John Venn and Charles Peirce: Probabilities as frequencies
Throughout the nineteenth century there were scientists and logicians who shared Laplace's view that we could, at least in principle, measure and compare degrees of certainty in scientific conclusions using the mathematics of probability. Thus, in the Philosophical Magazine, a distinguished journal for the mathematical sciences, we find a contributor declaring in 1851 that 'every definite state of belief concerning a proposed hypothesis, is in itself capable of being represented by a numerical expression, however difficult or impractical it may be to ascertain its actual value' (Donkin 1851: 354). Many placed much confidence in the capacity of inverse probabilistic reasoning to yield quantitative degrees of belief, and to reassure those who doubted the legitimacy of inductive and hypothetical reasoning. The economist and logician W. Stanley Jevons explained the fundamental idea clearly. Deduction, he said, allows us to reason from general scientific claims to particular conclusions which we can check experimentally, and we should understand induction as the inverse of deduction, for by means of it we reason from particular experimental data to general scientific conclusions. In deductive reasoning we sometimes use the mathematical theory of probability, and when we do so we are using it in a direct manner. For example, the degree of certainty that we have in a particular conclusion deduced from general scientific premisses will depend, in ways determined by the theory of probability, upon the degrees of certainty we attach to those premisses. Thus, Darwin's theory of evolution had deductive implications for what the fossil record would reveal, and confidence in that theory would have an effect upon confidence in what the fossil record would show. But, just as we need probability theory in assessing the credibility of conclusions of deductive arguments, so we need probability theory in assessing the credibility of conclusions of inductive arguments, only in this case we have to use it in an inverse manner. Using inverse probabilistic reasoning, we may argue 'from the known character of certain events . . . to the probability of a certain law or condition governing those events'. And, 'having satisfactorily accomplished this work, we may
[using direct probabilistic reasoning] calculate forwards to the probable character of future events happening under the same conditions' (Peirce 1931-58: 1.276).