Abstract
There are three fundamentally distinct ways of interpreting probability: personalistic, logical, and empirical. At least one of these conceptions of probability underlies any theory of statistical inference (or, to use Neyman’s phrase, ‘inductive behavior’). Each interpretation admits a variety of subinterpretations, but the three basic kinds of interpretation may be distinguished by the truth conditions associated with statements of probability.
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Notes
John Venn, The Logic of Chance, London, 1866.
Richard von Mises, Wahrscheinlichkeit, Statistik, und Wahrheit, J. Springer, Berlin, 1928. Translated as Probability, Statistics, and Truth, 2nd revised English edition, prepared by Hilda Geiringer, London and New York, 1957.
The concept of randomness in a collective has a long and interesting history discussed with clarity and precision by J. A. Coffa in his dissertation, Foundations of Inductive Explanation, University of Pittsburgh, 1972.
Among the distinguished contributors to this history are A. Wald, ‘Über die Wieder-spruchsfreiheit des Kollectivbegriffes’, Ergebnisse eines math. Kolloquiums, 1947, pp. 38–72;
A. Church, ‘The Concept of a Random Sequence’, Bull, of American Mathematical Society 46 (1940), 130–135;
and C. P. Schnoor ‘A Unified Approach to the Definition of Random Sequence’, Mathematical Systems Theory 5 (1971).
An important recent review article is Per Martin-Löf, ‘The Literature on von Mises’ Kollektives Revisited’, Theoria 35 (1969).
J. Neyman, First Course in Probability and Statistics, New York, 1950; Lectures and Conferences on Mathematical Statistics and Probability, Washington, D.C., 1952.
H. Cramér, Mathematical Methods of Statistics, Princeton, 1951. (First edition, Uppsala, 1945.)
K. R. Popper, ‘The Propensity Interpretation of Probability’, British Journal for the Philosophy of Science 10 (1959, 1960), 25–42.
Ian Hacking, The Logic of Statistical Inference, Cambridge, 1965. Hacking’s ‘chance’ corresponds roughly to Popper’s ‘propensity’.
For a recent work that makes much of the application of propensities to the single case, see D. H. Mellor, The Matter of Chance, Cambridge, 1971.
To these references we should now add D. A. Gillies, An Objective Theory of Probability, Methuen, 1973.
A general review of propensity theory is H. E. Kyburg, ‘Probabilities and Propensities’, British Journal for the Philosophy of Science, forthcoming.
For the views of Neyman, von Mises, and Cramer, see the works cited. Hans Reichen-bach’s views may be found in Experience and Prediction, Chicago 1938 and The Theory of Probability, Berkeley and Los Angeles 1949. The latter appeared first in German in 1934. It should be observed that Reichenbach also employs a concept of weight, analogous to (empirical) probability, which is applicable to the single case. Wesley Salmon’s views are most completely expressed in The Foundations of Scientific Inference, Pittsburgh 1967. Propensity theorists are prone to talk of the probability of the ‘single case’, but this only amounts to predicating distributively of each member of the class what is — on the frequency view — true of the whole class collectively.
This is not true of the view of Mellor.
These integrals are defined formally in the appendix. In most instances they amount to the familiar Riemann integral of calculus.
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© 1974 D. Reidel Publishing Company, Dordrecht, Holland
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Kyburg, H.E. (1974). The Probability Framework. In: The Logical Foundations of Statistical Inference. Synthese Library, vol 65. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-2175-3_1
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DOI: https://doi.org/10.1007/978-94-010-2175-3_1
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