Sam Cumming is Assistant Professor of Philosophy at the University of California, Los Angeles.

Preface to Second Edition xiv

Preface to Revised Edition xv

Acknowledgments xvi

Introduction to Teacher 1

1 A Brief Introduction to Key Terms 5

1.1 Arguments 5

1.1.1 What is a Statement? 6

1.1.2 Premises and Conclusion 6

1.2 Putting Arguments into a Standard Format 7

1.3 Multiple Conclusions 9

1.4 Deductive Validity 10

1.5 Soundness 13

1.6 Missing Premises and Conclusions 13

2 Argument Forms and Propositional Logic 17

2.1 Formal Validity 17

2.2 Quotation Marks 19

2.3 Metalinguistic Variables 21

2.4 Non-formal Validity 23

2.5 The Need for Propositional Logic 24

2.5.1 Symbolic Notation 25

2.6 The Type/Token Distinction 26

3 Conjunction 31

3.1 Logical Conjunction 31

3.2 Distinguishing Deductive from Non-deductive Aspects of Conjunction 33

3.3 Phrasal Logical Conjunctions 34

3.4 Series Decompounding 34

3.5 Using 'Respectively' 35

3.6 Symbolizing Logical Conjunctions 35

4 Negation 42

4.1 Logical Negation 42

4.2 Some Other Negative Expressions 43

4.3 A Point about Methodology 45

4.4 A Point on Ambiguity 45

4.5 Symbolizing Logical Negations 45

4.6 Ambiguity and the Need for Groupers 46

4.7 Review of Symbols 47

4.8 Using 'Without' 48

4.9 Argument Forms Continued 48

4.10 Symbolizing Logical Negations Continued 51

5 Truth Tables 56

5.1 Well-formed Formulas 56

5.2 Scope 57

5.3 Main Connective 58

5.4 Truth Tables 59

5.4.1 Truth Table Analyses of Statements 61

5.4.2 Truth Table Analyses of Arguments 64

6 Disjunction 68

6.1 Logical Disjunction 68

6.2 Disjunction and Negation 69

6.3 Iterations and Groupers 71

6.4 Inclusive versus Exclusive 'Or' 73

6.5 Symbolizing Logical Disjunctions Continued 76

7 Conditionals 79

7.1 Conditionals with Constituent Statements 79

7.2 Conditionals without Constituent Statements 80

7.3 Logical Conditionals 80

7.4 Symbolizing Conditionals in PL 82

7.5 Necessary and Sufficient Conditions 82

7.6 Only If 84

7.7 Unless 86

7.8 Since, Because 88

7.9 Conditionals and Groupers 89

7.10 If and Only If 90

7.11 A Revised Grammar for Well-formedness in PL 91

7.12 Summarizing Truth Tables 99

7.12.1 Validity 99

7.12.2 Contradiction, Tautology, Contingency 102

7.12.3 Consistency 104

7.12.4 Logical Equivalence 105

8 Truth Trees 109

8.1 Reviewing Validity 109

8.2 Tree Trunks and Compound and Atomic Statements 110

8.3 Truth Tree Rules 111

8.3.1 Non-branching Rules 111

8.3.2 Branching Rules 112

8.4 Strategies 114

8.5 Truth Trees and Invalidity 117

8.6 Propositional Logic and Counter-examples (Counter-models) 121

8.7 Logical Properties and Relations Revisited 123

8.7.1 Consistency 123

8.7.2 Contradiction, Tautology, Contingency 124

8.7.3 Logical Equivalence 126

9 Property Predicate Logic 129

9.1 Limits of Propositional Logic 129

9.2 Singular Terms 130

9.3 Property Predicates 132

9.4 Quantifiers 134

9.4.1 Simple Existential Quantifier Statements 135

9.4.2 Symbolizing Simple Existential Statements 135

9.4.3 Simple Universal Quantifier Statements 137

9.4.4 Negations of Existentials 138

9.5 Complex Predicates 139

9.6 Well-formedness in PPL 142

9.7 Quantifiers Modifying General Terms 145

9.7.1 Existential Quantifiers and General Terms 145

9.7.2 Universal Quantifiers and General Terms 147

10 Evaluating Arguments in Property Predicate Logic 155

10.1 Quantifiers and Scope 156

10.2 The Truth Tree Method Extended 157

10.2.1 Quantifier Exchange Rule (QE) 157

10.2.2 Universal Quantifier Rule (UQ) 158

10.2.3 Existential Quantifier Rule (EQ) 161

10.3 Super Strategy 164

10.4 Property Predicate Logic and Counter-examples (Counter-models) 166

10.5 PPL Logical Equivalences and Non-equivalences 168

10.6 Other Logical Properties and Relations 170

10.6.1 Consistency 170

10.6.2 Logical Equivalence 170

10.6.3 Contradiction, Logical Truth, Contingency 171

11 Property Predicate Logic Refinements 172

11.1 Literal Meaning 172

11.2 'Any' as an Existential 173

11.3 Restrictive Relative Clauses 175

11.4 Pronouns Revisited 176

11.4.1 Deixis and Anaphora 176

11.4.2 Quantification and Anaphora 177

11.5 Only 180

11.6 Restrictive Words in English 182

11.7 Evaluating Symbolizations of English in Logical Notation 185

12 Relational Predicate Logic 191

12.1 Limits of Property Predicate Logic 191

12.2 Convention 1: Number 193

12.3 Convention 2: Order 194

12.4 Convention 3: Active/Passive Voice 195

12.5 Convention 4: Single Quantifiers 197

12.6 Variables 199

12.6.1 Convention 5: Variables and Quantifiers 200

12.6.2 Convention 6: Variables and Property Predicates 200

12.6.3 General Comments about Variables 201

13 Relational Predicate Logic with Nested Quantifiers 207

13.1 Multiply General Statements 209

13.2 Universal Quantifier Procedure 212

13.3 Existential Quantifier Procedure 213

13.4 Double Binding Variables 213

13.4.1 Kicking Out 216

13.5 Systematic and Analytic Procedures 217

13.6 A Grammar for Well-formedness in RPL 218

13.7 Nested Quantifiers, Variables, and Scope 220

13.8 Order and Scope Refinements 221

13.8.1 The Order and Scope Procedure 224

13.9 Summary of the Overall Procedure for Symbolizing English Statements with Nested Quantifiers into RPL 226

14 Extending the Truth Tree Method to RPL 229

14.1 RPL Arguments without Quantifiers 229

14.2 RPL Arguments without Nested Quantifiers 230

14.3 RPL Arguments with Nested Quantifiers 232

14.4 Choosing Singular Terms to Instantiate 233

14.5 Infinite Truth Trees for RPL Arguments 234

14.6 Summary of Truth Tree Strategies 236

14.7 Relational Predicate Logic and Counter-examples (Counter-models) 239

15 Negation, Only, and Restrictive Relative Clauses 244

15.1 Negation 244

15.2 'Only' as a Quantifier 246

15.3 Restrictive Relative Clauses 249

15.3.1 The Quantificational Restrictive Relative Clause Procedure 250

15.4 Quantifiers and Anaphora 252

15.4.1 Repair Algorithm 254

15.5 Anaphora and Restrictive Relative Clauses 257

15.6 Anaphora Across Sentences 262

15.7 Quantification in English 265

16 Relational Predicate Logic with Identity 268

16.1 Limits of Relational Predicate Logic 268

16.2 Extending the Truth Tree Method to RPL= 270

16.2.1 Identity-out Rule 270

16.2.2 Identity-in Rule 271

16.3 Sameness and Distinctness in English 273

16.3.1 'Only' Again 273

16.3.2 Words of Distinction: Except, But, Other (than), Besides, Else 274

16.4 Numerical Adjectives 276

16.4.1 At Least n 276

16.4.2 At Most n (No More than n) 279

16.4.3 Exactly n 281

16.4.4 Counting Pairs 283

16.4.5 Combinatorics (optional) 283

16.5 Definite Descriptions 284

16.5.1 The Definite Description Quantifier Procedure 288

16.5.2 Definite Descriptions as Anaphors 289

16.5.3 Plural Definite Descriptions 289

17 Verbs and their Modifiers 294

17.1 Prepositional Phrases 294

17.2 The Event Approach 296

17.3 Indirect Support of the Event Approach 298

17.3.1 Fixing Referents and Binding Anaphoric Pronouns 298

17.3.2 Quantification over Events 299

17.3.3 Conversational Inferences and Events 300

17.3.4 Methodological Reflections 300

17.4 Adverbial Modification 301

17.5 Problems with the Event Approach 304

Appendix 308

A1 Conjunction 308

A1.1 Prepositional Phrases 308

A1.2 Conversational Inferences and Deductive Validity 309

A1.3 Relative Clauses 311

A2 Negation and Disjunction 314

A2.1 Modalities and Negation 314

A2.2 Disjunction and Conversational Inferences 315

A3 Conditionals 315

A3.1 Explication of the Material Conditional Truth Table 315

A3.1.1 Paradoxes of implication 318

A3.1.2 Conditionals and conversational inferences 318

A3.1.3 Paradoxes of implication revisited 320

A3.2 'If 's and 'Then's without Conditionality 321

A4 Property Predicate Logic 321

A4.1 Only 321

A4.2 Conversational Inferences 322

A4.2.1 Existential import 322

A4.2.2 Scalar inferences 323

A4.3 More on Literal Meaning 324

A4.4 Adjectival Modification and Predication 325

A4.5 A Non-standard Quantifier - Most 329

A5 Relational Predicate Logic 330

A5.1 Passive Voice: Another Argument for Variables 330

A5.1.1 Passive voice for nested quantifier procedure 332

A5.2 Properties of Relations 333

A5.2.1 Symmetry, asymmetry, non-symmetry 333

A5.2.2 Transitivity, intransitivity, non-transitivity 334

A5.2.3 Total reflexivity, reflexivity, irreflexivity, and non-reflexivity 335

A6 Relational Predicate Logic with Identity 337

A6.1 'Only' and Existential Import 337

A6.2 Descriptions and Anaphora 338

A6.3 Plural Anaphora 339

A6.3.1 Plural definite descriptions as anaphors 344