“From many years of experience with its applications in hundreds of real problems, our views on the foundations of probability theory have evolved into somethin

“From many years of experience with its applications in hundreds of real problems, our views on the foundations of probability theory have evolved into something quite complex, which cannot be described in any such simplistic terms as ‘pro-this' or ‘anti-that'. For example, our system of probability could hardly be more different from that of Kolmogorov, in style, philosophy, and purpose. What we consider to be fully half of probability theory as it is needed in current applications – the principles for assigning probabilities by logical analysis of incomplete information – is not present at all in the Kolmogorov system. “Yet, when all is said and done, we find ourselves, to our own surprise, in agreement with Kolmogorov and in disagreement with his critics, on nearly all technical issues. As noted in Appendix A, each of his axioms turns out to be, for all practical purposes, derivable from the Polya–Cox desiderata of rationality and consistency. In short, we regard our system of probability as not contradicting Kolmogorov's; but rather seeking a deeper logical foundation that permits its extension in the directions that are needed for modern applications.”