Table of Contents
Preface ix
1 An Introduction to Arguments 1
1.1 Premises and a Conclusion 1
1.2 Deductively Valid and Inductively Strong 4
1.3 Soundness and Reliability 9
1.4 Some Argument Forms 11
1.4.1 Modus Ponens 11
1.4.2 Modus Tollens 13
1.4.3 Inductive Generalization 13
1.4.4 Proportional Syllogism 15
1.4.5 Induction by Confirmation 16
1.4.6 Analogical Argument 18
1.5 A Note about Reading Arguments 21
1.6 Exercises 23
1.7 Answers 28
2 The Induction by Confirmation 35
2.1 Halley's Comet 35
2.2 The Hypothesis, Prediction, and Data 36
2.2.1 The Hypothesis 36
2.2.2 The Prediction 37
2.2.3 The Data 39
2.3 The Induction by Confirmation 41
2.3.1 The Structure of the Argument 41
2.3.2 Halley's Argument 42
2.4 Other Conclusions 44
2.4.1 Why Do Humans Reason? 44
2.4.2 An Indeterminate Hypothesis 45
2.4.3 Premise 5 46
2.4.4 A Failed Hypothesis: Spontaneous Generation 47
2.5 The Inference 49
2.5.1 An Indirect Inference 49
2.5.2 Valid and Invalid Arguments 49
2.6 Exercises 53
2.7 Answers 61
3 More on the Induction by Confirmation 67
3.1 The Crucial Experiment 67
3.2 The Inference to the Best Explanation 71
3.2.1 The Best Hypothesis 73
3.2.2 The IBE and IC 76
3.3 Exercises 78
3.4 Answers 89
4 Mill's Methods 99
4.1 Necessary and Sufficient Conditions 100
4.1.1 Conditionals and Necessary and Sufficient Conditions 102
4.2 Mill's First Three Methods 104
4.2.1 The Method of Agreement 104
4.2.2 The Method of Difference 105
4.2.3 Necessary and Sufficient Conditions Again 107
4.2.4 The Joint Method of Agreement and Difference 109
4.3 Typhoid Mary 110
4.4 Mill's Fourth and Fifth Methods 114
4.4.1 The Method of Residues 114
4.4.2 The Method of Concomitant Variations 116
4.5 Exercises 118
4.6 Answers 125
5 Describing Populations 131
5.1 Variables and Their Values 132
5.2 Describing a Population with Two Variables 133
5.3 Difference in Proportions, Independence, and Association 134
5.3.1 Difference in Proportions 134
5.3.2 Independent and Associated 135
5.4 The Strengths of Positive and Negative Associations 136
5.5 Information about a Population 138
5.6 Measuring the Strength of an Association (Again) 140
5.7 Exercises 141
5.8 Answers 143
6 The Proportional Syllogism 151
6.1 Probability and Proportion 151
6.2 The Theory of Probability 152
6.2.1 The General Conjunction Rule 154
6.2.2 The Conditional Probability Rule 155
6.2.3 The Special Conjunction Rule 155
6.2.4 The Special Disjunction Rule 157
6.2.5 The General Disjunction Rule 158
6.3 Relative Risk 159
6.4 Calculating the Probability for Multiple Individuals 161
6.5 The Proportional Syllogism 164
6.5.1 A Note about Evaluating Arguments 167
6.6 Exercises 169
6.7 Answers 175
7 The Inductive Generalization 189
7.1 Introduction 189
7.2 Calculating the Probability of an Interval 191
7.3 The 95 Percent Interval 193
7.4 An Example of an Inductive Generalization 196
7.5 The Conclusion of an Inductive Generalization 200
7.6 Inductive Strength 202
7.7 Exercises 203
7.8 Answers 207
8 Bayes' Rule 213
8.1 Bayes'Rule 213
8.2 Example 1: Two Jars of Marbles 214
8.3 Example 2: The Stolen $1.3 Million 215
8.4 The Argument Using Bayes' Rule 218
8.5 Example 3: A Match Made with Bayes' Rule 220
8.6 Example 4: A Positive Mammogram 222
8.7 Exercises 223
8.8 Answers 227
Appendices
A A Brief Introduction to Deductive Logic 235
A.1 Some Rules of Deductive Logic 235
A.1.1 And and Or Rules 235
A.1.2 If-Then Rules 237
A.2 Categorical Syllogisms 239
A.3 Exercises 240
A.4 Answers 243
B Some Further Topics on Probability 247
B.1 Odds 247
B.2 Expected Value 250
B.3 Where Do Probabilities Come From? 254
B.3.1 Proportion of a Population 254
B.3.2 Frequency 255
B.3.3 Betting 255
B.4 Exercises 257
B.5 Answers 261
Index 267
