University of Wisconsin Milwaukee UWM Digital Commons Philosophy Faculty Books Philosophy 2017 Fundamental Methods of Logic Matthew Knachel University of Wisconsin - Milwaukee, [email protected] Follow this and additional works at: https://dc.uwm.edu/phil_facbooks Part of the Philosophy Commons Recommended Citation Knachel, Matthew, "Fundamental Methods of Logic" (2017). Philosophy Faculty Books. 1. https://dc.uwm.edu/phil_facbooks/1 This Book is brought to you for free and open access by UWM Digital Commons. It has been accepted for inclusion in Philosophy Faculty Books by an authorized administrator of UWM Digital Commons. For more information, please contact [email protected]. FUNDAMENTAL METHODS OF LOGIC Fundamental Methods of Logic Matthew Knachel UWM Libraries University of Wisconsin, Milwaukee This book is published by the UW-Milwaukee Library Digital Commons and is available free of charge at this site: http://dc.uwm.edu/phil_facbooks/1 This work is licensed under the Creative Commons Attribution 4.0 International License. For details, visit this site: https://creativecommons.org/licenses/by/4.0/legalcode Cover Art by Dan Williams Recommended citation: Knachel, Matthew, "Fundamental Methods of Logic" (2017). Philosophy Faculty Books. 1. http://dc.uwm.edu/phil_facbooks/1 ISBN: 978-0-9961502-2-4 For my girls, Rose and Alice Preface There’s an ancient view, still widely held, that what makes human beings special—what distinguishes us from the “beasts of the field”—is that we are rational. What does rationality consist in? That’s a vexed question, but one possible response goes roughly like this: we manifest our rationality by engaging in activities that involve reasoning—making claims and backing them up with reasons, acting in accord with reasons and beliefs, drawing inferences from available evidence, and so on. This reasoning activity can be done well and it can be done badly—it can be done correctly or incorrectly. Logic is the discipline that aims to distinguish good reasoning from bad. Since reasoning is central to all fields of study—indeed, since it’s arguably central to being human—the tools developed in logic are universally applicable. Anyone can benefit from studying logic by becoming a more self-aware, skillful reasoner. This covers a variety of topics at an introductory level. Chapter One introduces basic notions, such as arguments and explanations, validity and soundness, deductive and inductive reasoning; it also covers basic analytical techniques, such as distinguishing premises from conclusions and diagramming arguments. Chapter Two discusses informal logical fallacies. Chapters Three and Four concern deductive logic, introducing the basics of Aristotelian and Sentential Logic, respectively. Chapters Five and Six concern inductive logic. Chapter Five deals with analogical and causal reasoning, including a discussion of Mill’s Methods. Chapter Six covers basic probability calculations, Bayesian inference, fundamental statistical concepts and techniques, and common statistical fallacies. The text is suitable for a one-semester introductory logic or “critical thinking” course. The emphasis is on formal techniques and problem solving rather than analytical writing, though exercises of the latter sort could easily be incorporated. A note on tone, style, and content. This book is written by an American teacher whose intended audience is American undergraduates; it is based on my lectures, developed over many years. Like the lectures, it assumes that some members of the intended audience lack an antecedent interest in the subject and may have trouble developing and maintaining enthusiasm to study it. It tries to compensate for this by adopting a casual style, using first- and second-person constructions, and by shamelessly trafficking in cultural references, lame jokes, and examples involving American current events. The result is a logic textbook with a somewhat unusual tone and a sometimes- narrow cultural perspective. Neither familiarity with the relevant cultural references, nor amusement at the lame jokes, is a prerequisite for understanding the material, but I thought it prudent to offer an apologia at the outset. Caveat lector. An acknowledgment of debts. The following books have influenced my teaching, and hence the present work: Virginia Klenk’s Understanding Symbolic Logic, John Norton’s How Science Works, Ian Hacking’s Introduction to Probability and Inductive Logic, Darrell Huff’s How to Lie with Statistics, and Irving Copi and Carl Cohen’s Introduction to Logic. The influence of those last two books is particularly profound, as I note throughout this text. I am indebted to all my logic viii teachers over the years: Kurt Mosser, Michael Liston, Mark Kaplan, Richard Tierney, Steve Leeds, Joan Weiner, Ken Manders, Mark Wilson, and Nuel Belnap. Thanks to J.S. Holbrook for sending me examples of fallacies. For extensive logistical support, I’m indebted to Kristin Miller Woodward; I also thank her for arranging financial support through the UW-Milwaukee Library and Center for Excellence in Teaching and Learning, who have undertaken a project to encourage the development and adoption of open textbooks. My logic students over the years also deserve acknowledgment, especially those who have recently served as guinea pigs, learning from earlier drafts of this book. Without student feedback, there would be no book. Finally, and most importantly, I could not have completed this project without my wife Maggie’s constant support and forbearance. Contents Chapter 1 - The Basics of Logical Analysis 1 I. What is Logic? 1 II. Basic Notions: Propositions and Arguments 2 III. Recognizing and Explicating Arguments 3 Paraphrasing 4 Enthymemes: Tacit Propositions 6 Arguments vs. Explanations 7 IV. Deductive and Inductive Arguments 10 Deductive Arguments 11 Inductive Arguments 15 V. Diagramming Arguments 18 Independent Premises 18 Intermediate Premises 19 Joint Premises 20 Chapter 2 - Informal Logical Fallacies 29 I. Logical Fallacies: Formal and Informal 29 II. Fallacies of Distraction 31 Appeal to Emotion (Argumentum ad Populum) 31 Appeal to Force (Argumentum ad Baculum) 32 Straw Man 33 Red Herring 34 Argumentum ad Hominem 36 III. Fallacies of Weak Induction 30 Argument from Ignorance (Argumentum ad Ignorantiam) 40 Appeal to Inappropriate Authority 42 Post hoc ergo propter hoc 43 Slippery Slope 44 Hasty Generalization 45 IV. Fallacies of Illicit Presumption 46 Accident 46 Begging the Question (Petitio Principii) 48 Loaded Questions 49 False Choice 51 Composition 52 Division 53 V. Fallacies of Linguistic Emphasis 53 Accent 54 Quoting out of Context 55 Equivocation 57 Manipulative Framing 59 x Chapter 3 – Deductive Logic I: Aristotelian Logic 68 I. Deductive Logics 68 II. Classes and Categorical Propositions 69 The Four Types of Categorical Proposition 71 Universal Affirmative (A) 71 Universal Negative (E) 74 Particular Affirmative (I) 74 Particular Negative (O) 76 A Note on Terminology 76 Standard Form for Sentences Expressing Categorical Propositions 77 III. The Square of Opposition 79 Contradictories 80 Contraries 81 Subcontraries 81 Subalterns 82 Inferences 83 IV. Operations on Categorical Sentences 84 Conversion 84 Obversion 86 Contraposition 87 Inferences 91 V. Problems with the Square of Opposition 96 Existential Import 97 Problems for the Square 97 Solution? 98 Boolean Solution 99 VI. Categorical Syllogisms 102 Logical Form 103 The Venn Diagram Test for Validity 104 Chapter 4 – Deductive Logic II: Sentential Logic 117 I. Why Another Deductive Logic? 117 II. Syntax of SL 120 Conjunctions 121 Disjunctions 121 Negations 121 Conditionals 122 Biconditionals 122 Punctuation – Parentheses 123 III. Semantics of SL 126 Negations (TILDE) 127 Conjunctions (DOT) 128 Disjunctions (WEDGE) 128 Biconditionals (TRIPLE-BAR) 129 Conditionals (HORSESHOE) 130 Computing Truth-Values of Compound SL Sentences 133 xi IV. Translating from English into SL 137 Tilde, Dot, Wedge 137 Horseshoe and Triple-Bar 140 V. Testing for Validity in SL 143 Logical Form in SL 143 The Truth Table Test for Validity 144 Chapter 5 – Inductive Logic I: Analogical and Causal Arguments 152 I. Inductive Logics 152 II. Arguments from Analogy 153 The Form of Analogical Arguments 153 The Evaluation of Analogical Arguments 156 Number of Analogues 157 Variety of Analogues 157 Number of Similarities 158 Number of Differences 158 Relevance of Similarities and Differences 159 Modesty/Ambition of the Conclusion 159 Refutation by Analogy 160 III. Causal Reasoning 163 The Meaning(s) of ‘Cause’ 164 Mill’s Methods 165 Method of Agreement 165 Method of Difference 166 Joint Method of Agreement and Difference 167 Method of Residues 167 Method of Concomitant Variation 168 The Difficulty of Isolating Causes 169 Chapter 6 – Inductive Logic II: Probability and Statistics 175 I. The Probability Calculus 175 Conjunctive Occurrences 176 Disjunctive Occurrences 180 II. Probability and Decision-Making: Value and Utility 187 III. Probability and Belief: Bayesian Reasoning 193 IV. Basic Statistical Concepts and Techniques 202 Averages: Mean vs. Median 202 Normal Distributions: Standard Deviation, Confidence Intervals 203 Statistical Inference: Hypothesis Testing 206 Statistical Inference: Sampling 212 V. How to Lie with Statistics 218 Impressive Numbers without Context 218 Misunderstanding Error 219 Tricky Percentages 222 The Base-Rate Fallacy 223 Lying with Pictures 226 CHAPTER 1 The Basics of Logical Analysis I. What Is Logic? In Logic, the object of study is reasoning. This is