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A primer of probability logic. (English) Zbl 0910.68202
CSLI Lecture Notes. 68. Stanford, CA: CSLI, Center for the Study of Language and Information. 376 p. (1998).
It is a fact that there are few books devoted to a systematic and application-oriented approach to probability logic (PL). Two main reasons decided, the author to provide an elementary introduction to PL: (1) PL brings to human reasoning an important advance compared to modern non-probabilistic logic; (2) The traditional “truth-conditional” approach fails in application to conditionals because of the classical logic’s radical separation of pure and practical logic, the way in which “pure logic” conclusions fail to influence the practical decision and action. The author takes into account some of the important theoretical fields that PL influenced directly: philosophical logic; decision theory applied within philosophy of science, philosphy of action, economics, various branches of psychology; linguistics and the philosphy of language; computer science and artificial intelligence, attempting to model the human reasoning processes by non-monotonic reasoning, which is so closed to conditional logic.
The book contains nine chapters, nine appendices, answers to selected exercises, references etc. Chapter 2 (Probability and Logic) and Chapter 4 (Conditional Probabilities and Conditionalization) provide the gentle but necessary knowledge on the mathematical background of probability theory. Chapters 3 and 5 answer to the following question: given the probabilities of some premises, which is the probability of the corresponding conclusion based on a deductively valid inference? Chapter 3 (Deduction and Probability. I.) considers the “static” approach, not taking into account the fact that probabilities can change as a result of gaining information of adding new premises. Chapter 5 (Deduction and Probability. II.) introduces a “dynamic” dimension based on Bayes’ principle and associated with the non-monotonicity of the inference process. Chapter 6 (Probability Conditionals) and Chapter 7 (Derivations in the Formal Theory of Probability Conditionals) focus on the logic of conditional propositions, relying on the following (controversial) assumption: the probabilities of conditional propositions are conditional probabilities. This kind of probabilities lead to a logic of conditionals radically different from the logic that is based on the assumption that the truth value of an “if-then” proposition is the same as that of a material conditional. Chapter 8 (Truth, Triviality, and Controversy) discusses the Alan Hajek’s variant of David Lewis’ fundamental Triviality Argument: in certain technical assumptions, if the “right” measure of a conditional’s probability is a conditional probability then its probability cannot equal the probability of its being true, no matter how its truth might be defined. Chapter 9 (Practical Reason) gives a brief introduction to the theory of practical reason and shows how conclusions about propositions and their probabilities influence actions. This chapter ends with a philosophical discussion on the pragmatics of probability. Nine useful appendices are enclosed: A1 (Coherent Degrees of Confidence); A2 (Infinitesimal Probabilities and Popper Functions); A3 (David Lewis’ Triviality Result); A4 (The Problem of Embedded Conditionals); A5 (‘Statistical Reasonableness’ of Contraposition); A6 (Counterfactual Conditionals); A7 (Probabilistic Predicate Logic); A8 (Probabilistic Identity Logic); A9 (Approximate Generalizations).
The interest area for the readability of this book is prodigious: from philosophers to practitioners, from logicians to economists, and from mathematicians to linguists.
Reviewer: N.Curteanu (Iaşi)

##### MSC:
 68T27 Logic in artificial intelligence 68-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to computer science 03B48 Probability and inductive logic 03B65 Logic of natural languages 62Cxx Statistical decision theory 91B06 Decision theory
##### Keywords:
probability logic; logic of conditionals