Request for Comments: A Possible-Worlds Semantics for Kolmogorov’s Axiomatization of Probability Theory

Probability theory is being very successfully used in a growing number of fields of science and engineering. It is essential to several areas of computer science that I am interested in, including areas critical to cybersecurity and privacy such as cryptogaphy, biometrics, artificial intelligence and human-computer interaction.

The success of probability theory has been made possible by the elegant formalization that was provided by Kolmogorov in his 1933 treatise, which is universally accepted by practitioners as the foundation of modern probability theory.

Curiously, however, in spite of the practical success of probability theory and the universal acceptance of Kolmogorov’s formalization as its foundation, the meaning of the concept of probability has been an open question in philosophy for three centuries, and still is today, as can be seen in this entry of the Stanford Encyclopedia of Philosophy. To try to understand why this is so, I’ve looked at the interpretation of his own formalization that Kolmogorov provides in his treatise.

It turns out that Kolmogorov’s interpretation is inconsistent with his formalization. Later authors, while keeping the formalization, have moved away from the interpretation, but without providing a satisfactory alternative. In this paper I propose an interpretation where the elements of the set that Kolmogorov called E and is now called Ω are possible worlds, in the simplest sense of the term, without complications such as Kripke’s frames necessitated by modal logic. I also propose an intensional definition of the notion of event as a condition on the values of one or more random variables, the less intuitive traditional definition as a subset of Ω being then more easily understood as the extensional counterpart of the intensional definition.

Comments on the paper would be very welcome.

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