Abstract
The purpose of this paper is to compare the two most plausible interpretations of the meaning of ‘probability’ as that term occurs within the context of statistical laws: the frequency interpretation (elaborated by Hans Reichenbach and Richard von Mises, for example) and the propensity interpretation (first proposed by Karl Popper and recently discussed by Ian Hacking, among others). Both interpretations assume there is an important connection between probabilities and frequencies, but they fundamentally differ in their conceptions of the nature of that connection. Once this basic difference is made explicit, it becomes apparent that choosing between them poses a dilemma resulting from a certain tension between desiderata of epistemological and of systematic character, respectively. My concern, therefore, is to contribute toward the resolution of this dilemma.
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Hans Reichenbach, The Theory of Probability, University of California Press, Berkeley. 1949.
Richard von Mises, Mathematical Theory of Probability and Statistics, Academic Press, New York, 1964.
Karl Popper, ‘The Propensity Interpretation of the Calculus of Probability, and the Quantum Theory’, in Observation and Interpretation in the Philosophy of Physics (ed. by S. Korner), Dover Publications, Inc., New York, 1955; and Karl Popper, ‘The Propensity Interpretation of Probability’, British Journal for the Philosophy of Science 10 (1959) 25–42.
Ian Hacking, Logic of Statistical Inference, Cambridge University Press, Cambridge, 1965.
Reichenbach distinguished between two kinds of ‘probability’ sequences, i.e., normal and non-normal, while Mises distinguished between two kinds of ‘chance’ sequences, i.e., random and non-random (only the former being regarded as ‘probability’ sequences). Mises’ strategy is followed here, with the adoption of Reiehenbach’s concept of normality in lieu of Mises’ concept of randomness. The difficulties with Mises’ definition of randomness are discussed, e.g., by Ernest Nagel, Principles of the Theory of Probability, University of Chicago Press, Chicago, 1939, pp. 32–33.
Reichenbach, op. cit., p. 46.
Reichenbach, op. cit., pp. 141–51.
For an analysis of the differences between Hacking’s ‘long run’ and Popper’s ‘single ease’ constructions (and of the advantages of Popper’s approach), soe James H. Fctzer, ‘Dispositional Probabilities’, in Boston Studies in the Philosophy of Science, Vol. 8 (ed. by R. Buck and R. Cohen), D. Reidel Publ. Co., Dordrecht, Holland. 1971.
Cf. Harold Freeman, Introduction to Statistical Inference, Addison-Wesley Publishing Co., Reading, Mass., 1963, pp. 167–68.
For elaboration, sec James H. Fetzer, op. cit., esp. p. 476. The sequences accessible within our range of experience, of course, are always finite; consequently, this epistemological difference is exclusively a matter of principle and not a matter of practice.
Wesley C. Salmon, Statistical Explanation and Statistical Relevance, University of Pittsburgh Press, Pittsburgh, 1971, pp. 42–43 and pp. 106–08. For an appraisal of Salmon’s and Hempel’s approach to this and related issues, see James H. Fetzer, ‘Statistical Explanations’, in Boston Studies in the Philosophy of Science, Vol. XX (ed. by K. Schaffner and R. Cohen), D. Reidel Publ. Co., Dordrecht, 1974.
Cf. Salmon, op. cit., p. 42.
Cf. Salmon, op. cit., p. 106.
Carl G. Hempel, ‘Empiricist Criteria of Cognitive Significance: Problems and Changes’, in Aspects of Scientific Explanation The Free Press, New York, 1965, p. 117.
Reichenbach, op. cit., pp. 376–77.
The distinction intended is that between final causes and efficient causes. The frequency theory accounts for the occurrence of certain outcomes with specific limiting frequencies within a spatially separated and temporally extended trial sequence without accounting for each of the individual outcomes that collectively constitute that sequence; while the propensity theory accounts for both the individual outcomes and also the distinctive character of that sequence itself on the basis of dispositional properties which bring about the result of each of its trial elements.
This difference may be informally expressed by observing that on the frequency approach, ‘probabilities’ are both displayed and defined by frequencies; while on the propensity approach, frequencies display but do not also define ‘probabilities’. Notice that this logical difference persists with the hypothetical frequency construction, since the truth values of probability statements under this interpretation are completely determnied, in principle, by hypothetical sequences of trials. Consequently, the hypo-thetical frequency account may be characterized as ‘weakly’ intensional, in contrast to the propensity account, which, by virtue of its ontological commitments, is ‘strongly’ intensional.
Carl G. Hempel, ‘Aspccts of Scientific Explanation’, op. cit., pp. 339 40.
Although the hypothetical frequency formulation prima facie provides support for counterfactua) and subjunctive conditionals, therefore, the problem still remains of explaining why these statements characterizing what the limiting frequencies would be or would have been if these trial sequences were or had been infinite are true. By contrast with the propensity approach, in other words, the frequency approach provides no theoretical principles or structural properties that may be invoked to explain the attribution of these hypothetical limiting frequencies to the physical world.
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© 1974 D. Reidel Publishing Company, Dordrecht-Holland
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Fetzer, J.H. (1974). Statistical Probabilities: Single Case Propensities vs. Long-Run Frequencies. In: Leinfellner, W., Köhler, E. (eds) Developments in the Methodology of Social Science. Theory and Decision Library, vol 6. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-2259-0_14
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DOI: https://doi.org/10.1007/978-94-010-2259-0_14
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