Logical Methods

An accessible introduction to philosophical logic, suitable for undergraduate courses and above.

Rigorous yet accessible, Logical Methods introduces logical tools used in philosophy—including proofs, models, modal logics, meta-theory, two-dimensional logics, and quantification—for philosophy students at the undergraduate level and above. The approach developed by Greg Restall a...

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An accessible introduction to philosophical logic, suitable for undergraduate courses and above.

Rigorous yet accessible, Logical Methods introduces logical tools used in philosophy—including proofs, models, modal logics, meta-theory, two-dimensional logics, and quantification—for philosophy students at the undergraduate level and above. The approach developed by Greg Restall and Shawn Standefer is distinct from other texts because it presents proof construction on equal footing with model building and emphasizes connections to other areas of philosophy as the tools are developed.

Throughout, the material draws on a broad range of examples to show readers how to develop and master tools of proofs and models for propositional, modal, and predicate logic; to construct and analyze arguments and to find their structure; to build counterexamples; to understand the broad sweep of formal logic's development in the twentieth and twenty-first centuries; and to grasp key concepts used again and again in philosophy.

This text is essential to philosophy curricula, regardless of specialization, and will also find wide use in mathematics and computer science programs.

Features:

• An accessible introduction to proof theory for readers with no background in logic

• Covers proofs, models, modal logics, meta-theory, two-dimensional logics, quantification, and many other topics

• Provides tools and techniques of particular interest to philosophers and philosophical logicians

• Features short summaries of key concepts and skills at the end of each chapter

• Offers chapter-by-chapter exercises in two categories: basic, designed to reinforce important ideas; and challenge, designed to push students' understanding and developing skills in new directions

A superb introduction to philosophical logic written by two leading experts in the field. At once mathematically rigorous and philosophically rewarding, it will be very much appreciated by philosophy students and professional philosophers alike.

Hannes Leitgeb, Chair of Logic and Philosophy of Language, LMU Munich

How to Read This Book xi
I PROPOSITIONAL LOGIC
1 Introduction 3
1.1 Arguments and Trees 4
1.2 Sentences and Formulas 8
1.3 Appendix: Some Useful Background 13
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How to Read This Book xi
I PROPOSITIONAL LOGIC
1 Introduction 3
1.1 Arguments and Trees 4
1.2 Sentences and Formulas 8
1.3 Appendix: Some Useful Background 13
1.4 Key Concepts and Skills 16
1.5 Questions for You 17
2 Connectives: and & if 19
2.1 Conjunction 19
2.2 Conditional 23
2.3 Biconditional 29
2.4 Derived Rules 29
2.5 Key Concepts and Skills 31
2.6 Questions for You 31
3 More Connectives: not & or 35
3.1 Negation and Falsum 35
3.2 Disjunction 38
3.3 Our System of Proofs 41
3.4 Comments on Negation and Falsum 43
3.5 Key Concepts and Skills 45
3.6 Questions for You 45
4 Facts about Proofs & Provability 49
4.1 Facts about Provability 49
4.2 Normalization 53
4.3 The Subformula Property 62
4.4 Consequences of Normalization 68
4.5 Key Concepts and Skills 70
4.6 Questions for You 72
5 Models & Counterexamples 75
5.1 Models and Truth Tables 76
5.2 Counterexamples and Validity 79
5.3 Model-Theoretic Validity 83
5.4 Key Concepts and Skills 85
5.5 Questions for You 86
6 Soundness & Completeness 91
6.1 Soundness 92
6.2 Completeness 94
6.3 Proofs First or Models First? 101
6.4 Appendix: Heyting Algebras 103
6.5 Appendix: Some Three-Valued Logics 108
6.6 Key Concepts and Skills 110
6.7 Questions for You 111
II MODAL LOGIC
7 Necessity & Possibility 117
7.1 Possible Worlds Models 117
7.2 Counterexamples and Validity 122
7.3 Two Applications 129
7.4 More Modal Models 133
7.5 Appendix: Intuitionistic Kripke Models 141
7.6 Key Concepts and Skills 148
7.7 Questions for You 149
8 Actuality & 2D Logic 153
8.1 Actuality Models 154
8.2 Validity 155
8.3 Double Indexing 156
8.4 Real-World Validity 160
8.5 Key Concepts and Skills 163
8.6 Questions for You 163
9 Modal Natural Deduction 167
9.1 Natural Deduction for S4 167
9.2 Natural Deduction for S5 173
9.3 Features of S5 174
9.4 Soundness and Completeness 179
9.5 Key Concepts and Skills 183
9.6 Questions for You 184
III PREDICATE LOGIC
10 Proofs for Predicate Logic 189
10.1 Syntax 189
10.2 Natural Deduction for CQ 199
10.3 Features of Proofs with Quantifiers 205
10.4 Discussion 211
10.5 Key Concepts and Skills 213
10.6 Questions for You 213
11 Models for Predicate Logic 217
11.1 Defining Models 217
11.2 Features of Models 224
11.3 Counterexamples and Validity 226
11.4 The Power and Limits of Predicate Logic 234
11.5 Key Concepts and Skills 239
11.6 Questions for You 240
12 Coda 245
12.1 Quantified Modal Logic 245
12.2 Suggestions for Further Reading 248
12.3 Key Concepts and Skills 252
12.4 Questions for You 252
Glossary 255
References 261
Symbol Index 267
Subject Index 269
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