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This short book, geared towards undergraduate students of computer science and mathematics, is specifically designed for a first course in mathematical logic.A proof of Gödel's completeness theorem and its main consequences is given using Robinson's completeness theorem and Gödel's compactness theorem for propositional logic. The reader will familiarize himself with many basic ideas and artifacts of mathematical logic: a non-ambiguous syntax, logical equivalence and consequence relation, the Davis-Putnam procedure, Tarski semantics, Herbrand models, the axioms of identity, Skolem normal forms, nonstandard models and, interestingly enough, proofs and refutations viewed as graphic objects. The mathematical prerequisites are minimal: the book is accessible to anybody having some familiarity with proofs by induction. Many exercises on the relationship between natural language and formal proofs make the book also interesting to a wide range of students of philosophy and linguistics.
Table des matières
Introduction.- Fundamental Logical Notions.- The Resolution Method.- Robinson Completeness Theorem.- Fast Classes for DPP.- Godel Compactness Theorem.- Propositional Logic: Syntax.- Propositional Logic: Semantics.- Normal Forms.- Recap: Expressivity and Efficiency.- The Quantifiers 'There Exists' and 'For All'.- Syntax of Predicate Logic.- The Meaning of Clauses.- Godel Completeness Theorem for the Logic of Clauses.- Equality Axioms.- The Predicate Logic L.
Résumé
This short book, geared towards undergraduate students of computer science and mathematics, is specifically designed for a first course in mathematical logic.
A proof of Gödel's completeness theorem and its main consequences is given using Robinson's completeness theorem and Gödel's compactness theorem for propositional logic. The reader will familiarize himself with many basic ideas and artifacts of mathematical logic: a non-ambiguous syntax, logical equivalence and consequence relation, the Davis-Putnam procedure, Tarski semantics, Herbrand models, the axioms of identity, Skolem normal forms, nonstandard models and, interestingly enough, proofs and refutations viewed as graphic objects. The mathematical prerequisites are minimal: the book is accessible to anybody having some familiarity with proofs by induction. Many exercises on the relationship between natural language and formal proofs make the book also interesting to a wide range of students of philosophy and linguistics.
Texte suppl.
From the reviews:
“This is a short introduction to mathematical logic that covers basic material in 17 chapters … . The book is interspersed with several small references to various scholars involved in the development of logic, which provides for welcome interruptions in the formal exposition. … An important aspect of the book is a veritable multitude of exercises. … it is a very nice booklet that in view of this reviewer is an attractive choice for an introductory logic course for first year computer science students.” (Krzysztof R. Apt, Theory and Practice of Logic Programming, Vol. 12 (3), 2012)
“The book contains all the necessary means to understand any advanced text in logic, including the subjects covering Gödel’s incompleteness theorems. Although brief, this course seems to be an excellent introduction to modern mathematical logic, and, as such, we recommend it firstly to students of mathematics and computer science, and also to students of philosophy and linguistics … . The author’s beautiful, clear and approachable style makes this book also recommendable to a broader range of readers who are interested in modern trends in logic.” (Branislav Boričić, Zentralblatt MATH, Vol. 1235, 2012)
Commentaire
From the reviews:
"This is a short introduction to mathematical logic that covers basic material in 17 chapters ... . The book is interspersed with several small references to various scholars involved in the development of logic, which provides for welcome interruptions in the formal exposition. ... An important aspect of the book is a veritable multitude of exercises. ... it is a very nice booklet that in view of this reviewer is an attractive choice for an introductory logic course for first year computer science students." (Krzysztof R. Apt, Theory and Practice of Logic Programming, Vol. 12 (3), 2012)
"The book contains all the necessary means to understand any advanced text in logic, including the subjects covering Gödel's incompleteness theorems. Although brief, this course seems to be an excellent introduction to modern mathematical logic, and, as such, we recommend it firstly to students of mathematics and computer science, and also to students of philosophy and linguistics ... . The author's beautiful, clear and approachable style makes this book also recommendable to a broader range of readers who are interested in modern trends in logic." (Branislav Boricic, Zentralblatt MATH, Vol. 1235, 2012)
