David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Studia Logica 89 (2):257 - 283 (2008)
In this paper a theory of finitistic and frequentistic approximations — in short: f-approximations — of probability measures P over a countably infinite outcome space N is developed. The family of subsets of N for which f-approximations converge to a frequency limit forms a pre-Dynkin system $D \subseteq \wp (N)$ . The limiting probability measure over D can always be extended to a probability measure over $\wp (N)$ , but this measure is not always σ-additive. We conclude that probability measures can be regarded as idealizations of limiting frequencies if and only if σ-additivity is not assumed as a necessary axiom for probabilities. We prove that σ-additive probability measures can be characterized in terms of so-called canonical and in terms of so-called full f-approximations. We also show that every non-σ-additive probability measure is f-approximable, though neither canonically nor fully f-approximable. Finally, we transfer our results to probability measures on open or closed formulas of first-order languages
|Keywords||Philosophy Computational Linguistics Mathematical Logic and Foundations Logic|
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References found in this work BETA
John Earman (1992). Bayes or Bust? Bradford.
Kevin Kelly (1996). The Logic of Reliable Inquiry. Oxford University Press, USA.
Henry Ely Kyburg (1961). Probability and the Logic of Rational Belief. Middletown, Conn.,Wesleyan University Press.
Citations of this work BETA
Sylvia Wenmackers & Leon Horsten (2013). Fair Infinite Lotteries. Synthese 190 (1):37-61.
Frederik Herzberg (2010). The Consistency of Probabilistic Regresses. A Reply to Jeanne Peijnenburg and David Atkinson. Studia Logica 94 (3):331 - 345.
Leendert Huisman (2015). Reflecting on Finite Additivity. Synthese 192 (6):1785-1797.
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