Theory of Probability

Appendix

1 Concerning Various Aspects of the Different Approaches

In every field, and in particular in the calculus of probability, there is scope, both hypothetically and in fact, for a number of axiomatic approaches, each of which, to a greater or lesser degree, differs from the others in various respects. It does not suit our purpose to choose just one of these, merely illustrating – even if exhaustively – that particular one; nor are we interested in presenting a somewhat wide and diverse collection from which each person makes his choice with the aid of a pin. The way that seems more appropriate, and that in any case we shall try to follow, consists in sticking to one preferred approach as a reference point but at the same time illustrating both the variants within it that seem admissible, or necessary, and the approaches inspired by divergent views. This provides the framework for the necessary conceptual and formal comparisons.

From a conceptual standpoint our choice has already been made, and explained at some length, in Chapters 3, 4 and subsequently. At that time, we gave what might be called an axiomatic approach, but between then and now there is a difference in attitude that can be expressed (in the summary form of a single sentence) by saying that we must pass from an axiomatic approach to the theory of probability, to an axiomatic approach to the calculus of probability. This transition must not be taken as implying the existence of any distinction or separation ...