Probabilistic logic in a coherent setting. (English) Zbl 1040.03017

This book will be most appreciated by readers already familiar with current issues in the interpretation of probabilities. Following de Finetti, probabilities are characterised here as linear mappings over propositions, expressing “degrees of belief”. Thus there is no Boolean “event space” as there is in Kolmogorov probabilities. However, coherence is introduced as the requirement that the space of such “propositional events” can be extended to an additive Boolean algebra. So the essential difference is one of interpretation, and this is the main concern of the authors. They are particularly concerned with our understanding of conditional probabilities, with many “ordinary life” examples to illustrate their analysis.
The “logic” of the book title is perhaps misleading. Although there is some discussion of truth-values at various times, the book does not attempt to present formally any logical aspects of this interpretation of probability. Some interesting discussion on 3-valued system for example is evocative but not developed. However, there is much detail, often supplemented by specific examples and a large bibliography is continually invoked.
The structure of the book may be frustrating to some, though perhaps suited to very familiar readers. For example, chapters and sections are of very different lengths, and there is an informality that makes information such as formal definitions missing or hard to find. The style is characterised by long sentences, sometimes awkward English and excessive use of brackets. A continual use of bold and italic font increases the readers difficulties. For example, this single sentence about the 2-slit experiment has the authors’ formatting, and may or may not help understand quantum theory:
“In the previous example the two incompatible experiments are not (so to say) “mentally” incompatible if we argue in terms of the general meaning of probability (for example P(A\(|\)S1) is the degree of belief in A under the assumption – not necessarily an observation but just an assumed state of information – “S1 is true”); then for a coherent evaluation of P(A) we must necessarily rely only on the above value obtained by resorting to equation (18.7), even if such probability does not express any sort of “physical property” of the given event.” (p. 201)
Unusual fonts and brackets are no substitute for clear writing.


03B48 Probability and inductive logic
60A99 Foundations of probability theory
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations