Lecture Notes for Logic

Lecture Notes for Logic

J. Dmitri Gallow 1 Spring, 2015, New York University

1 These notes were designed to accompany the online software for Hurley's A Concise Introduction to Logic. Please feel free to use any portion of them for any purpose (with attribution, of course); if you have comments or corrections, please send them to jdmitrig@nyu.edu.

Contents

I Basic Concepts of Logic

1 Validity, Soundness, Strength, Cogency, and Conditionals

9

1.1 Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Finding Argumentative Structure . . . . . . . . . . . . . . . . . . . 11

1.3 Conditionals . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4 Deductive Validity . . . . . . . . . . . . . . . . . . . . . . . 15

1.5 Inductive Strength. . . . . . . . . . . . . . . . . . . . . . . . 17

1.6 Proving Invalidity, take 1 . . . . . . . . . . . . . . . . . . . . . 17

1.7 Formal Deductive Validity . . . . . . . . . . . . . . . . . . . . . 22

1.8 Proving Invalidity, take 2 . . . . . . . . . . . . . . . . . . . . . 24

2 Dialectics

27

2.1 Rules for Dialectics . . . . . . . . . . . . . . . . . . . . . . . 29

3 Informal Fallacies

35

3

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3.1 Fallacies of Irrelevance . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Fallacies Involving Ambiguity . . . . . . . . . . . . . . . . . . . . 41 3.3 Fallacies Involving Unwarranted Assumptions . . . . . . . . . . . . . . . 43

II Propositional Logic

4 The Language PL

49

4.1 Syntax for PL . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 Semantics for PL . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3 Translation from PL to English . . . . . . . . . . . . . . . . . . . . 58

4.4 Translation from English to PL . . . . . . . . . . . . . . . . . . . . 61

5 Logical Properties of PL

65

5.1 How to Construct a Truth-Table . . . . . . . . . . . . . . . . . . . 65

5.2 What a Truth-Table Represents . . . . . . . . . . . . . . . . . . . . 68

5.3 PL-Validity . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.4 PL-Tautologies, PL-Self-Contradictions, & PL-Contingencies . . . . . . . . . . 73

5.5 PL-Equivalence & PL-Contradiction . . . . . . . . . . . . . . . . . 75

5.6 PL-Consistency & PL-Inconsistency . . . . . . . . . . . . . . . . . . 78

5.7 Thinking About the Relationship Between the Logical Properties of PL . . . . . . . . 79

6 PL Derivations

85

6.1 The Basics . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.2 Rules of Implication . . . . . . . . . . . . . . . . . . . . . . . 86

6.3 Rules of Replacement . . . . . . . . . . . . . . . . . . . . . . . 94

6.4 Four Final Rules of Inference . . . . . . . . . . . . . . . . . . . . 103

6.5 PL-Derivability and the Logical Notions of PL. . . . . . . . . . . . . . . 111

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III Quantificational Logic

7 Beyond Propositional Logic

123

7.1 Correctness and Completeness . . . . . . . . . . . . . . . . . . . . 123

7.2 Arguments that PL is Not Correct . . . . . . . . . . . . . . . . . . . 124

7.3 Why PL is Not Complete . . . . . . . . . . . . . . . . . . . . . 131

8 Historical Interlude: Syllogistic Logic

133

8.1 Categorical Propositions . . . . . . . . . . . . . . . . . . . . . . 133

8.2 The Square of Opposition . . . . . . . . . . . . . . . . . . . . . 138

8.3 Categorical Syllogisms . . . . . . . . . . . . . . . . . . . . . . 144

8.4 Venn Diagrams . . . . . . . . . . . . . . . . . . . . . . . . 147

9 The Language QL

153

9.1 The Language QL . . . . . . . . . . . . . . . . . . . . . . . 153

9.2 Translations from QL into English . . . . . . . . . . . . . . . . . . 164

9.3 Translations from English into QL . . . . . . . . . . . . . . . . . . 168

10 Logical Properties of QL

171

10.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 171

10.2 QL-Validity . . . . . . . . . . . . . . . . . . . . . . . . . 172

10.3 QL-tautologies, QL-self-contradictions, & QL-contingencies . . . . . . . . . . 173

10.4 QL-Equivalence & QL-Contradiction . . . . . . . . . . . . . . . . . 175

10.5 QL-Consistency & QL-Inconsistency . . . . . . . . . . . . . . . . . . 177

11 QL Derivations

179

11.1 Substitution Instances . . . . . . . . . . . . . . . . . . . . . . 179

11.2 QL-Derivations . . . . . . . . . . . . . . . . . . . . . . . . 180

11.3 QL-Derivability and the Logical Notions of QL . . . . . . . . . . . . . . 193

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