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
For more information browse the book’s website at: https://logicasatool.wordpress.com
Table of Content
1 Understanding Propositional Logic 9
1.1 Propositions and logical connectives. Truth-tables and tautologies. 9
1.2 Propositional logical consequence. Logically correct inferences 28
1.3 Logical equivalence. Negation normal form of propositional formulae. 39
1.4 Addendum: Inductive definitions. Structural induction and recursion 46
2 Deductive Reasoning in Propositional Logic 59
2.1 Deductive systems: an overview 59
2.2 Axiomatic systems for propositional logic 65
2.3 Semantic tableaux 72
2.4 Natural deduction 83
2.5 Normal forms. Propositional resolution. 92
2.6 Addendum: The Boolean satisfiability problem and NP-completeness. 101
2.7 Addendum: Completeness of the propositional deductive systems 104
3 Understanding First-order Logic 113
3.1 First-order structures and languages. Terms and formulae of first-order logic. 114
3.2 Semantics of first-order logic 127
3.3 Basic grammar and use of first-order languages 143
3.4 Logical validity, consequence and equivalence in first-order logic 156
3.5 Syllogisms 173
4 Deductive Reasoning in First-order Logic 181
4.1 Axiomatic system for first-order logic 182
4.2 Semantic Tableaux for First-order Logic 190
4.3 Natural Deduction for first-order logic 204
4.4 Prenex and clausal normal forms 211
4.5 Resolution for first-order logic 218
4.6 Addendum: Soundness and completeness of the deductive systems for first-order logic 234
5 Applications: Mathematical Proofs and Automated Reasoning 245
5.1 Logical reasoning and mathematical proofs 246
5.2 Logical reasoning on sets, functions and relations 255
5.3 Mathematical induction and Peano Arithmetic 271
5.4 Applications: automated reasoning and logic programming 278
6 Answers and solutions to selected exercises 289
6.1 Answers and solutions: Section 1.1 289
6.2 Answers and solutions: Section 1.2 292
6.3 Answers and solutions: Section 1.3 294
6.4 Answers and solutions: Section 2.2 296
6.5 Answers and solutions: Section 2.3 298
6.6 Answers and solutions: Section 2.4 307
6.7 Answers and solutions: Section 2.5 312
6.8 Answers and solutions: Section 3.1 318
6.9 Answers and solutions: Section 3.2 321
6.10 Answers and solutions: Section 3.3 323
6.11 Answers and solutions: Section 3.4 325
6.12 Answers and solutions: Section 3.5 330
6.13 Answers and solutions: Section 4.1 331
6.14 Answers and solutions: Section 4.2 333
6.15 Answers and solutions: Section 4.3 350
6.16 Answers and solutions: Section 4.4 352
6.17 Answers and solutions: Section 4.5 354
6.18 Answers and solutions: Section 5.1 363
6.19 Answers and solutions: Section 5.2 364
6.20 Answers and solutions: Section 5.3 368
6.21 Answers and solutions: Section 5.4 371
About the author
Valentin Goranko is an associate professor at the Department of Applied Mathematics and Computer Science of the Technical University of Denmark. He has had over 25 years of University teaching and research experience: in particular, he has taught several courses partly based on the proposed book.
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