Non-Archimedean probability functions allow us to combine regularity with perfect additivity. We discuss the philosophical motivation for a particular choice of axioms for a non-Archimedean probability theory and answer some philosophical objections that have been raised against infinitesimal probabilities in general. _1_ Introduction _2_ The Limits of Classical Probability Theory _2.1_ Classical probability functions _2.2_ Limitations _2.3_ Infinitesimals to the rescue? _3_ NAP Theory _3.1_ First four axioms of NAP _3.2_ Continuity and conditional probability _3.3_ The final axiom of NAP (...) _3.4_ Infinite sums _3.5_ Definition of NAP functions via infinite sums _3.6_ Relation to numerosity theory _4_ Objections and Replies _4.1_ Cantor and the Archimedean property _4.2_ Ticket missing from an infinite lottery _4.3_ Williamson’s infinite sequence of coin tosses _4.4_ Point sets on a circle _4.5_ Easwaran and Pruss _5_ Dividends _5.1_ Measure and utility _5.2_ Regularity and uniformity _5.3_ Credence and chance _5.4_ Conditional probability _6_ General Considerations _6.1_ Non-uniqueness _6.2_ Invariance Appendix. (shrink)

The naïve idea of “size” for collections seems to obey both Aristotle’s Principle: “the whole is greater than its parts” and Cantor’s Principle: “1-to-1 correspondences preserve size”. Notoriously, Aristotle’s and Cantor’s principles are incompatible for infinite collections. Cantor’s theory of cardinalities weakens the former principle to “the part is not greater than the whole”, but the outcoming cardinal arithmetic is very unusual. It does not allow for inverse operations, and so there is no direct way of introducing infinitesimal numbers. Here (...) we maintain Aristotle’s principle, instead halving Cantor’s principle to “equinumerous collections are in 1–1 correspondence”. In this way we obtain a very nice arithmetic: in fact, our “numerosities” may be taken to be nonstandard integers. These numerosities appear naturally suited to sets of ordinals, but they depend, for generic sets, on a “labelling” of the universe by ordinals. The problem of finding a canonical way of attaching numerosities to all sets seems to be worth further investigation. (shrink)

We provide a solution to the well-known “Shooting-Room” paradox, developed by John Leslie in connection with his Doomsday Argument. In the “Shooting-Room” paradox, the death of an individual is contingent upon an event that has a 1/36 chance of occurring, yet the relative frequency of death in the relevant population is 0.9. There are two intuitively plausible arguments, one concluding that the appropriate subjective probability of death is 1/36, the other that this probability is 0.9. How are these two values (...) to be reconciled? We show that only the first argument is valid for a standard, countably additive probability distribution. However, both lines of reasoning are legitimate if probabilities are non-standard. The subjective probability of death rises from 1/36 to 0.9 by conditionalizing on an event that is not measurable, or whose probability is zero. Thus we can sometimes meaningfully ascribe conditional probabilities even when the event conditionalized upon is not of positive finite (or even infinitesimal) measure. (shrink)

Metaphysicians speak of laws of nature in terms of necessity and universality; scientists, in terms of symmetry and invariance. In this book van Fraassen argues that no metaphysical account of laws can succeed. He analyzes and rejects the arguments that there are laws of nature, or that we must believe there are, and argues that we should disregard the idea of law as an adequate clue to science. After exploring what this means for general epistemology, the author develops the empiricist (...) view of science as a construction of models to represent the phenomena. (shrink)

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 (...) the books: a measure that assigns the same infinitesimal probability to each number between zero and one. I will show that such a measure, while mathematically interesting, is pathological for use in confirmation theory, for the same reason that a measure that assigns an infinitesimal probability to each possible outcome in a countably infinite lottery is pathological. The pathology is that one can force someone to assign a probability within an infinitesimal of one to an unlikely event. (shrink)

We show that infinitesimal probabilities are much too small for modeling the individual outcome of a countably infinite fair lottery.

Regularity is the thesis that all contingent propositions should be assigned probabilities strictly between zero and one. I will prove on cardinality grounds that if the domain is large enough, a regular probability assignment is impossible, even if we expand the range of values that probabilities can take, including, for instance, hyperreal values, and significantly weaken the axioms of probability.

Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collectionAis properly included in a collectionBthen the ‘size’ ofAshould be less than the (...) ‘size’ ofB(part–whole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the part–whole principle to infinite sets in a coherent fashion (Katz, Benci, Di Nasso, Forti). Finally, I show how these new developments are important for a proper evaluation of a number of positions in philosophy of mathematics which argue either for the inevitability of the Cantorian notion of infinite number (Gödel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part–whole principle, envisaged by Bolzano (Kitcher). (shrink)

Kolmogorov''s axiomatization of probability includes the familiarratio formula for conditional probability: 0).$$ " align="middle" border="0">.

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 (...) kinds of probability, such as evidential probability. The formal analogue of this picture is the regularity constraint: a probability distribution over sets of possibilities is regular just in case it assigns probability 0 only to the null set, and therefore probability 1 only to the set of all possibilities. (shrink)

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 (...) kinds of probability, such as evidential probability. The formal analogue of this picture is the regularity constraint: a probability distribution over sets of possibilities is regular just in case it assigns probability 0 only to the null set, and therefore probability 1 only to the set of all possibilities. (shrink)

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.

It is possible that a fair coin tossed infinitely many times will always land heads. So the probability of such a sequence of outcomes should, intuitively, be positive, albeit miniscule: 0 probability ought to be reserved for impossible events. And, furthermore, since the tosses are independent and the probability of heads (and tails) on a single toss is half, all sequences are equiprobable. But Williamson has adduced an argument that purports to show that our intuitions notwithstanding, the probability of an (...) infinite sequence is 0. In this paper, I rebut his argument.No Abstract. (shrink)

Considered by many to be Abraham Robinson's magnum opus, this book offers an explanation of the development and applications of non-standard analysis by the mathematician who founded the subject. Non-standard analysis grew out of Robinson's attempt to resolve the contradictions posed by infinitesimals within calculus. He introduced this new subject in a seminar at Princeton in 1960, and it remains as controversial today as it was then. This paperback reprint of the 1974 revised edition is indispensable reading for anyone interested (...) in non-standard analysis. It treats in rich detail many areas of application, including topology, functions of a real variable, functions of a complex variable, and normed linear spaces, together with problems of boundary layer flow of viscous fluids and rederivations of Saint-Venant's hypothesis concerning the distribution of stresses in an elastic body. (shrink)

Let X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{X }$$\end{document} be a set of outcomes, and let I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{I }$$\end{document} be an infinite indexing set. This paper shows that any separable, permutation-invariant preference order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document} on XI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{X }^\mathcal{I }$$\end{document} admits an additive representation. That is: there exists a linearly ordered abelian group R\documentclass[12pt]{minimal} (...) \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R }$$\end{document} and a ‘utility function’ u:X⟶R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u:\mathcal{X }{{\longrightarrow }}\mathcal{R }$$\end{document} such that, for any x,y∈XI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x},\mathbf{y}\in \mathcal{X }^\mathcal{I }$$\end{document} which differ in only finitely many coordinates, we have x≽y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x}\succcurlyeq \mathbf{y}$$\end{document} if and only if ∑i∈Iu-u≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{i\in \mathcal{I }} \left[u-u\right]\ge 0$$\end{document}. Importantly, and unlike almost all previous work on additive representations, this result does not require any Archimedean or continuity condition. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document} also satisfies a weak continuity condition, then the paper shows that, for anyx,y∈XI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x},\mathbf{y}\in \mathcal{X }^\mathcal{I }$$\end{document}, we have x≽y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x}\succcurlyeq \mathbf{y}$$\end{document} if and only if ∗∑i∈Iu≥∗∑i∈Iu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^*\!\sum _{i\in \mathcal{I }} u\ge {}^*\!\sum _{i\in \mathcal{I }}u$$\end{document}. Here, ∗∑i∈Iu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^*\!\sum _{i\in \mathcal{I }} u$$\end{document} represents a nonstandard sum, taking values in a linearly ordered abelian group ∗R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^*\!\mathcal{R }$$\end{document}, which is an ultrapower extension of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R }$$\end{document}. The paper also discusses several applications of these results, including infinite-horizon intertemporal choice, choice under uncertainty, variable-population social choice and games with infinite strategy spaces. (shrink)

I introduce a mathematical account of expectation based on a qualitative criterion of coherence for qualitative comparisons between gambles (or random quantities). The qualitative comparisons may be interpreted as an agent’s comparative preference judgments over options or more directly as an agent’s comparative expectation judgments over random quantities. The criterion of coherence is reminiscent of de Finetti’s quantitative criterion of coherence for betting, yet it does not impose an Archimedean condition on an agent’s comparative judgments, it does not require the (...) binary relation reflecting an agent’s comparative judgments to be reflexive, complete or even transitive, and it applies to an absolutely arbitrary collection of gambles, free of structural conditions (e.g., closure, measurability, etc.). Moreover, unlike de Finetti’s criterion of coherence, the qualitative criterion respects the principle of weak dominance, a standard of rational decision making that obliges an agent to reject a gamble that is possibly worse and certainly no better than another gamble available for choice. Despite these weak assumptions, I establish a qualitative analogue of de Finetti’s Fundamental Theorem of Prevision, from which it follows that any coherent system of comparative expectations can be extended to a weakly ordered coherent system of comparative expectations over any collection of gambles containing the initial set of gambles of interest. The extended weakly ordered coherent system of comparative expectations satisfies familiar additivity and scale invariance postulates (i.e., independence) when the extended collection forms a linear space. In the course of these developments, I recast de Finetti’s quantitative account of coherent prevision in the qualitative framework adopted in this article. I show that comparative previsions satisfy qualitative analogues of de Finetti’s famous bookmaking theorem and his Fundamental Theorem of Prevision.The results of this article complement those of another article (Pedersen, Strictly coherent preferences, no holds barred, Manuscript, 2013). I explain how those results entail that any coherent weakly ordered system of comparative expectations over a unital linear space can be represented by an expectation function taking values in a (possibly non-Archimedean) totally ordered field extension of the system of real numbers. The ordered field extension consists of formal power series in a single infinitesimal, a natural and economical representation that provides a relief map tracing numerical non-Archimedean features to qualitative non-Archimedean features. (shrink)

This is part A of a paper in which we defend a semantics for counterfactuals which is probabilistic in the sense that the truth condition for counterfactuals refers to a probability measure. Because of its probabilistic nature, it allows a counterfactual ‘ifAthenB’ to be true even in the presence of relevant ‘Aand notB’-worlds, as long such exceptions are not too widely spread. The semantics is made precise and studied in different versions which are related to each other by representation theorems. (...) Despite its probabilistic nature, we show that the semantics and the resulting system of logic may be regarded as a naturalistically vindicated variant of David Lewis’ truth-conditional semantics and logic of counterfactuals. At the same time, the semantics overlaps in various ways with the non-truth-conditional suppositional theory for conditionals that derives from Ernest Adams’ work. We argue that counterfactuals have two kinds of pragmatic meanings and come attached with two types of degrees of acceptability or belief, one being suppositional, the other one being truth based as determined by our probabilistic semantics; these degrees could not always coincide due to a new triviality result for counterfactuals, and they should not be identified in the light of their different interpretation and pragmatic purpose. However, for plain assertability the difference between them does not matter. Hence, if the suppositional theory of counterfactuals is formulated with sufficient care, our truth-conditional theory of counterfactuals is consistent with it. The results of our investigation are used to assess a claim considered by Hawthorne and Hájek, that is, the thesis that most ordinary counterfactuals are false. (shrink)

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 (...) this remark with the example of the sets of odd and even numbers. Depending on the ultrafilter, either each of these sets has probability 1/2, or the set of odd numbers has a probability infinitesimally higher than 1/2 and the set of even numbers infinitesimally lower. The point of the current paper is simply that the amount of indeterminacy is much greater than acknowledged in FIL: there are sets of natural numbers whose probability is far more indeterminate than that of the set of odd or the set of even numbers. (shrink)

The 'best-system' analysis of lawhood [Lewis 1994] faces the 'zero-fit problem': that many systems of laws say that the chance of history going actually as it goes--the degree to which the theory 'fits' the actual course of history--is zero. Neither an appeal to infinitesimal probabilities nor a patch using standard measure theory avoids the difficulty. But there is a way to avoid it: replace the notion of 'fit' with the notion of a world being typical with respect to a theory.

Using formal systems to represent and reason about uncertainty.

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 (...) view is the result of two mistakes. The first mistake, which this essay calls the “numerical fallacy,” is to assume that a distinction that isn't represented by different numbers isn't represented at all in a mathematical representation. In this case, the essay claims that although the real numbers do not make all relevant distinctions, the full mathematical structure of a probability function does. The second mistake is that the hyperreals make too many distinctions. They have a much more complex structure than credences in ordinary propositions can have, so they make distinctions that don't exist among credences. While they might be useful for generating certain mathematical models, they will not appear in a faithful mathematical representation of credences of ordinary propositions. (shrink)

Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.

entry for the Entry for the Stanford Encyclopedia of Philosophy, 2007.

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 (...) a different type of infinite additivity. (shrink)

Non-Archimedean probability functions allow us to combine regularity with perfect additivity. We discuss the philosophical motivation for a particular choice of axioms for a non-Archimedean probability theory and answer some philosophical objections that have been raised against infinitesimal probabilities in general.

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 (...) seem to be left open by all this. I argue that this is a mistake. The employment of real numbers as measures of chance in standard probability theory brings with it commitments in the metaphysics of (objective) chance that are not only substantial but also mistaken. To measure chance properly we need to employ extensions of the real numbers that contain infinitesimals: positive numbers that are infinitely small. But simply using infinitesimals alone is not enough, as a number of arguments show. Instead we need to put three ideas together: infinitesimals, the non-locality of chance and flexibility in measurement. Only those three together give us a coherent picture of chance and its mathematical representation. (shrink)