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  1. Vieri Benci, Leon Horsten & Sylvia Wenmackers (2013). Non-Archimedean Probability. 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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  2. Vieri Benci, Leon Horsten & Sylvia Wenmackers (2012). Axioms for Non-Archimedean Probability (NAP). 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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  3. Marc Burock, An Outcome of the de Finetti Infinite Lottery is Not Finite.
    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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  4. Kenny Easwaran (2014). Regularity and Hyperreal Credences. 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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  5. Philip Kremer (2014). Indeterminacy of Fair Infinite Lotteries. 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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  6. Rupert Macey-Dare, Expected Loss Divisibility Theorem.
    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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  7. 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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  8. Alexander R. Pruss (2014). Infinitesimals Are Too Small for Countably Infinite Fair Lotteries. 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. Alexander R. Pruss (2012). Infinite Lotteries, Perfectly Thin Darts and Infinitesimals. 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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  10. Yaroslav Sergeyev (2009). Numerical Computations and Mathematical Modelling with Infinite and Infinitesimal Numbers. 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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  11. Sylvia Wenmackers (2012). Ultralarge and Infinite Lotteries. 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. Sylvia Wenmackers (2011). Philosophy of Probability: Foundations, Epistemology, and Computation. 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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  13. Sylvia Wenmackers & Leon Horsten (2013). Fair Infinite Lotteries. 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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