
    This article discusses how the concept of a fair finite lottery can best be extended to denumerably infinite lotteries. Techniques and ideas from non-standard analysis are brought to bear on the problem.
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  16. added 2011-08-25
    Philosophy of Probability: Foundations, Epistemology, and Computation.Sylvia Wenmackers - 2011 - Dissertation, University of Groningen
    This dissertation is a contribution to formal and computational philosophy. -/- In the first part, we show that by exploiting the parallels between large, yet finite lotteries on the one hand and countably infinite lotteries on the other, we gain insights in the foundations of probability theory as well as in epistemology. Case 1: Infinite lotteries. We discuss how the concept of a fair finite lottery can best be extended to denumerably infinite lotteries. The solution boils down to the introduction (...)
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  17. added 2011-05-03
    Expected Loss Divisibility Theorem.Rupert Macey-Dare - manuscript
    This paper proposes and analyses the following theorem: For every total actual loss caused to a claimant with given probabilities by a single unidentified member of a defined group, there is a corresponding total expected loss, divisible and separable into discrete component expected sub-losses, each individually "caused" by a corresponding specific member of that defined group. Moreover, for every total estimated loss caused to a claimant in the past or present or prospectively in the future with estimable probabilities by one (...)
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  18. added 2009-12-02
    Numerical Computations and Mathematical Modelling with Infinite and Infinitesimal Numbers.Yaroslav Sergeyev - 2009 - Journal of Applied Mathematics and Computing 29:177-195.
    Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not related to the non-standard analysis) is used to work with finite, infinite, and infinitesimal numbers numerically. This can be done on a new kind of a computer – the Infinity Computer – able to work with all these types of numbers. The new computational tools both give possibilities to (...)
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  19. added 2008-12-31
    An Outcome of the de Finetti Infinite Lottery is Not Finite.Marc Burock - unknown
    A randomly selected number from the infinite set of positive integers—the so-called de Finetti lottery—will not be a finite number. I argue that it is still possible to conceive of an infinite lottery, but that an individual lottery outcome is knowledge about set-membership and not element identification. Unexpectedly, it appears that a uniform distribution over a countably infinite set has much in common with a continuous probability density over an uncountably infinite set.
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