The Rule of Succession, Inductive Logic, and Probability Logic

Abstract

Since the later seventeenth century there has existed a theory of inductive probability, which attained an impressive mathematical development at the hands of Laplace; in the nineteenth century it was identified by its advocates as a branch of logic, dealing with a type of inference ostensibly a generalisation of the deductive variety. A fundamental principle of the new logic was the principle of indifference, or insufficient reason.2 This principle may be stated thus: if there is no reason relative to our information to suppose that among a class of events none is any more likely than any other, then their probabilities are equal, relative to that information. This principle, combined with the additivity and normalisation rules for probability functions, generates the so-called classical definition of probability over the set of events definable within a finite possibility space of n elements, where all the n elementary events are indifferently favoured, so to speak, by the conditions defining that space. In Laplace's hands, the principle of indifference seemed to generate certain probability assignments sensitive to purely numerical features of the outcomes of a repeated experiment. An example is the formula called by John Venn the Rule of Succession: r+I