## Theory of Probability

1

Introduction

1.1 Why a New Book on Probability?

There exist numerous treatments of this topic, many of which are very good, and others continue to appear. To add one more would certainly be a presumptuous undertaking if I thought in terms of doing something better, and a useless undertaking if I were to content myself with producing something similar to the 'standard' type. Instead, the purpose is a different one: it is that already essentially contained in the dedication to Beniamino Segre

[who about twenty years ago pressed me to write it as a necessary document for clarifying one point of view in its entirety.]

Segre was with me at the International Congress of the Philosophy of Science (Paris 1949), and it was on the occasion of the discussions developed there on the theme of probability that he expressed to me, in persuasive and peremptory terms, a truth, perhaps obvious, but which only since appeared to me as an obligation, difficult but unavoidable.

'Only a complete treatment, inspired by a well-defined point of view and collecting together the different objections and innovations, showing how the whole theory results in coherence in all of its parts, can turn out to be convincing. Only in this way is it possible to avoid the criticisms to which fragmentary expositions easily give rise since, to a person who in looking for a completed theory interprets them within the framework of a different point of view, they can seem to lead unavoidably to contradictions.'

These are Segre's words, or, at least, the gist of them.

It follows that the requirements of the present treatment are twofold: first of all to clarify, exhaustively, the conceptual premises, and then to give an essentially complete exposition of the calculus of probability and its applications in order to establish the adequacy of the interpretations deriving from those premises. In saying 'essentially' complete, I mean that what matters is to develop each topic just as far as is necessary to avoid conceptual misunderstandings. From then on, the reader could follow any other book without finding great difficulty in making those modifications that are needed in order to translate it, if such be desired, according to the point of view that will be taken here. Apart from these conceptual exigencies, each topic will also be developed, in terms of the content, to an extent sufficient for the treatment to turn out to be adequate for the needs of the average reader.

1.2 What are the Mathematical Differences?

1.2.1. If I thought I were writing for readers absolutely innocent of probabilistic-statistical concepts, I could present, with no difficulty, the theory of probability in the way I judge to be meaningful. In such a case, it would not even have been necessary to say that the treatment contains something new and, except possibly under the heading of information, that different points of view exist. The actual situation is very different, however, and we cannot expect any sudden change.

My estimation is that another fifty years will be needed to overcome the present situation, but perhaps even this is too optimistic. It is based on the consideration that about thirty years were required for ideas born in Europe (Ramsey, 1926; de Finetti, 1931) to begin to take root in America (even though B.O. Koopman (1940) had come to them in a similar form). Supposing that the same amount of time might be required for them to establish themselves there, and then the same amount of time to return, we arrive at the year 2020.

It would obviously be impossible and absurd to discuss in advance concepts and, even worse, differences between concepts to whose clarification we will be devoting all of what follows; however, much less might be useful (and, anyway, will have to suffice for the time being). It will be sufficient to make certain summary remarks