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References
I.Degree of Confirmation, Vol. V, No 18, 1954; II.“Content” and “Degree of Confirmation”: A Reply to Dr. Bar-Hillel Vol. VI, No 22, 1955; III.Carnap, Bar-Hillel, Popper:Content and Degree of Confirmation. Vol. VII, No 27, 1956; IV.A Second Note on Degree of Confirmation. Vol. VII, No 28, 1957; V.A Third Note on Degree of Corroboration or Confirmation. Vol. VIII, No 32, 1958. See also: VI.Probability Magic or Knowledge out of Ignorance. Dialectica, Vol. 11, No 3/4, 1957.
Professor Popper seems not to be unaware of the fact when he writes: “If, as in our case,h is a statistical hypothesis, ande the report of the results of statistical tests ofh, thenC(h,e) is a measure of the degree to which these tests corroborateh, exactly as in the case of a non-statistical hypothesis. It should be mentioned, however, thatas opposed to the case of a non-statistical hypothesis, it might be at times quite easy to estimate the numerical values of E(h,e) and even of C(h,e) if h is a statistical hypothesis”. (V, p. 298, italics mine.)
Professor Popper writes: “It does not matter, in the context of the present paper, whether the objective interpretation (which he adopts-K.S.) is a purely statistical interpretation or a propensity interpretation” (V, p. 297). Which shows that the probabilities in V are statistical.
There is a slight mistake in V, p. 299. The postulated equidistribution ought to ascribe the probability\(\frac{1}{{n + 1}}\left( {and not\frac{1}{n}} \right)\) to each possible outcome. In view of this the value ofE(calculated on the following page) for a positively verified universal law ought to be\(\frac{n}{{n + 2}} = 1 - \frac{2}{{n + 2}}\left( {and not:\frac{{n - 1}}{{n + 1}} = 1 - \frac{2}{{n + 1}}} \right)\). Of course, for largen the difference is negligible.
It would perhaps be better, in order to avoid confusion, to use a different notation. Thus, if we designate the conditional probability byP(e,h) orP(e/h) we could designate the probability ofe calculated under the assumption thath is true by, say,P h (e).
The expression itself seems to be suggested by a simplified form of Bayes theorem:\(P(h,e) = P(h) \cdot \frac{{P(e,h)}}{{P(e).}}.\). It is easy to see that if the measure of confirmation is to be an increasing function ofP(h,e) the absolute probability ofh being kept constant, then this measure ought to be an increasing function ofP(e,h) and a decreasing one ofP(e). But I have already tried to call attention to the ct that the application of Bayes theorem implies thatp is a random variable — an assumption do not want to make.
Cf. the last two paragraphs on the page 299 in V. They constitute, so far as I know, the only suggestion concerning the choice of δ ifn is already fixed.
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Allatum est die 30 Julii 1959
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Szaniawski, K. A note on confirmation of statistical hypotheses. Stud Logica 10, 111–118 (1960). https://doi.org/10.1007/BF02120411
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DOI: https://doi.org/10.1007/BF02120411