Logic: a Brief Course

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Springer Science & Business Media, Mar 29, 2012 - Mathematics - 130 pages
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.
 

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Contents

1 Introduction
3
2 Fundamental Logical Notions
7
3 The Resolution Method
12
4 Robinsons Completeness Theorem
19
5 Fast Classes for DPP
27
6 Godels Compactness Theorem
31
Syntax
35
Semantics
40
Part II Predicate Logic
55
11 The Quantifiers There Exists and For All
56
12 Syntax of Predicate Logic
63
13 The Meaning of Clauses
70
14 Godels Completeness Theorem for the Logic of Clauses
79
15 Equality Axioms
89
16 The Predicate Logic L
95
17 Final Remarks
117

9 Normal Forms
47
Expressivity and Efficiency
53

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