We argue that in spite of their apparent dissimilarity, the methodologies employed in the a priori and a posteriori assessment of probabilities can both be justified by appeal to a single principle of inductive reasoning, viz., the principle of symmetry. The difference between these two methodologies consists in the way in which information about the single-trial probabilities in a repeatable chance process is extracted from the constraints imposed by this principle. In the case of a posteriori reasoning, these constraints inform (...) the analysis by fixing an a posteriori determinant of the probabilities, whereas, in the case of a priori reasoning, they imply certain claims which then serve as the basis for subsequent probabilistic deductions. In a given context of inquiry, the particular form which a priori or a posteriori reason may take depends, in large part, on the strength of the underlying symmetry assumed: the stronger the symmetry, the more information can be acquired a priori and the less information about the long-run behavior of the process is needed for an a posteriori assessment of the probabilities. In the context of this framework, frequency-based reasoning emerges as a limiting case of a posteriori reasoning, and reasoning about simple games of chance, as a limiting case of a priori reasoning. Between these two extremes, both a priori and a posteriori reasoning can take a variety of intermediate forms. (shrink)

My title is intended to recall Terence Fine's excellent survey, Theories of Probability [1973]. I shall consider some developments that have occurred in the intervening years, and try to place some of the theories he discussed in what is now a slightly longer perspective. Completeness is not something one can reasonably hope to achieve in a journal article, and any selection is bound to reflect a view of what is salient. In a subject as prone to dispute as this, there (...) will inevitably be many who will disagree with any author's views, and I take the opportunity to apologize in advance to all such people for what they will see as the narrowness and distortion of mine. (shrink)