Summary

Uncertainty is a fundamental and unavoidable feature of daily life; in order to deal with uncertaintly intelligently, we need to be able to represent it and reason about it. In this book, Joseph Halpern examines formal ways of representing uncertainty and considers various logics for reasoning about it. While the ideas presented are formalized in terms of definitions and theorems, the emphasis is on the philosophy of representing and reasoning about uncertainty; the material is accessible and relevant to researchers and students in many fields, including computer science, artificial intelligence, economics (particularly game theory), mathematics, philosophy, and statistics.

Halpern begins by surveying possible formal systems for representing uncertainty, including probability measures, possibility measures, and plausibility measures. He considers the updating of beliefs based on changing information and the relation to Bayes' theorem; this leads to a discussion of qualitative, quantitative, and plausibilistic Bayesian networks. He considers not only the uncertainty of a single agent but also uncertainty in a multi-agent framework. Halpern then considers the formal logical systems for reasoning about uncertainty. He discusses knowledge and belief; default reasoning and the semantics of default; reasoning about counterfactuals, and combining probability and counterfactuals; belief revision; first-order modal logic; and statistics and beliefs. He includes a series of exercises at the end of each chapter.

Choice Review

Any individual who has taught probability knows that certain errors, confusions, and paradoxes arise without the proper mastery and application of the axiomatic theory. Unfortunately, embracing these axioms requires sometimes forbidding reasonable questions, sometimes offering useless answers. For example, patients cannot meaningfully request their personal probabilities of recovery, and concerning whether they currently have a given disease, the probability simply equals 0 or 1--but, who knows which amount? Conventional probability deals with frequencies but avoids modeling issues such as knowledge, belief, possibility, and plausibility--all the domain of modal logic. This book builds a marriage of modal logic and probability for a more robust model for natural human reasoning than either alone supports. The author does not expect experts in probability to have preparation in formal logic or vice-versa, so undergraduates benefit both ways from the very patient treatment of fundamentals. Readers who know probability learn to ask again what once they learned not to. A wealth of ingenious examples problematize (overly) familiar and seemingly elementary concepts (e.g., the second-ace puzzle features a conventionally correct calculation that produces a counterintuitive answer--many chapters later, a new framework reveals a hidden interpretation mandating a different calculation that supports the original intuition). Summing Up: Recommended. Upper-division undergraduates and above; researchers and faculty. --David V. Feldman, University of New Hampshire