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Logic as a Tool

Book Description

Written in a clear, precise and user-friendly style, Logic as a Tool: A Guide to Formal Logical Reasoning is intended for undergraduates in both mathematics and computer science, and will guide them to learn, understand and master the use of classical logic as a tool for doing correct reasoning.  It offers a systematic and precise exposition of classical logic with many examples and exercises, and only the necessary minimum of theory.

The book explains the grammar, semantics and use of classical logical languages and teaches the reader how grasp the meaning and translate them to and from natural language.  It illustrates with extensive examples the use of the most popular deductive systems -- axiomatic systems, semantic tableaux, natural deduction, and resolution -- for formalising and automating logical reasoning both on propositional and on first-order level,  and provides the reader with technical skills needed for practical derivations in them.  Systematic guidelines are offered on how to perform logically correct and well-structured reasoning using these deductive systems and the reasoning techniques that they employ. 

•Concise and systematic exposition, with semi-formal but rigorous treatment of the minimum necessary theory, amply illustrated with examples
•Emphasis both on conceptual understanding and on developing practical skills
•Solid and balanced coverage of syntactic, semantic, and deductive aspects of logic
•Includes extensive sets of exercises, many of them provided with solutions or answers
•Supplemented by a website including detailed slides, additional exercises and solutions

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright
  4. Dedication
  5. Preface
    1. Aims
    2. Summary of the content and main features
    3. For the instructor
  6. Acknowledgements
    1. Attributions for the photos used in the book
  7. Introduction
  8. An Appetizer: Logical Paradoxes and Self-Reference
  9. Chapter 1: Understanding Propositional Logic
    1. 1.1 Propositions and logical connectives: truth tables and tautologies
    2. Exercises
    3. 1.2 Propositional logical consequence: logically correct inferences
    4. Exercises
    5. 1.3 Logical equivalence: negation normal form of propositional formulae
    6. Exercises
    7. 1.4 Supplementary: Inductive definitions and structural induction and recursion
    8. Exercises
  10. Chapter 2: Deductive Reasoning in Propositional Logic
    1. 2.1 Deductive systems: an overview
    2. 2.2 Axiomatic systems for propositional logic
    3. Exercises
    4. 2.3 Semantic Tableaux
    5. Exercises
    6. 2.4 Natural Deduction
    7. Exercises
    8. 2.5 Normal forms and Propositional Resolution
    9. Exercises
    10. 2.6 Supplementary: The Boolean satisfiability problem and NP-completeness
    11. 2.7 Supplementary: Completeness of the propositional deductive systems
    12. Exercises
  11. Chapter 3: Understanding First-order Logic
    1. 3.1 First-order structures and languages: terms and formulae of first-order logic
    2. Exercises
    3. 3.2 Semantics of first-order logic
    4. 3.3 Basic grammar and use of first-order languages
    5. Exercises
    6. 3.4 Logical validity, consequence, and equivalence in first-order logic
    7. Exercises
    8. 3.5 Syllogisms
    9. Exercises
  12. Chapter 4: Deductive Reasoning in First-order Logic
    1. 4.1 Axiomatic system for first-order logic
    2. Exercises
    3. 4.2 Semantic Tableaux for first-order logic
    4. Exercises
    5. 4.3 Natural Deduction for first-order logic
    6. Exercises
    7. 4.4 Prenex and clausal normal forms
    8. Exercises
    9. 4.5 Resolution for first-order logic
    10. Exercises
    11. 4.6 Supplementary: Soundness and completeness of the deductive systems for first-order logic
    12. Exercises
  13. Chapter 5: Applications: Mathematical Proofs and Automated Reasoning
    1. 5.1 Logical reasoning and mathematical proofs
    2. Exercises
    3. 5.2 Logical reasoning on sets, functions, and relations
    4. Exercises
    5. 5.3 Mathematical Induction and Peano Arithmetic
    6. Exercises
    7. 5.4 Applications: automated reasoning and logic programming
    8. Exercises
  14. Chapter 6: Answers and Solutions to Selected Exercises
    1. Section 1.1
    2. Section 1.2
    3. Section 1.3
    4. Section 1.4
    5. Section 2.2
    6. Section 2.3
    7. Section 2.4
    8. Section 2.5
    9. Section 3.1
    10. Section 3.2
    11. Section 3.3
    12. Section 3.4
    13. Section 3.5
    14. Section 4.1
    15. Section 4.2
    16. Section 4.3
    17. Section 4.4
    18. Section 4.5
    19. Section 4.6
    20. Section 5.1
    21. Section 5.2
    22. Section 5.3
    23. Section 5.4
  15. References
  16. Index
  17. End User License Agreement