On Purely Probabilistic Theories of Scientific Inference

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Abstract
This paper derives a mathematical expression giving the development of the probability of a scientific hypothesis with the number of confirming tests, as determined by Bayes's theorem, in a special case in which all the tests are “independent” of one another. The simple expression obtained shows clearly how the various factors influence the growth of the probability. The result is used to set a numerical lower bound on the probabilities representing the a priori beliefs of humans in generalizations that become accepted. By making a comparison with the predictions of a “logical atomic” model in the case of physical laws, it is argued that humans have significant a priori “knowledge” in a weak sense.

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