Probability Logic and Borel's Denumerable Probability

Abstract

Borel's concept of denumerable probability is described by means of three of his illustrative problems and their solution. These problems are then reformulated in contemporary terms and solved from the viewpoint of probability logic. A section compares Kolmogorov set-theoretic probability with probability logic. The concluding section describes a highly adverse criticism of Borel's conception for its not using something like Kolmogorov theory (introduced two decades later) and, in support of Borel, this criticism is countered from the standpoint of quantifier probability logic.

Notes

The principle that the probability of the conjunction of independent events is equal to the product of their probabilities.

See Hailperin 2007, section 3, for definition of a quantifier language probability model.

Mengen aus betrachten wir also im allgemeinen nur als ”ideelle Ereignisse”, welchen in der Erfahrungswelt nichts entspricht. Wenn aber eine Deduktion, die die Wahrscheinlichkeiten solcher ideelen Ereignisse gebraucht, zur Bestimmung der Wahrscheinlichkeit eines reellen Ereignisses aus führt, wird diese Bestimmung offensichtlich automatisch auch vom empirischen Standpunkte aus einwandfrei sein. (Kolmogorov 1933 p. 16).

The present Theorem 3.4 is replacing the inadequate treatment of Theorem 3.4 in Hailperin 2000. Other items taken over here from this paper have also benefited from further consideration and maturation.