Combinations of probability and logic have been a topic of interest since the seminal work of Boole , and they have always occupied a place of relevance in foundational studies of probability. However, the direction of development has been considerably different depending on whether the point of view is based on logic or based on statistical theory. This book places itself undeniably in the former category, emphasizing the aspects of logical inference, model theoretical variations, and logic-related problems such as propositional, first-order, and modal presentations of probabilistic logic theories.
In terms of logical inferences, the book considers, mostly, probability statements as restrictions that, given a class of general models, constrain the class of models that satisfy the restriction. In terms of model theoretical variations, besides the usual possible world semantics for propositional probabilistic logics, the book also considers infinitary languages for first-order probabilistic logics and nonstandard model theory. A modal logic view of probability is also discussed with some detail. All those topics are briefly reviewed in the introduction (chapter 1) as a preparation for the remaining chapters of the book.
A quite long and interesting review of the history of the developments of probabilistic logics is presented in chapter 2. The presentation starts with Leibnitz and his medieval predecessors, through Bernoulli, Laplace, Boole, and may others, but with the goal of arriving at modern interactions of probability and logical entailment. This chapter alone is a nice text for teachers and researchers of the interactions of probabilities and logic to have around.
Propositional probabilistic logic, without the iterations of probability operators, is the theme of chapter 3. Here, the logical view of probabilistic inference is developed as an articulated theory in light of the developments and interests of formal logic in the last 80 years. A sound and complete axiomatization is presented that employs an infinitary inference rule in the combination of finite axioms and inference rules. The authors also show that such a logic does not possess compactness properties. Completeness over other classes of models is also described.
The computational properties of probabilistic logic satisfiability (PSAT), however, are not covered in the same detail. The authors fail to mention that the NP-completeness of this problem was originally shown by Georgakopoulos and colleagues , and that deterministic algorithms for deciding PSAT based on linear algebra were proposed in later works [3,4].
The first-order approach to probabilistic logic and iterations of probability operators are discussed next. Again the treatment is, consistently, centered on probabilistic inference. While undecidability is unavoidable in this setting, the authors show that the infinitary axiomatization of propositional probability logic can be extended to the first-order case, providing a sound and complete axiomatization. It is important to note that this approach is based on the application of probabilities over possible worlds, not on domains. Based on such possible world models, modal probabilistic logic and discrete linear time probabilistic logic are briefly discussed.
The final chapters are devoted to the extension of the logics presented involving, among other things, infinitary languages, special probability operators, qualitative probabilities, intuitionistic probabilities, and conditional probabilities. Also, some applications of probabilistic logics are discussed. Obviously, these presentations do not have the depth of the previously discussed systems and serve as an appetizer for researchers and specialists interested in those topics.
Overall, this is a book that presents a logical view of probabilistic inference and, despite its biases and choice of presentation, should be of interest for students and researchers who are interested in such a foundational view of the interactions of logic and probabilities.