Studia Logica 102 (4):811-848 (2014)
AbstractI 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
Similar books and articles
An Extension Theorem and a Numerical Representation Theorem for Qualitative Comparative Expectations.Arthur Paul Pedersen - forthcoming - Studia Logica.
When Coherent Preferences May Not Preserve Indifference Between Equivalent Random Variables: A Price for Unbounded Utilities.Teddy Seidenfeld, Mark Schervish & Joseph Kadane - unknown
The Significance of the Ergodic Decomposition of Stationary Measures for the Interpretation of Probability.Jan Plato - 1982 - Synthese 53 (3):419-432.
The Significance of the Ergodic Decomposition of Stationary Measures for the Interpretation of Probability.Jan Von Plato - 1982 - Synthese 53 (3):419 - 432.
A Note on the Decidability of de Finetti's Coherence.Francesco Corielli - 1995 - Theory and Decision 38 (1):121-129.
Comparative Probability, Comparative Conﬁrmation, and the “Conjunction Fallacy”.Branden Fitelson - unknown
De Finetti Coherence and Logical Consistency.James M. Dickey, Morris L. Eaton & William D. Sudderth - 2009 - Notre Dame Journal of Formal Logic 50 (2):133-139.
Comparative Probability for Conditional Events: A New Look Through Coherence.Giulianella Coletti, Angelo Gilio & Romano Scozzafava - 1993 - Theory and Decision 35 (3):237-258.
A Conflict Between Finite Additivity and Avoiding Dutch Book.Teddy Seidenfeld & Mark J. Schervish - 1983 - Philosophy of Science 50 (3):398-412.
Probability as a Measure of Information Added.Peter Milne - 2012 - Journal of Logic, Language and Information 21 (2):163-188.
Preferences Representable by a Lower Expectation: Some Characterizations. [REVIEW]Andrea Capotorti, Giulianella Coletti & Barbara Vantaggi - 2008 - Theory and Decision 64 (2-3):119-146.
Added to PP
Historical graph of downloads
Citations of this work
Comparative Probabilities.Jason Konek - 2019 - In Richard Pettigrew & Jonathan Weisberg (eds.), The Open Handbook of Formal Epistemology. PhilPapers Foundation. pp. 267-348.
Infinitesimal Probabilities.Vieri Benci, Leon Horsten & Sylvia Wenmackers - 2016 - British Journal for the Philosophy of Science 69 (2):509-552.
Hyperintensionality and Normativity.Federico L. G. Faroldi - 2019 - Cham, Switzerland: Springer Verlag.
Infinitesimal Probabilities.Sylvia Wenmackers - 2016 - In Richard Pettigrew & Jonathan Weisberg (eds.), The Open Handbook of Formal Epistemology. PhilPapers Foundation. pp. 199-265.
References found in this work
Causal Necessity: A Pragmatic Investigation of the Necessity of Laws.Brian Skyrms - 1980 - Yale University Press.
Statistical Reasoning with Imprecise Probabilities.Peter Walley - 1991 - Chapman & Hall.
Studies in Inductive Logic and Probability.Rudolf Carnap & Richard C. Jeffrey (eds.) - 1971 - University of California Press.