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Reviewer: Calin Lucaciu

The author makes an important scientific contribution to the theory of knowledge and automatic decision making. The book will be a reference on fundamental research as well as a useful instrument for scientists, philosophers, and advanced students. The book's structure is constructive, facilitating a clear transmission of the author's ideas. Bacchus uses two plans of exposition: the epistemological plan justifies his theory in a wide, philosophical perspective, and the formal, mathematical plan gives the reader a valuable instrument. The book may be too short to fulfill the author's goals, but it reports a research result and requires the reader to take a good look at the bibliography. The author proposes two types of logics, which accept probabilities for subjective heuristics (in this case, measuring the degrees of belief of individuals) and for objective heuristics (giving a quantitative measure of knowledge). Finally, he constructs a dualistic logic that offers tools to treat unitary, stochastic, and particular logical values. To give an additional degree of consistency to his theory, the author outlines a system of deduction from objective to subjective knowledge. The first chapter is mostly philosophical—Bacchus makes an interesting effort to justify the need for and relevance of a logic of stochastic knowledge. The power of this logic, on principle, does not exceed that of the first-order logics. This argumentation is valuable but does not emphasize a fundamental concept: a measure for subjectivity and objectivity. For the author, subjectivity is strictly related to “particular individuals” and objectivity to “sets of individuals.” A more precise measure would be the statistical relevance. Subjectivity can be characterized as statistical irrelevance and objectivity as statistical relevance. The author admits that individuals have logically bit-like behavior, believing a statement to be either true or false. The stochastic logical comportment of individuals is more elastic, relating to the “objective state of the world.” I see here a particle-wave dualistic vision. Bacchus chooses Quine's epistemological strategy: local modifications in a theory must have only local effect. Three structurally identical chapters develop a logic for propositional probabilities, a logic for statistical probabilities, and then, by generalizing the first two logics with nonrigid variables and the introduction of an expectation operator, a logic with unitary resources on propositional and statistical logics. The logic of propositional probabilities “is an extension of ordinary first-order logic, with a probability operator which can be used to denote the probability of a formula.” The probability operator is defined over the Boolean algebra of the atomic propositions and two logical operators: conjunction and negation. Negation is not a proper connective, because it is a unary operator that can be associated as a simple modifier in the logic's syntax: disjunction should be introduced as a proper second connective. This point is more important in the proof theory associated with the logic. Finally, Bacchus proves that a logic of belief can be viewed as a particular case of the logics of propositional probabilities. “Probabilities defined over sets of individuals model statistical probabilities.” The author's logic admits a “variable binding statistical operator” and a set of user-defined measuring functions instead of the propositional probability operator. The semantics is adapted to the philosophy of multiple logics by fixing the measures to a unique value, to emphasize the objective character of this logic. Individuals are organized into vectors, but neither the properties of these vectors nor the structure that they form is presented. In this part the reader can learn, through an axiom system, to use probabilities as a logical tool. The combination of the two previous logics offers a mathematical and logical instrument for a stochastic default reasoning system. The key here is the expectation operator, which, according to the new logic's semantics, must have an expert behavior over the (possibly infinite) belief systems of individuals. As a direct application of the developed logics, Bacchus proposes a non-monotonic reasoning system and a formalism for default inferences from stochastic knowledge. The “direct inference principle” gives a numerical measure for the individual's degree of belief through the expectation over a set of statistical probabilities related to the individual's certitudes. As the second step in the practical use of his logics, the author defines a “non-monotonic reasoning framework.” This valuable work is conducive, because of the evidence given, to immediate implementation in the automata decision domain. The author's goals are partially satisfied. On the epistemological plane, the proposed integration (“epistemological adequacy”) is satisfied structurally, leaving some concepts open . On the formal plane, an interesting result is obtained, but the book lacks a full mathematical justification of the solutions.