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Reviewer: Magnus Steinby

This is a revised edition of a book that originally appeared in 1986 [1]. It is first of all a solid general introduction to mathematical logic, which primarily emphasizes the proof-theoretic aspect of this discipline. Second, it introduces the reader to type theory, and demonstrates how this can be used for formalizing mathematical theories. The material is divided into seven chapters: (1) Propositional Calculus (2) First-Order Logic (3) Provability and Refutability (4) Further Topics in First-Order Logic (5) Type Theory (6) Formalized Number Theory (7) Incompleteness and Undecidability There is also a set of supplementary exercises, a summary of theorems, a bibliography, a list of figures, and an index, as well as a very informative preface that guides the reader through the book. In chapter 1, the author carefully proves the soundness, completeness, and independence of a Hilbert-style axiom system for propositional logic. This discussion also serves as a general introduction to logistic systems. Other topics considered are the normal forms of propositions, complete sets of connectives, and resolution. The central notions and results of first-order predicate logic are presented in chapter 2. Again, an axiom system is developed in detail. Gödel’s completeness theorem is proved in the abstract setting due to Smullyan, but a more direct proof is also presented. As an interesting application of the compactness theorem, it is shown how the four color theorem can be extended from finite to infinite graphs. In the last section, the main results are proved for first-order logic with equality. Chapter 3 is devoted to some more practical methods for proving theorems in first-order logic. Thus, natural deduction and semantic tableaux are reviewed, and Gentzen’s cut-free system is presented. The idea of refutation proofs is also introduced in this chapter, and there are sections on Skolemization, resolution, Herbrand’s theorem, and unification. In the brief fourth chapter, the author discusses duality, introduced by an amusing anecdote, Craig’s interpolation theorem, and Beth’s definability theorem. In chapter 5, the author makes an eloquent case for type theory as the foundation of mathematics, arguing that axiomatic set theory is more popular with mathematicians simply because it is better known. In any case, type theory should appeal to computer scientists, who have an even greater need to make explicit distinctions between objects of different types. A typed &lgr;-calculus is developed, as a system in which mathematical ideas can be expressed as simply and naturally as possible. In chapter 6, the theory of natural numbers is formalized within this type theory, augmented by an axiom of infinity. The author demonstrates that, in this system, the Peano postulates are derivable, and also demonstrates how the naturals can be well ordered. Recursive functions and relations are also briefly considered. In chapter 7, finally, some of the most profound results of logic are presented for this system: it is essentially incomplete, truth is not definable within the system itself, its decision problem is undecidable, and so forth. The presentation of the text is systematic and polished. Formal definitions are preceded by helpful motivating discussions and examples. There are also a great number of exercises. The notation could be a source of some initial irritation: some symbols are perhaps not the most commonly used, and it may take some time to get accustomed to the parenthesis-stripped formulas, with square dots. Although not a typical “logic for computer science” text, matters of efficiency and questions of artificial intelligence are duly considered; the book can be recommended to computer scientists as an introduction to type theory and &lgr;-calculus. It is not an easy book, and as a text it is best suited for very good students, but the persevering reader is amply rewarded. Online Computing Reviews Service